Ranis N. Ibragimov

Generation of internal tides by an oscillating background flow along a corrugated slope

The process of internal wave generation by the interaction of an oscillatory background flow (U0 cos (ω0t), V0 sin (ω0t), W0 sin (ω0t)) over a uniform slope is investigated. The stratification is assumed to be uniform and the fluid of infinite depth. Analytical solutions are obtained which give the energy flux in the radiating internal wave field. Since waves are generated not only at the fundamental frequency ω0, but also at all the harmonic frequencies less than the buoyancy frequency, the energy flux for both low and high frequency waves is considered. The acoustic limit approximation (the limiting case in which the tidal excursion, U0/ω0, is much less than the scale of the topography) gives a reasonable approximation to the energy flux.

Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications

We study the nonlinear incompressible fluid flows within a thin rotating spherical shell. The model uses the two-dimensional Navier-Stokes equations on a rotating three-dimensional spherical surface and serves as a simple mathematical descriptor of a general atmospheric circulation caused by the difference in temperature between the equator and the poles. Coriolis effects are generated by pseudoforces, which support the stable west-to-east flows providing the achievable meteorological flows rotating around the poles. This work addresses exact stationary and non-stationary solutions associated with the nonlinear Navier-Stokes. The exact solutions in terms of elementary functions for the associated Euler equations (zero viscosity) found in our earlier work are extended to the exact solutions of the Navier-Stokes equations (non-zero viscosity). The obtained solutions are expressed in terms of elementary functions, analyzed, and visualized.

Applications of Lie group analysis in Geophysical fluid dynamics

This book introduces an effective method for seeking local and nonlocal conservation laws and exact solutions for nonlinear two-dimensional equations which provide a basic model in describing internal waves in the ocean. The model consists of non-hydrostatic equations of motion which uses the Boussinesq approximation and linear stratification. The Lie group analysis is used for constructing non-trivial conservation laws and group invariant solutions. It is shown that nonlinear equations in question have remarkable property to be self-adjoint. This property is crucial for constructing physically relevant conservation laws for nonlinear internal waves in the ocean. The comparison with the previous analytic studies and experimental observations confirrms that the anisotropic nature of the wave motion allows to associate some of the obtained invariant solutions with uni-directional internal wave beams propagating through the medium. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams are considered. The behavior of the invariant solutions near the critical latitude is investigated.

Effects of rotation on stability of viscous stationary flows on a spherical surface

We study the incompressible viscous fluid flows within a thin rotating atmospheric shell. The model uses the two-dimensional Navier–Stokes equations on a spherical surface and serves as a simple mathematical description of a general atmospheric circulation caused by the difference in temperature between the equator and the poles. Linearized stability of a particular stationary flow is considered. Under the assumption of no friction and a distribution of temperature dependent only upon latitude, the stationary flow models a zonal distribution of pressure corresponding to atmospheric currents parallel to the circles of latitude. We prove analytically that the stationary flow is asymptotically stable in the time evolution of the Navier–Stokes equations. When the spherical surface is truncated between two symmetrical rings near the North and South Poles, the asymptotic stability of the stationary flow is verified numerically.

Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids

The nonlinear solutions of the two-dimensional Boussinesq equations describing internal waves in rotating stratified fluids were obtained as group invariant solutions. The latter nonlinear solutions correspond to the rotation transformation preserving the form of the original nonlinear equations of motion. It is shown that the obtained class of exact solutions can be associated with the spherically pulsating patterns observed in uniformly stratified fluids. It is also shown that the obtained rotationally symmetric solutions are bounded functions that can be visualized as spinning patterns in stratified fluids. It is also shown that the rotational transformation provides the energy conservation law together with other conservation laws for which the spinning phenomena is observed. The effects of nonlinearity and the Earth’s rotation on such a phenomenon are also discussed.

Energy Balance Associated With a Mixing Process at the Interface of a Two-Layer Longitudinal Atmospheric Model

Several examples illustrating the energy balance associated with a mixing process at the interface of a planar dynamical model describing two-phase perfect fluid circulating around a circle with a sufficiently large radius within a central gravitational field are presented. The model is associated with the spatial and temporal structure of the zonally averaged global-scale atmospheric longitudinal circulation around the Earth. The fluid is supposed to be irrotational and pressure on a outer layer is constant. It is postulated that the total fluid depth is small compared to the radius of the circle and the gravity vector is directed to the center of the circle. Under these assumptions, this problem can be associated with a spatial and temporal structure of the zonally averaged global-scale atmospheric longitudinal circulation around equatorial plane. The model is the subject to the rigid lid approximation to the external boundary conditions for the outer fluid layer. One of the novelties in this work is the derivation of the nonlinear shallow water model by means of the average velocity. This introduction simplifies essentially further potential studies of mixing criteria associated with nonlinear mathematical models representing shallow water equations.