The Kelvin Waves in Vortex Dynamics and Their Stability

The rotating m-waves of Kelvin form a class K_m, m ⩾ 2, of m-fold symmetric regions D of constant vorticity ω₀ ≡ 1 which are uniformly rotating with an angular velocity (bifurcation parameter) Ω in the range Ω _m-1 < Ω ⩽ Ω _m, Ω _m ≡ {(m -1)}/{2m}. The class K₂ corresponds to the rotating ellipse of Kirchoff. We present a numerical method for the determination of the stability-characteristics of the class K_m. The method, based on Burbea's theory of the stability of vortex-motions, uses conformal mappings to construct the spectrum of a certain crucial "stability-operator" B. Numerical results show the existence of a critical value Ω _cr(m) = {(3Ω _m + Ω _m-1) }/{4} with the following property: Given any Ω ɛ ( Ω _m-1, Ω _m], there exists a steady state D ɛ K_m, unique up to rotation, magnification and reflection, whose angular velocity is Ω. Moreover, this state is (secularly) stable if and only if Ω _cr (m) < Ω ⩽ Ω _m. Other stability-characteristics of the class K_m are determined. The entire results of Love for the particular class K₂ are obtained here as special cases.