On the hydrostatic limit

Together with Roberta Bianchini, we have uploaded an arXiv preprint concerning the hydrostatic limit of stably stratified fluids.

The hydrostatic approximation consists in neglecting vertical accelerations when computing the pressure in the (inhomogeneous) incompressible Euler equations, which reduces then to the following hydrostatic balance formula (with standard notations) $$ \partial_z P = -g\rho. $$ This tremendously simplifies the problem of pressure reconstruction, avoiding the otherwise necessary elliptic boundary-value problem on the full 3-dimensional domain. Hence the hydrostatic approximation is at the heart of the so-called Primitive Equations which are ubiquitely used in large-scale simulations of oceanic flows.

From the perspective of asymptotic modeling, the hydrostatic approximation is formally valid in the shallow-water regime, where the typical horizontal wavelength is much larger than the depth of the layer. Yet when trying to provide rigorous foundations to the analysis, one is confronted with the fact that the hydrostatic approximation is in some sense singular: we lose the control of one variable (the vertical velocity) in the process. As a matter of fact, the initial-value problem for the equations with the hydrostatic approximation is ill-posed¹ in finite-regularity spaces, due to the emergence of high-frequency instabilities emerging from some arbitrarily small initial data (not satisfying the Rayleigh criterion).

Yet the above analysis does not take into account the stabilizing effect of stratification, when the density of the fluid augments with depth (in the presence of a vertical gravity field). In this framework the initial-value problem for the equations with the hydrostatic approximation in finite-regularity spaces is completely open.

This is why most works on these equations, and in particular the ones concerned with the theoretical justification of the hydrostatic approximation,² use the regularization effect of viscosity and diffusivity, which is argued to be physically relevant to model effective eddy correlations, typically used in Large Eddy Simulations.

Our work with Roberta Bianchini differs from preceding works through the following ingredients.

1. We do not consider viscosity contributions but only diffusivity effects. Specifically, we consider the isopycnal parameterization proposed by Gent and McWilliams.³
2. We heavily rely on the assumption of stable stratification. In particular, we use the reformulation of the systems by means of isopycnal coordinates, which allows us to provide careful energy estimates that are far from being evident in the original (Eulerian) coordinate system.
3. We consider strong limits in high-regularity spaces, keeping track of all relevant parameters. This allows to provide some insights on the interplay between the effects of stratification, shear velocities, and eddy diffusivity.