PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON ITYDRODYNAMICS / YOKOHAMA / 7-9 SEPTEMBER 2OOO HYDRODYNAMICSV

Theory andApplications

Edited by Y.Goda, M.Ikehata& K.Suzuki Yoko ham a Nat i onal Univers ity

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ICHD2000Local OrganizingCommittee lYokohama I 2000 Modeling of Strongly Nonlinear Internal Gravity Waves

WooyoungChoi Theoretical Division and Center for Nonlinear Studies Los Alamos National Laboratory, Los Alamos, NM 87545,USA

Abstract being able to account for full nonlinearity of the problem but this simple idea has never been suc- We consider strongly nonlinear internal gravity cessfully realized. Recently Choi and Camassa waves in a multilayer fluid and propose a math- (1996, 1999) have derived various new models for ematical model to describe the time evolution of strongly nonlinear dispersive waves in a simple large amplitude internal waves. Model equations two-layer system, using a systematic asymptotic follow from the original Euler equations under expansion method for a natural small parame- the sole assumption that the waves are long com- ter in the ocean, that is the aspect ratio between pared to the undisturbed thickness of one of the vertical and horizontal length scales. Analytic fluid layers. No small amplitude assumption is and numerical solutions of the new models de- made. Both analytic and numerical solutions of scribe strongly nonlinear phenomena which have the new model exhibit all essential features of been observed but not been explained by using large amplitude internal waves, observed in the any weakly nonlinear models. This indicates the ocean but not captured by the existing weakly importance of strongly nonlinear aspects in the nonlinear models. Differences between large am- internal wave propagation. An interesting point plitude surface and internal solitary waves are ad- to make is that the including strong nonlinear- dressed. ity does not simply add new nonlinear terms but change dispersive characters, from linear to non- 1 Introduction linear. Despite of recent progress, the two-layer sys- Two fundamental physical mechanisms, nonlin- tem is often too simplified to be useful for prac- earity and dispersion, play important roles in tical applications. For instance, the two-layer the evolution of long internal gravity waves com- model has not only its limited despcription of monly observed in the ocean, more generally in smoothly stratified ocean but also the lack of any stratified fluids. Under the assumption of higher-order baroclinic wave modes. Therefore weakly nonlinear and weakly dispersive, the bal- the theory needs be extended to the more general ance between these two effects leads to a re- situation relevant for real oceanic applications. markable phenomenon of solitons, very localized Here we will study a continuously stratified fluid disturbances propagating without any change in approximated by a stack ofseveral homogeneous form. However there have been a number of layers. observations of large amplitude internal waves for which the weakly nonlinear assumption is far from being realistic. For example, in the north- 2 Governing Equations western coastal area of the Unites States, Stan- For an inviscid and incompressible fluid of den- ton & Ostrovsky (1998) observed the train of sily pi, the velocity components in Cartesian tidally-generated internal waves in the form of coordinates (ut,wt) and the pressure pt (i : solitary waves depressing the 7m deep pycnocline 1,' . . N) satisfy the continuity equation and up to 30m. Evidently the weakly nonlinear mod- , the Euler equations: els like the Korteweg-de Vries (KdV) equation commonly used in the community of geophysi- ui, wi" (1) cal fluid dynamics are completely irrelevant in * 0, such cases and new models need to be devel- u6 * uiu6* * wiui" -pi.rlpi, (2) oped. The theory to be desired is of course one u61luiwi**w6w4" -ptr/pt-9, (3) where .L is a typical wavelength' For finite- amplitude waves' we also assume that u;lus: OG;'lh): O(1)' (10) : where LIs is a characteristic speed chosen as [/g scalings in (9)-(10), we Gh)t|". Based on the non-dimensionalize all physical variables as

n : Lr*, z : hz*, 1: (LlUs)t., ) (tl) Ci: hei, pi: biu&)pi, l ui : Uout, wi : eUowi' )

and assume that all variables with asterisks are O(1) in e. by integrating (i)-(2) across the i-th laver Figure 1: A multi-laYer sYstem the boundary con- ?lt+r < z < rli) and imposing ditions (4)-(6)' we obtain the layer-mean equa- acceleration and sub- where g is the gravitational tions (Wu J.981): scripts with respect to space and time represent For a stable stratiflcation' (12) puriiul difierentiation. uu+ (Hrtr),:0, Ht:nt- Tr+t, pi < pi+\ is assumed . The boundary condi- the i- -HIFG, (13) iions at the upper and lower interfaces of (Hfia)t+ (ntun;),: th layer require the continuity of normal velocity is defined as and pressure: where the layer-mean quantity / 2:r1i@'t)' (4) nit+ucrlir:wi, at (r,z,t)dz, (14) !(n,t1: ; I::.,f : at 2 : r1aa1(t't)' (5) \t+t1l uirli+lr ui, and we have dropped the asterisks for simplic- pi = pi-r at 7 : r1i(x',t) (6) ity. The quantities ufi and p1r prevent closure the location In (4)-(6), nt@,t) (d : 1,''' , N) is oi the .yste* of layer-mean equations (12)-(13)' given by of ttr" tpp"t interface of the i-th layer The following analysis will therefore focus on ex- pressing these quantities in terms of 14 and ul to nr: et-'i n,, (7) close the sYstem. j=t The vertical momentum equation (3) for the be written as i-th i-th layer can where h1 is the undisturbed thickness of the interfacial displacement' Iayer and (1(r,t) is the : -L - "l*0, * u'iwir r wi'wi"f' (15) be Pt, TLe position of the bottom, nv+t(n,t) can written as Hence, we can seek an asymptotic expansion of e2, N f : (ua,wi,pt) in Powersof : (n+r -Dni: (r+r - h, (8) ?N+1 (16) j=1 !(r,z,t):1(o) a e27$)*Ot'+''

(1,,11(z,t) is the bottom topography and :0, 1,' ' " where where /(-) : O(1) for m h is the total dePth. Ilom (15)-(16), after imposing the pressure (6)' continuity across the interface given by the 3 Nonlinear DisPersive Model leading-order Pressrue Pr(o) is (o): -e - rt)+ Pi@,t), (17) Under the assumption that the thickness of each p Iayer is much smaller than the characteristic where P;(r, t) : paa(x,r1l,J) is the pressureat wavelength, we have the following scaling relation of the ith layer' For i : 1' from the continuity equation the upper surface betweerrzl arrd wi, p1 : since the upper boundary is free, we impose (1), at' z :4r, instead of (6), where P1 is wtlut : O(hil L) : O(e) 4

Hru,-U: H;'E'idt+ O(eA), (2L) 4 Hydrostatic Approximation so that the layer-mean horizontal mormentum When we neglectdispersive effects (say ai : bi: equation (13) fn dlmbnsionlessform becomes 0), we recoverthe hydrostatic equationsfor mul- (i : il**Etdu - -P;; + O(ea). (22) tilayersystem L,." , N): At O(e2),from (15)-(16) and (19), the equa- H6+ (HtU),:0, Ht=qa-rli+r, (30) tion for pr(1)is du'41*Tilfra, -l /ort + SL Hj, * grtt+t, puLi) -L.ulo'+ui(o)ui(o) +wn@wn@ll \-r)r i=l rt : aa@,t)(z- \e+i 'f ba(r,t)' (23) = -Prrlh,(31)

In equation (23), a; and bl are definedas where (29) has been used. ' a6(x,t):-u;6 *fr;Ttir, - (6tn)' , (24) ba(x,t): -D;2rh+t, (25) 5 One-layer System where we'have used E; : vo(o) + O(e2)' After For the'.case of a honogeneous 'layer (1V - 1), integrating (23) with respect to z and imqosing equations (27)-(28) reduce to the system of equa- the dyna,mic boundary condition in (6), we ob- tions derivedby Green& Naghdi (1976)by using tain pa(1); the director sheet theory. Without any topogra- pressure - - - phy- at the bottom gnd exlernal at the p'(L): t"nl@ nr+t)2(no nn*r)"] &&eesurface (ez : h,: 0), equations(27)-(28) + urlk- q+L).-(a - ru,)] . (zo) possessthe solitary wave solution given by (r(x) : aseeh2(kx), X : n - ct, (32) Fbom'(lD'and (26), we can find ffi;',the right-hand-si{g term of the horizontal moderr- .,. tum equatiohlp2-);and the set cif equations k2h?:,,=3o 4 :t'* !. (g3) $nat 4(ht *a)^ gfu : h1'l for the i-th, lay"g;,can,befgund, in dimensional rorm. as where a irnd c. a,re wave;a,mplitude and'speed, r6sliectively. uar+(n6a);' *J,,',:l'l' *r: rti-qi+t, For weakly nonlinear unidirectional waves of \ /Z .,, ,t,, e!T) !.. -. | ., a/h - O(ez), (27)-(28) can be r€drfced to the a4*ilafr,6*.fffii.1*to$*q/,fy,1r,,. r,,; KdV equation.for (1: : qn*,,. G,rl", + t:.n?!:) *.bn) A2 i ?.+,,!j"*n L , (rr+ 3, +' o. (34) (28) herr4 fterfu, uJ,Cg,+: ,455 1.5 seemto be generic in strongly nonlinear t.4 f;j;J:

1.3 ./'t/ 6 Two-layer System 1.2 As shownin Choi & Camassa(1999), with rigid ,t lid approrimatio-n((1 : 0),.thecoupled system of (27)-(28) for i : !;2 can be reduced,for traveling 0.2 wavesat the inte:face, to ll2x)-: Figure 2: Wave speed (clt/gTi versuswave arn- sCSlczbrHz+pzTi - gbz- piHflz] plitude (a/h1) curvesfor surfacesolitary waves: ,\Jo),r^, -, strongly nonlinea,rtheory given by (33); - - -, - weakly nonlinear (KdV) theory given by where-EI1 ht - Czand Hz: hz * (2. This pa.r- (35). ticula^rform of equation has been obtained ea,r- lier by Miyata (1985)using conservation laws, for steayflows. 0.8 When replacing(2 by waveamplitude a in (36), the numerator,has to vanish and this gives wave 0.6 speedc in terms of wave amplitude o as r "' (n, - ")(hz + a) ''0.4 _ (r'lte.7\ &: hrh"-@/d"r where cs is the linear first baroclinic wave speed 0.2 given by z shthz(n--;-':- p) ,d^\ co':" ;- iJb, 0 Prn2+ Pzu 12t.056 For uni-directional weakly nonlinear waves of alh2: o(e2), (27)L(28)for i : 1,2 can be fur- Figure 3: Surface solitary wave solutions (- ther simplifiedto the KdV equationfor (2 as be- -) given by (32) for af h1 :(0.2, 0.41 0.6, 0.8) fore: compa,red with KdV solitary waves (- - -) of the * qCzn* ctezez,I c2e2*x:0 (39) same amplitude. Here we show only half of the ezt , wave profile. where 3co mhZ- pzh? It is interesting to notice that the solita,ry wave "t:- (+u///n\ 2 pthrhlrlnfitr' solution of the KdV equations has the sa,rreform conh?hz + pzhth\ as that of the GN equations give by (32) with "2:, (+r/| t1\ 6 ^h"-r -h' 3a c a K-ni:,t,, t*. (rD//^p\ 4hr' ffi: zhr' The solitary pa.veBo\ution of (39) is givenby (32) with is the srnall amplitude limit of (33). It ACt C1 which lS'.: c = co * 142) As shown in figure 2, the solita,ry wave speed 86,,,. Va. from the strongly nonlinear theory increases with Figures 4 and 5 clearly show that strongly non- wave amplitude at a much slower rate than that linea,r solitary waves are slower and broader than from the KdV theory. In figure 3, one can see wea,kly nonlinear waves of the same amplitude. that the strongly nonlinear solita,ry wave solu- In fact Choi & Ca,rrassa (1999) have demon- tion given by (32) is certainly wider than the strated that the strongly nohlinear theory yields KdV soliton of the same amplitude. In $6, we excelle[t; 4greepgpt twith' available expeimental will make the simila.r observations for interfacial data or numerical solutions of the Euler equa- solita"ry waves in a two-layer system and these tions. 456 t.8

1.6 T,5

v 1.4

t.2 0.5

I I 0 a 0 x Figure 4: Wave speed (c/c6) versus wave am- plitude (af h2) atrves for internal solita,ry wave Figure 6: A front solution of (36) for maximum (nlpz : 0.99, h1fh2: 5): -, strongly non- wave amplitude given by (43) for p1f p2: 0.99 linear theory given by (37); - - -, weakly nonlin- and fuf h2:5. ear (KdV) theory given by @2). which is often contributed to the effects of viscos- ity. On the other hand, for internal waves, the 1.5 front-like solutions of (36) hold the conservation 1.25 law for energy.

\z 0.75 7 Discussion

0.5 For small aspect ratio of the thickness of each 0.25 fluid layer to typical wavelength, we have derived a strongly nonlinear model to describe the evolu- 0 0st:rs20tion of finite amplitude long internal waves in a multi-layer system. The limiting behaviours of highest waves are Figure 5: Internal solitary wave solutions (- completely different between surface and inter- -) of (36) for hlpz:0.99, h1lh2: 5 and nal waves. As shown by Stokes (1880), surface a/h2 :(0.4, 0.8, 1.2, 1.6) compared with KdV waves of maximum amplitude have a sharp peak solitary waves (- -) of the same amplitude given of 120o,which cannot be captured by the present by @2). long wave theory. Therefore one can see that the GN theory for long surface waves ceases to be valid when the waves become too high. On the For a two-layer system, since wave amplitude other hand, as wave amplitude increases, inter- cannot be greater than the total depth, there is a nal solitary waves becomes broader and broader, maximum wave amplitude beyond which no soli- leading to internal bore. Therefore the long wave tary wave solution exists. As wave amplitude ap- theory for internal waves is expected to be valid proaches to the limiting value, the solitary wave even for large wave amplitude. This explains why becomes broader and finally degenerates into a the strongly nonlinear theory for internal waves front-like, or internal bore, solution, as shown in gives much better agreement with experimental figure 6. In this interesting limit, the amplitude data or numerical solutions of the Euler equa- and speed can be found, from (36) and (37), as tions (Choi & Camassa 1999) compared to that . .1 ht - h2\pr/ p2), for surface waves. *^^ -_ -i (43) (;iJl-' This work was supported by the U.S. Depart- ment of Energy. c*2: s(h1+ nzlL-!! .d1. e4) L + \Pr/P2)2 References For an inviscid homogeneous layer, front solu- tions are impossible without the loss of energy [1] Choi, W. & Camassa,R.: Weakly nonlinear 457 internal waves in a two-fluid system. J. Fluid Mech. 3L3, (1996), 83-103.

[2] Choi, W. & Camassa, R.: F\rlly nonlinear internal waves in a two.fluid system. J. Fluitl' Mech. 396, (1999), 1-36.

[3] Green, A. E. & Naghdi, P. M.: A derivation of equations for wave propagation in water of variable depth. .L Fluid Mech.78, (1976), 237-246.

[4] Miyata, M.: An internal solitary wave of large amplitude. La mer 23, (1985), 43-48.

[5] Stanton, T. P. & Ostrovsky L. A.: Obser- vations of highly nonlinea,r internal solitons over the continental shelf.. Geophys. Res. Lett. 25, (1998), 2695-2698.

[6] Stokes, G. G.: Supplement to a paper on the theory of oscillatory waves. Mathematical E Physical Papers 1, (1880), 314-326.

[7] Wu, T. Y. : Long waves in ocean and coastal waters. J. Engng Mech. Diu. ASCE lO7, (1981),50L-522.

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