Vol. 4, No. 3 (2015): Regular Issue

This Special Issue collects together works on analytic solutions and numerical simulation of fluid dynamics in the ocean and atmosphere. The contributed papers address a variety of problems in geophysical fluid dynamics including formation of coherent structures in random hydrodynamic flows, hyperbolicity in the ocean, mesoscale surface and deep vortices in the ocean, formation of vocalized atmospheric vortices and motion of tropical cyclones, convective instability and nonlinear structures in systems with a multi-component convection, instability development in shear stratified flows and others.

It is shown that, in parametrically excited stochastic dynamic systems described by partial differential equations, spatial structures (clusters) can appear with probability one, i.e., almost in every system realization, due to rare events happened with probability approaching to zero. The problems of such type arise in hydrodynamics, magnetohydrodynamics, physics of plasma, astrophysics, and radiophysics.

This paper deals with the problem of stochastic structure formation in random hydrodynamic flows. In particular, starting from an analysis of the steady-state probability density, it considers coherent structures of vortex formation (vortex genesis) in stochastic quasi-geostrophic flows, which are related to rotation and random topography of the bottom.

Some manifestations of hyperbolicity in the ocean, the important concept in dynamical systems theory, are discussed. It is shown how to identify hyperbolic points, hyperbolic trajectories and their stable and unstable manifolds solving advection equations for passive scalars in a satellite-derived AVISO velocity field and computing finite-time Lyapunov exponents by the singular-value decomposition method. To validate our simulation we use available tracks of oceanic drifters following near surface currents in some areas in the Northwestern Pacific Ocean. The tracks illustrate how drifters “feel” the presence of hyperbolic points, hyperbolic trajectories and stable and unstable manifolds and change abruptly their trajectories when approaching a hyperbolicity region.

Within the framework of the hydromechanical model (HMM), proposed by one of the authors, a tropical cyclone (TC) motion is defined by a largescale wind field and a TC intensity. The model contains parameters describing TC and its interaction with wind field. The diagnostic, quasi-prognostic and prognostic calculations of TC movement are carried out. Diagnostic and quasi-prognostic calculations mean that an objective analysis of a large scale wind field and an objective analysis of a TC intensity is used during a TC whole lifetime. In case of diagnostic calculations, model parameters (constants for each TC) are defined from the best coincidence between the real and calculated track of a TC during a TC whole lifetime; for quasiprognostic calculations they are defined during the preliminary “preprognostic” period. Diagnostic calculations show that the HMMrather correctly describes peculiarities of a TC motion. Quasi-prognostic calculations show that model parameters may be rather correctly defined during a preliminary “preprognostic” period. The results of the diagnostic, quasi-prognostic and prognostic calculations are presented.

We study the influence of deep vortices on the ocean surface in terms of sea-surface elevation, a quantity related to a fluid stream function. We use several mathematical and numerical models, from the most idealized configurations (point vortices) to realistic ones (finite volume vortices). We determine analytically the surface influence of vortices at rest (steady signature) and in motion (dynamical signature). Then, using a nonlinear, numerical hydrodynamicmodel for oceanic vortices, we determine the growth with time of a dynamical signature for drifting vortices without steady signature. We conclude on the possibility to detect several types of oceanic vortices with surface measurements, using the results from our theory and experiments.

The classical Rayleigh theory of convective instability of a viscous and heat conductive rotating atmospheric layer is generalized to the case of phase transitions of water vapor both for the precipitation convection (PC) and for the nonprecipitation (NPC) one. A principal difference is stated between moist convection and Rayleigh convection, on the one hand, and PC and NPC, on the other hand. In particular, the instability region on the plane of model parameters turned out to generally consist of two subregions, in one of which the localized axisymmetric disturbances with a tropical cyclone (hurricane) structure have the highest growth rate. In case of PC the ascending motions on the axis of symmetry correspond to such disturbances, in case of NPC a spontaneous growth of localized vortices both with ascending and descending motions on the axis is possible. Under other parameters values in case of PC spatially periodic cloud structures (convective rolls or closed cloud cells) have the highest growth rate and in case of NPC–mesoscale systems of convective rolls or mesoscale cloud clusters with annular cloud structures.

We consider 3D multi-component convection in a horizontally infinite layer of an uncompressible fluid slowly rotating around a vertical axis. A family of CGLE type amplitude equations is derived by multiple-scaled method in the neighborhood of Hopf bifurcation points. We numerically simulate a case of the three-mode convection at large Rayleigh numbers. It was shown that the convection typically takes a form of hexagonal structures for a localized initial conditions. The rotation of the system prevents the spread of the convective structures on the entire area. The approach to the modeling of the Saturn’s polar hexagon on the basis of amplitude equations is discussed.

Weakly stratified flows of the class under study have a wide 3D spectrum of the most unstable waves with very close growth rates and phase velocities so that their individual critical layers merge into a common one. The analysis of evolution equations for those waves has shown that throughout a weakly nonlinear stage of development their amplitudes grow explosively. During the first (three-wave) phase, the most rapidly growing are low-frequency waves whereas at the next phase, when numerous and diverse higher-order wave interactions come into play, the growth of highfrequency waves is accelerated and they overtake low-frequency waves. The results obtained are illustrated by numerical calculations for some ensembles of waves.

A transient free surface flow past a two-dimensional semi-infinite flat plate in the fluid of a finite depth is considered in the linear approximation. It is assumed that the fluid is inviscid and incompressible and the flow is irrotational. The plate is suddenly submerged at relatively small depth below the free surface into the fluid uniformly moving with a constant velocity. The linearized problem is solved for relatively small Froude numbers F < 1 using the Laplace and Fourier transforms, as well as the Wiener– Hopf technique. It is shown that eventually at large time, the transient solution approaches asymptotically the steady-state solution. Peculiarities of the solution obtained are discussed and illustrated graphically.

Two parallel plates with shear moving velocity in opposite direction is introduced as external excitations to induce cavitating flow between them, and a developed scheme based on Lattice Boltzmann method is used to simulate and analyze the evolution of the cavitation or phase transition. First, the principles and simulation process of Lattice Boltzmann method and potential models for single component multiphase flow are introduced, including a special model to the moving boundary conditions. Then, the numerical simulations of evolution of phase transition, induced by shear motions of two parallel plates, are carried out in detail, and the complicated pattern formation of cavitating flows are analyzed in such micro- and multiphase dynamic system and some new results are obtained. In particular, the influences of main parameters, such as initial density and moving velocity, on the cavitation and flow pattern are studied further. The results show that the shear moving motion of two parallel plates could induce the cavitation, and the cavitation and cavitating flow pattern could be controlled availably and efficiently by the main parameters listed above. Further, the method and analysis could be extended to flowing liquid, and an idea of drag reduction utilizing the cavitation due to phase transition in such liquid is proposed.