No Arabic abstract
Large eddy simulations (LES) are employed to investigate the role of time-varying currents on the form drag and vortex dynamics of submerged 3D topography in a stratified rotating environment. The current is of the form $U_c+U_t sin(2pi f_t t)$, where $U_c$ is the mean, $U_t$ is the tidal component and $f_t$ is its frequency. A conical obstacle is considered in the regime of low Froude number. When tides are absent, eddies are shed at the natural shedding frequency $f_{s,c}$. The relative frequency $f^*=f_{s,c}/f_t$ is varied in a parametric study which reveals states of high time-averaged form drag coefficient. There is a two-fold amplification of the form drag coefficient relative to the no-tide ($U_t=0$) case when $f^*$ lies between 0.5 and 1. The spatial organization of the near-wake vortices in the high drag states is different from a Karman vortex street. For instance, the vortex shedding from the obstacle is symmetric when $f^*=5/12$ and strongly asymmetric when $f^*=5/6$. The increase in form drag with increasing $f^*$ stems from bottom intensification of the pressure in the obstacle lee which is linked to changes in flow separation and near-wake vortices.
The large-scale structures in the ocean and the atmosphere are in geostrophic balance, and a conduit must be found to channel the energy to the small scales where it can be dissipated. In turbulence this takes the form of an energy cascade, whereas one possible mechanism in a balanced flow at large scales is through the formation of fronts, a common occurrence in geophysical dynamics. We show in this paper that an iconic configuration in laboratory and numerical experiments for the study of turbulence, that of the Taylor-Green or von Karman swirling flow, can be suitably adapted to the case of fluids with large aspect ratios, leading to the creation of an imposed large-scale vertical shear. To this effect we use direct numerical simulations of the Boussinesq equations without net rotation and with no small-scale modeling, and with this idealized Taylor-Green set-up. Various grid spacings are used, up to $2048^2times 256$ spatial points. The grids are always isotropic, with box aspect ratios of either $1:4$ or $1:8$. We find that when shear and stratification are comparable, the imposed shear layer resulting from the forcing leads to the formation of multiple fronts and filaments which destabilize and further evolve into a turbulent flow in the bulk, with a sizable amount of dissipation and mixing, and with a cycle of front creation, instability, and development of turbulence. The results depend on the vertical length scales for shear and for stratification, with stronger large-scale gradients being generated when the two length scales are comparable.
To investigate the formation mechanism of energy spectra of internal waves in the oceans, direct numerical simulations are performed. The simulations are based on the reduced dynamical equations of rotating stratified turbulence. In the reduced dynamical equations only wave modes are retained, and vortices and horizontally uniform vertical shears are excluded. Despite the simplifications, our simulations reproduce some key features of oceanic internal-wave spectra: accumulation of energy at near-inertial waves and realistic frequency and horizontal wavenumber dependencies. Furthermore, we provide evidence that formation of the energy spectra in the inertial subrange is dominated by scale-separated interactions with the near-inertial waves. These findings support oceanographers intuition that spectral energy density of internal waves is the result of predominantly wave-wave interactions.
Kraichnan seminal ideas on inverse cascades yielded new tools to study common phenomena in geophysical turbulent flows. In the atmosphere and the oceans, rotation and stratification result in a flow that can be approximated as two-dimensional at very large scales, but which requires considering three-dimensional effects to fully describe turbulent transport processes and non-linear phenomena. Motions can thus be classified into two classes: fast modes consisting of inertia-gravity waves, and slow quasi-geostrophic modes for which the Coriolis force and horizontal pressure gradients are close to balance. In this paper we review previous results on the strength of the inverse cascade in rotating and stratified flows, and then present new results on the effect of varying the strength of rotation and stratification (measured by the ratio $N/f$ of the Brunt-Vaisala frequency to the Coriolis frequency) on the amplitude of the waves and on the flow quasi-geostrophic behavior. We show that the inverse cascade is more efficient in the range of $N/f$ for which resonant triads do not exist, $1/2 le N/f le 2$. We then use the spatio-temporal spectrum, and characterization of the flow temporal and spatial scales, to show that in this range slow modes dominate the dynamics, while the strength of the waves (and their relevance in the flow dynamics) is weaker.
In the paper taking the assumption of the slowness of the change of the parameters of the vertically stratified medium in the horizontal direction and in time, the evolution of the non-harmonic wave packages of the internal gravity waves has been analyzed. The concrete form of the wave packages can be expressed through some model functions and is defined by the local behavior of the dispersive curves of the separate modes near to the corresponding special points. The solution of this problem is possible with the help of the modified variant of the special-time ray method offered by the authors (the method of geometrical optics), the basic difference of which consists that the asymptotic representation of the solution may be found in the form the series of the non-integer degrees of some small parameter. At that the exponent depends on the concrete form of representation of this package. The obvious kind of the representation is determined from the principle of the localness and the asymptotic behavior of the solution in the stationary and the horizontally-homogeneous case. The phases of the wave packages are determined from the corresponding equations of the eikonal, which can be solved numerically on the characteristics (rays). Amplitudes of the wave packages are determined from the laws of conservation of the some invariants along the characteristics (rays).
Numerical simulations are made for forced turbulence at a sequence of increasing values of Reynolds number, R, keeping fixed a strongly stable, volume-mean density stratification. At smaller values of R, the turbulent velocity is mainly horizontal, and the momentum balance is approximately cyclostrophic and hydrostatic. This is a regime dominated by so-called pancake vortices, with only a weak excitation of internal gravity waves and large values of the local Richardson number, Ri, everywhere. At higher values of R there are successive transitions to (a) overturning motions with local reversals in the density stratification and small or negative values of Ri; (b) growth of a horizontally uniform vertical shear flow component; and (c) growth of a large-scale vertical flow component. Throughout these transitions, pancake vortices continue to dominate the large-scale part of the turbulence, and the gravity wave component remains weak except at small scales.