Rossby Waves on Non-zonal Currents: Structural Stability of Critical Layer Effects
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Pure and Applied Geophysics
Rossby Waves on Non-zonal Currents: Structural Stability of Critical Layer Effects VLADIMIR G. GNEVYSHEV,1
SERGEI I. BADULIN,1,2
Abstract—The problem of the propagation of linear Rossby waves in horizontally inhomogeneous non-zonal flows is studied. The explicit solution within the geometric optics (WKBJ) approximation is found to be identical to the exact Cauchy problem solution for the case of a constant horizontal velocity shear.The effect of the short-wave transformation of Rossby waves near the so-called critical layer is detailed for the arbitrary direction of nonzonal flow. In the general case, this transformation can occur in two ways: (1) as an adhering, a monotonic approaching of wave packets to the critical layer for an infinitely long time. The sign of the intrinsic frequency of the packet remains the same all the time; (2) as an adhering with overshooting when the wave packet, first, crosses its critical layer at finite wavenumber. The wave changes the sign of the intrinsic frequency when overshooting the critical layer and then keeps the sign when it is adhering to this layer asymptotically similarly to the previous scenario. The latter regime does not exist for zonal flows, that degenerates the short-wave dynamics of Rossby waves in this special case. On the contrary, the anisotropy of the dispersion relation permits both positive and negative frequencies in a non-zonal flow. It allows for the effective use of the concept of waves of negative energy for the analysis of the stability of large-scale currents. Keywords: Rossby waves, zonal and non-zonal flows, WKBJapproximation (geometric optics), enstrophy, critical layer, adhering overshooting phenomenon.
1. Introduction The planetary Rossby waves are an essential link in the dynamics of the atmosphere and the World ocean. Understanding of mechanisms of their interaction with large-scale general circulation is crucial
1
Department of Physics, Shirshov Institute of Oceanology, Russian Academy of Sciences, Nakhimovsky pr. 36, Moscow 117997, Russia. E-mail: [email protected] 2 Center for Advanced Studies, Skolkovo Institute of Science and Technology, Bolshoy Boulevard 30, bld. 1, Moscow 121205, Russia. 3 Institute of Earth Sciences, Saint Petersburg State University, 13B, Universitetskaya Emb., St. Petersburg 199034, Russia.
and TATYANA V. BELONENKO3
for the description of magnitudes and signatures of global energy, momentum, and mass transfer. Study of the Rossby waves of infinitely small amplitudes (linear waves) is a necessary step to this understanding when universal tools of wave dynamics and examples of peculiar wave effects can help without resorting to massive computer consuming simulations. The refraction of the linear waves from step-like inhomogeneities can be treated, first, within the quite general problem of the waves on a tangential discontinuity (Lifshitz and Pitaevskii 1987). The analysis of the problem for waves of different types, acoustic, water gravity, internal (e.g. Fabricant 1987) discovers a p
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