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LONG INTERFACIAL WAVES IN A VORTEX SHEET
E. Barthelemy, J.-P. Germain
To cite this version:
E. Barthelemy, J.-P. Germain. LONG INTERFACIAL WAVES IN A VORTEX SHEET. Journal de
Physique Colloques, 1989, 50 (C3), pp.C3-199-C3-204. �10.1051/jphyscol:1989330�. �jpa-00229471�
LONG INTERFACIAL WAVES IN A VORTEX SHEET
E. BARTHELEMY and J.-P. GERMAIN
Institute de Mécanique de Grenoble, BP. 53X, F-38041 Grenoble Cedex, France
Résumé - En présence de courants de cisaillement les propriétés des ondes non-linéaires à l'interface de deux liquides de densité différente sont sensiblement affectées . Dans le cadre analytique de la théorie dite de l'eau peu profonde, on détermine les célérités critiques des ondes baroclines . Ensuite on calcule l'équation différentielle régissant la dénivellation de l'interface (KdV) . Les coefficients de cette équation dépendent de la stratification , du rapport des hauteurs de couches et de la vitesse du courant . Finalement les divers régimes de propagation pour les ondes d'interface sont obtenus .
Abstract - Interfacial non-linear waves propagating in a two layer medium are qualitatively changed due to the action of a vortex-sheet. In the frame-work of the so-called shallow water theory, linear wave speeds of the barocline modes are determined . The differential equation describing the interface elevation is derived . The coefficients of this equation are functions of the stratification, the depth ratio between the two layers and the magnitude of the shear velocity . Finally the different propagation regimes of the waves are obtained .
I) Introduction
The study of non-linear waves in a Kelvin-Helmholtz or vortex-sheet flow applies to a variety of problems . Amongst them we can quote the understanding of shear flow instabilities as well as the description of ocean internal waves in the presence of mean undisturbed currents . The first studies of interfacial waves in shear flow using linear theory were conducted by Kelvin and Helmholtz whose works are reported in LAMB'S textbook (LAMB (1934) ). They payed particular attention to calculating celerities of linear waves disturbing the interface of two semi-infinite flowing layers of homogeneous density and to the condition of their existence . More recently NAYFEH & SARIC (1972) have calculated the characteristics of the weakly non-linear waves by Stokes expansions to the third order in the same physical context. THORPE (1978) has undertaken experiments on interface instabilities through an analysis of the waves characteristics (celerities, wave shapes and slopes ...) . ART ALE & LEVI (1988) have established the KdV partial differential equation describing the interface of two finite layers of homogeneous density in the presence of shear flow . Their main concern was to explain certain oceanographical features in the Strait of Gibraltar . In the present investigation, the long wave supporting system is studied analytically with the simplification of a two layer stratification of homogeneous density in an infinite channel and of finite depth . The expressions of interfacial progressive waves as well as the celerities are obtained for the shallow water approximation (e = A/d2
; e «1 , A is the wave amplitude and d2 the total depth ) and for the long wave limit ( a = d2 fk ; a « 1 , where X represents a horizontal characteristic wave length ) . We shall give the critical depths of the two layers for wich the KdV balance (e = o2) does not stand anymore for a given shear velocity .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989330
C3-200
JOURNAL DE
PHYSIQUE11) Problem Formulation -
We consider two layers of incompressible fluids and of finite thickness
.
These fluids are inviscid and of homogeneous densities, respectively p l for the lower layer and p2 for the upper layer, filling an infinite flume with a horizontal bottom.
Inorder to have a stable configuration we impose p l > p2 and state p=p2/pI ( p c l )
.
The phenomena we wish to describe are two-dimensional, the x axis coincides with the horizontal channel bottom and the positive y axis is oriented vertically upwardsH*(x,t) represents the free surface vertical coordinate and h*(x,t) that of the interface
.
At rest, dl et d2 refer to'the depth of the lower layer and the total depth of the system respectively and m = (d2-
dl)/dl.
The undisturbed basic state shear flow for wich the constant velocities are respectivelyvl*
and v2* is perturbated by progressive long waves.
The Euler description will be used
.
The velocity potentials cpl*(x,y,t) and cpz*(x,y,t) of each layers ( i = 1,2 indicates the layer dealt with ) satisfy the following equations ,*
(i)*(1) A T = O ,
with the continuity condition of the normal velocity component at the interface ,
(i) '
a p - a h 8 ah.
a?'
at = * ,(2)
a y . . .
a t a x
ax
the kinematic condition expressing the impermeability of the free surface ,
.
(2)'a H * aH
av .
(3)
. .
a t y = H ,ay'
at'ax ax
and for a water proof channel bottom ,
ape)'
(4)
-=o
a t y = O.
a y
The Bernoulli's relation at the free surface is
.
*( 5 ) Q 2 = 0 a t y = H , and at the interface ,
The asterisks denote the dimensional variables and fonctions, to be afterwards nondimensionalised by the following scaling parametres :
length: d2
time :
m)
where g represents the gravity acceleration pressure : p1gd2After the classical distorsion of small parameter o used in shallow water theory ( FRlEDRICHS & HYERS, 1951 ), the equations (1) to (7) in terms of the new variables a = a(x-ct) and
P
= y are expressed in the form :naturally to using such variables .
111) Solution
-
We seek finite amplitude solutions wich are expandable in power series such as :
t i )
eCi) -
2"-i ( i )(15) cp ( a , P ) = - +
C
0 c p ~ , - ~ ( a , P ) where I $ ( ~ ) = V ~ ~ ; 0 n = 1The choice of the parity of the series simplifies the computations considerably. Since the series for the potential function are written in the form of an undisturbed flow and a perturbation, we assume a strong shear velocity, i.e, the mass flux associated to the wave has no effect on this basic state. We admit in addition that the frame of reference is the frame of nil mass flux :
(19) Vl = -mV2
The reductive perturbation method leads to the following results - At order n=l :
( i )
(20) cp = B ,(a) .
The horizontal component of the velocity is uniform on a vertical line The expression of the free surface height is :
(21) dB2
H 2 = - ( V 2 - ~ o ) - , aa and at the interface
.
-
At order n=3 :By treating the dynamic condition, the free surface and the bottom kinematic conditions and finally the kinematic condition at the interface in consecutive order we obtain the fourth degree equation for co ,
In the case of four real solutions , two fast modes (barotropic) and two slow modes (baroclinic) are present. Later on, the two celerities of the baroclinic mode will be written as co+ and co- , such as co+ 2 co-. As soon as the solutions become imaginary for a specific choice of m, p and V2 , the propagating waves are unstable, i.e., progressive solutions cease to exist
.
In Figure 1. neutral stability curves are given for different stratifications
.
For physical parametres located below the curves the propagating waves at the interface are in this case stable.
It is easily verified that for V1 = V2 = 0 we recover the classical longC3-202 JOURNAL DE PHYSIQUE
wave celerities obtained by LAMB (1934) and by KOOP & BUTLER (1981), amonst other. Furthermore when the relative density difference (1-p) is small, the baroclinic celerities either obtained with a free surface or with a rigid lid are identical within a very small error. For oceanographical applications, this leads us to the use of the previous approximation to simplify our derivations
.
Figure 1. Neutral stability curves - At order n=5 :
We are able to determine the first correction term of the celerity given by the identity (18)
.
From now on we will use the rigid lid approximation.
Following the same procedure as for order n=3, we obtainThis ordinary differential equation for h;? is the KdV equation for progressive waves . The different coefficents are defined by
Fig.2b Propagation regimes for interfacial solitary waves as a function of m and V2 (the velocity of the basic undisturbed state) (celerity y)-, (1 - p ) = 0.001)
.
( I ) : A 2 0 ; (2) : A 5 0C3-204 JOURNAL DE PHYSIQUE
Solutions are either periodic ( cno'idal waves ) or aperiodic ( solitary waves ) . Solitary waves take the following expression
where c is the celerity and A the amplitude at the first order such as , (36) c = c 0 + - AM
3 N
It is worth noting that the ratio N L cancels for certain choices of the physical parametres . The only solution physically acceptable is then the basic flow of our problem
.
The roots m of N/L for V2 and p fixed determine possible propagation regimes for solitary waves at the interface ( Figure-2a&b).
In the four zones defined by the curve N/L = 0 in Figure-2a the solitary waves have a fixed sign amplitude. When V2 = 0, the critical depth for wich N/L=O is m = 1, a result already quoted by YAMASAKI & KAKUTANI (1978)REFERENCES :
[I] LAMB H. Hvdrodvnamics Cambridge University Press (1934) [2] NAYFEH A.L. and SARIC W.S., J. Fluid Mech.
55
part2 (1972) 311 [3] THORPE S.A. , J. Fluid Mech. 85 part1 (1978) 7[41 FRIEDRICHS and HYERS, Comm. Pure and Appl. Math, 2 (1951) 571 [S] KOOP C.G. & BUTLER G., J. Fluid Mech.
112
(1981), 225[6] KAKUTANI T. and YAMASAKI N. , J. Physical Soc. of Japan
