ERROR ESTIMATION OF PSEUDOSPECTRAL METHOD FOR SOLVING THE BAROTROPIC VORTICITY EQUATION

Vol. 35, No. 1, 2005, 137-149

ERROR ESTIMATION OF PSEUDOSPECTRAL METHOD FOR SOLVING THE BAROTROPIC

VORTICITY EQUATION

Abdur Rashid¹

Abstract. This paper is concerned with pseudospectral method us- ing Legendre polynomials. The pseudospectral approximation for the barotropic vorticity equation is considered. The stability and rate of con- vergence are obtained. As a consequence, it is shown that this method achieves spectral accuracy if the solution to the barotropic vorticity equa- tion is smooth.

AMS Mathematics Subject Classification (2000): 65N35; 65N12

Key words and phrases: Barotropic vorticity equation, pseudospectral method, stability, convergence

1. Introduction

The barotropic vorticity equation model is very important in the re- search of meteorological science and applied mathematics. Many scientists pay attention to the research of numerical methods of this equation. The early works were mainly concerned with finite difference methods [2, 4, 8, 9, 13].

The spectral method has a convergence rate of ”infinite” order, i.e., the error decays faster than algebraically when the solution is infinitely smooth. This method has become one of the most powerful tools for the numerical solution of nonlinear partial differential equations arising in the fluid dynamics [1, 5, 10]. Many authors provide various spectral schemes and analyze the errors.

Usually, only nonlinear problems in Cartesian coordinates are considered. But in meteorological science and some other fields [7, 14], one also has to deal with the problem defined on the spherical surface.

Since spectral methods usually entail too much computation work, some- times it is even impossible to implement them strictly [3, 6]. The pseudospectral methods are preferred over spectral methods in practice, due to their numerical efficiency. However, there is a price to pay in the form of aliasing error which cause poor accuracy and instability in resolution demanding problems. There- fore, various techniques have already been developed for de-aliasing [11, 12].

In this paper, by taking the Barotropic Vorticity equation as an example, the pseudospectral method for solving partial differential equations on spherical sur- face is discussed. An interpolation procedure, which is different from that in the ordinary sense, is proposed. Based on such an interpolation, a pseudospectral scheme is constructed. Its generalized stability and convergence are analyzed rigorously.

1Department of Mathematical Sciences, COMSATS Institute of Information Technology, Defence Road, Off Raiwind Road, Lahore, Pakistan, e-mail: rashid−[email protected]

2. The Pseudospectral Scheme

LetS be the unit spherical surface S =n

(λ, θ) : 0≤λ <2π,−π

2 ≤θ≤ π 2 o

,

where λand θ are the longitude and latitude coordinate on the spherical sur- face, respectively. Let ξ(λ, θ, t), ψ(λ, θ, t), and Ω > 0 be the vorticity, the stream function and angular velocity of the earth respectively. The gradient, the jacobian operator and the laplacian are as follows:

∇u= 1

cosθ

∂u

∂λ,∂u

∂θ

, J(u, v) = 1 cosθ

∂u

∂λ

∂v

∂θ −∂u

∂θ

∂v

∂λ

, 4u= 1

cosθ

∂

∂θ

cosθ∂u

∂θ

+ 1

cos²θ

∂²u

∂λ². The barotropic vorticity equation onS is as follows:







∂ξ

∂t(t) +J(ξ(t), ψ(t))−2Ω_∂λ^∂ ψ(t) = 0, (λ, θ)∈S, t∈(0, T],

−4ψ(t) =ξ(t), (λ, θ)∈S, t∈(0, T], ξ(λ, θ,0) =ξ₀(λ, θ), (λ, θ)∈S,

(1)

where the initial valueξ0(λ, θ) is given. For fixedψ, we require

(2) µ(ψ(t)) =

Z Z

S

ψ(λ, θ, t)dS≡0.

Now, we are going to construct the pseudospectral scheme for (1). First we introduce some approximation subspaces and define an interpolation operator.

For a non-negative integern ≥0, denoted by Ln(x), the nth degree Legendre polynomial is defined on [−1,1]. Recall the orthogonality relation

(Li(x), Lj(x)) = 2

2i+ 1δij, ∀i, j≥0, where (f, g) = R1

−1f(x)g(x)dx. Also recall thatL⁰_n(x) satisfies the recurrence relation

L⁰_n(x) =

n−1

X

k=0

(2k+ 1)Lk(x).

For an integerm(|m| ≤n), the associated Legendre polynomials are defined as Lm,n(x) =

s

(2n+ 1)(n−m)!

2(n−m)! (1−x²)^m/2 d^m

dx^mLn(x), 0≤m≤n, Lm,n(x) =L_−m,n(x), −n≤m≤0.

Then the spherical harmonic functionsY_m,n(λ, θ) are

(3) Y_m,n(λ, θ) = 1

√2πe^imλL_m,n(sinθ), n≥ |m|.

Let N be any positive integer. We define the trail function space for pseudospec- tral approximation as

(4) Ve_N =





 v:v=

N

X

n=0

X

|m|≤n

bv_m,nY_m,n(λ, θ)





 .

Furthermore, define VN as the real subspace ofVeN, and V_N⁰ as the subspace of V_N, whose average on the spherical surface vanishes.

Next, we consider the interpolation fromC(S) ontoV_N, whereC(S) is the space containing the set of all continuous functions on S . Letx_j(0≤j ≤N) be the N+ 1 roots of the Legendre polynomialL_N+1(x). Clearly,x_j ∈[−1,1].

Let

w_j = 1

(1−x²_j)[L⁰_N+1(xj)]², 0≤j≤N,

which are theN+ 1 weights in the Legendre-Gauss quadrature formula associ- ated with theN+ 1 roots. Define SN as a set of grid points onS,

(5) S_N =

(λ_l, θ_j) :λ_l= 2lπ

2N+ 1,0≤l≤2N;θ_j =sin⁻¹x_j,0≤j≤N

. Then we define the interpolation operatorI_N, from C(S) onto V_N, as follows

(6) INv=

N

X

n=0

X

|m|≤n

vm,nYm,n(λ, θ), where

vm,n= 2π 2N+ 1

2N

X

l=0 N

X

j=0

wjv(λl, θj)Y_m,n^∗ (λl, θj).

Moreover, we introduce the discrete inner product (·,·)N as

(7) (u, v)N = 2π 2N+ 1

2N

X

l=0 N

X

j=0

wju(λl, θj)v^∗(λl, θj), ∀u, v∈C(S).

It is obvious that vm,n = (v, Ym,n)N. The degree of freedom of SN is (2N+ 1)(N+1), while the dimension ofV_N isPN

n=0(2n+1) = (N+1)², thus generally INv(λl, θj)6=v(λl, θj), ConsequentlyIN is not the interpolation operator in the ordinary sense. It is actually the projection fromC(S) ontoVN corresponding to the discrete inner product (·,·)N.

Finally, we consider the finite difference discretization in the temporal direc- tion. Let τ be the step size in timet, and

Rτ=

t=kτ : 1≤k≤ T

τ

.

Define

u_bt(λ, θ, t) = 1

2τ[u(λ, θ, t+τ)−u(λ, θ, t−τ)], u(λ, θ, t) =b 1

2[u(λ, θ, t+τ) +u(λ, θ, t−τ)].

The fully discrete Legendre pseudospectral scheme for solving (1)-(2) is to findη(t)∈VN,φ(t)∈V_N⁰, approximating toξ(t) andψ(t) respectively.











η_bt(t) +INJ(η(t), φ(t))−2Ω_∂λ^∂ φ(t) = 0, onS×Rτ,

−4φ(t) =η(t), onS×Rτ,

µ(φ(t)) = 0, onS,

η(0) =INξ0, η(τ) =IN

ξ0+τ_∂t^∂ξ(0)

onS.

(8)

Remark 2.1. The Fourier coefficients of unknown functionsη(t+ 2τ) andφ(t+ 2τ) can actually be evaluated explicitly. Besides, in calculating the nonlinear terms, Fast Fourier Transform technique can also be used in the λ direction.

The scheme (8) is of second order accuracy in the temporal direction, except for the calculation for an extra initial valueη(τ), its total computational work remain the same as two-level scheme

1

τ[η(t+τ)−η(τ)] +INJ(η(t), φ(t))−2Ω ∂

∂λφ(t) = 0.

3. Lemmas

Let D(S) be the set of all infinitely differentiable functions defined on S, which are regular at two polesθ=±^π₂. D⁰(S) is the dual space ofD(S). For anyu∈D⁰(S), we can defined the generalized derivatives, generalized gradient and Laplacian operators as in [15]. The inner product andL²-norm are defined as follows:

(u, v) = Z Z

S

uvds= Z 2π

0

Z π/2

−π/2

u(λ, θ)v(λ, θ)cosθdθdλ, kuk= (u, u)¹². The symbolL² denotes the set of all functions for which this integral is finite:

L²(S) ={u∈D⁰(S) :kuk<∞}. Furthermore, define

H¹(S) =

u:u, 1 cosθ

∂u

∂λ,∂u

∂θ ∈L²(S)

be the space equipped with the following semi-norm and norm as (9) |u|1=

1 cosθ

∂u

∂λ

2

+

∂u

∂θ

2!¹₂

, kuk1= (kuk²+|u|²₁)¹².

For the integerr≥0,H^r(S) can be defined similarly. In particular, the norm ofH²(S) is equivalent to (kuk²+k4uk²)^1/2.

For any real number r ≥ 0, we define H^r(S) as the complex interpolation between the two spaces H^[r](S) and H^[r+1](S), where [r] is the integral part of r. The Y_m,n are the eigenfunctions of the spherical Laplace operator 4, corresponding to the eigenvalues n(n+ 1). Thus in the space normH^r(S), the normkuk²_r is equivalent to [6]

(10)

∞

X

n=0

X

|m|≤n

n^r(n+ 1)^r|ub_m,n|²,

where bu_m,n are the Fourier coefficients related to the spherical harmonic func- tionsYm,n. Particularly,H⁰(S) =L²(S) andkuk0=kuk. It can be verified for any u, v∈H²(S)

(11) −(4u, v) = (∇u,∇v).

TheL²(S) orthogonal projection PN :L²(S)−→VN is such mapping that for any u∈L²(S)

(u−PNu, v) = 0, ∀v∈VN, or equivalently

P_Nu(λ, θ) =





N

X

l=−N

X

|m|≤N

ub_m,n



Y_l,m(λ, θ).

Throughout this paper we shall usecto denote a general positive constant independent ofτ andN. It can be of different values in different cases.

Lemma 1. If 0≤β≤r, then for allu∈H^r(S), ku−PNukβ≤cN^β−rkukr

Proof. By (10)

ku−PNuk²_β ≤ c

N

X

m=−N

∞

X

n=N+1

n^β(n+ 1)^β|bum,n|²+

+ c X

|m|>N

∞

X

n=|m|

n^β(n+ 1)^β|bum,n|²

≤ c

N

X

m=−N

∞

X

n=N+1

n^β(n+ 1)^β|bum,n|²+

+ c X

|m|>N

∞

X

n=N+1

n^β(n+ 1)^β|bu_m,n|²

≤ cN^2β−2r

∞

X

m=−∞

∞

X

n=N+1

n^r(n+ 1)^r|ubm,n|²

≤ cN^2β−2rkuk²_r

2

Lemma 2. (Inverse Inequality) If0≤r≤β, then∀u∈VN, kuk_β≤cN^β−rkuk_r

Proof. Let

u=

N

X

m=−N N

X

n=|m|

ub_m,nY_m,n(λ, θ).

Then by (10)

kuk²_β≤c

N

X

m=−N N

X

n=|m|

n^β(n+ 1)^β|ubm,n|²

≤cN^2β−2r

N

X

m=−N N

X

n=|m|

n^r(n+ 1)^r|ubm,n|²

≤cN^2β−2rkuk²_r

2

Lemma 3. [11] Ifu∈C(S)andv∈VN, then:

(i)I_Nv=v, (ii) (I_Nu, v) = (I_Nu, v)_N = (u, v)_N. Lemma 4. [5] For allu∈C(S)we have

kINuk=kINukN ≤ kukN, ku−INukN = inf

∀v∈VN

ku−vkN. Lemma 5. [5] For allu∈VN

(i)

1 cosθ

∂u

∂λ _N

=

1 cosθ

∂u

∂λ

, (ii)

∂u

∂θ _N

=

∂u

∂θ ,

Lemma 6. Assume 0≤r≤β andβ >1. Then for all u∈H^β(S), ku−INukr≤cN^1+r+ε−βkukβ

whereε is an arbitrary small number.

Proof. First we consider the caser= 0. By the embedding theorem on spher- ical surface, we have

(12) kuk_∞≤ckuk_1+ε, ∀ε >0

ThusH^β(S)⊂C(S) andINuis well defined. It follows from Lemma 4 that ku−I_Nuk ≤ ku−P_Nuk_N ≤cku−P_Nuk_∞.

BecauseINu∈VN,PNu∈VN, we have form Lemma 3 that

ku−I_Nuk ≤ ku−P_Nuk+kINu−P_Nuk=ku−P_Nuk+kINu−P_NukN

≤ ku−PNuk+ku−INukN+ku−PNukN

≤3ku−P_Nuk_∞.

The combination of (12) and Lemma 2 leads to

ku−P_Nuk_∞≤cku−P_Nuk_1+ε≤cN^1+ε−βkuk_β,

where ε is an arbitrary small number. If r >0, then we have from Lemma 2 that

kINu−P_Nukr≤cN^rkINu−P_Nuk

≤cN^r(ku−PNuk+ku−INuk)

≤cN^1+r+ε−βkukβ,

which, together with Lemma 1, implies the conclusion of this lemma. 2

Lemma 7. [14] There exists a positive constantcsuch thatkuk²≤c|u|²₁,∀u∈ H¹(S) withµ(u) = 0

Lemma 8. [3] Suppose thatE(t)andρ(t)are two non-negative, non-decreasing functions defined on R_τ, b₁, b₂ and c are non-negative constants, t₁ ∈R_τ, and the following conditions are fulfilled:

(i)E(t)≤ρ(t) +cPt−τ

t⁰=τ(E(t⁰) +N^b1E^b2+1(t⁰)), ∀t∈R_τ (ii) ρ(t1)e^2ct≤N^−b1/b2

Then, for all t∈Rτ,t≤t1, we have

E(t)≤ρ(t)e^2ct.

4. The Stability

We now analyze the generalized stability of scheme (8). Suppose that η(0) has the error eη0, while the right-hand side of first and second equation of

(8) have errors fe1 and fe2 respectively. They induce the error of η(t) and φ(t) denoted byη(t) ande φ(t). Thene















ηe_bt(t) +INJ(η(t),φ(t)) +e INJ(eη(t), φ(t)) +INJ(eη(t),φ(t))e

−2Ω_∂λ^∂ (eφ(t)) =fe1(t),

−4φ(t) =e η(t) +e fe2(t), µ(eφ(t)) = 0,

η(0) =e ηe0, eη(τ) =eη1. (13)

By taking theL² inner product with 2bη(t) on both sides of the first equation ofe (13), we obtain

(14) kη(t)ke ²

b t+

3

X

i=1

Fi(t)−4Ω ∂

∂λφ(t),e bη(t)e

= 2

fe1(t),bη(t)e , where

F1(t) =

INJ(η(t),φ(t)),e 2bη(t)e , F2(t) =

INJ(eη(t), φ(t)),2bη(t)e , F₃(t) =

I_NJ(eη(t),φ(t)),e 2bη(t)e , ∂

∂λφ(t),e b η(t)e

= 1 2

∂

∂λφ(t),e η(te +τ)

+1 2

∂

∂λφ(t),e η(te −τ)

. By taking theL² inner product with φ(t) on both sides of the second equatione of (13), and obtain

|eφ(t)|²₁≤ 1

2ckφ(t)ke ²+c(keη(t)k²+kfe₂(t)k²).

Thus Lemma 7 leads to

(15) |φ(t)|e ²₁≤c(kη(t)ke ²+kfe2(t)k²).

Moreover, by Lemma 7 and (15)

(16) kφ(t)ke ²₂≤c(kφ(t)ke ²+k4φ(t)ke ²)≤c(keη(t)k²+kfe2(t)k²).

Now we are going to estimate |Fj(t)|, j= 1,2,3. Assume thatεis suitably small number, andkuk1,∞ =kukC¹(S) = sup_x∈S|u(x)|1, kuk∞ =kukC(S), we have from (16) that

|F1(t)| ≤2

J(η(t),φ(t)),e b eη(t)

≤2√

2kη(t)k_1,∞kbη(t)k|e φ(t)|e 1

≤ckη(t)k_1,∞kbη(t)ke

keη(t)k²+kfe2(t)k²1/2

|F1(t)| ≤ kbeη(t)k²+ckη(t)k_1,∞

keη(t)k²+kfe2(t)k² . (17)

Furthermore by (12) and Lemma 5, we get

|F2(t)| ≤2

J(η(t), φ(t)),e b η(t)e

≤ckφ(t)k1,∞kbeη(t)k

1 cosθ

∂η(t)e

∂λ _N

+

∂eη(t)

∂θ _N

≤2√

2kφ(t)k1,∞kbeη(t)k |eη(t)|₁

|F2(t)| ≤ckbeη(t)k²+kφ(t)k²_1,∞keη(t)k². (18)

On the other hand, it follows from (12) and Lemma 2 that kφ(t)ke 1,∞≤ckφ(t)ke 2+ε≤cN^εkφ(t)ke 2. Thus from the above estimate and (16) we get

|F3(t)| ≤2√

2kφ(t)ke _1,∞kbeη(t)k |eη(t)|₁

≤cN^1+εkbeη(t)kkeη(t)kkeφ(t)k₂

|F3(t)| ≤ kbη(t)ke ²+cN^2(1+ε)

kη(t)ke ⁴+kfe2(t)k⁴ . (19)

Finally, we have

4Ω ∂φe

∂λ(t),b eη(t)

!

≤4Ωkbη(t)ke φ(t)e

₁≤ckbη(t)ke ²+c|φ(t)|e ²₁

≤ckbη(t)ke ²+c

kη(t)ke ²+kfe2(t)k² . (20)

(21) 2

fe1(t),b η(t)e

≤2kfe1(t)kkbeη(t)k ≤ kbeη(t)k²+kfe1(t)k² By substituting the above estimates (17)−(21) into (14), we get (22) keη(t)k²

b t≤c

bη(t)e

2

+Mkeη(t)k²+cN^2(1+ε)kη(t)ke ⁴+G(t), where

M =ckηk_1,∞+ckφk²_1,∞+c, G(t) =c(1 +kηk1,∞)

fe₂(t)

2

+cN^2(1+ε) fe₂(t)

4

+c fe₁(t)

2

. In fact

bη(t)e

2

≤ 1 2

kη(te +τ)k²+kη(te −τ)k² . Summing up (22) for allt⁰∈Rτ,τ ≤t⁰≤t−τ, we have

kη(t)ke ²≤ kη(0)ke ²+keη(τ)k²+cτ

t−τ

X

t⁰=τ

kbeη(t⁰)k²+M τ

t−τ

X

t⁰=τ

keη(t⁰)k² (23)

+cN^2(1+ε)τ

t−τ

X

t⁰=τ

kη(te ⁰)k²+ +τ

t−τ

X

t⁰=τ

kG(t⁰)k².

Let

E(t) =keη(t)k²,

ρ(t) =keη(0)k²+keη(t)k²+ 2

t−τ

X

t⁰=τ

G(t⁰), then (23) implies

E(t)≤ρ(t) +cτ

t−τ

X

t⁰=τ

h

E(t⁰) +cN^2(1+ε)E²(t⁰)i .

Finally, the application of Lemma 8 (with b1 = 2(1 +ε), b2 = 1) leads to the following theorem

Theorem 1. Suppose that in scheme (8) the mesh size τ and N⁻¹ are suffi- ciently small and there existt1∈Rτ such that

ρ(t1)e^2ct≤N^−2(1+ε),

εbeing a suitably small number. Then we have for all t∈Rτ, t≤t1, we have E(t)≤ρ(t)e^2ct.

5. Convergence

In this section we discuss the convergence of the scheme (8). Letξ^N(t) = PNξ(t), ψ^N(t) =PNψ(t). It follows from (1) that











ξ_b^N_t (t) +INJ(ξ^N(t), ψ^N(t))−2Ω_∂λ^∂ ψ^N(t) =P2

j=1gej(t),

−4ψ^N(t) =ξ^N(t), µ(ψ^N(t)) = 0,

ξ^N(0) =P_Nξ₀, ξ^N(τ) =I_N

ξ₀+τ_∂t^∂ξ(0) , (24)

where

eg1(t) =∂ξ^N(t)

∂t −ξ_bt^N(t), eg₂(t) =I_NJ ξ^N(t), ψ^N(t)

−P_NJ(ξ(t), ψ(t)).

Furthermore, letξ(t) =e η(t)−ξ^N(t) andψ(t) =e φ(t)−ψ^N(t), then by (1) and (8), we get















ξe_bt(t) +INJ(ξ^N(t),ψ(t)) +e INJ(eξ(t), ψ^N(t)) +INJ(eξ(t),ψ(t))e

−2Ω_∂λ^∂ ψ(t) =e −P2

j=1egj(t),

−4ψ(t) =e ξ(t),e µ(ψ(t)) = 0,e

ξ(0) = (Ie _N −P_N)ξ₀, ξ(τe ) = (I_N−P_N)

ξ₀+τ_∂t^∂ξ(0) . (25)

Take the inner product with 2b

ξ(t) ande b

ψ(t) in the first and second equatione of (25) respectively. Then, by a procedure similar to the proof of Theorem 1 we get an estimate for the error ξ(t). Hence in order to get the convergence rate,e we need only to estimatekegj(t)k². Let

ρ1(t) = ξ(0)e

2

+ ξ(τ)e

2

+τ d

t−τ

X

t⁰=τ

keg1(t⁰)k²+keg2(t⁰)k² .

According to Taylor’s formula

∂ξ^N(t)

∂t −ξ_bt^N(t)

≤

∂ξ(t)

∂t −ξ_bt(t)

≤τ^3/2 Z t+τ

t

∂³ξ

∂t³(t⁰)

2

dt⁰

!^1/2

we get

τ

t−τ

X

t⁰=τ

kg1(t⁰)k²≤cτ⁴kξ(t)k²_H3(0,T;L²(S)). We separateeg₂(t) aseg₂(t) =A₁(t) +A₂(t) +A₃(t),

A1(t) = (IN −PN)J(ξ^N(t), ψ^N(t)), PNJ(ξ^N(t)−ξ(t), ψ^N(t)), A₂(t) =P_NJ(ξ(t), ψ^N(t)−ψ(t)).

By Lemma 6, Lemma 1 and embedding theorem on spherical surface, it is not difficult to see that

|A₁(t)| ≤cN^1+ε−r

J(ξ^N(t), ψ^N(t))

_r≤cN^1+ε−r ξ^N(t)

_1+r

ψ^N(t)) _1+r

≤cN^1+ε−rkξ(t)k_1+rkψ(t))k_1+r.

Also, it follows from Lemma 1 and inverse inequality that

|A₂(t)| ≤

J(ξ^N(t)−ξ^N(t), ψ^N(t)) ≤

ξ^N(t)−ξ(t) ₁

ψ^N(t)) 1,∞

≤cN^1−rkξ(t)k_rkψ(t))k_r,

|A3(t)| ≤cN^1−rkξ(t)k_rkψ(t)k_r.

By substituting the above estimate into the expressionρ1(t) we get kge2(t)k²≤ kA1(t)k²+kA2(t)k²+kA3(t)k²

≤cN^2(1+ε−r)kξ(t)k²_1+rkψ(t))k²_1+r+cN^2(1−r)kξ(t)k²_rkψ(t)k²_r,

τ

t−τ

X

t⁰=τ

keg₂(t⁰)k²≤cN^2(1+ε−r)

kξ(t)k⁴_C(0,T;Hr+1(S))+kψ(t))k⁴_C(0,T;Hr+1(S))

+cN^2(1−r)

kξ(t)k⁴_C(0,T;Hr+1(S))+kψ(t)k⁴_C(0,T;Hr+1(S))

,

ξ(0)e

2

≤cN^2(ε−r)kξ0k²_r+1, ξ(τ)e

2

≤cτ⁴kξ(t)k²_H2(0,T;H¹(S)). Consequently

ρ1(t)≤d

τ⁴+N^2(1+ε−r) ,

where d is a positive constant depending only on the norm of ξ and ψ in the spaces mentioned above. Finally, we reach the following theorem for convergence rate

Theorem 2. Assume that the exact solution of (1) satisfies the following smoothness, ξ ∈ H³(0, T;L²(S)) T C(0, T;H^r+1(S)) T C(0, T;H^r(S)) T H²(0, T;H¹(S)), ψ ∈ C(0, T;H^r+1(S))TC(0, T;H^r+1(S)) with r >2. Then there exist a positive constantd, such that for all t∈R_τ

kξ(t)−η(t)k²≤d(τ⁴+N^2(1+ε−r)) whereεis suitably small.

References

[1] Boyd, J. P., Chebyshev and Fourier Spectral Methods, 2nd edition. New York:

Dover Publications INCm Mineola 2001.

[2] Buffoni, G., Griffa, A., The finite difference barotropic vorticity equation in ocean circulation modeling: basic property of the solutions. Dynamics of Atmospheres and Oceans, 15 (1990), 1-33.

[3] Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., Spectral Methods in Fluid Dynamics. Berlin: Springer-Verlag 1988.

[4] Chjan, C. L., Exact solution of an energy-entrophy theory for the barotropic equa- tion on a rotating sphere. Physica A: Statistical Mechanics and its Application, 290 (2001), 131-158.

[5] Guo, B. Y., Spectral Methods and Their Applications. Singapore: World Scien- tific 1998.

[6] Guo, B. Y., Cao Weiming, A spectral method for fluid flow with low Mach number on spherical surface. SIAM J. Numer. Anal. 32 (1995), 1764-1777.

[7] Haltiner, G. J., Numerical weather prediction. New York: John Wiley and Sons 1971.

[8] McWilliam, J., The emergence of isolated vortices in turbulents flows. J. Fluid Mec. 146 (1984), 21.

[9] Pedlosky, J., Geophysical Fluid dynamics, 2nd edition. Berlin: Springer 1987.

[10] Peyret, R., Spectral Methods in Incompressible Viscous Flow. New York:

Springer-Verlag 2002.

[11] Rashid, A., Ghulam Mustafa, The Fourier Pseudospectral Method with Restraint Operator for Burgers Equations. Journal of University of Science and Technology of China 33 (2003), 547-554.

[12] Rashid, A., Yuan, Li, Convergence Analysis of Pseudospectral Method with Re- straint operator for Fluid Flow with Low Mach Number. Appl. Comput. Math.

2 (2003), 135-142.

[13] William, L. B., A new class of steady solution of the barotropic vorticity equation.

Dynamics of Atmospheres and Oceans 4 (1980), 67-99.

[14] Zen Qing-cun, Physical Mathematical basic of Numerical prediction, Vol. 1. Bei- jing: Science press 1979.

[15] Lion, J. L., Magen E., Nonhomogenous Boundary value problems and applica- tions, Vol.1. Berlin: Springer-Verlag 1972.

Received by the editors February 25, 2005.