On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity (Q6595128)

The authors consider the evolution of heterogeneous incompressible flows under the influence of gravity, described by the system \N\[\N\begin{cases} \partial_t \rho + (\mathbf{u}+\mathbf{u}_\star)\cdot\nabla_{\mathbf{x}} \rho + (w+w_\star)\partial_z \rho = 0,\\\N\rho\big(\partial_t \mathbf{u} + ((\mathbf{u}+\mathbf{u}_\star)\cdot\nabla_{\mathbf{x}}) \mathbf{u} + (w+w_\star)\partial_z \mathbf{u}\big) +\nabla_x P =0,\\\N\rho\big(\partial_t w + (\mathbf{u}+\mathbf{u}_\star)\cdot\nabla_{\mathbf{x}} w +(w+w_\star)\partial_z w + \partial_z P\big) + g\rho =0,\\\N\nabla_{\mathbf{x}}\cdot\mathbf{u}+\partial_z w =0,\\\NP\vert_{z=\zeta} - P_{\mathrm{atm}}=0,\\\N\partial_t \zeta + (\mathbf{u} + \mathbf{u}_\star)\vert_{z=\zeta} \cdot\nabla_{\mathbf{x}} \zeta - (w+w_\ast)\vert_{z=\zeta}=0,\\\Nw\vert_{z=-H}=0. \end{cases}\tag{1}\N\]\NHere, \(t\) and \((\mathbf{x},z)\) are the time and horizontal-vertical space variables, the vector field \((\mathbf{u},w)\in\mathbb{R}^d\times\mathbb{R}\) is the (horizontal and vertical) velocity, \(\rho>0\) the density and \(P\in\mathbb{R}\) the incompressible pressure, all defined in the spatial domain\N\[\N\Omega_t =\{(\mathbf{x},z):\ \mathbf{x}\in\mathbb{R}^d,\ -H<z<\zeta(t,\mathbf{x}\}\N\]\Nwhere \(\zeta(t,x)\) describes the location of a free surface, and \(H\) is the depth of the layer at rest. The gravity field is assumed to be constant and vertical, \(g>0\) is the gravity acceleration constant.\N\NThe advection terms associated with the \textit{bolus velocity} \((\mathbf{u}_\star,w_\star)\in\mathbb{R}^d\times\mathbb{R}\) are motivated by the works of Gent and McWilliams, who proposed them to account for the contribution of geostrophic eddy correlations to the effective transport velocities in noneddy- resolving (large-scale) models.\N\NThe goal of this work is to rigorously justify, in the case of regular and stably stratified flows, the convergence of system (1) in the shallow water regime to the hydrostatic equations \N\[\N\begin{cases} \partial_t \rho + (\mathbf{u}+\mathbf{u}_\star)\cdot\nabla_{\mathbf{x}} \rho + (w+w_\star)\partial_z \rho = 0,\\\N\rho\big(\partial_t \mathbf{u} + ((\mathbf{u}+\mathbf{u}_\star)\cdot\nabla_{\mathbf{x}}) \mathbf{u} + (w+w_\star)\partial_z \mathbf{u}\big) +\nabla_x P =0,\\\N\partial_t \zeta + (\mathbf{u} + \mathbf{u}_\star)\vert_{z=\zeta} \cdot\nabla_{\mathbf{x}} \zeta - (w+w_\ast)\vert_{z=\zeta}=0,\\\NP=P_{\mathrm{atm}} + g\int_z^\zeta \rho(z',\cdot) \, \mathrm{d} z',\\\Nw=- \int_z^\zeta \nabla_{\mathbf{x}}\cdot \mathbf{u}(z',\cdot) \, \mathrm{d} z'. \end{cases}\tag{2}\N\]\NThis is accomplished by rewriting the system using isopycnal coordinates and performing suitable energy estimates in Sobolev spaces. The results heavily rely on the assumption of stable stratification, which allows to overcome the regularity issues for these PDEs in the absence of viscosity terms.\N\NSpecifically, Theorem 2.1 guarantees the wellposedness of hydrostatic system (2), formulated in isopycnal coordinates, in suitable Sobolev spaces \(H^{k,s}(\Omega)^{1+d}\) with lifespan \([0,T^h]\) depending on the initial data and stratification assumption. Theorem 2.2 then asserts that, as the shallowness parameter \(\mu\to 0\), solutions to the non-hydrostatic system (1) exist on \([0,T^h]\) and converge therein to (2), with rate \(\mu\). The main difficulty consists in providing uniform-in-\(\mu\) controls of the relevant energy norms; the strategy adopted by the authors is inspired by [\textit{B. Desjardins} et al., Water Waves 3, No. 1, 153--192 (2021; Zbl 1501.35293)].