Hydrodynamics V
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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 OFFPRINT IAHR1- S=== @ a' AIRH 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<I, (9) Py(t,t) the.'knowq,atposphericpressure. By'substit;rting For l ( i ( fl, (27)-(28) determines the evolu- (16)-(17)into (2), one obtains tion of 2-l[-un]nowns, E4 and.q;, while Pais given by the following recursionformula obtainedfrom u[o):ulo)@,t) if urLo):o (17) (26): ,at 1:ou, and - - Notice that condition (18) is automatically sat- Pi+r : Pt,i pr (ono Lon*o" uor,n). (29) isfied if we assumethat the flow is initi{tlly'ino- tational (Choi & Camassa1996). From ({), *e Theseequations have been derived under the sole ca,nnovv obtain the leading-order vertical'velodity assumption that the waves a,re long compa,red ut(o) tutt.*tot the kinematicboundary cbndition wlth the'water depth and we'have imposedno (5) at the interfaceas assumption on wave amplitude. The kinematic equation (27) is exact, while the dyna,rnicequa- .[o) : -1uaf))Q - rto+r,)* D$1r+t, (19) ti6ns (28)-(29) have non-hidioStatic contribu- tioirs coriect up to O(e2). where D; stands for material derivative, Even for rigidJid approximation ((r : 0), the ' Dt: 0t t un(016, (20) system :ot (27\-(28) is still tralid if Pr is regarded as unknown pressureat the top bounda,ry. Fbom (16) and (18), it can be easily shownthat 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- .,.