derivation of the nonlinear schrödinger equation for shoaling wave-groups

Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/83/004534-11 $ 3.70/0 Vol. 34, July 1983 �9 Birkhtiuser Verlag Basel, 1983

Derivation of the nonlinear Schr6dinger Equation for shoaling wave-groups

By Michael Stiassnie, Ralph M. Parsons Laboratory, Massachusetts Institute of Technology Cambridge, MA 02139 USA*

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

The aim of the present note is to provide a simple mathematical model for shoaling of nonlinear wave-groups, namely: modulated wave trains and wave packets. As an introduction we recall some well-known results obtained by linear wave theory.

The free-surface elevation r/for a shoaling linear water-wave field is given by:

co

rl(x,t) = S ~i(ho,co)'exp i{ i k(ho,co)dx -- co t} dco (1.1) 0 -xQo

where x - is the horizontal coordinate, and t - the time. The wavenumber function k and amplitude spectrum a are given by:

k t h ( k h o ) = o92/9; ~ = ~o~ "[th(kho) ' ( 1 + 2kho/sh(2kho))] -~/2,

and depend on the water depth ho (x) and the wave frequency co. Only real and positive values of k are considered and ao~ is the infinite-depth complex amplitude spectrum at the reference point xo~.

The term "wave-groups" means that we have wave-fields with narrow spec- tra in mind. Thus, first, we choose a frequency coo in the domain [091, coN] assuming that (con - co~)/coo = o (1). Second, we substitute in Eq. (1.1) the expressions:

COl m (_DO co = c o o ( l + e 7 ) , where s = o ( 1 ) and Y l = - - - O ( 1 ) ,

(.,O N - - (.D O ~'N = = 0 (1) .

k (ho, 09) = k o (ho) + COO

(~O~o/~ko) �9 Te + R ( h o , 7 e ) ' ( 7 ~ ) 2,

* On leave from the Department of Civil Engineering. Technion, I.I.T. Haifa 32000, Israel.

Vol. 34, 1983 Derivation of the nonlinear Sehr6dinger equation 535

where k o t h (k o ho) = cog~g; to obtain:

x

t/(x, t) = exp i { j" k o d x - 09 0 t} Xe~

"~i "~ a(ho,Y)'expi{c~ +(Tel 2 i Rdx}d7 (1.2)

where

,) z = e :,~ (OO~o/Sko) . (1.3 a)

Third, we assume a rather mild depth variation, so that h o = h o (0, where

= e 2 x. (1.3 b)

Finally, eq. (1.2) yields

t / = expi{ i k o d x - ~ ~v A(~,Y) ei'~176 dy (1.4) Xoc ~1

where (4, Y) = (ho, exp i J" R (4) dO.

The importance of the above results lies in the rather natural way by which the scaled variables z and ~, see eq. (1.3), are introduced.

In what follows we will, when necessary, distinguish between two types of boundary conditions at xoo. The first type, for which the amplitude-spectrum ~oo is a function of co in the ordinary sense of the word, corresponds to wave- packets which tend to zero as I~1--' ~ , The second type has its amplitude-

N spectrum given by a finite sum of Dirac delta functions (a~ = ~ an~ fi (7 - 7,),

n = l

where ?. are rational numbers) and corresponds to modulated wave-trains periodic in z.

In order to discuss the shoaling problem for nonlinear wave groups, (and thus allow for weakly nonlinear interactions among the various components), we turn in the next section to Whitham's modulation equations.

2. Modulation equations

Considering the 2-D problem of wave-groups propagating over water of slowly varying depth h o, the following five unknowns are usually chosen as dependent variables: the wave amplitude a, the wave frequency o, the wave number k, the average water depth h, and the current velocity U. To determine these unknowns we start from Whitham's set of modulation equations Whitham (1974), p. 556. Pseudo-phase consistency relation:

~U ~ g k a2 ~-~ + ~x [g(h - ho) + 2 s h ( 2 k h o ) + O(e4)] = O. (2.1 a)

536 M. Stiassnie Z A M P

Mass conservation equation:

(h - ho) St

+ o U + + o ( e ' ) = o .

Wave-action conservation equation: ,10[ s t + 0 ( " + b--;x G' - -

Consistency condition:

~k ~co

(2.1 b)

+ 0 (e'*)] = O. (2.1 c)

(2.1 d)

where, a = [g k t h(kho)] 1/2 is the linear dispersion relation, tr' = Sa/~k, and e is a typical wave steepness, e = O (a k).

Following Whitham (1974, p. 562) and including higher order dispersive terms which arise from the quadratic part of the Lagrangian G = co - a (see Whitham p. 526), the dispersion relation is given by:

g k s D a" ~2a g k2 (h - ho) + a 2 + 0 (e 4) (2.1 e)

co = t~ + k U + 2 o- c h 2 (k ho) ~ 2 a ~x 2

where D = (9 t h 4 (k ho) - 10 t h E (k ho) + 9)/8 t h 3 (k ho). Here, and in what follows, we assume that the small modulation parameter,

which is introduced via the boundary condition at x| (see e in the previous section), is of the same order of magnitude as the typical wave steepness. We allow for arbitrary total depth changes, but require mild bottom slopes, of the order of 8 2 at most.

3. I n d u c e d m e a n f l o w

To make the variation explicit and to facilitate the derivation we introduce the same multiple scale variables, z and 4, as used by Djordjevic' and Redekopp (1978), see also eqs. (1.3),

E "c = e f2' ~ = e2x. (3.1 a,b) (4) '

Where f2' is an averaged group velocity, to be defined in the sequel. Rewriting eqs. (2.1c) and (2.1 d) with the new coordinates (eqs. 3.1), and

averaging them over z gives:

O -t - - a z = constl = B; N = const2 = f2. (3.2a, b) (7

Vol. 34, 1983 Derivation of the nonlinear Schr6dinger equation 537

Here we assume that the behavior of the solution as a function of z is the same as that of the linear boundary condition at x~o. Namely, decaying for [z[ ~ oo in the case of wave-packets and with a constant period in the case of modulated wave-trains.

The bars indicate averaging over the appropriate domain in z (finite for modulated wave-trains and infinite for wave packets) and B, ~2 are the averaged wave-action flux and the so-called carrier frequency, respectively. Note that for wave-packets B = 0. We also define K, the carrier wave-number:

K t h (K ho) = Q2/g . (3.3)

Rewriting eqs. (2.1 a) and (2.1 b) with the new independent variables, yields:

gk z7 U} 2sh(2kho) a J --

g ka2 } 2sh(2kho ) = O,

~--~ { l Ig (h - ho) +

~ + s-~ g (h - ho) +

N oU+-~aaj-(h-ho) +e oU+~-~aa2 = 0 .

(3.4a)

(3.4b)

Neglecting, for the time being, the second terms in the above equations, we obtain:

_( kh0 gk ') h - h ~ \2sh(2kho) + 2a Jgho-(I2') 2 +f2'h~ 2 , (3.5a)

_ (g2 k 9_k__(2''] a 2 ~2' U = \--2-~a + 2sh(2kho) ,1 gh o - (0') 2 + (2' gG2g ho +_ (0') 2G~ (3.5 b)

where, G 1 and G2 are functions of 4, which emerged as a result of the integration. Now, averaging eqs. (3.4) and substituting (3.5) yields:

[ gK g ( OKho . gKe" 7a 2 s h (2K ho) g h o - (O') z \ 2 s h (2 K ho) ' 2 f2 /A

ho Gx + (2' G 2 + g O' g ho - (f2') 2 = c~ (3.6 a)

m [gK h_o_ ( (~_2K gK~'~' .'~]-~ gh o (2') 2 Ik-2- f f + 2s-~(2Kho),jj

gG2 + g2'G1 + h o g2' = const4 (3.6 b)

g ho _ (Q,)2

where, consistent with the order of approximation k and o have been replaced by K, g2.

538 M. Sfiassnie ZAMP

Fol lowing Stiassnie and Peregrine (1980) we assume zero averaged mass flow (thus, c o n s h = 0) and choose such a reference level that h - ho = 0 in deep water (which, in turn, sets const 3 = 0). Having fixed these tow constants we solve eqs. (3.6) for G 1 and G2 and then return to eqs. (3.5) to obtain the final results for the induced mean flow:

(g2 K g K O' "]. a 2 -- a 2 g K o 2 (3.7)

U (z, ~) = \-~--~- + 2 s h (2 K ho)] 9 ho - (f2') 2 2 f2 h o '

m

#_Kh o gKs a 2 - - a 2 K a 2 (3.8)

h ( z , r ho(~) = \ 2 s h ( 2 K h o ) + 2 0 J o h o - ( O ' ) 2 2 s h ( 2 K h o ) "

Consistent with our level of approximat ion we have

a-- ~ O / 0 --y for modula ted wave, t rains , = 0--- 7 B = / ~ a o o, (3.9)

(0, for wave-packets .

2 and O, is given in ap- A simple example, showing the calculations of a~

pendix A.

4. Nonlinear Schriidinger equation

To derive the N L S we rewrite eq. (2.1 c) as follows

2 a ~a a ' a 2 ~k 2 a ' a ~a a ' a 2 8a a 2 8a' a at a 8t + - - - - O. (4.1) a ~x 0 .2 ~x + - a - ~ x =

Applying eq. (2.1 d), which gives mutual cancellation of the second and fourth terms in the above equation, and dividing by 2 a/a yields

~a ~a a ~o-' --St + a ' ~x + 2 8 x - - = 0. (4.2)

The Taylor series expansion of o-(k) in the vicinity of k = K is

~'~,,

o- = f2 + 0 ' . (k - K) + -~ - - (k - - K) 2 + o ( ~ 2 ) . (4.3)

Using this series we rewrite eq. (4.2) as well as the dispersion relation (2.1e):

8a O' ~a K) ~a O" a ~ (k - K) a ~f2' e t + ~x + O"- (k - ~xx + 2 Ox + ~ 8--~- = 0, (4.4 a)

o9 = Q + O ' . (k - K) + - ~ - . ( k - K) 2 + ~x a2 O" 82a a ---i. (4.4 b) 2 a 8x 2 + ~z


Here g K3 9 t h 4 ( K h o ) - 10th2(Kho) + 9

c~1= 2Q 8tha(Kho)

- - \ - ~ + 2 s h ( 2 K ho) j " g h o - ( (2' ) 2

2sh(2Kho) + 2f2 ] "2f2[gh o - (f2')Z]chZ(Kho) ' (4.5 a)

~2 = ~ ~1

g k 2 g K 3

2 f2 ho 4 0 s h (2 K ho) c h 2 (K ho)

g K 3 9 t h 4 (Kho) - 10 t h 2 ( K ho) + 9 + 2---~ 8 t h 3 (K ho). (4.5 b)

Referring to eq. (2.1 d) we define a phase function 0 so that

00 00 o - f2 = - S t ' k - - K - 0 x " ( 4 . 6 )

Substituting eq. (4.6) into eqs. (4.4 a) and (4.4 b) we obtain

~a I2' 8a f2" f 020 00 ~a \ a 012' 0 t + ~xx + -~- ~a ~5x2 + 2 ~xx ~xx) + ~ 0---~ = 0, (4.7a)

Ot + ~ + - ~ \\-~x] a OxZJ + el + a2 a2 = 0. (4.7b)

Multiplying eq. (4.7 a) by the imaginary unit i, adding to it ( - a) times eq. (4.7 b) and then multiplying the sum by e i~ we get

+ i ( A , , + A , ~ ) + - ~ - A , , ~ - - a l [ A I 2 A = e z - ~ A (4.8) \oxj where A = a e *~ is a complex wave envelope. Alternatively, using the scaled coordinates (3.1 a, b) we have

2f2---- 7 O---~A + iA,r + 2 - ~ A , * * f2' [AI2A = ~ 7e-2~-~A" (4.9)

The last equation is almost the same as that obtained by Djordjevic' and Redekopp (1978) (Note that their A is g/2f2e times the one in eq. 4.9). In principle, after solving eq. (4.9) one can go back to eqs. (3.7) and (3.8) and obtain the induced mean current velocity U(z, ~) and the mean free surface h (z, 0 - ho (0 right away. These induced mean flow quantities are of great practical interest, since they are probably related to such phenomena as surf beats, long shore cellular structure, and harbor resonance.

540 M. Stiassnie Z A M P

5. Comparison with Djordjevic' and Redekopp

Djordjevic' and Redekopp used a multiple scale method and started their derivation from the Laplace equation for the Velocity potential and the nonlinear free-surface boundary condition. Their approach is basicly the same as the one used by Davey and Stewartson (1974) for water of constant depth. Equation (4.9) has a known function of ~ on its r.h.s, whereas previous authors did not attempt to determine this term explicitly (except for wave-packets, for which it is identically zero). The way in which eq. (4.9) enables to obtain the well-known monochromatic wave-train solution is shown in appendix B.

In order to obtain eq. (4.9) from Djordjevic' and Redekopp's results we start from their eqs. (2.11) and (2.12) given, in our notation, by:

i ~f2' f2" ~-281 2(2' ~ A + i A, r + 2 - ~ A, ,, 0-----7--Ihl z a

83 = 8~, qho,,A + f f ~b (~)A, (5.1)

e-2 f 821A12 + (2 (0 (5.2)

qho,~ = 20 [1 - Oho/(O') 2] where

9 K3 9-12th2(Kho) + 13 th4(Kho) - 2th6(Kho) 81 = 2f2 8 th3(Kho) , (5.3.a)

8z = ~-O + sech 2 (K h o , (5.3.b)

K 2 83 = ~ sech2 (K ho). (5.3.c)

Note that the last term on the r.h.s, of eq. (5.1) does not appear in Djordjevic' and Redekopp. In the sequel, we will show the necessity of including the term &~5 (0" t in the wave-induced mean current velocity potential., ~o o, which is, thereupon, given by:

q~o = e (~ (z, ~) + ~2 q3 (4)" t + s 2 ~02o (z, 4) + O (~3). (5.4)

To second order in e, the wave induced mean current velocity is given by

0 2 8z Ial z ~2 Q (5.5) U = ~ ~01o,~ = 2 f2 f2' [1 - 9 ho/(f2') 2] + 0---7-

To find Q, we impose a lateral boundary condition of zero averaged (over z) mass flow, which is appropriate for an impervious beach, as follows:

g K ~-~ = O. (5.6) ho 0 + ~--~


From eqs. (5.6) and (5.5) we obtain

Q ( r _2a-- 2 ~gKQ' 92~2 1 (5.7) (2 f2 h o + 2 f2 [1 - O hol(f2')2]J"

Substituting eq. (5.7) into eq. (5.5) we recover eq. (3.7). Integration of eq. (5.2) with respect to z yields

e-292f12 ,2 SIAl 2dz + Q ( r (5.8) qh0 = 2 y 2 [ f ~ g-~o/(f2) ]

The first and second terms in the above equation grow monotonically, and boundlessly with time. Secular terms of this nature are bound to cause troubles in higher order derivation and should be suppressed. The addition of the term e 2 q5 (4) t to eq. (5.4) seems to be the proper way to achieve this goal. Substituting eq. (5.8) into (5.4) gives:

92fl2 i l A x !(IAI2 ~ ) d t _ ~ - 7 = 2 a [ 1 xo

xo ~?'-"7 - - t "~ 82 p + e2 ~ t + g2 ~O2o" (5.9)

Thus, suppressing secular terms in t we get:

~-2 92 f12a2 ~-2 g K ga' a2 = + Q = (5.10)

2 f2 [1 - 9 ho/(f2')] 2 f2 h o

Following Djordjevic' and Redekopp's derivation, but including ~, one can recover eq. (3.8) for the average water surface level. Finally, substitution of eq. (5.7) for Q, and eq. (5.10) for ~, into eq. (5.1), leads to an equation identical to eq. (4.9). Note that by suppressing secular terms in x one can also show that

P = - g# ~ dx/f2'. Anyhow, this seems to be of less importance, since P does not XO

contribute anything to our third order derivation, in contrast to q~ which con- tributes to the lowest order average water surface level.

6. Simple cases

A simpler and dimensionless form of eq. (4.9) is obtained by means of the transformations

\ ~ f A - e x p i ~i ~ ; d , (6.1a)

T = O'c = eg2 ( i dx -O' - ' (6.1b)

542 M. Stiassnie ZAMP

X = I ? 2 S 2 ( - ~ de : e2f22 I ~ d x , (6.1c)

which yield __ g3 ~'-2' ~1

i~',x + O, rr + ~1~'120 = o; ~ (x ) = f27Q,, (6.2)

The dimensionless parameter p is a monotonic increasing function of K h0, having the values zero and one for K h 0 = 1.363 and K h 0 ~ ov respectively. The free surface elevation is given by:

~/= e Re{~ exp i(i~ K d x - Q t ) } . (6.3)

In order to complete the mathematical statement of the "shoaling wave-group" problem one has to supplement eq. (6.2) with a boundary condition at X = 0 (x = xo~). We suggest to focus the attention to the following two, relatively simple, model problems.

(i) Modulated wave-trains

Here, following the approach of Stiassnie & Kroszynski (1982), we consider a system initially composed of a carrier-wave and a symmetric "side-band" distur- bance. The water surface elevation of such a system at x = xo~ is given by

rl(t, x J = eK~ I Re {e - ~ t + fie -~t~l +r~)a~-~] + fle-iUl-Y~a'-~l}. (6.4)

The appropriate boundary condition for ~k is

~b (X = 0) = 1 + 2 ~ el~ cos (V T), (6.5)

which is periodic in T with period 2 rc y-1. 2 (eK2ol)2(1 + 2~ 2) for Applying the results of Appendix A one finds that a~o =

this case. Note also that/~ = 0 corresponds to the monochromat ic wave-train solution, presented in Appendix B.

(ii) Wave packets

For this case we follow Grimshaw (1979) and take as input a deep water soliton, which is an exact solution of eq. (6.2) for # = 1. The appropriate boundary condition becomes:

(X = 0) = ~- sech (8 T/V~). (6.6)

Analytical solutions of these two model problems are currently studied and will be reported in future publications.

This work was supported by the European Research Office, U.S. Army, under contract no. DAJA37-B2-C-0300.


2 and f2. Appendix A: Calculat ion of a~

Referring to the nota t ion of section 1 we consider the following simple l inear-group in infinitely deep water, N = 3 : T z = - 1 , 72 = 0 , ? 3 = 1, a l ~ = a3o ~ = q, and a2~ = Q. F rom eq. (1.2) we have

tl = e x p i {kox -- COot } �9 {qe ic-'~176 + Q + q e i(r176176 (A.1)

periodic in -c with period 2 ;c/co o. If we rewrite this equat ion in the form

t / = a~o exp i {k o x - COo t + aoo} (A.2)

we have

ao~2 = Qz + 4 q2 cos z (09 0 z) + 4 q Q cos R cos (COo r);

~0~co . CO = CO~ q- ~ OZ

Averaging over -c yields

a~ = QZ + 2 q2

2 q sin R c.os (COo'C) ~o = t g - 1 Q + 2 q cos R cos (coo z)J

(A.3 a)

(A.3 b)

f2 = r o . (A.4)

Appendix B: The monochromat ic wave-train solution.

For this case A = A(~), a 2 = [A[ 2, and from eq. (4.9) we have

1 ~f2' - - A + i [2'A,r = ~-2(a 1 + :r (B.1)

2" de

Recalling that A = a e i~ the imaginary and real parts of eq. (B.1), respectively, are

1 012' f2,0a _ f2, 00 ~. 0r a + ~--~ = 0, ~-~ = (~1 + e 2 ) e - Z a ; (B.2a, b)

where (~1 + 0r is given in eq. (4.5 b). The solut ions of these eqs. are

(2' a z const , k = K (~1 + ~2) = a 2 ' f2'

(B.3 a, b)

as they should be.

References

[1] A. Davey and K. Stewartson, On three-dimensional packets of surfaces waves. Proe. R. Soc. A338, 101-10 (1974).

[2] V. D. Djordjevic' and L. G. Redekopp, On the deveolpment of packets of surface gravity waves moving over an uneven bottom. J. AppI. Maths. Phys. 29, 950-62 (1978).

544 M. Stiassnie ZAMP

[3] R. Grimshaw, Slowly varying solitary waves, II. Nonlinear Schri~dinger equation. Proc. R. Soc. A368, 377-88 (1979).

[4] M. Stiassnie and U. I. Kroszynski, Long time evolution o f an unstable water wave train. J. Fluid Mech. 116, 207-25 (1982).

[5] M. Stiassnie and D. H. Peregrine, Shoaling o f finite-amplitude surface waves on water o f slowly- varying depth. J. Fluid Mech. 97, 783-805 (1980).

[6] G. B. Whitham, Linear and nonlinear waves. Wiley-Interscience, New York 1974.

Abstract

A nonlinear Schr6dinger equation with varying coefficients, describing the evolution of surface wave groups moving over an uneven bottom, is derived from Whitham's modulation equations. The derivation yields new expressions for the wave-induced mean flow field.

Zusammenfassung

Ausgehend von Whitham's Modulationsgleichungen wird eine nichtlineare Schrrdinger- Gleichung mit verfinderlichen Koeffizienten hergeleitet, welche die Entwicklung von Oberfl~iehen- weUengruppen beschreibt, die sich fiber einen unebenen Boden fortpflanzen. Die Herleitung ergibt neue Ausdriicke fiir das wellen-induzierte mittlere Str6mungsfeld.

(Received: October 8, 1982; revised: March 23, 1983)

derivation of the nonlinear schrödinger equation for shoaling wave-groups

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