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.
where is the unperturbed or basic flow. The perturbation velocity has the wave-like solution (real part understood). Using this knowledge, and the streamfunction representation for the flow, the following dimensional form of the Taylor–Goldstein equation is obtained:
where denotes the Brunt–Väisälä frequency. The eigenvalue parameter of the problem is . 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.
No-slip boundary conditions
The relevant boundary conditions are, in case of the no-slip boundary conditions at the channel top and bottom and