303

Journal of Hydrodynamics Ser.B, 2006,18(3): 303-309 sdlj.chinajournal.net.cn

GENERAL CAUCHY PROBLEM FOR THE LINEAR SHALLOW -WATER EQUATIONS ON AN EQUATORIAL BETA-PLANE*

SHEN Chun School of Mathematics and Information, Yantai Normal University, Yantai 264025, China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China, E-mail:shenchun3641@sina.com SHI Wei-hui Department of Mathematics, Shanghai University, Shanghai 200444, China (Received Dec. 29, 2004) ABSTRACT: Based on the theory of stratification, the well-posedness of the initial value problem for the linear shallow-water equations on an equatorial beta-plane was discussed. The sufficient and necessary conditions of the existence and uniqueness for the local solution of the equations were presented and the existence conditions for formal solutions of the equations were also ven. For the Cauchy problem on the hyper-plane{ , the local analytic solution were worked out and a special case was discussed. Finally, an example was used to explain the variety of formal solutions for the ill-posed problem.

gi}0=t

KEY WORDS: stratification, initial value problem, well-posedness, shallow-water equations 1. INTRODUCTION Over the last two decades, studies in the tropical meteorology and ocean have dramatically increased, such as the EI Nino/Southern Oscillation (ENSO), the large-scale heat-induced circulation of the tropical atmosphere and nonlinear effects of the equatorial waves . The linear shallow-water model on an equatorial beta-plane was widely used in these studies. Matsuno first applied this model to study quasi-geostrophic motions in the equatorial area. Gill et al. employing this model with highly idealized forcing and friction, investigated the response of the

tropical atmosphere to a given distribution of heating. Chao pointed out the semi-geostrophic adaptation and evolution motions in the tropical atmosphere and ocean by using the model.

]41[ −

]5[

]7,6[

]9,8[

Studies on the equations, at present, are mainly in the form of the Weber function, i.e., the parabolic cylinder function. This paper addresses itself to the general Cauchy problem for the equations by using stratification theory , which is a new method to transfer the problem for solving the equations into the related topological problem. And the complete and specific solution could be offered and a series of conclusions about the Boussinesq-type equations could be reached .

]11,10[

]1512[ −

Given a hyper-surface and the initial conditions on it, does the problem have a unique solution, infinite solutions or no solution at all? This question has considerable relevance to the problem of numerical weather prediction and 4D data assimilations. The objective of the present paper is to discuss the well-posedness of the initial value problem on the hyper-surface ( ){ }yxht ,= and the corresponding discriminating ways. And the construction of the local solution space for the equations is also investigated.

* Project supported by the National Natural Science Foundation of China (Grant No: 90411006, 40175014) and the Key

Foundation of Shanghai Municipal Commission of Science and Technology (Grant No: 02DJ14032). Biography: SHEN Chun (1975-), Male, Ph.D.

304

This paper is organized as follows. In Section 2, the non-dimensional linear shallow-water equations on an equatorial beta-plane are presented. The analysis of the well-posed property is preformed and the related theorem is proved in Section 3. In Section 4, the expressions of the local unique analytic solution are presented for the Cauchy problem on the hyper-plane . An example for the ill-posed problem is given in Section 5 and conclusions are drawn in Section 6.

{ 0=t }

2. MODEL AND BASIC EQUATIONS

The simplest model suitable for our purposes is the linear shallow-water model, i.e., a layer of incompressible fluid of homogeneous density with a free surface under hydrostatic balance. We can get various characteristics of large scale motions of the tropical atmosphere and ocean by applying the equatorial beta-plane approximation to this model.

The equations of momentum and mass conservation are given by

0

0

0

u hyv gt xv hyu gt y

h u vHt x y

β

β

⎧∂ ∂⎪ − + =∂ ∂⎪

⎪∂ ∂⎪ + + =⎨ ∂ ∂⎪⎪ ⎛ ⎞∂ ∂ ∂⎪ + +⎜ ⎟∂ ∂ ∂⎪ ⎝ ⎠⎩

=

)

(1)

where is the horizontal velocity, h is the small deviation from the top surface, the mean value of which is denoted by

( vu,

H , β is the Rossby parameter and taken as a constant. Take the time and length scales as follows:

[ ] [ ]1 12 21 , cT L

cβ β⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(2)

where is the square of the velocity of pure gravity waves.

( gHc =2 )

The Eqs. (1) can be transformed into a non-dimensional form:

0u yvt x

φ∂ ∂− + =

∂ ∂,

0v yut y

φ∂ ∂+ + =

∂ ∂,

0u vt x yφ∂ ∂ ∂

+ + =∂ ∂ ∂

(3)

where ( )yx, is the non-dimensional distance with x eastwards and measured northwards from the

equator,

y( )vu, is proportional to the horizontal

velocity and φ is proportional to the geopotential height. 3. METHOD OF SOLUTIONS

In order to solve the general Cauchy problem for the Eqs. (3), we will apply the theory of stratification as follows. Some signs and definitions of the notations used in this paper can be found in Refs.[10,11]. 3.1 Rewriting the equations

For convenience and conciseness, we denote ,2

+×= RRV 3RZ = and rewrite as ( )tyx ,,( )321 ,, xxx , the unknown functions ( )φ,,vu as

( )321 ,, uuu . Thus, Eqs.(3) can be regarded as a

subset of the Ehresmann space . Using the local coordinates of the Ehresmann space, Eqs. (3) are rewritten as

( ZVJ ,1 )

0

1 3

1 3 1 2 22 3

2 3 2 2 1

1 2 31 1 2 3

: 0

:

: 0

f p p x u

f p p x u

f p p p

⎧ + − =⎪

+ + =⎨⎪ + + =⎩

(4)

and Eqs.(4) can be denoted with

( ) ( ) ( )11 2 3, , ,D V f f f J V Z= ⊆

3.2 Calculating the gradual association

According to the definition of the gradual association, the gradual association of the equations ( )D can be given as

* ,lD D= ∪ ),,( ZVJD ll ⊆ ),,2,1,0,1( −=l

VD =−1 , , ),(0

0 ZVJD =

305

)(1 jfVD = , ),),((12 jji ffeVD =

),),(,),((

1121 jjijiiik ffefeVDk −

=

(5) 1 2( 1,2,3; , , 1,2,3; 1)kj i i i k= = ≥

We can prove that is true for any , namely, the pre-gradual association is identical with the gradual association for the equations

( ) 11 −− =∂ kkkk DD

k( )D .

3.3 The initial conditions We consider the initial value problem on the

hyper-surface . The initial conditions

( ){ } 3,: Ryxht ⊆=Σ( ) ( DIhh ,, 2 )Δ∈ ωγσ are defined as

follows:

,: 32 Rh →Δσ

( ) ( ),),(,,)()( 2121 ξξξξξξσ hx jh ==

),(: 0

2 ZVJh →Δγ ,

( ),)(),()( ξξξγ ijh ux= (6) ( , 1,2,3)i j = where with ,: 2 RRh → ωCh ∈ ( ) 00,0 =h ,

and ( ) 221, Δ∈= ξξξ are the coordinates of the gravity center. 3.4 Fundamental theorem and its proof

Fundamental Theorem: If the initial conditions ( hh )γσ , are given, the initial value problem for the

equations on the hyper-surface ( )D

( ){ } 3: ,t h x y RΣ = ⊆

is

( )( ) ( )0 , , 1,2,3i i

D

u u x y iΣ

⎧⎪⎨

= =⎪⎩ (7)

(1) The equations have a unique stable solution if and only if

( )D

2 2

1 21 δ δ− − ≠ 0 (8)

(2) The equations ( )D have an infinite

number of solutions if and only if

( ) ( ) ( )

2 21 2

3 3 31 1, 1 2 2, 1 3, 1

1 0

0k k k

δ δ

δ ϕ δ ϕ ϕ− − −

⎧ − − =⎪⎨

+ + =⎪⎩ (9)

(3) The equations ( )D have no solution if and

only if:

( ) ( ) ( )

2 21 2

3 3 31 1, 1 2 2, 1 3, 1

1 0

0k k k

δ δ

δ ϕ δ ϕ ϕ− − −

⎧ − − =⎪⎨

+ + ≠⎪⎩ (10)

where ( ,2,1=∂

)∂= lh

ll ξ

δ and the formulae to

calculate ( )3,kiϕ are as follows:

when ,1=k

( ) ( )31,0 2 2 3 1x u uϕ = − (11a)

( ) ( )32,0 2 1 3 2x u uϕ = − − (11b)

( ) ( ) ( )33,0 1 21u uϕ = − − 2 (11c)

when ,2≥k

( ) ( )1 13 2 3

1, 1 2 3 31k kk x p pϕ − −− = − (12a)

( ) ( )1 13 1 3

2, 1 2 3 32k kk x p pϕ − −− = − − (12b)

( ) ( ) ( )1 13 1 2

3, 1 3 31k kk p pϕ − −− = − − 2 (12c)

where

( ) ,l

ii

ulu

ξ∂∂

= ( )iji

jl

pp l

λ

λ ξ

∂=

∂,

( ), 1,2,3; 1,2; 1i j l kλ= = ≤ −

Proof: For the initial value problem of the

equations ( )D above-mentioned, all the calculations are preformed in the third open set of the opening coverings { }( )3,2,1=iU i of

306

( ) ( )( )* 12, 1 2, ,k

kW V Z G TJ V Z−− ⊆

Let 3U∈τ , τ being generated by the

following two tangent vectors of ( )ZVJ k ,1−

( ) ( )( )1 11,0, , 1 , 1ii j

u p λδ=η (13a)

( ) ( )( )2 20,1, , 2 , 2ii j

u p λδ=η (13b)

at the point

( ) ( ) ( ) 1,

: , (i kj j

, )p p x p J Vλ λτ τ −= ∈ Z ,

( ), 1,2,3; 1; 1i j k kλ= ≤ − ≥

)

Once the initial conditions ( hh γσ , are given,

the sufficient and necessary conditions for determining the solutions corresponding to ( )hh γσ ,

are the embedding series ∞C { }kγ ,

( ),,: 2 ZVJ kk →Δγ

( ) 1 2 3( ( ), ( ), ( ), ( ),k ix x x uγ ξ ξ ξ ξ= ξ

,( ))i

jp λ λ

ξ ( ), ,kJ V Z∈

( ), 1,2,3; ; 1i j k kλ= ≤ ≥

1

(14)

such that

( ) ( )( ),,0* ZVI kk ∈∀= ωωγ

( ) ( ),,2 ZVJD kkk ⊆⊆Δγ

,1 hk

k σγα =− ,0 hkk γγα =

1kk k kα γ γ− = − (15)

With the series { }kγ , we can write each coefficient

of the solutions of the equations , which is in the

form of power series.

( )D

Now we consider the case of , and the first-order gradual association of the equations

1=k( )D

is DD =1 . According to the above depiction, 1γ is determined by hγ and satisfies

( ) ( )( ),,0 1*1 ZVI∈∀= ωωγ ( )1 1Dγ ξ ∈ (16)

where

( )1 1 2 2 3 3d d d di i ii iu p x p x p xω = − + + ,

( )1,2,3i = Then we can get the following equations for ( ) :3,2,13 =ip i

( )

( )

( )

31 33 1 3 1,0

32 33 2 3 2,0

31 2 31 3 2 3 3 3,0

p p

p p

p p p

δ ϕ

δ ϕ

δ δ ϕ

⎧ − =⎪⎪ − =⎨⎪

− − + =⎪⎩

(17)

where ( ) ( )3,2,13

0, =iiϕ can be calculated by Eq.(11).

If the Eqs.(17) have solutions, then we can get 1γ as

( ) ( ) ( )( )ξξγξγ ijh p,1 = ( )3,2,1, =ji ,

in which ( )ξip3 ( )3,2,1=i are determined by Eqs.(17) and the others can be calculated by

( ) ( ) li

iil plup δξ ⋅−= 3 ( )1,2,3; 1,2i l= =

In general cases, suppose we have got 1−kγ ,

which satisfies the following conditions: ,1

11 hk

k σγα =−−

−1

0 1 ,kk hα γ γ−

− =

( ) ( )( ),,,0 1*

1 ZVIkk −− ∈∀= ωωγ

( ) ( )11 2 1 ,k

k kD J V Zγ −− −Δ ⊆ ⊆

(18) Based on ( )1, −kh γσ , repeating the above

procedure, we can get the equations for

307

ikp3

( 3,2,1=i )

1

)

as follows:

( )

( )

( )

31 31 1, 13 3

32 32 2, 13 3

31 2 31 2 3,3 3 3

k k

k k

k k k

k

k

k

p p

p p

p p p

δ ϕ

δ ϕ

δ δ ϕ

−

−

−

⎧ − =⎪⎪ − =⎨⎪

− − + =⎪⎩

(19)

where can be calculated by Eqs.(12).

( ) ( 3,2,131, =− ikiϕ

If the Eqs. (19) have solutions, we can get kγ as

( ) ( ) ( )( )ξξγξγ λi

kjkk p,1 ,−= ( ), 1,2,3i j =

where are determined by Eqs.(19) and the others can be calculated by

( )ξikp3

( 3,2,1=i )

( ) ( ) l

ij

ij

ilj plpp δξ λλλ ⋅−= 3

( ), 1,2,3; 1,2; 1i j l kλ= = = −

So we transfer the discussion on the solution

space of the equations to the discussion on the solution space of the Eqs. (19). Thus we can draw the conclusion described in Fundamental Theorem.

( )D

4. CAUCHY PROBLEM ON THE HYPER-PLANE { }0=tIf the initial conditions are given on the

hyper-plane , it is easy to conclude that

from

{ 0=t }01 2

22

1 ≠−− δδ 01 =δ and 02 =δ . Thus the following corollary under consideration can be obtained from Fundamental Theorem.

Corollary: If the initial conditions are given on the hyper-plane { } , then the Cauchy problem is well-posed.

30 Rt ⊆=

If the Cauchy problem for the equations ( )D is

well-posed, the series can be

got by elevating from the

initial conditions

( ZVJ kk ,: →Σγ )

))

( ZVJh ,: 0→Σγ( hh γσ , and it satisfies

,1 hk

k σγα =− ,0 hkk γγα =

,11 −− = kkkk γγα ( )k kDγ Σ ⊆ (20)

and the unique analytic solution of the well-posed Cauchy problem can be obtained from the series { }kγ .

Especially, we know that 01 =δ and 02 =δ for the Cauchy problem

( )( ) ( )0 , , 1,2,3i t i

D

u u x y i=

⎧⎪⎨

= =⎪⎩ (21)

Thus the initial conditions ( )hh γσ , are

( )1 2, ,0 ,hσ ξ ξ=

1 2 1 1 2 2 1 2( , ,0, ( , ), ( , )h u u ,γ ξ ξ ξ ξ ξ ξ=

3 1 2( , ))u ξ ξ (22) Substituting 01 =δ and 02 =δ into the equations (17) yields

(23)

( ) ( )( ) ( )( ) ( ) ( )

313 1,0 2 2 3

323 2,0 2 1 3

333 3,0 1 2

1

2

1 2

p u u

p u u

p u u

ϕ ξ

ϕ ξ

ϕ

⎧ = = −⎪⎪ = = − −⎨⎪

= = − −⎪⎩ So ( )3,2,13 =ip i can be determined by Eqs.(23)

and ( )2,1;3,2,1 == lip il can be calculated by

( ) i

l

ii

il p

ulup 3⋅

∂∂

==ξ

( )3,2,1=i

and

( )lpp ij

ilj λλ = ( )1;2,1;3,2,1, −=== klji λ

can also be got by continuing the process.

( )0,0ij

ij pp λλ = can be obtained by letting

( ) ( )0,0, 21 =ξξ . Thus the unique analytic solution can be calculated by

( ) 31 2

0, ,

!

ij

ik k

pu x y t x y t

λ λλ λ

λ λ

∞

= =

= ∑ ∑

308

321 λλλλ ++= !!!! 321 λλλλ = (24)

Particularly, we consider the Cauchy problem

( )0 0 0 0 0, ,t t t

D

u u v v 0φ φ= = =

⎧⎪⎨

= = =⎪⎩ (25)

where 000 ,, φvu are real constants. The unique analytic solution can be obtained by

( ) ( )2

0 2 0 2 10 02 ! 2 1 !

n

n nn n

tu u A v Bn

+∞ ∞

+= =

= ++∑ ∑

2 1ntn

(26a)

( ) ( )2 1 2

0 2 1 0 20 02 1 ! 2

n n

n nn n

t tv u A v Bn n

+∞ ∞

+= =

= ++∑ ∑ !

(26b)

( )2

2 10 0

1

dd 2 !

nn

n

A tuy n

φ φ∞

−

=

= − −∑

( )2 1

20

0

dd 2 1

nn

n

B tvy n

+∞

= +∑ ! (26c)

where and , the functions of only, can be determined respectively by

nA nB y

2

2 12 1 2 2

d ,d

nn n

AA yAy

−+ = − +

,122 −= nn yAA ,10 =A 1A y= − (27a)

22

2 2 2 1 2

d ,d

nn n

BB yBy+ += − +

,212 nn yBB =+ ,10 =B 1B y= (27b) 5. EXAMPLE FOR THE ILL-POSED

PROBLEM The existence conditions of formal solutions and

the methods of constructing formal solutions can be given from Fundamental Theorem for the ill-posed problem.

For example, if the initial values are given on the

hyper-plane { } 3Rxt ⊆= , then it can be concluded from Fundamental Theorem that there are infinitely many solutions to the problem with

( ) ( ) 031,3

31,1 =+ −− kk ϕϕ for all . Otherwise, there

is no solution at all. Consider the following problem: Nk ∈

( )2

0 01, 0,2t x t x t x

D

u u v uφ φ= = =

⎧⎪⎨

= = = −⎪⎩

0 y (28)

There are an infinite number of solutions to the problem. In fact, we have

( )

( )

20

2 20 0

1exp2

01 1exp2 2

n

n

u u c t x y

v

u y c t x yφ φ

⎧ ⎛ ⎞= + − −⎜ ⎟⎪ ⎝ ⎠⎪⎪ =⎨⎪ ⎛ ⎞⎪ = − + − −⎜ ⎟⎪ ⎝ ⎠⎩

(29)

where and can be any real number and any natural number respectively.

c n

6.CONCLUSIONS

It is easily seen that the linear shallow-water equations on an equatorial beta-plane belong to the Cauchy-Kovalevskaya-type equations. But with stratification theory, we can get more information about it, such as the construction of the solution space and the calculation method for the well-posed problem. From the above results and discussions, the following conclusions can be drawn:

(1) Once the hyper-surface

is given and the condition holds,

then the Cauchy problem for the equations

( ){ } 3,: Ryxht ⊆=Σ01 2

22

1 ≠−− δδ( )D is

well-posed and the unique analytic solution can be constructed.

(2) Once the hyper-surface

is given and the condition holds,

then the Cauchy problem for the equations

( ){ } 3,: Ryxht ⊆=Σ01 2

22

1 =−− δδ( )D is

ill-posed and infinitely many formal solutions can be got if the initial values satisfy

( ) ( ) ( ) 031,3

31,22

31,11 =++ −−− kkk ϕϕδϕδ

for all Nk ∈ .

309

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