On the hydrostatic approximation in rotating stratified flow - A. Wirth (Nonlinear Processes in Geophysics)
Hydrostatic models were and still are the workhorses for realistic simulations of ocean dynamics, especially for climate applications. Introducing a Fourier space projection method and using the Heisenberg–Gabor limit, a formalism is developed to systematically evaluate the role of flatness, stratification, rotation and friction for the fidelity of the hydrostatic approximation. The hydrostatic approximation is formally first order in γ=H/L<svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="42pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="5695540f6a11504171dfc14e10380a1d"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="npg-32-261-2025-ie00001.svg" width="42pt" height="14pt" src="npg-32-261-2025-ie00001.png"/></svg:svg>, where H is the vertical and L the horizontal scale of the phenomenon considered. For linear (low-amplitude) and unforced stratified rotating flow, the dynamics can be separated into balanced flow and wave motion. It is shown that for the linear balanced motion the hydrostatic approximation is exact and for wave motion it is second order, obtaining the leading prefactors. The fidelity of the hydrostatic approximation therefore also relies on the ratio of the amplitude of wave motion to balanced motion. This ratio adds considerably to the quality of the hydrostatic approximation for larger-scale flows in the atmosphere and the ocean.