Dispersion diagrams of linear damped waves on the equatorial beta plane

The linear equations of motion are solved to obtain dispersion diagrams with Rayleigh friction ($-\gamma u$) and Laplacian friction ($\nu {\partial }_{xx}u$), the latter being solved numerically. Laplacian friction is more efficient at eliminating the short-wavelength Rossby waves, whereas Rayleigh friction is more effective at dissipating long-wavelength Rossby waves. For Rayleigh friction, short-wavelength Rossby waves do not exist at periods longer than the damping time scale ($1/\gamma$); for Laplacian friction, they do not exist for wavenumbers less than $\sqrt{3}/\left(2\sqrt[3]{\nu }\right)$. For both damping forms, the imaginary wavenumber ($k_i$) no longer separates the upper branch (gravity waves) from the lower branch (Rossby waves) of the real wavenumber ($k_r$), and the waves are damped at all frequencies. The two solutions of $k_r$ do not overlap, and for Rayleigh damping, they meet at $\sqrt{0.5-{\gamma }^2}$, which roughly corresponds to $\sim$13.7 days for very low friction. For very high Rayleigh damping, which is an unrealistic scenario, only gravity waves exist in the dispersion diagram. The consequence of adding friction, even if negligible, is that the discontinuity evident in the inviscid solution at the critical latitude no longer holds, or the critical latitude ceases to exist.