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. E-mail: ismael@unam.mx

Received November 7, 2014; accepted June 1, 2015

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 esfera 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 es entender mejor la solución de la ecuación de vorticidad barotrópica sobre la variedad S² y su utilidad para identificar las propiedades de las 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.

ABSTRACT

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 differentiable manifold. When the differentiable manifold S 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 solutions for the barotropic vorticity equation.

Keywords: Rossby-Haurwitz waves, modons, hydrodynamics equation on manifolds, unit sphere, mathematical analysis of barotropic model.

1. Introduction

Let S² = {x ∈ R³ :| x | = 1} denote the unit sphere in R³. The large-scale dynamics of the 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-dimensional form as:

wheredenotes the stream function, μ = sinthe longitude, ø the latitude, and θ the colatitude. Δ is the Laplace-Beltrami operator on a sphere and J(, h) is the Jacobian.

The following is a solution for Eq. (1) on the sphere proposed by Thompson (1982):

where are the spherical coordinates relative to a rotated pole N with coordinates (λ₀, μ₀) with respect to the original system, and Y_v is an eigenfunction of the operator Laplace-Beltrami with eigenvalue x_v. Verkley(1984) generalized Thompson's solution and demonstrated that Y_v could be a set of eigenfunctions that contains more than only spherical harmonics. Then Eq. (2) describes a configuration in which the structure Y_v moves through the zonal flow -ωμ with constant velocity c_v and without changing size and shape. The pole of the primed system N that moves along a latitude at a constant angular velocity c_v is given by

where x_v is an eigenvalue for the spectral problem ΔY_v = —x_vY_v. In particular, for spherical harmonics Y of degree n corresponding to the eigenvalue X_v = Xn = n(n + 1), Eq.(2) 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 (Rossby, 1939; Haurwitz, 1940). The solution modon is constructed to divide the sphere S² into two regions (Tribbia, 1984; Verkley, 1984, 1987, 1990; Neven, 1992): an inner region S, centered around the pole NV, and an outer region S_o separated from the inner region by a boundary circle in which q and its normal derivative ' are continuous. Modons are considered appropriate to describe some types of atmospheric blocking events (Verkley, 1990).

Hydrodynamic equations on manifolds were studied by Ebin and Marsden (1970), Szeptycki (1973a, b), Avez and Bamberger (1978), Ghidaglia et al. (1988), Temam (1987) and Ilyin (1993). The existence, unicity and regularity of the solution for the evolution equation (Eq. (1)) on S² were proven by Szeptycki (1973a, b), Avez and Bamberger (1978), Ilyin (1993) and Skiba (2012). Ebin and Marsden (1970) 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 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 differentia-ble 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.

2. Structure of functions on the manifold S²

In 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 S is then a pair where Ω is an open subset of S², and is a homeomorphism of Ω onto some open subset of R². Let us consider the two charts of class C^p 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 denote the coordinate function, which maps from (x₁, x₂, x₃) to angles (λ, θ) or to (λ,μ). The domain of _K^—1 is the open set defined by λ ∈ (—π, π) and θ ∈ (0,π) (this excludes the poles). The inverse map^—1 yields the parameterization x₁ = cos λ sin θ, x₂ = sin λ sin θ, x₃ = cos θ and its variationyields the parameterization (λ', θ')=(cos λ' sin θ', cos θ', sin λ' sin θ'). The domain of in the open set defined by λ' ∈ (—π, π) and θ' ∈ (0, π). The charts and correspond to poles N and N' on the sphere S². N' might be taken as the point in the old system and as the angleλ' in this new north pole, so that the new international date line is the half circle of the old equator in the x₁x₂-plane, on the front where x₁ ≥ 0 (Richtmyer, 1981; Skiba, 1989; Pérez-García, 2001). The international date line, for the chart is the half circle in the x₁ x₃ -plane. The chart covers the sphere except for the set Γ_l , and the chart similarly covers the sphere with the exception of a set Γ_K. Hence the two charts together cover S² and they constitute an atlas.

The local coordinates associated with the chart are functions , such that forμ'(p)) are local coordinates with respect to the chart(Fig. 1).

To construct the map a bijection with inverse defined as it is seen that every is

open. Hence each is an open subset of S² and cover S².

Given two charts of the atlas with , the transition maps outline open sets of R² into R², where This determines a differentiable structure for S², and is a diffeomorphism. It is then said that the atlas is of class C^k if the transition functions are of C^k.

Let x be any point of and the coordinate of then is a continuous function on two variables. Now, if such that , and since we have the relations

This is the transformation formula betwen the two local coordinate systems defined on To obtain the relations between the unprimed and primed coordinate of any point Q on the sphere, Verkley (1984) 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 and set as the Jacobian matrix of map so we can verify that

Then det. Hence, it is said that if manifold S² is oriented for every pair of intersecting local coordinate neighbourhoods, det.

Indeed we can regard the coordinate as a device to decide which of many functions on S are to be differentiable. Since is just a set, it makes no sense to ask that be differentiable (Matsushima, 1972; Loomis and Sterberg, 1990). However, we can consider the map Then is a function defined on an open R², and we know what it means for such a function to be differentiable or smooth (see Fig. 1). Consider now what happens when we change coordinates to some other chart, lets say for convenience, assuming that Then it is possible forto be differentiable but is not. To compare both, let where the map is a bijection between open subsets of R². Then a suficient condition for to be differentiable if is, is that is also differentiable. We often write for the composite function .

Lets take a curve with (0) = p. In a local chart r is given by. On the manifold S², one can define a vector U tangent to the parametrized curve r at any point p on the curve. The tangent vector U is given by a column vector u whose components are (0), (i = 1, 2), with the initial condition (0) = p. If we use another coordinate system corresponding to the chart ( bythen the tangent vector U is given by a column vector v with components According to the chain rule, the column vectors u and v are related by . The expression is the partial differential operator in the direction of the tangent vector.

The space T_pS² is called the tangent space of S² at p, and T_pS² is a two-dimensional vector space. For each u ∈ T_pS² we shall write where uⁱ_l are the contravariant components of u. It is well known that on the manifold S² an inner product is defined at each tangent space T_pS². Now lets present a basis in which we denote the coordinate system corresponding to the chart by and for any define the vectors by so that they are independent since be the outward normal to S² in R³; without any loss of generality we may assume that the vectors T_pS².

We will denote the vector space of a vector field on S by A tangent vector field on S² is a smooth map u: S² → T S² such that, for any. At the chart , for the vector functions u and v ∈ have components respectively, being these the components of u_p as the vectors of the unitary base indicated by and in the directions 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 | v | _g(p) ∈ R⁺. The g on S² is a smooth (2, 0)-tensor field on S² such that for any p ∈ S², g_p: T_p(S²) x T_p(S²) → R is a scalar product on the tangent space T_p(S²), and in any chart of S², its components

form a symmetric matrix, with its inverse denoted by (g^ij) = (g_ij)^-1, and g = det(g_ij) = 1 - μ². The length of a tangent vector v ∈ T_pS² is defined as usual, | v |= g_p (v, v)^1/2 = (v ^. v)^1/2. Moreover, the inner product on T S² is given by u ^. v = g_ijuⁱv^J, 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 p, the gradient of , is given by the vector field grad : S² → T_pS², for which

If u ∈ Γ(S²), the divergence of u is the function on S² which on the chart is given by

A linear connection D on S² is a map D: T(S²) x called the covariant derivative and the usual notation for D(U, V) is D_UV. Let be a chart and as one can observe, the vectors can be nonconstant. An easy notation is set (e.g. Hebey, 2000). There are smooth functions such that for any i, j, and any p ∈

where are the Christoffel symbols, defined by On the chart , we have

The 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

This operator satisfies some properties: Δ_g, is sel-fadjoint, symmetric and non-negative (Aubin, 1998). Thus, the operators div, grad and Δ_g 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 dv_g 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 = dv which is defined on all of S². For every chart ip_e) which is consistently oriented with S², the coordinate expresion for , where Φ_l is a partition of unity subordinate to the covering

On the Riemannian manifold (S², g), at the chart ; a volume form n = dv_g defines a Lebesgue measure on S² by Then

where defines a Lebesgue measure on R².

Let denote the set of infinitely differentiable functions of compact support the functions are smooth, together with the periodic boundary condition at with period . If we define the usual Hilbert space L²(S²) to be the completion of with respect to the inner product

and norm , g* is the complex conjugate of function g. Let (S², g) be the compact Riemannian manifold and dv_g the Riemannian volume element. Then functional spaces (Sobolev and the Holder spaces) can be deined on S² as well (Skiba, 2012). For each p ∈ R with we associate a Banach space

with respect to the norm

and ess sup 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 Diaz and Tello, 1999; Hebey, 2000).

We now turn to the eigenvalue problems for Δ_g: We usually seek to find all eigenvalues for which there is an eigenfunction Y such that Then, which information about geometry of (S², g) is encoded by the eigenvalues?. The structure of eigenfunctions: L^p norms and relations to RH waves or modons.

Global harmonic analysis is the study of the spectral theory of the Laplacian Δ_g 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 {Y_j} of smooth eigenfunctions and the spectrum of is a discrete set . Recent developments show that the non-zero eigenvalues also contain substantial geometric and analytic information. The solution modon constructed by Tribbia (1984), Verkley (1984, 1987, 1990) and Neven (1993) proposed the use of eigenfunctions {Y_j} as basic geometric structures.

The space of spherical harmonics of degree n on S², which coincides with the eigenspace of operator - corresponding to the eigenvalue , is denoted by H_n. Self-adjoint operators have the property that its eigenfunctions with different eigenvalues are orthogonal, which implies that the eigenspaces H_n are orthogonal and have 2n+1dimensions. 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 (Hobson, 1931) and is understood as the development of a function in terms of this series of spherical harmonics.

The spaces H_n and H_k (n ≠ k) are mutually orthogonal in L²(S²). Then there is the orthogonal projection Y_n : L²(S²) — H_n , and so smooth functions ∈ L²(S²) on the sphere S² have a development in spherical harmonics,

where is the homogeneous spherical polynomial of degree n from H_n, and is the Fourier coefficient of . The 2n + 1 spherical harmonics

of degree n and zonal number form an orthonormal basis in H_n. Here the numbers C_nm are the normalizers in L²(S²), given by C_nm = and are the associated Legendre functions given by

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; Bihlo and Popoych, 2012) arises. Even if a manifold S² is not a Lie group, there may be an action •; : G x 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 (Richtmyer, 1981). When S²G/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 (Helgason, 1984).

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 Stein and Weiss (1971). 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 (Narcowich et al., 2006). 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; Hebey, 2000).

Let be an atlas of S² and the streamfunction of class C^r. We can consider that the map is the streamfunction defined on an open and that it is of class C^r.

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

whereJ (c,q)qis the jacobian, is a tangent velocity vector, grad div grad , is the relative vorticity, q = Δ + 2μ is the absolute vorticity and k is a unit outward normal vector. The velocity vector field u having the components 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 (Simmons et al., 1983; Skiba, 1997). The four types of exact solutions to Eq. (1) known up to now are described below:

• The zonal flows and Rossby-Haurwitz (RH) waves (Haurwitz, 1949), called classical solutions, diferentiated from the generalized solutions which are not so smooth.

• The first generalized solutions of Eq. (6), kown as modons, were originally constructed by Tribbia (1984) and Verkley (1984, 1987, 1990) by using two eigenfunctions for the Laplace operator of different degrees.

• Later on, Neven (1992) gave generalized solutions in the form of a quadrupole modon.

• Wu and Verkley (1993) constructed generalized global solutions composed of two RH waves (Pérez-García and Skiba, 1999).

In the present work, zonal flows, homogeneous spherical polynomials flows, RH waves, and modons on the manifold S² are considered.

Let us consider the zonal flows, homogeneous spherical polynomials flows and Rossby-Haurwitz (RH) waves.

Proposition (zonal flow). Let be an atlas of S² and the streamfunction of class C^r. Then the zonal flow map Ψ_l= of C^r defined as

is an exact solution of the vorticity Eq. (6) for any b_n.

Proof. The demonstration, obtained from Eq. (6), is quite trivial.

Proposition (homogeneous polynomials). Let be an atlas of S² and the streamfunction of class C^r. Then the homogeneous spherical polynomial map = of degree n ≥ 2 defined as

is an exact solution to the vorticity Eq. (6), where a_m can be a complex factor and

Proof. Given we define If, in addition, we have the following expression

we have

from BVE (Eq. (6)). It follows that .

Proposition (Rossby-Haurwitz waves). Let be an atlas of S² and the stream-function of class . Then, the map = of , defined as

with n ≥ 1; is called Rossby-Haurwitz (RH) waves. It is an exact solution of the vorticity Eq. (6) if the angular phase speed of the RH wave

Here ω is the super-rotation velocity and each H_n corresponds to the eigenvalue . Proof. Here is expressed by

where. We can notice that

.

y which implies that

Furthermore:

where ; so that from BVE (Eq. (6))

so that

and thus the proposition is proved.

The streamfunction of the stationary RH(2,5) wave

with the parameters defined by (m, n) = (2, 5), a = .007 and is given in Figure 2.

3.2 Generalized solutions

Denote the spherical distance between two points of S² by d(.,.). Let N' be the north pole of the chart coordinate . Then a disk or inner region S, on the sphere is defined as , such that . The solution modon is constructed (Tribbia, 1984; Verkley, 1984, 1987, 1990; Neven, 1993) to divide the sphere S² into two regions: an inner region S_i centered around the pole N', and an outer region S_o separated from the inner region by a boundary circle , on which , q and are continuous.

For S_i, a solution of the Eq. (2) form is chosen with an eigenfunction Y , which has its singularity in the outer region. The same type of solution is chosen for the outer region, but such that Y has its singularity in the inner region. Then both solutions are combined as smoothly as possible on the boundary circle (Tribbia, 1984; Verkley, 1984, 1987).

To construct the Verkley (1984) modon or the Neven (1992) cuadrupole modon on the manifold S², it is interpreted as:

Proposition. Let be an atlas of S², and the streamfunction of C^r. Then

Proof. Let and be two real-value functions of class C^r defined on the differential manifolds S². We define their sum by setting for any chart Since the sum of two functions of class C^r on S² are functions of class C^r, the proof of this formula can be obtained by the expression

where

Decompose now the streamfunctions into an eigenfunction part and a zonal part where -ωμ is a solid-body rotation and D a constant. In chart with coordinates , the north pole N' moves along a circle of constant latitude with constant angular velocity c_v. In the primed coordinates, Verkley (1984, 1987) modons have the form

which consists of a dipole and a monopole component:

where μ₀ = sen ø₀, μ_a = sen ø_a. The functions f^d(μ) and f^m(μ) are defined as

and

The fact that

is a solution to Eq. (6) is due to the work of Verkley (1984), which I will not reproduce in this paper. Y_v is an eigenfunction of the Laplace-Beltrami operator and is the eigenvalue of Y_v. The Legendre functions are solutions to the Legendre differential equation of hypergeometric type, where is a Legendre function of the first kind and is a Legendre function of the second kind for order m such that ν is the complex degree. The explicit expresion for and with -1 < μ < 1 can be found in Abramowitz and Stegun (1965) or Verkley (1984).

By using a grid of 5 x 5° upon the local coordinate associated with the chart , the Verkley 1984 modon was numerically generated. Using Eqs. (4) and (5) a workable Gaussian mesh of (128, 64) points upon the geographical coordinate group was also generated. This mesh was mapped onto the local coordinates system associated to the chart . The values of were interpolated on the Gaussians points (128, 64) by implementing a nine-point Lagrange interpolation scheme. The resulting function, i.e. the Verkley 1984 equatorial modon viewed on the geographical coordinate group is shown in Figure 3a. This small modon was defined by the following parameters:

k = 10, a = 10, μ_a = sin 66.14°, μ₀ = 0, λ₀ = 270° and D₀ = 0.

A numerical spectral model was used to simulate this small-size modon in Pérez-García and Skiba (1999), and in Skiba and Pérez-García (2009) a numerical spectral method for normal mode stability study of ideal flows on a rotating sphere was tested f or this isolated steady modon constructed by Verkley (1984).

Studies done by Illari (1984) and Crum and Stevens (1988) 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 S_i in which q is constant, and an outer region S_o separated from the inner region by the boundary circle , on which and q are both constant, i.e., = d and q = b.

In the primed coordinates the Verkley (1990) modon has the form

where solid-body rotation terms y can be expressed in primed coordinates using Eqs. (4-5), such that in chart the eigenfuctions at the outer region and inner region are being and e_i constants. Certain requirements of continuity must be met to generate these functions over the circle . We require continuity in and the first and second derivative in

To construct the Verkley 1990 uniform modon on the manifold S², it is interpreted as:

Proposition. Consider an atlas  be the stream-function of C^r. Then

Proof. Let and be two real-value functions of class C^r defined on the differential manifolds S². We define their sum by setting for any chart . Since the sum of two functions of class C^r on S² are functions of class C^r, the proof of the proposition follows from:

because

According to Verkley (1990), to express the modon in a more explicit manner, the functional forms Y_o and Y_i of the eigenfuntions are:

where A_s and B_s are constant. The special functions were given by Verkley (1990).

We have also reproduced numerically the uniform modon constructed by Verkley (1990) using the parameters , and when the modon center is in the point (Fig 3b).

4. Conclusions

The 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 was constructed for S², where every chart corresponds to a geographical coordinate group that covers most of S². The transition maps and 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 p: S² → R, for RH waves which are sufficiently smooth, and for Wu-Verkley waves and modons which are differentiably weak. RH waves as solutions p of the barotropic vorticity equation on the manifolds S² are presented at the local coordinate associated with a chart , The way in which the modon solution can be constructed in the chart , where is C², is investigated too. In the chart coordinate , the Verkley (1984, 1987) modons have the form

To construct the Verkley (1990) uniform modon on the manifold S², it is interpreted as

when the modon center is in the point However, to contruct the verkley (1984, 1987, 1990) with N' a point arbitrary on the sphere S² a collection of pairs (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).

The author is grateful to A. Aguilar in the preparation of the figures. N. Aguilar, A. Alazraki, and C. Amescua gave valuable suggestions for improving the manuscript.

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