Exact solutions of the vorticity equation on the sphere as a manifold

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Ramírez" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Sandra M." "apellidos" => "Ramírez" ] ] ] 2 => array:2 [ "autoresLista" => "Eric J. Alfaro" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Eric J." "apellidos" => "Alfaro" ] ] ] 3 => array:2 [ "autoresLista" => "David B. Enfield" "autores" => array:1 [ 0 => array:2 [ "nombre" => "David B." "apellidos" => "Enfield" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0187623617300048?idApp=UINPBA00004N" "url" => "/01876236/0000002800000003/v2_201709141210/S0187623617300048/v2_201709141210/en/main.assets" ] "itemAnterior" => array:19 [ "pii" => "S0187623617300024" "issn" => "01876236" "doi" => "10.20937/ATM.2015.28.03.02" "estado" => "S300" "fechaPublicacion" => "2015-07-01" "aid" => "73850" "copyright" => "Universidad Nacional Autónoma de México" "documento" => "article" "crossmark" => 0 "licencia" => "http://creativecommons.org/licenses/by-nc-nd/4.0/" "subdocumento" => "fla" "cita" => "Atmósfera. 2015;28:161-78" "abierto" => array:3 [ "ES" => true "ES2" => true "LATM" => true ] "gratuito" => true "lecturas" => array:2 [ "total" => 510 "formatos" => array:3 [ "EPUB" => 41 "HTML" => 343 "PDF" => 126 ] ] "en" => array:11 [ "idiomaDefecto" => true "titulo" => "On the distinct interannual variability of tropical cyclone activity over the eastern North Pacific" "tienePdf" => "en" "tieneTextoCompleto" => "en" "tieneResumen" => array:2 [ 0 => "es" 1 => "en" ] "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "161" "paginaFinal" => "178" ] ] "contieneResumen" => array:2 [ "es" => true "en" => true ] "contieneTextoCompleto" => array:1 [ "en" => true ] "contienePdf" => array:1 [ "en" => true ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0005" "etiqueta" => "Fig. 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 1219 "Ancho" => 942 "Tamanyo" => 151623 ] ] "descripcion" => array:1 [ "en" => "Observed number of tropical cyclones (TCs) reaching at least tropical storm strength in the NHC best track dataset over the Eastern North Pacific (ENP) during the period 1965-2013 for (a) the peak season (June through October) and (b) the whole year (January-December)" ] ] ] "autores" => array:2 [ 0 => array:2 [ "autoresLista" => "Haikun Zhao" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Haikun" "apellidos" => "Zhao" ] ] ] 1 => array:2 [ "autoresLista" => "Graciela B. Raga" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Graciela B." "apellidos" => "Raga" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0187623617300024?idApp=UINPBA00004N" "url" => "/01876236/0000002800000003/v2_201709141210/S0187623617300024/v2_201709141210/en/main.assets" ] "en" => array:18 [ "idiomaDefecto" => true "titulo" => "Exact solutions of the vorticity equation on the sphere as a manifold" "tieneTextoCompleto" => true "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "179" "paginaFinal" => "190" ] ] "autores" => array:1 [ 0 => array:3 [ "autoresLista" => "Ismael Pérez-García" "autores" => array:1 [ 0 => array:3 [ "nombre" => "Ismael" "apellidos" => "Pérez-García" "email" => array:1 [ 0 => "ismael@unam.mx" ] ] ] "afiliaciones" => array:1 [ 0 => array:2 [ "entidad" => "Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Circuito de la Investigación Científica s/n, Ciudad Universitaria, 04510 México, D.F." "identificador" => "aff0005" ] ] ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0015" "etiqueta" => "Fig. 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 546 "Ancho" => 1891 "Tamanyo" => 268273 ] ] "descripcion" => array:1 [ "en" => "Streamfunction isolines of equatorial <a class="elsevierStyleCrossRef" href="#bib0190">Verkley modon (1984)</a> with (a) k = 10., a = 10.,φa= 66.14°, λ0 = 270.0° and φ0 = 0.°; the uniform Verkley modon (1990) at (b) σ = 8.06, φa = φa=5π12,λ0=180.0°, φ0 = 180.0° and φ0 = 45.0°. The curve points are the spherical coordinates relative to a rotated pole N’(270.0°, 0.°) at (a) N’(180.0°, 45.°) at (b) with respect to the original system." ] ] ] "textoCompleto" => "1IntroductionLet S2 ={x ∈ R3: | x |= 1} denote the unit sphere it R3. The large-scale dynamics of tire atmosphere on the rotating sphere S2 can approximately be governed by the non-linear barotropic vorticity equation (BVE), which can be written in the non-di-mensional from as:<elsevierMultimedia ident="eq0005"></elsevierMultimedia>where Ψ(λ,μ) denotes the stream function, μ=sinφ=cosθ,−π≤λ≤π,−π2≤φ≤π2,0<θ<π,λ the longitude, φ the latitude, and θ the colatitude. Δ is the Laplace-Beltrami operator on a sphere and J(Ψ, h) is the Jacobian.The following is a solution for Eq. <a class="elsevierStyleCrossRef" href="#eq0005">(1)</a> on the sphere proposed by <a class="elsevierStyleCrossRef" href="#bib0180">Thompson (1982)</a>:<elsevierMultimedia ident="eq0010"></elsevierMultimedia>where (λ’, μ’) are the spherical coordinates relative to a rotated pole N’ with coordinates (λ0, μ0) with respect to the original system, and Yν is an eigenfunction of the operator Laplace-Beltrami with eigenvalue χν<a class="elsevierStyleCrossRef" href="#bib0190">Verkley(1984)</a> generalized Thompson's solution and demonstrated that Yν could be a set of eigenfunctions that contains more than only spherical harmonics. Then Eq. <a class="elsevierStyleCrossRef" href="#eq0010">(2)</a> describes a configuration in which the structure Yν moves through the zonal flow -ωμ with constant velocity cν and without changing size and shape. The pole of the primed system N’ that moves along a latitude at a constant angular velocity cν is given by<elsevierMultimedia ident="eq0015"></elsevierMultimedia>where χν is an eigenvalue for the spectral problem ΔYν= −χνYν. In particular, for spherical harmonics Y (λ’, μ’) of degree n corresponding to the eigenvalue χν = χn= n(n + 1), Eq.<a class="elsevierStyleCrossRef" href="#eq0010">(2)</a> is a Rossby-Haurwitz (RH) wave. RH waves have proven to be very useful to describe the large-scale wave structure of atmospheric circulation in middle latitudes (<a class="elsevierStyleCrossRef" href="#bib0125">Rossby, 1939</a>; <a class="elsevierStyleCrossRef" href="#bib0050">Haurwitz, 1940</a>). The solution modon is constructed to divide the sphere S2 into two regions (<a class="elsevierStyleCrossRef" href="#bib0185">Tribbia, 1984</a>; <a class="elsevierStyleCrossRef" href="#bib0190">Verkley, 1984</a>, <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>, <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>; <a class="elsevierStyleCrossRef" href="#bib0095">Neven, 1992</a>): an inner region Si centered around the pole N’, and an outer region So separated from the inner region by a boundary circle in which Ψ, q and its normal derivative Ψ’ are continuous. Modons are considered appropriate to describe some types of atmospheric blocking events (<a class="elsevierStyleCrossRef" href="#bib0200">Verkley, 1990</a>).Hydrodynamic equations on manifolds were studied by <a class="elsevierStyleCrossRef" href="#bib0045">Ebin and Marsden (1970)</a>, <a class="elsevierStyleCrossRef" href="#bib0165">Szeptycki (1973a</a>, <a class="elsevierStyleCrossRef" href="#bib0170">b</a>), <a class="elsevierStyleCrossRef" href="#bib0020">Avez and Bamberger (1978)</a>, Ghidaglia et al. (1988), Temam (1987) and Ilyin (1993). The existence, unicity and regularity of the solution for the evolution equation (Eq. <a class="elsevierStyleCrossRef" href="#eq0005">(1)</a>) on S2 were proven by <a class="elsevierStyleCrossRef" href="#bib0165">Szeptycki (1973a</a>, <a class="elsevierStyleCrossRef" href="#bib0170">b</a>), <a class="elsevierStyleCrossRef" href="#bib0020">Avez and Bamberger (1978)</a>, Ilyin (1993) and <a class="elsevierStyleCrossRef" href="#bib0155">Skiba (2012)</a>. <a class="elsevierStyleCrossRef" href="#bib0045">Ebin and Marsden (1970)</a> dealt with the motion of an incompressible fluid on manifolds under a differential geometric point of view. Problems from the transition map between the charts are transferred to those of finding geodesics on the group of all volume-preserving diffeomorphisms, to which the methods of global analysis and infinite-dimensional geometry can be applied.In this paper we study the manifolds S2 in terms of the stream function Ψ for an RH wave which is sufficiently smooth and for Wu-Verkley waves and modons which are weakly differentiable of higher orders. Section 2 deals with the compact differentiable manifold S2 and the way in which functions are constructed on this manifold. Section 3 shows the types of solutions that will be considered. Another aim of this paper is to deepen the understanding of the BVE solution on the manifold S2 and its usage for deriving the properties of solutions to the manifold (S2, g). The paper concludes with a summary in section 4.2Structure of functions on the manifold S2In this section we review some basic facts concerning to the manifold S2. We should recall that the unit sphere S2 is a compact and connected differentiable manifold. Indeed, because S2 is compact it is not possible to cover it with only one chart. A chart of S2 is then a pair (Ω,φ) where Ω is an open subset of S2, and φ is a homeomorphism of Ω onto some open subset of R2. Let us consider the two charts {(Ωι,φι), (Ωκ, φκ)} of class Cp for S2 where every chart corresponds to a geographical coordinate group. It is possible to define a coordinate chart that covers most of S2 by using the standard spherical coordinate map. Let φι denote the coordinate function, which maps from (x1, x2, x3) to angles (λ, θ) or to (λ, μ). The domain of φι−1 is the open set defined by λ ∈ (−π, π) and θ ∈ (0, π) (this excludes the poles). The inverse map φι−1 yields the parameterization x1 = cos λ sin θ, x2 = sin λ sin θ, x3 = cos θ and its variation φκ−1 yields the parameterization φκ−1 (λ’, θ’)=(cos λ’ sin θ’, cos θ’, sin λ’ sin θ’). The domain of φκ−1 in the open set defined by λ’ ∈ (–π, π) and θ’ ∈ (0, π). The charts Ωι,φι and Ωκ,φκ correspond to poles N and N’ on the sphere S2. N’ might be taken as the point (λ0=−π2φ0=0) in the old system and as the angle λ’ in this new north pole, so that the new international date line is the half circle Γκ={p∈S2:−π2<λ(p)<π2,θ=π2,} of the old equator in the x1x2- plane, on the front where x1 ≥ 0 (<a class="elsevierStyleCrossRef" href="#bib0120">Richtmyer, 1981</a>; <a class="elsevierStyleCrossRef" href="#bib0135">Skiba, 1989</a>; <a class="elsevierStyleCrossRef" href="#bib1115">Pérez-García, 2001</a>). The international date line, for the chart (Ωι, φι) is the half circle Γι={p∈S2:−π2<φ(p)<π2,λ=± π} in the x1x3 –plane. The chart covers (Ωι, φι) covers the sphere except for the set Γι, and the chart (Ωκ, φκ) similarly covers the sphere with the exception of a set Γκ. Hence the two charts {(Ωι, φι),(Ωκ, φκ)} together cover S2 and they constitute an atlas.The local coordinates associated with the chart (Ωι, φι) are functions φι: Ωι,→R2, such that for p∈S2, φι(p)=(φι,1(p),φι,2(p))=(xι1(p),  xι2(p))=(λ(p),μ(p))and φκ(p)=(xκ1(p),xκ2(p))=(λ'(p);μ'(p)) are local coordinates with respect to the chart (Ωκ, φκ) (<a class="elsevierStyleCrossRef" href="#fig0005">Fig. 1</a>).<elsevierMultimedia ident="fig0005"></elsevierMultimedia>To construct the map φℓ:S2→U⊂Rℓ2,ℓ=ι, κ a bijection with inverse φι−1:Uι→S2 defined as φι−1(xι1,xι2)=(1−(xι2)2cos xι1, 1−(xι2)2 sinxι1,xι2), and the φκ−1(xκ1,xκ2)=(1−(xκ2)2cos xκ1,xκ2,1−(xκ2)2sinxκ1), it is seen that every Uℓ is open. Hence each Ωℓ is an open subset of S2 and Ωι∪Ωκ cover S2.Given two charts of the atlas {(Ωι, φι), (Ωκ, φκ)} with Ωκ ∩ Ωι≠ø, the transition maps φικ=φκ∘φι−1: Uικ→Uκι outline open sets of R2 into R2, where Uικ=φιΩι ∩ Ωκ and Uκι=φκΩκ ∩ Ωι. This determines a differentiable structure for S2, and φικ=φκ ∘ φι−1: is a diffeomorphism. It is then said that the atlas is of class Ck if the transition functions are of Ck.Let x be any point of Uικ, and (φικ1(x),φικ2(x)) the coordinate of φικ (x); then φικi(x) is a continuous function on two variables. Now, if p ∈ Ωι ∩ Ωκ such that x = φι (p), and since φι(p) ∈ Uικ, we have the relations<elsevierMultimedia ident="eq0020"></elsevierMultimedia><elsevierMultimedia ident="eq0025"></elsevierMultimedia>This is the transformation formula betwen the two local coordinate systems (xι1,xι2) and (xκ1,xκ2) defined on Ωκ ∩ Ωι. To obtain the relations between the unprimed and primed coordinate of any point Q on the sphere, <a class="elsevierStyleCrossRef" href="#bib0190">Verkley (1984)</a> examined the spherical triangle NQN’ and the application of the cosine rules to this triangle, deriving explicit expressions for the transformation between the two coordinate systems as given by (4) and (5).Let Ωι ∩ Ωκ≠ø and set J φικ as the Jacobian matrix of map φικ, so we can verify that<elsevierMultimedia ident="eq0030"></elsevierMultimedia>Then det J φικ > 0. Hence, it is said that if manifold S2 is oriented for every pair Ωι, Ωκ of intersecting local coordinate neighbourhoods, det J φικ > 0.Indeed we can regard the coordinate as a device to decide which of many functions ψ on S2 are to be differentiable. Since Ωι is just a set, it makes no sense to ask that ψ: Ωι→ R be differentiable (<a class="elsevierStyleCrossRef" href="#bib0085">Matsushima, 1972</a>; <a class="elsevierStyleCrossRef" href="#bib0080">Loomis and Sterberg, 1990</a>). However, we can consider the map Ψι=ψ ∘ φι−1:φι (Ωι)→R Then ψ ∘ φι−1 is a function defined on an open φι(Ωι)⊂R2, and we know what it means for such a function to be differentiable or smooth (see <a class="elsevierStyleCrossRef" href="#fig0005">Fig. 1</a>). Consider now what happens when we change coordinates to some other chart, lets say (Ωκ, φκ) for convenience, assuming that Ωι = Ωκ, Then it is possible for ψ ∘ φι−1 to be differentiable but ψ ∘ φκ−1 is not. To compare both, let ψ ∘ φι−1=ψ ∘φκ−1 ∘ (φκ ∘ φι−1) where the map φκ∘φι−1:φι(Ωι)→φκ(Ωκ) is a bijection between open subsets of R2. Then a sufficient condition for ψ ∘ φι−1 to be differentiable if φ ∘ φι−1 is, is that ’φκ ∘ φι−1 is also differentiable. We often write Ψ for the composite function ψ ∘ φι−1Lets take a curve τ : (−1, 1) → S2 with τ (0) = p. In a local chart τ is given by xιi=τi(t). On the manifold S2, one can define a vector U tangent to the parametrized curve τ at any point p on the curve. The tangent vector U is given by a column vector u whose components uιi are dτidt0, (i = 1, 2), with the initial condition τ (0) = p. If we use another coordinate system corresponding to the chart (Ωκ, φκ) by xκi then the tangent vector U is given by a column vector v with components υκi. According to the chain rule, the column vectors u and v are related by υκi=uιj∂xκi∂xιj. The expression uιi∂∂xιj is the partial differential operator in the direction of the tangent vector.The space TpS2 is called the tangent space of S2at p, and TpS2 is a two-dimensional vector space. For each u ∈ TpS2 we shall write u=uιi∂∂xιj=u1∂∂λ+u2∂∂θ, where uιi are the contravariant components of u. It is well known that on the manifold S2 an inner product is defined at each tangent space TpS2. Now lets present a basis in which we denote the coordinate system corresponding to the chart (Ωι, φι) by {xιi}=(λ,μ), and for any ψ: S2 → R define the vectors (∂∂xιj)p by (∂∂xιj)pψ=(∂ψ∘φι−1∂xιi)φι(p), so that they are independent since (∂∂xιj)pxιj=δij Let (nˆ=(1−(xι2)2cosxι1,1−(xι2)2sinxι1,xι2) be the outward normal to S2 in R3; without any loss of generality we may assume that the vectors eλ=∂nˆι∂xι1,   eμ=∂nˆι∂xι2 form a basis for TpS2.We will denote the vector space of a vector field on S2 by Γ(S2) A tangent vector field on S2 is a smooth map u: S2 → T S2 such that, for any x ∈ S2, u(x) ∈ TxS2. At the chart (Ωι, φι), for x ∈ Ωι, the vector functions u ∈ TxS2 and v ∈ Γ(S2) have components u=up1eλ+up2eμ and v=υp1eλ+υp2eμ, respectively, being these up upi=upxιi=uλ,uμ the components of up as the vectors of the unitary base indicated by eλ, and eμ in the directions λ and μ, respectively.Let us recall that an oriented Riemannian manifold is a pair (S2, g) where S2 is the oriented compact manifold and g a Riemannian metric on S2, which assigns a length vgp∈R+. The g on S2 is a smooth (2, 0)-tensor field on S2 such that for any p ∈ S2, gp: Tp(S2) × Tp(S2) → R is a scalar product on the tangent space Tp(S2), and in any chart (Ωι, φι) of S2, its components<elsevierMultimedia ident="eq0035"></elsevierMultimedia>form a symmetric matrix, with its inverse denoted by (gij) = (gij)-1, and g = det(gij) = 1 – μ2. The length of a tangent vector v ∈ TpS2 is defined as usual, v=gpv,v12=v·v12 Moreover, the inner product on T S2 is given by u . v = gijuiνj for u, v ∈ TS2.Let (S2, g) be the smooth Riemannian manifolds of S2. Let us now recall some operators arising in partial differential equations on the sphere as manifold. Given the scalar function ψ : S2 → R, the gradient of ψ, is given by the vector field grad ψ: S2 → TpS2, for which<elsevierMultimedia ident="eq0040"></elsevierMultimedia>where eˆλ =11−μ2eλ and eˆμ=1−μ2 eμ.If u ∈ Г(S2), the divergence of u is the function on S2 which on the chart (Ωι, φι) is given by<elsevierMultimedia ident="eq0045"></elsevierMultimedia>A linear connection D on S2 is a map D: T(S2) × Г(S2) → T(S2) called the covariant derivative and the usual notation for D(U, V) is DUV. Let (Ωι, φι) be a chart and as one can observe, the vectors (eλ=∂∂λ,eμ=∂∂μ) can be nonconstant. An easy notation is set ∇i=D∂∂xi (eg. <a class="elsevierStyleCrossRef" href="#bib0055">Hebey, 2000</a>). There are smooth functions Γijk:Ωι→R such that for any i, j, and any p ∈Ωι,<elsevierMultimedia ident="eq0050"></elsevierMultimedia>where Γijk are the Christoffel symbols, defined by Γijk=12∑l=12gkl(∂glj∂xιi+∂gil∂xιj+∂gij∂xιl).On the chart (Ωι, φι), we have Γ111=0,Γ112=−2μ1−μ2,Γ121=cot θ=μ1−μ2,Γ122=0,Γ221=0 and Γ222=0The fundamental operator which we study is the Laplacian Δ, then for real or complex valued functions, Δ is the Laplace-Beltrami operator on S2 and it is given by<elsevierMultimedia ident="eq0055"></elsevierMultimedia>This operator satisfies some properties: Δg, is selfadjoint, symmetric and non-negative (Aubin, 1998). Thus, the operators div, grad and Δg on the manifold S2 have the conventional meaning.Let (S2, g) be a compact oriented Riemannian manifold, with metric g. The metric and the orientation are combined to give a volume element dυg on S2, which can be used to integrate functions on (S2, g). In order to apply the integral calculus on the oriented manifold S2, we define a volume element to be a two-form ω = dυ which is defined on all of S2. For every chart (Ωℓ, φℓ) which is consistently oriented with S2, the coordinate expresion for ω=dυl is Φldxl1Λdxl2 where Φℓ is a partition of unity subordinate to the covering Ωl, l=ι,κOn the Riemannian manifold (S2, g), at the chart (Ωι, φι); a volume form η = dυg defines a Lebesgue measure on S2 by dυg=η=gdxl1 Λdxl2 Then<elsevierMultimedia ident="eq0060"></elsevierMultimedia>wheredxl=dxl1dxl2 defines a Lebesgue measure on R2.Let C∞(S2) denote the set of infinitely differentiable functions of compact support ψ(x). At μ = ± 1 the functions are smooth, together with the periodic boundary condition at λ with period 2π. If we define the usual Hilbert space L2(S2) to be the completion of C∞(S2) with respect to the inner product<elsevierMultimedia ident="eq0065"></elsevierMultimedia>and norm f={∫s2f2dυg}12,  g* is the complex conjugate of function g. Let (S2, g) be the compact Riemannian manifold and dυg the Riemannian volume element. Then functional spaces (Sobolev and the Holder spaces) can be defined on S2 as well (<a class="elsevierStyleCrossRef" href="#bib0155">Skiba, 2012</a>). For each p ∈ R with 1≤ p < ∞ we associate a Banach space<elsevierMultimedia ident="eq0070"></elsevierMultimedia>with respect to the norm<elsevierMultimedia ident="eq0075"></elsevierMultimedia>and ess sup | f | < ∞ if p = ∞. L2(TS2) represent the Hilbert space of the vector fields U : S2 → TS2 endowed with the inner product in L2(S2) induced by g in Tp(S2) (see <a class="elsevierStyleCrossRef" href="#bib0040">Díaz and Tello, 1999</a>; <a class="elsevierStyleCrossRef" href="#bib0055">Hebey, 2000</a>).We now turn to the eigenvalue problems for Δg: We usually seek to find all eigenvalues γ for which there is an eigenfunction Y such that ΔgY = –γY. Then, which information about geometry of (S2, g) is encoded by the eigenvalues?. The structure of eigenfunctions: Lp norms and relations to RH waves or modons.Global harmonic analysis is the study of the spectral theory of the Laplacian Δg on a compact Riemannian manifold (S2, g), and its relation to the global geometric structure. Since (S2, g) is compact, there exists an orthonormal basis {Yj} of smooth eigenfunctions and the spectrum of Δg is a discrete set {γ0 = 0 < γ1 ≤ γ2 ≤ γ3 ≤ ...}. Recent developments show that the non-zero eigenvalues also contain substantial geometric and analytic information. The solution modon constructed by <a class="elsevierStyleCrossRef" href="#bib0185">Tribbia (1984)</a>, Verkley (1984, <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>, <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>) and Neven (1993) proposed the use of eigenfunctions {Yj} as basic geometric structures.The space of spherical harmonics of degree n on S2, which coincides with the eigenspace of operator – Δg corresponding to the eigenvalue γn =χn=n(n+1), is denoted by Hn. Self-adjoint operators have the property that its eigenfunctions with different eigenvalues are orthogonal, which implies that the eigenspaces Hn are orthogonal and have 2n+1 dimensions. On the sphere, the homogeneous harmonic polynomials span the set of all polynomials, which in turn are dense in L2. Our spherical harmonics therefore span L2. If we take a basis within each eigenspace then this collection will give a basis for L2 of the sphere. The harmonics spherical term was introduced by Kelvin on potentials studies (<a class="elsevierStyleCrossRef" href="#bib0065">Hobson, 1931</a>) and is understood as the development of a function in terms of this series of spherical harmonics.The spaces Hn and Hk (n ≠ k) are mutually orthogonal in L2(S2). Then there is the orthogonal projection Yn : L2(S2) → Hn, and so smooth functions Ψ ∈ L2(S2) on the sphere S2 have a development in spherical harmonics,<elsevierMultimedia ident="eq0080"></elsevierMultimedia>where Ynλ,μ=∑m= −nnΨnm Ynmλ,μ is the homogeneous spherical polynomial of degree n from Hn, and Ψnm= < Ψ,Ynm> is the Fourier coefficient of Ψ. The 2n + 1 spherical harmonics<elsevierMultimedia ident="eq0085"></elsevierMultimedia>of degree n and zonal number m (–n ≤ m ≤ n) form an orthonormal basis in Hn. Here the numbers Cnm are the normalizers in L2(S2), given byCnm=2n+14πn−m!n+m!12 and Pnm are the associated Legendre functions given by<elsevierMultimedia ident="eq0090"></elsevierMultimedia>Considering that an oriented compact Riemannian manifold is a pair (S2, g) where S2 is the oriented compact manifold and g a Riemannian metric on S2, we can define in it covariant derivatives and various notions of curvature. When a manifold also has a group structure (so that multiplication and inversion are smooth), a very interesting structure called a Lie group (Bihlo, 2007; <a class="elsevierStyleCrossRef" href="#bib0030">Bihlo and Popoych, 2012</a>) arises. Even if a manifold S2 is not a Lie group, there may be an action : G × S2 → S2 of a Lie group G on S2, and under certain conditions S2 can be viewed as a “quotient” G/K, where K is a subgroup of G (<a class="elsevierStyleCrossRef" href="#bib0120">Richtmyer, 1981</a>). When S2 ≅ G/K as above, certain notions on G can be transported to S2, then we say that S2 is a homogeneous space. As an example of the last point we could mention the theory of spherical harmonic expansion on the S2, which is a homogeneous space for the rotation group O(n+1). The surface spherical harmonics are eigenfunctions for the Laplace-Beltrami operator, which is a rotation invariant (<a class="elsevierStyleCrossRef" href="#bib0060">Helgason, 1984</a>). Harmonic analysis is concerned with the representation of functions as the superposition of basic waves, the study and generalization of the notions of Fourier series as well as the Fourier transforms.Elements of harmonic analysis on the sphere can be found at <a class="elsevierStyleCrossRef" href="#bib0160">Stein and Weiss (1971)</a>. After introducing the manifold S2 and the Riemannian manifolds (S2, g), a general type of spaces (Besov and Triebel-Lizorkin spaces) on the sphere may also be introduced (<a class="elsevierStyleCrossRef" href="#bib0090">Narcowich et al., 2006</a>). Using the power of a Laplace operator, the Sobolev space on Riemannian manifolds can also be incorporated as a field currently undergoing great development (Aubin, 1998; <a class="elsevierStyleCrossRef" href="#bib0055">Hebey, 2000</a>).3Exact solutions to the barotropic vorticity equation on the manifold S2Let {(Ωι, φι)},ℓ=ι, κ be an atlas of S2 and ψ: S2 → R the streamfunction of class Cr. We can consider that the map Ψ=ψ∘φι−1:φιΩι→R and ψ∘φι−1 is the streamfunction defined on an open Uι = φι (Ωι) ⊂ R2 and that it is of class Cr.To simulate the time evolution of a two-dimensional nondivergent and inviscid flow for a rotating sphere, S2 is governed by a non-linear barotropic vorticity equation, which can be written in the non-dimensional form as<elsevierMultimedia ident="eq0095"></elsevierMultimedia>where Jc,q=∂c∂λ∂q∂μ−∂c∂μ∂q∂λ=k×∇c·∇q=u·∇q is the jacobian, u = k×∇c=uλ,uμ=−1−μ2∂c∂μ,11−μ2∂c∂λ is a tangent velocity vector, gradc=∇c=11−μ2∂c∂λ,1−μ2∂c∂μ,c=Ψ,ξ=ΔΨ=div grandΨ, is the relative vorticity q =ΔΨ + 2 μ is the absolute vorticity and k is a unit outward normal vector. The velocity vector field u having the components (uλ,uμ) is solenoidal: ∇ · u = 0 Throughout decades the nonlinear barotropic vorticity equation has been successfully used to describe low frequencies and large-scale barotropic processes of atmospheric dynamics. Despite the simplicity to this nonlinear equation, it contains the principal elements that describe the complexity of atmospheric behavior (<a class="elsevierStyleCrossRef" href="#bib0130">Simmons et al., 1983</a>; <a class="elsevierStyleCrossRef" href="#bib0140">Skiba, 1997</a>). The four types of exact solutions to <a class="elsevierStyleCrossRef" href="#eq0005">Eq. (1)</a> known up to now are described below:<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005">•The zonal flows and Rossby-Haurwitz (RH) waves (Haurwitz, 1949), called classical solutions, differentiated from the generalized solutions which are not so smooth.</li><li class="elsevierStyleListItem" id="lsti0010">•The first generalized solutions of Eq. <a class="elsevierStyleCrossRef" href="#eq0095">(6)</a>, kown as modons, were originally constructed by <a class="elsevierStyleCrossRef" href="#bib0185">Tribbia (1984)</a> and Verkley (1984, <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>, <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>) by using two eigenfunctions for the Laplace operator of different degrees.</li><li class="elsevierStyleListItem" id="lsti0015">•Later on, <a class="elsevierStyleCrossRef" href="#bib0095">Neven (1992)</a> gave generalized solutions in the form of a quadrupole modon.</li><li class="elsevierStyleListItem" id="lsti0020">•<a class="elsevierStyleCrossRef" href="#bib0205">Wu and Verkley (1993)</a> constructed generalized global solutions composed of two RH waves (<a class="elsevierStyleCrossRef" href="#bib0100">Pérez-Garcia and Skiba, 1999</a>).</li></ul>In the present work, zonal flows, homogeneous spherical polynomials flows, RH waves, and modons on the manifold S2 are considered.3.1Classical solutionsLet us consider the zonal flows, homogeneous spherical polynomials flows and Rossby-Haurwitz (RH) waves.Proposition (zonal flow). Let {(Ωι, φι)}, ℓ = ι, κ be an atlas of S2 and the streamfunction ψ : S2 → R of class Cr. Then the zonal flow map Ψι=ψ∘φι−1:Uι⊂Rι2→R of Cr defined as<elsevierMultimedia ident="eq0100"></elsevierMultimedia>is an exact solution of the vorticity Eq. <a class="elsevierStyleCrossRef" href="#eq0095">(6)</a> for any bn.Proof. The demonstration, obtained from Eq. <a class="elsevierStyleCrossRef" href="#eq0095">(6)</a>, is quite trivial.Proposition (homogeneous polynomials).Let {(Ωι, φι)},ℓ = ι, κ be an atlas of S2 and the streamfunction ψ : S2 → R of class Cr. Then the homogeneous spherical polynomial map Ψι=ψ∘φι−1:Uι⊂Rι2→R of degree n ≥ 2 defined as<elsevierMultimedia ident="eq0105"></elsevierMultimedia>is an exact solution to the vorticity Eq. <a class="elsevierStyleCrossRef" href="#eq0095">(6)</a>, where am can be a complex factor and<elsevierMultimedia ident="eq0110"></elsevierMultimedia>is the angular phase speed.Proof. Given Ψι ∈ Hn, we define Ψιλ, μ,  t= Ψn(λ−ct, μ)=∑m=−nnamYnmλ−ct, μ, then ∂Ψn∂t=−2Ψ′, and ∂Ψn∂λ=Ψ′, where Ψ′=∑m=−nnimamYnm(λ−ct, μ). If, in addition, we have the following expression<elsevierMultimedia ident="eq0245"></elsevierMultimedia> we have <elsevierMultimedia ident="eq0115"></elsevierMultimedia>from BVE (Eq. <a class="elsevierStyleCrossRef" href="#eq0095">(6)</a>). It follows that c =−2χn.Proposition (Rossby-Haurwitz waves). Let {(Ωι, φι)}, ℓ = ι, κ be an atlas of S2 and the streamfunction ψ : S2 → R of class C∞.Then, the map Ψι=ψ∘φι−1:Uι⊂Rι2→R of C∞ define as<elsevierMultimedia ident="eq0120"></elsevierMultimedia><elsevierMultimedia ident="eq0250"></elsevierMultimedia>with n ≥ 1 is called Rossby-Haurwitz (RH) waves.It is an exact solution of the vorticity Eq. <a class="elsevierStyleCrossRef" href="#eq0095">(6)</a> if the angular phase speed of the RH wave<elsevierMultimedia ident="eq0130"></elsevierMultimedia>Here ω is the super-rotation velocity and each Hn corresponds to the eigenvalue χn = n(n+1).Proof. Here Ψι is expressed by<elsevierMultimedia ident="eq0135"></elsevierMultimedia>where Ψnλ−ct, μ=∑m=−nnamYλ−ct, μ. We can notice that<elsevierMultimedia ident="eq0140"></elsevierMultimedia>which implies that<elsevierMultimedia ident="eq0145"></elsevierMultimedia>Furthermore:<elsevierMultimedia ident="eq0150"></elsevierMultimedia>where Ψ′=∑m=−nnimamYnm(λ−ct, μ); so that from BVE (Eq. <a class="elsevierStyleCrossRef" href="#eq0095">(6)</a>)<elsevierMultimedia ident="eq0155"></elsevierMultimedia>Hence<elsevierMultimedia ident="eq0160"></elsevierMultimedia>so that<elsevierMultimedia ident="eq0165"></elsevierMultimedia>and thus the proposition is proved.The streamfunction of the stationary RH(2,5) wave<elsevierMultimedia ident="eq0170"></elsevierMultimedia>with the parameters defined by (m, n) = (2, 5), a = .007 and ω=23(χ3−2) is given in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>.<elsevierMultimedia ident="fig0010"></elsevierMultimedia><a class="elsevierStyleCrossRef" href="#bib0105">Pérez and Skiba (2001)</a> and <a class="elsevierStyleCrossRef" href="#bib0145">Skiba and Pérez (2006)</a> developed a numerical spectral method for the normal mode instability study of the arbitrary steady flow of an ideal nondivergent fluid on a rotating sphere, and <a class="elsevierStyleCrossRef" href="#bib0145">Skiba and Pérez (2006)</a> tested this method for the RH(2,5) wave. <a class="elsevierStyleCrossRef" href="#bib0115">Pérez-García (2014)</a> constructed a basic flow regarded as a sum of a zonally symmetric flow (<a class="elsevierStyleCrossRef" href="#eq0100">Eq. 7</a> and a Rossby-Haurwitz wave component (<a class="elsevierStyleCrossRef" href="#eq0120">Eq. 9</a>.3.2Generalized solutionsDenote the spherical distance between two points of S2 by d(.,.). Let N’ be the north pole of the chart coordinate (Ωι, φκ). Then a disk or inner region Si on the sphere is defined as Si=DN′,φa={s ∈ S2| dN′,s<φa}, such that 0<φa≤π2 The solution modon is constructed (<a class="elsevierStyleCrossRef" href="#bib0185">Tribbia, 1984</a>; <a class="elsevierStyleCrossRef" href="#bib0190">Verkley, 1984</a>, <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>, <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>; Neven, 1993) to divide the sphere S2 into two regions: an inner region Si centered around the pole N’, and an outer region So separated from the inner region by a boundary circle ∂DN′,φa={s ∈ S2| dN′,s<φa}, on which ψ, q and ψ’ are continuous.For Si a solution of the Eq. <a class="elsevierStyleCrossRef" href="#eq0010">(2)</a> form is chosen with an eigenfunction Y (λ’, μ’) which has its singularity in the outer region. The same type of solution is chosen for the outer region, but such that Y (λ’, μ’) has its singularity in the inner region. Then both solutions are combined as smoothly as possible on the boundary circle ∂DN′,φa (<a class="elsevierStyleCrossRef" href="#bib0185">Tribbia, 1984</a>; <a class="elsevierStyleCrossRef" href="#bib0190">Verkley, 1984</a>, <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>).To construct the <a class="elsevierStyleCrossRef" href="#bib0190">Verkley (1984)</a> modon or the <a class="elsevierStyleCrossRef" href="#bib0095">Neven (1992)</a> cuadrupole modon on the manifold S2, it is interpreted as:Proposition. Let {(Ωℓφℓ)}, ℓ = ι,κ, be an atlas of S2, and ψ = ψ1 + ψ2 : S2 → R the streamfunction of Cr. Then<elsevierMultimedia ident="eq0175"></elsevierMultimedia>Proof. Let ψ1 and ψ2 be two real-value functions of class Cr defined on the differential manifolds S2. We define their sum by setting Ψι=(ψ1+ψ2)∘φι−1=ψ1∘φι−1+ψ2∘φι−1 for any chart (Ωι,φι) Since the sum of two functions of class Cr on S2 are functions of class Cr, the proof of this formula can be obtained by the expression<elsevierMultimedia ident="eq0180"></elsevierMultimedia>where φικ(λ, μ)=(φικ1 (λ, μ), φικ2 (λ, μ))=(λ′(λ, μ),μ′(λ, μ))Decompose now the streamfunctions into an eigenfunction part (ψ1∘φκ−1) (λ′, μ′)=Yν(λ′,  μ′) and a zonal part (ψ2∘φι−1) (λ, μ)=−ωμ+D where –ωμ is a solid-body rotation and D a constant. In chart (Ωκ, φκ) with coordinates (λ’, μ’), the north pole N’ moves along a circle of constant latitude with constant angular velocity cν. In the primed coordinates, Verkley (1984, <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>) modons have the form<elsevierMultimedia ident="eq0185"></elsevierMultimedia>which consists of a dipole and a monopole component:<elsevierMultimedia ident="eq0190"></elsevierMultimedia>where μ0 = sen φ0μa = sen φa. The functions fd(μ) and fm(μ) are defined as<elsevierMultimedia ident="eq0195"></elsevierMultimedia>and<elsevierMultimedia ident="eq0200"></elsevierMultimedia>where b=k2+14+2α(α+1)−2 and<elsevierMultimedia ident="eq0205"></elsevierMultimedia>The fact that<elsevierMultimedia ident="eq0210"></elsevierMultimedia>is a solution to Eq. <a class="elsevierStyleCrossRef" href="#eq0095">(6)</a> is due to the work of <a class="elsevierStyleCrossRef" href="#bib0190">Verkley (1984)</a>, which I will not reproduce in this paper. Yν is an eigenfunction of the Laplace-Beltrami operator and χν = –ν(ν + 1) is the eigenvalue of Yν. The Legendre functions Hμ=Pνmμ and Hμ=Qνmμ are solutions to the Legendre differential equation of hypergeometric type, where Pνm(μ) is a Legendre function of the first kind and Qνm(μ) is a Legendre function of the second kind for order m such that ν is the complex degree. The explicit expresion for Pνmμ and Qνmμ with –1 < μ < 1 can be found in <a class="elsevierStyleCrossRef" href="#bib0005">Abramowitz and Stegun (1965)</a> or <a class="elsevierStyleCrossRef" href="#bib0190">Verkley (1984)</a>.By using a grid of 5 × 5° upon the local coordinate associated with the chart (Ωκ,φκ), the <a class="elsevierStyleCrossRef" href="#bib0190">Verkley, 1984</a> modon was numerically generated. Using <a class="elsevierStyleCrossRef" href="#eq0020">Eqs. (4)</a> and <a class="elsevierStyleCrossRef" href="#eq0025">(5)</a> a workable Gaussian mesh of (128, 64) points upon the geographical coordinate group (Ωι, φι) was also generated. This mesh was mapped onto the local coordinates system associated to the chart (Ωκ,φκ). The values of Ψκ were interpolated on the Gaussians points (128, 64) by implementing a nine- point Lagrange interpolation scheme. The resulting function, i.e. the <a class="elsevierStyleCrossRef" href="#bib0190">Verkley, 1984</a> equatorial modon viewed on the geographical coordinate group (Ωι, φι), is shown in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3a</a>. This small modon was defined by the following parameters:<elsevierMultimedia ident="eq0215"></elsevierMultimedia><elsevierMultimedia ident="fig0015"></elsevierMultimedia>A numerical spectral model was used to simulate this small-size modon in <a class="elsevierStyleCrossRef" href="#bib0100">Pérez-García and Skiba (1999)</a>, and in <a class="elsevierStyleCrossRef" href="#bib0150">Skiba and Pérez-García (2009)</a> a numerical spectral method for normal mode stability study of ideal flows on a rotating sphere was tested for this isolated steady modon constructed by <a class="elsevierStyleCrossRef" href="#bib0190">Verkley (1984)</a>.Studies done by <a class="elsevierStyleCrossRef" href="#bib0070">Illari (1984)</a> and <a class="elsevierStyleCrossRef" href="#bib0035">Crum and Stevens (1988)</a> noted that the values of isentropic potential vorticity are relatively low and uniform in the blocking region. In our following argument we consider that Verkley's modon (1990) provides a better and more uniform description of atmospheric blocking. Our interest lays within the fields that build this phenomenon. These solutions are characterized by a region Si in which q is constant, and an outer region So separated from the inner region by the boundary circle ∂D, on which Ψ and q are both constant, i.e., Ψ = d and q = b.In the primed coordinates the <a class="elsevierStyleCrossRef" href="#bib0200">Verkley (1990)</a> modon has the form<elsevierMultimedia ident="eq0220"></elsevierMultimedia>where solid-body rotation terms can be expressed in primed coordinates using <a class="elsevierStyleCrossRefs" href="#eq0020">Eqs. (4-5)</a>, such that in chart (Ωι, φι) the eigenfuctions at the outer region and inner region are Δ′Yo=−χoYo; Δ′Yi=ei being χo and ei constants. Certain requirements of continuity must be met to generate these functions over the circle ∂D. We require continuity in and the first and second derivative in φ’ at φa.To construct the <a class="elsevierStyleCrossRef" href="#bib0200">Verkley, 1990</a> uniform modon on the manifold S2, it is interpreted as:Proposition. Consider an atlas {(Ωℓ, φℓ)}, ℓ = ι,κ, of S2 and let ψ = ψ1 + ψ2 : S2 → R be the stream-function of Cr. Then<elsevierMultimedia ident="eq0225"></elsevierMultimedia>Proof. Let ψ1 and ψ2 be two real-value functions of class Cr defined on the differential manifolds S2. We define their sum by setting Ψκψ1+ψ2∘φκ−1=ψ1∘φκ−1+ψ2∘φκ−1 for any chart (Ωκ, φκ). Since the sum of two functions of class Cr on S2 are functions of class Cr, the proof of the proposition follows from:<elsevierMultimedia ident="eq0255"></elsevierMultimedia>because λ,μ=φκιλ′,μ′According to <a class="elsevierStyleCrossRef" href="#bib0200">Verkley (1990)</a>, to express the modon in a more explicit manner, the functional forms Yo and Yi of the eigenfuntions are:<elsevierMultimedia ident="eq0230"></elsevierMultimedia>where As and Bs are constant. The special functions Sσmθ′=Pσm−cos θ′ and Tm(θ’) were given by <a class="elsevierStyleCrossRef" href="#bib0200">Verkley (1990)</a>.We have also reproduced numerically the uniform modon constructed by <a class="elsevierStyleCrossRef" href="#bib0200">Verkley (1990)</a> using the parameters φa=5π12,  σ=8.06;  ωo=0.028, and when the modon center is in the point λo=180°,φo=π4 (<a class="elsevierStyleCrossRef" href="#fig0015">Fig 3b</a>).4ConclusionsThe exact solutions of the barotropic vorticity equation on the rotating unit sphere as a compact differentiable manifold without boundary, which are zonal flows, homogeneous spherical polynomial flows, Rossby-Haurwitz waves and generalized solutions named modons, were represented in this paper. A concrete notion of local chart, change of charts, and atlas for the manifolds S2 was developed. An atlas {(Ωι, φι), (Ωκ, φκ)} was constructed for S2, where every chart corresponds to a geographical coordinate group that covers most of S2. The transition maps φικ=φκ∘φι−1:φιΩι∩Ωκ→φκΩι∩Ωκ and φκι=φι∘φκ−1:φκΩκ∩Ωι→φιΩκ∩Ωι were also constructed, and the exact solutions of the barotropic vorticity equation in a manifold context were studied. This work also formulates on the manifolds S2 in terms of the stream function ψ: S2 → R, for RH waves which are sufficiently smooth, and for Wu-Verkley waves and modons which are differentiably weak. RH waves as solutions ψ∘φι−1 of the barotropic vorticity equation on the manifolds S2 are presented at the local coordinate associated with a chart (Ωι, φι). The way in which the modon solution ψ∘φκ−1 can be constructed in the chart (Ωκ, φκ), where ψ is C2, is investigated too. In the chart coordinate (Ωκ, φκ), the Verkley (1984, <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>) modons have the form<elsevierMultimedia ident="eq0235"></elsevierMultimedia>To construct the <a class="elsevierStyleCrossRef" href="#bib0200">Verkley (1990)</a> uniform modon on the manifold S2, it is interpreted as<elsevierMultimedia ident="eq0240"></elsevierMultimedia>when the modon center is in the point λo = 270; φo = 0. However, to contruct the <a class="elsevierStyleCrossRef" href="#bib0190">verkley (1984</a>,<a class="elsevierStyleCrossRef" href="#bib0195">1987</a>, <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>) with N’ a point arbitrary on the sphere S2 a collection of pairs (Ωι, φι) (i > 2) is needed. Our viewpoint here was to understand the solution of the barotropic vorticity equation on the manifold S2 and its use to derive properties of the solutions to the Riemannian manifold (S2, g)." "textoCompletoSecciones" => array:1 [ "secciones" => array:9 [ 0 => array:3 [ "identificador" => "xres901033" "titulo" => "Resumen" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:3 [ "identificador" => "xres901032" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0010" ] ] ] 2 => array:2 [ "identificador" => "xpalclavsec882097" "titulo" => "Keywords" ] 3 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 4 => array:2 [ "identificador" => "sec0010" "titulo" => "Structure of functions on the manifold S" ] 5 => array:3 [ "identificador" => "sec0015" "titulo" => "Exact solutions to the barotropic vorticity equation on the manifold S" "secciones" => array:2 [ 0 => array:2 [ "identificador" => "sec0020" "titulo" => "Classical solutions" ] 1 => array:2 [ "identificador" => "sec0025" "titulo" => "Generalized solutions" ] ] ] 6 => array:2 [ "identificador" => "sec0030" "titulo" => "Conclusions" ] 7 => array:2 [ "identificador" => "xack299707" "titulo" => "Acknowledgments" ] 8 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2014-11-07" "fechaAceptado" => "2015-06-01" "PalabrasClave" => array:1 [ "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Keywords" "identificador" => "xpalclavsec882097" "palabras" => array:5 [ 0 => "Rossby-Haurwitz waves" 1 => "modons" 2 => "hydrodynamics equation on manifolds" 3 => "unit sphere" 4 => "mathematical analysis of barotropic model" ] ] ] ] "tieneResumen" => true "resumen" => array:2 [ "es" => array:2 [ "titulo" => "Resumen" "resumen" => "El propósito de este trabajo es representar las soluciones exactas de la ecuación de vorticidad barotrópica sobre la esfera unitaria S2 en rotación como una variedad, que son flujos zonales, ondas Rossby-Haurwitz y soluciones generalizadas llamadas modones. Se relacionan los métodos modernos de la teoría de funciones con la esfeoa definida como una variedad compacta y diferenciable. Cuando ésta se ha comprendido de forma correcta, se esclarece la noción abstracta de mapa local, cambio de mapa y atlas. Uno de los objetivos de este trabajo ns entender mejor lo solución de la ecuación de vorticidad barotrópica sobre la varieded S2 y su utilidad para identificar las propiedades de lac soluciones en la variedad Riemanniana (S2, g). Por lo tanto, estará disponible un tipo más general de espacio que también puede contener información geométrica y analítica sustancial sobre las soluciones a la ecuación de vorticidad barotrópica." ] "en" => array:2 [ "titulo" => "Abstract" "resumen" => "The purpose of this paper is to represent the exact solutions of the barotropic vorticity equations (BVE) on the rotating unit sphere S2 as a manifold, which are zonal flows, Rossby-Haurwitz waves and generalized solutions named modons. Modern methods of the function theory are connected to the sphere defined as a compact diferentiable manifold. When the differentiable manifold S2 is well understood, the abstract notion of local chart, change of chart, and atlases becomes evident. One of the aims of this paper is to better understand the solution of the barotropic vorticity equation on the manifold S2 and its usefulness to identify the properties of the solutions on the Riemannian manifold (S2, g). Therefore, a more general type of space will be available, which can also contain substantial geometric and analytic information about sotutions for the barotropic vorticity equation." ] ] "multimedia" => array:53 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Fig. 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 853 "Ancho" => 1392 "Tamanyo" => 168427 ] ] "descripcion" => array:1 [ "en" => "The set {(Ωv, φι), (Ωκ, φκ)} forms an atlas for S2: Ωι ∪ Ωκ = S2." ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Fig. 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 475 "Ancho" => 948 "Tamanyo" => 146568 ] ] "descripcion" => array:1 [ "en" => "Isolines of the streamfunction of Rossby-Haurwitz waves (2, 5)." ] ] 2 => array:7 [ "identificador" => "fig0015" "etiqueta" => "Fig. 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 546 "Ancho" => 1891 "Tamanyo" => 268273 ] ] "descripcion" => array:1 [ "en" => "Streamfunction isolines of equatorial <a class="elsevierStyleCrossRef" href="#bib0190">Verkley modon (1984)</a> with (a) k = 10., a = 10.,φa= 66.14°, λ0 = 270.0° and φ0 = 0.°; the uniform Verkley modon (1990) at (b) σ = 8.06, φa = φa=5π12,λ0=180.0°, φ0 = 180.0° and φ0 = 45.0°. The curve points are the spherical coordinates relative to a rotated pole N’(270.0°, 0.°) at (a) N’(180.0°, 45.°) at (b) with respect to the original system." ] ] 3 => array:6 [ "identificador" => "eq0005" "etiqueta" => "(1)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∂ΔΨ∂t+J Ψ, ΔΨ+2μ=0" "Fichero" => "STRIPIN_si2.jpeg" "Tamanyo" => 1772 "Alto" => 21 "Ancho" => 191 ] ] 4 => array:6 [ "identificador" => "eq0010" "etiqueta" => "(2)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψλ,μ,t=Yνλ′,μ′−ωμ+D" "Fichero" => "STRIPIN_si4.jpeg" "Tamanyo" => 2037 "Alto" => 20 "Ancho" => 233 ] ] 5 => array:6 [ "identificador" => "eq0015" "etiqueta" => "(3)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "cν=ω−2(ω+1)χν" "Fichero" => "STRIPIN_si5.jpeg" "Tamanyo" => 984 "Alto" => 25 "Ancho" => 108 ] ] 6 => array:6 [ "identificador" => "eq0020" "etiqueta" => "(4)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "φικ1(xι1,xι2)=xκ1(xι1,xι2)=λ′(λ,μ)=tan−1sin(λ−λ0)μocos(λ−λo)−μ1−μo21−μ2" "Fichero" => "STRIPIN_si30.jpeg" "Tamanyo" => 4903 "Alto" => 60 "Ancho" => 441 ] ] 7 => array:6 [ "identificador" => "eq0025" "etiqueta" => "(5)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "φικ2(xι1,xι2)=xκ2(xι1,xι2)=μ′(λ,μ)=μμo+1−μ21−μo2cos(λ−λo)" "Fichero" => "STRIPIN_si31.jpeg" "Tamanyo" => 4740 "Alto" => 30 "Ancho" => 515 ] ] 8 => array:5 [ "identificador" => "eq0030" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "det J φικ=∂φικ1∂xι1∂φικ1∂xι2∂φικ2∂xι1∂φικ2∂xι2=1" "Fichero" => "STRIPIN_si36.jpeg" "Tamanyo" => 3724 "Alto" => 89 "Ancho" => 200 ] ] 9 => array:5 [ "identificador" => "eq0035" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "gij=1−μ2001" "Fichero" => "STRIPIN_si67.jpeg" "Tamanyo" => 1746 "Alto" => 49 "Ancho" => 136 ] ] 10 => array:5 [ "identificador" => "eq0040" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "grad ψ=∑k, l2(gkl∂Ψ∂xιl)∂∂xιl=11−μ2∂Ψ∂λeˆλ+1−μ2∂Ψ∂μeˆμ" "Fichero" => "STRIPIN_si69.jpeg" "Tamanyo" => 4441 "Alto" => 50 "Ancho" => 377 ] ] 11 => array:5 [ "identificador" => "eq0045" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "div u =1g∑i∂∂xιi(uig)=11−μ2∂uλ∂λ+∂uμ1−μ2∂μ" "Fichero" => "STRIPIN_si72.jpeg" "Tamanyo" => 3563 "Alto" => 36 "Ancho" => 335 ] ] 12 => array:5 [ "identificador" => "eq0050" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "D∂∂xιi(∂∂xιj)(p)=Γijk(p)(∂∂xιk)p" "Fichero" => "STRIPIN_si76.jpeg" "Tamanyo" => 2271 "Alto" => 31 "Ancho" => 186 ] ] 13 => array:5 [ "identificador" => "eq0055" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Δgψ=−gij(∂2ψ∂xιi∂xιj−Γijk∂ψ∂xιk)=1(1−μ2)∂2Ψ∂λ2+∂∂μ(1−μ2)∂Ψ∂μ" "Fichero" => "STRIPIN_si81.jpeg" "Tamanyo" => 4425 "Alto" => 32 "Ancho" => 411 ] ] 14 => array:5 [ "identificador" => "eq0060" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∫S2ψ dυg=∑l∫φlΩlΦlgψ°φl−1 dxl" "Fichero" => "STRIPIN_si86.jpeg" "Tamanyo" => 2858 "Alto" => 36 "Ancho" => 276 ] ] 15 => array:5 [ "identificador" => "eq0065" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "<f,g> = ∫S2fg*  dυg" "Fichero" => "STRIPIN_si88.jpeg" "Tamanyo" => 1599 "Alto" => 19 "Ancho" => 164 ] ] 16 => array:5 [ "identificador" => "eq0070" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Lp(S2)=ψ:S2→R;ψ measurable and∫s2fp<∞," "Fichero" => "STRIPIN_si90.jpeg" "Tamanyo" => 3430 "Alto" => 20 "Ancho" => 384 ] ] 17 => array:5 [ "identificador" => "eq0075" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "fp=∫s2fpdυg1p" "Fichero" => "STRIPIN_si91.jpeg" "Tamanyo" => 1611 "Alto" => 32 "Ancho" => 154 ] ] 18 => array:5 [ "identificador" => "eq0080" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψλ,μ=∑n=1∞Ynλ,μ" "Fichero" => "STRIPIN_si92.jpeg" "Tamanyo" => 1994 "Alto" => 46 "Ancho" => 158 ] ] 19 => array:5 [ "identificador" => "eq0085" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ynmλ,μ=CnmPnmμeimλ" "Fichero" => "STRIPIN_si95.jpeg" "Tamanyo" => 1749 "Alto" => 18 "Ancho" => 190 ] ] 20 => array:5 [ "identificador" => "eq0090" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Pnmμ=1−μ2m22nn!dn+mdμn+m(μ2−1)n" "Fichero" => "STRIPIN_si98.jpeg" "Tamanyo" => 2492 "Alto" => 34 "Ancho" => 228 ] ] 21 => array:6 [ "identificador" => "eq0095" "etiqueta" => "(6)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∂ΔΨ∂t+JΨ,ΔΨ+2μ=0" "Fichero" => "STRIPIN_si101.jpeg" "Tamanyo" => 1772 "Alto" => 21 "Ancho" => 186 ] ] 22 => array:6 [ "identificador" => "eq0100" "etiqueta" => "(7)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψιλ,μ=∑n=0NbnPn0(μ)" "Fichero" => "STRIPIN_si106.jpeg" "Tamanyo" => 1980 "Alto" => 49 "Ancho" => 158 ] ] 23 => array:6 [ "identificador" => "eq0105" "etiqueta" => "(8)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψιλ,μ,t=∑m=−nNamYnmλ−ct, μ∈ Hn" "Fichero" => "STRIPIN_si108.jpeg" "Tamanyo" => 2791 "Alto" => 48 "Ancho" => 286 ] ] 24 => array:5 [ "identificador" => "eq0110" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "c=−2χn" "Fichero" => "STRIPIN_si109.jpeg" "Tamanyo" => 577 "Alto" => 23 "Ancho" => 58 ] ] 25 => array:5 [ "identificador" => "eq0245" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "ΔΨι=−χnΨn;∂ΔΨι∂μ=−χn∂Ψn∂μ;ΔΨι+2μ=−χnΨι+2μ; ∂ΔΨι∂t=cχnΨ′" "Fichero" => "STRIPIN_si114.jpeg" "Tamanyo" => 4243 "Alto" => 24 "Ancho" => 505 ] ] 26 => array:5 [ "identificador" => "eq0115" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Δ∂Ψι∂t=χncΨ′=J−χnΨn+2μ,Ψn=−2∂Ψn∂λ=−2Ψ′" "Fichero" => "STRIPIN_si115.jpeg" "Tamanyo" => 3120 "Alto" => 21 "Ancho" => 387 ] ] 27 => array:6 [ "identificador" => "eq0120" "etiqueta" => "(9)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψιλ, μ, t=−ωμ+∑m=−nnamYnm" "Fichero" => "STRIPIN_si118.jpeg" "Tamanyo" => 2427 "Alto" => 45 "Ancho" => 228 ] ] 28 => array:5 [ "identificador" => "eq0250" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "λ−ct,μ∈H0⊕Hn" "Fichero" => "STRIPIN_si119.jpeg" "Tamanyo" => 1322 "Alto" => 14 "Ancho" => 155 ] ] 29 => array:6 [ "identificador" => "eq0130" "etiqueta" => "(10)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "c=ω−2(ω+1)χn." "Fichero" => "STRIPIN_si120.jpeg" "Tamanyo" => 952 "Alto" => 25 "Ancho" => 107 ] ] 30 => array:5 [ "identificador" => "eq0135" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψιλ, μ, t=−ωμ+Ψnλ−ct, μ" "Fichero" => "STRIPIN_si121.jpeg" "Tamanyo" => 1933 "Alto" => 14 "Ancho" => 249 ] ] 31 => array:5 [ "identificador" => "eq0140" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "ΔΨι=2ωμ−χnΨn=−χnΨι+2ω−χnωμ" "Fichero" => "STRIPIN_si123.jpeg" "Tamanyo" => 2503 "Alto" => 14 "Ancho" => 318 ] ] 32 => array:5 [ "identificador" => "eq0145" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "ΔΨι+2μ=−χnΨι−χn(ω−2(ω+1)χn)μ =−χnΨι+[(2−χn)ω+2]μ." "Fichero" => "STRIPIN_si124.jpeg" "Tamanyo" => 4049 "Alto" => 25 "Ancho" => 480 ] ] 33 => array:5 [ "identificador" => "eq0150" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∂∂λΔΨι=−χn∂Ψι∂λ,Δ∂Ψι∂t=χncΨ′,∂∂μΔΨι=(2−χn)ω−χn∂Ψι∂μ" "Fichero" => "STRIPIN_si125.jpeg" "Tamanyo" => 3672 "Alto" => 24 "Ancho" => 429 ] ] 34 => array:5 [ "identificador" => "eq0155" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Δ∂Ψι∂t=−J (Ψι,ΔΨι+2μ)=∂Ψι∂μ∂ΔΨι∂λ−∂Ψι∂λ ∂ΔΨι∂μ−2∂Ψι∂λ=∂Ψι∂μ−χn∂ΔΨι∂λ−∂Ψι∂λ(2−χn)ω−χn∂Ψι∂μ−2∂Ψι∂λ." "Fichero" => "STRIPIN_si127.jpeg" "Tamanyo" => 7801 "Alto" => 52 "Ancho" => 756 ] ] 35 => array:5 [ "identificador" => "eq0160" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "χncΨ′=−∂Ψι∂λ(2−χn) ω−2∂Ψι∂λ=                −(2−χn) ω+2Ψ′=                −−ωχn+2(ω+1)Ψ′" "Fichero" => "STRIPIN_si128.jpeg" "Tamanyo" => 4915 "Alto" => 80 "Ancho" => 247 ] ] 36 => array:5 [ "identificador" => "eq0165" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "c=ω−2(ω+1)χn" "Fichero" => "STRIPIN_si129.jpeg" "Tamanyo" => 902 "Alto" => 25 "Ancho" => 102 ] ] 37 => array:6 [ "identificador" => "eq0170" "etiqueta" => "(11)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψ(λ, μ)=−ωμ+a Pnm(μ) cos(mλ)" "Fichero" => "STRIPIN_si130.jpeg" "Tamanyo" => 2099 "Alto" => 16 "Ancho" => 245 ] ] 38 => array:5 [ "identificador" => "eq0175" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψι(λ, μ)=(ψ1∘φκ−1)(λ′,μ′)+(ψ2∘φι−1)(λ, μ)." "Fichero" => "STRIPIN_si136.jpeg" "Tamanyo" => 3197 "Alto" => 19 "Ancho" => 334 ] ] 39 => array:5 [ "identificador" => "eq0180" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψι(λ, μ)=(ψ∘φι−1)(λ, μ)=ψ1∘φι−1(λ, μ)+ψ2∘φι−1(λ, μ)                  =(ψ1∘φκ−1)∘(φκ∘φι−1)(λ, μ)+(ψ2∘φι−1) (λ, μ)                  =(ψ1∘φκ−1)∘φικ(λ, μ)+(ψ2∘φι−1) (λ, μ)                  =(ψ1∘φκ−1)(λ′, μ′)+(ψ2∘φι−1) (λ, μ)" "Fichero" => "STRIPIN_si138.jpeg" "Tamanyo" => 11786 "Alto" => 93 "Ancho" => 431 ] ] 40 => array:5 [ "identificador" => "eq0185" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "(ψ1∘φκ−1) (λ′,  μ′)=X(λ′,  μ′)=Xd(μ′)cos λ′+Xm(μ′)→Yν(λ′,  μ′)" "Fichero" => "STRIPIN_si142.jpeg" "Tamanyo" => 4258 "Alto" => 19 "Ancho" => 489 ] ] 41 => array:5 [ "identificador" => "eq0190" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Xd(μ′)=(cν−ω)1−μa21−μ02fd(μ′)Xm(μ′)=(cν−ω)μ01−μa2fm(μ′)" "Fichero" => "STRIPIN_si143.jpeg" "Tamanyo" => 5112 "Alto" => 54 "Ancho" => 287 ] ] 42 => array:5 [ "identificador" => "eq0195" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "fdμ=−bB1,1,μ+1+b1−μ21−μa212, if μ≥μaP1,1, −μ,  if μ<μa" "Fichero" => "STRIPIN_si144.jpeg" "Tamanyo" => 5446 "Alto" => 81 "Ancho" => 395 ] ] 43 => array:5 [ "identificador" => "eq0200" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "fm(μ)=−bB0,  1,  μ +1 + bμ−μa1−μa2        −P(0, 1, −μa)+bB(0, 1, μa),    if μ≥μa−P(1,1,−μ), if μ <μa " "Fichero" => "STRIPIN_si145.jpeg" "Tamanyo" => 6952 "Alto" => 88 "Ancho" => 392 ] ] 44 => array:5 [ "identificador" => "eq0205" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Br,s,μ=PαrμPαSμa;  Pr,s,μ=P−0.5+ikrμP−0.5+ikS−μa" "Fichero" => "STRIPIN_si147.jpeg" "Tamanyo" => 3401 "Alto" => 33 "Ancho" => 302 ] ] 45 => array:5 [ "identificador" => "eq0210" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψ(λ, μ, t)=Yν(λ′, μ′)−ωμ+D" "Fichero" => "STRIPIN_si148.jpeg" "Tamanyo" => 1862 "Alto" => 16 "Ancho" => 233 ] ] 46 => array:5 [ "identificador" => "eq0215" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "k=10, α=10, μa=sin 66.14°, μ0=0, λ0=270° and D0=0" "Fichero" => "STRIPIN_si155.jpeg" "Tamanyo" => 3574 "Alto" => 14 "Ancho" => 441 ] ] 47 => array:5 [ "identificador" => "eq0220" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψκλ′,μ′=Yo λ′,μ′−ωoμ+DoatSoYi λ′,μ′−ωiμ+DiatSi" "Fichero" => "STRIPIN_si156.jpeg" "Tamanyo" => 4587 "Alto" => 49 "Ancho" => 302 ] ] 48 => array:5 [ "identificador" => "eq0225" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψκλ′,μ′=ψ1∘φκ−1λ′,μ′+ψ2∘φκ−1λ,μ" "Fichero" => "STRIPIN_si158.jpeg" "Tamanyo" => 3262 "Alto" => 20 "Ancho" => 353 ] ] 49 => array:5 [ "identificador" => "eq0255" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψκλ′,μ′=ψ1∘φκ−1λ′,μ′+ψ2∘φκ−1λ′,μ′                        =ψ1∘φκ−1λ′,μ′+ψ2∘φι−1∘φι∘φκ−1λ′,μ′                        =ψ1∘φκ−1λ′,μ′+ψ2∘φι−1λ,μ" "Fichero" => "STRIPIN_si160.jpeg" "Tamanyo" => 8672 "Alto" => 70 "Ancho" => 447 ] ] 50 => array:5 [ "identificador" => "eq0230" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Yoλ′,μ′=A0Sσ1θ′ cos λ′+B0Sσ0θ′Yiλ′,μ′=AiT1θ′ cos λ′+BiT0θ′" "Fichero" => "STRIPIN_si162.jpeg" "Tamanyo" => 5218 "Alto" => 45 "Ancho" => 280 ] ] 51 => array:5 [ "identificador" => "eq0235" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψιλ, μ=ψ1∘φκ−1λ′, μ′+ψ2∘φι−1λ, μ" "Fichero" => "STRIPIN_si170.jpeg" "Tamanyo" => 3341 "Alto" => 20 "Ancho" => 348 ] ] 52 => array:5 [ "identificador" => "eq0240" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ψκλ′, μ′=ψ1∘φκ−1λ′, μ′+ψ2∘φι−1λ, μ" "Fichero" => "STRIPIN_si171.jpeg" "Tamanyo" => 3338 "Alto" => 20 "Ancho" => 361 ] ] ] "bibliografia" => array:2 [ "titulo" => "References" "seccion" => array:1 [ 0 => array:2 [ "identificador" => "bibs0005" "bibliografiaReferencia" => array:41 [ 0 => array:3 [ "identificador" => "bib0005" "etiqueta" => "Abramowitz and Stegun, 1965" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Handbook of mathematical functions" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:2 [ 0 => "M. 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Ismael Pérez-García

Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Circuito de la Investigación Científica s/n, Ciudad Universitaria, 04510 México, D.F.

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            "entidad" => "Centro de Ciencias de la Atm&#243;sfera&#44; Universidad Nacional Aut&#243;noma de M&#233;xico&#44; Circuito de la Investigaci&#243;n Cient&#237;fica s&#47;n&#44; Ciudad Universitaria&#44; 04510 M&#233;xico&#44; D&#46;F&#46;"
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          "en" => "<p id="spar0025" class="elsevierStyleSimplePara elsevierViewall">Streamfunction isolines of equatorial <a class="elsevierStyleCrossRef" href="#bib0190">Verkley modon &#40;1984&#41;</a> with &#40;a&#41; <span class="elsevierStyleItalic">k</span> &#61; 10&#46;&#44; <span class="elsevierStyleItalic">a</span> &#61; 10&#46;&#44;<span class="elsevierStyleItalic">&#966;<span class="elsevierStyleInf">a</span></span>&#61; 66&#46;14&#176;&#44; <span class="elsevierStyleItalic">&#955;</span><span class="elsevierStyleInf">0</span> &#61; 270&#46;0&#176; and <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> &#61; 0&#46;&#176;&#59; the uniform Verkley modon &#40;1990&#41; at &#40;b&#41; &#963; &#61; 8&#46;06&#44; <span class="elsevierStyleItalic">&#966;<span class="elsevierStyleInf">a</span></span> &#61; &#966;a&#61;5&#960;12&#44;&#955;0&#61;180&#46;0&#176;&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> &#61; 180&#46;0&#176; and <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> &#61; 45&#46;0&#176;&#46; The curve points are the spherical coordinates relative to a rotated pole <span class="elsevierStyleItalic">N&#8217;</span>&#40;270&#46;0&#176;&#44; 0&#46;&#176;&#41; at &#40;a&#41; <span class="elsevierStyleItalic">N</span>&#8217;&#40;180&#46;0&#176;&#44; 45&#46;&#176;&#41; at &#40;b&#41; with respect to the original system&#46;</p>"
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    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">1</span><span class="elsevierStyleSectionTitle" id="sect0020">Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">Let <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#61;&#123;<span class="elsevierStyleItalic">x</span> &#8712; <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">3</span>&#58; &#124; <span class="elsevierStyleItalic">x</span> &#124;&#61; 1&#125; denote the unit sphere it <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">3</span>&#46; The large-scale dynamics of tire atmosphere on the rotating sphere <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> can approximately be governed by the non-linear barotropic vorticity equation &#40;BVE&#41;&#44; which can be written in the non-di-mensional from as&#58;<elsevierMultimedia ident="eq0005"></elsevierMultimedia>where &#936;&#40;<span class="elsevierStyleItalic">&#955;&#44;&#956;</span>&#41; denotes the stream function&#44; &#956;&#61;sin&#966;&#61;cos&#952;&#44;&#8722;&#960;&#8804;&#955;&#8804;&#960;&#44;&#8722;&#960;2&#8804;&#966;&#8804;&#960;2&#44;0&#60;&#952;&#60;&#960;&#44;&#955; the longitude&#44; <span class="elsevierStyleItalic">&#966;</span> the latitude&#44; and <span class="elsevierStyleItalic">&#952;</span> the colatitude&#46; &#916; is the Laplace-Beltrami operator on a sphere and <span class="elsevierStyleItalic">J</span>&#40;&#936;&#44; <span class="elsevierStyleItalic">h</span>&#41; is the Jacobian&#46;</p><p id="par0010" class="elsevierStylePara elsevierViewall">The following is a solution for Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0005">&#40;1&#41;</a> on the sphere proposed by <a class="elsevierStyleCrossRef" href="#bib0180">Thompson &#40;1982&#41;</a>&#58;<elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0015" class="elsevierStylePara elsevierViewall">where &#40;<span class="elsevierStyleItalic">&#955;</span>&#8217;&#44; <span class="elsevierStyleItalic">&#956;</span>&#8217;&#41; are the spherical coordinates relative to a rotated pole <span class="elsevierStyleItalic">N&#8217;</span> with coordinates &#40;&#955;<span class="elsevierStyleInf">0</span>&#44; &#956;<span class="elsevierStyleInf">0</span>&#41; with respect to the original system&#44; and <span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> is an eigenfunction of the operator Laplace-Beltrami with eigenvalue <span class="elsevierStyleItalic">&#967;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span><a class="elsevierStyleCrossRef" href="#bib0190">Verkley&#40;1984&#41;</a> generalized Thompson&#39;s solution and demonstrated that <span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> could be a set of eigenfunctions that contains more than only spherical harmonics&#46; Then Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a> describes a configuration in which the structure <span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> moves through the zonal flow <span class="elsevierStyleItalic">-&#969;&#956;</span> with constant velocity <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> and without changing size and shape&#46; The pole of the primed system <span class="elsevierStyleItalic">N&#8217;</span> that moves along a latitude at a constant angular velocity <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> is given by<elsevierMultimedia ident="eq0015"></elsevierMultimedia>where <span class="elsevierStyleItalic">&#967;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> is an eigenvalue for the spectral problem &#916;<span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span><span class="elsevierStyleItalic">&#61; &#8722;</span>&#967;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span><span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span>&#46; In particular&#44; for spherical harmonics <span class="elsevierStyleItalic">Y</span> &#40;<span class="elsevierStyleItalic">&#955;&#8217;&#44; &#956;&#8217;</span>&#41; of degree <span class="elsevierStyleItalic">n</span> corresponding to the eigenvalue <span class="elsevierStyleItalic">&#967;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> &#61; <span class="elsevierStyleItalic">&#967;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span><span class="elsevierStyleItalic">&#61; n</span>&#40;<span class="elsevierStyleItalic">n &#43;</span> 1&#41;&#44; Eq&#46;<a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a> is a Rossby-Haurwitz &#40;RH&#41; wave&#46; RH waves have proven to be very useful to describe the large-scale wave structure of atmospheric circulation in middle latitudes &#40;<a class="elsevierStyleCrossRef" href="#bib0125">Rossby&#44; 1939</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0050">Haurwitz&#44; 1940</a>&#41;&#46; The solution modon is constructed to divide the sphere <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> into two regions &#40;<a class="elsevierStyleCrossRef" href="#bib0185">Tribbia&#44; 1984</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0190">Verkley&#44; 1984</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0095">Neven&#44; 1992</a>&#41;&#58; an inner region <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> centered around the pole <span class="elsevierStyleItalic">N</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">&#8217;</span></span><span class="elsevierStyleItalic">&#44;</span> and an outer region <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">o</span></span> separated from the inner region by a boundary circle in which &#936;&#44; <span class="elsevierStyleItalic">q</span> and its normal derivative &#936;<span class="elsevierStyleItalic">&#8217;</span> are continuous&#46; Modons are considered appropriate to describe some types of atmospheric blocking events &#40;<a class="elsevierStyleCrossRef" href="#bib0200">Verkley&#44; 1990</a>&#41;&#46;</p><p id="par0020" class="elsevierStylePara elsevierViewall">Hydrodynamic equations on manifolds were studied by <a class="elsevierStyleCrossRef" href="#bib0045">Ebin and Marsden &#40;1970&#41;</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0165">Szeptycki &#40;1973a</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0170">b</a>&#41;&#44; <a class="elsevierStyleCrossRef" href="#bib0020">Avez and Bamberger &#40;1978&#41;</a>&#44; Ghidaglia <span class="elsevierStyleItalic">et al&#46;</span> &#40;1988&#41;&#44; Temam &#40;1987&#41; and Ilyin &#40;1993&#41;&#46; The existence&#44; unicity and regularity of the solution for the evolution equation &#40;Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0005">&#40;1&#41;</a>&#41; on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> were proven by <a class="elsevierStyleCrossRef" href="#bib0165">Szeptycki &#40;1973a</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0170">b</a>&#41;&#44; <a class="elsevierStyleCrossRef" href="#bib0020">Avez and Bamberger &#40;1978&#41;</a>&#44; Ilyin &#40;1993&#41; and <a class="elsevierStyleCrossRef" href="#bib0155">Skiba &#40;2012&#41;</a>&#46; <a class="elsevierStyleCrossRef" href="#bib0045">Ebin and Marsden &#40;1970&#41;</a> dealt with the motion of an incompressible fluid on manifolds under a differential geometric point of view&#46; Problems from the transition map between the charts are transferred to those of finding geodesics on the group of all volume-preserving diffeomorphisms&#44; to which the methods of global analysis and infinite-dimensional geometry can be applied&#46;</p><p id="par0025" class="elsevierStylePara elsevierViewall">In this paper we study the manifolds <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> in terms of the stream function &#936; for an RH wave which is sufficiently smooth and for Wu-Verkley waves and modons which are weakly differentiable of higher orders&#46; Section 2 deals with the compact differentiable manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and the way in which functions are constructed on this manifold&#46; Section 3 shows the types of solutions that will be considered&#46; Another aim of this paper is to deepen the understanding of the BVE solution on the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and its usage for deriving the properties of solutions to the manifold &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41;&#46; The paper concludes with a summary in section 4&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">2</span><span class="elsevierStyleSectionTitle" id="sect0025">Structure of functions on the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span></span><p id="par0030" class="elsevierStylePara elsevierViewall">In this section we review some basic facts concerning to the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#46; We should recall that the unit sphere <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is a compact and connected differentiable manifold&#46; Indeed&#44; because <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is compact it is not possible to cover it with only one chart&#46; A chart of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is then a pair &#40;&#937;&#44;<span class="elsevierStyleItalic">&#966;</span>&#41; where &#937; is an open subset of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; and &#966; is a homeomorphism of &#937; onto some open subset <span class="elsevierStyleItalic">of R</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#46;</span> Let us consider the two charts &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44;<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#44; &#40;&#937;<span class="elsevierStyleInf">&#954;</span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">&#954;</span>&#41;&#125; of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">p</span></span> for <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> where every chart corresponds to a geographical coordinate group&#46; It is possible to define a coordinate chart that covers most of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> by using the standard spherical coordinate map&#46; Let <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span> denote the coordinate function&#44; which maps from &#40;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">2</span>&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">3</span>&#41; to angles &#40;<span class="elsevierStyleItalic">&#955;&#44; &#952;</span>&#41; or to &#40;<span class="elsevierStyleItalic">&#955;&#44; &#956;</span>&#41;&#46; The domain of &#966;&#953;&#8722;1 is the open set defined by <span class="elsevierStyleItalic">&#955;</span> &#8712; &#40;&#8722;&#960;&#44; &#960;&#41; and <span class="elsevierStyleItalic">&#952;</span> &#8712; &#40;0&#44; &#960;&#41; &#40;this excludes the poles&#41;&#46; The inverse map &#966;&#953;&#8722;1 yields the parameterization <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">1</span></span> &#61; cos <span class="elsevierStyleItalic">&#955;</span> sin <span class="elsevierStyleItalic">&#952;</span>&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span> &#61; sin <span class="elsevierStyleItalic">&#955;</span> sin <span class="elsevierStyleItalic">&#952;</span>&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span> &#61; cos <span class="elsevierStyleItalic">&#952;</span> and its variation &#966;&#954;&#8722;1 yields the parameterization &#966;&#954;&#8722;1 &#40;<span class="elsevierStyleItalic">&#955;&#8217;&#44; &#952;</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">&#8217;</span></span>&#41;&#61;&#40;cos <span class="elsevierStyleItalic">&#955;&#8217;</span> sin <span class="elsevierStyleItalic">&#952;&#8217;&#44;</span> cos <span class="elsevierStyleItalic">&#952;&#8217;&#44; sin &#955;&#8217; sin &#952;&#8217;</span>&#41;<span class="elsevierStyleItalic">&#46;</span> The domain of &#966;&#954;&#8722;1 in the open set defined by <span class="elsevierStyleItalic">&#955;&#8217;</span> &#8712; &#40;&#8211;&#960;&#44; &#960;&#41; and <span class="elsevierStyleItalic">&#952;&#8217;</span> &#8712; &#40;0&#44; &#960;&#41;&#46; The charts &#937;&#953;&#44;&#966;&#953; and &#937;&#954;&#44;&#966;&#954; correspond to poles N and N<span class="elsevierStyleItalic">&#8217;</span> on the sphere <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#46; N<span class="elsevierStyleItalic">&#8217;</span> might be taken as the point &#40;&#955;0&#61;&#8722;&#960;2&#966;0&#61;0&#41; in the old system and as the angle <span class="elsevierStyleItalic">&#955;&#8217;</span> in this new north pole&#44; so that the new international date line is the half circle &#915;&#954;&#61;&#123;p&#8712;S2&#58;&#8722;&#960;2&#60;&#955;&#40;p&#41;&#60;&#960;2&#44;&#952;&#61;&#960;2&#44;&#125; of the old equator in the <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">2</span>- plane&#44; on the front where <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span> &#8805; 0 &#40;<a class="elsevierStyleCrossRef" href="#bib0120">Richtmyer&#44; 1981</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0135">Skiba&#44; 1989</a>&#59; <a class="elsevierStyleCrossRef" href="#bib1115">P&#233;rez-Garc&#237;a&#44; 2001</a>&#41;&#46; The international date line&#44; for the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; is the half circle &#915;&#953;&#61;&#123;p&#8712;S2&#58;&#8722;&#960;2&#60;&#966;&#40;p&#41;&#60;&#960;2&#44;&#955;&#61;&#177;&#8201;&#960;&#125; in the <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">3</span> &#8211;plane&#46; The chart covers &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; covers the sphere except for the set &#915;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; and the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41; similarly covers the sphere with the exception of a set &#915;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#46; Hence the two charts &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#44;&#40;&#937;<span class="elsevierStyleInf">&#954;</span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41;&#125; together cover <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and they constitute an atlas&#46;</p><p id="par0035" class="elsevierStylePara elsevierViewall">The local coordinates associated with the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; are functions &#966;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#58; &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44;&#8594;<span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span>&#44; such that for p&#8712;S2&#44;&#8201;&#966;&#953;&#40;p&#41;&#61;&#40;&#966;&#953;&#44;1&#40;p&#41;&#44;&#966;&#953;&#44;2&#40;p&#41;&#41;&#61;&#40;x&#953;1&#40;p&#41;&#44;&#8201;&#8201;x&#953;2&#40;p&#41;&#41;&#61;&#40;&#955;&#40;p&#41;&#44;&#956;&#40;p&#41;&#41;and &#966;&#954;&#40;p&#41;&#61;&#40;x&#954;1&#40;p&#41;&#44;x&#954;2&#40;p&#41;&#41;&#61;&#40;&#955;&#39;&#40;p&#41;&#59;&#956;&#39;&#40;p&#41;&#41; are local coordinates with respect to the chart &#40;&#937;<span class="elsevierStyleInf">&#954;</span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41; &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>&#41;&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0040" class="elsevierStylePara elsevierViewall">To construct the map &#966;&#8467;&#58;S2&#8594;U&#8834;R&#8467;2&#44;&#8467;&#61;&#953;&#44;&#8201;&#954; a bijection with inverse &#966;&#953;&#8722;1&#58;U&#953;&#8594;S2 defined as &#966;&#953;&#8722;1&#40;x&#953;1&#44;x&#953;2&#41;&#61;&#40;1&#8722;&#40;x&#953;2&#41;2cos&#8201;x&#953;1&#44;&#8201;1&#8722;&#40;x&#953;2&#41;2&#8201;sinx&#953;1&#44;x&#953;2&#41;&#44; and the &#966;&#954;&#8722;1&#40;x&#954;1&#44;x&#954;2&#41;&#61;&#40;1&#8722;&#40;x&#954;2&#41;2cos&#8201;x&#954;1&#44;x&#954;2&#44;1&#8722;&#40;x&#954;2&#41;2sinx&#954;1&#41;&#44; it is seen that every U<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#8467;</span></span> is open&#46; Hence each &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#8467;</span></span> is an open subset of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and &#937;&#953;&#8746;&#937;&#954; cover <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#46;</p><p id="par0045" class="elsevierStylePara elsevierViewall">Given two charts of the atlas &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#44; &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41;&#125; with &#937;&#954;&#8201;&#8745;&#8201;&#937;&#953;&#8800;&#248;&#44; the transition maps &#966;&#953;&#954;&#61;&#966;&#954;&#8728;&#966;&#953;&#8722;1&#58;&#8201;U&#953;&#954;&#8594;U&#954;&#953; outline open sets of <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span> into <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span><span class="elsevierStyleItalic">&#44;</span> where U&#953;&#954;&#61;&#966;&#953;&#937;&#953;&#8201;&#8745;&#8201;&#937;&#954; and U&#954;&#953;&#61;&#966;&#954;&#937;&#954;&#8201;&#8745;&#8201;&#937;&#953;&#46; This determines a differentiable structure for <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; and &#966;&#953;&#954;&#61;&#966;&#954;  &#8728;  &#966;&#953;&#8722;1&#58; is a diffeomorphism&#46; It is then said that the atlas is of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">k</span></span> if the transition functions are of <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">k</span></span><span class="elsevierStyleItalic">&#46;</span></p><p id="par0050" class="elsevierStylePara elsevierViewall">Let <span class="elsevierStyleItalic">x</span> be any point of <span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;&#954;</span></span>&#44; and &#40;&#966;&#953;&#954;1&#40;x&#41;&#44;&#966;&#953;&#954;2&#40;x&#41;&#41; the coordinate of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;&#954;</span></span> &#40;<span class="elsevierStyleItalic">x</span>&#41;&#59; then &#966;&#953;&#954;i&#40;x&#41; is a continuous function on two variables&#46; Now&#44; if <span class="elsevierStyleItalic">p</span> &#8712; &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span> &#8745; &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span> such that <span class="elsevierStyleItalic">x</span> &#61; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span> &#40;<span class="elsevierStyleItalic">p</span>&#41;&#44; and since <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#40;<span class="elsevierStyleItalic">p</span>&#41; &#8712; <span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;&#954;</span></span>&#44; we have the relations<elsevierMultimedia ident="eq0020"></elsevierMultimedia><elsevierMultimedia ident="eq0025"></elsevierMultimedia></p><p id="par0055" class="elsevierStylePara elsevierViewall">This is the transformation formula betwen the two local coordinate systems &#40;x&#953;1&#44;x&#953;2&#41; and &#40;x&#954;1&#44;x&#954;2&#41; defined on &#937;&#954;&#8201;&#8745;&#8201;&#937;&#953;&#46; To obtain the relations between the unprimed and primed coordinate of any point <span class="elsevierStyleItalic">Q</span> on the sphere&#44; <a class="elsevierStyleCrossRef" href="#bib0190">Verkley &#40;1984&#41;</a> examined the spherical triangle <span class="elsevierStyleItalic">NQN&#8217;</span> and the application of the cosine rules to this triangle&#44; deriving explicit expressions for the transformation between the two coordinate systems as given by &#40;4&#41; and &#40;5&#41;&#46;</p><p id="par0060" class="elsevierStylePara elsevierViewall">Let &#937;&#953;&#8201;&#8745;&#8201;&#937;&#954;&#8800;&#248; and set <span class="elsevierStyleItalic">J &#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;&#954;</span></span> as the Jacobian matrix of map <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;&#954;</span></span>&#44; so we can verify that<elsevierMultimedia ident="eq0030"></elsevierMultimedia></p><p id="par0065" class="elsevierStylePara elsevierViewall">Then det <span class="elsevierStyleItalic">J &#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;&#954;</span></span> &#62; 0&#46; Hence&#44; it is said that if manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is oriented for every pair &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span> of intersecting local coordinate neighbourhoods&#44; det <span class="elsevierStyleItalic">J &#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;&#954;</span></span> &#62; 0&#46;</p><p id="par0070" class="elsevierStylePara elsevierViewall">Indeed we can regard the coordinate as a device to decide which of many functions <span class="elsevierStyleItalic">&#968;</span> on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> are to be differentiable&#46; Since &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span> is just a set&#44; it makes no sense to ask that <span class="elsevierStyleItalic">&#968;</span>&#58; &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#8594; <span class="elsevierStyleItalic">R</span> be differentiable &#40;<a class="elsevierStyleCrossRef" href="#bib0085">Matsushima&#44; 1972</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0080">Loomis and Sterberg&#44; 1990</a>&#41;&#46; However&#44; we can consider the map &#936;&#953;&#61;&#968;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1&#58;&#966;&#953;&#8201;&#40;&#937;&#953;&#41;&#8594;R Then &#968;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1 is a function defined on an open &#966;&#953;&#40;&#937;&#953;&#41;&#8834;R2&#44; and we know what it means for such a function to be differentiable or smooth &#40;see <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>&#41;&#46; Consider now what happens when we change coordinates to some other chart&#44; lets say &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41; for convenience&#44; assuming that &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span> &#61; &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; Then it is possible for &#968;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1 to be differentiable but &#968;&#8201;&#8728;&#8201;&#966;&#954;&#8722;1 is not&#46; To compare both&#44; let &#968;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1&#61;&#968;&#8201;&#8728;&#966;&#954;&#8722;1&#8201;&#8728;&#8201;&#40;&#966;&#954;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1&#41; where the map &#966;&#954;&#8728;&#966;&#953;&#8722;1&#58;&#966;&#953;&#40;&#937;&#953;&#41;&#8594;&#966;&#954;&#40;&#937;&#954;&#41; is a bijection between open subsets of <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span>&#46; Then a sufficient condition for &#968;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1 to be differentiable if &#966;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1 is&#44; is that &#8217;&#966;&#954;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1 is also differentiable&#46; We often write &#936; for the composite function &#968;&#8201;&#8728;&#8201;&#966;&#953;&#8722;1</p><p id="par0075" class="elsevierStylePara elsevierViewall">Lets take a curve &#964; &#58; &#40;&#8722;1&#44; 1&#41; &#8594; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> with &#964; &#40;0&#41; &#61; <span class="elsevierStyleItalic">p&#46;</span> In a local chart &#964; is given by x&#953;i&#61;&#964;i&#40;t&#41;&#46; On the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; one can define a vector <span class="elsevierStyleBold">U</span> tangent to the parametrized curve &#964; at any point <span class="elsevierStyleItalic">p</span> on the curve&#46; The tangent vector <span class="elsevierStyleBold">U</span> is given by a column vector <span class="elsevierStyleBold">u</span> whose components u&#953;i are d&#964;idt0&#44; &#40;<span class="elsevierStyleItalic">i</span> &#61; 1&#44; 2&#41;&#44; with the initial condition &#964; &#40;0&#41; &#61; <span class="elsevierStyleItalic">p</span>&#46; If we use another coordinate system corresponding to the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41; by x&#954;i then the tangent vector <span class="elsevierStyleBold">U</span> is given by a column vector <span class="elsevierStyleBold">v</span> with components &#965;&#954;i<span class="elsevierStyleItalic">&#46;</span> According to the chain rule&#44; the column vectors <span class="elsevierStyleBold">u</span> and <span class="elsevierStyleBold">v</span> are related by &#965;&#954;i&#61;u&#953;j&#8706;x&#954;i&#8706;x&#953;j&#46; The expression u&#953;i&#8706;&#8706;x&#953;j is the partial differential operator in the direction of the tangent vector&#46;</p><p id="par0080" class="elsevierStylePara elsevierViewall">The space <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is called the <span class="elsevierStyleItalic">tangent space of S</span><span class="elsevierStyleSup">2</span><span class="elsevierStyleItalic">at p</span>&#44; and <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is a two-dimensional vector space&#46; For each <span class="elsevierStyleBold">u</span> &#8712; T<span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> we shall write u&#61;u&#953;i&#8706;&#8706;x&#953;j&#61;u1&#8706;&#8706;&#955;&#43;u2&#8706;&#8706;&#952;&#44; where u&#953;i are the contravariant components of <span class="elsevierStyleBold">u</span>&#46; It is well known that on the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> an inner product is defined at each tangent space <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#46;</span> Now lets present a basis in which we denote the coordinate system corresponding to the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; by &#123;x&#953;i&#125;&#61;&#40;&#955;&#44;&#956;&#41;&#44; and for any <span class="elsevierStyleItalic">&#968;</span>&#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">R</span> define the vectors &#40;&#8706;&#8706;x&#953;j&#41;p by &#40;&#8706;&#8706;x&#953;j&#41;p&#968;&#61;&#40;&#8706;&#968;&#8728;&#966;&#953;&#8722;1&#8706;x&#953;i&#41;&#966;&#953;&#40;p&#41;&#44; so that they are independent since &#40;&#8706;&#8706;x&#953;j&#41;px&#953;j&#61;&#948;ij Let &#40;n&#710;&#61;&#40;1&#8722;&#40;x&#953;2&#41;2cosx&#953;1&#44;1&#8722;&#40;x&#953;2&#41;2sinx&#953;1&#44;x&#953;2&#41; be the outward normal to <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> in <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">3</span>&#59; without any loss of generality we may assume that the vectors e&#955;&#61;&#8706;n&#710;&#953;&#8706;x&#953;1&#44;&#8201;&#8201;&#8201;e&#956;&#61;&#8706;n&#710;&#953;&#8706;x&#953;2 form a basis for T<span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleItalic"><span class="elsevierStyleSmallCaps">S</span></span><span class="elsevierStyleSup"><span class="elsevierStyleSmallCaps">2</span></span><span class="elsevierStyleSmallCaps">&#46;</span></p><p id="par0085" class="elsevierStylePara elsevierViewall">We will denote the vector space of a vector field on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> by &#915;&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; A tangent vector field on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is a smooth map <span class="elsevierStyleBold">u</span>&#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">T S</span><span class="elsevierStyleSup">2</span> such that&#44; for any <span class="elsevierStyleItalic">x</span> &#8712; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleBold">u</span>&#40;<span class="elsevierStyleItalic">x</span>&#41; &#8712; <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">x</span></span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#46; At the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#44; for <span class="elsevierStyleItalic">x</span> &#8712; &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; the vector functions <span class="elsevierStyleBold">u</span> &#8712; <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">x</span></span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span> and <span class="elsevierStyleBold">v</span> &#8712; &#915;&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span>&#41; have components u&#61;up1e&#955;&#43;up2e&#956; and v&#61;&#965;p1e&#955;&#43;&#965;p2e&#956;&#44; respectively&#44; being these up upi&#61;upx&#953;i&#61;u&#955;&#44;u&#956; the components of <span class="elsevierStyleBold">u</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> as the vectors of the unitary base indicated by <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#955;</span></span>&#44; and <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#956;</span></span> in the directions <span class="elsevierStyleItalic">&#955;</span> and <span class="elsevierStyleItalic">&#956;</span>&#44; respectively&#46;</p><p id="par0090" class="elsevierStylePara elsevierViewall">Let us recall that an oriented Riemannian manifold is a pair &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41; where <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is the oriented compact manifold and <span class="elsevierStyleItalic">g</span> a Riemannian metric on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; which assigns a length vgp&#8712;R&#43;&#46; The <span class="elsevierStyleItalic">g</span> on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is a smooth &#40;2&#44; 0&#41;-tensor field on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> such that for any <span class="elsevierStyleItalic">p</span> &#8712; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span>&#58; <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; &#215; <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; &#8594; <span class="elsevierStyleItalic">R</span> is a scalar product on the tangent space <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41;&#44; and in any chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; its components<elsevierMultimedia ident="eq0035"></elsevierMultimedia>form a symmetric matrix&#44; with its inverse denoted by &#40;<span class="elsevierStyleItalic">g</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">ij</span></span>&#41; &#61; &#40;<span class="elsevierStyleItalic">g</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span>&#41;<span class="elsevierStyleSup">-1</span>&#44; and <span class="elsevierStyleItalic">g</span> &#61; <span class="elsevierStyleItalic">det</span>&#40;<span class="elsevierStyleItalic">g</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span>&#41; &#61; 1 &#8211; <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleSup">2</span>&#46; The length of a tangent vector <span class="elsevierStyleBold">v</span> &#8712; <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is defined as usual&#44; v&#61;gpv&#44;v12&#61;v&#183;v12 Moreover&#44; the inner product on <span class="elsevierStyleItalic">T S</span><span class="elsevierStyleSup">2</span> is given by <span class="elsevierStyleBold">u</span> &#46; <span class="elsevierStyleBold">v</span> &#61; <span class="elsevierStyleItalic">g</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span><span class="elsevierStyleItalic">u</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleItalic">&#957;</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">j</span></span> for <span class="elsevierStyleBold">u</span>&#44; <span class="elsevierStyleBold">v</span> &#8712; <span class="elsevierStyleItalic">TS</span><span class="elsevierStyleSup">2</span>&#46;</p><p id="par0095" class="elsevierStylePara elsevierViewall">Let &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41; be the smooth Riemannian manifolds of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#46; Let us now recall some operators arising in partial differential equations on the sphere as manifold&#46; Given the scalar function <span class="elsevierStyleItalic">&#968;</span> &#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">R</span>&#44; the <span class="elsevierStyleItalic">gradient of &#968;</span>&#44; is given by the vector field <span class="elsevierStyleItalic">grad &#968;</span>&#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span>&#44; for which<elsevierMultimedia ident="eq0040"></elsevierMultimedia>where e&#710;&#955;&#8201;&#61;11&#8722;&#956;2e&#955; and e&#710;&#956;&#61;1&#8722;&#956;2  e&#956;&#46;</p><p id="par0100" class="elsevierStylePara elsevierViewall">If <span class="elsevierStyleBold">u</span> &#8712; &#1043;&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41;&#44; the divergence of <span class="elsevierStyleBold">u</span> is the function on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> which on the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; is given by<elsevierMultimedia ident="eq0045"></elsevierMultimedia></p><p id="par0105" class="elsevierStylePara elsevierViewall">A linear connection <span class="elsevierStyleItalic">D</span> on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is a map D&#58; <span class="elsevierStyleItalic">T</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; &#215; &#1043;&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; &#8594; <span class="elsevierStyleItalic">T</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; called the <span class="elsevierStyleItalic">covariant derivative</span> and the usual notation for <span class="elsevierStyleItalic">D</span>&#40;<span class="elsevierStyleItalic">U&#44; V</span>&#41; is <span class="elsevierStyleItalic">D</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">U</span></span><span class="elsevierStyleItalic">V&#46;</span> Let &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; be a chart and as one can observe&#44; the vectors &#40;e&#955;&#61;&#8706;&#8706;&#955;&#44;e&#956;&#61;&#8706;&#8706;&#956;&#41; can be nonconstant&#46; An easy notation is set &#8711;i&#61;D&#8706;&#8706;xi &#40;eg&#46; <a class="elsevierStyleCrossRef" href="#bib0055">Hebey&#44; 2000</a>&#41;&#46; There are smooth functions &#915;ijk&#58;&#937;&#953;&#8594;R such that for any <span class="elsevierStyleItalic">i</span>&#44; <span class="elsevierStyleItalic">j&#44;</span> and any <span class="elsevierStyleItalic">p</span> &#8712;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44;<elsevierMultimedia ident="eq0050"></elsevierMultimedia></p><p id="par0110" class="elsevierStylePara elsevierViewall">where &#915;ijk are the Christoffel symbols&#44; defined by &#915;ijk&#61;12&#8721;l&#61;12gkl&#40;&#8706;glj&#8706;x&#953;i&#43;&#8706;gil&#8706;x&#953;j&#43;&#8706;gij&#8706;x&#953;l&#41;&#46;On the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#44; we have &#915;111&#61;0&#44;&#915;112&#61;&#8722;2&#956;1&#8722;&#956;2&#44;&#915;121&#61;cot&#8201;&#952;&#61;&#956;1&#8722;&#956;2&#44;&#915;122&#61;0&#44;&#915;221&#61;0 and &#915;222&#61;0</p><p id="par0115" class="elsevierStylePara elsevierViewall">The fundamental operator which we study is the Laplacian &#916;&#44; then for real or complex valued functions&#44; &#916; is the Laplace-Beltrami operator on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and it is given by<elsevierMultimedia ident="eq0055"></elsevierMultimedia></p><p id="par0120" class="elsevierStylePara elsevierViewall">This operator satisfies some properties&#58; &#916;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span>&#44; is selfadjoint&#44; symmetric and non-negative &#40;Aubin&#44; 1998&#41;&#46; Thus&#44; the operators <span class="elsevierStyleItalic">div&#44; grad</span> and &#916;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> on the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> have the conventional meaning&#46;</p><p id="par0125" class="elsevierStylePara elsevierViewall">Let &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41; be a compact <span class="elsevierStyleItalic">oriented Riemannian</span> manifold&#44; with metric <span class="elsevierStyleItalic">g</span>&#46; The metric and the orientation are combined to give a volume element <span class="elsevierStyleItalic">d</span>&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; which can be used to integrate functions on &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41;&#46; In order to apply the integral calculus on the oriented manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; we define a volume element to be a two-form &#969; &#61; <span class="elsevierStyleItalic">d</span>&#965; which is defined on all of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#46; For every chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#8467;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#8467;</span></span>&#41; which is consistently oriented with <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; the coordinate expresion for &#969;&#61;d&#965;l is &#934;ldxl1&#923;dxl2 where &#934;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#8467;</span></span> is a partition of unity subordinate to the covering &#937;l&#44;&#8201;l&#61;&#953;&#44;&#954;</p><p id="par0130" class="elsevierStylePara elsevierViewall">On the Riemannian manifold &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41;&#44; at the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#59; a volume form <span class="elsevierStyleItalic">&#951;</span> &#61; <span class="elsevierStyleItalic">d</span>&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> defines a Lebesgue measure on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> by d&#965;g&#61;&#951;&#61;gdxl1&#8201;&#923;dxl2 Then<elsevierMultimedia ident="eq0060"></elsevierMultimedia>wheredxl&#61;dxl1dxl2 defines a Lebesgue measure on <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span>&#46;</p><p id="par0140" class="elsevierStylePara elsevierViewall">Let C<span class="elsevierStyleSup">&#8734;</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; denote the set of infinitely differentiable functions of compact support <span class="elsevierStyleItalic">&#968;</span>&#40;<span class="elsevierStyleItalic">x</span>&#41;&#46; At <span class="elsevierStyleItalic">&#956;</span> &#61; &#177; 1 the functions are smooth&#44; together with the periodic boundary condition at <span class="elsevierStyleItalic">&#955;</span> with period 2&#960;&#46; If we define the usual Hilbert space <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup">2</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; to be the completion of C<span class="elsevierStyleSup">&#8734;</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; with respect to the inner product<elsevierMultimedia ident="eq0065"></elsevierMultimedia></p><p id="par0145" class="elsevierStylePara elsevierViewall">and norm f&#61;&#123;&#8747;s2f2d&#965;g&#125;12&#44;&#8201;&#8201;g&#42; is the complex conjugate of function <span class="elsevierStyleItalic">g&#46;</span> Let &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g&#41;</span> be the compact <span class="elsevierStyleItalic">Riemannian</span> manifold and <span class="elsevierStyleItalic">d&#965;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> the Riemannian volume element&#46; Then functional spaces &#40;Sobolev and the Holder spaces&#41; can be defined on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> as well &#40;<a class="elsevierStyleCrossRef" href="#bib0155">Skiba&#44; 2012</a>&#41;&#46; For each <span class="elsevierStyleItalic">p</span> &#8712; <span class="elsevierStyleBold">R</span> with 1&#8804; <span class="elsevierStyleItalic">p</span> &#60; &#8734; we associate a Banach space<elsevierMultimedia ident="eq0070"></elsevierMultimedia></p><p id="par0150" class="elsevierStylePara elsevierViewall">with respect to the norm<elsevierMultimedia ident="eq0075"></elsevierMultimedia></p><p id="par0155" class="elsevierStylePara elsevierViewall">and <span class="elsevierStyleItalic">ess sup</span> &#124; f &#124; &#60; &#8734; if <span class="elsevierStyleItalic">p</span> &#61; &#8734;&#46; <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup">2</span>&#40;<span class="elsevierStyleItalic">TS</span><span class="elsevierStyleSup">2</span>&#41; represent the Hilbert space of the vector fields <span class="elsevierStyleItalic">U</span> &#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">TS</span><span class="elsevierStyleSup">2</span> endowed with the inner product in <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup">2</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; induced by <span class="elsevierStyleItalic">g</span> in <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span><span class="elsevierStyleItalic">&#40;S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#41;</span> &#40;see <a class="elsevierStyleCrossRef" href="#bib0040">D&#237;az and Tello&#44; 1999</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0055">Hebey&#44; 2000</a>&#41;&#46;</p><p id="par0160" class="elsevierStylePara elsevierViewall">We now turn to the eigenvalue problems for &#916;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span>&#58; We usually seek to find all eigenvalues &#947; for which there is an eigenfunction <span class="elsevierStyleItalic">Y</span> such that &#916;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span><span class="elsevierStyleItalic">Y</span> &#61; &#8211;&#947;Y&#46; Then&#44; which information about geometry of &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41; is encoded by the eigenvalues&#63;&#46; The structure of eigenfunctions&#58; <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">p</span></span> norms and relations to RH waves or modons&#46;</p><p id="par0165" class="elsevierStylePara elsevierViewall">Global harmonic analysis is the study of the spectral theory of the Laplacian &#916;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> on a compact Riemannian manifold &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41;&#44; and its relation to the global geometric structure&#46; Since &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41; is compact&#44; there exists an orthonormal basis &#123;<span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#125; of smooth eigenfunctions and the spectrum of &#916;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> is a discrete set &#123;&#947;<span class="elsevierStyleInf">0</span> &#61; 0 &#60; &#947;<span class="elsevierStyleInf">1</span> &#8804; &#947;<span class="elsevierStyleInf">2</span> &#8804; &#947;<span class="elsevierStyleInf">3</span> &#8804; &#46;&#46;&#46;&#125;&#46; Recent developments show that the non-zero eigenvalues also contain substantial geometric and analytic information&#46; The solution modon constructed by <a class="elsevierStyleCrossRef" href="#bib0185">Tribbia &#40;1984&#41;</a>&#44; Verkley &#40;1984&#44; <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>&#41; and Neven &#40;1993&#41; proposed the use of eigenfunctions &#123;<span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#125; as basic geometric structures&#46;</p><p id="par0170" class="elsevierStylePara elsevierViewall">The space of spherical harmonics of degree <span class="elsevierStyleItalic">n</span> on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; which coincides with the eigenspace of operator &#8211; &#916;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> corresponding to the eigenvalue &#947;<span class="elsevierStyleInf">n</span> &#61;&#967;<span class="elsevierStyleInf">n</span>&#61;<span class="elsevierStyleItalic">n</span>&#40;<span class="elsevierStyleItalic">n</span>&#43;1&#41;&#44; is denoted by <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleBold">n</span></span>&#46; Self-adjoint operators have the property that its eigenfunctions with different eigenvalues are orthogonal&#44; which implies that the eigenspaces <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleBold">n</span></span> are orthogonal and have 2<span class="elsevierStyleItalic">n</span>&#43;1 dimensions&#46; On the sphere&#44; the homogeneous harmonic polynomials span the set of all polynomials&#44; which in turn are dense in <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup">2</span>&#46; Our spherical harmonics therefore span <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup">2</span>&#46; If we take a basis within each eigenspace then this collection will give a basis for <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup">2</span> of the sphere&#46; The harmonics spherical term was introduced by Kelvin on potentials studies &#40;<a class="elsevierStyleCrossRef" href="#bib0065">Hobson&#44; 1931</a>&#41; and is understood as the development of a function in terms of this series of spherical harmonics&#46;</p><p id="par0175" class="elsevierStylePara elsevierViewall">The spaces <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleBold">n</span></span> and <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleBold">k</span></span> &#40;<span class="elsevierStyleItalic">n</span> &#8800; <span class="elsevierStyleItalic">k</span>&#41; are mutually orthogonal in <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup">2</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41;&#46; Then there is the orthogonal projection <span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> &#58; <span class="elsevierStyleItalic">L</span><span class="elsevierStyleInf">2</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; &#8594; <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleBold">n</span></span>&#44; and so smooth functions &#936; &#8712; <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup">2</span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#41; on the sphere <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span> have a development in spherical harmonics&#44;<elsevierMultimedia ident="eq0080"></elsevierMultimedia>where Yn&#955;&#44;&#956;&#61;&#8721;m&#61;&#8201;&#8722;nn&#936;nm&#8201;Ynm&#955;&#44;&#956; is the homogeneous spherical polynomial of degree <span class="elsevierStyleItalic">n</span> from <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleBold">n</span></span>&#44; and &#936;nm&#61;&#8201;&#60;&#8201;&#936;&#44;Ynm&#62; is the Fourier coefficient of &#936;&#46; The 2<span class="elsevierStyleItalic">n</span> &#43; 1 spherical harmonics<elsevierMultimedia ident="eq0085"></elsevierMultimedia>of degree <span class="elsevierStyleItalic">n</span> and zonal number <span class="elsevierStyleItalic">m</span> &#40;&#8211;<span class="elsevierStyleItalic">n</span> &#8804; <span class="elsevierStyleItalic">m</span> &#8804; <span class="elsevierStyleItalic">n&#41;</span> form an orthonormal basis in <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleBold">n</span></span>&#46; Here the numbers <span class="elsevierStyleItalic">C</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">nm</span></span> are the normalizers in <span class="elsevierStyleItalic">L</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span>&#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span>&#41;&#44; <span class="elsevierStyleItalic">given by</span>Cnm&#61;2n&#43;14&#960;n&#8722;m&#33;n&#43;m&#33;12 and Pnm are the associated Legendre functions given by<elsevierMultimedia ident="eq0090"></elsevierMultimedia></p><p id="par0180" class="elsevierStylePara elsevierViewall">Considering that an oriented compact Riemannian manifold is a pair &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41; where <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is the oriented compact manifold and <span class="elsevierStyleItalic">g</span> a Riemannian metric on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; we can define in it covariant derivatives and various notions of curvature&#46; When a manifold also has a group structure &#40;so that multiplication and inversion are smooth&#41;&#44; a very interesting structure called a Lie group &#40;Bihlo&#44; 2007&#59; <a class="elsevierStyleCrossRef" href="#bib0030">Bihlo and Popoych&#44; 2012</a>&#41; arises&#46; Even if a manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is not a Lie group&#44; there may be an action &#58; G &#215; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> of a Lie group G on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; and under certain conditions <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> can be viewed as a &#8220;quotient&#8221; G&#47;K&#44; where K is a subgroup of G &#40;<a class="elsevierStyleCrossRef" href="#bib0120">Richtmyer&#44; 1981</a>&#41;&#46; When <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8773; G&#47;K as above&#44; certain notions on G can be transported to <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; then we say that <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is a homogeneous space&#46; As an example of the last point we could mention the theory of spherical harmonic expansion on the <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; which is a homogeneous space for the rotation group O&#40;n&#43;1&#41;&#46; The surface spherical harmonics are eigenfunctions for the Laplace-Beltrami operator&#44; which is a rotation invariant &#40;<a class="elsevierStyleCrossRef" href="#bib0060">Helgason&#44; 1984</a>&#41;&#46; Harmonic analysis is concerned with the representation of functions as the superposition of basic waves&#44; the study and generalization of the notions of Fourier series as well as the Fourier transforms&#46;</p><p id="par0185" class="elsevierStylePara elsevierViewall">Elements of harmonic analysis on the sphere can be found at <a class="elsevierStyleCrossRef" href="#bib0160">Stein and Weiss &#40;1971&#41;</a>&#46; After introducing the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and the Riemannian manifolds &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">g</span>&#41;&#44; a general type of spaces &#40;Besov and Triebel-Lizorkin spaces&#41; on the sphere may also be introduced &#40;<a class="elsevierStyleCrossRef" href="#bib0090">Narcowich <span class="elsevierStyleItalic">et al&#46;&#44;</span> 2006</a>&#41;&#46; Using the power of a Laplace operator&#44; the Sobolev space on Riemannian manifolds can also be incorporated as a field currently undergoing great development &#40;Aubin&#44; 1998&#59; <a class="elsevierStyleCrossRef" href="#bib0055">Hebey&#44; 2000</a>&#41;&#46;</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">3</span><span class="elsevierStyleSectionTitle" id="sect0030">Exact solutions to the barotropic vorticity equation on the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span></span><p id="par0190" class="elsevierStylePara elsevierViewall">Let &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#125;<span class="elsevierStyleItalic"><span class="elsevierStyleSmallCaps">&#44;&#8467;&#61;</span>&#953;</span>&#44; <span class="elsevierStyleItalic">&#954;</span> be an atlas of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and <span class="elsevierStyleItalic">&#968;</span>&#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">R</span> the streamfunction of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span><span class="elsevierStyleItalic">&#46;</span> We can consider that the map &#936;&#61;&#968;&#8728;&#966;&#953;&#8722;1&#58;&#966;&#953;&#937;&#953;&#8594;R and &#968;&#8728;&#966;&#953;&#8722;1 is the streamfunction defined on an open <span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span> &#61; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span> &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; &#8834; <span class="elsevierStyleItalic">R</span><span class="elsevierStyleSup">2</span> and that it is of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span><span class="elsevierStyleItalic">&#46;</span></p><p id="par0195" class="elsevierStylePara elsevierViewall">To simulate the time evolution of a two-dimensional nondivergent and inviscid flow for a rotating sphere&#44; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> is governed by a non-linear barotropic vorticity equation&#44; which can be written in the non-dimensional form as<elsevierMultimedia ident="eq0095"></elsevierMultimedia></p><p id="par0200" class="elsevierStylePara elsevierViewall">where Jc&#44;q&#61;&#8706;c&#8706;&#955;&#8706;q&#8706;&#956;&#8722;&#8706;c&#8706;&#956;&#8706;q&#8706;&#955;&#61;k&#215;&#8711;c&#183;&#8711;q&#61;u&#183;&#8711;q is the jacobian&#44; u  &#61; k&#215;&#8711;c&#61;u&#955;&#44;u&#956;&#61;&#8722;1&#8722;&#956;2&#8706;c&#8706;&#956;&#44;11&#8722;&#956;2&#8706;c&#8706;&#955; is a tangent velocity vector&#44; <span class="elsevierStyleItalic">grad</span>c&#61;&#8711;c&#61;11&#8722;&#956;2&#8706;c&#8706;&#955;&#44;1&#8722;&#956;2&#8706;c&#8706;&#956;&#44;c&#61;&#936;&#44;&#958;&#61;&#916;&#936;&#61;div&#8201;grand&#936;&#44; is the relative vorticity <span class="elsevierStyleItalic">q &#61;</span>&#916;&#936; &#43; 2 <span class="elsevierStyleItalic">&#956;</span> is the absolute vorticity and <span class="elsevierStyleBold">k</span> is a unit outward normal vector&#46; The velocity vector field <span class="elsevierStyleBold">u</span> having the components &#40;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#955;</span></span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#956;</span></span>&#41; is solenoidal&#58; &#8711; &#183; <span class="elsevierStyleBold">u</span> &#61; 0 Throughout decades the nonlinear barotropic vorticity equation has been successfully used to describe low frequencies and large-scale barotropic processes of atmospheric dynamics&#46; Despite the simplicity to this nonlinear equation&#44; it contains the principal elements that describe the complexity of atmospheric behavior &#40;<a class="elsevierStyleCrossRef" href="#bib0130">Simmons <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1983</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0140">Skiba&#44; 1997</a>&#41;&#46; The four types of exact solutions to <a class="elsevierStyleCrossRef" href="#eq0005">Eq&#46; &#40;1&#41;</a> known up to now are described below&#58;<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005"><span class="elsevierStyleLabel">&#8226;</span><p id="par0205" class="elsevierStylePara elsevierViewall">The zonal flows and Rossby-Haurwitz &#40;RH&#41; waves &#40;Haurwitz&#44; 1949&#41;&#44; called classical solutions&#44; differentiated from the generalized solutions which are not so smooth&#46;</p></li><li class="elsevierStyleListItem" id="lsti0010"><span class="elsevierStyleLabel">&#8226;</span><p id="par0210" class="elsevierStylePara elsevierViewall">The first generalized solutions of Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0095">&#40;6&#41;</a>&#44; kown as modons&#44; were originally constructed by <a class="elsevierStyleCrossRef" href="#bib0185">Tribbia &#40;1984&#41;</a> and Verkley &#40;1984&#44; <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>&#41; by using two eigenfunctions for the Laplace operator of different degrees&#46;</p></li><li class="elsevierStyleListItem" id="lsti0015"><span class="elsevierStyleLabel">&#8226;</span><p id="par0215" class="elsevierStylePara elsevierViewall">Later on&#44; <a class="elsevierStyleCrossRef" href="#bib0095">Neven &#40;1992&#41;</a> gave generalized solutions in the form of a quadrupole modon&#46;</p></li><li class="elsevierStyleListItem" id="lsti0020"><span class="elsevierStyleLabel">&#8226;</span><p id="par0220" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#bib0205">Wu and Verkley &#40;1993&#41;</a> constructed generalized global solutions composed of two RH waves &#40;<a class="elsevierStyleCrossRef" href="#bib0100">P&#233;rez-Garcia and Skiba&#44; 1999</a>&#41;&#46;</p></li></ul></p><p id="par0225" class="elsevierStylePara elsevierViewall">In the present work&#44; zonal flows&#44; homogeneous spherical polynomials flows&#44; RH waves&#44; and modons on the manifold S<span class="elsevierStyleSup">2</span> are considered&#46;</p><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">3&#46;1</span><span class="elsevierStyleSectionTitle" id="sect0035">Classical solutions</span><p id="par0230" class="elsevierStylePara elsevierViewall">Let us consider the zonal flows&#44; homogeneous spherical polynomials flows and Rossby-Haurwitz &#40;RH&#41; waves&#46;</p><p id="par0235" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proposition &#40;zonal flow&#41;</span>&#46; Let &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#125;&#44; <span class="elsevierStyleItalic">&#8467; &#61; &#953;&#44; &#954;</span> be an atlas of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span> and the streamfunction <span class="elsevierStyleItalic">&#968;</span> &#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span> &#8594; <span class="elsevierStyleItalic">R</span> of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span><span class="elsevierStyleItalic">&#46;</span> Then the zonal flow map &#936;&#953;&#61;&#968;&#8728;&#966;&#953;&#8722;1&#58;U&#953;&#8834;R&#953;2&#8594;R of <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span> defined as<elsevierMultimedia ident="eq0100"></elsevierMultimedia>is an exact solution of the vorticity Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0095">&#40;6&#41;</a> for any <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span>&#46;</p><p id="par0245" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proof&#46;</span> The demonstration&#44; obtained from Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0095">&#40;6&#41;</a>&#44; is quite trivial&#46;</p><p id="par0250" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proposition &#40;homogeneous polynomials&#41;</span>&#46;</p><p id="par0255" class="elsevierStylePara elsevierViewall">Let &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#125;&#44;<span class="elsevierStyleItalic">&#8467; &#61; &#953;&#44; &#954;</span> be an atlas of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and the streamfunction <span class="elsevierStyleItalic">&#968;</span> &#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">R</span> of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span><span class="elsevierStyleItalic">&#46;</span> Then the homogeneous spherical polynomial map &#936;&#953;&#61;&#968;&#8728;&#966;&#953;&#8722;1&#58;U&#953;&#8834;R&#953;2&#8594;R of degree n &#8805; 2 defined as<elsevierMultimedia ident="eq0105"></elsevierMultimedia>is an exact solution to the vorticity Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0095">&#40;6&#41;</a>&#44; where <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> can be a complex factor and<elsevierMultimedia ident="eq0110"></elsevierMultimedia></p><p id="par0260" class="elsevierStylePara elsevierViewall">is the angular phase speed&#46;</p><p id="par0265" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proof&#46;</span> Given &#936;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span> &#8712; <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span>&#44; we define &#936;&#953;&#955;&#44;&#8201;&#956;&#44;&#8201;&#8201;t&#61;&#8201;&#936;n&#40;&#955;&#8722;ct&#44;&#8201;&#956;&#41;&#61;&#8721;m&#61;&#8722;nnamYnm&#955;&#8722;ct&#44;&#8201;&#956;&#44; then &#8706;&#936;n&#8706;t&#61;&#8722;2&#936;&#8242;&#44; and &#8706;&#936;n&#8706;&#955;&#61;&#936;&#8242;&#44; where &#936;&#8242;&#61;&#8721;m&#61;&#8722;nnimamYnm&#40;&#955;&#8722;ct&#44;&#8201;&#956;&#41;&#46; If&#44; in addition&#44; we have the following expression<elsevierMultimedia ident="eq0245"></elsevierMultimedia> we have <elsevierMultimedia ident="eq0115"></elsevierMultimedia>from BVE &#40;Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0095">&#40;6&#41;</a>&#41;&#46; It follows that c&#8201;&#61;&#8722;2&#967;n&#46;</p><p id="par0270" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proposition &#40;Rossby-Haurwitz waves&#41;</span>&#46; Let &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#125;&#44; <span class="elsevierStyleItalic">&#8467; &#61; &#953;&#44; &#954;</span> be an atlas of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and the streamfunction <span class="elsevierStyleItalic">&#968;</span> &#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">R</span> of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">&#8734;</span>&#46;Then&#44; the map &#936;&#953;&#61;&#968;&#8728;&#966;&#953;&#8722;1&#58;U&#953;&#8834;R&#953;2&#8594;R of <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">&#8734;</span> define as<elsevierMultimedia ident="eq0120"></elsevierMultimedia><elsevierMultimedia ident="eq0250"></elsevierMultimedia>with <span class="elsevierStyleItalic">n &#8805;</span> 1 is called Rossby-Haurwitz &#40;RH&#41; waves&#46;</p><p id="par0275" class="elsevierStylePara elsevierViewall">It is an exact solution of the vorticity Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0095">&#40;6&#41;</a> if the angular phase speed of the RH wave<elsevierMultimedia ident="eq0130"></elsevierMultimedia></p><p id="par0280" class="elsevierStylePara elsevierViewall">Here <span class="elsevierStyleItalic">&#969;</span> is the super-rotation velocity and each <span class="elsevierStyleBold">H</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> corresponds to the eigenvalue <span class="elsevierStyleItalic">&#967;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> &#61; <span class="elsevierStyleItalic">n</span>&#40;<span class="elsevierStyleItalic">n</span>&#43;1&#41;&#46;</p><p id="par0285" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proof</span>&#46; Here &#936;<span class="elsevierStyleInf">&#953;</span> is expressed by<elsevierMultimedia ident="eq0135"></elsevierMultimedia>where &#936;n&#955;&#8722;ct&#44;&#8201;&#956;&#61;&#8721;m&#61;&#8722;nnamY&#955;&#8722;ct&#44;&#8201;&#956;&#46; We can notice that<elsevierMultimedia ident="eq0140"></elsevierMultimedia>which implies that<elsevierMultimedia ident="eq0145"></elsevierMultimedia></p><p id="par0290" class="elsevierStylePara elsevierViewall">Furthermore&#58;<elsevierMultimedia ident="eq0150"></elsevierMultimedia>where &#936;&#8242;&#61;&#8721;m&#61;&#8722;nnimamYnm&#40;&#955;&#8722;ct&#44;&#8201;&#956;&#41;&#59; so that from BVE &#40;Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0095">&#40;6&#41;</a>&#41;<elsevierMultimedia ident="eq0155"></elsevierMultimedia></p><p id="par0295" class="elsevierStylePara elsevierViewall">Hence<elsevierMultimedia ident="eq0160"></elsevierMultimedia>so that<elsevierMultimedia ident="eq0165"></elsevierMultimedia>and thus the proposition is proved&#46;</p><p id="par0300" class="elsevierStylePara elsevierViewall">The streamfunction of the stationary RH&#40;2&#44;5&#41; wave<elsevierMultimedia ident="eq0170"></elsevierMultimedia>with the parameters defined by &#40;<span class="elsevierStyleItalic">m</span>&#44; <span class="elsevierStyleItalic">n</span>&#41; &#61; &#40;2&#44; 5&#41;&#44; <span class="elsevierStyleItalic">a</span> &#61; &#46;007 and &#969;&#61;23&#40;&#967;3&#8722;2&#41; is given in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>&#46;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia><p id="par0305" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#bib0105">P&#233;rez and Skiba &#40;2001&#41;</a> and <a class="elsevierStyleCrossRef" href="#bib0145">Skiba and P&#233;rez &#40;2006&#41;</a> developed a numerical spectral method for the normal mode instability study of the arbitrary steady flow of an ideal nondivergent fluid on a rotating sphere&#44; and <a class="elsevierStyleCrossRef" href="#bib0145">Skiba and P&#233;rez &#40;2006&#41;</a> tested this method for the RH&#40;2&#44;5&#41; wave&#46; <a class="elsevierStyleCrossRef" href="#bib0115">P&#233;rez-Garc&#237;a &#40;2014&#41;</a> constructed a basic flow regarded as a sum of a zonally symmetric flow &#40;<a class="elsevierStyleCrossRef" href="#eq0100">Eq&#46; 7</a> and a Rossby-Haurwitz wave component &#40;<a class="elsevierStyleCrossRef" href="#eq0120">Eq&#46; 9</a>&#46;</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">3&#46;2</span><span class="elsevierStyleSectionTitle" id="sect0040">Generalized solutions</span><p id="par0310" class="elsevierStylePara elsevierViewall">Denote the spherical distance between two points of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> by <span class="elsevierStyleItalic">d</span>&#40;&#46;&#44;&#46;&#41;&#46; Let <span class="elsevierStyleItalic">N&#8217;</span> be the north pole of the chart coordinate &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span><span class="elsevierStyleSmallCaps">&#41;&#46;</span> Then a disk or inner region <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> on the sphere is defined as Si&#61;DN&#8242;&#44;&#966;a&#61;&#123;s&#8201;&#8712;&#8201;S2&#124;&#8201;dN&#8242;&#44;s&#60;&#966;a&#125;&#44; such that 0&#60;&#966;a&#8804;&#960;2 The solution modon is constructed &#40;<a class="elsevierStyleCrossRef" href="#bib0185">Tribbia&#44; 1984</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0190">Verkley&#44; 1984</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>&#59; Neven&#44; 1993&#41; to divide the sphere <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> into two regions&#58; an inner region <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> centered around the pole <span class="elsevierStyleItalic">N&#8217;&#44;</span> and an outer region <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">o</span></span> separated from the inner region by a boundary circle &#8706;DN&#8242;&#44;&#966;a&#61;&#123;s&#8201;&#8712;&#8201;S2&#124;&#8201;dN&#8242;&#44;s&#60;&#966;a&#125;&#44; on which <span class="elsevierStyleItalic">&#968;</span>&#44; <span class="elsevierStyleItalic">q</span> and <span class="elsevierStyleItalic">&#968;&#8217;</span> are continuous&#46;</p><p id="par0315" class="elsevierStylePara elsevierViewall">For <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> a solution of the Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a> form is chosen with an eigenfunction <span class="elsevierStyleItalic">Y</span> &#40;<span class="elsevierStyleItalic">&#955;&#8217;</span>&#44; <span class="elsevierStyleItalic">&#956;&#8217;</span>&#41; which has its singularity in the outer region&#46; The same type of solution is chosen for the outer region&#44; but such that <span class="elsevierStyleItalic">Y</span> &#40;<span class="elsevierStyleItalic">&#955;&#8217;</span>&#44; <span class="elsevierStyleItalic">&#956;&#8217;</span>&#41; has its singularity in the inner region&#46; Then both solutions are combined as smoothly as possible on the boundary circle &#8706;DN&#8242;&#44;&#966;a &#40;<a class="elsevierStyleCrossRef" href="#bib0185">Tribbia&#44; 1984</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0190">Verkley&#44; 1984</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>&#41;&#46;</p><p id="par0320" class="elsevierStylePara elsevierViewall">To construct the <a class="elsevierStyleCrossRef" href="#bib0190">Verkley &#40;1984&#41;</a> modon or the <a class="elsevierStyleCrossRef" href="#bib0095">Neven &#40;1992&#41;</a> cuadrupole modon on the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; it is interpreted as&#58;</p><p id="par0325" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proposition</span>&#46; Let &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic"><span class="elsevierStyleSmallCaps">&#8467;</span></span></span><span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic"><span class="elsevierStyleSmallCaps">&#8467;</span></span></span>&#41;&#125;&#44; <span class="elsevierStyleItalic"><span class="elsevierStyleSmallCaps">&#8467;</span></span> &#61; <span class="elsevierStyleItalic">&#953;</span>&#44;<span class="elsevierStyleItalic">&#954;</span><span class="elsevierStyleSmallCaps">&#44;</span> be an atlas of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; and <span class="elsevierStyleItalic">&#968;</span> &#61; <span class="elsevierStyleItalic">&#968;</span><span class="elsevierStyleInf">1</span> &#43; <span class="elsevierStyleItalic">&#968;</span><span class="elsevierStyleInf">2</span> &#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">R</span> the streamfunction of <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span><span class="elsevierStyleItalic">&#46;</span> Then<elsevierMultimedia ident="eq0175"></elsevierMultimedia></p><p id="par0330" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proof</span>&#46; Let <span class="elsevierStyleItalic">&#968;</span><span class="elsevierStyleInf">1</span> and <span class="elsevierStyleItalic">&#968;</span><span class="elsevierStyleInf">2</span> be two real-value functions of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span> defined on the differential manifolds <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#46; We define their sum by setting &#936;&#953;&#61;&#40;&#968;1&#43;&#968;2&#41;&#8728;&#966;&#953;&#8722;1&#61;&#968;1&#8728;&#966;&#953;&#8722;1&#43;&#968;2&#8728;&#966;&#953;&#8722;1 for any chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44;<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; Since the sum of two functions of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span> on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> are functions of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span><span class="elsevierStyleItalic">&#44;</span> the proof of this formula can be obtained by the expression<elsevierMultimedia ident="eq0180"></elsevierMultimedia>where &#966;&#953;&#954;&#40;&#955;&#44;&#8201;&#956;&#41;&#61;&#40;&#966;&#953;&#954;1&#8201;&#40;&#955;&#44;&#8201;&#956;&#41;&#44;&#8201;&#966;&#953;&#954;2&#8201;&#40;&#955;&#44;&#8201;&#956;&#41;&#41;&#61;&#40;&#955;&#8242;&#40;&#955;&#44;&#8201;&#956;&#41;&#44;&#956;&#8242;&#40;&#955;&#44;&#8201;&#956;&#41;&#41;</p><p id="par0335" class="elsevierStylePara elsevierViewall">Decompose now the streamfunctions into an eigenfunction part &#40;&#968;1&#8728;&#966;&#954;&#8722;1&#41;&#8201;&#40;&#955;&#8242;&#44;&#8201;&#956;&#8242;&#41;&#61;Y&#957;&#40;&#955;&#8242;&#44;&#8201;&#8201;&#956;&#8242;&#41; and a zonal part &#40;&#968;2&#8728;&#966;&#953;&#8722;1&#41;&#8201;&#40;&#955;&#44;&#8201;&#956;&#41;&#61;&#8722;&#969;&#956;&#43;D where &#8211;<span class="elsevierStyleItalic">&#969;&#956;</span> is a solid-body rotation and <span class="elsevierStyleItalic">D</span> a constant&#46; In chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41; with coordinates &#40;<span class="elsevierStyleItalic">&#955;&#8217;&#44; &#956;&#8217;</span>&#41;&#44; the north pole <span class="elsevierStyleItalic">N&#8217;</span> moves along a circle of constant latitude with constant angular velocity c<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span>&#46; In the primed coordinates&#44; Verkley &#40;1984&#44; <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>&#41; modons have the form<elsevierMultimedia ident="eq0185"></elsevierMultimedia>which consists of a dipole and a monopole component&#58;<elsevierMultimedia ident="eq0190"></elsevierMultimedia></p><p id="par0340" class="elsevierStylePara elsevierViewall">where <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf">0</span> &#61; sen <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf">a</span> &#61; sen <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">a</span></span>&#46; The functions <span class="elsevierStyleItalic">f</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">d</span></span>&#40;<span class="elsevierStyleItalic">&#956;</span>&#41; and <span class="elsevierStyleItalic">f</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">m</span></span>&#40;<span class="elsevierStyleItalic">&#956;</span>&#41; are defined as<elsevierMultimedia ident="eq0195"></elsevierMultimedia>and<elsevierMultimedia ident="eq0200"></elsevierMultimedia></p><p id="par0345" class="elsevierStylePara elsevierViewall">where b&#61;k2&#43;14&#43;2&#945;&#40;&#945;&#43;1&#41;&#8722;2 and<elsevierMultimedia ident="eq0205"></elsevierMultimedia></p><p id="par0350" class="elsevierStylePara elsevierViewall">The fact that<elsevierMultimedia ident="eq0210"></elsevierMultimedia>is a solution to Eq&#46; <a class="elsevierStyleCrossRef" href="#eq0095">&#40;6&#41;</a> is due to the work of <a class="elsevierStyleCrossRef" href="#bib0190">Verkley &#40;1984&#41;</a>&#44; which I will not reproduce in this paper&#46; <span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> is an eigenfunction of the Laplace-Beltrami operator and <span class="elsevierStyleItalic">&#967;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span> &#61; &#8211;&#957;&#40;&#957; &#43; 1&#41; is the eigenvalue of Y<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#957;</span></span>&#46; The Legendre functions H&#956;&#61;P&#957;m&#956; and H&#956;&#61;Q&#957;m&#956; are solutions to the Legendre differential equation of hypergeometric type&#44; where P&#957;m&#40;&#956;&#41; is a Legendre function of the first kind and Q&#957;m&#40;&#956;&#41; is a Legendre function of the second kind for order <span class="elsevierStyleItalic">m</span> such that &#957; is the complex degree&#46; The explicit expresion for P&#957;m&#956; and Q&#957;m&#956; with &#8211;1 &#60; <span class="elsevierStyleItalic">&#956;</span> &#60; 1 can be found in <a class="elsevierStyleCrossRef" href="#bib0005">Abramowitz and Stegun &#40;1965&#41;</a> or <a class="elsevierStyleCrossRef" href="#bib0190">Verkley &#40;1984&#41;</a>&#46;</p><p id="par0355" class="elsevierStylePara elsevierViewall">By using a grid of 5 &#215; 5&#176; upon the local coordinate associated with the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span>&#44;</span><span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41;&#44; the <a class="elsevierStyleCrossRef" href="#bib0190">Verkley&#44; 1984</a> modon was numerically generated&#46; Using <a class="elsevierStyleCrossRef" href="#eq0020">Eqs&#46; &#40;4&#41;</a> and <a class="elsevierStyleCrossRef" href="#eq0025">&#40;5&#41;</a> a workable Gaussian mesh of &#40;128&#44; 64&#41; points upon the geographical coordinate group &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; was also generated&#46; This mesh was mapped onto the local coordinates system associated to the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span>&#44;</span><span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41;&#46; The values of &#936;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span> were interpolated on the Gaussians points &#40;128&#44; 64&#41; by implementing a nine- point Lagrange interpolation scheme&#46; The resulting function&#44; i&#46;e&#46; the <a class="elsevierStyleCrossRef" href="#bib0190">Verkley&#44; 1984</a> equatorial modon viewed on the geographical coordinate group &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;<span class="elsevierStyleSmallCaps">&#44;</span> is shown in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3a</a>&#46; This small modon was defined by the following parameters&#58;<elsevierMultimedia ident="eq0215"></elsevierMultimedia></p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><p id="par0360" class="elsevierStylePara elsevierViewall">A numerical spectral model was used to simulate this small-size modon in <a class="elsevierStyleCrossRef" href="#bib0100">P&#233;rez-Garc&#237;a and Skiba &#40;1999&#41;</a>&#44; and in <a class="elsevierStyleCrossRef" href="#bib0150">Skiba and P&#233;rez-Garc&#237;a &#40;2009&#41;</a> a numerical spectral method for normal mode stability study of ideal flows on a rotating sphere was tested for this isolated steady modon constructed by <a class="elsevierStyleCrossRef" href="#bib0190">Verkley &#40;1984&#41;</a>&#46;</p><p id="par0365" class="elsevierStylePara elsevierViewall">Studies done by <a class="elsevierStyleCrossRef" href="#bib0070">Illari &#40;1984&#41;</a> and <a class="elsevierStyleCrossRef" href="#bib0035">Crum and Stevens &#40;1988&#41;</a> noted that the values of isentropic potential vorticity are relatively low and uniform in the blocking region&#46; In our following argument we consider that Verkley&#39;s modon &#40;1990&#41; provides a better and more uniform description of atmospheric blocking&#46; Our interest lays within the fields that build this phenomenon&#46; These solutions are characterized by a region <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> in which <span class="elsevierStyleItalic">q</span> is constant&#44; and an outer region <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">o</span></span> separated from the inner region by the boundary circle &#8706;<span class="elsevierStyleItalic">D</span>&#44; on which &#936; and <span class="elsevierStyleItalic">q</span> are both constant&#44; i&#46;e&#46;&#44; &#936; &#61; <span class="elsevierStyleItalic">d</span> and <span class="elsevierStyleItalic">q</span> &#61; <span class="elsevierStyleItalic">b</span>&#46;</p><p id="par0370" class="elsevierStylePara elsevierViewall">In the primed coordinates the <a class="elsevierStyleCrossRef" href="#bib0200">Verkley &#40;1990&#41;</a> modon has the form<elsevierMultimedia ident="eq0220"></elsevierMultimedia></p><p id="par0375" class="elsevierStylePara elsevierViewall">where solid-body rotation terms can be expressed in primed coordinates using <a class="elsevierStyleCrossRefs" href="#eq0020">Eqs&#46; &#40;4-5&#41;</a>&#44; such that in chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; the eigenfuctions at the outer region and inner region are &#916;&#8242;Yo&#61;&#8722;&#967;oYo&#59;&#8201;&#916;&#8242;Yi&#61;ei being &#967;<span class="elsevierStyleSup"><span class="elsevierStyleItalic">o</span></span> and <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> constants&#46; Certain requirements of continuity must be met to generate these functions over the circle &#8706;<span class="elsevierStyleItalic">D&#46;</span> We require continuity in and the first and second derivative in <span class="elsevierStyleItalic">&#966;</span>&#8217; at <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">a</span>&#46;</p><p id="par0380" class="elsevierStylePara elsevierViewall">To construct the <a class="elsevierStyleCrossRef" href="#bib0200">Verkley&#44; 1990</a> uniform modon on the manifold S<span class="elsevierStyleInf">2</span>&#44; it is interpreted as&#58;</p><p id="par0385" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proposition</span>&#46; Consider an atlas &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#8467;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#8467;</span></span>&#41;&#125;&#44; <span class="elsevierStyleItalic">&#8467;</span> &#61; <span class="elsevierStyleItalic">&#953;</span><span class="elsevierStyleBold">&#44;</span><span class="elsevierStyleItalic">&#954;</span><span class="elsevierStyleBold">&#44;</span> of <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> and let <span class="elsevierStyleItalic">&#968;</span> &#61; <span class="elsevierStyleItalic">&#968;</span><span class="elsevierStyleInf">1</span> &#43; <span class="elsevierStyleItalic">&#968;</span><span class="elsevierStyleInf">2</span> &#58; S<span class="elsevierStyleSup">2</span> &#8594; R be the stream-function of <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span><span class="elsevierStyleItalic">&#46;</span> Then<elsevierMultimedia ident="eq0225"></elsevierMultimedia></p><p id="par0390" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Proof</span>&#46; Let <span class="elsevierStyleItalic">&#968;</span><span class="elsevierStyleInf">1</span> and <span class="elsevierStyleItalic">&#968;</span><span class="elsevierStyleInf">2</span> be two real-value functions of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span> defined on the differential manifolds <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#46; We define their sum by setting &#936;&#954;&#968;1&#43;&#968;2&#8728;&#966;&#954;&#8722;1&#61;&#968;1&#8728;&#966;&#954;&#8722;1&#43;&#968;2&#8728;&#966;&#954;&#8722;1 for any chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41;&#46; Since the sum of two functions of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span> on <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> are functions of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">r</span></span><span class="elsevierStyleItalic">&#44;</span> the proof of the proposition follows from&#58;<elsevierMultimedia ident="eq0255"></elsevierMultimedia>because &#955;&#44;&#956;&#61;&#966;&#954;&#953;&#955;&#8242;&#44;&#956;&#8242;</p><p id="par0395" class="elsevierStylePara elsevierViewall">According to <a class="elsevierStyleCrossRef" href="#bib0200">Verkley &#40;1990&#41;</a>&#44; to express the modon in a more explicit manner&#44; the functional forms <span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">o</span></span> and <span class="elsevierStyleItalic">Y</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> of the eigenfuntions are&#58;<elsevierMultimedia ident="eq0230"></elsevierMultimedia>where <span class="elsevierStyleItalic">A</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span> and <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span> are constant&#46; The special functions S&#963;m&#952;&#8242;&#61;P&#963;m&#8722;cos&#8201;&#952;&#8242; and <span class="elsevierStyleItalic">T</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">m</span></span>&#40;<span class="elsevierStyleItalic">&#952;</span>&#8217;&#41; were given by <a class="elsevierStyleCrossRef" href="#bib0200">Verkley &#40;1990&#41;</a>&#46;</p><p id="par0400" class="elsevierStylePara elsevierViewall">We have also reproduced numerically the uniform modon constructed by <a class="elsevierStyleCrossRef" href="#bib0200">Verkley &#40;1990&#41;</a> using the parameters &#966;a&#61;5&#960;12&#44;&#8201;&#8201;&#963;&#61;8&#46;06&#59;&#8201;&#8201;&#969;o&#61;0&#46;028&#44; and when the modon center is in the point &#955;o&#61;180&#176;&#44;&#966;o&#61;&#960;4 &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Fig 3b</a>&#41;&#46;</p></span></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">4</span><span class="elsevierStyleSectionTitle" id="sect0045">Conclusions</span><p id="par0405" class="elsevierStylePara elsevierViewall">The exact solutions of the barotropic vorticity equation on the rotating unit sphere as a compact differentiable manifold without boundary&#44; which are zonal flows&#44; homogeneous spherical polynomial flows&#44; Rossby-Haurwitz waves and generalized solutions named modons&#44; were represented in this paper&#46; A concrete notion of local chart&#44; change of charts&#44; and atlas for the manifolds <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> was developed&#46; An atlas &#123;&#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#44; &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41;&#125; was constructed for <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; where every chart corresponds to a geographical coordinate group that covers most of S2&#46; The transition maps &#966;&#953;&#954;&#61;&#966;&#954;&#8728;&#966;&#953;&#8722;1&#58;&#966;&#953;&#937;&#953;&#8745;&#937;&#954;&#8594;&#966;&#954;&#937;&#953;&#8745;&#937;&#954; and &#966;&#954;&#953;&#61;&#966;&#953;&#8728;&#966;&#954;&#8722;1&#58;&#966;&#954;&#937;&#954;&#8745;&#937;&#953;&#8594;&#966;&#953;&#937;&#954;&#8745;&#937;&#953; were also constructed&#44; and the exact solutions of the barotropic vorticity equation in a manifold context were studied&#46; This work also formulates on the manifolds <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> in terms of the stream function <span class="elsevierStyleItalic">&#968;</span>&#58; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> &#8594; <span class="elsevierStyleItalic">R</span>&#44; for RH waves which are sufficiently smooth&#44; and for Wu-Verkley waves and modons which are differentiably weak&#46; RH waves as solutions &#968;&#8728;&#966;&#953;&#8722;1 of the barotropic vorticity equation on the manifolds <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> are presented at the local coordinate associated with a chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41;&#46; The way in which the modon solution &#968;&#8728;&#966;&#954;&#8722;1 can be constructed in the chart &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41;<span class="elsevierStyleSmallCaps">&#44;</span> where <span class="elsevierStyleItalic">&#968;</span> is <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">2</span>&#44; is investigated too&#46; In the chart coordinate &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#954;</span></span>&#41;<span class="elsevierStyleSmallCaps">&#44;</span> the Verkley &#40;1984&#44; <a class="elsevierStyleCrossRef" href="#bib0195">1987</a>&#41; modons have the form<elsevierMultimedia ident="eq0235"></elsevierMultimedia>To construct the <a class="elsevierStyleCrossRef" href="#bib0200">Verkley &#40;1990&#41;</a> uniform modon on the manifold <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span>&#44; it is interpreted as<elsevierMultimedia ident="eq0240"></elsevierMultimedia></p><p id="par0410" class="elsevierStylePara elsevierViewall">when the modon center is in the point <span class="elsevierStyleItalic">&#955;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">o</span></span> &#61; 270&#59; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">o</span></span> &#61; 0&#46; However&#44; to contruct the <a class="elsevierStyleCrossRef" href="#bib0190">verkley &#40;1984</a>&#44;<a class="elsevierStyleCrossRef" href="#bib0195">1987</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0200">1990</a>&#41; with <span class="elsevierStyleItalic">N&#8217;</span> a point arbitrary on the sphere <span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup">2</span> a collection of pairs &#40;&#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#44; <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#953;</span></span>&#41; &#40;i &#62; 2&#41; is needed&#46; Our viewpoint here was to understand the solution of the barotropic vorticity equation on the manifold S<span class="elsevierStyleSup">2</span> and its use to derive properties of the solutions to the Riemannian manifold &#40;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#44; g</span>&#41;&#46;</p></span></span>"
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Article information

ISSN: 01876236
Original language: English

DOI:10.20937/ATM.2015.28.03.03

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Year/Month	Html	Pdf	Total

2024 November	2	0	2
2024 October	22	13	35
2024 September	22	13	35
2024 August	18	7	25
2024 July	9	5	14
2024 June	17	8	25
2024 May	34	18	52
2024 April	29	2	31
2024 March	176	2	178
2024 February	35	6	41
2024 January	28	7	35
2023 December	18	7	25
2023 November	26	10	36
2023 October	15	12	27
2023 September	17	8	25
2023 August	12	9	21
2023 July	11	8	19
2023 June	28	11	39
2023 May	22	9	31
2023 April	23	5	28
2023 March	29	12	41
2023 February	7	13	20
2023 January	16	14	30
2022 December	29	8	37
2022 November	39	18	57
2022 October	20	17	37
2022 September	19	23	42
2022 August	26	11	37
2022 July	16	14	30
2022 June	15	12	27
2022 May	21	20	41
2022 April	23	28	51
2022 March	32	21	53
2022 February	27	11	38
2022 January	60	8	68
2021 December	29	10	39
2021 November	46	6	52
2021 October	21	14	35
2021 September	20	13	33
2021 August	14	5	19
2021 July	22	8	30
2021 June	12	4	16
2021 May	31	13	44
2021 April	41	23	64
2021 March	40	10	50
2021 February	13	9	22
2021 January	15	10	25
2020 December	10	8	18
2020 November	19	13	32
2020 October	9	7	16
2020 September	12	8	20
2020 August	16	3	19
2020 July	7	1	8
2020 June	16	7	23
2020 May	15	2	17
2020 April	10	4	14
2020 March	5	5	10
2020 February	16	11	27
2020 January	13	8	21
2019 December	11	5	16
2019 November	11	6	17
2019 October	4	1	5
2019 September	10	1	11
2019 August	7	2	9
2019 July	7	3	10
2019 June	26	29	55
2019 May	60	43	103
2019 April	34	2	36
2019 March	6	0	6
2019 February	6	3	9
2019 January	3	1	4
2018 December	1	2	3
2018 November	6	4	10
2018 October	8	6	14
2018 September	22	2	24
2018 August	5	4	9
2018 July	5	2	7
2018 June	4	0	4
2018 May	6	1	7
2018 April	2	2	4
2018 March	1	0	1
2018 February	1	0	1
2018 January	4	0	4
2017 December	3	0	3
2017 November	7	1	8
2017 October	3	3	6
2017 September	7	2	9
2017 August	5	0	5
2017 July	9	3	12
2017 June	3	1	4
2017 May	1	0	1
2017 March	3	1	4

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