I. Sibgatullin

Internal wave attractors examined using laboratory experiments and 3D numerical simulations

In the present paper, we combine numerical and experimental approaches to study the dynamics of stable and unstable internal wave attractors. The problem is considered in a classic trapezoidal setup filled with a uniformly stratified fluid. Energy is injected into the system at global scale by the small-amplitude motion of a vertical wall. Wave motion in the test tank is measured with the help of conventional synthetic schlieren and PIV techniques. The numerical setup closely reproduces the experimental one in terms of geometry and the operational range of the Reynolds and Schmidt numbers. The spectral element method is used as a numerical tool to simulate the nonlinear dynamics of a viscous salt-stratified fluid. We show that the results of three-dimensional calculations are in excellent qualitative and quantitative agreement with the experimental data, including the spatial and temporal parameters of the secondary waves produced by triadic resonance instability. Further, we explore experimentally and numerically the effect of lateral walls on secondary currents and spanwise distribution of velocity amplitudes in the wave beams. Finally, we test the assumption of a bidimensional flow and estimate the error made in synthetic schlieren measurements due to this assumption.

Energy cascade in internal wave attractors

One of the pivotal questions in the dynamics of the oceans is related to the cascade of mechanical energy in the abyss and its contribution to mixing. Here, we propose internal-wave attractors in the large-amplitude regime as a unique self-consistent experimental and numerical setup that models a cascade of triadic interactions transferring energy from large-scale monochromatic input to multi-scale internal-wave motion. We also provide signatures of a discrete wave turbulence framework for internal waves. Finally, we show how, beyond this regime, we have a clear transition to a regime of small-scale high-vorticity events which induce mixing.

Transitional regimes of penetrative convection in a plane layer

Hydrodynamical regimes are described for penetrative convection in a two-dimensional bounded plane layer of water at the temperature range close to the point of maximum density. Stress-free conditions on the horizontal and vertical boundaries of the domain are assumed. The point of maximum density is supposed to be located in the middle plane of the layer in conductive state. Steady and time-periodic solutions are found on large horizontal scales and the lengths of a spatial periodicity inside the layer are determined. Hydrodynamical peculiarities of time-periodic solution are described. In the domain equal to the periodicity cell corresponding to time-periodic solution, the sequence of hydrodynamical regimes with the increase of supercriticality is analyzed and the full sequence of bifurcations from conductive state to chaotic motion is described. The time-periodic solution loses the stability through the subcritical Neimark–Sacker bifurcation. First, the closed invariant curve has one stable period-2 cycle, so the bifurcation looks like period doubling. With a further increase of supercriticality the period-2 cycle transforms to a torus through the saddle-node bifurcation for maps. Then, loss of stability of quasiperiodic motion occurs through intermittency and chaotic mixing on the background of quasiperiodic solution. The window of quasiperiodicity was obtained in the interval of supercriticality corresponding to intermittent regimes, after which intermittency on the background of another quasiperiodic mode appears.

Scale effects in internal wave attractors

As a necessary preliminary step toward geophysically significant extrapolations, we study the scale effects in internal wave attractors in the linear and nonlinear regimes. We use two geometrically similar experimental set-ups, scaled to factor 3, and numerical simulations (a spectral element method, based on the Nek5000 open solver) for a range of parameters that is typically accessible in laboratory. In the linear regime, we recover the classical viscous scaling for the beam width, which is not affected by variations of the amplitude of the input perturbation. In the nonlinear regime, we show that the scaling of the width-to-length ratio of the attractor branches is intimately related with the energy cascade from large-scale energy input to dissipation. We present results for the wavelength, amplitude and width of the beam as a function of time and as a function of the amplitude of the forcing.

Magnetic fields of pulsars surrounded by accretion disks of finite extension

The problem of finding the magnetic field of a system consisting of a magnetized neutron star and a perfectly conducting accretion disk of finite extension is reduced to the solution of the integral Fredholm equation of the second kind. Unlike the pioneering works of Aly (1980), Riffert (1980), we admit that an arbitrary total electric current can flow in the disk with finite outer rim. Having regularized the Fredholm equation, we obtain a convenient method for the construction of the magnetic field. The distribution of the magnetic field along the disk is plotted for different values of the total current and the ratios between the outer and inner radii of the disk.

Some properties of two-dimensional stochastic regimes of double-diffusive convection in plane layer

The existence of a two-dimensional attracting manifold is established for trajectories emanating from the vicinity of static solution. The structure of this manifold is studied with the help of succession map of points in intersection of trajectories with some coordinate plane. The structure of the attracting manifold varies depending on Rayleigh numbers of heat and salinity. With growth of Rayleigh numbers of heat and salinity the structure of one-dimensional curve becomes more irregular and sophisticated. The convergence of Bubnov-Galerkin approximation with a large number of basic functions was demonstrated in norms of kinetic energy, dissipation function, and directly by norm evaluation of the residual (noncompensated terms in substitution of Bubnov-Galerkin approximation to the initial double-diffusive convection system).

Transitional regimes of penetrative convection in a plain layer

Penetrative convection in a plain layer of water is considered for the interval of temperatures containing the point of density maximum. When the unstable and stable layers are equal in the static conductive state, the development of convective instability is investigated from the formation of steady structures to chaotic motion. Scales of the periodicity cell for regimes with a strong nonlinear effect were chosen with particular attention. Specifics are shown for steady structures with small-scale vortices near upper boundaries and for periodic motions with synchronized oscillations of the lower parts of vertically elongated profiles of temperature that move symmetrically. With the increase of supercriticality, the motion loses reflection symmetry, becomes doubly periodic, and finally becomes quasi-periodic before transition to chaotic motion. Domains of hysteresis are investigated for which motions with different structure and heat fluxes coexist, and it is illustrated by the dependence of the Nusselt number on supercriticallity.

Abyssal Mixing in the Laboratory

One of the important questions in the dynamics of the oceans is related to the cascade of mechanical energy in the abyss and its contribution to mixing. Here, we propose a unique self-consistent experimental and numerical set up that models a cascade of triadic interactions transferring energy from large-scale monochromatic input to multi-scale internal wave motion. We show how this set-up can be used to tackle the open question of studying internal wave turbulence in a laboratory, by providing, for the first time, explicit evidence of a wave turbulence framework for internal waves. Finally, beyond this regime, we highlight a clear transition to a cascade of small-scale overturning events which induce mixing.

HIGH-RESOLUTION SIMULATION OF INTERNAL WAVES ATTRACTORS AND IMPACT OF INTERACTION OF HIGH AMPLITUDE INTERNAL WAVES WITH WALLS ON DYNAMICS OF WAVES ATTRACTORS

Internal wave attractors have received great attention since its discovery in 1995 by Leo Maas ([1, 2]). Now convectional theory describing the formation of attractors is generally accepted, and the principal interest of researchers in recent years is focused on nonlinear interactions. A number of theories had been proposed and now nonlinear interaction due to triadic resonance is accepted as a principle cause of instability of attractors, in which case the parent wave of large amplitude gives birth to two daughter waves, such that conditions of triadic resonance are fulfilled for all the three waves [3]. All this presumes that wave-wall interaction participate in the process only by focusing-defocusing of energy on the inclined boundaries. Here we address interactions of large amplitude internal waves with the boundaries in real laboratory conditions, where Prandtl-Schmidt number is equal to 700. We show that large amplitude waves produce folded structures which are clearly visible on density-gradient images, as if produced by “kneading the dough”. These structures are not quickly dissipated due to high Schmidt number and with time they propagate to the interior of the domain, as was shown by our numerical simulation. These structures have visible impact on the instability of attractor and the whole picture of turbulent motion for large amplitudes. The mechanism of interaction of these structures with internal gravity waves structures is the subject of further research. Numerical simulation is quite challenging due to high Prandtl-Schmidt number and small scale of the folded structures in highly nonlinear regimes. Also study is performed on large time scales. As a consequence most of conventional computational approaches give unreliable results. We have applied spectral element approach base on code nek5000 by Paul Fischer. It allowed us to carefully follow the fine space structures on large time scales.