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Study of internal wave breaking dependence on stratification

Mixing effect in a stratified fluid is considered and examined. Euler equations for incompressible fluid stratified by a gravity field are applied to state a mathematical problem and describe the effect. It is found out that a system of Euler equations is not enough for a formulation of correct generalized problem. Some complementary relations are suggested and justified. A numerical method is developed and applied for study of processes of vortex destruction and mixing progress in a stratified fluid. The dependence of vortex destruction on a stratification scale is investigated numerically and it is shown that the effect increases with the stratification scale. It is observed that the effect of vortex destruction is absent when the fluid density is constant. Some simple mathematical explanation of the effect is suggested.


Introduction
The system of Euler equations for an incompressible fluid stratified by a gravity field is considered. It is known that internal gravity waves can propagate within fluids stratified by density in a gravity field. Internal wave breaking is one of interesting nonlinear effects, which is characterized by disintegration of waves with formation of typical spots with intensive small-scale convection inside them. The spots are often termed convective or turbulent ones because the convection inside them looks like turbulence. Static stability in the fluid becomes recovered with the course of time, but the density inside the spot remains different from the surrounding stratification. The phenomenon of internal wave breaking going with formation of a convective spot often is termed "internal wave mixing".
The effect of internal wave breaking is often observed in the ocean. The phenomenon is registered in the atmosphere [5] as well. Broken down internal waves reorganize stratification with time. Therefore, it is impossible to understand and explain stratification of the ocean or the atmosphere without taking into account the effect. Intensive intermixing of water by internal waves creates good nutrient medium for plankton in the ocean. Regions of the ocean, where wave breaking happens, are generally places of heightened biological productivity. It stimulates growing interest to them and to the physical effect, being cause of heightened biological productivity.
Many papers are dedicated to experimental study of the phenomenon [6,7,8,9,10]. In McEwan's experimental papers, the phenomenon has been studied by means of thin laser measurements. Internal waves were excited in a laboratory tank with sizes of 25 cm × 50 cm × 25cm filled with a stratified fluid. Stratification is formed by dependence of water saltiness on height. The liquid density in the tank varied with height only by 4%. Therefore, it is possible to term this stratification as a weak one. The composite experimental equipment including two lasers has allowed to watch very small-scale density fluctuations originated owing to vortex destruction in the stratified fluid and to discover many details of the effect under study. Observation of small-scale structures had been based on dependence of refraction index of light on liquid density. McEwan has discovered and studied different stages of evolution of wave breaking and mixing [9], [10]: overturning, development of interliving microstructure, restoring of static stability. According to experiments [9], [10], the initial phase of evolution of a nonlinear wave finishes formation of tongues of more heavy liquid inside more light, and vice versa. Further, the effect of overturning develops, and tongues disintegrate into waves of much smaller scales. The static stability is restored in course of time, but the fluid stratification becomes changed.
Let us discuss reasons of formation of tongues. The velocity field within a local vortex is inhomogeneous, and top speeds are achieved at some distance from the vortex center. It leads to formation of the tongue of a heavy liquid, permeating into strata of a light liquid, and of the tongue of a light fluid, permeating into strata of a heavy liquid. Different fluid particles cover different path lengths for the same time, and difference of path lengths increases with time. If we neglect influence of gravity, such fluid movement within a vortex of an elliptic shape leads to distortion of isopycnic lines (lines of equal density) and to reeling of isopycnic lines onto an ellipse. The ellipse semiaxes are determined by distances from the vortex center up to particles having top speeds in the vortex. Some thin layer structure in which stratums of light and heavy fluids are alternating is an outcome of local rotating fluid movement. Imposed gravity field gives birth to wave processes, and it entails breaking down of the regular thin layer structure. So, we observe mixing of the fluid.
Due to the phenomenon mechanism, whatever weak the stratification would be, the density gradients will become arbitrary large with the course of time. Therefore, when we start mathematical modeling of the phenomenon, it is natural to presume a solution of nonlinear fluid equations may be nonsmooth and may contain discontinuities. It seems to be of interest to study nondifferentiable, generalized solutions of hydrodynamic equations of a stratified incompressible fluid placed into a gravity field.
Further, the fluid is supposed to be an ideal one. Estimates show that dissipative effects are important at late stages of evolution. Therefore, the nondissipative problem is of certain interest. To be sure, taking into account of dissipative terms should increase solution smoothness.
The smoothing action of dissipation is explained by inhibition of waves with scales smaller than some minimally possible scale dependent on dissipative constants. At numerical modeling, the smoothing action of dissipation shows one's worth only if the grid step is less than the minimally possible scale. Often it is not advantageous or impossible to increase resolution up to a minimally possible scale. It is one more argument for study of a nondissipative problem.
Processes of mixing in viscous two-layer 3D− fluids with unstable stratification were numerically explored in [11], [12] (see also the literature quoted there). Authors for raising productivity of computations have utilized parallelizing of calculations. The present paper essentially differs from [11], [12] because in mentioned papers the instability develops at the expense of unstable stratification (a heavy liquid has been initially placed above a light one). In the present paper, the stratification is continuous and stable; but in time the instability develops in itself, that is, under other mechanism than in [11], [12]. The instability is shaped by an initial vortex as it occur in occur in experiments of [9], [10]. Furthermore, the problem under consideration is a two-dimensional one, and dissipative processes are not taken into account, excepting (3.6). As against the mentioned papers, we are interested not only in simulation outcomes, but also in substantiation of mathematical statement of the problem and in admissible numerical methods. It is explained by nontrivial items in the problem. Researched questions are important for viscous liquid as well.
Monographies [13], [14], and papers [15], [16], [17], [18], [19], [20] are dedicated to mathematical research of linear equations for small-amplitude internal waves. As for nonlinear Euler equations for incompressible fluid, correctness of many problems is not proved yet [21]. Multimode approach [2], in a finite-mode model directly related to McEwan experiment demonstrates important role of nonlinear dispersion [3] We study nonlinear equations and show that a system of Euler equations is not enough for statement of a correct generalized problem in case of a stratified fluid. The effects of wave overturning and fluid mixing are simulated numerically.
The study of dependence of phenomenon of vortex destructions on a stratification scale is another purpose of the research. By means of numerical experiments, we shall demonstrate the decreasing of overturning effects with diminution of stratification scale H. On the contrary, the effect grows with increasing stratification scale H. Nevertheless, it have been discovered that if we take a limiting case when the fluid density is a constant, the effect of vortex destruction is absent.

Basic system of equations
We suppose a fluid behavior is described by Euler equations for an incompressible fluid Here ρ is the density, p is the pressure, u and w are the horizontal and vertical mass velocities of the fluid, t is time, x, z are the horizontal and vertical coordinates respectively, g is the free fall acceleration.
It is supposed as well that the nonperturbed fluid is stratified in density exponentially: The chosen value of H approximately corresponds to the one for McEwan's experiments [9], [10].
In considered McEwan's experiments, the density stratification was linear, but the stratification scale was much greater than the size of the tank; therefore, the linear and exponential stratifi- 3 Statement of the generalized problem

Preliminary analysis
One of tested methods of numerical integration of Euler equations (2.1) was based on Galerkin method [22]. The solution was searched for as a series in terms of functions dependent on vertical coordinate z and belonging to a complete orthogonal system of functions, the expansion coefficients were dependent on horizontal coordinate x and time t. From Euler equations, by known recipe we have deduced the equations for expansion coefficients [3]. We term these equations as a set of equations for wave modes. The set of equations for wave modes have been solved numerically, and then the velocity and density have been calculated using the series.
In such a way we successfully simulated an initial phase of evolution of the nonlinear wave, which was finished with formation of tongue of a heavy liquid inside a light one, and vice-versa.
A typical scale of the tongues is about 2 cm. Formation of such tongues is (observed) watched in laboratory experiments as well. In laboratory experiments, these tongues further disintegrate into waves of very small scales. However, we have miscarried this effect, though formally it is possible to obtain solution of the problem for all time.
In-depth study of the problem has given the discovery that it is impossible to simulate correctly the nonlinear process under study within the framework of a standard Galerkin method.
The reason is that partial derivatives do not exist for functions with discontinuities. In such cases we should search for the nondifferentiable solution as a generalized, nonclassical one. In 1954, Lax, considering quasilinear equations, has shown that nonlinear equations admit definition of different, nonequivalent classes of generalized solutions [23], [24], [25], [26], (see also [27]).
It is explained by the fact the main equations are crashed at discontinuity surfaces, so we need some additional conditions to match solutions on discontinuity surfaces. However, the con- but Lax's argumentation is applicable to them as well. Analysis has shown that the conventional Galerkin method does not allow achieving simultaneous conservation of mass, momentum and energy for an obtained nondifferentiable solution. May be it is possible to correct somehow conventional Galerkin method that would allow to simulate correctly solutions with discontinuities.
However, the authors cannot offer acceptable adjusting of a Galerkin method. At the same time, finite-difference methods give major freedom that permits choosing a numerical scheme ensuring warranted realization of fundamental conservation laws. if we linearize equations (2.1) with respect to a background density, we come to known equations of the theory of small-amplitude internal waves for which the law of wave energy conservation is (particular in [4,28]): Here the function φ(x, z, t) is defined by the formula ρ (x, z, t) = ρ 0 (z) (1 + φ(x, z, t)). In (3.1) the term containing velocities means a kinetic energy, and the term containing 1 2 g H φ 2 is a potential energy. Functional (3.1) is of great importance in mathematical theory of equations for smallamplitude waves: proofs of theorems of existence, uniqueness, stability for the linearized hydrodynamic equations essentially uses conservation of functional H lin over solutions.
The linearized fluid equations also possess other conservative functional of energy We will name this functional a functional of hydrodynamic energy [29]. In (3.2), the term containing velocities means the kinetic energy and the term containing g z φ is the potential energy.
Here both conservative energy functionals are written to underline their distinction. Different formulas for energy functional are possible because energy is determined up to a constant.
If we add any other conservative functional to the energy functional, we obtain a conservative functional again, which we may term an energy functional, but the expression for energy density becomes other. Therefore, different formulas for energy functional exist.
We have selected the form of wave energy functional (3.1) so that the energy to be zero at lack of waves. For this reason, functional (3.1) is named wave energy functional. Functional (3.2) is more known, but not possesses necessary mathematical properties.

Wave energy functional for nonlinear equations
Having the purpose to develop a theory for nonlinear equations, it is naturally to extend wave energy functional (3.1) to nonlinear case. The following theorem is valid.

Theorem 1. Let a solution to equations (2.1) be differentiable. Then the functional
turns into the functional (3.1), when we take the small-amplitude limit.
Proof. is obvious at ρ > 0. To prove nonnegativity of remaining summands of the integrand, we consider We substitute ρ = ρ 0 (z) (1 + η). To be sure, the requirement ρ > 0 is equivalent to η > −1. For derivative with respect to variable η, it is valid: Unique minimum of the function a(η) corresponds to η = 0. It is enough to be convinced that in the limit of small-amplitude waves the functional under study turns into (3.1).
We term the functional (3.3) as a generalized functional of wave energy because in a smallamplitude limit it turns into wave energy functional (3.1) for linearized equations. At lack of wave disturbances, the integrand in 3.3) is equal zero. The idea of functional (3.3) form, is taken from the Hamilton formalism explained in [4]. Nevertheless, nonnegativity of the integrand, also some other important mathematical properties of the functional are discovered anew.
Let us generalize (3.3) to a case when a solution may be nondifferentiable. The functionals of mass and hydrodynamic energy from which the functional is constructed (3.3), should be conserved for nondifferentiable solutions; it follows from their physical sense. However the func- d Ω is not conserved in a case if the density field is nondifferentiable. Therefore, we demand that the following inequality have to be fulfilled The sign "≤" is chosen because the sign ">" in (3.4) leads to an unstable problem. We will use The relations should be fulfilled for any For the basic functions (velocity components and density), it is convenient to introduce the stream , where ψ is a curve inW 1 2 (Ω) parameterized with time t [25], [30], [31].
However, the above integral relations are not sufficient for unambiguous definition of a generalized solution.
To be sure, we should supply the equations (3.7) with the relation (3.5) and the equation (3.6) in an integral form: Here S(ω) is a surface of star domain ω with smooth boundary. Below, we formulate working definition of the generalized solution.
We construct here a definition of a generalized solution, which should be acceptable from physical point of view and mathematically correct. From the previous reviewing, it is evident that (3.7) is not enough to find uniquely a physically justified generalized solution. It is necessary to supply (3.7) with requirements (3.5), (3.8), or with equivalent ones. If A hypothetical case when requirements (3.9) are fulfilled but the solution belongs to a class of functions, for which the surface integral in (3.5) is not defined, deserves special examination. In this case, it is possible to try to interpret a surface integral in (3.5) as a pair of functionals ( x and z components). It is important that these functionals exist for any star-shaped domain ω with smooth boundary S(ω) and the functionals are additive ones. That is, if ω = ω 1 ∪ ω 2 , ω 1 ∩ ω 2 = ∅, then a functional over surface S(ω) is equal to sum of functionals over surfaces S(ω 1 ) and S(ω 2 ), and relations (3.10) and (3.5) are fulfilled. One can easily be convinced that if requirements (3.9) and (3.10) are fulfilled, then such functionals can always be defined. Therefore, for definition of the generalized solution, we demand realization of (3.9), (3.10); at the same time, relation (3.5) is not utilized in explicit form. Analogous analysis reveals that condition (3.11) has to play an important role in definition of the generalized solution, but it is possible to not use (3.8) resulted in (3.11).

Remark 1. For Navier-Stokes equations, the energy conservation law is not included into the system of constitutive equations, but the solution belongs the classV 2 (Q T ) [32]; and the energy relation follows
Navier-Stokes equations for functions belonging toV 2 (Q T ) [32], [33]. In our case, an energy relation does not follow from integral relations (3.7), and the energy relation is imposed independently. Per se, we apply the condition of conservativeness (conservation of mass, energy) widely used in gas dynamics.

In fact only the condition (3.11) important for inhomogeneous liquid, is novel. The condition (3.11) is
important for viscous liquid as well. Fluid of constant density ρ ≡ 1 was studied in known monographs [32], [33].

Simulation of vortex destruction
We take the initial condition: MackEwan's experiments [9], [10]. The horizontal scale is approximately twice more than the vertical one. The amplitude of a vertical velocity is of 5 cm/s, the amplitude of a level velocity is of 16 cm/s. The top speed of internal wave propagation in the tank within a linear approximation is c = 10 cm/s. That is, the fluid velocity amplitude exceeds the limit of linear wave propagation speed.
The solution procedure was grounded on numerical integration of the system of Euler equations (2.1) by finite-difference method. The fluid density for t = 3 s, 7 s, 8 s is shown in fig. 1, 2, 3.  typical water density. We see that the vortex breakdown and formation of small-scale convection is absent in a strictly homogeneous medium, and only the vortex has become more symmetric.

Study of dependence of a vortex destruction on a stratification scale
Perhaps, it is explained by the fact that any function ψ(R), R = (x 2 + z 2 ) gives stationary solution to equations for homogeneous fluid. Therefore, the vortex takes a symmetric form in time.
Numerical simulations reveal that the phenomenon of breakdown becomes more and more brightly with increase of stratification scale H. Nevertheless, the phenomenon of vortex destruction is absent, when the fluid density is strictly a constant. It points out that continuous limit from a stratified fluid to the case of a fluid of constant density is absent for our initial conditions.
The approximation of constant density is out of the physical sense in such cases, even if density variations are very small.

Explanation of the effect
Below we try to explain why evolution of a vortex in fluid of strict-constant density qualitatively differs from evolution of a vortex in fluid whose density is almost constant and varies within very narrow range.
Moreover, the characteristic method also shows that all evolution of the starting vortex is only vortex deformation, but vortex destruction is impossible.
In case of weakly stratified fluid, the equation for ξ is like this The equation for ψ is the same, as (5.2). The right-hand terms containing square brackets are small against the first right-hand term in Q. Therefore, we ignore them and we consider only the approximation ρ Q ≈ g ∂ρ ∂x . We discussed at beginning of the paper that a starting vortex in a stratified fluid by all means shapes tongues of light and heavy fluids which become very thin in time. Therefore, very thin density structures arise in time. So, after some evolution, the term |g ∂ρ ∂x | becomes essential. This source Q is of small scales and the one generates new vortices of smaller scales. These new small-scale vortices arise in weakly stratified fluid, and they should generate vortices of smaller scales in due time by means of the mechanism they arisen themselves.
Therefore, we have a cascade process of generation of vortices of more and more small scales. It gives a stationary solution. For example, we can take At the following step, we calculate the source Let stratification be linear and Here ξ 0 corresponds to (5.4) We also note that other right-hand terms in Q in (5.3) take into account inertial forces arising in flow moving with acceleration. This inertia forces are equivalent to presence of some gravity field. That is, the effect of cascade vortex destruction in a nonuniform fluid can start without gravity field, but when we apply accelerated movement to vortex motion.

Basic outcomes and conclusion
The system of Euler equations for an incompressible stratified fluid was studied. A non-negative nonincreasing functional extending wave energy functional of the theory of linearized Euler equations is suggested. The functional value is conserved over differentiable solutions, and diminishes if density discontinuities are present. Functional properties have allowed using it for analysis of statement of a generalized problem. It is shown that Euler equations are not enough for statement of a correct generalized problem in case of a stratified fluid in the gravity field. Some auxiliary conditions are formulated and justified. One of them is an energy conservation law, and the second is a special requirement for density (3.6). The statement of a generalized problem is formulated. The finite-difference problem-solving procedure is developed. Existence of a weak solution is proved.
The phenomenon of destruction of starting vortex in conditions of McEwan's laboratory experiments [9], [10] has been simulated and studied. Qualitative concurrence of the simulated effect with the one observed in laboratory experiments, is good, but the last stage of restoring of smooth stratification is absent due to usage of an ideal liquid model. Some new data about the phenomenon, difficult for laboratory measurement, are obtained, and evolution of the flow function is studied. It is shown that large gradients appear in the velocity field at the expense of nonlinear effects and they may play an important role in development of instability.
By means of numerical experiments, dependence of the solution on a stratification scale H is studied. It is revealed that the effects of vortex destruction and formation of small-scale convection grows with H for strongly nonlinear waves. On the contrary, when we decrease H, the effect of wave destruction becomes weak. The wave breakdown is starting from the density field and some later formation of small-scale structures in the flow function is observed. The effect of vortex destruction is not observed when the fluid density is strictly a constant.