Stability of Modons and Wu-Verkley Waves

Abstract

The modons and Wu-Verkley (WV) waves are weak solutions of the barotropic vorticity equation governing the motion of an ideal and unforced fluid on a sphere. Since the pioneering work by Larichev and Reznik, vortex pairs known as modons have attracted a great deal of attention because of their potential applications in geophysical fluid dynamics and plasma physics. Indeed, multiple attempts were made to use the geometric structure and the relative stability of modons for explaining the phenomena of atmospheric blocking. In this connection, modon stability was an important issue in all modon applications. Also, using the unstable and stable manifolds of a WV wave, Wu showed numerically that a non-stationary solution of forced and dissipative vorticity equation can cyclically approach one of the two main atmospheric regimes, namely the zonal circulation and blocking-like circulation.

In this chapter, the stability of WV waves and three different types of modons by Verkley is studied. The structure of these solutions is briefly described in Sect. 6.1. A conservation law for infinitesimal perturbations of a stationary modon and WV wave is derived in Sect. 6.2 and used in Sect. 6.3 to obtain a necessary condition (Theorem 6.3.1) for the exponential (normal mode) instability of such solutions. We will show that the new condition imposes a restriction on the spectral distribution of the energy of both unstable and decaying modes through the value of mean spectral number χ of the mode amplitude where χ is the square of Fjörtoft’s average spectral number. The new instability conditions specify the spectral structure of unstable disturbances which must belong to a hypersurface in the phase space of perturbations. Unlike LP flows and RH waves, the conditions for the normal mode instability of the WV waves and modons depend not only on the degree of the basic solution, but also on the spectral distribution of the mode energy in the inner and outer regions of the solution. The only exception is the modon with uniform absolute vorticity in the inner region. Note that some results on the stability obtained in this chapter can also be applied to quadrupole modons by Neven.

In Sect. 6.4, the instability conditions are used to evaluate the maximum growth rate of unstable modes of the WV waves and modons (Theorem 6.4.1). It is also shown in this section that the amplitude of any unstable, decaying or non-stationary mode is orthogonal to the basic solution in the $\mathbb{H}_{0}^{1}$-inner product (Theorem 6.4.2). The results obtained allow testing the numerical algorithms and program packages developed for the linear stability study. Note again that the average spectral number by Fjörtoft appears both in the instability conditions and in the estimates of the maximum growth rate of unstable modes.

The case of two dipole modons moving along the same latitudinal circle is considered in Sect. 6.5, and the distance between these modons is estimated. The results are used in Sect. 6.6 to prove the instability of dipole modons in the sense of Liapunov. The mechanism of instability is explained.