Internal waves in marginally stable abyssal stratified flows

Abstract. The problem on internal waves in a weakly stratified two-layer fluid is studied semi-analytically. We discuss the 2.5-layer fluid flows with exponential stratification of both layers. The long-wave model describing travelling waves is constructed by means of a scaling procedure with a small Boussinesq parameter. It is demonstrated that solitary-wave regimes can be affected by the Kelvin–Helmholtz instability arising due to interfacial velocity shear in upstream flow.


continuously stratified, near two-layer fluids was studied numerically and analytically by Almgren, Camassa, & Tiron (2012). 10 They demonstrated that the wave-induced shear can locally reach unstable configurations and give rise to local convective 11 instability. This is in good qualitative agreement with the laboratory experiments performed by Grue et al. (2000). It seems that 12 such a marginal stability of long internal waves could explain the formation mechanism of a very long billow trains in abyssal 14 2 Basic Equations 15 We consider a 2D motion of inviscid two-layer fluid which is weakly stratified due to gravity in both layers. The fully nonlinear 16 Euler equations describing the flow are 17 ρ(u t + uu x + vu y ) + p x = 0, (2) 18 ρ(v t + uv x + vv y ) + p y = −ρg, 21 where ρ is the fluid density, (u, v) is the fluid velocity, p is the pressure and g is the gravity acceleration. We assume that the  where the square brackets denote the discontinuity jump at the interface between the layers. Non-disturbed parallel flow has 1 no vertical velocity and elevation (i.e. v = 0, η = 0) but the horizontal velocity u = u 0 (y) may be piece-wise constant, 3 In this stationary case, the fluid density ρ = ρ 0 (y) and pressure p = p 0 (y) should be coupled by the hydrostatic equation 4 dp 0 /dy = gρ 0 . We consider the density profile depending exponentially on height, 5 ρ 0 (y) =    ρ 1 exp (−N 2 1 y/g) (−h 1 < y < 0), ρ 2 exp (−N 2 2 y/g) (0 < y < h 2 ), (9) 6 where N j = const is the Brunt -Väisälä frequency in the j-th layer, and constant densities ρ 1 and ρ 2 are related as ρ 2 < ρ 1 .

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The special case N j = 0 (j = 1, 2) gives a familiar two-fluid system with piece-wise constant density ρ = ρ j in the j-th layer.

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Further we consider a steady non-uniform flow, hence we have η t = 0 and u t = v t = ρ t = 0 in Eqs.
(2) -(4). We introduce 9 the stream function ψ by standard formulae u = ψ y , v = −ψ x , hence the mass conservation implies the dependence ρ = ρ(ψ), 10 and pressure p can be found from the Bernoulli equation 12 Seeking for a solitary-wave solutions, we require that the upstream velocity of the fluid (u, v) tends to (u j , 0) as x → −∞. In 13 this case, boundary conditions (6) transform to the conditions for the stream function as

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It is known (Yih, 1980) that system (2) -(5) can be reduced in a stationary case to the non-linear Dubreil-Jacotin -Long Here, the function H(ψ) = ρ(ψ)b(ψ) involves the Bernoulli function b(ψ) and the density function ρ(ψ), so that H is specified 19 by the upstream condition. More exactly, the density function is determined by the relation ρ(ψ) = ρ 0 (ψ/u j ) in the j-th layer, 20 3 As a consequence, we can rewrite the DJL equation (12) as follows: where j = 1 is related to the lower layer, and j = 2 to the upper layer. Further, in accordance with relations (7) and (10), the 5 continuity of pressure p provides non-linear boundary condition for stream function ψ 7 Using the explicit form of functions ρ(ψ) and b(ψ), condition (14) can be also rewritten in detail as follows: We reformulate this boundary condition in view of conservation of the total horizontal momentum in a steady two-layer flow, 10 which has integral formulation where the integrand functions Ψ j are , 17 and constant C depends on the parameters of the upstream flow as follows: It is important here that the integral relation (15) is equivalent to the boundary condition (14) which is rather simple. This 20 equivalence can be checked immediately by differentiation the relation (15) with respect to the variable x, so the integrals can 21 be evaluated explicitly due to Eq.(13). Equation (15) will be used later instead of (14) by the construction model differential Now we introduce scaled independent variablesx,ȳ and scaled unknown functionsη,ψ in order to reformulate the basic 2 equations in the dimensionless form. Namely, the fixed ratio h 1 /π is used as an appropriate length scale for x, y, η, and 3 normalized volume discharges u j h j /π serve as the units for the stream function; thus, we have separately in the lower layer (j = 1) or in the upper layer (j = 2). The number π is only introduced here due to the specific 6 form of trigonometric modal functions which are typical for the exponential density (9). Scaling procedure with this density 7 profile uses the Boussinesq parameters σ 1 , σ 2 and the Atwood number µ defined by the formulae Here, constants σ j characterize the slope of the density profile in continuously stratified layers, and parameter µ determines 10 the density jump at interface. 11 Following Turner (1973), we introduce densimetric (or internal) Froude number which presents scaled fluid velocity u j in the j-th layer, defined with reduced gravity acceleration g j = (ρ 1 − ρ 2 )g/ρ j . In 14 addition to the Froude numbers F j , it is also convenient to use the pair of the Long's numbers λ j given by the formula The Long's numbers λ j are coupled with the Boussinesq parameters σ 1 , σ 2 , the Atwood number µ and the Froude numbers F j 17 by the relations Finally, we introduce the ratio of undisturbed thicknesses of the layers r = h 1 /h 2 . By that notation, we locate the bottom 20 asȳ = −π, and relationȳ = π/r defines the rigid lid. Thus, we obtain the equations for scaled stream functionψ and non-21 dimensional wave elevationη as follows (bar is omitted throughout what follows): in the lower layer −π < y < η(x), and in the upper layer η(x) < y < π/r. Kinematic boundary conditions (11) can be rewritten now as follows: Correspondingly, Eq. (14) providing continuity of pressure at interface y = η(x) leads to nonlinear boundary condition 2 and the dimensionless version of integral relation (15) takes the form πr 2 (λ 2 2 + σ 2 2 ) is chosen here so that the horizontal upstream flow given by the solution 12 satisfies momentum relation (22).

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The model of fully nonlinear travelling waves in a two-layer irrotational flows, with the interface y = η(x) between the fluids 14 with constant densities ρ 2 in the upper layer and ρ 1 > ρ 2 in the lower layer, can be specified as follows. In this limit case, at 15 least formally, the Boussinesq parameters σ j and Long's numbers λ j vanish: for stationary harmonic wave-packets Here k is the non-dimensional wave-number, a is the amplitude of interfacial wave, and W (y) is the modal eigenfunction 6 which describes deformation of streamlines within the fluid layers. For the given Long's numbers λ 1 , λ 2 and the Boussinesq 7 parameters σ 1 , σ 2 , we also introduce non-dimensional values 9 where k 1 = rk and k 2 = k are dimensionless wave-numbers specified for each layer. According to these notations, dispersion where functions Cot j (j = 1, 2) are denoted as follows: 13 14 In fact, function ∆ takes such a combined form since modal function W (y) depends on y trigonometrically or hyperbolically, 15 if the radicand term λ 2 j − k 2 j − 1 4 π 2 σ 2 j in (26) is positive or negative. Explicit formulae for these modal eigenfunctions W (y) 16 are given in Appendix A.

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Spectrum of stationary harmonic waves, defined on the (F 1 , F 2 )-plane, is formed by the Froude points (F 1 , F 2 ) so that 18 dispersion function ∆(k; F 1 , F 2 ), which is even in k, has at least one pair of real roots ±k. Wave modes differ by the number 19 of these pairs, and this number can change only by passing of the root across the value k = 0. Therefore, the modal bounds 20 should satisfy the equation ∆(0; F 1 , F 2 ) = 0; these bounds are defined by separate branches of the curve where parameters λ j should be coupled with the Froude numbers F j using the formulae (17). 23 We emphasize that parameters σ j characterize the slope of density profile in continuously stratified layers, and µ defines the 24 density jump at the interface. As usual, all these parameters are small in the case of low stratification. However, the interfacial The 2.5-layer model starts with the hypotheses that the Boussinesq parameters σ 1 , σ 2 and the Atwood number µ are of the 8 same order, so we can use a single small parameter σ by setting 10 The limit passage σ → 0 is singular because the Long's numbers λ j involve the ratios σ j /µ in formulae (17). However, 11 condition (30) allows us to simplify the spectral portrait, hence modal curve (27) defining the critical wave speeds takes the with the yellow, and the third mode is marked with the pink color. It is important that this spectrum differs essentially from the 3 ordinary 2-layer spectrum (29), even the flow is characterized with a pair of the Froude numbers F 1 , F 2 , defined by the same 4 manner. We specially note that the 2.5-layer spectrum extents infinitely on the spectral plane by involving unbounded Froude 5 numbers F j . with the parameter σ gives the equation 11 in the lower layer −π < y < η(ξ), and 12 σψ ξξ + ψ yy + λ 2 2 r 2 (ψ − y) = 1 2 σ ( σψ 2 ξ + ψ 2 y − r 2 ) (33) 13 in the upper layer η(ξ) < y < π/r. Kinematic boundary conditions (11) can be rewritten now as follows: 14 ψ(ξ, −π) = −π, ψ(ξ, η(ξ)) = 0, ψ(ξ, π/r) = π.

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The final form of dispersive term ψ (1) is much more complicated, therefore this coefficient is given in Appendix A. terms with the powers higher than the first power of σ. By that, equation (22) reduces to the first-order ordinary differential 2 equation for the wave elevation η(x) and is written as Here function D is given by the formula where α 1 and α 2 should be taken as since we have at the leading order in σ the relations λ j = 1/ √ πF j (j = 1, 2) obtained under condition (30). Denominator Q in 10 (37) has a complicated form, therefore this function is given in Appendix C. Solitary-wave solutions of Eq. (37) are given in 11 the implicit form by the formula where parameter a determines non-dimensional amplitude of the wave.
14 Small-amplitude waves can be modelled by simplified weakly nonlinear version of the Eq.(37) having the form where the coefficients D 0 = D(0; F 1 , F 2 ) and with q * = Q(0), a = a 1 and θ 2 = a 1 /a 2 < 1, and the bore (internal front) corresponds to the double root a = a 1 = a 2 , it has 1 the following profile  In this paper we have considered the problem on permanent internal waves at the interface between exponentially stratified 8 fluid layers. An ordinary differential equation describing large amplitude solitary waves has been obtained using the long-wave 9 scaling procedure. Parametric range of solitary waves is characterized, including regimes of broad plateau-shape solitary waves 10 and internal fronts. It is demonstrated that these solitary wave regimes can be affected by the Kelvin -Helmholtz instability 11 generated due to the velocity shear at the interface. 12