shear–gravity wave

A combination of gravity waves and a Helmholtz wave on a surface of discontinuity of density and velocity.
If the densities of the lower and upper layers respectively are ρ and ρ′ and the velocities U and U′, the phase speed c of the shear–gravity wave is

where g is the acceleration of gravity and L the wavelength. The motion is unstable if and only if the bracketed quantity is negative; the density difference thus contributes to stability and the velocity difference to instability. Applications have been made to atmospheric frontal surfaces and inversions; perhaps the most successful of these is to the phenomenon of billow clouds. Reasonable atmospheric values for the parameters yield stationary wavelengths of the order of 1 km.
Drazin, P. G., and W. H. Reid 1981. Hydrodynamic Stability. Cambridge University Press, . 14–22.

The variation (usually the directional derivative) of a vector field along a given direction in space.

1. The characteristic of a system if sufficiently small disturbances have only small effects, either decreasing in amplitude or oscillating periodically; it is asymptotically stable if the effect of small disturbances vanishes for long time periods.