Journal of Fluid Mechanics

The critical layer for internal gravity waves in a shear flow

John R.  Booker a1 and Francis P.  Bretherton a2
a1 Institute of Geophysics and Planetary Physics, La Jolla, California
a2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge

Article author query
booker jr   [Google Scholar] 
bretherton fp   [Google Scholar] 


Internal gravity waves of small amplitude propagate in a Boussinesq inviscid, adiabatic liquid in which the mean horizontal velocity U(z) depends on height z only. If the Richardson number R is everywhere larger than 1/4, the waves are attenuated by a factor $\exp\{-2\pi(R - \frac{1}{4})^{\frac{1}{2}}\}$ as they pass through a critical level at which U is equal to the horizontal phase speed, and momentum is transferred to the mean flow there. This effect is considered in relation to lee waves in the airflow over a mountain, and in relation to transient localized disturbances. It is significant in considering the propagation of gravity waves from the troposphere to the ionosphere, and possibly in transferring horizontal momentum into the deep ocean without substantial mixing.

(Published Online March 28 2006)
(Received April 25 1966)