Stability of Rossby-Haurwitz (RH) Waves

It is well known that the Rossby-Haurwitz (RH) waves, being exact solutions of the barotropic vorticity equation (BVE) for an ideal and unforced fluid, represent one of the main features of the meteorological fields. Therefore, the stability properties of the RH wave are also of considerable interest for a better understanding of the nature of low-frequency variability of large-scale atmospheric processes, as well as for developing data assimilation methods. It should be noted that in most cases, stability conditions have been derived for the parallel shear flows or zonal flows whose velocity is a function of only one variable. The first conditions for the linear instability of two-dimensional flows such as stationary RH waves, Wu-Verkley (WV) waves and modons were derived only recently. In this book, we try to give a unified approach to the normal mode instability study of such stationary BVE solutions as the Legendre polynomial (LP) flow, RH wave, WV wave, and modons. This chapter is devoted to the stability of the Rossby-Haurwitz waves and LP flows. In Sect. 5.1, we derive a conservation law for arbitrary perturbations of LP flow and RH wave. Invariant sets (M − n , M + n $$\mathbf{M}_{+}^{n}$$ , M 0 n ∖ H n , and H n ) $$\mathbf{H}_{n})$$ , quotient spaces, and norms of perturbations are considered in Sect. 5.2. The law for arbitrary perturbations from invariant sets is used in Sect. 5.3 to show that the kinetic energy of a perturbation of a RH wave is hyperbolically related with the mean spectral number of the perturbation. The geometric interpretation of variations in the energy of perturbations is given in Sect. 5.4. It is shown in Sect. 5.2 that a RH wave of H 1 ⊕ H n (n ≥ 2) is stable to any perturbation from the invariant set H n (see (5.2.2)). In Sect. 5.5, we demonstrate the Liapunov instability of any non-zonal RH wave of subspace H 1 ⊕ H n ( n ≥ 2 $$n \geq 2$$ ) in the invariant set M − n . The mechanism of instability is also explained. Note that so far there have been obtained no results on the stability of RH wave in the invariant set M + n (n ≥ 2). This problem is not trivial. In Sects. 5.6 and 5.7, the conservation law for perturbations is used to derive a simple necessary condition for the normal mode instability of steady RH waves and LP flows, respectively. The condition imposes a restriction on the spectral distribution of the energy of unstable modes; to wit, the average spectral number by Fjörtoft of the amplitude of any growing normal mode of a steady RH wave of H 1 ⊕ H n $$\mathbf{H}_{1} \oplus \mathbf{H}_{n}$$ (n ≥ 2) or LP flow aP n (μ) (n > 2) must be equal to n ( n + 1 ) $$\sqrt{n(n + 1)}$$ . Thus, the new instability condition specifies the spectral structure of each growing disturbance (a perturbation may be unstable only if it belongs to the set M 0 n ∖ H n ) and depends only on the degree n of the spherical functions representing the basic wave. The bounds of the maximum growth rate of unstable normal modes are also estimated, and the orthogonality of the amplitude of any unstable mode to the basic flow is shown in the inner products of the Hilbert spaces ℍ 0 0 $$\mathbb{H}_{0}^{0}$$ and ℍ 0 1 $$\mathbb{H}_{0}^{1}$$ . The results are especially helpful for testing the computational algorithms (and ultimately, the program packages) used for the numerical linear stability study. In the case of the LP (zonal) flows, the new instability condition complements the famous conditions by Rayleigh–Kuo and Fjörtoft. Also it will be proved that for n ≥ 3, a normal mode of the LP flow P n (μ) and zonal RH wave − ω μ + a P n μ $$-\omega \mu + aP_{n}\left (\mu \right )$$ is stable if its zonal wavenumber m satisfies the condition m ≥ n $$\left \vert m\right \vert \geq n$$ (Theorem 5.7.3). It should be specially noted that both the instability conditions and the estimates of the growth rate of unstable modes obtained in Sects. 5.6 and 5.7 use the mean spectral number by Fjörtoft. Therefore, one can say that this parameter is of paramount importance in the linear instability of harmonic waves on a sphere.