Vortex generation in a continuously stratified rotating fluid in the case of displacement of a basin bottom region (Q2714028)

The motion is considered of a plane layer \((-\infty < x, y < +\infty\), \(- H < z < 0)\) of an ideal incompressible heavy fluid rotating with angle velocity \(l/2\). In the nonperturbed state, the density \(\rho_0(z)\) of the fluid varies continuously with \(\rho_0(z)' < 0\) for all \(z\in [-H,0]\). The displacement of the basin bottom is governed by the formula \(z = - H + h(x,y,z)\), where \(h\) vanishes at \(t\leq 0\) and \(h\equiv h_0(x,y)\) for \(t\geq T\). It is assumed that \(\max| h| \ll H\) and \(h\to\infty\) as \(R = \sqrt{x^2 + y^2}\to\infty\).NEWLINENEWLINE For describing the evolution of the fluid, the system of equations NEWLINE\[NEWLINE \begin{gathered} \frac{\partial u}{\partial t} - lv= - \frac{1}{\rho_0}\frac{\partial p}{\partial x},\quad \frac{\partial v}{\partial t} + lu = - \frac{1}{\rho_0}\frac{\partial p}{\partial y},\quad \frac{\partial w}{\partial t} = - \frac{1}{\rho_0}\left(\frac{\partial p}{\partial z} + \rho g\right), \\ \frac{\partial \rho}{\partial t} - \frac{\rho_0N^2}{g}w = 0, \quad \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \end{gathered} NEWLINE\]NEWLINE is considered together with the following boundary and initial conditions: NEWLINE\[NEWLINE \begin{gathered} \frac{\partial\zeta}{\partial t} = w,\qquad p - \rho_1g\zeta = 0\quad (z = 0), \qquad w = \frac{\partial h}{\partial t}\quad (z = - H), \\ u = v = w = \rho = \zeta = 0\qquad (t = 0). \end{gathered} NEWLINE\]NEWLINE Here \(u\), \(v\), \(w\) are the horizontal components and vertical component of the velocity vector; \(p\), \(\rho\) are the pressure and density of the fluid; \(z = \zeta(x,y,t)\) denotes the free surface of the fluid; \(N\) is the Brunt--Väisälä frequency; \(\rho_1 = \rho_0(0)\).NEWLINENEWLINE The analysis of the model under consideration is based on an integral representation of solutions to the stationary problem in the case of exponentially stratified fluid in density. The authors expose calculation results which indicate dependence of the intensity of geostrophic fields on the parameters of the problem. The results presented are important for studying the effects that accompany seismic tsunami generation in ocean.