Ann. Univ. Fe r r a r a - Sez. VII - Sc. Mat. Vol. XLII I , 65-119 (1997)

Small Amplitude Steady Internal Waves in Stratified Fluids.

G U I L L A U M E J A M E S (*)

SUNTO - Si studiano le onde interne gravitazionali, bidimensionali stazionarie, propagan- tisi in un fluido stratificato, incomprensibile, non viscoso contenuto in una striscia delimitata da due ret te orizzontali situate in un piano verticale. Nell'intorno del nu- mero critico di Froude, il teorema della variet~ centrale assicura che piccole soluzio- ni sono parametrizzate da due coordinate verificanti un sistema di equazioni diffe- renziali ordinarie non lineari. Si stabiliscono espressioni di calcolo semplici per i coefficienti quadratici e cubici della forma normale di questo sitema ridotto. Per una famiglia atre parametr i di stratificazioni, tali coefficienti sono calcolati numerica- mente e si vede che il coefficiente quadratico pu6 diventare piccolo. In questocaso, vi

la possibilit~ di propagazione di onde non usuali. Nella seconda parte del lavoro si analizza il caso in cui una stratificazione regolareconverge ad un profilo costante a t rat t i avente un discontinuitY. Si osserva formalmente che piccole onde propagantisi alla superficie di separazione di due fluidi omogenei sono limiti di ,,traveling waves, nella regione in cui la densit~ varia rapidamente.

ABSTRACT - We study the weakly non linear solutions of the D u b r e i l - J a c o t i n - L o n g el- liptic equation in a strip, which describes two dimensional gravity internal waves propagating steadily in a stratified fluid. In the neighborhood of the first critical value of the Froude number, the center manifold theorem ensures that small sol- utions are parametr ized by two coordinates which verify a system of nonlinear ordi- nary differential equations. We compute numerically the coefficients of the normal form of this reduced system for a three parameters family of stratifications and show that the quadratic coefficient (the most important) may become small. In that case, nonusual waves such as fronts can propagate. The last par t of our work studies the case when a smooth stratification converges towards a piecewise constant profile having one discontinuity. We observe formally that the small waves which propagate at the interface of two homogeneous fluids are limits at leading order of waves trav- elling in the region where the smooth density varies rapidly.

(*) Indirizzo dell 'autore: I.N.L.N., U.M.R.C.N.R.S. 6618, 1361 route des lucioles, 06560 Valbonne, France.

AMS subject classifications: 76V05, 35J25, 34G20, 35A05, 35B32, 34C20.

66 G U I L L A U M E J A M E S

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

This paper studies internal gravity waves in an inviscid incompressible stratified fluid confined between two horizontal rigid boundaries. We suppose that the flow is two dimensional and restrict our attention to waves which propagate steadily with a velocity c. Moreover, one makes the assumption that the fluid has smooth variations of density and is nondiffusive. In that case, the density gradient is orthogonal to the velocity field at each point and thus the density is constant along the streamlines (see figure 1).

Looking at the flow in a frame moving at velocity c, this problem can be for- mulated as a nonlinear elliptic partial differential equation satisfied by a stream function ~f and designed as DJL (Dubreil-Jacotin - Long) equation (see [8], [17]). However, one uses in general the DJLY (Dubreil-Jacotin - Long - Yih) equation which is slightly simpler (see [26]). For a given stratification (density at rest), a natural scaling shows that the parameter of the problem is

k - gd C 2 "

It is known since [12] that the weakly non linear solutions of these equa- tions can be rigorously studied through dynamical system techniques. In this approach, one often considers these equations as an evolution problem (E) in an infinite dimensional Banach space, where the unbounded space variable plays the role of time. A center manifold reduction theorem (see [18], [25], [4] for infinite dimensional spaces) is then used to describe the dynamics of sol- utions around an equilibrium which corresponds to the layer at rest. It en- sures that the small solutions of (E) lie on a integral C k center manifold of fi- nite dimension. Thus small solutions can be parametrized by a finite number of coordinates which verify a system of non linear ordinary differential equations (R). This approach was used in [13] where it was proved that the set of small solutions changes qualitatively for a sequence (k n)n >~ 0 of critical values of 2. In the cases when 2 ~ k0 and k ~ k 1 (codimension 1 singularities), the reduced

y

----, ~ .----~ a . . . ~ ---~ - .~"-~__.~. . ._.~

) 0 rim p(x,y)

Fig. 1. - Solitary wave in a stratified fluid.

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 67

equation (R) was written in a ,,normal form, which contained only its essential terms. When ;t ~ ;to, the small amplitude waves consist generically in two ,~conjugate flows- including the case of the layer at rest (see [5]), in periodic waves and in solitary waves connecting a conjugate flow to itself. Moreover, global existence results for periodic and solitary waves have been obtained in [6] for special classes of stratifications, using variational methods. These re- sults have been completed by Amick (see [1]) who obtained an unbounded con- nected branch of solitary wave solutions (;t, 9) for a general stratification, using global bifurcation techniques.

In the present paper, we pursue in different directions the study of the small amplitude solutions when ;t ~ ;t 0. We first establish simple expressions of the quadratic and cubic coefficients of the normal form of (R). These formu- las allow very precise numerical calculations even though the stratification varies rapidly in a thin band. Moreover, we show that for a concave stratifica- tion the small amplitude solitary waves are waves of depression when ;t is suf- ficiently close to ;t 0.

Furthermore, we consider a three parameter family of stratifications which converges in some limit cases towards a constant, a linear or a piecewise constant profile. One then shows numerically that the quadratic coefficient of (R) may vanish (codimension 2 singularity). We also observe a codimension 3 singularity when both quadratic and cubic coefficients vanish, but this situ- ation occurs when the density of the lower layer is so high with respect to the density of the upper one that it might not be physically realistic. In the case of the codimension 2 singularity, we refer to the studies of the normal form in [19], [14] for showing that there exist other types of travelling waves than the ones described above. For most of the parameter values, this singularity leads to existence of fronts. However, we show in certain cases that solitary waves of elevation and depression coexist, and solitary waves with an algebra- ic decay at infinity appear, although for apparently non realistic parameter values. Finally, we observe numerically that the problem seems to be robust when the density converges towards a piecewise constant profile, which moti- vates the last part of the paper.

In that part, we study in a general context the case when a smooth stratifi- cation converges towards a piecewise constant profile Q ~ having one discon- t inuity. The motivation of this study is to compare two models of the situation when travelling waves propagate in two unmiscible superposed layers of dif- ferent fluids between two plates. The first model consists in considering two layers of constant densities separated by a free interface (see [2], [3], [19]). The second model is the case of a continuously stratified fluid which density is almost constant in two regions and varies rapidly in a thin band. This problem has already been studied in [23] for the codimension 1 singularity ;t ~ ;t 0 by the

68 GUILLAUME JAMES

mean of variational methods. In this work, Turner considered a sequence of smooth stratifications converging towards a piecewise constant one and a re- lated sequence of small amplitude periodic solutions of the second problem. He then extracted a subsequence of these solutions converging towards a sol- ution of the first problem having the same amplitude and period, and stated that the same was possible for solitary waves. These results have been com- pleted in [2], where it was proved that the first model possesses an unbounded connected set of solitary wave solutions (4, w), w being the deviation of streamlines from the trivial flow in semi - lagrangian coordinates. These sol- utions are also obtained as extracted limits of solutions of the second problem. In the present paper, we verify formally that the small interfacial waves which exist in the fwst problem for the codimension 1 singularity/t ~ ~ 0 (see [3], [19]) are the limit at leading order of ,,interfacial- waves of the second problem. In the neighborhood of the codimension 2 singularity for which fronts exist, we show that this result holds when ~t < 2 0. Thus when ~ ~ ~ o there seems to be a good connection between the two models. However, these are only formal con- vergence results since the center manifold may shrink when the stratification Q becomes discontinuous. Our asymptotical study is consequently a first step towards a final goal, which is to show the existence in a suitable space of a cen- ter manifold which size is uniform when Q--*Q ~, and also to establish the smoothness of the center manifold reduction with respect to Q.

We now state the outline of the paper. In section 2, we write DJL equation in the form of an evolution problem. Section 3 deals with the linearized prob- lem and a center manifold reduction is performed in section 4. In section 5, one describes briefly a numerical scheme for computing eigenvalues and eigenvec- tors of our problem, and apply this method to a three parameters family of stratifications in section 6, where we compute the coefficients of the normal form. In section 7, one describes the small solutions of the reduced equation in the case of codimension 1 and 2 singularities. These results are applied in sec- tion 8 to the case when the stratification converges towards a piecewise con- stant profile. At last, the proofs of some results are included in two appendix.

2. - F o r m u l a t i o n o f t h e p r o b l e m .

Let us consider travelling waves in a stratified fluid layer contained in a horizontal channel of finite width. The scales for length, velocity and density are respectively the depth of the layer, the velocity c of waves and the density at the bottom. We consider a reference frame moving with the waves, oriented so that the velocity of the fluid at rest has components (1, 0). In this (x, y) co-

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 69

ordinate system, the flow is stationary. We then introduce the stream function y; through ( ~ , - ~ ) = (u, v), W(x, 0) = 0, where U = (u, v) denotes the vel- ocity. The streamlines are the level lines of W since V~f. U = 0 and clearly ~(x, y) -- ~o(Y) = Y when the fluid is at rest. Moreover we fLX the flux of the flow by W(x, 1 ) = 1 which corresponds to the case when u tends towards 1 at infinity. Assuming u > 0 in all the domain, the nondiffusivity hypothesis V o. U = 0 reads 0 = Q ( ~ ) where we suppose 0~Ck+l [0 , 1], for some k large enough with 0 ( 0 ) = 1, 0 ( 1 ) = q e (0, 1) and Q' <0. Similary, Bernoulli's law for stationary stratified flows can be written:

IUI 2 (1) P + )~OY + 0 - H(~) .

2

We now suppose that the function r is the same as for the fluid at rest, and make the same assumption for the Bernoulli function H, which is true in par- ticular when U tends towards (1, 0) at infinity with some hypothesis on Vv. For the layer at equilibrium, normalizing the pressure Po by P0 (0) = 0 leads by integrating Euler's equations to:

I ( 1 ) (2) H(~,) = - ,~ O dy + O(~) ,~f + -~ �9

0

Differentiating (1) with respect to x and y yields, after simple calculations from the Euler equations, to DJL equation (see [17] or [8]):

O' IVY3[ 2 O' H ' (3) A~p + )~y - - (V:) + - - (~v) = (y3) .

There is another choice of variable ~, incorporating V~, which cancels the Vy: term and leads to DJLY equation (see for instance [26] or [23] ). We do not use this variable for keeping an explicit H, unlike what was done in [13]. Hence is solution of the nonlinear elliptic boundary value problem in an infinite strip:

1 2 f (~ ) = (~Y~ + 2 ) f(~f) ' (x, y ) e R x (0, 1),

(4) ~ ( x , 0 ) = 0 ,

~(x, 1) = 1,

where f = r162 is a given function of ~. In order to reformulate the problem as a dynamical system, we define the spaces D = ( H 2 ( O , 1 ) A H o I ( 0 , 1 ) ) • H i ( 0 , 1), X = H I ( 0 , 1 ) x L2(0, 1) which are Banach spaces with respect to

70 GUILLAUME JAMES

t h e norms II(?z1, U2)II D = ]]Ul]IH 2 + ]]?~211H1 a n d I I (u l , u 2 ) H x = IlUl]IH 1 + ]]U211L2 wi th D continuously embedded in X.

We then consider the operator:

Fa: D---) X ,

(~1, ~p2) ~..~(~f12 ' d2~ldy 2 + f ( y + ~ f ~ ) [ d~fldy 21 (d~f~) 2 ~ +)-~P~- -~f212])

defined in a neighborhood of 0 in D. Then we can rewrite (4) with the function !P= (~f - y, ~ l , /~x) which is searched to be in C~ D) fq CI(R, X) and sol- ution of the following evolutionary system in X:

(5) d ~ _ Fx(tP), dx

where F~. is a regular map from D to X with a nonlinear part regular from D to D (it is the most propitious case to apply the center manifold theorem proved in [25]). This problem possesses the trivial solution W0 = 0 corresponding to the layer at rest. The operator F~ has some additional properties which will be very useful. The first one is due to the reflexional symmetry x ~ - x of the problem (4), which implies that F;. anticommutes with the symmetry:

:1) Moreover, considering a subdomain t2 limited by two vertical sections of the channel and the rigid boundaries, Euler's theorem ensures that:

f [p n + Qu(u.n)] ds = - ;t l Q V y d x d y , ~Q t~

where n is the outside unitary vector normal to 3Y2. As a direct consequence, 1

the quantity J = f ( p + ~u2) dy is independent of x and the problem has the 0

first integral:

(6) J( ~ , 2, ~)) =

1 i[ 1 ] = H ( y + u 2 1 ) + ~ ( y + v 2 1 ) ( l + 2 ~ 1 v + ~ v - ~ f ~ ) - , ~ ( y + ~ f 1 1 ) y d y . 0

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 71

3. - Study of the l inearized operator.

The linearized operator L~ of F~ at 'P= 0 is given by:

(7) L).(~O1, ~f12) = (~P2, de~ l fd~dyl + ~-f~fll)- dy 2

Thus L~ maps continuously D into X and clearly L~: D r X - - ) X is closed. Now if belongs to the resolvent set of L~, then ( ~ I - L ~ ) -1 ~ ( X , D) and

(~I - L z ) x ~ x = I n - ~ x o ( ~ l - - L t ) ~ : l n

is compact, since the embedding I: D - ~ X is compact. Consequently, the spec- trum X~ of L~ consists of isolated eigenvalues a ~ C of finite multiplicities, with no accumulation point at finite distanee. Then a ~ X~ if and only if

(8) L~ W= a~g

for some ~ge D - { 0 }. Since L~ anticommutes with S, it is clear that Xx is in- variant under a ~-~- a. Moreover, equation (8) can be written:

(0v2~)' + ( a 2 0 - 2 0 ' ) ~ 1 = 0 , (9)

~01(0) = 0 ,

~01(1) = 0 .

The three last equations form an eigenvalue problem of Sturm-Liouville type for a 2 with a parameter 2, and consequently a 2 e R. It follows that a e R U iR and one can prove easily that a e R* when it = 0 by multiplying the second equation by F 1 and integrating from 0 to 1. By the mean of variational calculus technics (see for instance [7], p. 397-429) using the fact that 0 > 0, one can show that the eigenvalues form an increasing sequence (an)~ ~>o accumulating only at infinity, which ensures that only a finite number of eigenvalues _+ an are purely imaginary (in what follows we set 5tea~>10 and 5m(an)>>-0). Moreover, these techniques allow us to describe the evolution of the spectrum 2:~ as it ranges over R. Indeed, the property 0 ' < 0 implies that increasing it yields to decrease a~ for all n e N, and consequently brings the eigenvalues towards the imaginary axis. Hence there exists a strictly increasing sequence (it n)n ~ O, which elements are called the critical values of the parameter it, such that an( i t~)= 0. Let us sum up the evolution of X~ in figure 2.

In what follows, we restrict our attention to the case it ~ it o. Until the end of this section, we will give some results concerning Lxo which will be useful in the next section for the study of the nonlinear problem.

72 GUILLAUME JAMES

Ik_

Fig. 2. - Behaviour of X~ in the complex plane when ~ increases.

Our fwst step is to calculate the generalized eigenspace associated with oo = 0 when 2 =2o. Now ~1 in (9) is a solution of the system

(10) { ( ~ ' ) ' - ~ o ~ '

~(0) = 0,

r = 0 ,

~ = 0 ,

which is still a problem of Sturm-Liouville type, and thus the eigenvectors cor- responding to the first eigenvalue ;[o don't vanish over (0, 1). Consequently, we f'LX the solution ~ of (10) in satisfying ~o(1/2)= 1 . After straightforward calculations, one finds that the generalized kernel E0 of L~0 is spanned by:

*:(:) where L~ 0 V= 0, L~ o W = V, S V = V, S W = - W.

It is well known that eigenvalues and eigenvectors have explicit expres- sions in the case of an exponential stratification ~o(y) = e -YY (y > 0). In particu- lar, )[o = (l /Y)( z2 + Y2/4) and ~(y) = e (~/2)(v- 1/~)sin ( z y ) .

For determining the range of Lx 0 and calculating a pseudo inverse, consid- er U = (Ul, u 2 ) e D and B = (bl, b 2) EX. The equation L ~ o U = B yields to:

(11)

{ ue=b~,

(Qul ' ) ' - 2 o ~ ' u l = - ob2,

um(O) = O,

Ul(1) = 0.

SMALL A M P L I T U D E STEADY I N T E R N A L WAVES ETC. 73

An obvious necessary condition to solve this system is:

1

(12) I ~ocfb2 dy = 0 0

(multiply by cf the second equation and integrate by parts). I t is a classical re- sult that it is also sufficient (see appendix A). Let us introduce the scalar pro-

1 1

duct ( ~ , O)x = f QF 1 q~ 1 dy + f Qy, 2 ~ 2 dy. Then the range of L~ o is just W l in 0 0

X. Now, we can define a pseudo-inverse of L~o by:

(/ ) L~-oi(bl, b2) = K(y , s) ob2d~, bl ,

where K is the symmetric Green's kernel:

K(y, s) = I cp(y) rp.,(s) for 0 <~ s ~< y

t cp2(y) q)(s) for y < ~ s ~ < l ,

with q~2�9176 1] defined by:

1 ~ 2 ( 0 ) - - - ,

~ ' ( 0 ) Y

h Oq~2

1 ~ 2 ( 1 ) ---- _ _ ,

qcf'(1)

y � 9 1),

for some h �9 (0, 1).

4. - C e n t e r m a n i f o l d r e d u c t i o n

The formulation of DJL equation as a dynamical system has the advantage to allow the use of the center manifold reduction theorem. In this perspective, we define Z = { 0 } • H I ( 0 , 1)r D and write equation (5) in the form:

d ~ (13)

dx - L~. o ' Y + M ( ' / O + / ~ N ( W ) ,

where/~ = ; t - )~o and M, N � 9 ~) verify M(0) = N ( 0 ) = 0, DM(O) = O.

74 GUILLAUME JAMES

More explicitely, if we set ~ = (~)1, ~)2):

M(~g) = (0, M2o (~o(~ 2)) + Mo.~ (~(2 2)) + M~o(~ ~ ~)) + Mr , (~ 1, ~)t 2)) "{" o(11 ~14)),

N ( ~ ) = (0, N1 (~)1) + N2 (y~ ~2)) + o(l ly, 1 I[~r~)),

where:

f ( d'~D1 / 2 - - f ' d~)l + Xof' ~ , M2o(~(12)) = - - ~ -~-y ] ' f l dy

Mo2(F(22)) = f 2 - ~,,~,

1 _ f , _ f , , F , 2 M3o(~3)) = ~ ( F~2F~ 1~1 + ~ o f " ~ ) ,

M~2(~,, ~(2 2)) = f ' - - ~ - ~ 1 ~ ,

N I ( ~ 1) --f~fl 1,

N 2 ( ~ 2~) = f ' ~'~.

Since X~. o M iR = {0} and dimEo = 2, the purely imaginary part of the spec- t rum consists of a finite number of isolated eigenvalues with a finite dimen- sional generalized eigenspace. Fur thermore , introducing ~9 = { 0 } x L 2 (0, 1 ) c X, we prove in appendix A that there exists w o > 0 and c > 0 such that for any w e R such that [w I /> Wo, iw belongs to the resolvent set of L~o and:

(14) II(i~ - L~o )-111~,-~x ~ ~ I~l

(see also [13], [25] ). Under these conditions, a center manifold reduction is available and the following theorem is proved in [25] since M and N take their values in D with a zero first component:

THEOREM 4.1. There exists a neighborhood Q x -V of the origin in D x R and a mapping OECk(Eo x R, D) with O(0, H ) = 0 , VH~\9 and DO~,(O, O) = 0 such that for all tt E ~ the following properties hold:

i) The set M~,= { ~ +O(~g, ff) e D / ~ e E o M ~ } is a local integral C k manifold of (13).

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 75

ii) I f 'IJ is a solution of(13) such that tIJ(x) ~ ~V x e R, then ~belongs to M, . Moreover, its coordinate ~ verifies in Eo a reduced equation of the form:

dq' (15) dx - L~'~ ~ +G(~, it)

(G(0, #) = 0, DG~(0, 0) = 0).

iii) Conversely, i f ~ is a solution of (15) such that ~(x) �9 Eo • t-2 for all x e R, then tiC(x) = ~(x) + O(~(x),/~) /s solution of (13).

iv) At last, for all ~ �9 E0, Sq~(~, it) = O(S ~,/~)

(for the property iv), the reader can refer to [9]). Consequently, for/~ suffi- ciently small, all the solutions of (13) that stay close enough to zero are ob- tained via equation (15). We thus have reduced the problem of finding small amplitudes solutions of (13) to the same problem for an ODE in R 2.

The study of this equation can be simplified using the theory of normal forms determined by the structure of L?. o i~:o and by the symmetries acting on Fx. For a general description of these tools, the interested reader can refer to [9]. In our case we have:

L~01Eo = ( ; 10), SIEo = (10 O1)

and the following result holds (see for instance [10] or [13]):

THEOREM 4.2. Fix an order p <. k. Then the function �9 of theorem 1 may be chosen such that, setting ~(x) = A(x) V + B(x) W, equation (15) reads:

(16)

dA

dx

dB

dx

= B ,

= P(A, it) + R(A, B, it),

ivhere P is a polynomial in A of degree <. p which coefficients are regular functions of It in a neighborhood of O. Moreover, P(A, t t ) = a ~ A + T(A,/~) with T(A, /~) = O(A 2) and R = o( IAI + IBI F is even in B. The system (16)/s called the normal form of order p of the reduced system (15).

76 GUILLAUME JAMES

The truncated system obtained by setting R = 0 is then integrable and its solutions (A, B) verify:

(17) 2 2 - Q(A, tt) = constant ,

where ~ Q = 2 P. Taking p = 3, the reduced equation in normal form reads:

d A = B ' da:

(18) dB -~ =(all + O ( l ~ f ) ) ~ +(a~+O(l~,f))A 2 +(o~+O([l ,I))A 3 + R(A, B, ~,),

where the coefficients a/j depend on 0. Coefficients of normal forms can be cal- culated in a systematic way, which is r described in [9]. Given the Tay- lor expansion of ~ in terms of A, B, ~:

(19) ~'=AV+BW+ ~, ~p~APBql~r + p+q~l,r~O p+q+r<~3

+O(IA [ + IBI + It t l)(IA [ + IBI) '~

(~1oo = ~o,o = 0, S4~r, = ( - 1)q ~ ) , the idea is to identify the powers of A, B, in d~/dx obtained by differentiating (19) and using (18) with the ones calcu-

lated by inserting (19) into (13). Let us apply this method to the coefficients we are interested in. Identifying terms of order Art leads after simple calculations to:

L~o ~1o, -- aH W + Rio,,

with Rlol = (0, -fcf). Thus all W + Rio, e range L~o and the results of section 3 ensure that (a,1W + Rlol, W)x = 0, which yields immediatly to:

1

f e ' ~2dy (20) a , , - o < 0

1

f Qef2 dy o

In the same way, we obtain for the terms of order A e:

(21) L~ o 'ib2oo = aeo W + R~oo,

where R2oo = (0, ( f /2) ~r ,2 _~f, ~ , _ ). a f ' ~2 ). In what follows, we denote by w the fL,'st component of ~2oo. We show in appendix B that the solvability con-

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 77

dition (a2o W + R2o0, W)x = 0 leads to:

(22)

1

f Qcf '~ dy 30 0-2o--

2 1 f qcp2 dy

o

This coefficient is fundamental in the present problem, and can be derived through various types of analysis (see in particular [1] and also [23]). As we will see in section 7, the local existence of elevation or depression solitary waves depends on the sign of a2o, which is not obvious to determine. However, the following lemma is proved in appendix B:

LEMMA 4.1. I f Q is concave then a2o < O.

Furthermore, the general solution of equation (21) is given by

O2oo = L~ 1 (aeo W + R2o0) + a 2o0 V,

where L~ ~ is the pseudo-inverse given in section. The constant a2o0 is deter- mined by looking at the AB and B2 terms. Identification at order AB leads to :

L~ o OllO = 2 ~2o0.

Then the condition S~11o = - OllO gives easily:

~ i i 0 : ( 0 , 2w).

Identifying the B 2 terms, one obtains:

L~o 0020 = OllO + Ro2o,

where Ro2o=(O,(f/2)cf2). The solvability condition (Ono+Ro2o, W)x=O then determines the constant a2o0.

At last, identification of order A 3 terms yields to:

L~o ~3o0 = a3oW+ a2o ~110 + A,(cf, w) + A2(qo(3)),

where we define:

A l ( C f , w ) = (O,fcf 'w' + f ' ( c f w ) ' - 2~of ' (pw) ,

A2(~(3)) = ~ (0 , ( f ' cp ) ' cp-2of"cp 3)

78 GUILLAUME JAMES

The condition

(23) (a~0 W + a2o 1~110 + Al(q), W) + A2(cp(s)), W)x = 0

then leads after lengthy calculations described in appendix B to:

1 (24) aao : 1

YQr 0

( 1 1 l 1 1 1 2 )

-2o5fQw'4dy-'~ o o i

where the function I is given in (0,1) by:

Y (25) I ( y ) - 1 | Q(3~v ,8 _ 2oe0 ~v2) ds

~(Y) o J

and can be continuously extended to [0, 1 ].

5. - N u m e r i c a l c o m p u t a t i o n o f c o e f f i c i e n t s o f the n o r m a l form.

In most of the cases, the knowledge of the sign of ae0 is sufficient to study qualitatively the set of small amplitude solutions of equation (18), but one has to calculate a~0 when ~0 becomes small. The values of these two coefficients are given by equations (22) and (24) once ~v has been calculated. To this pur- pose, we have to compute numerically the first mode of the eigenvalue problem:

{ q~,, + f ~ ' = )~f~,

(26) ~v(0) = 0,

~(1) =o,

with the normalyzing condition r f (1 /2 )= 1. To achieve this, we use the Tau method of Chebychev polynomials approximation for which we refer the read- er to [21]. The first step is to transport the problem (26) on [ - 1, 1] by the change of variable ~ = 2 y - 1. We then seek an approximate solution ~v N under the form:

N (27) CfN(~) = ~, a~Tn(~ ) , n=0 where Tn denotes the n-th Chebychev polynomial and N ~ N is sufficiently

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 79

large. The Tau method consists first to insert ~ N in the ODE and obtain N - 1 linear equations for a0, . . . , a N by identifying the N - 1 first modes. Taking ac- count of the two boundary conditions then leads to a generalized eigenvalue problem:

(28) L A = ~ , M A ,

where L, MeMN+I(R) and A = (ao...aN) t. This method is ve ry accurate since er rors decrease more rapidly than any power of 1 / N as N--* + ~ when 0 e C ~ [0, 1] (one can verify this phenomenon in the case of an exponential stratification, for which the exact solution is explicit). The problem (28) is then solved using a matrix eigenvalue algorithm, namely a QZ method (see [20]).

6. - Appl icat ion to a 3 parameter fami ly o f strat i f icat ions.

Let us consider q e (0, 1), h e [0, 1], }' > 0 and the density:

(29) O ( Y ) = a tanh [ y ( y - h)] + f l ,

where

q - 1

a = tanh (}'h) + tanh (}'(1 - h)) ' f l =

qtanh (}'h) + tanh (},(1 - h))

tanh (}'h) + tanh (}'(1 - h))

The graph of 0 is depicted in figure 3. As it results from the center manifold theorem (just in considering more

parameters), the center manifold reduction is smooth with respect to the pa- rameters q, h, ~,, as well as the reduced system, hence the coefficients of the normal form are smooth functions of q, h, }'. Le t us examine some limit cases for our parameters q, h, }'.

p(y)

!

q . . . . . . . . . . . . . .

o h y

Fig. 3. - Density profile at rest.

80 GUILLAUME JAMES

When X---) 0, 0 converges uniformly towards the linear stratification r = (q - 1 ) y + 1 independent of h. For X = 0, lemma 4.1 ensures that aeo < 0 for all q � 9 (0, 1).

An important limit case is also X ~ ~-oo. Here ~) converges towards the piecewise constant stratification:

1 f f O ~ < y < h 0 ~(Y) = '

q , i f h < y < ~ l .

Thus the case when X is large can be in terpre ted as the situation in which two fluid layers of almost constant densities mix up in a very thin band that be- comes thinner as X increases. In section 8, we compare in a more general con- text this situation with the case of two fluids of constant densities separated by an interface (see [2], [3], [19]). Here we only mention an important proper ty of the two-layer model related to the e leva t ion num be r :

(30) e(h , q) - 1 q

h 2 (1 - h) 2 '

q �9 (0, 1 ) being the density ratio of the two layers and h �9 (0, 1 ) the depth of the lower fluid over the depth of the channel. I t is a well known result (see for instance [3], [19]) that when e ~ 0 and ,l ~ ~ with:

1(1 q) ; t ~ : ( h , q ) - 1 q h- +

- 1 - h

there exist interfacial solitary waves connecting the rest state to itself which are respectively waves of elevation or depression depending of e > 0 or e < 0, with an amplitude in O( I ;t - A $ I ). We will see in section 7 that in our situation the sign of aeo has the same influence on the type of solitary waves.

The case q = 0 arises when the density at the bottom of the layer is far higher than at the top. I t is not obvious that this situation has a physical inter- est for average values of X (for example, the case of a layer of infinite depth seems to be more relevant to modelize atmospheric permanent waves). How- ever, one can wonder if for a large X some of our results are related to the problem of an homogeneous fluid with a free surface (treated in [16] for in- stance). This question is not simple since this lat ter model requires the pres- sure to be constant on the free surface. F rom a numerical point of view, one observes that an accurate computation of ~ requires a matrix size N + 1 that increases as q-- )0 due to a boundary layer effect (see figure 4).

The case q = 1 corresponds to a weakly stratified fluid. When q = 1, D JL equation becomes Laplace's equation in an infinite strip and the only bounded

S M A L L AM} LtT<~[ E S T E A D Y I N T E R N A L W A V E S E T C , $ 1

9(Y} g ~ O,@Ol N~80 ... . . .

~ ~ 0,0001 N~2~ .....

0 0

Fis 4. - Graph of ~unction r %~~ 7 = 1 and s = 0o25..

smooth solution is the trfvfal one~ In what %llows, we b i e f l y s tudy this singu lap lhnfto F o r q near I and h, near 1/2~ let us introduce the small p a r a m e t e r s ~: = 1 g~ U = h, 1,/2 and observe that @(y) = 1 + cd(?7) where dO/) = do + ~d~ +

~} O(v~)o Defining f = #20~ the eis problem (26) leads t.o:

(s~l)

( r e( d r ' )' - fd '

r : 0 ,

r 0 o

{{:) --- 0

]Vo]l<m@~g" per turba t ion theory of linear opera tors %r a simple isolated eigen- value (see [11])~ we then consider the follow[hA expans%ns in powers of ~:-:

- zo + Go + *~%~ + o( [~,[ + [,[ )2

where ~{o and 9(~ 0 are respectively the []rst eigenva]ue and the corresponding

normalized ei&<envector obtained by setting ~ =<j = 0 in (31)o The fact that

2 o ~ ~(0~ ~)/~: diverg~es as c--~ 0 ~s consistent with what was mentionned above~

Fu~J~ermorG one has:

1

,s d y 3 o f r o

0

8~e GU, n ,LA[IME JAIVIE S

k 0 �9 t h "~"~ ......

,, i'/i ,;, ,.., .......

0" s 1 5 ~2

I0 .//: ,/,e I /: / / /: // //

o ........ i; .....

~ J

Fig. 5. - Graph of A~ 9~r ~.-= 1/~ and various values of Vo

q

The p r o p e r t y ~Z~; (.y) =~ d~; ( 1 ......... y ) c lear ly implies 9 o (~') = g o ( 1 - y ) a n d then a~o .... O( t~' I ~ I ~71 )- This r e su l t indie~i:es Why the cu~Je

(% :-: {(~,~,, ~Dh,~o(h, q, r ) :~ o}

tha~ we c,omp@,e mm~eNcally pas ses throu?A'h the poinI:. (h , g) : ( l /E , ~ ) fbr &ll y,

Af te r the s t udy of t hese l imit cases, let us examine w h a t occurs in all the paramete; ; ~ space t h rough numer ica l calculus~ F ix ing h. ~ (0, 1)~ the g r a p h of t, he f i rs t c i t [ c~I value A o(h, g, V) as a :fhnction of q is s h o ~ o in f igure 5 for' d ig 9~rent va lues of y~

Thus one obsen ;e s t h a t 2 0 has a finite l imit as q 0. Moreover , A 0 t ends to--. wa rds i ~:? ~s ~s a p p r o a c h e s @ ,~ (we p rove this ~'esult in a m e r e gene ra l con tex t in sectic:n 8). Consequen t ly , one has s o ~ t/l~, w h e n q is small and y lar?(e~ Thus the f i rs t bifl~;rcation occurs when

gh~

C 2

h.~ = <Lh being the depth of the <lower fluid~> at rest Hence gh.~/c ~ is close to its c~:'itica} value fbr the pK~btem of ~ m~ h o m o g e n e o u s ltuid wi th a f ree surf%~ce [.reared in [~[6]~

Moreover , the d e p e n d a n c y e f t , he ct~rve q~. in 7 fs show~ Jn f i~ r r e 6. T h e cu~we C:, ge t s c loser to the curve ~'(A, q) = 0 as V--~ + '~. Moreove[ ~ C~, t akes @e aspec t of a stra4tJ~t line as y--~ 0 and d i s a p p e a r s w h e n V = 0o tn w h a t f%l-

<MXI , I AM!>I , IT( I)Iq ~ T l { ' t l ) t IN'I ' I.]I>NAI, W.',&I.],'-: ICI'('. N8

V",!~ 0"5 i ,

,.\

0

O

y = ~ o , y ~ 2 J )

y = 3 , 4 ___ y = 2 . . . . y ~ I . . y = 0 . 1 ..... y = 0 . 2 -

a ~ o < 0

..... i % 0 5 I

Ki<a. & - l)el~en(tai~c 5- {}f' th{~ cui 've (2 in 7.

to~s, ;;(~ ;::~il ~/ {7') l he lowesl xahm {}t'q Peached on the cu rve ( ' . t,'(}r ; c t q

and ,'i ~:- ( q , ( 7 ) 1 ), {me can (lc/5ne the funct ion D{}(},, q) by (t2{>(h.(7, q), q, 7) - O. Tlx, ]lu~aeric;d COml}utation of az{} in{licates tha i t'or all ),, (r2{} > {} in the do-

rmii l~ I 7 d ( i ] ~ o ( i }}5:

f

i r t , , , ( 7 ) < q < 1 ,

I 0 < D < te~(;', ?)

and ~'~::~ < (t it~ ~(). 1 1 :~ ' \H: . T h u s for fixed h and q with c (h , q) r 0, (l~,,(/t, q, ,,) has t}~' * i ~ ~}t' : when 7 is suf t ic ien t ly lar~e. This p r o p e r t y will be jus t i f ied and anatS.<od f2~ctIar i~ sec t ion S.

W~ n~x~ }mxo to s t u d y the siKn of r162 ?, 7) w h e n (r =: 0. To this p u t -

�84184

Vi~. T. (;raph of (t:~,,(D~gT, qL q. 7) fop vm'ious values of 7.

8 4 G[ I L L A U M E JAMES

o.~

O.O.SS

0.0~ / /

0025

0.02

0.015

0.01

o

0

/ /

! /

/ /:" /

/ / / / /

- '" z . . . . / ..... / /

J ' / ~r

ii .....

K i .......

/ i

~ ~ h

}rig. & Interseetions of the eup~es (%o = 0 and a,ao ~" 0 in the (h, q) p lane

pose, we compute numerically for different values of y the graph of e~0 (h~(~j, q)~ q, 7) fo r q e (q,~,,(7), 1)~ ~}%~en 7 ~ 1.6, one o b s e r v e s t h a t a;~o > Oo For dill%rent larger vNues of 7, the graphs are shown in figure 7.

O~e o b s e r v e s that the ~ 'aph of a:.~o gets closer to the graph of

g ( g ) .... 24 .\/~ + q2

as 7 -:~ + ~, as proved analytically in section & Moreover, a:~o may vanish and become negaf~ve Ibr small values of qo Figure 8 shows in the (h,, q) plane how (%, c r o s s e s the curve of equation e~0(h,, q, ?/) = ()~

Although t/he parameter values for which a2o ~ 0 and a:~o < 0 may not be physically realistic, we analyse in section which types of small solutions do

they lead~

7. - Small solutions of the reduced equation.

For a ger~eral stratification e, one has generically a~0 ~ (). This case has al- ready been mentioned in [13] and the corresponding situation for the normal

fo~u is eom~detely treated in [I0]~ Taking p = 2 in (16), the reduced equation

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 85

in normal form reads:

(32)

dA B , dx

dB = a2o A + (aeo + O( I~I))A ~ + R(A, B,/~) ,

dx

with a~ = allft + O(~t 2) and R(A, B, it) = O( [A[ 3 + [A[B 2 + B*) even in B. One then consider the integrable truncated system:

(33)

d~ dx

d~ _ _ = ~ + a ~ ~. dx

The phase portrait of bounded solutions of (33) is shown in figure 9 for a2o > 0 with

_~o(tt ) = [ 3al1 ~--~ ] + 0 ( ~ 2) and B o ( / ~ ) V ~ I an/~13~ = + O( I/~ I~r2) �9 aeo

The phase portraits in the case when aeo < 0 are obtained from the above ones through the symmetry (A, B) ~ ( - A , B). Bounded solutions of (33) consist of the following solutions up to a phase shift in x:

- two stationary points,

JL / JL ~ A Y

r

~'o

a) CASE P.<O b) CASE p > O

Fig. 9. - Bounded solutions for small tt and ar > 0.

h

86 GUILLAUME JAMES

- a family of periodic solutions with .4 even in x which can be parametrized by a frequency o la01 , o E (0, 1),

- an homoclinic solution with -4 even in x.

I t is proved in [10] (see also [12]) that considering one of these solutions, the full system (32) has a solution of the same type if/~ is small enough. More- over, since the phase space is two dimensional, the phase portrait of the small solutions of (32) is a perturbed version of figure 9 for small g. The stationary point A = B = 0 corresponds to the layer at rest and the other equilibrium rep- resents a flow independent of x with no vertical component (called ,,conjugate flow-, see [5]). Moreover, periodic orbits correspond to periodic waves, and ho- moclinic orbits to solitary waves.

We now discuss in which cases solitary waves of elevation or depression ap- pear (a similar local result has been proved in [1] in a more complicated way). To this purpose, one considers an homoclinic solution (A(x,/~), B(x , it)) of (32) and the corresponding solution F(x, y, Z) of equation (4). According to equa- tion (19), one has:

~(x , y, it) = y + A(x , t~) c;(y) + R(A(x , it), B(x , /~), tt)(y),

where R ~ C k ( R 3 , H2(O, 1)) and R ( A , B , t t ) = O ( A 2 + B 2 + IA/~I). For c e [ 0, 1], one defines

F~.~ = {(x, y ) e R • [0, 1] /~(x, y, ~ ) = c},

which is a streamline corresponding to a solitary wave. Using the implicit function theorem, the equation F(x, y,/~) = c then leads to:

where

(34)

Y = Yc (A(x, ~), B(x,/~), tt)

yc(A, B , ~) = c - q~(c) A + o(ll(A, B, ~)11) .

We now introduce r(z) = lim A(x , it) = O( lit I ) and

y| (/~) = lim Yc (A(x, Z), B(x, tt), /~) = c - q~(c) r(/~) + o(tt).

Then, for (x, y) e F t , z:

y - y~(~) = ~(c) (r(/~) - A ( x , / ~ ) ) + o(~)

uniformly in x e R. Thus homoclinic orbits correspond to waves of elevation when ae0 > 0 and of depression for a20 < 0 since r - A has the same sign as aeo. I t follows by lemma 4.1 that in the case of a concave stratification the small solitary waves are waves of depression when ~ ~ ~ o. We mention at last that a

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 87

global existence result for elevation or depression solitary waves has been proved in [1], depending upon a2o > 0 or aeo < 0.

The amplitudes A that correspond to homoclinic orbits have an explicit form at first order in/~. In the case when/x < 0, the homoclinic solution for which A is even in x reads (see [10], [12]):

(35) A ( x , it) all 3/~ - + O(/~ 2) a2o 1 + cosh (V~-~lttX)

uniformly in x E R. In the case when tt and aeo are small, the reduced equation has other small

solutions in addition to those depicted above (see [19], [14], [3]). In what fol- lows, we study completely this codimension 2 singularity, but for the sake of conciseness we only give the main ideas of the proofs.

We first consider a one parameter family of densities (~)h) smooth in h which satisfies the following assumption:

(A) there exists a single root hoE(0 , 1) of a2o(h) = 0 and a3o(ho) > 0 .

(for each ? E R, the family of density profiles of section 6 clearly verifies (A) when q is not too small). Taking p = 4 in (16), one obtains:

(36)

dA =B,

dx

dB - - = O2o~,h)A + a (u ,h )A ~ + b ~ , h ) A ~ + c(,u,h)A 4 + M ( A , B , t~, h) , dx

where

a~(/~, h) = all(ho)/x + O(Itt I( I/tl + Ih - hol ) )

aa20 a(/~, h) = - - ~ - ( h o ) ( h - ho) + a21(ho) tt + O(( It~ I + Ih - ho I)2),

b(tt, h) = a~ao(ho) + O(I/~ I + I h - h o I) ,

c(/~, h) = a40(ho) + O( I/xl + I h - ho I),

M ( A , B , t~, h) = M ( A , - B , ix, h) = O ( I A I ( A 2 + B2) 2 + B 6)

are regular functions of their arguments. In order to simplify the system (36),

88 GUILLAUME JAMES

one makes the change of variables A = a /~rb , B = f l / V ~ which leads to:

(37) dfl a~a + ea2 + a3 + la4 + R l ( a , fl, aS, e) , dx

where e = a l v a , 1 = (aao(ho))/a3o(ho) 3/2 and R , ( a , fl, o20, e) = O(a4(]a~ I + Is]) + [a i (a 2 +fle)e + f i e ) is even in ft. The map (/~, h) ~ ( a ~ , e) is a diffeo- morphism in a neighborhood of (0, h0) since all (ho)(3aeo/~h)(ho) ~ 0 and thus one considers a~o and e as two independent small parameters.

The equilibria of (37) have the form (a, fl) = (,J, 0). When v ~ 0, they satis- fy an equation of the type:

(38) + + 3 + o([ 13(]v I + ]o 1 + 1 1)) = o .

At different steps of this section, it is convenient to use (v, e) instead of (a~, e) as small parameters. Using the implicit function theorem in the neighborhood of (v, e) = 0, equation (38) then yields to:

a~ = e(v, e) , (39)

with

e(v, e) = - ev - v 2 - lv s + O([vla(Ivl + le l ) ) .

So near the origin the system (37) has from one to three equilibria, depending on the values of a~, e. In particular, the origin is the only stationary point when aS > a~(e) with:

(~(e) e~ 1 = _ _ + - s ~ + O ( e 4 ) .

4 8

When aS = a2c(e), there is an other equilibrium (re(e), 0) and

e 3 v ~(s) = ls2 + O( [s[3) .

2 8

For a~ < a~(e), there are two nonzero equilibria v and v*, which can be ex- pressed in function of (v, e). One shows using (39) that the equilibrium (v*(v, e), 0) of (37) satisfies

(40) v * ( v , e ) - v = ( v c ( e ) - v ) 2 + / v + - e + O ( [ v l + [ e [ ) 2 . 2

We now study the nature of equilibria. Clearly the equilibrium (a, fl) = 0 of

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 89

(37) is hyperbolic when a2o > 0 and elliptic if a~ < 0. Moreover, the translation (a, fl) = (5 + v,/~) leads to the reduced system:

(41)

d~

F(5, 3, v, e). dx

The function F is even in/~ and has the form:

F(5, fl, v, e) =

4

E an(v, e) an + O( [ 5 ['5 + fl2( [ a [ + [v[ )(( [ S l + [v[ )~ + fl2) + ~ ) , n = l

where

al(v , e ) = 2 v ( v - v c ( e ) ) ( l + 3 1 v - 3 le +O(([v l + [el)2)) 2 4

a2(v, e) = e + 3v + 6lv 2 + O(v2( [v[ + le[) ) ,

aa(v, e) = 1 + 4lv + O(Iv[([v[ + [el ) ) ,

a4(v, e) = l + O( [v[ + [e[).

For small (v, e), an equilibrium (v, 0) of (37) is consequently hyperbolic if v(v - vc(e)) > 0 and elliptic if v(v - vc(e)) < O.

Our aim is now to study the stable and unstable manifolds of the hyperbolic equilibria of (37). If (a(x), fl(x)) is a solution of (37), then ( a ( - x), - f l ( - x)) is a solution too and consequently the unstable manifold of each equilibrium is the image of the stable manifold by the symmetry (a, fl) ~ (a, -fl) . We now prove the following:

PROPOSITION 7.1. Suppose that 0 is an hyperbolic equi l ibrium of (37) (respectively (41)). There exists a neighborhood ~ (resp. ~?) o f (a, a~, e) = 0 (resp. (5, v, e) = O) and a regular func t ion n (resp. ~t) such that i f ( a , a~, e) �9 "~ (resp. (St, v, e) ~ ~) and (a, fl) �9 W'~(O) (resp. (~, fl) �9 Ws(0)) then fl~ = n(a , a'~, e) (resp. ~2 = ~t(~, v, e) ). Moreover:

(42) ( 2 )

n(a , a2o, e ) = a 2 a 2 + - - e a + l a 2 + O( ,ala) 3 2

9O

(43)

with

GUILLAUME JAMES

) ~(5, v, e) = ~2 c~(v, e) 5 i + 0(~ 4) , i=

2 4

2 cl(v, e) = - -e + 2v + 4lv 2 + 0@2(Ivl + [e[)) ,

3

1 c 2 o , , = - + 2 i v + o ( I , l ( I , I +

2

c3(v, e) = _2 l + O ( [ v I + lel)" 5

PROOF. We first observe that the system (36) has the first integral:

(44) f ( A , B,I~, h) = J ( A V + B W + ~p(AV+BW,/~) , )~0 +/~, ~oh)

where J is the first integral of the system (5) given in section 2. Since J is in- variant by S and S commutes with (/), it follows that f is even in B. Moreover, one obtains after expanding the two sides of equation (44) in powers of A, B and identifying the B 2 term:

1

~2f (0, 0, /~, h) = - Igq~2dy ~ O. ~B 2

0

Then it is easy to construct a first in tegra l j (a , fl, ae0, e) of (37) even in fl satis- fying j (0 , 0, o~, e) = 0 and (~2j/~f12)(O, 0, a~, e) = 2. For 5/> 0, one then de- fine the regular function:

m(a , 5, O~o, e) = j (a , V"5, a~, e)

and construct a regular extension of m in the neighborhood of 0 (i.e. for 5 < 0) with the Taylor expansion o f j truncated at a sufficiently large order. Obvious- ly, this process is possible since j is even in ft. One then has m(0, 0, a~, e) = 0, (5~n/aS)(0, 0, o~, e) = 1 and m(a, fie, o~, e) is a first integral of (37). We now consider the equation:

m(a , 5, a~, e) = 0 ,

which is solved in a neighborhood ~ of (a , o20, e) = 0 by the implicit function

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 91

theorem, leading to

5 = n(a, ~2o, e).

Now if (a , fl) ~ WS(0) one has m(a, f12, a~, e) = 0 and thus for (a , ~ , e) e

(45) f12= n(a, ~2o, e).

In order to calculate n, one chooses for the system (37) an initial condition (a0, rio) on WS(0) and one differentiates equation (45) with respect to x. Then for all a in an open neighborhood I~. ~ of 0:

3n ~-a (a , o~, e) = 2(a20 a + E{22 -4- (23 A- O(a4))

which leads to equation (42). The other part of the proof concerning the sys- tem (41) is obtained in the same way.

As a first conclusion, we know the equations of W ~ and W u in the neighbor- hood of the hyperbolic equilibria of (37). In what follows, this allows us to find the small homoclinie and heteroclinic solutions of (37). We first prove the following

PROPOSITION 7.2. In the neighborhood of (v, e) = O, the system (37) has two heteroclinic orbits joining the origin with a critical point (v, O) i f and only i f v = vf(e) with:

2 v:(~) = - - ~ + O(E2).

3 (46)

In that case,

(47) .~ =.~(~)= _2: + o(1~1~). 9

PROOF. I f there exists such an heteroclinic orbit, then

n(v, a~(v, E), ~)= O.

For v ~ 0, equation (42) shows that this relation reads:

1 1 - e + - v + O(v 2) = 0 . 3 2

Equation (46) is then given by the implicit function theorem and equation (47) is obtained with (39). Conversely, suppose that a~ = a~(e). When e is small,

92 GUILLAUME JAMES

2 n (a,a~(e) ,r

Vr(r I

a

Fig. 10. - Graph of n(-, ay(e), s) for s < 0.

(vf(e), 0) is an hyperbolic equilibrium of (37) since al(vf(s) , ~ ) = e 2 / 9 + O( [ t [ ~) > 0. Then differentiating equation (45) with respect to x on its stable manifold gives (3n/3a)(vf(e), ay(e), c ) = 0, whieh leads to

n(a, ay(e), e)=a2(a- vf(t))2( 1 + O ( [ a l + l e [ ) ) .

For a fixed s ~ 0, the graph of this function for a ~ 0 is shown in fig- ure 10.

Then, according to proposition 7.1, W'+(0) and WU(0) are shown in figure 11.

Using the results of proposition 7.2, the s tudy of the solutions which are homoclinic to 0 is straightforward. Indeed, one observes that for ~ ;~ 0, (3n/~a~)(a, a~, e) = a2(1 + O( la lS) ) > 0 in the neighborhood of (a , a~, ~) = 0. Then for ae0 > ay(s) one has n(a, a~, E )> 0 and no solution homoclinic to 0 exists. When 0 < a ~ < ay(e), the same variationnal argument shows

\ )

Fig. 11. - Heteroelinic orbits for a~ = a~(~) and ~ < 0.

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 93

n ,Q,~,c, B

I ' ~oX../ o o~

Fig . 12. - Orb i t homocl in ic to 0 fo r 0 < o20 < o f ( s ) a n d e < 0.

easily that n(ao, o2, e ) = 0 for some a0(o~, e ) e ( 0 , vf(e)), leading to the existence of an homoclinic orbit (see figure 12).

We now prove the existence of other heteroclinic solutions (not connecting the origin):

\ PROPOSITION 7.3. In the neighborhood of (v, e) = O, the sys tem (37) has tCwo heteroclinic orbits jo in ing an hyperbolic equil ibrium (v, O) wi th (v*(v , e), O) i f and only i f e = e~(v) with:

(48) e s ( V ) = -- ~- lv 2 + o ( I v l 3 ) . 5

This condition is equivalent to e = si(O2o), where

(49) 3

= + o(o o)

and a~<0 .

PROOF. We use the same method as in proposition 7.2 by considering the equation

~ ( v * ( v , ~) - v, v, ~) = O,

where v*(v , e) is given by equation (40). At last, it is lengthy but straightforward to determine for which values of

(a z, e) an orbit homoclinic to a nonzero hyperbolic equilibria exists (it suffices to study the variations of ~ with 5 and to use equation (39)).

As a conclusion, the different kinds of phase portraits of the system (37) are depicted in figure 13.

94 GUILLAUME JAMES

2

C

E

Fig. 13. - Small amplitude solutions around the codimension 2 singularity for aao (ho) > 0 (here 1 > 0).

The equations of the curves are:

re : ao = a~(e),

r f o~ = o~(~),

Fs: E = ~f(a~).

Using the original parameters (/~, h), these equations become:

F c : /~ =

F f : /~ =

4 Wao (ho) an (ho)

9a3o (ho) au (ho)

I"8 : h - ho = l ( 3 a l l ( h o ) a4o(ho) ( ~a,2o / ~ h )( ho ) a~o ( ho )

~h (h~ - h~ § O( Ih - ho [8),

~a2o (ho)2( h _ ho)2 + O( Ih - h o 13), ~h

a2t(ho)//~ + o(/D. ]

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 95

One can mention that heteroclinic solutions have an explicit expression at fLrst order. For (/~, h) ~ Ff and (gaeo/~h)(ho)(h - ho) > 0, one has for example:

(50) A(x , it) = - 2 a l l ( h ~ ~ + O([ /~])

a3o(ho) 1 + e - ~ x

uniformly in x ~ R (see [19]). Coming back to equation (4), these are front sol- utions which connect two parallel flows at different heights. More precisely, they connect a rest state with an other conjugate flow when parameters lie on Ff and two y-dependent flows when they lie on F~.

One can notice that the crests of the solitary waves corresponding to homo- clinic orbits become flatter and longer as (it, h) approaches Ff or Fs, due to the influence of a very close stationary point. In the case of gravity interfacial waves between two layers, this broadening phenomenon has been conjecturat- ed in [2], numerically observed in [24] and analytically studied in [19]. At last, numerical observations indicating this phenomenon in a continuously strati- fied fluid can be found in [22].

We consider finally a family of densities (r smooth in h, satisfying the following assumption:

(B) there exists a single root hoe (0, 1) of aeo(h)= 0 and aao(h0) < 0.

(notice that the sign of a3o is opposite to the one given in assumption (A)). For example, the density Q(y, q, h) introduced in section 6 verifies (B) for 7 �9 [1.8, 6] and q----qm(7) (see figures 7, 8). The corresponding situation for the normal form has been treated in [14] to which we refer the reader for the proof of the following results. We sketch the different kinds of phase portraits of small solutions of equation (36) for h ~ h0 and t t - -0 at figure 14.

The equation of the curve F is:

tt = 4 a3o (ho) all (ho)

aa2o ~h (h~ - h~ + O([h - ho 13).

The phase portraits for (aa2o/~h)(ho) > 0 are obtained from the above ones by making the symmetry (A, B) ~ ( - A , B). One observes that two homoclinic orbits exist when parameters lie under the curve F with/~ ;~ 0, yielding to the coexistence of solitary waves of elevation and of depression. Once more, these homoclinic solutions have at first order an explicit expression. One has for

96 GUILLAUME JAMES

T

~ ~ :! o h_ho

Fig. 14. - Small amplitude solutions around the codimension 2 singularity for a~o(ho) < 0 (here ( 3a2o/Sh)(ho) < 0).

example when M < O:

a l l ( h ~ l c o s h ( ~ # x ) s + 0 ( { # { ) A ( x , M, s ) = - - ~ - - ~ ) +- 2

for a rrxed s given by

a~o(h) 8----

~ V ~ (ho) ~ ~/-~(ho) The homoclinic orbits obtained when # = 0 or when the parameters lie on /" correspond to solitary waves having an algebraic decay at in f in i t y . One has for example when M = 0:

12 aeo(h) A ( x , h) = + O(h - ho) ~,

aao(ho) 9 + re(h) x 2

with

a~o ( h ) 2 m ( h ) = - 2 - -

Wao (ho)

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 97

8. - C o m p a r i s o n w i t h the t w o layer f lu id mode l .

The aim of this section is to compare formally for ;t ~ )~ o the limit case when converges towards:

1 i f O < . y < h 0 ~(Y) = ' h, q e (0, 1)

q if h < y ~ < l ,

and the model of two homogeneous fluids with a density ratio q separated by an interface (see [3], [19]). In what follows, we restrict our attention to smooth densities satisfying Q(0) = 1 and Q(1) = q (what matters in fact is that Q(1) I>

1 > 0 for all ~)). After showing some convergence results for the eigenvalue problem, we study the behaviour of the normal form coefficients as ~)-o ~)~ .

We first introduce a weak formulation of the eigenvalue problem (10). Let w ~ 0 be in/-/ol (0, 1) and 20 �9 R satisfying:

1 1

(51) Yv �9 1), [~w' v' dy - ~o f Q(wv)' dy = O. o o

Using C ~ test functions v with supports in (0, h) and (h, 1), one shows easily that for Q =Q ~ the eigenvectors w are proportionnal to:

f Y if O<~y<.h h '

(52) u~ (y) = 1 - y ~- _--~ if h<~y<~l.

Then taking w = v = u~ in equation (51) leads to:

1 ( 1 q )

~ $ - 1 - q h + 1 - h

which is the critical value of ~. for the problem of two homogeneous layers (see [3], [19]). In what follows, we call u the eigenvector of (51) normalized with u(h) = 1 in the ease when ~ is smooth. Moreover, we define

Uoo

u~ (1/2)

We now prove some convergence results as Q--* ~) | Proposition 8.2 and a re- sult similar to proposition 8.1 have been proved in [23] and [2] for a less gener- al class of smooth stratifications, decreasing in [h, h + l /y ] and being constant

98 GUILLAUME JAMES

elsewhere. Using these results, the limit of Oeo (normalized differently) has been calculated in [1].

PROPOSITION 8.1. There are pos i t ive n u m b e r s R, C1, C2, Ca such that f o r

a n y smoo th 2 w i th 1 1 2 - 2 ~ [[L2 <~ R the f o l l ow ing holds:

i) Ilu - u = 11~1 -< c i 112 - 2 = t l~

i i) lifo - ~ = IIH, -< c ~ 112 = 2 = It/+

PROOF. By subtracting the two weak formulations (51) for 2 and 2 ~ and observing that for all v e H ~ ( 0 , 1):

(53) 1 1

f 2 o~ (uv) ' d y = 12 ~ (u~ v) ' d y = (1 - q) v(h) 0 0

one finds for all veHol(0 , 1):

(54) 1

] 2 ( u ' - u ' ) v ' d y =

0

1 1

= f ( 2 = - 2) u~ v ' d y + 2 o f ( 2 = 2 = ) (uv) , d y + (40 - ~t~ ) (1 - q ) v ( h ) .

0 0

Then choosing v = u - u~ and using HSlder's inequality, one obtains:

1

' Lf - I q[[u--U~[PL2"~[[U~[IL=H2--2~[IL211U--U~[IL2+~O ( 2 - - 2 | u ~ ) ) ' d y . d

Using the identity:

1 1

~(2 - 2 ~ ) (u (u - u~ )) ' dy = 2 ~(2 - 2 o~ ) ( u ' - u " )(u - u| ) dy + 0 0

1 1

+ ~(2 - 2 ~ ) u " (u - uoo ) dy + ~(2 - 2 ~ ) u| ( u ' - u " ) dy 0 0

together with Poincarf 's inequality Ilu - u| IlL 2 ~< llu' - u " IlL 2 then yields to:

qllu' - u " ilL= < 112 - 2 = I1~=[(~.o + 1) l lu" IlL= +,~ol lu= I1~=] +

+ 2 ~ o l l u - u = I1~o 112 - 2 = 11~2.

S M A L L A M P L I T U D E S T E A D Y I N T E R N A L WAVES E T C . 99

Since H i ( 0 , 1) is continuously embedded in C~ 1], one obtains af ter precis- ing the constants:

[ (1 _1 q]lu' - u" IlL 2 <~ lie - o ~ IIL~ (40 + 1) Max h ' 1 - h

+ 2,~ o I lu - u ~ I1. , l ie - e ~ I1,.".

A straightforward application of Poincar6's inequality gives consequently:

( 5 5 ) (q-4;~olle-e~lL~)llu-u~ll., <<-

[ ( 1 ' ) 1 - - + 4 o Ik, - o ~ I1,.~. ~<2 ( ) . o + l ) M a x h ' 1 - h

One then gives an upper bound for 2 o independent of Q. We use the proper ty (see [7], p. 398):

f ~v'2 dy 2 o = M i n o v e H o l ( 0 , 1 )

1

- f e ' v 2 d y 0

which implies:

1

f u ~ d y (56) 2 o ~< o

1

- f o ' u ~ dy 0

Now, integrating by parts and using (N3):

1 1

o o" 1

Then for I1• - ~ ~ IlL 2 small enough, one obtains - f o ' u ~ d y >~ (1 - q)/2. Hence (56) implies: o

2 (57) ;t o ~< M =

h(1 - h)(1 - q)

Consequently, equation (55) indicates that for ILo-(~ ~ 111.2 sufficiently small there exists C1 > 0 such that:

Ilu - u ~ I1. , < c , lie - e ~ IIL~

and the assertion i) is proved.

100 G U I L L A U M E J A M E S

Moreover, observing that cf = u /u (1 /2 ) and applying a triangular inequali- ty, one finds:

1 1 I ifull,r + IIq;- ~ ooll,,, < u ( l / 2 ) u~ (1/2)

When I1~-0 ~ IlL 2 is sufficiently small:

liull., -< Ilu - u~ I1.1 + Ilu~ I1., -< 211u~ I1.~,

1 1 ~<C4 lu(1/2) - u~ (1/2) u(1/2) u~ (1/2)

and then

119 - ~ ~ II., -< ( 2 < Ilu~ rl., +

which proves ii). At last, one obtains by setting v = u~ in (54):

(1 - q ) (2o - ) ~ ) =

1

= [ r

u~(1 /2 )

Using i) and

IIQ - ~ ~ ILL.:

- - ilu - u~ II.,.

~< c4 rru - u~ Boo

1 ) Ilu - Uoo II,-,' ~ C~ IIQ - Q oo ILL', u ~(1 /2 )

1 l

' - u : ) u " d y + ] ( Q - Q ~ ) u "2 d y - '~o ~ (Q - ~ | ) u u s d y -

0 0

1 1

- ~ o f (Q - Q ~ ) ( u ' - u " ) u ~ d y - 2o f ( Q - e ~) u " u ~ d y . 0 0

(57), the following estimates are easily obtained for small

and the proof of iii) is straightforward.

I/.uu ,ul.. f .

SMALL A M P L I T U D E STEADY I N T E R N A L WAVES ETC. 101

Now let (~ . ) be a one parameter family of densities smooth in y satisfying the hypothesis:

(H) For any closed interval C with h ~ C, tim Q y = 0 ~ in C 1 ( C ) .

(the stratification of section 6 clearly verifies (H)). Under this assumption, lim I1~) - q ~ IlL z = 0 and proposition 8.1 implies that ~, u converge uniformly

towards V ~, u~ as y - ~ + ~ and lim ;t o = ;t ~- We then have a s t ronger con- vergence result: ~ ~ § ~

PROPOSITION 8.2. I f (Qy) satisf ies (H), then:

i) IIqJ'llL ~ is u n i f o r m l y bounded as y--> + ~ ,

ii) f o r any closed interval C wi th h~tC, cp-->cf | in C1(C) as ~,-~ + ~ .

PROOf. Defining C, = [0, h - e ]U [h +~ , 1], (H) can be expressed as follows:

(H) V e > 0 , lira 0 r = 0 | in CI(C~). y--* + oo

One proves easily that q r converges towards Q ~ in L2(0 , 1) as r--~ + oo when (H) is satisfied and thus proposition 8.1 can be used when y is large. In what follows, we omit the dependence of ~ and ;t o in ~ for the sake of simplicity. Since of pC1[0 , 1] with o f ( 0 ) = c p ( 1 ) = 0 , there exists some 0 e ( 0 , 1 ) w i t h ~ ' ( 0 ) =0 . Then for all y e [ 0 , 1]:

Y

Consequently, integrating by pal~s:

- - O y ( y ) c f ( y ) + Q y ( O ) q ~ ( O ) + Iq~' lds . q o

Proposition 8.1 clearly implies that ;to, IlcfllL~, I1~' IlL 1 are uniformly bounded as y--) + ~ and thus i) is straightforwaz-d. Moreover, one has clearly using (10):

1 I "1 -< - IQ I +;to l. q

102 GUILLAUME J A M E S

Then applying proposition 8.1 and i), there exists a l such that for ~, large enough:

Consequently, Yy �9 [0, h - e]:

Y

(58) I r = J ~ " d s [ ~ < a l l l o r - o | I

Integrat ing from 0 to y then leads to:

~(Y) - c~1 aDo 7 - o ~ NCI(Ce)Y < r y <~ ~(y) + C~ 1 IO0 r - - O ~ I[CI(Ce)Y

and thus for 7 large enough:

~ (y) - C1110r - O ~ ilL 2 - a l H O t - O ~ IICI(Ce)Y <~

Cf ' (0 ) y ~< Cf ~ (y) + C 1 II~)r - O Qr IlL 2 § ( ~ l I IOy - - O ~ IICI(C~)Y"

Since ~ ~(y) = ~ ' ( 0 ) y, one obtains by sett ing y = h - e:

(59) I ~ ' ( o ) - ~ u I ~ - ~ l l o r - o ~ II~,(c~) + - ~ l l o r - o ~ I1~

for some a 2, a s > O. Consequently, one obtains using (58) and (59) af ter a tri- angular inequality:

Sup Icf'(Y) - ~o'~1 ~ a~llor - O ~ IIL~ + ~ l l o r -- O ~ I1~'(~). y e [ 0 , h - s ]

The case when y � 9 [h + e, 1] can be t rea ted in the same way. One obtains finally:

(60) Supl~'(y)-~:l<asIlor-o~llL~+a~llor-Ooollcl(c~) yeCE

and the proof of ii) follows easily from proposition 8.1. We now establish the asymptotic behaviour of the normal form coeffi-

cients.

PROPOSITION 8.3. If (Or) satisfies (H), then the coefficients all, aeo and

SMALL A M P L I T U D E STEADY I N T E R N A L WAVES ETC. 103

a3o have f in i te l imi ts al~, a2~ and a3~ when y ~ + ~ with:

1 - q av a l l = - 3

h + q(1 - h)

9 e 1 a 2 ~ -

2 h + q ( 1 - h ) u=(1 /2 )

(1 + X/q) 3 az~ = 24 when a2~ = 0

V ~ + q 2

(e is the elevation number defined in equation (30)).

PROOF. A simple integration by parts in equation (20) yields to:

1

fe ' r a l l = - - 2 0

1

f ~q~2 dy 0

Using propositions 8.1 and 8.2, a straightforward application of the dominated convergence theorem then proves that an and a~zo converge respectively towards:

1 1

' '~ dy f e ~ cf ~ q~ = dY 3 J e = cf o~ 0

all = - 2 1 , a,z~ - 2 1 f o = cf~ dy f o = q ~ dy

0 0

which leads to the explicit form given at proposition 8.3. We now study the be- haviour of a30 as y--* + Qr For y e ( 0 , 1), one has:

I (y ) -

Y

1 f ~) y(3 of,3 _ 2Wzo ~v2) d s . ~(Y) o

By dominated convergence, the function I converges simply in (0, 1) towards:

Y

_ 1 f I ~ ( y ) Q = (3r - 2a~z~ r ~ ) d s . r ~ ( Y ) o

J

We now prove that IIIIIL~ is uniformly bounded as 7--* + ~ . Since II~IIL',

104 G U I L L A U M E J A M E S

Ikv'llL | and a2o are uniformly bounded, there exists ct 7 > 0 such that:

(61) I l e , ( 3 ~ '3 - 2a2o q~) ] lL ~ ~< a 7.

We then consider three cases:

1) If y � 9 (0, h/2), it follows from (60):

Integrating from 0 to 1 leads easily to:

y 1

and when ~ is sufficiently large, one has:

y 2

~(Y) ~ -

2) In the same way, one has for y � 9 [(h + 1)/2, 1):

1 - y 2

~(Y) ~

3) In the case when y � 9 ( h / 2 , ( h + 1)/2), one has for y large enough:

y 2

~(y) Min r [ h / 2 , ( h + 1 ) / 2 1

One then define:

a s = 2 M a x ( 1 1 1 ) - - , - - o q~" (0) q)" (1) ' Min q~

[ h / 2 , ( h + 1 ) / 2 ]

Now if y �9 (0, (h + 1 )/2] it follows from (61):

Y I I ( y ) l <<- a7 - - <<- a 7 a s .

of(y)

As the same way, if y e [(h + 1)/2, 1):

1

r Qy(3~ - 2a2or 2) ds I ~< a 7 a s

Y

and consequently [ [ ~ [ L | is bounded independently of y.

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 105

As a conclusion, one can apply the dominated convergence theorem to the coefficient a3o given by equation (24). Hence in the case when a2~ = 0 and ~--* + ~, aa0 tends towards:

f 1 ,4 l ~ I ~ d . , O dy+3 y] f q ~ of 5 d y o o

o

The calculation of a3~ can be simplified observing that a2~ = 0 implies 1/h 2 =

q/(1 - h) e and then q ~ ~'~ is constant. It follows immediatly that I~ is inde- pendent of y and one obtains the result of proposition 8.3 for aa~, which com- pletes the proof.

Thus al~ < 0, a3~ > 0 when a2~ = 0 and a2~ has the sign of e. As a first con- clusion, this asymptotic analysis justifies the numerical results obtained for high 7 in section 6. It follows immediatly that for fLxed h and q with e(h, q) ~ 0

and 7 large enough, the small solutions of (4) are characterized for )L ~ )~ o by the small solutions of the reduced equation (32).

As we said previously, the limit case r q ~ has been studied by Amick and Turner in [23], [2] for the less general class of smooth stratifications de- scribed above. Their approaches are completely different, and it is interesting to emphasize what differs in the results given by their methods. After a change of coordinates, Turner considers in [23] a weak formulation of DJLY equation. It is remarkable that for Q =Q ~ this formulation is equivalent to equations that describe travelling waves in two superposed homogeneous flu- ids. These equations apply to a stream function and consist of Laplace equa- tion in two strips separated by an unknown interface. Moreover, the pressure has to be continuous across this interface, which is tangent to the velocity field. After a lengthy analysis, it is shown in [23] that when e ~ 0 and ~ is large enough (particularly for ~ = + ~), there exists travelling waves of arbitrary small amplitudes and arbitrary large periods (infinite for solitary waves). Moreover, the corresponding value of )~ tends towards )~ o as the waves become small, and )~ < )[ o.

For a smooth stratification, the center manifold reduction is a much sim- pler tool and gives apparently more results since it determines for small tt all the small solutions, with explicit expressions at first order. Moreover, this method has been employed in the case of two homogeneous layers (see [3], [19]), where in addition heteroclinic solutions were found for small elevation numbers.

However, Turner's work gives supplementary interesting results. Indeed, if ~, I> ~ for ~ large enough, the bounds for the amplitudes of the solutions and their periods are independent of y, whereas the neighborhood Q • ~ of theo-

106 GUILLAUME JAMES

rem 4.1 may shrink as 7--) + or Moreover, the periodic solutions which exist for ~) = 0 ~ are obtained as extracted limits of sequences of solutions having the same amplitude and period, related to the family of smooth densities con- verging towards r (Turner stated that the same is possible for solitary waves). This is a first convergence result for the nonlinear problem as y---~ + oo.

On the other hand, Amick and Turner proved in [2] the existence of a glob- al branch of solitary waves when 0 = ~) ~ and e ~ 0. Though the method is dif- ferent from the one used in [23], these solutions are also extracted limits of se- quences of solutions of the regular problem.

In what follows, we compare for 2 ~ 2 5 and 7 ---> + :r the limiting forms of the streamline ~0(x, y ) = h with the interface of the two layer model. Of course, these are only formal convergence results since we only study the sol- utions at first order.

For h, q fixed with e(h, q) ~ 0 and 2 ~ 2 5 , it follows from the center mani- fold reduction in [19] that for small amplitude waves the interface between the two fluids of constant densities can be approximated by:

Y~ (x, 2) = h + a~ (x, 2)

where a~ satisfies the reduced equation:

d2a~ (62) - alT (2 - 2 5 ) a~ - a2~ u ~ ( 1 / 2 ) a~

dx 2

which is the stationary Korteweg-De Vries equation integrated once. Now one considers a smooth stratification 0 r satisfying (H). When it ~ 2 o(7), the small solutions of (4) can be approximated by

~ r ( x , y , 2) = y +AT(x, 2) q~(y),

with

d2Ar - all(y)(2 -2o(y))A~ + a2o(7) A~. dx 2

Let us define the - interface, Yr(x, 2) through:

~f y(x, Y~(x, 2), 2) = h .

So, when y is large the graph of Y~ defines in R x [0, 1] two regions in which is almost constant. Using equation (34), Yr is given at leading order by:

~'y(x, 2) = h - q~(h) A~(x, 2) = h - - 1 Ar (x , 2) . u( 1/2 )

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 107

One then introduces

1 a~(x, 2) - - - A t ( x , ~),

u(1/2)

which verifies:

(63) d2 a~

dx 2 2

- - a l l (~') (~, - - ~, o ( ~ ) ) a r - - a2o (~') u ( 1 / 2 ) a t .

For a fixed 2 ~ ;t 0, this equation is jus t a reversible per turba t ion of (62) as 7 ~ + ~ . Consequently, for all interfacial wave Y~ (x, A) of period T ~< + and ~ I> Y0()~, T), there exists ~'r(x, ~) having the same period and:

Yr(x , )~)--) ~ '~(x, 2) in L ~ ( R ) as 7---) + r162

As a first conclusion, for e(h, q) ~ O and ;t sufficiently close to ) ~ , the small amplitude interfacial waves between two homogeneous fluids are the limit at first order of waves propagat ing in a continuously stratif ied fluid, the stratifi- cation of which tending towards ~) ~.

In the same way, we now s tudy the codimension 2 singulari ty in the limit ---) Q ~. In what follows, we assume q to be fixed and suppose

1 h = ho ~ - - -

I+V~ where a2~ (h0 ~ ) = 0. We now introduce the p a r a m e t e r

tt = 2 - )~ ~ (h)

and assume tt ~ 0 with tt < 0. In what follows, we denote by u~ (1/2, ho ~ ) the value of the eigenfunction defined in (52) taken at y = 1/2 for h = h0 ~ . Since a2~ (h0 ~ ) = 0, the center manifold reduction in [19] on the limit problem with two superposed homogeneous layers leads to the following equation which is jus t equation (62) with an additionnal cubic term:

d2a~ (64) - - - a17 (h~ ) tta~ -

dx 2

-a2~ ( h ) u ~ (1/2, h~ ) a ~ + a~ ( h~ ) u~ (1/2, h~ )2 a ~

(it suffices to observe tha t u~(1 /2 , h ) = u ~ ( 1 / 2 , h ~ ) + O ( I h - h ~ r = (1 + Vq) /2 + O( I h - ho ~ I )). This equation is the s ta t ionary modified Ko- r teweg - De Vries equation in tegra ted once. We now find convenient to use ~ , aug(h)) instead of (it, h) as bifurcation parameters . According to the re-

108 GUILLAUME JAMES

\

___~~> a~

F O

Fig. 15. - Small solutions of (64) for tt < 0 and (/~, a~ ) -- 0.

suits of section 7, the phase portraits of small solutions of (64) in the (a=, a " ) plane are depicted in figure 15.

The equations of Fc and F f are

1 Fc:/~ = a ~ 2,

4 a ~ (h0 ~ ) a17 (ho ~ )

2 r f = 2

9a3~ (ho ~ ) a17 (ho ~ )

One can mention a resul t proved in [2] and [15], which gives as a necessary condition for the existence of front interfacial waves connecting the res t state to a conjugate flow:

l+V - _ ~ (ho ~) .

1-V

Thus 2 is independent of h on the curve Ff corresponding to the exact reduced system.

As above, we now compare equation (64) with the reduced equation corre- sponding to a smooth stratification. To this purpose, we consider a two param- eters family of stratifications (Qr.h) which satisfies the following assump- tions:

i) for all fixed he(O, 1), (Q~,h) satisfies (H) when 7--~ + ~ ,

ii) for all fLxed h~ e (0, 1) and all function h(7) such that lim h| (~)y,h(~)) satisfies (H) with h = h ~ 7-,+~

h(7) =

S M A L L A M P L I T U D E S T E A D Y I N T E R N A L W A V E S E T C . 1 0 9

(the family of stratifications of section 6 clearly verifies assumptions i) and ii)). One first observes that when y is large, the equation

a20(~, h) = 0

has a solution ho(y) with lim ho (y )= ho ~ �9 Indeed, for all e > 0 , aeo(y, ~,--~ + ~

ho ~ - e) > 0 and a~o(y, ho ~ + E) < 0 if y is large enough. According to the study in section 7, the amplitude A v then satisfies the equation

d2 Av - all (y, ho(y))(4 - 4o(~, h(a2~)))A v +

dx 2

Normalizing A as above, one obtains:

+a2o(y, ~ 2 h(a2o ) )A r + a3o(~, ho(~))A~.

where

~'v(x, 4, h) = h + a~,(x, 4, h) ,

d 2 a v (65) dx 2 - a n ( 7 , ho(~))~ + 4 ~ ( h ( a ~ ) ) - 4 o ( ~ , h(a .~ ) ) )a v -

-a-zo (7, h ( a ~ ) ) u ( i / 2 , ho(y))a~ + a~o (y, ho ( r ) )u ( I /2 , ho(y))2a~

(we denote by u(1/2, ho(~)) the value of the eigenfunction u corresponding to 0 v, ~(v) taken at y = 1/2). Now, observing that the family (0 v, ho(v)) satisfies (H) with h = ho ~ , propositions 8.1, 8.2 and 8.3 ensure that

lim u(1/2, h o ( r ) ) = u ~ ( 1 / 2 , ho~), y----) + ~r

lira a~j(y, ho(~ , ) )=a iT(h~) , 7,---* + ~o

for ( i , j ) = (1, 1),(3, 0). Hence the bifurcation diagram of equation (65) for /~ < 0 is jus t a perturbed version of figure 15 as ~ --) + ~ and the small sol- utions of (64) are limits in L ~ (R) of solutions of (65). Consequently, we still have a convergence of small interfacial waves at leading order for h ~ h0 ~ and 4--450(h) (4 < ~ ( h ) ) .

However, the results of section 7 show that for/1 > 0 the front solutions of (36) are given at leading order by the front solutions of the normal form of order 4. Indeed, the coefficients a4o and ae~ appear at first order in the equation of /'~. Hence the convergence of front interracial waves at leading order for /~ > 0 is not clear since we have no convergence result for a4o and a21 as ~---~ + ~ (computations are too lengthy to check

110 GUILLAUME JAMES

wether or not the p rope r ty we proved on a n , a2o, a30 is t rue also for

a4o and a21).

Appendix A. Linear operator.

We establish in this appendix some e lementa ry propert ies of the opera tor L;. o. In the first part , we prove the results of section 3 concerning the Fred- holm alternative for solving L~oU = B and the resolvent es t imate (14) is ob- tained in second part .

Le t U = (ul , u2) e D , B = (bl, be) e X and consider the equation L;.oU = B which leads to:

J U~. : bl, (qu{) ' - 2 o ~ ' U , = - Qb2,

ul(O) = 0 ,

u l (1 ) = 0 .

One observes that the derivative ~ ' of the normalized eigenvector of equation (10) doesn ' t vanish nei ther at y = 0 nor at y = 1, because (~, of ' ) would be in tha t case solution of an e lementary Cauchy problem that would imply q~ = 0 in [0, 1]. Moreover q ( O ) = 1 and q ( 1 ) = q ~ O , which allows us to define (peeC~ 1] by:

(66)

1 qJ2(O) - - - ,

q~'(O)

Y I 1 cf2(y) = of(y) - - d s

h ~q)2 '

1 qJ2(1) -

qcf ' (1) '

y e ( 0 , 1),

where h e (0, 1 ). Clearly ~ 2 verifies (Qcf ~)' - ;t o Q' ~v 2 = O in (0, 1 ), and ~vtf ~ - - r r 2 = 1/~). Thus we can express u, in t e rms of of, ~ 2 and b2 using a variation of constants method. E lemen ta ry calculations then lead to the part icular solution:

(67)

Y 1

u l ( y ) = q)(y) ]~b2cf2ds + q~2(Y) f Qb2cfds 0 Y

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 111

or equivalently:

1

Ul (y ) ---- f K(y, s) Qb2ds ,

o

where K is the symmetric Green's kernel:

K ( y , s) = I of(y) qs2(s),

t q~2(Y) q~(s),

if 0~<s~<y ,

if y < ~ s < ~ l .

Clearly u~ ( 1 ) = 0 and additionnaly Ul (0) = 0 when:

1

roof b2 dy = 0 0

which completes the proof. We now prove that the resolvent estimate (14) holds. I f I w I > 0, iw belongs

to the resolvent set of L~.o and for all B e W the problem (icoI - Lao) U = B has a unique solution U e D . Setting B = (0, b2) and U = (u, , u2) leads to:

u2 = iwul ,

(Ou; )' - ;~oQ ' ul - co2 Qu~

ul(O) = O,

u: (1) = 0 .

= ~ob2,

Multiplying the second equation by ul and integrating from 0 to 1, one finds after a simple integration by parts:

1 1

fQlul'l dy+ .ofO 0 0

1 1

' lUl 12dy + (2)2f~)lUl [ 2 d y = - f Q b 2 u l d y . 0 0

Using 0 < q <- 0 ~< 1, - m ~< 0 ' < 0 and HSlder's inequality, one obtains easily:

1 1

q f l Ul' ] 2 d y -{- (go9 2 - ~ o m ) f l ul 12dy ~ 1152 IlL 2 ]IUl IlL 2 0 0

112 G U I L L A U M E J A M E S

and then for I w] sufficiently large the following holds:

9

2 7

ql~ol

2

qlo~l

Then clearly IIU][x ~< (4/ql~o I ) l~l~ for [~ I sufficiently large and consequently

4 II( iw - L~ o) - 111~_~ x ~< - -

ql~ol

Appendix B. Normal form coefficients.

We prove in this appendix some results concerning the coefficients ar and a3o. The expressions of these coefficients given by the solvability conditions are transformed into much simpler forms (equations (22) and (24)). These new expressions have the advantage of being integrals of terms that stay in L 2 (0, 1 ) as r becomes discontinuous at some h c (0, 1 ) (see section 8), allow- ing theoretical investigations as well as precise numerical calculations. We ad- ditionnaly give a simple criterion (lemma 4.1) for having a20 < 0.

Equation (21) f'LrSt leads to:

(68) { ( Q w ' ) ' - ~oQ'W = g ,

w(0) = 0,

w(1) = 0,

where w is the first component of ~ and

o (o) (69) g = - a e o ~ ) ~ - __~_r cf 'cf+~0~) ~ cp 2.

We then prove the following lemma that we will use several times throughout this appendix:

LEMMA B.11. Let v ~ C2[0, 1] satis fying:

(70) (~)v')' - )[ o r v = 0.

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 113

Then:

Y Y Y

f g v d s = 3 a~ofo~v~+ R2(y), -~ f ocp'2v' d~ - RI(y) + 0 0 0

with:

R l = ) . o Q ' q ? 2 v - o ' q?' c f v - Q c f ' q )v ' , R2= - 2 q~'2v + lq~'2(O) v(O).

PROOF. According to (69) one has:

(71) Y Y Y Y ()' - -~ O ' q ) '2 v ds - 0 --~ q~v ds +

0 0 0 0

Y

5 ' ) ' +2o f o (--~- cP 2vds. 0

By integrating by parts, using (10) and performing a second integration by parts, one finds:

(72) Y

2o 0 --~ ~ 2 v d s = II + I2 + I3 + Rl + 4Re, 0

where

Y Y

0 0

Y Y

I2 = 2 f Qq~" q~' vds + 3 f Qq)'2v' ds , 0 0

f I~= f o ~ - ~' q~v~ + O' ~'2vds . 0 0

Y

Observing that I1 = f(ov') 'cp'q~ ~ , using successively (70) and (10) and inte- o

grating by parts yields then to:

Y Y

11 = - f o~" ~' vr - f ~ a8 - 2R2. 0 0

114 GUILLAUME JAMES

Inserting (72) into (71) gives consequently:

Y Y

fgvds:I4-a~ofQ~vds§247 0 0

where

Y Y Y

lf~,2v f f I4---- ~ Q ' d s + 2 Oq~'2v'ds+ oq~"cp'vds. 0 0 0

A simple integration by parts then leads to:

Y Y

s[ 1 ] i s ( cp " cp ' ) Ov + -~ O ' cp '2 v ds = - R2 - -~ Qcf '2 v ' ds ,

0 0

which completes the proof. Equation (22) is immediatly proved using lemma B.1. Indeed, taking v = q~

1

and using the solvability condition fgq~ dy = O, one finds: 0

1

3 J Oq~'Sdy Cleo- -

2 1 foq~edy

0 1

In order to study the sign of forp "~dy, one transforms this integral using inte- o

gration by parts and equation (10) again. One obtains by this way:

1 1 1

1 f ( rp ,2) ,oq)dy 20 f o,, ~ady_ 0r ''~ ey = T g 0 0 0

which leads easily to:

1 1 1

if ,~ Xof if ~,~ Ocz dY = T O" cp'q dy + -~ O' dy , 0 0 0

1

with 0 ' <0, q~>0 and ~.o~>0. Then if 0 " 4 0 one has f ~ q ~ ' a d y < O and thus lemma 4.1 is proved, o

We now establish the expression of O~ao given in equation (24). Let g-- 1 c~of~q~2dy; the solvability condition (23) reads:

0

(73) fi = I5 + I6 + I7,

SMALl . A M P L I T U D E STEADY I N T E R N A L WAVES ETC. 1 1 5

where

1 1 ~( ( ) ) (~) , o ' ' ~oT Is = - 0 q~' + O ~ q~ q~w' dy + 2,~o cf2wdy , 0 0

1 1 (~,)' ,~:~o v ~,~w~-~o~of~w~, 0 0

1 1

I v = - - ~ 0 7 c f ' eq~2dY+~ 0 --~- cf3('~oCf-cP')dY �9 0 0

We now successively use the following relations:

1

2 0 (f3(20~ cf ) d y 0

1 1

,~ f ( ) =-2~to O ' c f 3 d Y + - ~ ~ - ~ cp2dY - P 0 0

1 1 1

I (~ ~- I ' - 0 c f 2w 'dy = ~ cf2(Ow')'dY +2 0 q~'qnv d y , 0 0 0

1 o ' ) ' 2~of~(~ cf'Zwdy =

0

1 1 1 (~,)' 0 0 0

1 1 1 1

f p, , Q Q -~- c f w d y = - 2 o c f2wdy - Q ' c f ' ~ w d y - f o ' c p ' c f w ' d y -

0 0 0 0

I 1 1 1

- 2 f Qcf" cfw' dy = 2 f gqs' c fdy + f Q' c f '2wdy + f Qcf'2w' dy , 0 0 0 0

116 G U I L L A U M E J A M E S

which, using (68), leads to:

(74) 1 1

5 = 3 f Qq~'2w' d y - 2auo f Qq~wdy + R3, o o

where

(75)

1 fQ' R3 = - - r +

~o 0

1 1 1 f(~ +2 gq)' rpdy + ~o -~- q)'2rp2dy-22o ~) cp' q~3dy . 0 0 0

In order to simplify Ra, one inserts equation (69) in (75) and obtains after some calculations:

(76) 1 1 1 1

5 = 3 fQ '2w'dy-2 of wdy+ fQ' '3 dy+a ofQ ' 2dy 0 0 0 0

which is much simpler than equation (73). However, some difficulties arise from this equation when ~) becomes discontinuous at y = h (one can observe numerically that this discontinuity propagates to w). In fact, we show in what follows that the singular terms cancel in this case and give an expression of 5 still valid for a discontinuous density (see section 8).

For eliminating w from (76), we use (68) and the results of appendix A. One has:

(77) Y Y

w(y) = - of(y) f gcf2ds + q~2(Y) f gcfds + a ~ c f 0 0

where cf2, g are respectively given in equations (66), (69) and a 2oo is deter- mined by a solvability condition in section 4. Besides, equation (76) shows that the value of a 20o has no influence on 5. Taking successively v = cf and v = ~ 2 in lemma B.1 leads to:

(78) w(y)= ~cf ' (0 )+a2oo c f + ~ ' ~ + I s + I g ,

SMALl, AMPLITUDE STEADY INTERNAL WAVES ETC. 117

with:

( ; ; 3 Is = - q~ 5o Oqg"zcf~ds - aeOo Qcfcf2ds

Io = q: 2 Qcf 'ads - Oq~2 ds .

One then report (78) into (76), obtaining a complicated formula for 5. Elemen- tary integrations by parts then allow to reduce the number of terms and one obtains at this step:

(79) 1 1

=sf f - -~ Qcf'4 d y - a 2 o Ocf' cf2 dy + Ilo + Ill, 0 0

where:

I[ i ] fl[ i ] I~o = 9 f oq)'2q)~ Qrp'ads d y - 6aeo oq~'2rp oq)ecl~ gy , 0 0 0

111 -- - 6aeoJ 0cf~ Qcf 'ads dy + 4a2~~ oq~q~ Ocfe ds dy .

Using equation (66), (79) can be written:

1 1 1

f f 2 Oq)'4dy a20 Ocf '~f2dy+3 q~'2Idy+Ii2, 0 0 0

with

(80)

I(0) = 3q~iy(O) i , Y

I(y) t)(3 cf 'a - 2a2o cf 2) ds, cf )o

I(1) 3qr 2,

ye (0, 1),

sl[ ] ~o(3cf ' a - 2a2or r ds r0(3 ~ ,a_ 2aeocf2) ds dy . 112 = 0 J QCfl20J

118 GUILLAUME JAMES

Observing that

~)(3~ ,a

Y Y

h QCf2o

12 ) ' 1 I z ,3 _ 2aeoCZ2) ds = q~q~2 --~ 2 Q

one finds:

1 1 ] I2

= - d _ y 112 ~

0

and at length equation (24) is proved.

Acknowledgments. I wish to thank G. Iooss, K. Kirchg~issner and F. Dias for fruitful discussions.

R E F E R E N C E S

[1] C. J. AMICK, Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids, Ann. Scuola Norm. Sup. Pisa S6r. (4), 11 (1984), pp. 441-499.

[2] C. J. AMICK - R. E. L. TURNER, A global theory of internal solitary waves in two fluid systems, Trans. AMS, 298 (1986), pp. 431-481.

[3] C. J. AMICK - R. E. L. TURNER, Small internal waves in two fluid systems, Arch. Rat. Mech. Anal., 198 (1989), pp. 111-139.

[4] C. J. AMICK - R. E. L. TURNER, Center manifolds in equations from hydrodynam- ics, Nonlinear Differential Equations and Applications, I (1994), pp. 47-90.

[5] T. B. BENJAMIN, A unified theory of conjugate flows, Philos. Trans. Roy. Soc. Lon- don, A 269 (1971), pp. 587-643.

[6] J. BONA - D. K. BOSE - R. E. L. TURNER, Finite amplitude steady waves in strati- fied fluid, J. Math. Pures Appl., 62 (1983), pp. 389-439.

[7] R. COURANT - D. HILBERT, Methods of mathematical physics, Scientific Publishers, Vol. 1 (1966).

[8] M. L. DUBREIL-JACOTIN, Sur les thdorOmes d'existence relatifs aux ondes perma- nentes pdriodiques & deux dimensions dans les liquides hdtdrog~nes, J. Math. Pures Appl. (9), 19 (1937), pp. 43-67.

[9] G. Iooss - M. ADELMEYER, Topics in bifurcation theory and applications, Ad- vanced Series in Nonlinear Dynamics, World Scientific, Vol. 3 (1991).

[10] G. Iooss - K. KIRCHG')[SSNER, Water waves for small surface tension - an approach via normal form, Proceedings of the Royal Society of Edinburgh, 122 A (1992), pp. 267-299.

[11] T. KATO, Perturbation theory for linear operators, Springer Verlag, 1966. [12] K. KIRCHG)i~SSNER, Wave solutions of reversible systems and applications, Journal

of Differential Equations, 45 (1982), pp. 113-127.

SMALL AMPLITUDE STEADY INTERNAL WAVES ETC. 119

[13] K. KIRCHGti.SSNER - K. LANKERS, Structure of permanent waves in density- stratified media, Meccanica, 28 (1993), pp. 269-276.

[14] P. KIRRMANN, Reduktion nichtlinearer eUiptischer Systeme in Zielendergebieten unter Vervendung Von optimaler Regularitdt in HSlder Rdumen, PhD Thesis, Universit~it Stuttgart, 1991.

[15] O. LAGET - F. DIAS, Numerical computation of capillary-gravity interfacial soli- tary waves, J. Fluid Mech., 349 (1997), pp. 221-251.

[16] T. LEvI-CIVITA, Ddtermination rigoureuse des ondes permanentes d'ampleur finie, Math. Annalen, 93 (1925), pp. 264-314.

[17] R. R. LONG, Some aspects of the flow of stratified fluids, Tellus, 5 (1953), pp. 42-57.

[18] t . MIELKE, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Meth. Appl. Sci., 10 (1988), pp. 51-66.

[19] A. MIELKE, Homoclinic and heteroclinic solutions in two phase flow, Advanced Series in Nonlinear Dynamics, Proc. IUTAM/ISIMM Symposium on Structure and Dynamics of Nonlinear Waves in Fluids, World Scientific, 7 (1995), pp. 353- 362.

[20] C. B. MOLER - G. W. STEWART, An algorithm for generalized matrix eigenvalue problems, SIAM J. Numer. Anal., Vol. 10, No. 2 (1973), pp. 241-256.

[21] S. A. ORSZAG, Accurate solution of the Orr-Sommerfeld stability squation, J. Fluid Mech., Vol. 50, Part 4 (1971), pp. 689-703.

[22] B. TURKINGTON, A. EYDELAND - S. WANG, A computational method for solitary in- ternal waves in a continuously stratified fluid, Studies in Applied Mathematics, 85 (1991), pp. 93-127.

[23] R. E. L. TURNER, Internal waves in fluids with rapidly varying density, Ann. Scuola. Norm. Sup. Pisa C1. Sci., 8 (1981), pp. 513-573.

[24] R. E. L. TURNER - J. M. VANDEN-BROECK, Broadening of interfacial solitary waves, Physics of Fluids, Vol. 31, No. 9 (1988), pp. 2486-2490.

[25] A. VANDERBAUWHEDE - G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported 1 (1992), New Series, C. Jones, U. Kirchgraber, H. Walther eds., Springer Verlag, pp. 125-163.

[26] C. S. YIH, Exact solutions for steady two dimensional flow of a stratified fluid, J. Fluid Mech., 9 (1960), pp. 161-174.

Pervenuto in Redazione il 30 dicembre 1997.