Barotropic instability of a viscous boundary jet

The barotropic instability of a boundary jet on a beta plane is considered with emphasis on the effect of internal viscosity. An eigenvalue problem for the disturbance equations and its inviscid version are solved by the aid of numerical methods, and instability characteristics are determined as functions of the Reynolds numberR for various values of the beta-parameter. Typical disturbance structures (eigenfunctions) are also computed. Numerical examples show that the minimum critical Reynolds numberR _cr for instability is smaller than 100. At a Reynolds number of the order of hundreds, there appears a second mode of instability in addition to the first unstable mode originating atR _cr ; a kind of ‘resonance’ between the first and second eigenvalues occurs at the particular value ofR. The neutral stability curves are accordingly multi-looped. Although each of the two unstable modes asymptotically approaches its inviscid counterpart asR→∞, the asymptotic approach to the inviscid limit is rather slow and the effect of varyingR is conspicuous even atR∼O (10⁴). It is thus demonstrated that the Reynolds number is an essential stability parameter for real boundary jets. The main part of the material contained in this paper was presented at 1981-Autumn Assembly of the Oceanographical Society of Japan.