On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity

Abstract

This article is concerned with rigorously justifying the hydrostatic limit for continuously stratified incompressible fluids under the influence of gravity. The main distinction of this work compared to previous studies is the absence of any (regularizing) viscosity contribution added to the fluid-dynamics equations; only thickness diffusivity effects are considered. Motivated by applications to oceanography, the diffusivity effects in this work arise from an additional advection term, the specific form of which was proposed by Gent and McWilliams in the 1990s to model the effective contributions of geostrophic eddy correlations in non-eddy-resolving systems. The results of this paper heavily rely on the assumption of stable stratification. We establish the well-posedness of the hydrostatic equations and the original (non-hydrostatic) equations for stably stratified fluids, along with their convergence in the limit of vanishing shallow-water parameter. These results are obtained in high but finite Sobolev regularity and carefully account for the various parameters involved. A key element of our analysis is the reformulation of the systems using isopycnal coordinates, enabling us to provide meticulous energy estimates that are not readily apparent in the original Eulerian coordinate system.

KEYWORDS:

• Hydrostatic limit
• stratified fluids
• isopycnal diffusivity

2020 Mathematics Subject Classification:

• 35Q35
• 76B03
• 76B70
• 76M45
• 76U60
• 86A05

Acknowledgments

VD thanks Eric Blayo for revealing to him the work of Gent and McWilliams, Mahieddine Adim for his careful proofreading, as well as the Centre Henri Lebesgue, program ANR-11-LABX-0020-0. RB is partially supported by PRIN 2022HSSYPN and the GNAMPA group of INdAM. VD and RB thank Charlotte Perrin for identifying the relationship between Gent and McWilliams eddy-diffusivity contributions and the BD entropy, and the anonymous referee for pointing out many additional relevant references.

Notes

1 Let us point out that our analysis would hold (and would in fact be simpler) without the contributions ${\mathit{u}}_{\star }$ and $w_{\star }$ in the evolution equations for the velocity. We add these terms because we believe they are important from a modeling point of view, and would play a crucial role in a refined analysis of the large-time behavior and/or less regular (weak) solutions.

2 The effective velocities are defined slightly differently in [33, 34] compared with (1.8)(1.8) $$\begin{array}{cc}{\partial }_{t}h+{\nabla }_{\mathit{x}}·(h\mathit{u})\hfill & ={\nabla }_{\mathit{x}}·(\kappa {\nabla }_{\mathit{x}}h),\hfill \\ {\partial }_t(h\mathit{u})+{\text{div}}_{\mathit{x}}(\varrho \mathit{u}\otimes (\mathit{u}-\kappa \frac{{\nabla }_{\mathit{x}}h}{h}))+h{\nabla }_{\mathit{x}}\psi \hfill & =0,\hfill \end{array}$$(1.8) , since the analogous fully continuous equations read

$\begin{array}{cc}{\partial }_{t}h+{\nabla }_{\mathit{x}}·(h\mathit{u})\hfill & ={\nabla }_{\mathit{x}}·(\kappa {\nabla }_{\mathit{x}}\psi ),\hfill \\ {\partial }_t(h\mathit{u})+{\text{div}}_{\mathit{x}}(\varrho \mathit{u}\otimes (\mathit{u}-\kappa \frac{{\nabla }_{\mathit{x}}\psi }{h}))+h{\nabla }_{\mathit{x}}\psi \hfill & =0,\hfill \end{array}$

where ψ is defined in (1.9)(1.9) $$\psi (t,\mathit{x},r)=g{\int }_{{\rho }_0}^r\mathrm{\text{min}}(1,r\prime /r)h(t,\mathit{x},r\prime )\text{dr}\prime .$$(1.9) . The multilayer shallow water system can be viewed as a system of several equations of the form (1.6)(1.6) $$\begin{array}{cc}{\partial }_t\varrho +{\nabla }_{\mathit{x}}·(\varrho \mathit{u})\hfill & ={\nabla }_{\mathit{x}}·(\kappa {\nabla }_{\mathit{x}}\varrho ),\hfill \\ {\partial }_t(\varrho \mathit{u})+{\text{div}}_{\mathit{x}}(\varrho \mathit{u}\otimes (\mathit{u}-\kappa \frac{{\nabla }_{\mathit{x}}\varrho }{\varrho }))+{\nabla }_{\mathit{x}}p\hfill & ={\text{div}}_{\mathit{x}}\mathbb{S}({\nabla }_{\mathit{x}}\mathit{u}),\hfill \end{array}$$(1.6) , coupled through the pressure contributions. Alternatively it may be interpreted as a semi-discretized (with respect to the density variable) version of the system (1.8)(1.8) $$\begin{array}{cc}{\partial }_{t}h+{\nabla }_{\mathit{x}}·(h\mathit{u})\hfill & ={\nabla }_{\mathit{x}}·(\kappa {\nabla }_{\mathit{x}}h),\hfill \\ {\partial }_t(h\mathit{u})+{\text{div}}_{\mathit{x}}(\varrho \mathit{u}\otimes (\mathit{u}-\kappa \frac{{\nabla }_{\mathit{x}}h}{h}))+h{\nabla }_{\mathit{x}}\psi \hfill & =0,\hfill \end{array}$$(1.8) , as rigorously shown in [35].

3 Incidentally, notice that it was proven that smooth solutions to the hydrostatic equations in the homogeneous framework can develop singularities in finite time; see [44, 45], and [46] in the presence of rotation. Again, such a result is not known in the stably stratified setting. Our study will be limited to local-in-time solutions.

4 Notice the different scaling between the horizontal and vertical velocity fields. There, λ is a reference horizontal length.

5 We could scale also the ϱ-coordinate. Adjusting accordingly the other variables, we can set without loss of generality ${\rho }_0=1$. In the following we shall not discuss the dependency with respect to ρ₁, and in particular the physically relevant limit of small density contrast, $\frac{{\rho }_1-{\rho }_0}{{\rho }_0}\ll 1$; see [59] and references therein.

6 As highlighted by the anonymous referee, another reasonable approach, inspired by the specific form of the viscosity contributions in the shallow water equations advocated in [19] and analyzed using the energy method in [60], would be to consider the following system

$\begin{array}{cc}{\partial }_{t}h+{\nabla }_{\mathit{x}}·((\underset{¯}{h}+h)(\underset{¯}{\mathit{u}}+\mathit{u}))\hfill & =\kappa {\Delta }_{\mathit{x}}h,\hfill \\ {\partial }_t\mathit{u}+((\underset{¯}{\mathit{u}}+\mathit{u}-\kappa \frac{{\nabla }_{\mathit{x}}h}{\underset{¯}{h}+h})·{\nabla }_{\mathit{x}})\mathit{u}+\frac{1}{\varrho }{\nabla }_{\mathit{x}}\psi \hfill & =\nu \frac{{\text{div}}_{\mathit{x}}((\underset{¯}{h}+h){\nabla }_{\mathit{x}}\mathit{u})}{\underset{¯}{h}+h}.\hfill \end{array}$ (3.2)

Notice that after straightforward computations the system (3.2) can be reformulated as

$\begin{array}{cc}{\partial }_{t}h+{\nabla }_{\mathit{x}}·((\underset{¯}{h}+h)(\underset{¯}{\mathit{u}}+\mathit{u}))\hfill & =\kappa {\Delta }_{\mathit{x}}h,\hfill \\ {\partial }_t\mathit{u}+((\underset{¯}{\mathit{u}}+\mathit{u}-(\kappa +\nu )\frac{{\nabla }_{\mathit{x}}h}{\underset{¯}{h}+h})·{\nabla }_{\mathit{x}})\mathit{u}+\frac{1}{\varrho }{\nabla }_{\mathit{x}}\psi \hfill & =\nu {\Delta }_{\mathit{x}}\mathit{u},\hfill \end{array}$ (3.3)

so that the difference with respect to (3.1)(3.1) $$\begin{array}{cc}{\partial }_{t}h+{\nabla }_{\mathit{x}}·((\underset{¯}{h}+h)(\underset{¯}{\mathit{u}}+\mathit{u}))\hfill & =\kappa {\Delta }_{\mathit{x}}h,\hfill \\ {\partial }_t\mathit{u}+((\underset{¯}{\mathit{u}}+\mathit{u}-\kappa \frac{{\nabla }_{\mathit{x}}h}{\underset{¯}{h}+h})·{\nabla }_{\mathit{x}})\mathit{u}+\frac{1}{\varrho }{\nabla }_{\mathit{x}}\psi \hfill & =\nu {\Delta }_{\mathit{x}}\mathit{u},\hfill \end{array}$$(3.1) is minor, and our analysis can easily be extend to (3.3). Specifically, the only significant difference arises in the energy estimates of Lemma 3.5. Because the first contribution of (v) in the proof vanishes, the assumption on the control of ${\nu }^{1/2}{\Vert {\nabla }_{\mathit{x}}h(t,·)\Vert }_{L^{\infty }(\Omega )}$ is no longer necessary. As a consequence, Proposition 3.6 holds without assuming $\nu \le \kappa$. This leads to the interesting problem of the large-time well-posedness of the initial-value problem for (3.3) when κ = 0, $\nu >0$, which we believe can be addressed in the spirit of the BD and κ entropies described in Section 1.2, but leave open for future studies.

7 We point out that the only term requiring the above smallness condition (4.45)(4.45) $$C {\kappa }^{-1/2} (C_0{M}_0+{\left|\underset{¯}{\mathit{u}}\prime \right|}_{L_{\varrho }^{\infty }})\le \frac{1}{16}{\rho }_0{h}_{\star }$$(4.45) on the initial data is (the time integral of) R₂, and more precisely the product ${\Vert P_{\text{nh}}\Vert }_{H^{s,k}}{\Vert {\nabla }_{\mathit{x}}\eta \Vert }_{H^{s,k}}$, where both terms are only square-integrable in time.