On the motion of various vertical modes of transient, very long waves Part I. Beta Plane Approximation

^*Publication No. 182 from the Department of Meteorology and Oceanography, The University of Michigan, Ann Arbor, Michigan. Research supported by the National Science Foundation under Grant No. GA-16166.

Abstract

The motion and structure of very long, transient waves are investigated by solving the perturbation problem using a basic state characterized by no motion and a constant lapse rate. It has been assumed that the motion is adiabatic and frictionless, and a beta-plane geometry has been used.

Using the method of separation of variables it has been shown that the solution consists of a discrete set of vertical structure functions each characterized by an equivalent depth which decreases as the eigenvalue increases. The existence of the smallest eigenvalue depends on the use of a boundary condition of co w ≠ 0 at the surface of the earth, while the higher eigenvalues are influenced only slightly if the boundary condition w = 0 is used.

The phase speed corresponding to the smallest eigenvalue is in good agreement with the results obtained from observational studies from 500 mb data or for the first vertical mode, and the wave corresponding to this eigenvalue behaves in a way similar to waves in a homogeneous fluid with a free surface. For the larger eigenvalues we obtain phase speeds which agree with the motion observed for the higher vertical modes. It is finally shown that the equations based on Burger's assumption give good results for the higher vertical modes, but not for the mode corresponding to the smallest eigenvalue.

Notes

^*Publication No. 182 from the Department of Meteorology and Oceanography, The University of Michigan, Ann Arbor, Michigan. Research supported by the National Science Foundation under Grant No. GA-16166.