On spurious instabilities on the β-planes with no mean flows

Abstract. Recent numerical calculations that suggest the existence of instabilities of the shallow water equations on the β-planes (both mid-latitudes and on the equator) are shown to be associated with spurious instabilities rather than genuine unstable modes. The spurious unstable modes arise in an analysis of the necessary condition for the existence of complex roots to the cubic dispersion relation but it is shown here that these complex roots are not associated with solutions (eigenvalues and eigenfunctions) of the corresponding eigenvalue equation.