Relaxing the sharp density stratification and columnar motion assumptions in layered shallow water systems

The bilayer shallow water system is a standard model for the description of internal waves in density-stratified flows in situations where the density distribution is such that the fluid can be approximately described as two layers with almost-constant densities separated by a thin a pycnocline; see e.g.  [27, Chap. 6]. In addition to this sharp stratification assumption, the formal derivation of the bilayer shallow water system relies on two additional ingredients. Firstly, the internal pressure is assumed to be hydrostatic, that is the pressure-gradient force balances the external force due to gravity. Secondly, the flow velocity is assumed to be columnar, that is the horizontal velocity of fluid particles is constant with respect to the vertical variable within each layer. Of course the validity of bilayer shallow water system relies on the expectation that, if originally approximately satisfied, these three assumptions remain accurate on a relevant timescale as the flow evolves.

The rigorous justification of the hydrostatic assumption in the shallow water regime —that is when the typical horizontal wavelength of the flow is large with respect to the vertical depth of the layer— has been rigorously analyzed either in situations of homogeneous density [5, 36, 23, 37], with smooth density distributions [42, 41], or in the bilayer framework [11, 21, 20]. In the bilayer framework and assuming that the pressure is hydrostatic, the columnar assumption is propagated exactly by the flow. By this we mean that the bilayer shallow water system produces exact solutions to the hydrostatic (incompressible) Euler equations with a density and horizontal velocity distributions which are piecewise constant with respect to the vertical variable. This statement is made explicit below.

In this work we investigate solutions to the hydrostatic Euler equations in the vicinity of such solutions, that is relaxing the sharp stratification as well as columnar motion assumptions. We prove that for initial data suitably close to the bilayer framework, the emerging solutions to the hydrostatic Euler equations remain close to the solution predicted by the bilayer shallow water system on a relevant timescale.

This task is made difficult in part because we lack good stability estimates for the hydrostatic Euler equations in the presence of density stratification. For that matter, as in our previous works [9, 1], we rely on the regularizing properties of thickness-diffusivity terms proposed by Gent and McWilliams [25] so as to model the effective contributions of geostrophic eddy correlations in non-eddy-resolving systems.

Specifically, the hydrostatic Euler equations we consider take the form

where the Montgomery potential $\Psi$ is given by

Here, the equations are formulated using isopycnal coordinates (in particular we assume that the fluid is stratified in the sense that the two-dimensional fluid domain is foliated through lines of equal density, namely isopycnals). The variable $h$ represents the deviation of the infinitesimal depth of isopycnals from the reference value $1$, and $u$ is the deviation of the horizontal velocity of the fluid particles from the reference value $\underline{u}$, and $\underline{\rho}$ their density. The unknowns $h,u$ depend on the time $t$, the horizontal space $x$, and the variable $r\in(-1,0)$ referring to the isopycnal line at stake, while $\underline{u}$ and $\underline{\rho}$ are given and depend only on $r$. The derivation of these equations from the more standard formulation in Eulerian coordinates is described for instance in [9].¹¹1Let us point out that in [9] we choose to label isopycnal lines using the value of the density of fluid particles: $\varrho=\underline{\rho}(r)$. Here we use a different convention, so as to set the reference infinitesimal depth of isopycnals at value $\underline{h}(r)=1$. Notice that the change of variable $\varrho=\underline{\rho}(r)$ is bijective in the stably stratified situation, i.e. when $r\mapsto\underline{\rho}(r)$ is strictly decreasing, but that our choice in this work allows to consider stratifications that are not strictly monotonic such as homogeneous and layered configurations. Incidentally, let us mention that in (1.1) we have set the gravity acceleration to $g=1$ and the total depth of the fluid domain at rest to $\int_{-1}^{0}\underline{h}(r)\operatorname{{\rm d}\!}r=1$ through suitable rescaling. Finally $\kappa>0$ is the thickness diffusivity coefficient and $u_{\star}=-\kappa\frac{\partial_{x}h}{1+h}$ is often referred to as the “bolus velocity”.

The corresponding bilayer shallow water system reads

Here, $H_{s}$ (resp. $H_{b}$) represents the deviation of the depth of the upper (resp. lower) layer from the reference constant value $\underline{H}_{s}$ (resp. $\underline{H}_{b}$), and $U_{s}$ (resp. $U_{b}$) is the deviation of the horizontal velocity within the upper (resp. lower) layer from the reference constant value $\underline{U}_{s}$ (resp. $\underline{U}_{b}$). Notice $H_{s},H_{b},U_{s},U_{b}$ depend only on the time and horizontal space variables, bringing about a relative ease of use of the bilayer model. We denote $\rho_{s}$ (resp. $\rho_{b}$) the constant density of fluid particles in the upper (resp. lower) layers. Finally, $\kappa>0$ is again the thickness diffusivity coefficient.

As mentioned above, solutions to (1.3) provide exact solutions to (1.1). Specifically, if we denote

where $(\rho_{s},\rho_{b},\underline{H}_{s},\underline{H}_{b},\underline{U}_{s},\underline{U}_{b},H_{s},H_{b},U_{s},U_{b})$ is a solution to (1.3), then $(\underline{\rho}_{\rm bl},\underline{u}_{\rm bl},h_{\rm bl},u_{\rm bl})$ is a solution to (1.1)-(1.2). In this work we shall compare these solutions with the ones emerging from profiles satisfying (1.4) only approximately. Specifically, our results are twofold.

1. i.

   We prove that strong solutions to the bilayer shallow water system (1.3) emerge from sufficiently regular initial data satisfying some hyperbolicity conditions.

2. ii.

   We prove that strong solutions to the hydrostatic Euler equations (1.1) emerge from profiles close to the piecewise constant profiles given by (1.4), and that these solutions remain close to the bilayer solutions.

Notice that to accommodate our aim of comparing solutions with a piecewise constant density distribution with a solution with a continuous density distribution, we need to consider deviations that can be pointwise large in small regions, that is the pycnocline. This demands to weaken the topology measuring the size of deviations in (ii). Yet a control in the strong $L^{\infty}$ topology associated with a Banach algebra turns out to be necessary to secure suitable convergence estimates. Our strategy then relies on the construction of a refined approximate solution, which is proved to be close to the corresponding solution to the hydrostatic Euler equation in a strong topology, and close to the bilayer solution in a weaker topology. Hence this refined approximate solution improves the description of the exact solution within the pycnocline.

A second important remark is that while the contribution of thickness diffusivity is essential to our stability estimates, we wish to control and compare solutions on a time interval which is uniform with respect to the thickness diffusivity parameter, $0<\kappa\leq 1$. The dependency on the thickness parameter will appear only as a restriction on the size of admissible deviations. Concerning (i) this is made possible by the well-known fact that the bilayer shallow-water system is well-posed (under some hyperbolicity conditions) when $\kappa=0$ (see [40]). Yet obtaining the corresponding result for $\kappa>0$ is not straightforward and demands to use finely the structure of the thickness diffusivity parameters, following the “two-velocity” strategy developed in the context of the BD-entropy (see e.g. [16]). Concerning (ii) we use the existence of the bilayer solution and consequently the existence of the refined approximate solution to bootstrap the control of sufficiently close solutions to the hydrostatic Euler equations on the relevant timescale. For that purpose we strongly use the regularizing effect of thickness diffusivity contributions, but any non-uniformity with respect to the diffusivity parameter, $\kappa$, can be balanced through the smallness of the deviations.

Let us point out that the result (ii) applies to any given (sufficiently regular with respect to the time and horizontal space variables) solution to the hydrostatic Euler equations. Hence our work provides the same stability estimates around other solutions, constructed for instance in the framework of multiple layers and/or simple waves.

Several existing works discuss the matter of modeling thin pycnoclines through the bilayer framework. Let us first recall that in the bilayer framework, non-hydrostatic pressure contributions trigger strong Kelvin–Helmholtz instabilities that in particular prevent the well-posedness of the initial-value problem in the absence of any additional regularizing ingredients; see [32]. Such regularizing ingredients include interfacial tension as proved in [34] but this is not expected to be te physically relevant mechanism. In [10], Bogucki and Garrett describe and model a scenario of interface-thickening due to mixing triggered by shear instabilities, up to a situation where the Richardson number in the interface becomes compatible with the celebrated Miles [39] and Howard [30] stability ion. Recall however that the bilayer shallow water system does not suffer from shear-induced instabilities when shear velocities are sufficiently small. Consistently, the authors in [19] discuss the simultaneous limits of sharp stratification together with shallow water (which can be considered as a low-frequency or hydrostatic pressure limit). In our work we impose the hydrostatic pressure assumption thus taming shear instabilities, although as pointed out previously the stability of continuously stratified hydrostatic Euler equations is poorly understood in the presence of density variations (see [43, 12, 28, 38] in the homogeneous framework). Notice also that shear-induced instabilities disappear when restricting the framework to purely traveling waves. In this framework, James [31] (improving upon [44, 4]) was able to rigorously justify the sharp stratification limit: writing the bilayer and continuously stratified problems in a unified formulation, James proved the existence of internal traveling waves associated with density stratifications in a small neighborhood (according to the $L^{2}$ topology) of the bilayer framework which converge towards the bilayer solution in the limit of sharp stratification. Our results are in the same spirit: system (1.1) is our unified formulation, and the topology controlling the limit of sharp stratification in our work is the $L^{1}_{r}$ topology. However, our results admit non-trivial dynamics thanks to the hydrostatic assumption and the presence of thickness diffusivity.

The propagation of internal waves with thin pycnoclines in relation with bilayer models was also investigated through experiments. In particular Grue et al. [29] set up precise experiments generating large-amplitude solitary waves and reported the dynamical development of rolls on the trailing side of the largest considered waves in accordance with the mechanism promoted by Bogucki and Garrett, while bilayer models provide very accurate predictions otherwise. Almgren, Camassa and Tiron [3] investigate thoroughly this matter through careful numerical simulations and analytical results of asymptotic bilayer models, analyzing the triggering of shear-induced instabilities in the region of maximal displacement as well as their advection into stable regions of the flow. White and Helfrich [46] consider internal bores generated by a dam-break, and compare continuously stratified and bilayer models with numerical experiments. From their findings they suggest an improvement on existing bilayer theories. In [18], Camassa and Tiron optimize bilayer models (specifically calibrating the top and bottom densities and the position of the sharp interface) and compare the analytical predictions of the optimized bilayer models with respect to the numerically computed continuously stratified solutions —considering infinitesimally small waves, internal bores and solitary waves— showing excellent agreement even in situations of relatively thick pycnoclines. Furthermore they propose a new asymptotic model taking into account thin pycnoclines in view of reconstructing analytically local properties of traveling waves within the pycnocline, which is similar in spirit with the “refined approximate solution” that we introduce in this work. Notice the authors consider that “fully time-dependent models governing the evolution of the pycnocline thickness probably constitute one of the most relevant extensions of the model [they] have introduced”, and we believe that our work provides a partial answer in that respect.

This manuscript is structured as follows. In Section 2 we study the bilayer shallow water systems. We first recall some known results without thickness diffusivity, and then consider the system with thickness diffusivity contributions. The main result is Proposition 2.7 which provides for any sufficiently regular initial data satisfying some hyperbolicity criterion the existence and control of strong solutions to the bilayer system on a time interval which is uniform with respect to $0<\kappa\leq 1$, while Proposition 2.8 states the strong convergence as $\kappa\searrow 0$. In Section 3 we study the hydrostatic Euler equations. We provide first some stability estimates with respect to perturbations of the equations and of the data using suitable distances. As a second step we introduce refined approximate solutions associated with some given reference exact solution and close-by profiles. Building upon these approximate solutions, we prove Proposition 3.8 which controls the difference between the reference solution and exact solutions to the hydrostatic Euler equations emerging from close-by profiles. Together, Proposition 2.7 and Proposition 3.8 provide the announced result that for profiles suitably close to the bilayer framework and satisfying some hyperbolicity criterion the emerging solutions to the hydrostatic Euler equations remain close to the solution predicted by the bilayer shallow water system on a relevant timescale. The rigorous statement is displayed in Section 4, completed with a discussion on analogous statements in the one-layer and multilayer frameworks.

Let us introduce some notations for functional spaces used in this work.

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  The spaces $L^{p}(\mathbb{R})$ are the standard Lebesgue spaces endowed with the usual norms denoted $\|\cdot\|_{L^{p}}$.

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  The spaces $W^{k,p}(\mathbb{R})$ for $k\in\mathbb{N}$ are the $L^{p}$-based Sobolev spaces endowed with the usual norms denoted $\|\cdot\|_{W^{k,p}}$.

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  The spaces $H^{s}(\mathbb{R})$ for $s\in\mathbb{R}$ are the $L^{2}$-based Sobolev spaces endowed with the usual norms denoted $\|\cdot\|_{H^{s}}$.

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  Given $I$ a real interval and $X$ a Banach space, $L^{p}(I;X)$ (respectively $C^{n}(I;X)$) the space of $p$-integrable (respectively $n$-continuously differentiable) $X$-valued functions, endowed with their usual norms.

• •

  When useful, we provide insights on the variables at stake in aforementioned functional spaces by means of subscripts. For instance for $f:(x,r)\in\mathbb{R}\times(-1,0)\mapsto f(x,r)\in\mathbb{R}$ we may denote

  $$\|f\|_{L^{\infty}_{r}H^{s}_{x}}=\operatorname*{ess\,sup}\big{(}\big{\{}\|f(\cdot,r)\|_{H^{s}}\ ,\ r\in(-1,0)\}).$$

• •

  We sometimes also use subscripts to provide information on the interval at stake in functional spaces. For instance for $T>0$ and $f:(t,x)\in[0,T]\times\mathbb{R}\mapsto f(t,x)\in\mathbb{R}$ we may denote

  $$\|f\|_{L^{\infty}_{T}H^{s}_{x}}=\operatorname*{ess\,sup}\big{(}\big{\{}\|f(t,\cdot)\|_{H^{s}}\ ,\ t\in[0,T]\}).$$