The Adjustment of a Non-balanced Velocity Field towards Geostrophic Equilibrium in a Stratified Fluid

The previous work by ROSSBY and CAHN on the adjustment of a non-balanced velocity field towards geostrophic equilibrium has been generalized to incorporate the corresponding process in a stratified but incompressible fluid. It is shown that the character of the process essentially depends upon the width of the current, the velocity profile in the vertical and the vertical density gradient. As long as the vertical density gradient is small the adjustment of the mean motion of the fluid (∫₀^D₀Udz) exactly corresponds to the one described by CAHN. On the other hand vertical variations of the unbalanced motion will set up internal oscillations which have considerable larger (vertical) amplitudes than is found for the adjustment of the mean motion. Furthermore, the speed of propagation of the gravity inertia waves thus generated (c_K) is considerably smaller than that of the waves generated during the adjustment of the mean motion (c₀). For values of the vertical density gradient as found in the oceans c_K < 0.05 c₀. In the final state of equilibrium the mean motion of the fluid will be approximately the same as before the adjustment, while a considerable smoothing of the vertical gradients of the velocity field will take place, the more the broader the original current is.