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.

[1] 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.

Taylor and S. Goldstein, who derived the equation independently from each other in 1931.

[2] The equation is derived by solving a linearized version of the Navier–Stokes equation, in presence of gravity

and a mean density gradient (with gradient-length

is the unperturbed or basic flow.

The perturbation velocity has the wave-like solution

for the flow, the following dimensional form of the Taylor–Goldstein equation is obtained: where

If the imaginary part of the wave speed

is positive, then the flow is unstable, and the small perturbation introduced to the system is amplified in time.

Note that a purely imaginary Brunt–Väisälä frequency

The relevant boundary conditions are, in case of the no-slip boundary conditions at the channel top and bottom