HARMONIC WAVE SOLUTIONS OF THE NON-LINEAR VORTICITY EQUATION FOR A ROTATING VISCOUS FLUID

Abstract

With the assumption of non-divergent horizontal flow, harmonic wave solutions of the non-linear hydrodynamic equations for a rotating, viscous fluid are obtained both for the plane and for the sphere. These solutions yield waves which may be damped, amplified, or remain unchanged, depending on the longitudinal and latitudinal extents of the waves, and also on the vertical profile of the stream function. For plane flow, the vertical profile of the velocity components is shown to be a sinusoidal function, or a hyperbolic sine and cosine of z, proportional to or independent of z, the elevation. For spherical flow, the vertical profile is shown to be proportional to a Bessel function of order n + ½, and n, an integer, is the degree of a surface spherical harmonic. Harmonic waves are amplified, neutral or damped, depending respectively on whether the argument of the Bessel function is imaginary, zero or real. No steady wave is possible in spherical viscous flow, although quasi-stationary waves, which may be defined as waves with zero wave-velocity but changing wave-amplitude, may occur under certain circumstances.