waves in a rotating stratified shallow water

IL NUOVO CIMENTO VOL. 104 B, N. 3 Settembre 1989

Waves in a Rotating Stratified Shallow Water.

MOHAMED ATEF A. E. HELAL(*) and SOAD E. BADAWI(*)

Department of Mathematics, Faculty of Science, Cairo University - Giza, Egypt

(ricevuto il 9 Marzo 1988; manoscritto revisionato ricevuto il 6 Aprile 1989)

Summary. - - In this paper, a study of nonlinear stratified rotating fluids is introduced with the shallow-water theory. The mathematical model for two layers of perfect fluids in a rotating circular cylinder is given as well. A distortion for shallow water and slowly rotating fluids is established. A detailed study of the particular case for progressive waves on both layers is presented and the analytical solution has been obtained; this solution is believed to be new.

PACS 03.40.Gc - Fluid dynamics: general mathematical aspects.

1. - I n t r o d u c t i o n .

Since Lord Kelvin(i), a considerable amount of work has been done on periodic oscillations, in a shallow rotat ing fluid for a circular cylinder. Brillouin and Coulomb (2) gave an original s tudy on the linear problem. The physical phenomena are presented in the ouvrage of Bjerkens et al. (3). Miles (4) has considered the centrifugal forces for the same problem. Also Saint-Guily in a series of papers has studied the same problem (~8). Greenspan's famous book (9) is

(*) Present address: Department of Mathematics, College of education for girls, Scientific sections, A1 Sitteen st., A1 Malaz, Riyadh 11113, Saudi Arabia. (1) Lord M. KELVIN: Philos. Mag., 10, 109 (1880). (2) M. BRILLOUIN and J. COULOMB: Oscillation d'un liquide p~sant dans un bassin cylinderique en rotation (Gautier-Villars, Paris, 1933). (~) V. BJERKNES, J. BJERKNES, H. SOLBERG and T. BERGERON: Hydrodynamique Physique, avec l'application ~ la m~t~orologie dynamique (Presses Univ. France, 1934). (') J. W. MILES: Phys. Fluids, 2, 279 (1959). (~) B. SAINT-GUILY: La Houille Blanche, 5, 629 (1962). (6) B. SAINT-GUILY: J. Mec., 2, 425 (1963). (7) B. SAINT-GUILY: J. Mec., 4, 295 (1965). (8) B. SAINT-GUILY: Detsch. Hydrog. z., 23, 16 (1970).

17 - I1 Nuovo Cimento B. 245

246 MOHAMED ATEF A.E. HELAL and SOAD E. BADAWI

one reference where a wide treatment of linear and nonlinear problems is given. Suberville (10) in his thesis has presented a complete study of linear problems for surface and internal waves in rotating circular and rectangular cylinders. LeBlond and Mysak (11) in their book cover considerable material related to this subject. Helal (~) has studied the nonlinear surface waves for a rotating circular cylinder. In addition, a further study for rotating fluids around an island has been given by Helal (ls). Mei et al. (14) have paid a great attention to the dynamics of Oceans and their results are presented in their book.

Nonlinear internal waves for rotating stratified fluids(1~17) are of great interest as they clarify geophysical and oceanographical phenomena, especially the giant waves (1~).

Mathematicians have shown a great interest in the study of nonlinear problems(lS.~9), with particular emphasis on solitary waves(2~ and KdV equations (29.

In this work we study the nonlinear stratified fluids in a rotating circular cylinder, in which the shallow-water perturbation technique is used. This paper contains an analytical and numerical study of the problem and it can be considered as a first part of the study for such problems. Fur ther work including numerical results and applications for the internal waves will be reported in a forthcoming paper.

2. - T h e p r o b l e m .

Consider a circular cylinder with radius R at which we have two immiscible, incompressible perfect fluids. The cylinder rotates around its axis of symmetry with angular velocity De.

(9) H. P. GREESPAN: The Theory of Rotating Fluids (Cambridge University Press, Cambridge, 1968). (lo) j. L. SUBERVmLE: Thesis (USMG, Grenoble, France, 1974). (11) p. H. LEBLOND and L. A. MYSAK: Waves in The Oceans (Elsevier, Amsterdam, 1978). (lz) M. A. HELAL: Ph. D. Thesis (Cairo University, Cairo, 1981). (13) M. A. HELAL: Miami 6th International Conference on Alternative Energy Sources (1983). (14) C. C. MEI: The Applied Dynamics of Ocean Surface Waves (John Wiley, New York, N. Y., 1983). (15) L. A. MYSAK: CISM Lectures, Springer Verlag, 286, 81 (1984). (16) L. A. MYSAK: CISM Lectures, Springer Verlag, 286, 129 (1984). (17) j. PEDLOSKY: Geophysical Fluid Dynamics (Springer Verlag, New York, N.Y., 1982). (18) G. B. WHITHAM: Linear and Nonlinear Waves (John Wiley, New York, N.Y., 1974). (19) M. A. HELAL and J. M. MOULINES: Tellus, 33, 488 (1981). (20) j. W. MILES: J. Fluid Mech., 106, 131 (1981). (z~) L. A. OSTROVSKIY: Oceanology, 18, 109 1978).

WAVES IN A ROTATING STRATIFIED SHALLOW WATER 247

In this work we aim at finding the shape of the free surface and the interface. Throughout this paper the following notations represent the given definitions (fig. 1) M(r, O, z); the cylindrical coordinates of the point M; R the radius of the

Fig. 1.

Z 9r

- M

H2

' I I t { l

2R (" "1

~ z : f ( r , O ~t)

j z = h ( r , O ~ t )

basin; H1, H2 thickness at rest of the upper and lower layers; ~1, p2 constant densities of the upper and lower layers; p(1), p(2) pressures of the upper and lower layers; V(~)(~ri), ~ ) , V~ ~)) the three components of the velocity vector (the index on the component defines the layer: 1 for the upper layer, 2 for the lower layer); z =f i r , 0, t) free surface equation, f being one of the unknown functions of our problem; z = h(r, O, t) the interface equation, h is unknown and should be determined as a part of the solution; t~r the angular velocity of the system; g the acceleration of gravity.

3. - B a s i c e q u a t i o n s .

3"1. Hydrodynamic equations. - As we are dealing with a circular cylinder basin, the equations can be expressed in vector form(TM) as follows (using cylindrical coordinates):

The dynamic equations

(3.1) ~V(i---~)+(V(i).V)V(i)+2~IcAV(i)+~r 3t

+ grad (gz) = 1 gradp(i) (i = 1, 2).

The continuity equation

(3.2) div V (~) = 0 (i = 1, 2),


where

V (i) = V(r {) < + V [ ) ~o + V i {) ~ (i = 1, 2),

O M = r~r + zOz ,

and {~r, ~0, ~z} are the unit vectors in a cylindrical coordinate system. Equations (3.1) and (3.2) can be rewri t ten in their scaler form as

(3.3) aVe) V~)2 V(~ ~) DV(~) V(o ~) aVe) av(~) 4 - V ( i ) - - r _ _

at r + Dr + r ~ - - ~ az

1 3p (~ - 2~9c V(o ~) - ~ r = (i = 1, 2),

~ Dr

(3.4) . . . . DV(o i) + --r --o F V (~ '~" o + + V ~ i ) ~ + Dt r r Dr r DO

+2~cV7)= 1 DP (~) ( i = 1,2), p~r D0

_ _ _ V(o ~) aV~ i) aV~ i) 1 Dp (i) (i 1, 2), (3.5) DV~~ F V(~ i) DV~) ~ e V~ ~) - - - g = at ar r D0 az p~ Dz

av(~ ~) __(~. ov(~ ~) a v i i) (3.6) r - - ~ - + v r ' + - ~ - + r - ~ - = O ( i= 1,2).

3"2. Boundary conditions. - We assume that all the boundaries of the domain are impermeable, including the free surface and the interface which are unknown.

i) At the bottom (3.7) V~ 2) = 0 for z = O.

ii) On the vertical surface of the cylinder

(3.8) V~ ) = 0 ( i = 1 , 2 ) for r = R .

iii) On the free surface z =f(r, O, t) we have two conditions, the first describes the impermeability, i.e.

(3.9) {V~' a f (~) a f v~i) _--~'~ = o l

a-t - v~ Tr - r a o j ~ , o , o


and the second condition describes its isobarity, i .e .

(3.10) p(1)(r, O, Z = f ( r , O, t), t) -= Pat = const,

where P~t represents the atmospheric pressure which can be taken as zero without any loss of generality.

iv) On the interface z = h(r, O, t), we have also two equivalent conditions, the first of which describes its impermeability:

{ (3.11) V~)_ Oh _ v( i ) ah _ V~ ~) 3t - ~ 3 r r ~=h(r,O,t) -- 0 (i = 1, 2)

and the second condition of isobarity has the form

(3.12) {p(1)(r, O, z, t) = p(2)(r, O, z, t)}~=h(~,0,t) �9

Equations (3.3)-(3.12) describe completely the governing equations for the nonlinear problem.

4 . - T h e s h a l l o w - w a t e r t h e o r y .

From now on, we consider the case of basin with very large horizontal length compared with its vertical height, applying the shallow-water approximation theory (12,19), which designates the following new distorted parameters:

(4.1) ~ = ~r, 0 = 0 , ~ = z , ~= ~t and R = ~ R ,

where ~ is a small parameter which can depend on the ratio between the vertical height and the radius of the basin. Moreover, we are interested in the case of slow rotation, i .e .

(4.2) t ) c = ~ .

System (3.3)-(3.12) (after using the distorted parameters) can be wri t ten as

(4.3) 3u(i) v(i)2 (i) 3u(i) v(i) 3u(i) w(i) 3 u(i)

- 2et~v(~) _ d ) 2 ~ - p~ aT

(i = 1, 2),

(4.4) 3v(i ) u(i ) ?~(i ) (i)3v(i)+ v (~) 3v(~) ~ av (~)

+~u ~ _- = +w()_--:-~ + r 30 5 2

�9 $

+ 2~t~u (~) - ap(i)

(i = 1, 2),


3w (~ (~) 3w (i) v (~) w(i) ~w (~) 1 3P (~) (4.5) ~ ~ + ~u ~ + ~ ~w(i) ~ ~ ~ ~i ~ g (i = 1, 2),

[_ 3U (i) 8V(~)~ + ~ 3W (i) (4.6) ~ t r - - ~ r + u(i) + - ~ - ] 3~ = 0 (i = 1, 2) ,

(4.7) {W(2)}~=O = O,

(4.8) {U(~)}~.~R=O

(4.9) p(~) (LO,~=f ,v )=O,

(i = 1, 2),

(4.10)

(4.11)

ao / J~.i

P(~)(L -0, -z = h, ~) = p(2)ff., -0, -~ = h, z) ,

�9 [ 3 h + (oah . v ( i )~h

(4.12a) {V(r i), V(o i), V~ i) } --- {u (~ v (i), w (i)) (i = 1, 2),

Expanding the dependent variables in the neighbourhood of the undis turbed uniform s ta te in general asymptotic series will lead to two sets of identical equations and, due to the pari ty of the sys tem of equations, we take the more suitable expansions for the dependent variables in the form

(4.13)

V(r i) = U(i) = Z ~ U~in)(e, ~, ~, ~), ~=1

vii) = v(i) = y. ~vi i i (e , ~, ~, ~), n=l

v i i ) = w ( i ) ~ o2n+1,1,,(/) [,~ O, ~, T) ~ ~V2n+lk~ , n=O

p(O = ~ ~p~in)(O ' ~, ~, v) (i = 1, 2) , n~O

where V (i) = u (i) ~r + v (i) g0 + w (i) ~z,

h = E ~'~ h~(L -o, ~), n=O

f = ~ ~2"kn(~" , -0, ~), n=o


where the value of ho, fo, P~> and p~2) have the forms (from studying the stat ionary s tate (ID)

(4.14)

~2 2 (2~ _ ~ 2 ) , fo = (HI + H2) + - ~

D e h0 = H~ + ~gg (2~ 2 - ~ 2 ) ,

p~l) _-- ? lg(H1 + H2 - ~) + ~1--!----(2~ 2 - ~2 ) ,

p~2) = g(~l H~ + ~2H2) - ~2gz + ~ - ~ (2r 2 - ~2) .

For simplification of writing, we omit the bar sign ( - ) throughout the res t of the paper.

The boundary conditions on the free surface and on the interface can be expanded in the form

(4.15) {F}~=zo+~={F}~o+~A~3FI . (r [ 32F) , +-w., i i,= + . . . .

Expanding all the boundary conditions in a similar form like that of (4.15) and making use of eqs. (4.13), (4.14) in the sys tem of equations (4.3)-(4.12), this will lead to an ordered set of equations to be solved in the sequel.

5. - Ord ers o f a p p r o x i m a t i o n s .

5"1. First-order approximation

r ~wti> - - ~ - z - O

{wt2>}z:o = 0 ,

this leads to

(5.1) wt ~) = 0

{wtl)}z=fo ~-- 0 ;

(i -- 1, 2),

(i = 1, 2)

5"2. Second-order approximation. - For this approximation we have the following equations:

1 ~P~) = 0 (i = 1, 2) pi ~z


and from the conditions of isobarity on both free surface and interface we get

and

{P~) + f~ ~ } z = f o =Pat=O

from which it can be shown that

(5.2) P~) = P~ gf2,

+ h2 ap~2) ~1) az j.=~

p~2) __ g { ~ l f 2 + h2(32 - ~Zl)) �9

5"3. The third-order approximation. - This approximation gives

(5.3) au~ i) 2tgv~i)_ 1 aP~ i) az ~i ar '

(5.4) av~ ~) + ZOu~i) = 1 aP~ ~) (i = 1, 2) az pir a0

(5.5) au~ i) av~ ~) aw~) ~

r ar + uii) + - ~ - + '--~-z = "

with boundary conditions

(5.6)

{w~2)}z=o = O,

[ (1) [af2 ,1) afoul .

[ \ av Or ]J~=ho (i = 1, 2).

At this stage of approximation we can deduce that u~ i), v~ i) are functions of r, 0 and z only (12).

Integrating eq. (5.5) with respect to z and making use of the boundary conditions (5.6), we get

(5.7) w~ 2)= [au~2) 2) 1 av~ 2) - z l--~-r + u~2) + r -~ -} ' r

WAVES IN A ROTATING STRATIFIED SHALLOW WATER

(5.8) W~ I) = (U~ 1) -- U~ 2)) -- ho \--~--r + r + -r --~-] -

253

- (z - ho) \ 3 r r r - ~ - ) "

From eqs. (5.2), (5.6)-(5.8) we obtain the following system of equations:

S

I 3p~') 3p~ 1) g(~.-~l) V ~.ru~) .3u~) 3v?]] - ho(-~-r + u~) + 1 -~- ) j 3r 3z = L g r r '

" 3 (1) {P2- PI~ P~ {3P~ 2) 3p~I)]__gHI(P2__R1)(3U~I)U~ 1) 1 3V~1)~

1 3~P~ i) , 2g) 3P~ i)]

1~1 32P~ i) 2~ 3P~i) l ( i= 1,2),

where the differential operator [] is defined by

3 2 [ ] - 3 ~ + 4 ~ 2 .

Elimination of u~ i), v~ ~ and p~2) from system S gives (cf. appendix A)

3 2 (5.9) P2-ff~ [] ([] - H l g V e ) p ~ D =

= 2~9 3 3 2 - - m) V2]~ p~l), ]

where V 2 is the Laplacian operator in polar coordinates. The PDE for p~2) can be obtained in a similar way like that followed above:

~ 2 (5.10) ~1 ~ [] [] p~2)

Solving eqs. (5.9) and (5.10) we get p~l) and p~2) and consequently all other unknowns can be obtained.


6. - P r o g r e s s i v e w a v e s .

In this section we pay attention to the s tudy of progressive waves (~). I t has been shown that all unknowns p~), u~ ~) and v~ ~) are functions of r, 0 and z only.

In the present case the parameters 0 a n d , are combined as 0 = 0 - o~ where oJ is considered as the pulses.

The unknowns p~) can be expanded as a Four ier series in 0 as follows:

(6.1) p~O _- ~ {a~)(r) cos nO + b~)(r) sin nO} (i = 1, 2 ) . nffiO

Substi tuting p~) into (5.3) and (5.4) af ter some manipulation, we get two PDE for u~ ~) and v~ ~), the general solutions of which are given by

(6.2) u~ i) = ~ {a(~)(r) cosnO +fl~)(r) sinnO} rt=O

(i = 1, 2),

(6.3) v~ i) = ~ {y~)(r) cosnO + r s innO) , ~tffi0

where

(6.4)

a(~)Cr~ = - n [_oj[~)(r) + 2_~r b~)(r)] n , / (4 t92 - n 2 J ) Pi

~(i)(r~= - - - n [ojd~)(r)- 2--~-Dr a~)(r)] , ~n, , (4t) 2 - n 2 co 2) ~i

,n ,'(~)(r~ = - - ( 4 t ) 2 ~ - n oJ )Pi [ 2Dd~)(r) - 'n2c~ j

n2~ b(~) r ~ ( i ) ( r ~ - - - l ~ [ 2 D b ( ~ ) ( r ) r .o ( ) ] " J (4,0 2 - n 2 J ) pi

(i = 1, 2; n = 0, 1,2, .. .).

I t is to be noted that, due to the periodicity of solutions of (5.3) and (5.4), the quanti ty (402- n2~ 2) never vanishes.

I f we substi tute from eq. (6.1) into eq. (5.9) we get the following ordinary

(~) Sir H. LAMB: Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1932).


differential equation which determines the unknown coefficients a(~)(r) and b~)(r):

(6.5) ,.//fo(r) J~V)(r) + ,~fl(r) J~'(r) . �9 , ,fn'(r) r + ~ H2(r) - 7 - +

. . . . . f ' ( r ) ~ ~ , ~f~(r) + J ] 3 ( r ) - 7 - +~/U4(r) --7-- = 0 ,

where try(r) stands for a~)(r) or b~)(r), the coefficients t~#0(r), L#l(r), ~f2(r), t.~3(r) and ~/fft(r) are given in appendix B.

Also, since the basin vertical wall is impermeable, so the functions u~ ~) must vanish at the wall, i .e .

{u~i)}r:R = 0 (i = 1, 2),

which gives

~d(~)(R) - - ~ a~)(R) = 0

(6.6) (i = 1, 2),

J ~ ) ( R ) 2~ - - ~ b~)(R) = O.

By noting that the solution must be bounded at r = 0, then a~)(0) and b~)(0) should be bounded.

Now we are left with a fourth-order linear differential equation with variable coefficients, namely eq. (6.5). Our purpose is to obtain a bounded solution for this differential equation (6.5) which satisfies the boundary conditions (6.6). This is dealt with in the next section.

7. - De terminat ion o f the Fourier coef f ic ients a~)(r) and b~)(r).

In this section, we consideer the fourth-order DE:

u~k(r) d (4-k) ~fn(r) k:o r k d r ( 4 _ k ) = 0, n = 0, 1, 2, . . . ,

where the variable coefficients ~/~(r) are given in appendix B. Since r = 0 is a regular singular point for the ODE, then we assume the

following series expansion:

(7.1) ~fn(r) = r : ~ Bt(z , n ) r t (n = 0, 1, 2,...), /=0


where the four values of z are determined from the indicial equation to be

z = n + 2 , z = n , ~ = - n and z = - n + 2 ,

noting that Bo :/: 0 and B~ = 0. It is to be noted that, if we take, B~ r 0 and B0 = 0, we get exactly the same

solution like the previous one. Therefore, the solution has the form

(7.2) ~Fn(r) = A1 r n+2 ~ B2k (~ = n + 2, n) r 2~ + A2 ~ ( r ) + Aa ~ ( r ) + A4 ~ 4 ( r ) ,

n = 0, 1, 2,

where ~ ( r ) , ~ ( r ) and ~ ( r ) are the well-known logarithmic solutions. Such solutions are infinite at r = 0; therefore the bounded solution is given by

(7.3) ~ f , ( ~ = ~ B ~ ( n ) r 2k+~+2, n = O, 1, 2 , . . . , k=0

where A1 is absorbed in B2k. The coefficients B2k(n) satisfy the recurrence relation

(7.4)

and

(7.5)

Bk+4Zk+ 4 "4" Bk+2~k+ 2 "4- Bk vk = 0,

B2 = - ~ B o and B1 = 0 , Z2

k = 0 , 1 , 2 , . . . ,

X~, ~k and vk are given in appendix B. Despite the fact that the solution of the DE (6.5) has been already

determined, the pulse o~ is not assigned yet. To determine % the boundary conditions (6.6) (which are not used yet) must be taken into consideration.

8. - N u m e r i c a l a p p l i c a t i o n s .

In this section we give a numerical solution to the problem based on nondimensional data as it helps in considering different cases whenever physical data become available. The problem we are dealing with is briefly the fourth order DE (6.5), together with the boundary conditions (6.6) such that the solution is bounded at r = 0.

Our nondimensional parameters are

~=~_~ ~ = ~ ~ _ RDo

'00 ' ~0 ' 2 - ~ / ' ~ 0 '

H i = Pi Hi = H0 and ~i - - (i = 1, 2),

•o

WAV E S IN A ROTATING S T R A T I F I E D SHALLOW W A T E R 257

where l)0, ~0 and Ho are characteristic angular velocity, density and height, respectively.

In our applications we considered n = 1 ,,principal harmonics><

0 < ~ < 3 , 0<~o<1 .5 , 0 < R ~ < 2 ,

0.5 ~< H~ ~< 2.5 and -~1 + H2 = 3,

~1 = 0.99 and ~2 = 1.01.

The numerical solution is obtained by considering (6.5) under the condition of boundedness at r = 0, then applied to (6.6) leads to a set of curves clarifying the relation between R and ~ for different values of ~ (fig. 2). Such results enable us

2.0

1.0

0

Fig. 2.

I I

I

* - $ ~ * - * - $ - - , - $- - $-- :r , - - $ 4 . - - - w ~ 0 . 3

~, ~ -$ -~-~-, __, ~,_ ~ ~- *-- *--*--,_ ,_ ,_, _,_, ~.~-- --0.4

~ ./ , - * - * - * - * - * . ~ - * ' - * - * - * - - , - * , . - - - - = 0 7

/ , / 1.0

I L I I I I I I I I I I I L L I I I I I I I I I I I I I I 3 1 ] . O 2.0 ~ 0

to proceed for finding a relation between ~ and ~ for R = 1/2 and 1 at different values of H1 and H~ (cf. fig_ 3). This last relation leads to assigning the corresponding values of ~o and t) for each fixed R, so as to use these values to get the curves for atl)(r), ,(the Fourier coefficients of p2>> (fig. 4 and fig. 5).

These results are very useful in the geophysical applications, especially those for obtaining frequencies and amplitudes for the tidal waves in both closed (semiclosed) seas and oceans.

As a matter of fact their are not exactly circular seas, but can be considered approximately as circular, e.g., the black sea.

Finally, we should say that both analytical and numerical results for the internal waves can be deduced in a similar way and this may be presented in a future work.

2 5 8 MOHAMED A T E F A . E . H E L A L and SOAD E. B A D A W I

Fig. 3.

1.0 H~ = 0 . 5

H~ = 2.5 = 1/2

, . " l t l ' - = t l ' - ~ - " ' ~ - * - -'~t,- . ~ . , __ . , . __ ,__ .~__ , - , - . __ ,__ .__ ,__ ,__ . , __ . , __ .__ , - ,

l i l l l l i ' l i l i l i l l l i l i i l i l l l l i l i i , l i l l i l l i l i l l l i , l

1.0 * , - - '* ' - - R = 1 / 2 Hz = 2 . * " * " . . , - '*"- " "

, i l i l l i l l i l l i l i l l l t l i l t l l l i i l l i l l l l l l l l i l i l f l l l i

1.0 H1 = 1 .5

~,=15 , . , p - * - - R =1/2

, . * - ' "

, _ - ~ - , - -*- R' =1

,,,,,,,,I,,,, .... ,I,,,,,,,,,I,,,,,,,,,I,,,,,,L,I

0.5 1.0 1.5 2.0 2.5 ~ 3.0

o I 0"~ ) H'I =I R = I

~ = 2 ~ = I _,.,.,-*'*'* =0.52 . , , , * - * ' * " * ~

~ ~ " ~ s ~ ~ ~ .[, ~ ; . ~ ~_~4 ,k .# , "{'- . ~ r " - ' ~ ' -F~ " r " , i i i I I I

Fig. 4.

0 0.2 0.4 0.6 0.8 1.0 7

Fig. 5.

I #'~=2 ~=~ '~= 0,605

j. , ~ . . ,4,.~i,.,4r "~*******~ ~* �9 T . . . . - . . . r - , . . . . . . -,- - -

0,1 0.2 0,3 0.4 'F 0,5


9 . - C o n c l u s i o n .

The problem of stratified rotating fluids in its very simple domain has been presented in this work. A nonlinear system of equations is considered, the shallow-water approximation theory is applied to the system and a new set of equations is obtained corresponding to each order of approximations. Our main concern is devoted to progressive waves, the surface and the interface have been determined as solutions of the resulting DE.

Throughout the work, many complicated calculations are involved in order to get the exact solution in a compact form (a lot of these calculations has not been presented because they are quite lengthy). It is a fruitful area of further research work and many points need to be tackled, e.g., considering a rectangular cylinder which is more difficult to study especially at the corners. Also we clarify that the type of distortion presented here does not lead to solitons, but on using another type of distortion (29 it can lead to the well-known KdV equation.

Finally, it is useful to verify our results experimentally (which is not our domain of interest) and is still open.

A P P E N D I X A

In this appendix we explain how eqs. (5.9) and (5.10) can be derived from the system S given in the text. Applying the operator [] on the first equation of the system S, we get

\ [_0__~ ( 3 Du~e) +lDu~2) + l _~o Dv~e))] =

= - g(~2 - ~1) [ ] u~ 2) - ho ~ r r

- , t )2r[ 1 (32P~ 2) 2~ ~P~2)I] =-g~2-PlJ-7-[-7\a-7gg~-~ r ~ ]J

g(pe-~l)ho[3 [~2p~2) , 2~ 3p~2)~ 1 (3~p~2) 2~ aP~) 1 ~2 [~-rr[3-r-~z ~ r ~-0 ]+r \~-7-~z + - r ~ ] +

+1 3 (i 32p~2) zo3P~2) I] 7gg\ r a0a~ 8 r / J "

Making use of the third and fourth equations in S, and after little manipulation, we get

[] 8P~1) - [] 3P~2) - g(~2- P1)[D2r { 32p~2) + 2.(2 3P~e)~ • ~ 3 v2~(e)-

260 MOHAMED ATEF A.E. HELAL and SOAD E. BADAW!

A p p l y i n g the s a m e p r o c e d u r e to t he second equa t ion of the s y s t e m S, w e g e t

(A.2) p2 [] Op~2) ~ - ~ (p2 DP~ ~) - Hlg($2 ~- P1) V2P~)} �9

E q u a t i o n s (A.1) and (A.2) g ive i m m e d i a t e l y eqs. (5.9) and (5.10).

A P P E N D I X B

In th is a p p e n d i x we w r i t e down the coeff icients of the O D E (6.5) and the coeff ic ients of eq. (7.4) as follows:

u/Eo(r) = H1 g2(p2 - p1) h0(r),

~/Y/l(r) -- Hlg(g~ - p1)[t) 2 r 2 + 2gho],

+ r2(n2 o~ 2 - 4~2) p2 g(/-/i + ho),

+ (n 2 ~0 2 - 4~2)[r 4 p2 ~2 _ g(H~ + ho) p2 r2] ,

w h e r e Q2

ho(r) = H2 + - ~ (2r 2 - R 2)

and

Zk = al[(z + k)(z + k - 1)(~ + k - 2)(z + k - 3)] +

+ bl[(z + k)(a + k - 1)(z + k - 2)] + dl[(e + k)(z + k - 1)] + e~(e + k) + f ~ ,

~k = a2[(~ + k)(~ + k - 1)(~ + k - 2)(~ + k - 3)] +

+ b2[(z + k)(z + k - 1)(z + k - 2)] + d2[(~ + k)(z + k - 1)] + e2(z + k) +fe ,

vk = d3[(a + k)(z + k - 1)] + e3(z + k) + f 3 ,

W A V E S I N A R O T A T I N G S T R A T I F I E D S H A L L O W W A T E R

w h e r e

t)2R2~ = al = g l g 2 ( ~ 2 - p1) H 2 - - - ~ g ] ~ ,

b1=2 o p,

dl = - el = - (2n 2 + 1) ~ ,

f l = (n 4 - 4n2) ~ ,

1 b 1 H , a ~ = ~ 2 = ~ l g ~ 2 - ~ 1 ) 192 ,

d 2 = g { H ~ ( P 2 - ~ l ) t ~ 2 [ ( 1 - n 2 ) - - - ~ - ] - P 2 ( 4 D e - n 2 ~ H ~ + H 2 4g ]J '

= 1 2t) ( H i + 192R2\] g{Hl~2--P1)f22(~----~)-P2(4192-n2~2) g 2 - - ' ~ ) I , e2

2 ' f x=n2g{H~(P2-~ i )192(2+- -2~)+~2(4192-n2m2) ( H i + H 2 4g ] j

d3 -- - ~2(4~ 2 - n 2 ~ ) ~-2 ,

e3 = 3d3 ,

f8 = P2( 4~2 - n2 ~J) 2 ~ + 4 + ~2 _ n 2 co2 . 09

261

�9 R I A S S U N T O (*)

In questo articolo si introduce uno studio di fluidi rotanti stratificati non lineari con la teoria delle acque poco profonde. Si da anche il modello matematico per due strati di fluidi perfett i in un cilindro circolare rotante. Si determina una distorsione per fluidi debolmente rotanti ed acque basse. Si presenta uno studio dettagliato del caso particolare per onde progressive su entrambi gli strati e si e ot tenuta la soluzione analitica; si crede che tale soluzione sia nuova.

(*) Traduzione a cura della Redazione.

18 - Il Nuovo Cimento B.


BoanbI BO BpamaIome~cH crpaTH~RUaposannoi neray6oso/~ eoae.

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(*) Hepeaec)eno pec)atc~ueCt.

waves in a rotating stratified shallow water

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