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Modulation of an internal gravity wave packet in a stratified shear flow
WAVE MOTION 3 (1981) 81-103 @ NORTH-HOLLAND PUBLISHING
COMPANY
MODULATION OF AN INTERNAL STRATIFIED SHEAR FLOW
GRAVITY
WAVE PACKET IN A
R. GRIMSHAW Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
Received 4 August 1980
Modulations of an internal gravity wave packet in a stratified shear flow are discussed in the weakly nonlinear and weakly dispersive context. It is shown that the modulations are described by a variable coefficient nonlinear Schradinger equation when the modulations are confined to the direction of wave propagation. Transverse modulations couple the nonlinear Schriidinger equation to the mean flow equations. For long waves, it is shown that the modulation equations may be somewhat simplified. An Appendix describes the equations governing long wave resonance.
1. Introduction
For many dynamical systems, the modulation of a wave packet in the direction of wave propagation due to dispersive and weakly nonlinear effects is described by the nonlinear Schrodinger equation [3]. In homogeneous media, when the nonlinear Schrodinger equation has constant coefficients, there are N-soliton solutions and the equation is exactly integrable through the inverse scattering transform technique. For internal gravity waves propagating in a horizontal channel the nonlinear Schrijdinger equation describing a modulated wave packet was derived by Borisenko etal. [4], Grimshaw [5] and Thorpe [16]. In this paper we extend this to consider the evolution equation describing modulated internal gravity waves propagating on a shear flow in a horizontal channel. We shall allow the background shear flow to be weakly inhomogeneous, and we shall also consider the modifications necessary when there are modulations transverse to the direction of wave propagation. Grimshaw [8] has given a general theory for modulated wave packets in inhomogeneous media. For modulations in the direction of wave propagation, the evolution equation for the amplitude is the variable coefficient nonlinear Schrodinger equation aB
ig+A
(1.1)
$+p[~~2~=~.
Here A and k are real-valued functions of s. In inhomogeneous media wave packets propagate along rays, whose spatial direction is defined by the local group velocity, s is a time-like co-ordinate which varies along a ray, and 6 is a co-ordinate which is constant along a ray, but whose spatial projection defines the spatial direction of wave propagation. lBj2 is the wave action flux along the ray. The scaling required to produce equation (1.1) scales s with u-’ where ‘u’ is a small parameter measuring wave amplitude, and 6 scales with a -I; the background medium varies on scales of ae2. The coefficient A describes dispersion about the dominant wavenumber in the wave packet and the coefficient p describes the nonlinear effects due to the interaction of the primary wave with the mean flow and second harmonic. For modulations which are transverse to the direction of wave propagation (1.1) is modified and coupled in a rather complicated way to 81
82
R. Grimshaw / Modulation of waue packet in stratified flow
mean flow equations. The general theory of Grimshaw [8] was for a system of equations without boundary conditions. For the case of internal gravity waves on a shear flow in a horizontal channel, we shall show in this paper how the theory of Grimshaw [8] may be extended to produce an evolution equation (1.1) for modulations in the direction of wave propagation, and the modification of (1.1) when there are transverse modulations. In Section 2 we present the equations of motion for internal gravity waves in a stratified shear flow using the generalized Lagrangian-mean formulation of Andrews and McIntyre [l]. This has the advantage of simplifying considerably the treatment of the free surface and the mean flow equations, and also seems to be generally technically simpler to handle than the more usual Eulerian formulation. At the end of Section 2 the equations are cast into an operator symbolism, so that the comparison with the general theory of Grimshaw [8] can be used. In Section 3 the evolution equation for the amplitude is derived, and in Section 4 the nonlinear terms are evaluated. In Section 5 we present a brief discussion of equations such as (1.1). Of critical importance is the relative signs of A and CL,and we show that for long waves A and p have opposite signs. In the Appendix we derive the modification to (1.1) necessary when there is a long wave resonance (i.e. the group velocity of the wave packet coincides with the phase speed of a free long wave).
2. The generalized Lagrangian-mean
formulation
We propose to present the equations of motion in the generalized Lagrangian mean (GLM) formulation of Andrews and McIntyre [l] as their formulation has conceptual advantages, and, at least for the present problem, simplifies much of the analysis. A derivation of the equations in the GLM formulation for an inviscid, incompressible, stably statified fluid has been given by Grimshaw [7], so we shall only give an outline here. Let xi be Eulerian Cartesian co-ordinates and r be the time; here the Latin index ‘i’ takes the values 1,2 or 3 and the summation convention is used. Let Uibe the velocity components, p be the density and p be the pressure. We shall use non-dimensional variables based on a length scale L (a typical wave-length), a time scale N;‘, where N1 is a typical value of the Brunt-VaisiilH frequency; a velocity scale NIL and a pressure scale plgL, where p1 is a typical value of the density. These scales produce the non-dimensional Boussinesq parameter U, N:Lg-’ , which is small in the Boussinesq approximation. Then if 4 is any field variable we put 4(X:, t) = &xl, t)+4’(xl,
t, e),
(2.1)
where 4’ is periodic in the phase 8 with period 2~ and zero mean, and the phase 8 is itself a function of xl, t. The decomposition (2.1) effectively defines an averaging operator (a) where
J(xl, t)=(c#J(xi, t))=l
2n
2lT I 0 4d0*
(2.2)
It is conceptually useful to envisage 8 being replaced by 8 + I++, and defining the averaging operator with respect to the ensemble label 4, rather than the physical variable 8 (cf. [l 11,or [2]). & is the Eulerian mean of 4, and 4’ the Eulerian perturbation. Next let Xi be generalized Lagrangian co-ordinates, and ti(xi, t) be the particle displacements defined so that Xl =Xi+[i*
(2.3)
R. Grimshaw / Modulation of wave packet in stratified flow
This equation defines a transformation @(X, t) = (4 (Xi+ 69
from xi to xi. We then define a Lagrangian-mean
83
operator by (2.4)
t)>-
In physical terms, I$, the Eulerian mean, is the average over the phase at a fixed place, while BL, the Lagrangian mean, is an average over the phase of the waves following the fluid motion. As shown by Andrews and McIntyre [l] this latter notion is made precise by requiring that (6)
= 09
(2.5)
whence it follows that Xi is a co-ordinate which moves with the Lagrangian mean velocity CL whenever the co-ordinate xl moves with the true velocity Ui. In contrast to (2.1) we put 4
C-G
1) =
6”C-G
t) +
dh
6 @I,
(2.6)
where 4, the Lagrangian perturbation, is periodic in the phase 6 (now a function of xi, t) with period 2~ and zero mean. Assuming that the particle displacements are O(a), where ‘a’ is a small parameter measuring wave amplitude, then it is readily shown that (2.7a) 6%
J+@,
(2.7b)
where (2.7~) 6” is usually called the ‘Stokes correction’. Finally we note the useful result [l]
d4 -
d =,,w, (dt >
(2.8a)
where (2.8b) Here d/dt is the material derivative following the fluid motion. To obtain the equations of motion in the GLM formulation, we first introduce the Jacobian
(2.9)
J = det{axi/axi}.
But then the continuity equation and conservation of mass together imply that J is a mean quantity which satisfies the equations dJ aa!T~;+J-=0, axi
_f=o.
(2.10)
Expanding (2.9) in powers of ei, and substituting into (2.10) it follows that aa” ,=$(f a&/&i
&t&!Cj))
-MO = 0,
+O(a4>,
(2.11a) (2.11b)
84
R. Grimshaw / Modulation of wave packet in stratified flow
(2.11c) The density p is a mean quantity 6‘ (i.e. 6 is zero) and satisfies the equation dp‘/dt
(2.12)
= 0.
The equation of motion is (2.13) where Sij is the Kronecker delta. Before separating this equation into its mean and perturbed multiply it by ax i/ax, and make the substitution
parts, we (2.14)
P=~‘+~P’+B(-~‘d~~/dt-~‘Si~),
In spite of the notation, up’ differs from the Eulerian perturbation pressure by terms of O(a’). We find that the equation of motion for the Lagrangian mean flow is (cf. [7]) dii; I 1 apL 1 ;L -+-Si3 dt ap axi u
(2.15a)
=
where Pi = -(i;j &$/t3xi),
(2.15b)
aj = dtj/d t.
(2.1%)
and
The equation of the motion for the perturbed
flow is
_Ld2&ap’ q --5;$(pL~~ a/T‘ +M’=
’ Z+iGj-L
dxi
,
0,
(2.16a)
where (2.16b) Here n = 63 is the vertical particle displacement. We shall suppose that the fluid is confined in a horizontal waveguide which is bounded above by a free surface Z’ = &/&XL, t) and below by the rigid boundary z’ = -h(x&, t). Here, and subsequently, we distinguish the vertical co-ordinate so that z’ = xi, z =x3 and w = ~3, and employ Greek indices for horizontal variables XL ((Y= 1,2), x, and u,. In the GLM formulation the free surface is a Lagrangian mean quantity, z = f‘(x,, t), and the Lagrangian perturbation, i, is identically zero. The boundary conditions at the free surface are
a.f"
_L ar” at+u_dx=GL
onz=5‘,
(2.17a)
OL FL=0
onz=r,
(2.17b)
R. Grimshaw / Modulation of waue packet in stratified flow
85
and UP’ -pLn -pL& dak/dt = 0
on z = [“.
(2.17~)
The last two equations follow from the condition that the pressure, given by (2.14), should vanish on the free surface. At the bottom boundary, the conditions are I?rL+&&r/&+*.*=O
onz=-h(x,),
(2.18a)
and =0
q+&ah/aXp+---
onz=-h(x,),
(2.18b)
where the omitted terms are 0(p2a2), and /3 is a small parameter which measures the gradients ah/%x,. In the mean flow equations (2.11a), (2.12) and (2.15a) the forcing terms due to the waves are O(a2), and so we decompose the mean flow into an 0( 1) basic flow, and an 0(a2) term. Further, we shall suppose that the basic flow varies on long horizontal length and time scales relative to the wave field, and so we put tif = uoi(Xh,T',Z)
+u~E:~,
(2.19a)
FL = p,,(X:, T’, z) + c~a2&,
(2.19b)
,jL = &XL,
T’, z) + au ‘&,
(2.19c)
~=&-,(X&,
T’, z)+a25;L,
(2.19d)
X& =&.a,
T’ = pt:
(2.19e)
where
Here tiki etc. are the wave induced O(u*) terms, and p is a small parameter which measures the length and time scales of the wave field relative to the basic flow. We shall also suppose that h = h(Xk). Since the horizontal scale is O(p-‘) relative to the vertical scale, it is convenient to redefine w. by flwo. The equations for the basic flow are then auop/ax:, + aw,/a2 = 0,
(2.20a)
Dpo/DT’ = 0,
(2.20b)
p. Duo,/DT’+
(2.2Oc)
l/~r ap,/aXh_ = 0,
p2po Dwo/DT’+
l/a apo/az + l/~rp~ = 0,
D/DT’=
uon
(2.20d)
where a/aT’+
a/ax:,
+
w.
a/az.
(2.20e)
The boundary conditions are w. + uoa
ah/ax& =0
alo/aT’ +
uop
onz=-h(XL)
afb/ax& = w.
on 2 = Lo(X&),
(2.21a) (2.21b)
and p0=0
on2 =lo(X&).
(2.21c)
R. Grimshaw / Modulation of wave packet in stratified flow
86
The equations for the perturbations & and p’ are (2.1 lb) and (2.16a), while the boundary conditions are (2.17~) and (2.18b). For the purposes of the next section it is convenient to recast these equations into the following matrix operator form. Let u be the 4-vector.
[I P’
fJ=
(2.22)
ZP,
77
where since (Y= 1,2, the second entry is a short-hand notation for two terms. Then (2.1 lb) and (2.16a) may be written in the form (2.23)
for-h
Lv+M=O
where L is the 4 x4-matrix
operator,
(2.24a) D2 poN2 + PO Dt2
0
1
where
L’ =
0 b
0
0
0
---1 ape
a $+&%DT’ax I (ZPOWO~ 1 ape --_ ( Duo7
U
Y
0
2PoWojg
u ax&
D/Dt = a/at +
uool
;+
ax&
Duo, PODT’
9
&-
(2.24b)
a cl
(2.24~)
a/ax,,
and (2.24d)
aN2 = - 1/p,, apo/a2. Here N is the Brunt-Vlislki
M=
frequency. The nonlinear term A4 is given by
-+2poi&3--+f
D a&
-+2poii:,---~dt+. D 877
Dt a+
(2.25)
**
Dt ax0 a&
**
where MO is defined by (2.11c), and M’ is defined by (2.16b). In the second term in M, only those terms which contribute to the subsequent analysis are displayed. The boundary conditions may be written in the form Bv+Mg=O
forz=lOand-h,
(2.26a)
R. Grimshaw / Modulation of waue packet in stratified fZow
87
where for z = &,,
-d&l
)[
Ba .-f- d;T’,xhJ =
(at’ax,‘az
0
0
1
0 1
0 0 +pB’+o(p2), 0 0
(2.26b)
(2.26~) and for z =-h,
=
0 ah/ax:, 0 B’=O
[
0
0
1
00.
0 0 -0 -0 00
1
0 +~B’+O(p2), 01
(2.26d)
(2.26e)
The nonlinear term MB is given by
(2.27)
while for .z = -h, MB is O(p2a2) and does not feature in the subsequent analysis. The perturbation equations thus consist of (2.23), with the boundary conditions (2.26a). Schematically they may be combined into the single equation (2.28a) where pR
=
(2.28b)
a/axR,
and xo=t,
XL = T’.
(2.28~)
Here 9 is an operator incorporating L and the boundary operators B, while A is the nonlinear term. Here, and in the next section we find it convenient to introduce the capital Latin index ‘I?’ which takes the values 0, 1 and 2; this is a device which enables us to incorporate the time variables and the horizontal space variables into the same notation in situations where it is not necessary to make a distinction.
88
R. Grimshaw
/ Modulation
of wave packet in stratified flow
3. Modulated waves We propose to discuss a modulated wave packet where the modulation occurs on scales long compared to the waves, and is described by co-ordinates
x, = EX,,
(3.1)
T=et,
where E is a small parameter. For a strongly inhomogeneous medium the modulation scale is identical with the scale of the basic flow and we put p = E (i.e. XR and Xi are identical); for a weakly inhomogeneous medium we put /3 = E* (cf. [8]). Subsequently we shall consider only this latter case, but we note that the former case may be recovered by the omission of the dispersive terms from the evolution equation for the wave amplitude. For the present we shall retain both parameters. The general procedure for obtaining the evolution equation for the amplitude of a modulated wave packet in a homogeneous medium is well known [3]. For internal gravity waves in the absence of a shear flow, and any other inhomogeneities, the evolution equation has been derived by Borisenko et al. [4], Grimshaw [S] and Thorpe [16]. When the medium is inhomogeneous a general procedure for obtaining the evolution equation has been developed by Grimshaw [8]. Although that procedure was described for a system of equations in the absence of boundary conditions, the modifications necessary to deal with boundary conditions are readily dealt with, and so we shall only give an outline here. Let 0 = f un(XR, Xk ; 2 ; E) exp(in8) + c.c., n=l
(3.2a)
f!+(X2).
(3.2b)
where
Here C.C. denotes the complex conjugate. The phase 13varies rapidly with respect to the background medium. The local frequency w and wavenumber K~ are defined by w = -aOlaT’,
K,
=
(3.3)
@/ax&.
Substituting into (2.28a) it follows that L?? niK,+E-++P+.-j$Xk,z ( R
v,+.AG=o, R
(3.4a)
>
where Jbl
exp(-inO) de.
(3.4b)
We shall further assume that if ‘a’ measures the wave amplitude, then VI is O(a), v2 is O(a*) etc. This assumption excludes such phenomena as second harmonic resonance; similarly the assumption that tiii etc. are O(a*) excludes long wave resonance. However, similar procedures to those described below may be used for these phenomena as well (cf. [5,6]). The equations describing long wave resonance are derived in the Appendix. We shall show later that JU~is 0(a3), and that these nonlinear terms balance dispersion when
R. Grimshaw / Modulation of wave packet in stratified flow
89
a = E. Expanding the operator 9 in E and p it follows that, for n = 1,
Lz+l+&-
apR a&
a23 au, a2th +p-apR 8x2 apR aps axR axs
+ZE
(3Sa)
where zo=z
(* IKR,
(3Sb)
;;xk,z).
Here the derivatives of 2 with respect to PR (i.e. with respect to the variable PR = a/&R) are evaluated at &= 0. To leading order in E and a, u1 is a null vector of _Yc,.We put UI=
UT(KR,
Z;
Xk)A(XR, XL) +O(U),
(3.6a)
where (3.6b)
&jr = 0.
Here r is a null vector of L&,,and A is the complex wave amplitude whose evolution we wish to describe. In order that r be a non-trivial null vector a dispersion relation W=
(3.7)
W(K,;X;p)
must be satisfied. From (3.3) we see that this is a partial differential equation for the phase 0; its solution will be described later in this section. Recalling the definition of the operator 9 in Section 2 (i.e. (2.24a), (2.26b) and (2.26d)), we see that
Pob*2/K2)af/az r= 1 (iK,/K2)af/aZ
[
where
(3.8a)
,
f
-f(&w*2$) +poK2(N2-m*2)f
(3.8b)
(3.8~)
on z = lo, f=O
=o,
(3.8d)
onz=-h,
and W*=W-K&a,
K ’ =
K,Kp.
(3.8e)
Here w* is the Doppler-shifted, or intrinsic, frequency. Eqs. (3.8b, c, d) define an eigenvalue problem for w once K~ has been specified, and the solution is given by the dispersion relation (3 -7) and the eigenfunction f. A sufficient condition that w be real for real K, is that the Richardson number (the minimum of
90
R. Grimshaw / Modulation of wave packet in stratified flow
%,,/c~z)-‘1 be greater than a everywhere, and we shall assume henceforth that this is the case. We shall also assume that there are no critical levels, and so w* does not vanish for any z in the flow domain. In general there will be a number (possibly infinite) of eigenvalues, each defining a wave mode. Henceforth we shall confine attention to just one such mode. In the first instance the null vector r is only defined when the dispersion relation (3.7) is satisfied. However we shall extend the definition of r to all KR by assuming that r is defined by (3.8a) where the eigenfunction f satisfies the boundary conditions for all KR, but only satisfies the differential equation (3.8b) when the dispersion relation (3.7) is satisfied. Much of the subsequent discussion depends on the properties of the operator L& For two 4-vectors ZIand w, we define the inner product N’K’(K,
{w, ~clv}= ILo w+Lov dz +[~+&a]~%,
(3.9)
-h
where the superscript “’ denotes the complex conjugate transpose of the 4-vector w, and Lo is L(iK& a/&z ; Xk, z), with a similar definition for &. Now Z0 is a self-adjoint operator and it may be shown that (3.10)
1% %u) = L%w, VI. The method of solution of (3.5a) leads to a sequence of equations of the form
(3.11)
L&u+g=o. A necessary and sufficient condition that this inhomogeneous
equation have a solution is (3.12)
jr, g) = 0.
The necessity follows immediately from (3.10), while the sufficiency may be established by converting (3.11) to an inhomogeneous form of (3.8b, c, d) and employing the method of variation of parameters. It is useful to define ~(KR,
XZ) = {r,=Y04.
(3.13)
Clearly 9 = 0 is equivalent to the dispersion relation (3.7). LSmay be eval uated from (3.8a, b, c, d) and we find that (3.14) Next we put o1 = arA + am:‘) ,
(3.15)
and substituting this expression into (3.5a) it follows that (3.16) The compatibility . a9
condition (3.12) now shows that
aA = O(E*, p, a*),
‘&ii& a&
(3.17a)
R. Grimshaw / Modulation of wave packet in stratified frow
91
or ie-,“F( g+ .O
v$
a1
(3.17b)
= O(Z, p, a2).
where (3.17c)
v, = a w/aK,.
Here, in deriving (3.17a), we have differentiated (3.13) with respect to KR. Eq. (3.17b) is obtained by differentiating the dispersion relation 9 = 0, while V, is the group velocity; this equation states that, at the leading order in E, modulations in A propagate at the group velocity. Further it may be shown that [8] EVl
(*) =
-Ei
s -$$+O(c2, p,2). R
(3.18)
R
For the higher order terms a similar procedure is used. At each stage, the compatibility condition (3.12) is applied. At the second order in E, we find that
+p{r,
The terms proportional aE*
_Y’r}A +{r, d~}+O(m3)
= 0.
(3.19)
to p may be recast into a simpler form, so that (3.19) becomes
a29
a*A
+{r,&}+O(ae3)=0.
(3.20)
The manipulations leading to (3.20) are similar to those described by Grimshaw [8], where we note that since .Yo is self-adjoint, and we are dealing with an inviscid fluid, the terms proportional to p describe the conservation of wave action. This latter conclusion also follows from [7], where the wave action equation was derived for a strongly inhomogeneous medium. Eq. (3.20) is the required evolution equation for the amplitude A, and in the next section, we shall evaluate the nonlinear term {r, Yaci}which is 0(a3). The first term in (3.20) describes the effects of dispersion about the dominant wavenumber, while the second term describes the evolution of the amplitude on the long time scale p-‘* , a balance between these two terms and the nonlinear term requires a = E and /3 = E*. In the remainder of this section we shall describe the form the evolution equation takes in ray co-ordinates [8]. These are obtained by solving the dispersion relation (3.7) for the phase 0, using the method of characteristics, or rays. Let s be a parameter along a ray. Then the ray equations are dT’/ds = 1,
dXh/ds = V,,
do/ds = aW/aT’,
dK,/ds = -a w/ax:.
(3.21a) (3.21b)
These are ordinary differential equations, which are to be solved subject to the initial conditions Xk =XZo([&)
for s =0
(3.22a)
and 0 = @,(&)
for s = 0.
(3.22b)
92
R. Grimshaw
/ Modulation
of wave packet in stratified j-low
Eq. (3.22a) describes an initial manifold on which [& are interior co-ordinates, while (3.22b) determines the initial phase on this manifold. We also assume that the dispersion relation (3.7) holds on the initial manifold. With these initial conditions, the solution of the ray equations is xlp = xlp (s, 5:).
(3.23)
This equation describes a transformation 9 = det[aXk/@$],
of co-ordinates
from (3.23) Xk to (s, 5;). Let
where .$ = s.
(3.24)
Then it can be shown that (3.20) becomes (3.25) where here, and henceforth, we shall omit the error terms. The entity 4 dCB/ao $A2 is proportional wave action flux along a ray tube. We define a new amplitude B by
B2 = -!$&42, and new co-ordinates
to the
(3.26)
by (3.27)
where e&(X&) is determined from (3.23). We then express (3.25) in terms of B and the co-ordinates &. The result is aB
iz+A,,
-_e+~_&~“’ yLo, at2 6
s and
(3.28a)
where
A
1 -A-a*w
=@= 2
aK,
aK,
at’ ax:
a& ax;
(3.28b) ’
In deriving the expression (3.28b) for A,@, use is made of (3.17b), and the expressions obtained by differentiating the dispersion relation 9 = 0 with respect to KR. Various specific cases for the ray co-ordinates are described by Grimshaw [8]. Here we shall just present one case when the basic flow is homogeneous, and so W (3.7) does not depend on Xk. In this case w and K, are constants, V, is a constant and the ray co-ordinates are s
=T’E~‘~
& = X, - V,T = E(x, - Vat).
3
Also, the phase 13is --of +
=--
A nR
1 2
a*w aK,
aKB
K,x,,
(3.29)
and, (3.30)
*
93
R. Grimshaw / Modulation of wave packet in stratified flow
4. The nonlinear terms As a prelude to the calculation of Ai (3.5a), we must first calculate the second harmonic 02 and the mean flow Gki etc. (2.19a, etc.). We consider the second harmonic u2 first, which satisfies the equation L?(2iKR, a/6)2; Xk, z)V2+&
= O(ca2).
(4.1)
If there is no second harmonic resonance (i.e. W(~K,; XX) # 2 W( KR ; XL)), then the linear operator LZ?(2iKR, a/&z ; Xk, z) in (4.1) is non-singular, and there is a unique solution to (4.1). The forcing term & is found from M (2.25) and MB (2.27); there is no contribution to MB at O(a*>, and M2 is given by
(4.2)
M2 = a2A2
Next we eliminate &2 and p; from (4.1) and obtain a single equation for 772,correct to O(a2), 772= a2A2f2,
(4.3a)
where (4.3b) and
1
d2 af
Po~f-$-&Pow
1
a
l
*2 ( ( af)2_f2). ) ;Tz g
(4.3c)
The boundary conditions are extracted in a similar manner. (4.4a) and f2 = 0
in z = -h.
(4.4b)
Eq. (4.3b) and the boundary conditions (4.4a, b) determine f2 uniquely. This determination of the second harmonic agrees with [17] in the Boussinesq approximation (a + 0). Next we shall calculate the contribution to Jtl due to the quadratic interaction of the second harmonic u2 with the primary harmonic ul; this contribution will be denoted by &i*’ . Before proceeding we note that it can be shown that there is no contribution at O(a3) due to cubic interactions at 0(a3). In calculating Ac$ we find that there is no contribution from MB at O(a3), and M’:’ is given by 2af2
G
Mi2’
= a3JA12A
a’f af -g+2f’g+Tf
1
iPo~*2K.(2f2f1+f~($-I))
Po,*2
(
-4f2
~_~f~)_~~(~_~)_~$(~_3
a’f, 1 a*2 --5fa,-a,’
aI af
,
(4.5a)
94
R. Grimshnw / Modulation of wave packet in stratified flow
where
( >2_f-g
I=
g
(4Sb)
The final step is the evaluation of the nonlinear term {r, A/u’:‘}in the evolution equation (3.20). We find that {r, A:“‘} = u3v21Aj2A,
(4.6a)
where
v24; w*2[ 2~I-212+-f(f2(($)2-~2f2))] POT
dr.
(4.6b)
Thus the result of the quadratic interaction of the second harmonic with the primary harmonic is a cubic nonlinearity in the amplitude A, which, as anticipated, is 0(a3). The mean flow equations are (2.11a), (2.12) and (2.15a), with the boundary conditions (2.17a, b) and (2.18a). In these equations the forcing terms Pi (2.15b) etc. are evaluated from v1 to 0(a2); substitution of (2.19) into the mean flow equations leads to equations for the wave induced quantities .Qkietc. The details are given by Grimshaw [7], and the result is, correct to O(ca’) (4.7a) D& _ L aUOp -) +s=&]Al’( PO DT+W2 a* m
(-
+&‘( a
aa
2p,~,w*( f2+f(92)] p,,(~V~-w*‘~f~-~(~)~],
_L
a~+~2
=
IAl’(
(4.7b)
(4.7c)
~(2p0(NZ-w*~)f~)-~(poN’)f’],
&& - po&N2 =0.
(4.7d)
where D/DT
=
a/aT + uoa a/aXcll.
(4.7e)
The boundary conditions are Dfi/DT
= K$
s;” -3j2=
(4.8a)
on z = lo,
IAl'{ ~r~+,~(2,*~-N~)$](~)~
on z = lo,
and fik=O
onz=-h.
(4.8~)
Here we have also used the fact that the basic flow depends only on z and the variables Xk, and to the order considered, derivatives with respect to XL do not appear in (4.7) or (4.8). These equations can be reduced
95
R. Grimshaw / Modulation of wave packet in stratified flow
to a single equation if we put C$ = D+/DT,
(4.9)
where fz can be interpreted as a mean vertical displacement. Then it follows from (4.7d) that & = &V*~Z, and (4.8a) shows that fk = jj2 at z = lo. Elimination of i& and p2 then leads to the equation
a G
D2 afj2 2a2fj2 -+pcJv x=~[ “DT2 az )
a
D2 ~\A~o$-K,$
-$A[‘( n
&1A122po-*(f2++432))
$-~oo*~J(
(4.10)
f2+$(-$)2)].
With respect to the variables XR, modulations in A propagate at the group velocity (3.17b), and so we seek a solution of (4.10a) (or (4.7)) with the same property. Thus in these equations we replace a/aT with - V, a/aXa, where it follows that D/DT = - Vg a/ax=,
where Vx = V, - uop.
(4.11)
‘Thus Vp* is the group velocity relative to the basic flow. The boundary conditions are (4.8~) which becomes jj2 = 0 at z = -h, and (4.8b). This latter condition involves the mean pressure p2 which must be determined from (4.7c). We put poN2ii2-IA/2(~(2~o(NZ-~
@2=
*2)f2)-;(pti2)f2)]
dz
where Q (independent of z) is the surface mean pressure. Then substitution differentiation with (4.7a) leads to the following equation for 0:
a2Q x=
D2 afj, D2 2AJZpoaZfZ+K & -I p°F2 Z-DT a2 +
a21A12 ax”,
po(N2-W*2)f2-pOII)-i-
&IA122p,o*(
(4.12)
+a(xR),
into (4.7b) and cross-
f2+$(j92)
*2
(4.13)
K
Finally the boundary condition (4.8b) becomes 772=
~+v/A/~(
~+u~(~u*~-N~)$)
(9’
on z = lo.
(4.14)
In general, (4.10) may be solved with a Fourier transform in X, [7]. However, it is convenient to first make the transformation ii2 =
(4+af2/adlA12+&
Q=[poV*2$
(4.15a) -%‘ow*KV*(
f2++($-)2)]
zzcJA12f
6
(4.15b) and (4.15c)
R. Grimshaw / Modulation of wave packet in stratified f7ow
96
Substitution into (4.10), (4.13) and (4.14) then shows that $ satisfies the equations
-
2poN2f-
af
- -
a (pow
a.2 a.2
*2)(
+L f2
(4.16a)
3’ K2
( az ))
9
(4.16b)
on z = lo, $=O
(4.16~)
onz=-h,
where v* =
K,
v,*/
(4.16d)
K,
while 6 is given by
(4.17a) ~$-ad/p~=O
$=O
(4.17b)
ont=[O,
(4.17c)
onz=-h,
and 6 is given by
(4.18) Here V” is the component of the relative group velocity Vx in the direction of K,, and so 4 represents the mean flow induced by modulations parallel to the direction of wave propagation, while 4 and 6 represent the response to modulations transverse to the direction of wave propagation. Note that sl, is completely given by (4.16) independently of [AI, and can be found uniquely if there is no long wave resonance (i.e. Volt, # KW~, where W. is the phase speed of a free long wave). $ and 6 must in general be found using Fourier transforms, and again can be found uniquely if there is no long wave resonance. Finally 2, is found from (4.7b) after substituting (4.12) and (4.15~) into (4.7b). Next we calculate the contribution to JGlrdue to the interaction of the primary harmonic zll with the wave induced mean flow. Denoting this contribution by My), we find that {r,Jti”‘)=.‘a[l_(:(
2p,N2f-$2+(-$(~)2+f2)(2pow*K,fikc~poN20*2~2))
dz (4.19)
97
R. Grimshaw / Modulation of wave packet in stratified flow
Using the decomposition
(4.15), this may be written in the form
{r,A\“‘} = c~~vo(A/~A + a3Ag,
(4.20a)
where
and s^ = If,” (2,0Ar~&+(-$($)~+p)(
~P,,o*K,&, +$A,w**)$)]
dr (4.20~)
Note that s^ depends only on 4 and ii,, and so responds only to modulations in amplitude transverse to the direction of wave propagation. The total nonlinear term is the sum of (4.6a) and (4.20a).
5. Discussion
The evolution equation for the amplitude is (3.28a) with nonlinear terms given by (4.6a) and (4.20a). Putting a = E, this equation is the nonlinear Schrodinger equation (5.la) where .=(vo+.2,/(393.
Here s^ is given by (4.20~) and responds only to modulations in amplitude transverse to the direction of wave propagation, through (4.17) and (4.18). The coefficients holPand Ydepend on s and & ; since t& = ES,, the dependence on s predominates. When s^ is zero, and APBand Y are constants, (5.la) has N-soliton solutions and is exactly integrable through the inverse scattering transform technique [18]. However for the variable coefficient nonlinear Schrodinger equation, there is no general method of solution unless there is a transformation which converts (5.la) into an equation with constant coefficients [8]. Perturbation methods must be used such as those which describe the slowly varying solitary wave [9,12]. However, (5. lb) has a ‘plane wave’ solution in which B is a function of s alone, and s^ is zero. It is given by B=Roexp[iR~[o’vds],
(5.2)
98
R. Grimshaw / Modulation of wave packet in stratified flow
where R0 is a Constant-This is unstable to modulations in the direction of wave propagation whenever VAii is positive where we have chosen the co-ordinate & to vary in this direction. There is also a more complicated criterion for instability to modulations transverse to the direction of wave propagation; this will not be displayed as it involves solving equation (4.17) for $. In general, there seems to be no simple way of determining the sign of the coefficients v and Aii. However, we shall show that in the long wave limit AI I is positive, and Y is negative, so that long waves are stable to modulations in the direction of wave propagation, although they may be unstable to transverse modulations. To obtain the long wave limit we let K + 0. Of course, the assumptions of the present theory prohibits this limit, but it is useful to use the approximation K =O to obtain information about the sign of v and Ari. As K + 0, the eigenfunction f, and the dispersion relation W (3.7) are given by (5.3a)
f=fo+C(K*),
(5.3b)
W=KWO(&;X~)-K~W~(KI,;X~~)+O(K~),
where i, = K,/K. Here f0 defines a long wave mode, and Substitution into (3.14) shows that
W0
is the phase speed of this long wave mode.
(5.4a) where (5.4b)
w,* = wo-KI,U~ol,
and hence WI has the same sign as W,*. Note that, since we are assuming that there are no critical layers, W,* cannot be zero. Now it follows from (5.3b) that the group velocity (3.17~) is given by
vm=&Wo+ aw,(&s_ *
I?,/&)
K&)-3K*&W~-K
ai
+
0(K4).
(5.5)
P
Now, from the definition (3.21) of the ray co-ordinates, d@/ds =
-CO+KK,Vm
=
-4K3Wl+O(K5).
and (5.5) it follows that (5.6)
Hence, with an error of O(K~), 0 is constant along a ray, and from (3.22b) is approximately O,(&). Consequently 0 is an interior co-ordinate on the initial manifold s = 0, and we may identify 5; with 0 (or & with 66) so that & varies in the direction of wave propagation; r2 is then chosen to be a co-ordinate transverse to the direction of wave propagation. With this choice of & we now use (5.5) to evaluate AolP (3.28b). Ai1 =
-3K3wl
h12=hzl=-~
1 A**=-2K
(5.7a)
+O(K’),
,aw a& -
-+
a& ax:
ho-&- aWo a&
(5.7b)
0(K4),
a& a& ->( ax; ax;
--
(5.7c)
The last two expressions (5.7b, c) have been simplified by assuming that 2, &/ax& = 0. These equations show that Ai1 is O(K~) relative to A22. Consequently if modulations in the direction of wave propagation are to remain of the same significance as modulations in the transverse direction, the transverse co-ordinate e2
R. Grimshaw / Modulation of wave packet in stratified flow
99
must be resealed
by a factor O(K’). It then follows from (4.17a) that 4 is Ok relative to $, and consequently 8 (4.20~) is Ok relative to v. (4.20b). Thus in calculating the nonlinear terms as K + 0, it is sufficient to calculate Y (5.lb). To find an approximate form for v2 we must findf2 from (4.3b). As K + 0 the homogeneous part of (4.3b) and (4.4a, b) reduce to the equation satisfied by fo; also the forcing term g2 (4.3c), and the forcing term in (4.4a) are O(1) with respect to K, and so we put
(5.8)
f2 = a2fo/K2+o(l).
If we now substitute (5.3a, b) and (5.8) into (4.3b) and (4.4a, b) and apply a compatibility condition for the O(1) term in (5.8), we find that (5.9a)
a2=$S/Wl, where
(5.9b) Next we substitute (5.3a, b) and (5.8) into (4.6b) to find v2. The result is
y2 a9/at0
-=-&-+0(K).
(5.10)
1
We note that the contribution of ~2 to v gives a term of the same sign as Ari, and so the second harmonic is a destabilizing term for long waves. Next we calculate I+Q from (4.16). As K + 0, V* + WO and so the homogeneous part of (4.16) is just the equation satisfied by fo. Also the forcing terms in (4.16) are O(1) with respect to K, and so we put $ =
U,&/K2
(5.11)
+ o(1).
If we now substitute (5.3a, b), (5.5) and (5.11) into (4.16), and apply a compatibility condition for the O(1) term in (5.11), we find that a,=-$/WI.
(5.12)
Then substituting into (4.20b) we can show that
-=VO ag/aW
s2
-+ 3Kwl
O(K).
(5.13)
Thus the coefficient Y (5.lb) is (5.14) Thus for long waves v and Ar1 have opposite signs, and long waves are modulationally stable. In contrast to the second harmonic, the contribution of the mean flow to v is stabilizing, and indeed v. is exactly twice ~2in magnitude. In summary, for long waves the evolution equation takes the approximate form aB
a2B
as
ah
i--3K3W~y+h22-+
a2B
S2
az; g-&$$d2B
=O,
(5.15)
100
R. Grimshaw / Modulation of wave packet in stratified flow
where AZ2is given by (5.7~) and & is EI~ and varies in the direction of wave propagation, while 52 is a transverse co-ordinate. Note that although v and hii have opposite signs, AZ2may have the same sign as Y and long waves would then be unstable to transverse modulations (cf. [15], for a discussion of the possible instabilities in nonlinear Schrodinger equations of the form (5.15)). For dispersion relations of the type (5.3b), the equation which describes long waves is the KortewegDe Vriesequation. In the present context the appropriate equation has been derived by Grimshaw [lo]. As K +o, 77 -
(5.16a)
G(xR)~o,
H = If a9/aw
)‘j2G,
(5.16b)
and (5.16~) Here V,o is V, evaluated as K + 0 from (5.5) and 8, & and 52 are the same co-ordinates specified above. The scaling required to produce this equation has 8 scaling with c-l, r2 with &-l, the amplitude with Ed, and the time scale for the evolution is C3. If, in (5.16c), we seek a solution of the form H = aB(s, &) exp(i0) +c.c.,
(5.17)
then it may be readily be shown that B will satisfy (5.15). This connection between long wave theory and modulated wave packets has previously been noted by Johnson [13] for water waves on a shear flow, (included in the present theory), and by Grimshaw [6] for continental shelf waves. Apart from the approximation K -* 0, it is difficult to obtain information about the sign of v except for special cases when the eigenfunction f is known explicitly, or when K is close to a value for which there is either second harmonic resonance, or long wave resonance, so that Vat, = KWO, where W. is the phase speed of a free long wave. Let f. be the eigenfunction of this long wave mode (f. is not necessarily the limit off as K + 0 in this context). In the vicinity of the resonance, 4 (4.16) is approximately given by aofo where Uois proportional to ( VoK, - K wo)-’ ; then v. (4.20b) is proportional to ao, and so is infinite at the long wave resonance. Near the long wave resonance Y is dominated by vo, and changes sign as K passes through the long wave resonance. Thus each long wave resonance is a boundary in wavenumber space of the modulational instability. Similarly, each second harmonic resonance is a boundary of the modulational instability.
Appendix.
Long wave resonance
When V,K, =KW~, or from (4.16d) and (5.4b), when V*= W,*, there is a long wave resonance. Eq. (5.16) for $ cannot be solved as the homogeneous part of this equation is satisfied by a free long wave mode with eigenfunction f. and phase speed Wo. Instead of the mean flow being an O(u2) wave induced quantity, there is a resonant interaction between the primary wave of wave number K, and group velocity V, and a long wave of phase speed Wo. This resonance was observed by McIntyre [14] for internal gravity waves in the absence of a shear flow, and the equations describing the resonance in that case were developed by Grimshaw [5]; similar equations were obtained by Grimshaw [6] for continental shelf waves. In this resonant case, the primary wave is again given by (3.2a) and the development of Section 3 is unchanged, so the evolution equation for the primary wave amplitude B is again (3.28a). In addition there is
101
R. Grimshaw / Modulation of wave packet in stratified flow
now a free long wave, so r&j2 = aoA,(XR, Xk)f&;
Xip)+ aoefj;.
64.1)
Here a0 is the amplitude parameter for the free long wave (the factor u2 is inserted on the left-hand side as the factor was previously inserted into the mean flow quantities in (2.19)), and A0 is the long wave amplitude whose evolution can be determined from (4.10). The resonance condition is
vat?,= wo+
64.2)
EX,
where x is a detuning parameter. For long wave resonance modulations in the direction of wave propagation dominate over transverse modulations. We choose & as a co-ordinate which varies in the direction of propagation, and t2 is then a transverse co-ordinate, and assume that A and A0 are functions of s, &, and 6; (~5~). It may then be shown that u2z& = -ao&
auOa dfo 1 + O(Uo&). w”*-p +foa2 )
(
(A.3)
Now the nonlinear term in the amplitude equation (3.28a) for B is {r, &} and in this context, the dominant contribution to this is {r, A\“‘} (4.19). Substituting (A.l) and (A.2) into (4.19) it follows that the leading term in {r, Mi} is {r, yacl)= aao@Bo,
(A.4a)
Bo= IZ~~~~oW*(~)‘dl(li2Ao
(A.4b)
where
and ~lsJ_~~ow*(~)2d=11’2=J~~( -2pow*lcwgh $($($)2+f2))
+[Po-*~(+2(~)2-f~)fo]z~,
dz
(A.4c)
The amplitude equation (3.28a) for B is thus i aB/as + A 1a2B/a& + pBBo = 0,
(A.%
where to balance the nonlinear term with the remaining terms we have put a0 = E2. The reason for introducing B. in (A.4a) rather than A0 will emerge below. To find the amplitude equation for A0 we substitute (A.l) into (4.10). Noting that D PC_ DT
v,$&++&$&) IX
as
(2
(A.6a)
102
R. Grimshaw / Modulation of wave packet in stratified frow
and
--a2 axi
where a&/ax:
-a& * a2 a*& a - (ax& >at1 ,zax -+W2), a45
(A.6b)
2+E
is parallel to t?, so that
a& v~--.= ax&
86; w,*(t2,-ax& >+
(A.7)
EX.
It follows that
(A.@ Here the omitted terms are also 0(&a& and occur due to the inhomogeneity of the basic flow; as these terms have not been displayed in (4.10), where they are of higher order than the terms retained we shall, for simplicity, not display them here. The balance of terms in (A.7) requires that uo& = a*; thus we have &=a 2/3 and a0 = u413. The boundary condition (4.14) is treated in a similar way, and we find that
uo( ki$)&(+‘ji;-~W~*~)+CT~U~(&--& =~m*(~~(&--$)~(
V$$32Aop,W:~+.
(~.0**-2o*~W~)(f*+-$($)‘)-
W$‘-$f*]
.-
onz =lo.
(A.9)
The boundary condition at z = -h is jj; + * * *= 0. Eq. (A.8) for 77; and the associated boundary conditions can only be solved if a compatibility condition is satisfied. This gives the required evolution for Ao: (A.lOa) where
(A.lOb)
R. Grimshaw / Modulauon of wave packet in stratified flow
103
Here Vaois the group velocity of the long wave mode (evaluated from (5.5) in the limit K + 0). The terms involving the inhomogeneity of the basic flow are accounted for by the transformation (A.4b); B. is the wave action for the long wave mode. We have also used (3.26) to convert IAl* into lBl* on the right-hand side. The equations describing long wave resonance are thus (AS) and (A.20a).
References [l] D.G. Andrews and M.E. McIntyre, “An exact theory of nonlinear waves on a Lagrangian-mean flow”, J. Fluid Mech. 89, 609-646 (1978). (1978). [2] D.G. Andrews and M.E. McIntyre, “On wave-action and its relatives”, J. Fluid Mech. 89,647-664 [3] D.J. Benney and A.C. Newell, “The propagation of nonlinear wave envelopes”, J. Math. Phys. 46, 133-139 (1967). [4] Yu.D. Borisenko, A.F. Voronovich, A.I. Leonov and Yu.Z. Miropolskiy, “Towards a theory of non-stationary weakly nonlinear internal waves in a stratified fluid”, Zzo. Atmos. Ocean. Phys. 12, 174-178 (1976) (English translation). [5] R. Grimshaw, “The modulation of an internal gravity-wave packet, and the resonance with the mean motion”, Studies in Applied Mathematics 56,241-266 (1977). [6] R. Grimshaw, “The stability of continental shelf waves I. Side band instability and long wave resonance”, J. Austral. Math. Sot. 2OJ3, 13-30 (1977). [7] R. Grimshaw, “Mean flows induced by internal gravity wave packets propagating in a shear flow”, Philos. Trans. Roy. Sot. 292A, 391-417 (1979). [8] R. Grimshaw, “The modulation of a weakly nonlinear wave packet in an inhomogeneous medium”, Math. Research Report No. 29, University of Melbourne (1979). [9] R. Grimshaw, “Slowly varying solitary waves II. Nonlinear Schrodinger equation”, Proc. Roy. Sot. 368A, 377-388 (1979).
[lo] R. Grimshaw, “Evolution equations for long, nonlinear internal waves in stratified shear flows”, Math. Research Report No. 1, University of Melbourne (1980). [ll] W.D. Hayes, “Conservation of action and modal wave action”, Proc. Roy. Sot. 32OA, 187-208 (1970). [12] D.J. Kaup and A.C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory”, Proc. Roy. Sot. 361A, 413-446 (1978). [13] R.S. Johnson, “On the modulation of water waves on shear flows”, Proc. R. Sot. 347A, 537-546 (1976). [14] M.E. McIntyre, “Mean motions and impulse of a guided internal gravity wave packet”, J. FZuid Mech. 60, 801-811 (1973). [15] A.C. Newell, “Soliton perturbations and nonlinear focussing”, in: Solitons and Condensed Matter Physics, Lecture Notes in
Solid-State Sciences 8, Springer, Berlin (1978) 52-67. [16] S.A. Thorpe, “On the stability of internal wave trains”, in: Deep-Sea Research, A Voyage of Discovery, Pergamon, Oxford (1977) 199-212. [ 171 S.A. Thorpe, “On the shape and breaking of finite amplitude internal gravity waves in a shear flow”, .I. Fluid Mech. 85,7-31 (1978). [18] V.E. Zakharov and A.B. Shabat, “Exact theory of two-dimensional self-focussing and one-dimensional waves in nonlinear media”, Soviet Physics JETP, 62-69 (1972) (English translation).
self-modulation
of
COMPANY
MODULATION OF AN INTERNAL STRATIFIED SHEAR FLOW
GRAVITY
WAVE PACKET IN A
R. GRIMSHAW Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
Received 4 August 1980
Modulations of an internal gravity wave packet in a stratified shear flow are discussed in the weakly nonlinear and weakly dispersive context. It is shown that the modulations are described by a variable coefficient nonlinear Schradinger equation when the modulations are confined to the direction of wave propagation. Transverse modulations couple the nonlinear Schriidinger equation to the mean flow equations. For long waves, it is shown that the modulation equations may be somewhat simplified. An Appendix describes the equations governing long wave resonance.
1. Introduction
For many dynamical systems, the modulation of a wave packet in the direction of wave propagation due to dispersive and weakly nonlinear effects is described by the nonlinear Schrodinger equation [3]. In homogeneous media, when the nonlinear Schrodinger equation has constant coefficients, there are N-soliton solutions and the equation is exactly integrable through the inverse scattering transform technique. For internal gravity waves propagating in a horizontal channel the nonlinear Schrijdinger equation describing a modulated wave packet was derived by Borisenko etal. [4], Grimshaw [5] and Thorpe [16]. In this paper we extend this to consider the evolution equation describing modulated internal gravity waves propagating on a shear flow in a horizontal channel. We shall allow the background shear flow to be weakly inhomogeneous, and we shall also consider the modifications necessary when there are modulations transverse to the direction of wave propagation. Grimshaw [8] has given a general theory for modulated wave packets in inhomogeneous media. For modulations in the direction of wave propagation, the evolution equation for the amplitude is the variable coefficient nonlinear Schrodinger equation aB
ig+A
(1.1)
$+p[~~2~=~.
Here A and k are real-valued functions of s. In inhomogeneous media wave packets propagate along rays, whose spatial direction is defined by the local group velocity, s is a time-like co-ordinate which varies along a ray, and 6 is a co-ordinate which is constant along a ray, but whose spatial projection defines the spatial direction of wave propagation. lBj2 is the wave action flux along the ray. The scaling required to produce equation (1.1) scales s with u-’ where ‘u’ is a small parameter measuring wave amplitude, and 6 scales with a -I; the background medium varies on scales of ae2. The coefficient A describes dispersion about the dominant wavenumber in the wave packet and the coefficient p describes the nonlinear effects due to the interaction of the primary wave with the mean flow and second harmonic. For modulations which are transverse to the direction of wave propagation (1.1) is modified and coupled in a rather complicated way to 81
82
R. Grimshaw / Modulation of waue packet in stratified flow
mean flow equations. The general theory of Grimshaw [8] was for a system of equations without boundary conditions. For the case of internal gravity waves on a shear flow in a horizontal channel, we shall show in this paper how the theory of Grimshaw [8] may be extended to produce an evolution equation (1.1) for modulations in the direction of wave propagation, and the modification of (1.1) when there are transverse modulations. In Section 2 we present the equations of motion for internal gravity waves in a stratified shear flow using the generalized Lagrangian-mean formulation of Andrews and McIntyre [l]. This has the advantage of simplifying considerably the treatment of the free surface and the mean flow equations, and also seems to be generally technically simpler to handle than the more usual Eulerian formulation. At the end of Section 2 the equations are cast into an operator symbolism, so that the comparison with the general theory of Grimshaw [8] can be used. In Section 3 the evolution equation for the amplitude is derived, and in Section 4 the nonlinear terms are evaluated. In Section 5 we present a brief discussion of equations such as (1.1). Of critical importance is the relative signs of A and CL,and we show that for long waves A and p have opposite signs. In the Appendix we derive the modification to (1.1) necessary when there is a long wave resonance (i.e. the group velocity of the wave packet coincides with the phase speed of a free long wave).
2. The generalized Lagrangian-mean
formulation
We propose to present the equations of motion in the generalized Lagrangian mean (GLM) formulation of Andrews and McIntyre [l] as their formulation has conceptual advantages, and, at least for the present problem, simplifies much of the analysis. A derivation of the equations in the GLM formulation for an inviscid, incompressible, stably statified fluid has been given by Grimshaw [7], so we shall only give an outline here. Let xi be Eulerian Cartesian co-ordinates and r be the time; here the Latin index ‘i’ takes the values 1,2 or 3 and the summation convention is used. Let Uibe the velocity components, p be the density and p be the pressure. We shall use non-dimensional variables based on a length scale L (a typical wave-length), a time scale N;‘, where N1 is a typical value of the Brunt-VaisiilH frequency; a velocity scale NIL and a pressure scale plgL, where p1 is a typical value of the density. These scales produce the non-dimensional Boussinesq parameter U, N:Lg-’ , which is small in the Boussinesq approximation. Then if 4 is any field variable we put 4(X:, t) = &xl, t)+4’(xl,
t, e),
(2.1)
where 4’ is periodic in the phase 8 with period 2~ and zero mean, and the phase 8 is itself a function of xl, t. The decomposition (2.1) effectively defines an averaging operator (a) where
J(xl, t)=(c#J(xi, t))=l
2n
2lT I 0 4d0*
(2.2)
It is conceptually useful to envisage 8 being replaced by 8 + I++, and defining the averaging operator with respect to the ensemble label 4, rather than the physical variable 8 (cf. [l 11,or [2]). & is the Eulerian mean of 4, and 4’ the Eulerian perturbation. Next let Xi be generalized Lagrangian co-ordinates, and ti(xi, t) be the particle displacements defined so that Xl =Xi+[i*
(2.3)
R. Grimshaw / Modulation of wave packet in stratified flow
This equation defines a transformation @(X, t) = (4 (Xi+ 69
from xi to xi. We then define a Lagrangian-mean
83
operator by (2.4)
t)>-
In physical terms, I$, the Eulerian mean, is the average over the phase at a fixed place, while BL, the Lagrangian mean, is an average over the phase of the waves following the fluid motion. As shown by Andrews and McIntyre [l] this latter notion is made precise by requiring that (6)
= 09
(2.5)
whence it follows that Xi is a co-ordinate which moves with the Lagrangian mean velocity CL whenever the co-ordinate xl moves with the true velocity Ui. In contrast to (2.1) we put 4
C-G
1) =
6”C-G
t) +
dh
6 @I,
(2.6)
where 4, the Lagrangian perturbation, is periodic in the phase 6 (now a function of xi, t) with period 2~ and zero mean. Assuming that the particle displacements are O(a), where ‘a’ is a small parameter measuring wave amplitude, then it is readily shown that (2.7a) 6%
J+@,
(2.7b)
where (2.7~) 6” is usually called the ‘Stokes correction’. Finally we note the useful result [l]
d4 -
d =,,w, (dt >
(2.8a)
where (2.8b) Here d/dt is the material derivative following the fluid motion. To obtain the equations of motion in the GLM formulation, we first introduce the Jacobian
(2.9)
J = det{axi/axi}.
But then the continuity equation and conservation of mass together imply that J is a mean quantity which satisfies the equations dJ aa!T~;+J-=0, axi
_f=o.
(2.10)
Expanding (2.9) in powers of ei, and substituting into (2.10) it follows that aa” ,=$(f a&/&i
&t&!Cj))
-MO = 0,
+O(a4>,
(2.11a) (2.11b)
84
R. Grimshaw / Modulation of wave packet in stratified flow
(2.11c) The density p is a mean quantity 6‘ (i.e. 6 is zero) and satisfies the equation dp‘/dt
(2.12)
= 0.
The equation of motion is (2.13) where Sij is the Kronecker delta. Before separating this equation into its mean and perturbed multiply it by ax i/ax, and make the substitution
parts, we (2.14)
P=~‘+~P’+B(-~‘d~~/dt-~‘Si~),
In spite of the notation, up’ differs from the Eulerian perturbation pressure by terms of O(a’). We find that the equation of motion for the Lagrangian mean flow is (cf. [7]) dii; I 1 apL 1 ;L -+-Si3 dt ap axi u
(2.15a)
=
where Pi = -(i;j &$/t3xi),
(2.15b)
aj = dtj/d t.
(2.1%)
and
The equation of the motion for the perturbed
flow is
_Ld2&ap’ q --5;$(pL~~ a/T‘ +M’=
’ Z+iGj-L
dxi
,
0,
(2.16a)
where (2.16b) Here n = 63 is the vertical particle displacement. We shall suppose that the fluid is confined in a horizontal waveguide which is bounded above by a free surface Z’ = &/&XL, t) and below by the rigid boundary z’ = -h(x&, t). Here, and subsequently, we distinguish the vertical co-ordinate so that z’ = xi, z =x3 and w = ~3, and employ Greek indices for horizontal variables XL ((Y= 1,2), x, and u,. In the GLM formulation the free surface is a Lagrangian mean quantity, z = f‘(x,, t), and the Lagrangian perturbation, i, is identically zero. The boundary conditions at the free surface are
a.f"
_L ar” at+u_dx=GL
onz=5‘,
(2.17a)
OL FL=0
onz=r,
(2.17b)
R. Grimshaw / Modulation of waue packet in stratified flow
85
and UP’ -pLn -pL& dak/dt = 0
on z = [“.
(2.17~)
The last two equations follow from the condition that the pressure, given by (2.14), should vanish on the free surface. At the bottom boundary, the conditions are I?rL+&&r/&+*.*=O
onz=-h(x,),
(2.18a)
and =0
q+&ah/aXp+---
onz=-h(x,),
(2.18b)
where the omitted terms are 0(p2a2), and /3 is a small parameter which measures the gradients ah/%x,. In the mean flow equations (2.11a), (2.12) and (2.15a) the forcing terms due to the waves are O(a2), and so we decompose the mean flow into an 0( 1) basic flow, and an 0(a2) term. Further, we shall suppose that the basic flow varies on long horizontal length and time scales relative to the wave field, and so we put tif = uoi(Xh,T',Z)
+u~E:~,
(2.19a)
FL = p,,(X:, T’, z) + c~a2&,
(2.19b)
,jL = &XL,
T’, z) + au ‘&,
(2.19c)
~=&-,(X&,
T’, z)+a25;L,
(2.19d)
X& =&.a,
T’ = pt:
(2.19e)
where
Here tiki etc. are the wave induced O(u*) terms, and p is a small parameter which measures the length and time scales of the wave field relative to the basic flow. We shall also suppose that h = h(Xk). Since the horizontal scale is O(p-‘) relative to the vertical scale, it is convenient to redefine w. by flwo. The equations for the basic flow are then auop/ax:, + aw,/a2 = 0,
(2.20a)
Dpo/DT’ = 0,
(2.20b)
p. Duo,/DT’+
(2.2Oc)
l/~r ap,/aXh_ = 0,
p2po Dwo/DT’+
l/a apo/az + l/~rp~ = 0,
D/DT’=
uon
(2.20d)
where a/aT’+
a/ax:,
+
w.
a/az.
(2.20e)
The boundary conditions are w. + uoa
ah/ax& =0
alo/aT’ +
uop
onz=-h(XL)
afb/ax& = w.
on 2 = Lo(X&),
(2.21a) (2.21b)
and p0=0
on2 =lo(X&).
(2.21c)
R. Grimshaw / Modulation of wave packet in stratified flow
86
The equations for the perturbations & and p’ are (2.1 lb) and (2.16a), while the boundary conditions are (2.17~) and (2.18b). For the purposes of the next section it is convenient to recast these equations into the following matrix operator form. Let u be the 4-vector.
[I P’
fJ=
(2.22)
ZP,
77
where since (Y= 1,2, the second entry is a short-hand notation for two terms. Then (2.1 lb) and (2.16a) may be written in the form (2.23)
for-h
Lv+M=O
where L is the 4 x4-matrix
operator,
(2.24a) D2 poN2 + PO Dt2
0
1
where
L’ =
0 b
0
0
0
---1 ape
a $+&%DT’ax I (ZPOWO~ 1 ape --_ ( Duo7
U
Y
0
2PoWojg
u ax&
D/Dt = a/at +
uool
;+
ax&
Duo, PODT’
9
&-
(2.24b)
a cl
(2.24~)
a/ax,,
and (2.24d)
aN2 = - 1/p,, apo/a2. Here N is the Brunt-Vlislki
M=
frequency. The nonlinear term A4 is given by
-+2poi&3--+f
D a&
-+2poii:,---~dt+. D 877
Dt a+
(2.25)
**
Dt ax0 a&
**
where MO is defined by (2.11c), and M’ is defined by (2.16b). In the second term in M, only those terms which contribute to the subsequent analysis are displayed. The boundary conditions may be written in the form Bv+Mg=O
forz=lOand-h,
(2.26a)
R. Grimshaw / Modulation of waue packet in stratified fZow
87
where for z = &,,
-d&l
)[
Ba .-f- d;T’,xhJ =
(at’ax,‘az
0
0
1
0 1
0 0 +pB’+o(p2), 0 0
(2.26b)
(2.26~) and for z =-h,
=
0 ah/ax:, 0 B’=O
[
0
0
1
00.
0 0 -0 -0 00
1
0 +~B’+O(p2), 01
(2.26d)
(2.26e)
The nonlinear term MB is given by
(2.27)
while for .z = -h, MB is O(p2a2) and does not feature in the subsequent analysis. The perturbation equations thus consist of (2.23), with the boundary conditions (2.26a). Schematically they may be combined into the single equation (2.28a) where pR
=
(2.28b)
a/axR,
and xo=t,
XL = T’.
(2.28~)
Here 9 is an operator incorporating L and the boundary operators B, while A is the nonlinear term. Here, and in the next section we find it convenient to introduce the capital Latin index ‘I?’ which takes the values 0, 1 and 2; this is a device which enables us to incorporate the time variables and the horizontal space variables into the same notation in situations where it is not necessary to make a distinction.
88
R. Grimshaw
/ Modulation
of wave packet in stratified flow
3. Modulated waves We propose to discuss a modulated wave packet where the modulation occurs on scales long compared to the waves, and is described by co-ordinates
x, = EX,,
(3.1)
T=et,
where E is a small parameter. For a strongly inhomogeneous medium the modulation scale is identical with the scale of the basic flow and we put p = E (i.e. XR and Xi are identical); for a weakly inhomogeneous medium we put /3 = E* (cf. [8]). Subsequently we shall consider only this latter case, but we note that the former case may be recovered by the omission of the dispersive terms from the evolution equation for the wave amplitude. For the present we shall retain both parameters. The general procedure for obtaining the evolution equation for the amplitude of a modulated wave packet in a homogeneous medium is well known [3]. For internal gravity waves in the absence of a shear flow, and any other inhomogeneities, the evolution equation has been derived by Borisenko et al. [4], Grimshaw [S] and Thorpe [16]. When the medium is inhomogeneous a general procedure for obtaining the evolution equation has been developed by Grimshaw [8]. Although that procedure was described for a system of equations in the absence of boundary conditions, the modifications necessary to deal with boundary conditions are readily dealt with, and so we shall only give an outline here. Let 0 = f un(XR, Xk ; 2 ; E) exp(in8) + c.c., n=l
(3.2a)
f!+(X2).
(3.2b)
where
Here C.C. denotes the complex conjugate. The phase 13varies rapidly with respect to the background medium. The local frequency w and wavenumber K~ are defined by w = -aOlaT’,
K,
=
(3.3)
@/ax&.
Substituting into (2.28a) it follows that L?? niK,+E-++P+.-j$Xk,z ( R
v,+.AG=o, R
(3.4a)
>
where Jbl
exp(-inO) de.
(3.4b)
We shall further assume that if ‘a’ measures the wave amplitude, then VI is O(a), v2 is O(a*) etc. This assumption excludes such phenomena as second harmonic resonance; similarly the assumption that tiii etc. are O(a*) excludes long wave resonance. However, similar procedures to those described below may be used for these phenomena as well (cf. [5,6]). The equations describing long wave resonance are derived in the Appendix. We shall show later that JU~is 0(a3), and that these nonlinear terms balance dispersion when
R. Grimshaw / Modulation of wave packet in stratified flow
89
a = E. Expanding the operator 9 in E and p it follows that, for n = 1,
Lz+l+&-
apR a&
a23 au, a2th +p-apR 8x2 apR aps axR axs
+ZE
(3Sa)
where zo=z
(* IKR,
(3Sb)
;;xk,z).
Here the derivatives of 2 with respect to PR (i.e. with respect to the variable PR = a/&R) are evaluated at &= 0. To leading order in E and a, u1 is a null vector of _Yc,.We put UI=
UT(KR,
Z;
Xk)A(XR, XL) +O(U),
(3.6a)
where (3.6b)
&jr = 0.
Here r is a null vector of L&,,and A is the complex wave amplitude whose evolution we wish to describe. In order that r be a non-trivial null vector a dispersion relation W=
(3.7)
W(K,;X;p)
must be satisfied. From (3.3) we see that this is a partial differential equation for the phase 0; its solution will be described later in this section. Recalling the definition of the operator 9 in Section 2 (i.e. (2.24a), (2.26b) and (2.26d)), we see that
Pob*2/K2)af/az r= 1 (iK,/K2)af/aZ
[
where
(3.8a)
,
f
-f(&w*2$) +poK2(N2-m*2)f
(3.8b)
(3.8~)
on z = lo, f=O
=o,
(3.8d)
onz=-h,
and W*=W-K&a,
K ’ =
K,Kp.
(3.8e)
Here w* is the Doppler-shifted, or intrinsic, frequency. Eqs. (3.8b, c, d) define an eigenvalue problem for w once K~ has been specified, and the solution is given by the dispersion relation (3 -7) and the eigenfunction f. A sufficient condition that w be real for real K, is that the Richardson number (the minimum of
90
R. Grimshaw / Modulation of wave packet in stratified flow
%,,/c~z)-‘1 be greater than a everywhere, and we shall assume henceforth that this is the case. We shall also assume that there are no critical levels, and so w* does not vanish for any z in the flow domain. In general there will be a number (possibly infinite) of eigenvalues, each defining a wave mode. Henceforth we shall confine attention to just one such mode. In the first instance the null vector r is only defined when the dispersion relation (3.7) is satisfied. However we shall extend the definition of r to all KR by assuming that r is defined by (3.8a) where the eigenfunction f satisfies the boundary conditions for all KR, but only satisfies the differential equation (3.8b) when the dispersion relation (3.7) is satisfied. Much of the subsequent discussion depends on the properties of the operator L& For two 4-vectors ZIand w, we define the inner product N’K’(K,
{w, ~clv}= ILo w+Lov dz +[~+&a]~%,
(3.9)
-h
where the superscript “’ denotes the complex conjugate transpose of the 4-vector w, and Lo is L(iK& a/&z ; Xk, z), with a similar definition for &. Now Z0 is a self-adjoint operator and it may be shown that (3.10)
1% %u) = L%w, VI. The method of solution of (3.5a) leads to a sequence of equations of the form
(3.11)
L&u+g=o. A necessary and sufficient condition that this inhomogeneous
equation have a solution is (3.12)
jr, g) = 0.
The necessity follows immediately from (3.10), while the sufficiency may be established by converting (3.11) to an inhomogeneous form of (3.8b, c, d) and employing the method of variation of parameters. It is useful to define ~(KR,
XZ) = {r,=Y04.
(3.13)
Clearly 9 = 0 is equivalent to the dispersion relation (3.7). LSmay be eval uated from (3.8a, b, c, d) and we find that (3.14) Next we put o1 = arA + am:‘) ,
(3.15)
and substituting this expression into (3.5a) it follows that (3.16) The compatibility . a9
condition (3.12) now shows that
aA = O(E*, p, a*),
‘&ii& a&
(3.17a)
R. Grimshaw / Modulation of wave packet in stratified frow
91
or ie-,“F( g+ .O
v$
a1
(3.17b)
= O(Z, p, a2).
where (3.17c)
v, = a w/aK,.
Here, in deriving (3.17a), we have differentiated (3.13) with respect to KR. Eq. (3.17b) is obtained by differentiating the dispersion relation 9 = 0, while V, is the group velocity; this equation states that, at the leading order in E, modulations in A propagate at the group velocity. Further it may be shown that [8] EVl
(*) =
-Ei
s -$$+O(c2, p,2). R
(3.18)
R
For the higher order terms a similar procedure is used. At each stage, the compatibility condition (3.12) is applied. At the second order in E, we find that
+p{r,
The terms proportional aE*
_Y’r}A +{r, d~}+O(m3)
= 0.
(3.19)
to p may be recast into a simpler form, so that (3.19) becomes
a29
a*A
+{r,&}+O(ae3)=0.
(3.20)
The manipulations leading to (3.20) are similar to those described by Grimshaw [8], where we note that since .Yo is self-adjoint, and we are dealing with an inviscid fluid, the terms proportional to p describe the conservation of wave action. This latter conclusion also follows from [7], where the wave action equation was derived for a strongly inhomogeneous medium. Eq. (3.20) is the required evolution equation for the amplitude A, and in the next section, we shall evaluate the nonlinear term {r, Yaci}which is 0(a3). The first term in (3.20) describes the effects of dispersion about the dominant wavenumber, while the second term describes the evolution of the amplitude on the long time scale p-‘* , a balance between these two terms and the nonlinear term requires a = E and /3 = E*. In the remainder of this section we shall describe the form the evolution equation takes in ray co-ordinates [8]. These are obtained by solving the dispersion relation (3.7) for the phase 0, using the method of characteristics, or rays. Let s be a parameter along a ray. Then the ray equations are dT’/ds = 1,
dXh/ds = V,,
do/ds = aW/aT’,
dK,/ds = -a w/ax:.
(3.21a) (3.21b)
These are ordinary differential equations, which are to be solved subject to the initial conditions Xk =XZo([&)
for s =0
(3.22a)
and 0 = @,(&)
for s = 0.
(3.22b)
92
R. Grimshaw
/ Modulation
of wave packet in stratified j-low
Eq. (3.22a) describes an initial manifold on which [& are interior co-ordinates, while (3.22b) determines the initial phase on this manifold. We also assume that the dispersion relation (3.7) holds on the initial manifold. With these initial conditions, the solution of the ray equations is xlp = xlp (s, 5:).
(3.23)
This equation describes a transformation 9 = det[aXk/@$],
of co-ordinates
from (3.23) Xk to (s, 5;). Let
where .$ = s.
(3.24)
Then it can be shown that (3.20) becomes (3.25) where here, and henceforth, we shall omit the error terms. The entity 4 dCB/ao $A2 is proportional wave action flux along a ray tube. We define a new amplitude B by
B2 = -!$&42, and new co-ordinates
to the
(3.26)
by (3.27)
where e&(X&) is determined from (3.23). We then express (3.25) in terms of B and the co-ordinates &. The result is aB
iz+A,,
-_e+~_&~“’ yLo, at2 6
s and
(3.28a)
where
A
1 -A-a*w
=@= 2
aK,
aK,
at’ ax:
a& ax;
(3.28b) ’
In deriving the expression (3.28b) for A,@, use is made of (3.17b), and the expressions obtained by differentiating the dispersion relation 9 = 0 with respect to KR. Various specific cases for the ray co-ordinates are described by Grimshaw [8]. Here we shall just present one case when the basic flow is homogeneous, and so W (3.7) does not depend on Xk. In this case w and K, are constants, V, is a constant and the ray co-ordinates are s
=T’E~‘~
& = X, - V,T = E(x, - Vat).
3
Also, the phase 13is --of +
=--
A nR
1 2
a*w aK,
aKB
K,x,,
(3.29)
and, (3.30)
*
93
R. Grimshaw / Modulation of wave packet in stratified flow
4. The nonlinear terms As a prelude to the calculation of Ai (3.5a), we must first calculate the second harmonic 02 and the mean flow Gki etc. (2.19a, etc.). We consider the second harmonic u2 first, which satisfies the equation L?(2iKR, a/6)2; Xk, z)V2+&
= O(ca2).
(4.1)
If there is no second harmonic resonance (i.e. W(~K,; XX) # 2 W( KR ; XL)), then the linear operator LZ?(2iKR, a/&z ; Xk, z) in (4.1) is non-singular, and there is a unique solution to (4.1). The forcing term & is found from M (2.25) and MB (2.27); there is no contribution to MB at O(a*>, and M2 is given by
(4.2)
M2 = a2A2
Next we eliminate &2 and p; from (4.1) and obtain a single equation for 772,correct to O(a2), 772= a2A2f2,
(4.3a)
where (4.3b) and
1
d2 af
Po~f-$-&Pow
1
a
l
*2 ( ( af)2_f2). ) ;Tz g
(4.3c)
The boundary conditions are extracted in a similar manner. (4.4a) and f2 = 0
in z = -h.
(4.4b)
Eq. (4.3b) and the boundary conditions (4.4a, b) determine f2 uniquely. This determination of the second harmonic agrees with [17] in the Boussinesq approximation (a + 0). Next we shall calculate the contribution to Jtl due to the quadratic interaction of the second harmonic u2 with the primary harmonic ul; this contribution will be denoted by &i*’ . Before proceeding we note that it can be shown that there is no contribution at O(a3) due to cubic interactions at 0(a3). In calculating Ac$ we find that there is no contribution from MB at O(a3), and M’:’ is given by 2af2
G
Mi2’
= a3JA12A
a’f af -g+2f’g+Tf
1
iPo~*2K.(2f2f1+f~($-I))
Po,*2
(
-4f2
~_~f~)_~~(~_~)_~$(~_3
a’f, 1 a*2 --5fa,-a,’
aI af
,
(4.5a)
94
R. Grimshnw / Modulation of wave packet in stratified flow
where
( >2_f-g
I=
g
(4Sb)
The final step is the evaluation of the nonlinear term {r, A/u’:‘}in the evolution equation (3.20). We find that {r, A:“‘} = u3v21Aj2A,
(4.6a)
where
v24; w*2[ 2~I-212+-f(f2(($)2-~2f2))] POT
dr.
(4.6b)
Thus the result of the quadratic interaction of the second harmonic with the primary harmonic is a cubic nonlinearity in the amplitude A, which, as anticipated, is 0(a3). The mean flow equations are (2.11a), (2.12) and (2.15a), with the boundary conditions (2.17a, b) and (2.18a). In these equations the forcing terms Pi (2.15b) etc. are evaluated from v1 to 0(a2); substitution of (2.19) into the mean flow equations leads to equations for the wave induced quantities .Qkietc. The details are given by Grimshaw [7], and the result is, correct to O(ca’) (4.7a) D& _ L aUOp -) +s=&]Al’( PO DT+W2 a* m
(-
+&‘( a
aa
2p,~,w*( f2+f(92)] p,,(~V~-w*‘~f~-~(~)~],
_L
a~+~2
=
IAl’(
(4.7b)
(4.7c)
~(2p0(NZ-w*~)f~)-~(poN’)f’],
&& - po&N2 =0.
(4.7d)
where D/DT
=
a/aT + uoa a/aXcll.
(4.7e)
The boundary conditions are Dfi/DT
= K$
s;” -3j2=
(4.8a)
on z = lo,
IAl'{ ~r~+,~(2,*~-N~)$](~)~
on z = lo,
and fik=O
onz=-h.
(4.8~)
Here we have also used the fact that the basic flow depends only on z and the variables Xk, and to the order considered, derivatives with respect to XL do not appear in (4.7) or (4.8). These equations can be reduced
95
R. Grimshaw / Modulation of wave packet in stratified flow
to a single equation if we put C$ = D+/DT,
(4.9)
where fz can be interpreted as a mean vertical displacement. Then it follows from (4.7d) that & = &V*~Z, and (4.8a) shows that fk = jj2 at z = lo. Elimination of i& and p2 then leads to the equation
a G
D2 afj2 2a2fj2 -+pcJv x=~[ “DT2 az )
a
D2 ~\A~o$-K,$
-$A[‘( n
&1A122po-*(f2++432))
$-~oo*~J(
(4.10)
f2+$(-$)2)].
With respect to the variables XR, modulations in A propagate at the group velocity (3.17b), and so we seek a solution of (4.10a) (or (4.7)) with the same property. Thus in these equations we replace a/aT with - V, a/aXa, where it follows that D/DT = - Vg a/ax=,
where Vx = V, - uop.
(4.11)
‘Thus Vp* is the group velocity relative to the basic flow. The boundary conditions are (4.8~) which becomes jj2 = 0 at z = -h, and (4.8b). This latter condition involves the mean pressure p2 which must be determined from (4.7c). We put poN2ii2-IA/2(~(2~o(NZ-~
@2=
*2)f2)-;(pti2)f2)]
dz
where Q (independent of z) is the surface mean pressure. Then substitution differentiation with (4.7a) leads to the following equation for 0:
a2Q x=
D2 afj, D2 2AJZpoaZfZ+K & -I p°F2 Z-DT a2 +
a21A12 ax”,
po(N2-W*2)f2-pOII)-i-
&IA122p,o*(
(4.12)
+a(xR),
into (4.7b) and cross-
f2+$(j92)
*2
(4.13)
K
Finally the boundary condition (4.8b) becomes 772=
~+v/A/~(
~+u~(~u*~-N~)$)
(9’
on z = lo.
(4.14)
In general, (4.10) may be solved with a Fourier transform in X, [7]. However, it is convenient to first make the transformation ii2 =
(4+af2/adlA12+&
Q=[poV*2$
(4.15a) -%‘ow*KV*(
f2++($-)2)]
zzcJA12f
6
(4.15b) and (4.15c)
R. Grimshaw / Modulation of wave packet in stratified f7ow
96
Substitution into (4.10), (4.13) and (4.14) then shows that $ satisfies the equations
-
2poN2f-
af
- -
a (pow
a.2 a.2
*2)(
+L f2
(4.16a)
3’ K2
( az ))
9
(4.16b)
on z = lo, $=O
(4.16~)
onz=-h,
where v* =
K,
v,*/
(4.16d)
K,
while 6 is given by
(4.17a) ~$-ad/p~=O
$=O
(4.17b)
ont=[O,
(4.17c)
onz=-h,
and 6 is given by
(4.18) Here V” is the component of the relative group velocity Vx in the direction of K,, and so 4 represents the mean flow induced by modulations parallel to the direction of wave propagation, while 4 and 6 represent the response to modulations transverse to the direction of wave propagation. Note that sl, is completely given by (4.16) independently of [AI, and can be found uniquely if there is no long wave resonance (i.e. Volt, # KW~, where W. is the phase speed of a free long wave). $ and 6 must in general be found using Fourier transforms, and again can be found uniquely if there is no long wave resonance. Finally 2, is found from (4.7b) after substituting (4.12) and (4.15~) into (4.7b). Next we calculate the contribution to JGlrdue to the interaction of the primary harmonic zll with the wave induced mean flow. Denoting this contribution by My), we find that {r,Jti”‘)=.‘a[l_(:(
2p,N2f-$2+(-$(~)2+f2)(2pow*K,fikc~poN20*2~2))
dz (4.19)
97
R. Grimshaw / Modulation of wave packet in stratified flow
Using the decomposition
(4.15), this may be written in the form
{r,A\“‘} = c~~vo(A/~A + a3Ag,
(4.20a)
where
and s^ = If,” (2,0Ar~&+(-$($)~+p)(
~P,,o*K,&, +$A,w**)$)]
dr (4.20~)
Note that s^ depends only on 4 and ii,, and so responds only to modulations in amplitude transverse to the direction of wave propagation. The total nonlinear term is the sum of (4.6a) and (4.20a).
5. Discussion
The evolution equation for the amplitude is (3.28a) with nonlinear terms given by (4.6a) and (4.20a). Putting a = E, this equation is the nonlinear Schrodinger equation (5.la) where .=(vo+.2,/(393.
Here s^ is given by (4.20~) and responds only to modulations in amplitude transverse to the direction of wave propagation, through (4.17) and (4.18). The coefficients holPand Ydepend on s and & ; since t& = ES,, the dependence on s predominates. When s^ is zero, and APBand Y are constants, (5.la) has N-soliton solutions and is exactly integrable through the inverse scattering transform technique [18]. However for the variable coefficient nonlinear Schrodinger equation, there is no general method of solution unless there is a transformation which converts (5.la) into an equation with constant coefficients [8]. Perturbation methods must be used such as those which describe the slowly varying solitary wave [9,12]. However, (5. lb) has a ‘plane wave’ solution in which B is a function of s alone, and s^ is zero. It is given by B=Roexp[iR~[o’vds],
(5.2)
98
R. Grimshaw / Modulation of wave packet in stratified flow
where R0 is a Constant-This is unstable to modulations in the direction of wave propagation whenever VAii is positive where we have chosen the co-ordinate & to vary in this direction. There is also a more complicated criterion for instability to modulations transverse to the direction of wave propagation; this will not be displayed as it involves solving equation (4.17) for $. In general, there seems to be no simple way of determining the sign of the coefficients v and Aii. However, we shall show that in the long wave limit AI I is positive, and Y is negative, so that long waves are stable to modulations in the direction of wave propagation, although they may be unstable to transverse modulations. To obtain the long wave limit we let K + 0. Of course, the assumptions of the present theory prohibits this limit, but it is useful to use the approximation K =O to obtain information about the sign of v and Ari. As K + 0, the eigenfunction f, and the dispersion relation W (3.7) are given by (5.3a)
f=fo+C(K*),
(5.3b)
W=KWO(&;X~)-K~W~(KI,;X~~)+O(K~),
where i, = K,/K. Here f0 defines a long wave mode, and Substitution into (3.14) shows that
W0
is the phase speed of this long wave mode.
(5.4a) where (5.4b)
w,* = wo-KI,U~ol,
and hence WI has the same sign as W,*. Note that, since we are assuming that there are no critical layers, W,* cannot be zero. Now it follows from (5.3b) that the group velocity (3.17~) is given by
vm=&Wo+ aw,(&s_ *
I?,/&)
K&)-3K*&W~-K
ai
+
0(K4).
(5.5)
P
Now, from the definition (3.21) of the ray co-ordinates, d@/ds =
-CO+KK,Vm
=
-4K3Wl+O(K5).
and (5.5) it follows that (5.6)
Hence, with an error of O(K~), 0 is constant along a ray, and from (3.22b) is approximately O,(&). Consequently 0 is an interior co-ordinate on the initial manifold s = 0, and we may identify 5; with 0 (or & with 66) so that & varies in the direction of wave propagation; r2 is then chosen to be a co-ordinate transverse to the direction of wave propagation. With this choice of & we now use (5.5) to evaluate AolP (3.28b). Ai1 =
-3K3wl
h12=hzl=-~
1 A**=-2K
(5.7a)
+O(K’),
,aw a& -
-+
a& ax:
ho-&- aWo a&
(5.7b)
0(K4),
a& a& ->( ax; ax;
--
(5.7c)
The last two expressions (5.7b, c) have been simplified by assuming that 2, &/ax& = 0. These equations show that Ai1 is O(K~) relative to A22. Consequently if modulations in the direction of wave propagation are to remain of the same significance as modulations in the transverse direction, the transverse co-ordinate e2
R. Grimshaw / Modulation of wave packet in stratified flow
99
must be resealed
by a factor O(K’). It then follows from (4.17a) that 4 is Ok relative to $, and consequently 8 (4.20~) is Ok relative to v. (4.20b). Thus in calculating the nonlinear terms as K + 0, it is sufficient to calculate Y (5.lb). To find an approximate form for v2 we must findf2 from (4.3b). As K + 0 the homogeneous part of (4.3b) and (4.4a, b) reduce to the equation satisfied by fo; also the forcing term g2 (4.3c), and the forcing term in (4.4a) are O(1) with respect to K, and so we put
(5.8)
f2 = a2fo/K2+o(l).
If we now substitute (5.3a, b) and (5.8) into (4.3b) and (4.4a, b) and apply a compatibility condition for the O(1) term in (5.8), we find that (5.9a)
a2=$S/Wl, where
(5.9b) Next we substitute (5.3a, b) and (5.8) into (4.6b) to find v2. The result is
y2 a9/at0
-=-&-+0(K).
(5.10)
1
We note that the contribution of ~2 to v gives a term of the same sign as Ari, and so the second harmonic is a destabilizing term for long waves. Next we calculate I+Q from (4.16). As K + 0, V* + WO and so the homogeneous part of (4.16) is just the equation satisfied by fo. Also the forcing terms in (4.16) are O(1) with respect to K, and so we put $ =
U,&/K2
(5.11)
+ o(1).
If we now substitute (5.3a, b), (5.5) and (5.11) into (4.16), and apply a compatibility condition for the O(1) term in (5.11), we find that a,=-$/WI.
(5.12)
Then substituting into (4.20b) we can show that
-=VO ag/aW
s2
-+ 3Kwl
O(K).
(5.13)
Thus the coefficient Y (5.lb) is (5.14) Thus for long waves v and Ar1 have opposite signs, and long waves are modulationally stable. In contrast to the second harmonic, the contribution of the mean flow to v is stabilizing, and indeed v. is exactly twice ~2in magnitude. In summary, for long waves the evolution equation takes the approximate form aB
a2B
as
ah
i--3K3W~y+h22-+
a2B
S2
az; g-&$$d2B
=O,
(5.15)
100
R. Grimshaw / Modulation of wave packet in stratified flow
where AZ2is given by (5.7~) and & is EI~ and varies in the direction of wave propagation, while 52 is a transverse co-ordinate. Note that although v and hii have opposite signs, AZ2may have the same sign as Y and long waves would then be unstable to transverse modulations (cf. [15], for a discussion of the possible instabilities in nonlinear Schrodinger equations of the form (5.15)). For dispersion relations of the type (5.3b), the equation which describes long waves is the KortewegDe Vriesequation. In the present context the appropriate equation has been derived by Grimshaw [lo]. As K +o, 77 -
(5.16a)
G(xR)~o,
H = If a9/aw
)‘j2G,
(5.16b)
and (5.16~) Here V,o is V, evaluated as K + 0 from (5.5) and 8, & and 52 are the same co-ordinates specified above. The scaling required to produce this equation has 8 scaling with c-l, r2 with &-l, the amplitude with Ed, and the time scale for the evolution is C3. If, in (5.16c), we seek a solution of the form H = aB(s, &) exp(i0) +c.c.,
(5.17)
then it may be readily be shown that B will satisfy (5.15). This connection between long wave theory and modulated wave packets has previously been noted by Johnson [13] for water waves on a shear flow, (included in the present theory), and by Grimshaw [6] for continental shelf waves. Apart from the approximation K -* 0, it is difficult to obtain information about the sign of v except for special cases when the eigenfunction f is known explicitly, or when K is close to a value for which there is either second harmonic resonance, or long wave resonance, so that Vat, = KWO, where W. is the phase speed of a free long wave. Let f. be the eigenfunction of this long wave mode (f. is not necessarily the limit off as K + 0 in this context). In the vicinity of the resonance, 4 (4.16) is approximately given by aofo where Uois proportional to ( VoK, - K wo)-’ ; then v. (4.20b) is proportional to ao, and so is infinite at the long wave resonance. Near the long wave resonance Y is dominated by vo, and changes sign as K passes through the long wave resonance. Thus each long wave resonance is a boundary in wavenumber space of the modulational instability. Similarly, each second harmonic resonance is a boundary of the modulational instability.
Appendix.
Long wave resonance
When V,K, =KW~, or from (4.16d) and (5.4b), when V*= W,*, there is a long wave resonance. Eq. (5.16) for $ cannot be solved as the homogeneous part of this equation is satisfied by a free long wave mode with eigenfunction f. and phase speed Wo. Instead of the mean flow being an O(u2) wave induced quantity, there is a resonant interaction between the primary wave of wave number K, and group velocity V, and a long wave of phase speed Wo. This resonance was observed by McIntyre [14] for internal gravity waves in the absence of a shear flow, and the equations describing the resonance in that case were developed by Grimshaw [5]; similar equations were obtained by Grimshaw [6] for continental shelf waves. In this resonant case, the primary wave is again given by (3.2a) and the development of Section 3 is unchanged, so the evolution equation for the primary wave amplitude B is again (3.28a). In addition there is
101
R. Grimshaw / Modulation of wave packet in stratified flow
now a free long wave, so r&j2 = aoA,(XR, Xk)f&;
Xip)+ aoefj;.
64.1)
Here a0 is the amplitude parameter for the free long wave (the factor u2 is inserted on the left-hand side as the factor was previously inserted into the mean flow quantities in (2.19)), and A0 is the long wave amplitude whose evolution can be determined from (4.10). The resonance condition is
vat?,= wo+
64.2)
EX,
where x is a detuning parameter. For long wave resonance modulations in the direction of wave propagation dominate over transverse modulations. We choose & as a co-ordinate which varies in the direction of propagation, and t2 is then a transverse co-ordinate, and assume that A and A0 are functions of s, &, and 6; (~5~). It may then be shown that u2z& = -ao&
auOa dfo 1 + O(Uo&). w”*-p +foa2 )
(
(A.3)
Now the nonlinear term in the amplitude equation (3.28a) for B is {r, &} and in this context, the dominant contribution to this is {r, A\“‘} (4.19). Substituting (A.l) and (A.2) into (4.19) it follows that the leading term in {r, Mi} is {r, yacl)= aao@Bo,
(A.4a)
Bo= IZ~~~~oW*(~)‘dl(li2Ao
(A.4b)
where
and ~lsJ_~~ow*(~)2d=11’2=J~~( -2pow*lcwgh $($($)2+f2))
+[Po-*~(+2(~)2-f~)fo]z~,
dz
(A.4c)
The amplitude equation (3.28a) for B is thus i aB/as + A 1a2B/a& + pBBo = 0,
(A.%
where to balance the nonlinear term with the remaining terms we have put a0 = E2. The reason for introducing B. in (A.4a) rather than A0 will emerge below. To find the amplitude equation for A0 we substitute (A.l) into (4.10). Noting that D PC_ DT
v,$&++&$&) IX
as
(2
(A.6a)
102
R. Grimshaw / Modulation of wave packet in stratified frow
and
--a2 axi
where a&/ax:
-a& * a2 a*& a - (ax& >at1 ,zax -+W2), a45
(A.6b)
2+E
is parallel to t?, so that
a& v~--.= ax&
86; w,*(t2,-ax& >+
(A.7)
EX.
It follows that
(A.@ Here the omitted terms are also 0(&a& and occur due to the inhomogeneity of the basic flow; as these terms have not been displayed in (4.10), where they are of higher order than the terms retained we shall, for simplicity, not display them here. The balance of terms in (A.7) requires that uo& = a*; thus we have &=a 2/3 and a0 = u413. The boundary condition (4.14) is treated in a similar way, and we find that
uo( ki$)&(+‘ji;-~W~*~)+CT~U~(&--& =~m*(~~(&--$)~(
V$$32Aop,W:~+.
(~.0**-2o*~W~)(f*+-$($)‘)-
W$‘-$f*]
.-
onz =lo.
(A.9)
The boundary condition at z = -h is jj; + * * *= 0. Eq. (A.8) for 77; and the associated boundary conditions can only be solved if a compatibility condition is satisfied. This gives the required evolution for Ao: (A.lOa) where
(A.lOb)
R. Grimshaw / Modulauon of wave packet in stratified flow
103
Here Vaois the group velocity of the long wave mode (evaluated from (5.5) in the limit K + 0). The terms involving the inhomogeneity of the basic flow are accounted for by the transformation (A.4b); B. is the wave action for the long wave mode. We have also used (3.26) to convert IAl* into lBl* on the right-hand side. The equations describing long wave resonance are thus (AS) and (A.20a).
References [l] D.G. Andrews and M.E. McIntyre, “An exact theory of nonlinear waves on a Lagrangian-mean flow”, J. Fluid Mech. 89, 609-646 (1978). (1978). [2] D.G. Andrews and M.E. McIntyre, “On wave-action and its relatives”, J. Fluid Mech. 89,647-664 [3] D.J. Benney and A.C. Newell, “The propagation of nonlinear wave envelopes”, J. Math. Phys. 46, 133-139 (1967). [4] Yu.D. Borisenko, A.F. Voronovich, A.I. Leonov and Yu.Z. Miropolskiy, “Towards a theory of non-stationary weakly nonlinear internal waves in a stratified fluid”, Zzo. Atmos. Ocean. Phys. 12, 174-178 (1976) (English translation). [5] R. Grimshaw, “The modulation of an internal gravity-wave packet, and the resonance with the mean motion”, Studies in Applied Mathematics 56,241-266 (1977). [6] R. Grimshaw, “The stability of continental shelf waves I. Side band instability and long wave resonance”, J. Austral. Math. Sot. 2OJ3, 13-30 (1977). [7] R. Grimshaw, “Mean flows induced by internal gravity wave packets propagating in a shear flow”, Philos. Trans. Roy. Sot. 292A, 391-417 (1979). [8] R. Grimshaw, “The modulation of a weakly nonlinear wave packet in an inhomogeneous medium”, Math. Research Report No. 29, University of Melbourne (1979). [9] R. Grimshaw, “Slowly varying solitary waves II. Nonlinear Schrodinger equation”, Proc. Roy. Sot. 368A, 377-388 (1979).
[lo] R. Grimshaw, “Evolution equations for long, nonlinear internal waves in stratified shear flows”, Math. Research Report No. 1, University of Melbourne (1980). [ll] W.D. Hayes, “Conservation of action and modal wave action”, Proc. Roy. Sot. 32OA, 187-208 (1970). [12] D.J. Kaup and A.C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory”, Proc. Roy. Sot. 361A, 413-446 (1978). [13] R.S. Johnson, “On the modulation of water waves on shear flows”, Proc. R. Sot. 347A, 537-546 (1976). [14] M.E. McIntyre, “Mean motions and impulse of a guided internal gravity wave packet”, J. FZuid Mech. 60, 801-811 (1973). [15] A.C. Newell, “Soliton perturbations and nonlinear focussing”, in: Solitons and Condensed Matter Physics, Lecture Notes in
Solid-State Sciences 8, Springer, Berlin (1978) 52-67. [16] S.A. Thorpe, “On the stability of internal wave trains”, in: Deep-Sea Research, A Voyage of Discovery, Pergamon, Oxford (1977) 199-212. [ 171 S.A. Thorpe, “On the shape and breaking of finite amplitude internal gravity waves in a shear flow”, .I. Fluid Mech. 85,7-31 (1978). [18] V.E. Zakharov and A.B. Shabat, “Exact theory of two-dimensional self-focussing and one-dimensional waves in nonlinear media”, Soviet Physics JETP, 62-69 (1972) (English translation).
self-modulation
of