Asymmetric behavior of surface waves induced by an underlying interfacial wave
AAsymmetric behavior of surface wavesinduced by an underlying interfacial wave ∗ Shixiao W. Jiang † Gregor Kovaˇciˇc ‡ Douglas Zhou § July 29, 2019
Abstract
We develop a weakly nonlinear model to study the spatiotemporal manifesta-tion and the dynamical behavior of surface waves in the presence of an underlyinginterfacial solitary wave in a two-layer fluid system. We show that interfacialsolitary-wave solutions of this model can capture the ubiquitous broadening oflarge-amplitude internal waves in the ocean. In addition, the model is capable ofcapturing three asymmetric behaviors of surface waves: (i) Surface waves becomeshort in wavelength at the leading edge and long at the trailing edge of an under-lying interfacial solitary wave. (ii) Surface waves propagate towards the trailingedge with a relatively small group velocity, and towards the leading edge with arelatively large group velocity. (iii) Surface waves become high in amplitude atthe leading edge and low at the trailing edge. These asymmetric behaviors can bewell quantified in the theoretical framework of ray-based theories. Our model isrelatively easily tractable both theoretically and numerically, thus facilitating theunderstanding of the surface signature of the observed internal waves.
Keywords. interfacial waves, surface waves, ray-based theory
Mathematics Subject Classification.
Internal waves with large amplitudes and long wavelengths are widely observed in coastalocean regions, and are believed to be important for transferring momentum, heat, andenergy in the ocean [28, 30]. Because of their strong turbulent mixing and breaking,they can influence many ocean processes, such as nutrient supply, sediment and pollu-tant transport, acoustic transmission, and interaction with man-made structures [3,20].Generated by tidal flow, internal waves usually can propagate thousands of kilometersfrom their source before dissipation, sloping, and breaking extinguish them [5, 28]. Inrecent decades, observation data for internal waves and their corresponding surface sig-nature have been recorded using in situ measurements and Synthetic Aperture Radar(SAR) in many coastal seas worldwide [3, 20, 38]. ∗ Received date, and accepted date (The correct dates will be entered by the editor). † Department of Mathematics, the Pennsylvania State University, University Park, PA 16802-6400,USA, ([email protected]). ‡ Mathematical Sciences Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NewYork 12180, USA, ([email protected]). § School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao TongUniversity, Shanghai 200240, China, ([email protected]). a r X i v : . [ n li n . PS ] J u l .W. Jiang, G. Kovaˇciˇc, and D. Zhou Traveling wave solutions : Traveling-wave solutions exhibiting oscillations arefound in the reduced models. For example, generalized solitary waves with non-decayingoscillations along their tails in addition to the solitary pulse were found in a long-wavemodel [18, 22, 39]. Besides the generalized solitary waves, multi-humped solitary waveswith a finite number of oscillations riding on the solitary pulse were found in a fully-nonlinear long-wave model [8].[b]
Ray-based theories : Many ray-based studies take a statistical viewpoint of SWsmodulated by a near-surface current induced by IWs [5, 6, 9, 24, 37]. These studiesinvoke phase-averaged models based on a wave-balance equation and ray-based theory[6, 9], which can be applied to the remote-sensing observations of IWs via their surfacemanifestations.[c]
Resonant excitations : When two different modes coexist in a fluid system, aresonant interaction becomes possible between the modes to aid in transferring energyin the ocean [41]. Class 3 triad resonance is regarded as being responsible for thesurface signature of the underlying IWs [1, 16, 23, 27, 38, 41]. Based on a class 3 triadresonance condition, many reduced models have been derived for the interfacial andsurface waves [17, 23, 27, 29, 32, 36, 37, 42]. A detailed discussion of surface signaturephenomena of IWs was presented in [14–17] using a coupled Korteweg–de Vries (KdV)and a linear Schr¨odinger model. Narrow rough regions containing surface ripples werefound and interpreted as a result of the energy accumulation in the localized boundstates of the Schr¨odinger equation [17].Different from previous works, we here develop a new reduced model to investi-gate the spatiotemporal manifestation of the small-amplitude SWs in the presence ofan interfacial solitary wave in the two-layer setting. Based on model simulations, wedemonstrate that our model is successful in characterizing many types of dynamicalbehavior of SWs, which can be well understood using the ray-based theories. In Sec.2, we derive a reduced model for the two-layer fluid system. In Sec. 3, we analyze thebasic properties of the model, including interfacial solitary-wave solutions and disper-sion relations. In Sec. 4, we present the numerical scheme and examine its numericalconvergence. In Sec. 5, we show the numerical results for the asymmetric behavior ofSWs and quantify this asymmetric behavior using the ray-based theories. Conclusionsand discussion are given in Sec. 6, and some mathematical details are presented in theAppendix. .W. Jiang, G. Kovaˇciˇc, and D. Zhou We first introduce Euler equations for two immiscible layers of potential fluids withunequal densities. The two layers of fluids are assumed to be inviscid, irrotational, andincompressible. The unequal densities for the upper layer and for the lower layer aredenoted by ρ and ρ , respectively. Here, ρ > ρ is assumed for the stable case. Thehorizontal and vertical coordinates are x and z , respectively. We focus on the evolutionof large-amplitude interfacial waves ξ between the two fluid layers, and their couplingwith the the overlaying free surface, ξ [see Fig. 1]. The velocity potential φ i ( i = 1 forthe upper layer and i = 2 for the lower layer) satisfies Laplace’s equation, φ ixx + φ izz = 0 . (2.1)The kinematic equations for the continuity of the normal velocity at the surface h + ξ ,the interface ξ , and the flat topography − h are given in the form ξ t + φ x ξ x = φ z , at z = h + ξ , (2.2) ξ t + φ x ξ x = φ z , at z = ξ , (2.3) ξ t + φ x ξ x = φ z , at z = ξ , (2.4) φ z = 0 , at z = − h , (2.5)where h and h are the undisturbed thicknesses of the upper and lower layers, respec-tively. The dynamical equations for the continuity of pressure at the surface and theinterface are the Bernoulli equations, φ t + 12 (cid:0) φ x + φ z (cid:1) + gξ = 0 , at z = h + ξ , (2.6) ρ (cid:18) φ t + 12 (cid:0) φ x + φ z (cid:1) + gξ (cid:19) = ρ (cid:18) φ t + 12 (cid:0) φ x + φ z (cid:1) + gξ (cid:19) , (2.7)at z = ξ . where g is the gravitational acceleration.For the small-amplitude approximation, we assume that the characteristic ampli-tude, a , of the IWs and SWs is much smaller than the thickness of the two fluid layers, a/h = α (cid:28) , h /h = O (1) . (2.8)For the long-wave approximation, we assume that the thickness of each fluid layer ismuch smaller than the characteristic wavelength, L , of the IWs and SWs, h /L = β (cid:28) , h /h = O (1) . (2.9)The two small parameters, α and β , control the nonlinear and dispersive effects, re-spectively. Based on the scaling (2.8)-(2.9), we may nondimensionalize all the physicalvariables by taking the original variables to be x = Lx ∗ , z = h z ∗ , t = ( L/U ) t ∗ , ( φ ,φ ) = ( aLU /h )( φ ∗ ,φ ∗ ) , ( ξ ,ξ ) = a ( ξ ∗ ,ξ ∗ ) , (2.10) .W. Jiang, G. Kovaˇciˇc, and D. Zhou ρ , φ ρ , φ h h ξ ξ z x C s near field far field far field z = 0 Figure 1: Sketch of the two-layer fluid system [see text].where U = √ gh is the characteristic speed of the gravity waves. Here, all the variableswith asterisks are assumed to be O (1) in α and β . In the dimensionless, starred variables,Laplace’s equation (2.1) is formulated as βφ ∗ ix ∗ x ∗ + φ ∗ iz ∗ z ∗ = 0 , for i = 1 , . (2.11)We first focus on the upper fluid layer. The analogous derivation for the lower fluidlayer follows a similar procedure. We seek an asymptotic expansion of φ ∗ in powers of β , φ ∗ = φ ∗ (0)1 + βφ ∗ (1)1 + O ( β ) , (2.12)which we use in the asymptotic analysis of the nondimensionalized problem of equations(2.1)-(2.7) for small values of the parameter β . From the leading O (1) term in Eqs. (2.1)and (2.2), φ ∗ (0)1 is found to be independent of the height z , φ ∗ (0)1 = φ ∗ (0)1 ( x ∗ ,t ∗ ) . (2.13)The O ( β ) terms in Eqs. (2.1) and (2.2) yield the equations φ ∗ (0)1 x ∗ x ∗ + φ ∗ (1)1 z ∗ z ∗ = 0 , αξ ∗ < z ∗ < h ∗ + αξ ∗ , with h ∗ = 1 ,φ ∗ (1)1 z ∗ = ξ ∗ t ∗ + αφ ∗ (0)1 x ∗ ξ ∗ x ∗ , at z ∗ = h ∗ + αξ ∗ . (2.14)The expression for φ ∗ (1)1 is obtained as φ ∗ (1)1 = D ξ ∗ Z − φ ∗ (0)1 x ∗ x ∗ Z , (2.15)where Z = z ∗ − h ∗ − αξ ∗ and D ξ ∗ ≡ ξ ∗ t ∗ + αφ ∗ (0)1 x ∗ ξ ∗ x ∗ . Combining expressions (2.13)and (2.15), to the first power of β , we can obtain the solution φ ∗ as φ ∗ = φ ∗ (0)1 + β (cid:18) D ξ ∗ Z − φ ∗ (0)1 x ∗ x ∗ Z (cid:19) + O ( β ) . (2.16) .W. Jiang, G. Kovaˇciˇc, and D. Zhou αξ ∗ to h ∗ + αξ ∗ with respect to z ∗ , imposing theboundary conditions (2.2) and (2.3), and substituting the expression (2.16) into equation(2.1), we obtain the kinematic equation for the upper fluid layer, η ∗ t ∗ + ( η ∗ (cid:101) u ∗ ) x ∗ − β ( h ∗ ) (cid:101) u ∗ x ∗ x ∗ x ∗ − β ( h ∗ ) ξ t ∗ x ∗ x ∗ + O ( αβ,β ) = 0 , (2.17)where η ∗ = h ∗ + αξ ∗ − αξ ∗ , (cid:101) u ∗ = φ ∗ (0)1 x ∗ . Upon substitution of the velocity potential φ ∗ (2.16) into the dynamical boundary con-dition (2.6), we obtain the dynamical equation governing the motion of the upper fluidlayer, (cid:101) u ∗ t ∗ + α (cid:101) u ∗ (cid:101) u ∗ x ∗ + gξ ∗ x ∗ + O ( αβ,β ) = 0 , (2.18)where the equation has been differentiated with respect to x ∗ once, the terms in the firstpower of β are retained [the O ( β ) terms happen to vanish in Eq. (2.18)], and terms of O ( αβ,β ) are dropped. From the velocity potential φ ∗ , (2.16), we obtain the horizontalvelocity φ ∗ x ∗ as φ ∗ x ∗ = (cid:101) u ∗ + βξ ∗ t ∗ x ∗ Z − β (cid:101) u ∗ x ∗ x ∗ Z + O ( αβ,β ) . (2.19)By averaging (2.19) over the depth, we obtain the layer-mean horizontal velocity for theupper fluid layer, u ∗ = (cid:101) u ∗ − βh ∗ ξ ∗ t ∗ x ∗ − β ( h ∗ ) (cid:101) u ∗ x ∗ x ∗ + O ( αβ,β );the corresponding inverse is (cid:101) u ∗ = u ∗ + 12 βh ∗ ξ ∗ t ∗ x ∗ + 16 β ( h ∗ ) u ∗ x ∗ x ∗ + O ( αβ,β ) , (2.20)where the layer-mean horizontal velocity is defined as u ∗ ( x ∗ ,t ∗ ) = 1 η ∗ (cid:90) h ∗ + αξ ∗ αξ ∗ φ ∗ x ∗ ( x ∗ ,z ∗ ,t ∗ ) dz ∗ . After substituting Eq. (2.20) for the horizontal velocity (cid:101) u ∗ , equations (2.17) and (2.18)provide Boussinesq-type equations governing the motion of the fluid in the upper layer.Repeating a similar procedure, we can obtain the governing equations for the lowerfluid layer. The final set of equations for the variables ( ξ ,ξ ,u ,u ), in the dimensionalform, is η t + ( η u ) x = 0 , η = h + ξ − ξ , (2.21) η t + ( η u ) x = 0 , η = h + ξ , (2.22) u t + u u x + gξ x − h u xxt + 12 h ξ ttx = 0 , (2.23) u t + u u x + gξ x + ρ r gη x (2.24) − ρ r h u xxt + ρ r h ξ ttx − h u xxt = 0 , .W. Jiang, G. Kovaˇciˇc, and D. Zhou ( a ) ( b ) −40 −20 0 20 40−0.9−0.6−0.30 X Ξ TWNMCCKdV −0.9 −0.6 −0.3 0481216 Ξ (0) Λ s TWNMCCKdV
Figure 2: (Color online) Comparison of interfacial solitary-wave solutions among ourTWN model [Eqs. (3.27)-(3.28)], the MCC model [13], and the KdV model [13]. ( a )Profiles of interfacial solitary-wave solutions with the same amplitude. ( b ) Effectivewidth Λ s versus wave amplitude Ξ (0). As the amplitude increases, the solitary-wavesolutions broaden and eventually develop a flat crest when the amplitude increases toapproximately ( h − h ) /
2. The TWN model and MCC model can capture the broad-ening of IWs whereas the KdV model cannot.where ρ r is the density ratio ρ /ρ , and u ( x,t ) = 1 η (cid:90) h + ξ ξ φ x ( x,z,t ) dz, u ( x,t ) = 1 η (cid:90) ξ − h φ x ( x,z,t ) dz, are the layer-mean horizontal velocities. We refer to the model (2.21)-(2.24) as the two-layer weakly-nonlinear model or TWN model. The TWN model can also be obtainedvia a direct reduction from a fully nonlinear model (to which we refer as MCC model)given in [7, 8, 12]. For the numerical examples throughout the paper, all the variablesand parameters are dimensionless and the parameters are fixed to be ( h ,h ,g,ρ ,ρ ) =(1 , , , , . h , the characteristic speed is √ gh , and the characteristic time is (cid:112) h /g [see similar dimensionless forms used in numerical simulations in [11, 13]]. Notethat, without loss of generality, our conclusions concerning broadening of IWs in Sec.3.1 and asymmetric behavior of SWs in Sec. 5 also hold true for other parameter regimesof ( h ,h ,g,ρ ,ρ ). In this section, we study some basic properties of the TWN model, including interfacialsolitary wave solutions and dispersion relations.
To study the behavior of the overlaying SWs when an interfacial solitary wave movesbeneath the surface, we first seek interfacial solitary-wave solutions of the TWN model(2.21)-(2.24). Then, we use these solitary-wave solutions as initial conditions for waveprofiles and layer-mean velocities for the subsequent numerical simulations in Sec. 5. .W. Jiang, G. Kovaˇciˇc, and D. Zhou C s , we assume the following ansatz for the surface elevation, internal elevation,upper-layer velocity, and lower-layer velocity, ( ξ i ,u i ) [ i = 1 , ξ i ( x,t ) = Ξ i ( X ) , u i ( x,t ) = U i ( X ) , X = x − C s t. (3.25)Substituting this ansatz into Eqs. (2.21)-(2.22) and integrating once with respect to X yields U i = C s ( H i − h i ) H i , with H = h + Ξ − Ξ and H = h + Ξ , (3.26)where we have assumed that H i → h i as X → ±∞ , and h ( h ) is the undisturbedthickness of the upper (lower) fluid layer, respectively. Substituting the horizontalvelocity (3.26) for U i into Eqs. (2.23)-(2.24) and integrating once with respect to X leads to the equations − C s h H XX H − C s h H XX (3.27)= C s h (cid:18) H − h (cid:19) + g ( H + H − h − h ) − C s h H X H , − ρ r C s h H XX H − ρ r C s h H XX − C s h H XX H (3.28)= C s h (cid:18) H − h (cid:19) + g ( H − h ) + ρ r g ( H − h ) − ρ r C s h H X H − C s h H X H , where we have assumed that H iX , H iXX → X → ±∞ for i = 1 ,
2. Since explicitsolutions to Eqs. (3.27)-(3.28) are difficult to establish, we numerically compute theirsolitary-wave solutions by applying the method in [26].In Fig. 2( a ), we show the numerical solutions of the TWN model for IWs withdifferent amplitudes. For comparison, we also show the corresponding MCC and KdVsolutions with the same amplitudes [13]. From Fig. 2( a ), we can see that the TWNmodel and the MCC model can capture the broadening of internal waves that is oftenobserved in the ocean. For instance, a single large internal wave in 340 meters of waterwas observed in the northeastern South China Sea by [21]. The typical wavelength ofthe observed internal wave is longer than the KdV solution of the same amplitude thatis used to fit this internal wave. It is worthwhile to mention that the broadening ofinterfacial solitary wave solutions can also be captured by other models [15, 25], notonly by the MCC-type models.To quantify this broadening, we introduce a measure of the effective width, Λ s , forthe interfacial solitary-wave solution [34], defined asΛ s = (cid:12)(cid:12)(cid:12)(cid:12) (0) (cid:90) ∞ Ξ ( X ) dX (cid:12)(cid:12)(cid:12)(cid:12) . (3.29)Meanwhile, the effective width of MCC and KdV solutions are provided in the reference[13]. Figure 2( b ) displays the comparison of effective width among the TWN solutions,MCC solutions, and KdV solutions. When the amplitude of waves is small, there is goodagreement of the effective widths among all solutions. However, when the amplitude ofthe waves becomes large, discrepancy grows rapidly among these three solutions. Whenthe amplitude increases to the limiting value, approximately ( h − h ) /
2, the TWN and .W. Jiang, G. Kovaˇciˇc, and D. Zhou ( a ) ( b ) wavenumber k d i s p e r s i o n r e l a t i o n ω k Ω k wavenumber k ph a s e v e l o c i t y phase velocity forfast-mode waves, c p phase velocity forslow-mode waves, C p Figure 3: (Color online) ( a ) Pure linear dispersion relations Ω k for slow-mode waves [Eq.(3.31)] and ω k for fast-mode waves [Eq. (3.32)]. ( b ) Phase velocities C p for slow-modewaves and c p for fast-mode waves.MCC solutions become much broader than the KdV solutions. The maximum amplitudeof our TWN model is approximately ( h − h ) /
2. Beyond the maximum amplitude, nosolitary waves can exist for IWs.
We now investigate the dispersion relation of the TWN model (2.21)-(2.24) using linearanalysis. By substituting the monochromatic solutions ( ξ i ,u i ) ∼ exp[i( kx − µ k t )] into thesystem (2.21)-(2.24), the pure linear dispersion relation in the absence of shear betweenthe frequency µ k and the wavenumber k can be obtained as (cid:18) ρ r k h h + 13 k h + 13 k h + 19 k h h + 112 ρ r k h h (cid:19) µ k (3.30) − (cid:18) gh + gh + 13 gk h h + 13 gk h h (cid:19) k µ k + (1 − ρ r ) g k h h = 0 . Here, the shear is the interface and velocity jump induced by an interfacial solitarywave. The same dispersion relation can be found in [8].Equation (3.30) has 4 real roots in the oceanic regime (the density ratio ρ r is closeto 1). At the leading order in 1 − ρ r , the dispersion relations of the two-mode waves,denoted by Ω k and ω k [see Fig. 3], can be approximated asΩ k = (1 − ρ r ) g h h k gh + gh + gk h h + gk h h , (3.31)and ω k = (cid:0) gh + gh + gk h h + gk h h (cid:1) k ρ r k h h + k h + k h + k h h + ρ r k h h . (3.32)In the following, the two kinds of waves that correspond to the dispersion relations Ω k ,(3.31), and ω k , (3.32), are referred to as the slow-mode waves and the fast-mode waves,respectively.The modulated dispersion relation ω k in the presence of shear can be obtained bysubstituting ( ξ i ,u i ) ∼ (Ξ i ,U i ) + exp[i( kx − ω k t )] into the system (2.21)-(2.24), where the .W. Jiang, G. Kovaˇciˇc, and D. Zhou T X The evolution of IWs’ profile − ξ Figure 4: (Color online) Spatiotemporal evolution of the interfacial solitary-wave so-lution ξ with the amplitude being − . ≤ T ≤ ξ ,ξ ,u ,u ) are (0 . , − . , . , − . T = 0 . X = 600 / .shear is induced by an interfacial solitary wave (Ξ i ,U i ) in above Sec. 3.1. The resultingequation is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ω k + kU ω k − kU kh + k Ξ − k Ξ − ω k + kU kh + k Ξ gk − kh ω k − ω k − k h ω k + kU ρ r gk (1 − ρ r ) gk − ρ r kh ω k − ρ r k h ω k − ω k − k h ω k + kU (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (3.33)where |·| denotes the determinant of the enclosed matrix. In the following Secs. 4 and 5,we will numerically study the TWN model in the right-moving frame with the nonlinearphase velocity C s , that is, T = t and X = x − C s t . Note that the solitary-wave solutions(Ξ i ,U i ) are steady in time T in this moving frame. Then, the modulated dispersionrelation ν , corresponding to the moving frame T and X , is given by ν k = ω k − C s k, (3.34)where ω k is the modulated dispersion relation in Eq. (3.33) corresponding to the restingframe t and x . Note that the dispersion relation ν k is independent of time T since(Ξ i ,U i ) are steady in time T . Moreover, the wavelengths of (Ξ i ,U i ) are relatively longwith respect to the characteristic wavelengths of fast-mode waves, so (Ξ i ,U i ) in thedispersion relation ν k can be locally treated as constant in X space. For numerical computations, we cast Eqs. (2.21)-(2.24) in the conservation form in theright-moving frame with the nonlinear phase velocity (soliton speed) C s as follows: η T − C s η X + ( η u ) X = 0 , (4.35) η T − C s η X + ( η u ) X = 0 , (4.36) M T − C s M X + (cid:18) u + gξ (cid:19) X = 0 , (4.37) .W. Jiang, G. Kovaˇciˇc, and D. Zhou M T − C s M X + (cid:18) u + gξ + ρ r gη (cid:19) X = 0 , (4.38)where M = u − h u XX − h ( η u ) XX , (4.39) M = u − ρ r h u XX − h u XX − ρ r h ( η u ) XX , (4.40)and T = t , X = x − C s t . The computational domain is set to be [ − M,M ], with peri-odic boundary conditions. Even for an initially narrowly localized perturbation wave,radiation can be quickly emitted towards the two boundaries x = − M and x = M . Toeliminate possible reflected waves from these boundaries, two buffer zones in the regions[ − M, − M/
2] and [ M/ ,M ] are established, and damping and diffusion terms are addedto absorb the outgoing radiation. For numerical integration, we use the fourth-orderRunge-Kutta method in time and the second-order collocation method in space [10].The Kelvin-Helmholtz (KH) instability is suppressed by applying a low-pass filter [31].(The wavenumbers for the KH instability are much larger than the wavenumbers of thetrapped right-moving SWs as introduced in the following Sec. 5. Thus, these KH unsta-ble wavenumbers are physically irrelevant in our computations.) In our simulations, wefix the parameter regime ( ρ ,ρ ,h ,h ,g ) = (1 , . , , ,
1) and the computational do-main M = 300. All the variables and parameters in our simulations are dimensionless.We first focus on the evolution of initially unperturbed interfacial solitary-wavesolutions. Figure 4 shows the spatiotemporal evolution of the IWs’ profile, ξ , for 0 ≤ T ≤ l norm error for the SWs’ profile defined as E l = (cid:88) j (cid:12)(cid:12) ξ ( X j ) − ξ ref1 ( X j ) (cid:12)(cid:12) ∆ X, (4.41)where the reference solution ξ ref1 is approximated by the numerical result obtained from avery small time step for the time accuracy test or from a very small spatial discretizationfor the spatial accuracy test. We can see from Fig. 5 that the scheme has fourth-orderaccuracy in time and second-order accuracy in space. In this section, we present our numerical results for the system (2.21)-(2.24) describingthe behavior of a SW packet in the presence of an underlying interfacial solitary wave,and then compare them to the results of our theoretical analysis using ray-based theories.First, we initialize the SWs’ height ξ to be a profile composed of a sufficiently-long-wavelength interfacial-solitary-wave solution and a localized perturbation, that is, theinitial condition [Fig. 6( b )] for ( ξ ,ξ ,u ,u ) is taken to be(Ξ + δ , Ξ ,U ,U ) , (5.42) .W. Jiang, G. Kovaˇciˇc, and D. Zhou ( a ) ( b ) −2 −1 −14 −12 −10 −8 −6 s l o p e = ∆ T E l ξ (0) = − . ξ (0) = − . ξ (0) = − . O ( ∆ T ) −2 −1 −9 −8 −7 −6 −5 −4 −3 s l o p e = ∆ X E l ξ (0) = − . ξ (0) = − . ξ (0) = − . O ( ∆ X ) Figure 5: (Color online) Numerical convergence examination of the scheme in time andspace. ( a ) The l norm error (4.41) as a function of the time step ∆ T under different-amplitude IWs. The green, red, and blue lines correspond to the IW amplitudes ξ (0)being − . − .
8, and − . T s = 4 and fix a spatial discretization ∆ X = 600 / . The referencesolution is computed with a very small time step ∆ T = 0 . /
32. The result in panel( a ) shows fourth-order time-accuracy of the scheme. ( b ) The l norm error (4.41) as afunction of the spatial discretization ∆ X under different-amplitude IWs. To examinethe convergence in space, we use a stopping time T s = 1 and fix the time step ∆ T = 0 . X = 600 / . The result in panel ( b ) showssecond-order accuracy in space of the scheme. .W. Jiang, G. Kovaˇciˇc, and D. Zhou ( a ) ( b )( c ) ( d )( e ) ( f ) ( g ) Figure 6: (Color online) ( a ) Spatiotemporal evolution of SWs’ profile ξ in the near fieldfor 0 ≤ T ≤ − ≤ X ≤ T = 0. The amplitude of the interfacial solitary wave is − . (cid:101) v g , when the right-moving fast-mode SW packet propagates towardsthe trailing (leading) edge. ( b ) The initial condition for wave profiles and horizontalvelocities ( ξ ,ξ ,u ,u ). ( c ) Zoomed-in version of panel ( a ) for 2060 ≤ T ≤ ≤ X ≤
6. The black line corresponds to the positive phase velocity, (cid:101) v p , and the redline corresponds to the negative group velocity, (cid:101) v g . Wave packets are traveling in thedirection of decreasing X towards the trailing edge. ( d ) Zoomed-in version of panel ( a )for 4450 ≤ T ≤ − ≤ X ≤
60. The magenta line corresponds to the positivephase velocity, (cid:101) v p , and the blue line corresponds to the positive group velocity, (cid:101) v g .Wave packets are traveling in the direction of increasing X towards the leading edge.( e )( f )( g ) Snapshot of SWs in the near field. In panels ( e ) and ( g ), the group velocities v g are negative. In panel ( f ), the group velocity v g is positive. .W. Jiang, G. Kovaˇciˇc, and D. Zhou (cid:0) Ξ , Ξ ,U ,U (cid:1) is the solitary-wave solutions described in §§ δ is a narrow SW packet with a narrow band of wavenumbers, δ = A ε [tanh( X − X + x ) − tanh( X − X − x )]cos( k ( X − X )) , (5.43)with A ε = 5 × − , X = 84, x = 2, and k = 5. For the interfacial solitary wave, Ξ ,the amplitude is − . ∼ C s = 0 . v g , the phase velocity v p , and the frequency ν correspondto the moving frame ( T,X ). On the other hand, the group velocity c g = ( v g + C s ), thephase velocity c p = ( v p + C s ), and the frequency ω (= ν + C s k ) correspond to the restingframe ( t,x ). The variable with the tilde, (cid:101) · , stands for the numerical measurement of thecorresponding quantity.Initially, left-moving radiation is quickly emitted from the near field and eventuallyabsorbed by our absorbing boundary condition in the buffer zones [dark stripe parallelto the yellow line in Fig. 6( a )]. After this initial transient, we can see that one SWpacket is trapped in the near field [dark stripes parallel to the red line and blue linein Fig. 6( a )]. These trapped waves are all right-moving waves, that is, their phasevelocities (cid:101) v p > c ) and tothe magenta line in Fig. 6( d )]. Thus, only the right-moving SWs that propagate in thesame direction as the underlying interfacial solitary wave remain trapped in the nearfield.We now study the spatiotemporal manifestation of these right-moving SWs in thenear field. From Fig. 6, we can observe three features of these right-moving SWs:(i) SWs become short in wavelength at the leading edge and long at the trailing edge.For 0 ≤ T ≤ (cid:101) k ∼
23 [Fig. 6( e )]. For 4300 ≤ T ≤ (cid:101) k ∼ . f )].(ii) SW packets propagate towards the trailing edge with a relatively small groupvelocity, and towards the leading edge with a relatively large group velocity. From Figs.6( a )( c )( e ), we can see that for 0 ≤ T ≤ (cid:101) v g ∼ − . ≤ T ≤ a )( d )( f )], the SW packets propagate towards the leading edge with a relativelylarge group velocity (cid:101) v g ∼ .
37. For 4800 ≤ T ≤ a )( g )], the SW packetsagain propagate towards the trailing edge with a relatively small group velocity.(iii) SWs become high in amplitude at the leading edge and low at the trailing edge.For 0 ≤ T ≤ ∼ × − at the leading edgeand then SWs propagate towards the trailing edge [Fig. 6( e )]. For 4300 ≤ T ≤ ∼ × − at the trailing edge and then propagatetowards the leading edge [Fig. 6( f )].To understand the dynamical behavior of these right-moving SWs, we first quan-tify the dispersion relation of these waves. Figure 7( a ) shows the logarithmic modulus,log | (cid:98) ξ ( k,ν ) | , with its magnitude color-coded, where (cid:98) ξ ( k,ν ) is the spatiotemporalFourier transform of ξ ( X,T ). For comparison, also shown are the pure linear dis-persion relation ν k = ω k − C s k [Eq. (3.32), red dashed-dotted curve in Fig. 7( a )] andthe modulated dispersion relation ν k = ω k − C s k [Eq. (3.33), black solid curve in Fig.7( a )]. For the modulated dispersion relation ν k , we take the amplitude of the interfacialsolitary wave to be (Ξ , Ξ ,U ,U ) = (0 . , − . , . , − . a ) that, for 15 ≤ k ≤
25, the modulated dispersion relation ν k can capturethe peak locations of the spectrum well, whereas the pure linear dispersion relation ν k deviates greatly. Therefore, these right-moving SWs can be well characterized by the .W. Jiang, G. Kovaˇciˇc, and D. Zhou ( c ) ( d )( a ) ( b ) Figure 7: (Color online) ( a ) The logarithmic modulus, log | (cid:98) ξ ( k,ν ) | , of SWs’ pro-file ξ for 3181 ≤ T ≤ − ≤ X ≤ ν k (red dashed-dotted curve) and the modulated disper-sion relation ν k (black solid curve). The green rectangle corresponds to the range ofwavenumbers and frequencies predicted by the ray-based theories [see text and panel( b ) for details]. ( b ) The phase portrait of the motion of wave packets in variables X and k . Along each cyan-colored curve, the frequency ν k remains constant. Inside theregion enclosed by the blue-level curve, the fast-mode waves are trapped. The redpoint P , ( X,k ) = (84 , P with a constant frequency. The distance of two neighboring greenpoints corresponds to the spatial width of the initial perturbation (5.43) and the twogreen-level curves correspond to the wave passing through the two green points withconstant frequencies ν = 1 .
46 and ν = 1 . . .
5, respectively. For the green rectangle in panel ( a ),the upper and lower bounds correspond to the two frequencies ν = 1 .
46 and ν = 1 . k = 26 . k = 15 .
5. Thearrows indicate the direction of movement of wave packets. ( c ) The temporal evolutionof peak location of the fast-mode waves in the near field. The green curve correspondsto the peak locations of the fast-mode waves from the TWN model (4.35)-(4.38). Theblue curve corresponds to the peak locations of the fast-mode waves from the effectivelinearized equations (1.46)-(1.49) in appendix A. The red curve corresponds to the peaklocations of the wave packet predicted by the ray-based theory (5.44)-(5.45). Theynearly overlap one another. ( d ) The temporal evolution of the maximal amplitude, A ,of the fast-mode waves. The green curve and the blue curve correspond to the heightof A for the fast-mode waves of the TWN model (4.35)-(4.38) and those of the effec-tive linearized equations (1.46)-(1.49) in appendix A, respectively. They overlap oneanother. The time period marked by the red line corresponds to the negative groupvelocity v g and the black line corresponds to the positive group velocity v g . The timeperiod marked by the magenta line corresponds to the positive peak location X and thecyan-line one corresponds to the negative peak location X . .W. Jiang, G. Kovaˇciˇc, and D. Zhou ν k and thereafter referred to as right-moving fast-modeSWs [see the definition of fast-mode waves in Sec. 3.2].Incidentally, there are two yellow spots on the pure linear dispersion relation ν k near k = 5, as can be observed faintly in Fig. 7( a ). However, the spectral power at these twoyellow spots on the linear dispersion relation ν k is six orders of magnitude lower thanthat at the modulated dispersion relation ν k . These two blurry yellow spots correspondto the wave spectra of radiation waves in the far field, which is not the interest of thiswork.To understand the asymmetric behavior (i)-(iii), we compare our numerical results ofthe TWN model with those of the effective linearized equations (1.46)-(1.49) in appendixA. Effective linearized equations (1.46)-(1.49) are obtained from the linearization ofour TWN model (4.35)-(4.38) in the presence of the interfacial solitary wave. Theirmathematical details are presented in appendix A. We also compare our TWN solutionswith the theoretical predictions of ray-based theories. Due to the slow varying in spaceand time of the phase of fast-mode waves, the governing equations of space-time raysfor the location X and the wavenumber k are given by [6, 44], dXdT = ∂ν k ∂k , (5.44) dkdT = − ∂ν k ∂X , (5.45)where ν k is the modulated dispersion relation in Eq. (3.33). Note that the modulateddispersion relation ν k does not explicitly depend on time T since the interfacial solitarywave is stationary in the moving frame. Clearly equations (5.44) and (5.45) constitutea Hamiltonian system with ν k as the Hamiltonian, X displacement and k momentum.Equation (5.44) states that the wave packet propagates at the group velocity.We now quantify the asymmetric behavior of these right-moving fast-mode SWs bycomparing the results of the TWN model, the results of the effective linearized equations,and theoretical predictions from ray-based theories:(i) First, we provide a theoretical prediction for the temporal evolution of thewavenumber (equivalently the wavelength) of fast-mode waves. By the ray-based the-ory, when propagating with the initial perturbation (5.43), fast-mode waves possessthe peak locations X and wavenumbers k between the two green-level curves in Fig.7( b ). For the minimal wavenumber, the theoretical prediction k = 1 . X = 0 on the red-level curve in Fig. 7( b ). This theoretical minimal wavenumber isin good agreement with the measured one (cid:101) k ∼ . T,X ) = (4500 ,
0) in Fig. 6( f ). Forthe maximal wavenumber, the theoretical prediction ranges from 15 . . b ). For the theoretical wavenumbers k = 15 . k = 26 .
7, the corresponding frequencies are ν = 1 .
895 and ν = 1 .
46, respectively. Thisrange of theoretical wavenumbers and frequencies [depicted by the green rectangle inFig. 7( a )] is in good agreement with the the range of the measured ones in the spectrumin Fig. 7( a ). Therefore, the temporal evolution of the wavenumber can be characterizedby the ray-based theory (5.44)-(5.45) for fast-mode waves. In particular, fast-mode SWsbecome short in wavelength at the leading edge [ X >
0] and long at the trailing edge[
X <
0] [see Fig. 7( b )].(ii) Next, we investigate the motion of the peak location X as a function of time T . Figure 7( c ) displays the temporal evolution of numerically measured peak locationsof the fast-mode waves for the TWN model (4.35)-(4.38) [green curve in Fig. 7( c )] aswell as the prediction using the effective linearized equations (1.46)-(1.49) [blue curve .W. Jiang, G. Kovaˇciˇc, and D. Zhou c )]. For comparison, also displayed are the peak locations of the wave packetspredicted by the ray-based theory (5.44)-(5.45) [red curve in Fig. 7( c )]. One canobserve that there is excellent agreement between the numerical results and theoreticalpredictions for the motion of the peak locations. This confirms that the wave packetmoves at the group velocity v g = ∂ν k /∂k . As predicted by the ray-based theory, for 0 ≤ T ≤ v g ∼ − .
045 is negative with a relatively small magnitude,whereas for 4300 ≤ T ≤ v g ∼ .
37 is positive with a relativelylarge magnitude. These two theoretical group velocities are in excellent agreement withthe measured ones, (cid:101) v g ∼ − .
045 and (cid:101) v g ∼ .
37, respectively. As reflected in the zig-zagpattern in Fig. 7( c ), we can observe that SW packets propagate towards the trailingedge with a relatively small group velocity, and towards the leading edge with a relativelylarge group velocity.(iii) Finally, we discuss the temporal evolution of the maximal amplitude, A , of fast-mode waves in the near field. Figure 7( d ) displays the maximal amplitude of the fast-mode waves in our TWN model (4.35)-(4.38) [green curve] and that predicted using theeffective linearized equations (1.46)-(1.49) [blue curve]. One can see very good agreementbetween them. In addition, one can observe from Fig. 7( d ) that the amplitude A isrelatively large for the negative group velocity v g (interval marked by the red color),whereas the amplitude A is relatively small for the positive group velocity v g (intervalmarked by the black color). Furthermore, for 4500 ≤ T ≤ A growsfor positive X (interval marked by the magenta color), whereas it decays for negative X (interval marked by the cyan color). Therefore, SWs become high in amplitude atthe leading edge ( X >
0) whereas low at the trailing edge (
X <
Using the long-wavelength and small-amplitude approximations, we have proposed atwo-layer, weakly nonlinear (TWN) model (2.21)-(2.24) describing the long-wave inter-actions between IWs and SWs. The TWN model captures the broadening of large-amplitude IWs that is a ubiquitous phenomenon in the ocean [21, 40]. In Sec. 5, wehave investigated the spatiotemporal manifestation and the dynamical behavior of right-moving fast-mode SWs in the near field in the presence of an underlying IW. From ournumerical results, the wavenumber, group velocity, and amplitude of fast-mode SWpackets of our TWN model (4.35)-(4.38) can always be well captured by the predictionsof the effective linearized equations (1.46)-(1.49) and the ray-based theory (5.44)-(5.45).The fast-mode waves behave as linear waves modulated by the underlying interfacialsolitary wave. Importantly, the behavior of the right-moving fast-mode waves is asym-metric at the leading edge vs. the trailing edge when an underlying IW is present:(i)
SWs become short in wavelength at the leading edge and long at the trailing edge[Fig. 7(b)]. (ii)
SW packets propagate towards the trailing edge with a relatively small group velocity,and towards the leading edge with a relatively large group velocity [Fig. 7(c)]. .W. Jiang, G. Kovaˇciˇc, and D. Zhou
SWs become high in amplitude at the leading edge and low at the trailing edge [Fig.7(d)].
In this work, we only focus on the spatiotemporal manifestation and dynamical be-havior of SWs under a small-amplitude initial perturbation. As a natural extension ofthe above results, it is interesting to study the SWs when the amplitude of the perturba-tion is large, that is, the nonlinearity becomes prominent. In particular, it is importantto understand how the nonlinearity and resonance affect the spatiotemporal manifesta-tion and dynamical behavior of the right-moving fast-mode SWs in the presence of anunderling IW.Class 3 triad resonance is believed to be responsible for the surface signature of theunderlying internal waves [17, 29, 35, 38]. The TWN model possesses two-mode waves,one slow and the other fast, and thus resonant interactions among different modescan occur during the energy exchange process [more details can be found in AppendixB]. The class 3 triad resonance condition [Eq. (2.50) in Appendix B], c g ( k ) = C p (0),shows that, for resonantly-interacting waves, the group velocity of fast-mode waves c g ( k ) and the phase velocity of slow-mode waves C p (0) are equal [27, 41]. From manyfield observations, a narrow band of SWs with the resonant wavenumber was foundto be located at the leading edge of an underlying IW and travel nearly at the samespeed as the underlying IW [35, 38]. The surface phenomena may be related to boththe triad resonance excitation and the three asymmetric types of behavior (i)-(iii).The allowance of triad resonance in the TWN model encourages us to investigate thespatiotemporal manifestation and dynamical behavior of SWs under large-amplitudeinitial perturbations in future work. This work is supported by NYU Abu Dhabi Institute G1301, NSFC Grant No. 11671259,11722107, and 91630208, and SJTU-UM Collaborative Research Program (D.Z.). Wededicate this paper to our late mentor David Cai.
A Effective linearized equations
In this section, we present the effective linearized equations of the TWN model (4.35)-(4.38). The variables ( ξ i ,u i ) in the TWN model are composed of two components,one being the interfacial solitary wave (Ξ i ,U i ) and the other the perturbation of fast-mode waves ( (cid:98) ξ i , (cid:98) u i ). By substituting ( ξ i ,u i ) = (Ξ i ,U i ) + ( (cid:98) ξ i , (cid:98) u i ) into the system (4.35)-(4.38) and collecting the linear terms with respect to ( (cid:98) ξ i , (cid:98) u i ), we can obtain the effectivelinearized equations for the fast-mode waves as follows, (cid:98) η T − C s (cid:98) η X + (cid:104) H (cid:98) u + U ( (cid:98) ξ − (cid:98) ξ ) (cid:105) X = 0 , (1.46) (cid:98) η T − C s (cid:98) η X + ( H (cid:98) u + U (cid:98) ξ ) X = 0 , (1.47) (cid:99) M T − C s (cid:99) M X + (cid:16) U (cid:98) u + g (cid:98) ξ (cid:17) X = 0 , (1.48) (cid:99) M T − C s (cid:99) M X + (cid:16) U (cid:98) u + g (cid:98) ξ + ρ r g (cid:98) η (cid:17) X = 0 , (1.49)where H = h + Ξ − Ξ , H = h + Ξ , .W. Jiang, G. Kovaˇciˇc, and D. Zhou ( a ) ( b ) k v e l o c i t y phase velocity forfast-mode waves, c p group velocity forfast-mode waves, c g phase velocity forslow-mode waves, C p ( k res , C p ( k = 0)) ( k res ,C p (0)) k v e l o c i t y group velocity forfast-mode waves, c g phase velocity forslow-mode waves, C p Figure 8: (Color online) ( a ) Comparison of the phase velocity for fast-mode waves c p ,the group velocity for fast-mode waves c g , and the phase velocity for slow-mode waves C p , with the parameters ( h ,h ,g,ρ ,ρ ) = (1 , , , , . k res satisfies the triad resonance condition (2.50) ( b ) Zoomed-in version of panel ( a ). (cid:98) η = h + (cid:98) ξ − (cid:98) ξ , (cid:98) η = h + (cid:98) ξ , (cid:99) M = (cid:98) u − h (cid:98) u XX − h (cid:16) H (cid:98) u + U (cid:98) ξ (cid:17) XX , (cid:99) M = (cid:98) u − ρ r h (cid:98) u XX − h (cid:98) u XX − ρ r h (cid:16) H (cid:98) u + U (cid:98) ξ (cid:17) XX . B Class 3 triad resonance condition
In this section, we briefly verify the existence of solutions to the three-wave-resonancecondition in the TWN model. If the dispersion relation allows the wavenumbers k i ( i =1 , ,
3) and the corresponding frequencies ω k ( k ), ω k ( k ), and Ω k ( k ) to satisfy theconditions, k − k = k ,ω k ( k ) − ω k ( k ) = Ω k ( k ) , these three waves constitute a class 3 resonant triad. Moreover, if the wavenumbers arespecified as k = k + ∆ k/ k = k − ∆ k/ k = ∆ k , where ∆ k (cid:28) k and ∆ k →
0, then theresonance condition reduces to c g ( k ) = C p (0) , (2.50)where the group velocity c g and the phase velocity C p are given by the equations c g ( k ) ≡ ∂ω k ( k ) ∂k , C p (0) ≡ Ω k (∆ k )∆ k (cid:12)(cid:12)(cid:12)(cid:12) ∆ k → . (2.51)Here, ω k corresponds to the dispersion relation of fast-mode waves in Eq. (3.32), andΩ k corresponds to the dispersion relation of slow-mode waves in Eq. (3.31). Many earlyresults [1,16,17,27,41] have confirmed that there exists a unique resonant wavenumber,denoted by k res , satisfying Eq. (2.50) in the two-layer fluid system. For the TWNmodel, one can observe from Fig. 8 that there exists a unique resonant wavenumber .W. Jiang, G. Kovaˇciˇc, and D. Zhou k res , satisfying the resonance condition (2.50), c g ( k res ) = C p ( k = 0). Therefore, class 3resonant triads exist among two fast-mode waves and one slow-mode wave for the TWNmodel. References [1] Mohammad-Reza Alam. A new triad resonance between co-propagating surfaceand interfacial waves.
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