Resonant excitation and nonlinear evolution of waves in the equatorial waveguide in the presence of the mean current.

We study interactions of planetary waves propagating across the equator with trapped Rossby or Yanai modes, and the mean flow. The equatorial waveguide with a mean current acts as a resonator and responds to planetary waves with certain wave numbers by making the trapped modes grow. Thus excited waves reach amplitudes greatly exceeding the amplitude of the incoming wave. Nonlinear saturation of the excited waves is described by an amplitude equation with one or two attracting equilibrium solutions. In the latter case spatial modulation leads to formation of characteristic defects in the wave field. The evolution of the envelopes of long trapped Rossby waves is governed by the driven complex Ginzburg-Landau equation, and by the damped-driven nonlinear Schrödinger equation for short waves. The envelopes of the Yanai waves obey a simple wave equation with cubic nonlinearity.

Resonant excitation of rossby waves in the equatorial waveguide and their nonlinear evolution.

Nonlinear interactions between the baroclinic Rossby waves trapped in the equatorial waveguide and the barotropic Rossby waves freely propagating across the equator are studied within the two-layer model of the atmosphere, or the ocean. It is shown that a barotropic wave can resonantly excite a pair of baroclinic waves with amplitudes much greater than its proper amplitude. The envelopes of the baroclinic waves obey Ginzburg-Landau-type equations and exhibit nonlinear saturation and formation of characteristic "domain-wall" and "dark-soliton" defects.

Breaking of balanced and unbalanced equatorial waves.

A clear-cut signature of a wave-breaking event is irreversible modification of the mean flow. In this paper, we provide examples of different breaking mechanisms and show that breaking scenario of equatorial waves in the beta-plane shallow water model is determined by the degree of balance between the zonal component of the Coriolis force and the pressure gradient. Our analysis is based on a specially designed numerical method which guarantees two essential conditions to simulate nonlinear equatorial waves: (i) the scheme converges toward weak solutions including shocks and (ii) preserves the steadiness of balanced stationary solutions. This allows for accurate diagnostics of Lagrangian invariants of motion such as passive tracer density or potential vorticity. For unbalanced waves, the lack of balance leads to shock formation in finite time. In shock fronts, the variation of the dissipation rate induces a nonadvective potential vorticity flux and violates the local potential vorticity conservation valid for smooth solutions. This dissipative breaking mechanism is generic for unbalanced waves and is associated with enhanced mixing. For long, balanced (Rossby) waves, breaking consists in appearance of recirculation regions. It results in the formation of propagating patterns, the equatorial modons, which trap fluid particles. Such breaking occurs during the propagation of Rossby wave packets with positive geopotential anomaly and is strengthened by decreasing fluid depth. The modons are robust and collide quasielastically with Kelvin waves.

Self-consistent finite-mode approximations for the hydrodynamics of an incompressible fluid on nonrotating and rotating spheres.

Self-consistent finite-mode approximations for both Euler and Navier-Stokes equations for vorticity on a sphere are constructed and extended to the case of a rotating sphere, aiming at application to ocean and atmosphere modeling. In the absence of dissipation they preserve the specific Hamiltonian structure of hydrodynamics and have, at each level of approximation, an appropriate number of integrals of motion, which is not the case for standard schemes.