Taylor–Goldstein equation

The Taylor–Goldstein equation is an ordinary differential equation used in the fields of geophysical fluid dynamics, and more generally in fluid dynamics, in presence of quasi-2D flows. It describes the dynamics of the Kelvin–Helmholtz instability, subject to buoyancy forces (e.g. gravity), for stably stratified fluids in the dissipation-less limit. Or, more generally, the dynamics of internal waves in the presence of a (continuous) density stratification and shear flow. The Taylor–Goldstein equation is derived from the 2D Euler equations, using the Boussinesq approximation.

The equation is named after G.I. Taylor and S. Goldstein, who derived the equation independently from each other in 1931. The third independent derivation, also in 1931, was made by B. Haurwitz.

The equation is derived by solving a linearized version of the Navier–Stokes equation, in presence of gravity ${\displaystyle g}$ and a mean density gradient (with gradient-length ${\displaystyle L_{\rho }}$), for the perturbation velocity field

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