A Fourier/Modal-Spectral-Element Method for the Simulation of High-Reynolds Number Incompressible Stratified Flows in Domains with a Single Non-Periodic Direction

We present the components of a high-order accurate Navier-Stokes solver designed to simulate high-Reynolds-number stratified flows. The proposed numerical model addresses some of the numerical and computational challenges that high-Reynolds-number simulations pose, facilitating the reproduction of stratified turbulent fluid dynamics typically observed in oceanic and atmospheric flows, namely the development of thin regions of high vertical shear, strongly layered turbulence at high Reynolds numbers and internal wave radiation. This Navier-Stokes solver utilizes a Fourier pseudo-spectral method in the horizontal direction and a modal spectral element discretization in the vertical. We adopt an implicit-explicit time discretization scheme that involves solving several one-dimensional Helmholtz problems at each time step. Static condensation and modal boundary-adapted basis functions result in an inexpensive algorithm based on solving many small tridiagonal systems. A series of benchmark studies is presented to demonstrate the robustness of the flow solver. These include two-dimensional and three-dimensional problems, concluding with a turbulent stratified wake generated by a sphere in linear stratification.