Non-linear waves interactions in rotating shallow water magnetohydrodynamics

<p>This study is devoted to the development of the nonlinear theory of the magneto-Poincare waves and magnetostrophic waves in rotating layers of astrophysical and space plasma in the shallow-water approximation. These waves determine the large-scale dynamics of the various astrophysical and space objects such as solar tachocline, as well as  magnetoactive atmospheres of exoplanets trapped by tides of a carrier star, neutron stars atmospheres and the flows in accretion disks of neutron stars. For this purpose we derived magnetohydrodynamic shallow water equations with a rotation and presence of an external vertical magnetic field. The system is obtained from conventional magnetohydrodynamic equations for incompressible inviscid heavy plasma layer with free surface in an external vertical magnetic field. The pressure is assumed to be hydrostatic, and the height of the plasma layer is considered to be much smaller than horizontal scales of the flow. The magnetohydrodynamic equations in the shallow-water approximation play equally important role in the space and astrophysical plasma flows like classical shallow-water equations in the fluid dynamics of a neutral fluid. The magnetohydrodynamic shallow water equations with an external vertical magnetic field are modified by supplementing them with the equation for the vertical component of the magnetic field and divergence-free condition for magnetic field contains its vertical component. Thus the velocity field remains two-dimensional while the magnetic field becomes three-dimensional. It is shown that the presence of a vertical magnetic field significantly changes the dynamics of the wave processes in astrophysical plasma compared to the neutral fluid and plasma layer in a horizontal magnetic field.  We have investigated the interaction of Magneto-Poincare waves and magnetostrophic waves in the magnetohydrodynamic shallow water flows in external vertical magnetic field and in horizontal (toroidal and poloidal) magnetic field. In the absence of the horizontal magnetic field the dynamics of plasma appears to be similar to the neutral fluid dynamics and it is shown that there are four-waves interactions in this case. Using the asymptotic multiscale method we obtained the non-linear amplitude equations for the three interacting Magneto-Poincare waves and magnetostrophic waves. The analysis of the amplitude equations shows that there are two types of instabilities for four different types of three-waves configurations. These instabilities occur in both cases: in the external vertical magnetic field and in the horizontal magnetic field. For all types of instabilities the growth rates are found. In the absence of the vertical magnetic field we obtained the non-linear amplitude equations for the four interacting waves. It is shown that the resulting system of equations has the first integrals that describe the mechanism of energy transfer among interacting waves of small amplitude. This work was supported by the Russian Foundation for Basic Research (project no. 19-02-00016).</p>

INFLUENCE OF ICE COVER ON KELVIN AND POINCARE WAVES

This work presents theoretical foundations of Kelvin and Poincare waves in the homogeneous ocean under an ice cover. The ice is considered as thin elastic plate of uniform thickness, with constant values of Young’s modulus, Poisson’s ratio, density, and compressive stress. The boundary conditions are such that the normal velocity at the bottom is zero, and at the undersurface of the ice the linearized kinematic and dynamic boundary conditions are satisfied. We present and analyze explicit solutions for the Kelvin and Poincare waves and the dispersion equations. The problem is examined in the context of a unified theory and without the hydrostatic assumption.

Multi-scale phenomena of rotation-modified mode-2 internal waves

We present high-resolution, three-dimensional simulations of rotation-modified mode-2 internal solitary waves at various rotation rates and Schmidt numbers. Rotation is seen to change the internal solitary-like waves observed in the absence of rotation into a leading Kelvin wave followed by Poincaré waves. Mass and energy is found to be advected towards the right-most side wall (for a Northern Hemisphere rotation), leading to increased amplitude of the leading Kelvin wave and the formation of Kelvin–Helmholtz (K–H) instabilities on the upper and lower edges of the deformed pycnocline. These fundamentally three-dimensional instabilities are localized within a region near the side wall and intensify in vigour with increasing rotation rate. Secondary Kelvin waves form further behind the wave from either resonance with radiating Poincaré waves or the remnants of the K–H instability. The first of these mechanisms is in accord with published work on mode-1 Kelvin waves; the second is, to the best of our knowledge, novel to the present study. Both types of secondary Kelvin waves form on the same side of the channel as the leading Kelvin wave. Comparisons of equivalent cases with different Schmidt numbers indicate that while adopting a numerically advantageous low Schmidt number results in the correct general characteristics of the Kelvin waves, excessive diffusion of the pycnocline and various density features precludes accurate representation of both the trailing Poincaré wave field and the intensity and duration of the Kelvin–Helmholtz instabilities.

Instabilities within Rotating mode-2 Internal Waves

We present high resolution, three dimensional simulations of rotation modified mode-2 internal solitary waves at various rotation rates and Schmidt numbers. Rotation is seen to change the internal solitary-like waves observed in the absence of rotation into a leading Kelvin wave followed by Poincaré waves. Mass and energy is found to be advected towards the right-most side wall (for Northern hemisphere rotation) which led to Kelvin-Helmholtz instabilities within the leading Kelvin wave that form above and below the pycnocline. These instabilities are localized within a region near the side wall and intensify in vigour with increasing rotation rate. Secondary Kelvin waves form further behind the wave from either resonance with radiating Poincaré waves or the remnants of the K-H instability. The first of these mechanisms is in accord with published work on mode-1 Kelvin waves. Both types of secondary Kelvin waves form on the same side of the channel as the leading Kelvin wave. Comparisons of equivalent cases with different Schmidt numbers indicate that while low Schmidt number results in the correct general characteristics of the modified ISWs, it does not correctly predict the trailing Poincaré wave field or the intensity and duration of the K-H instabilities.

Planetary (Rossby) waves and inertia–gravity (Poincaré) waves in a barotropic ocean over a sphere

The construction of approximate Schrödinger eigenvalue equations for planetary (Rossby) waves and for inertia–gravity (Poincaré) waves on an ocean-covered rotating sphere yields highly accurate estimates of the phase speeds and meridional variation of these waves. The results are applicable to fast rotating spheres such as Earth where the speed of barotropic gravity waves is smaller than twice the tangential speed on the equator of the rotating sphere. The implication of these new results is that the phase speed of Rossby waves in a barotropic ocean that covers an Earth-like planet is independent of the speed of gravity waves for sufficiently large zonal wavenumber and (meridional) mode number. For Poincaré waves our results demonstrate that the dispersion relation is linear, (so the waves are non-dispersive and the phase speed is independent of the wavenumber), except when the zonal wavenumber and the (meridional) mode number are both near 1.

Resonant excitation of trapped waves by Poincaré waves in the coastal waveguides

After having revisited the theory of linear waves in the rotating shallow-water model with a straight coast and arbitrary shelf/beach bathymetry, we undertake a detailed study of resonant interaction of free Poincaré waves with modes trapped in the coastal waveguide. We describe and quantify the mechanisms of resonant excitation of waveguide modes and their subsequent nonlinear saturation. We obtain the modulation equations for the amplitudes of excited waveguide modes in the absence and in the presence of spatial modulation and analyse their solutions. Different saturation regimes are exhibited, depending on the nature of the modes involved. The excitation is proved to be efficient, i.e. the saturated amplitudes of the excited waves considerably exceed the amplitude of the generator waves. Back-influence of the excited waveguide modes onto the open ocean results in a phase shift of the reflected Poincaré waves and possible energy redistribution between them. A comparison of rotating and non-rotating cases displays substantial differences in excitation mechanisms in the two cases.