We find the turbulent energy spectrum of weakly interacting 2D internal gravity waves using the full, non-hydrostatic dispersion relation. This spectrum is an exact solution of a regularized kinetic equation, from which the zero-frequency shear modes have been excised by a careful limiting process. This is a new method in wave kinetic theory. The turbulent spectrum agrees with the 2D oceanic Garrett--Munk spectrum for frequencies large compared to the Coriolis frequency and vertical scales small compared to the depth of the ocean. We show that this turbulent spectrum is the unique power law solution to the steady kinetic equation with a non-zero radial flux. Our solution provides an interesting insight into a turbulent energy cascade in an anisotropic system -- like isotropic turbulence it is self-similar in scale, but its angular part is peaked along the curve of vanishing frequency and is self-similar in frequency.
In this paper, we employ Lagrangian coherent structures (LCSs) theory for the three dimensional vortex eduction and investigate the effect of large-scale vortical structures on the turbulent/non-turbulent interface (TNTI) and entrainment of a gravity current. The gravity current is realized experimentally and different levels of stratification are examined. For flow measurements, we use a multivolume three-dimensional particle tracking velocimetry technique. To identify vortical LCSs (VLCSs), a fully automated 3D extraction algorithm for multiple flow structures based on the so-called Lagrangi…
We derive equations for the mean entropy and the mean internal energy in the low-Mach-number temperature stratified turbulence (i.e., for turbulent convection or stably stratified turbulence), and show that turbulent flux of entropy is given by ${\bf F}_s=\overlineρ \, \overline{{\bf u} s}$, where $\overlineρ$ is the mean fluid density, $s$ are fluctuations of entropy and overbars denote averaging over an ensemble of turbulent velocity field, ${\bf u}$. We demonstrate that the turbulent flux of entropy is different from the turbulent convective flux, ${\bf F}_c=\overline{T} \, \overlineρ \, \o…
We demonstrate via direct numerical simulations that a periodic, oscillating mean flow spontaneously develops from turbulently generated internal waves. We consider a minimal physical model where the fluid self-organizes in a convective layer adjacent to a stably stratified one. Internal waves are excited by turbulent convective motions, then nonlinearly interact to produce a mean flow reversing on timescales much longer than the waves' period. Our results demonstrate for the first time that the three-scale dynamics due to convection, waves, and mean flow is generic and hence can occur in many…
With the aim of assessing internal wave-driven mixing in the ocean, we develop a new technique for direct numerical simulations of stratified turbulence. Since the spatial scale of oceanic internal gravity waves is typically much larger than that of turbulence, fully incorporating both in a model would require a high computational cost, and is therefore out of our scope. Alternatively, we cut out a small domain periodically distorted by an unresolved large-scale internal wave and locally simulate the energy cascade to the smallest scales. In this model, even though the Froude number of the out…
A polymer in a turbulent flow undergoes the coil-stretch transition when the Weissenberg number, i.e. the product of the Lyapunov exponent of the flow and the relaxation time of the polymer, surpasses a critical value. The effect of internal friction on the transition is studied by means of Brownian dynamics simulations of the elastic dumbbell model in a homogeneous and isotropic, incompressible, turbulent flow and analytical calculations for a stochastic velocity gradient. The results are explained by adapting the large deviations theory of Balkovsky et al. [Phys. Rev. Lett., 2000, 84, 4765] …
This study starts with balances deduced by Baumert and Peters (2004, 2005) from results of stratified-shear experiments made in channels and wind tunnels by Itsweire (1984) and Rohr and Van Atta (1987), and of free-decay experiments in a resting stratified tank by Dickey and Mellor (1980). Using a modification of Canuto's (2002) ideas on turbulence and waves, these balances are merged with an (internal) gravity-wave energy balance presented for the open ocean by Gregg (1989), without mean-flow shear. The latter was augmented by a linear (viscous) friction term. Gregg's wave-energy source is in…
We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of convective--stably-stratified fluids. We demonstrate that in the limit of large Rayleigh number ($Ra\in [4\times 10^7,10^9]$) and large stratification (Brunt-Väisälä frequencies $f_N \gg f_c$, where $f_c$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region (Lecoanet \& Quataert 2013 MNRAS 430 (3) 2363-2376). Specific…
We present an experimental study of the mixing processes in a gravity current flowing on an inclined plane. The turbulent transport of momentum and density can be described in a very direct and compact form by a Prandtl mixing length model: the turbulent vertical fluxes of momentum and density are found to scale quadratically with the vertical mean gradients of velocity and density. The scaling coefficient, the square of the mixing length, is approximately constant over the mixing zone of the stratified shear layer. We show how, in different flow configurations, this length can be related to t…
Heavy particles suspended in a turbulent flow settle faster than in a still fluid. This effect stems from a preferential sampling of the regions where the fluid flows downward and is quantified here as a function of the level of turbulence, of particle inertia, and of the ratio between gravity and turbulent accelerations. By using analytical methods and detailed, state-of-the-art numerical simulations, settling is shown to induce an effective horizontal two-dimensional dynamics that increases clustering and reduce relative velocities between particles. These two competing effects can either in…
We study the joint probability distributions of separation, $R$, and radial component of the relative velocity, $V_{\rm R}$, of particles settling under gravity in a turbulent flow. We also obtain the moments of these distributions and analyze their anisotropy using spherical harmonics. We find that the qualitative nature of the joint distributions remains the same as no gravity case. Distributions of $V_{\rm R}$ for fixed values of $R$ show a power-law dependence on $V_{\rm R}$ for a range of $V_{\rm R}$, exponent of the power-law depends on the gravity. Effects of gravity are also manifested…
A problem of finding the linear theory satisfaction limits in propagation of the internal gravity waves is considered. It is evident that internal gravity waves excitation, propagation in actual practice is highly nonlinear phenomenon. However with some reasonable assumptions it is possible to linearize equations of internal waves generation and propagation. It is shown that in interesting for us wavelength range we can use linear approximation during study of internal gravity waves dynamics. Similarly it is easy to evaluate also influence of other corrections to the linear theory of internal …
The internal gravity waves far field exited by a non-local perturbation sources was considered. A separate wave mode asymptomatic presentation was constructed, describing the wave field key features depending on the source geometry.
A novel approach to identify internal interfacial layers, or IILs, in wall-bounded turbulent flows is proposed. Using a Fuzzy Cluster Method (FCM) on the streamwise velocity component, a unique and unambiguous grouping of the Uniform Momentum Zones is achieved, thus allowing the identification of the IILs. The approach overcomes some of the key limitations of the histogram-based IIL identification methods. The method is insensitive to the streamwise domain length, can be used on inhomogeneous grids, uses all the available flow field data, is trivially extended to three dimensions, and does not…
Spatial clustering and intermittency in the relative velocity of heavy particles of the same size settling in turbulent flows can be strongly affected by gravity. We present a model for the timescale of the fluid velocity gradient seen by particle pairs and propose an effective Kubo number based on this timescale to explain the mechanism of gravity-enhanced clustering. We explore the mechanisms of the gravity-induced reduction or enhancement of the intermittency in the particle radial relative velocity (RRV) at different Stokes numbers based on backward-in-time relative dispersion and preferen…
We study the dynamics and properties of a turbulent flame, formed in the presence of subsonic, high-speed, homogeneous, isotropic Kolmogorov-type turbulence in an unconfined system. Direct numerical simulations are performed with Athena-RFX, a massively parallel, fully compressible, high-order, dimensionally unsplit, reactive-flow code. A simplified reaction-diffusion model represents a stoichiometric H2-air mixture. The system being modeled represents turbulent combustion with the Damkohler number Da = 0.05 and with the turbulent velocity at the energy injection scale 30 times larger than the…
A key issue in stratified turbulence theory concerns the nature of the link between D(APE), the dissipation rate of available potential energy APE, and W_{r,turbulent}, the turbulent rate of change of background gravitational potential energy GPE_r, which are both controlled by molecular diffusion. For Boussinesq fluids with a linear equation of state, this link is simply W_{r,turbulent}=D(APE), widely interpreted as implying that GPE_r increases at the expense of APE, in contrast with the laminar case where GPE_r increases at the expense of internal energy (IE). This idea is revisited here by…
Convection over a wavy heated bottom wall in the air flow has been studied in experiments with the Rayleigh number $\sim 10^8$. It is shown that the mean temperature gradient in the flow core inside a large-scale circulation is directed upward, that corresponds to the stably stratified flow. In the experiments with a wavy heated bottom wall, we detect large-scale standing internal gravity waves excited in the regions with the stably stratified flow. The wavelength and the period of these waves are much larger than the turbulent spatial and time scales, respectively. In particular, the frequenc…
This work focuses on the mathematical modeling of wave dynamics in a stratified medium. Non-local absorbing boundary conditions are considered based on the two following assumptions: (i) a linear theory can be applied at large distances from perturbation sources; and (ii) there are no other sources of wave disturbance outside the mixing zone in the stratified medium. The boundary conditions considered in this paper allowed us to describe the diverging internal gravity waves generated by the mixing region in a stratified medium.
Mesoscale wave processes are a transport mechanism for the energy exchange between the troposphere and stratosphere, since the tropopause blocks such an exchange. It is believed that internal gravity waves (IGW) are the main wave process in the atmosphere. However, the explanation of the process of IGW occurrence cannot be considered sufficiently substantiated, because the reasons for the appearance of the selected air parcel, which is affected by the initial disturbance, are not yet clear. The data of the performed experiments on the detection of wave processes shows the presence of noticeabl…
We report the experimental observation of a robust horizontal mean flow induced by internal gravity waves. A wave beam is forced at the lateral boundary of a tank filled with a linearly stratified fluid initially at rest. After a transient regime, a strong jet appears in the wave beam, with horizontal recirculations outside the wave beam. We present a simple physical mechanism predicting the growth rate of the mean flow and its initial spatial structure. We find good agreement with experimental results.
In this paper, we consider a fundamental problem of describing the dynamics of internal gravity waves in the stratified ocean with shear flows. We develop an asymptotic representation of the wave fields in terms of the Green's functions. We explore the far field of the internal gravity waves generated by disturbing sources, and propose asymptotic algorithms for calculating its dynamics.
In this paper we consider fundamental processes of the disturbance and propagation of internal gravity waves in the ocean modeled as a vertically stratified, horizontally non-uniform, and non-stationary medium. We develop asymptotic methods for describing the wave dynamics by generalizing the spatiotemporal ray-tracing method (a geometrical optics method). We present analytical and numerical algorithms for calculating the internal gravity wave fields using actual ocean parameters such as physical characteristics of the sea water, topography of its floor, etc. We demonstrate that our mathematic…
We advance our prior energy- and flux-budget turbulence closure model (Zilitinkevich et al., 2007, 2008) for the stably stratified atmospheric flows and extend it accounting for additional vertical flux of momentum and additional productions of turbulent kinetic energy, turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). Main effects of IGW are following: the maximal value of the flux Richardson number (universal constant 0.2-0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratificati…
We present results of three-dimensional Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) of turbulent gravity currents with a Discontinuous Galerkin (DG) Finite Elements method. In particular, we consider the three-dimensional lock-exchange test case as a typical benchmark for gravity currents. Since, to the best of our knowledge, non-Boussinesq three-dimensional reference DNS are not available in the literature for this test case, we first perform a DNS experiment. The three-dimensional DNS allows to correctly capture the loss of coherence of the three-dimensional turbulent…
In this paper, we consider fundamental problems of the dynamics of internal gravity waves. We present analytical and numerical algorithms for calculating the wave fields for a set of values of the parameters, as observed in the ocean. We show that our mathematical models can describe the wave dynamics of the Arctic Basin, taking into account the actual physical characteristics of sea water, topography of its floor, etc. The numerical and analytical results show that the internal gravity waves have a significant effect on underwater sea objects in the Arctic Basin.
The condition of internal gravity waves (IGW) parametric excitation in the rotating fluid layer heated from above, with the layer vibration along the vertical axis or with periodic modulation in time of the vertical temperature distribution, is obtained. We show the dual role of the molecular dissipative effects that may lead not only to the wave oscillations damping, but also to emergence of hydrodynamic dissipative instability (DI) in some frequency band of IGW. This DI also may take place for the localized in horizontal plane tornado-like disturbances, horizontal scale of which does not exc…
Three-dimensional convection driven by internal heat sources and sinks (CISS) leads to experimental and numerical scaling-laws compatible with a mixing-length - or `ultimate' - scaling regime $Nu \sim \sqrt{Ra}$. However, asymptotic analytic solutions and idealized 2D simulations have shown that laminar flow solutions can transport heat even more efficiently, with $Nu \sim Ra$. The turbulent nature of the flow thus has a profound impact on its transport properties. In the present contribution we give this statement a precise mathematical sense. We show that the Nusselt number maximized over al…
Gravity waves (or their signatures) are detected in stars thanks to helio- and asteroseismology and they may play an important role in the evolution of stellar angular momentum. Moreover, the observational study of the CoRoT target HD51452 by Neiner and collaborators demonstrated the potential strong impact of rotation on the stochastic excitation of gravito-inertial waves in stellar interiors. Our goal is to explore the action of rotation on the stochastic excitation of gravity and gravito-inertial waves in stars. The dynamics of gravito-inertial waves in stellar interiors, both in radiation …
We calculate the flux of internal gravity waves (IGWs) generated by turbulent convection in stars. We solve for the IGW eigenfunctions analytically near the radiative-convective interface in a local, Boussinesq, and cartesian domain. We consider both discontinuous and smooth transitions between the radiative and convective regions and derive Green's functions to solve for the IGWs in the radiative region. We find that if the radiative-convective transition is smooth, the IGW flux depends on the exact form of the buoyancy frequency near the interface. IGW excitation is most efficient for very s…
Hydrodynamic waves propagate through stellar interiors, transporting energy and angular momentum. They can also advect fluid elements to produce mixing, but this effect has not been quantified from first principles. We derive the leading order non-linear wave mixing due to internal gravity waves in a thermally and compositionally-stratified fluid. We find that this scales as the fourth power of wave velocity, that it is suppressed by compositional stratification, and that it depends on the thermal and compositional diffusivities.
The problem of reconstructing non-harmonic internal gravity wave packets generated by a source moving in a stratified ocean is considered. The uniform asymptotic form of the internal gravity waves field generated by a source moving above the smoothly varying floor is constructed. The solution is proposed in terms of wave modes, propagating independently at the adiabatic approximation, and described as a non-integral degree series of a small parameter characterizing the stratified medium. A specific form of the wave packets, which can be parameterized in terms of model functions (Airy functions…
To understand how internal flow structures manifest themselves in the global heat transfer, we study the correlation between different flow modes and the instantaneous Nusselt number ($Nu$) in a two-dimensional square Rayleigh-Bénard convection cell. High-resolution and long-time direct numerical simulations are carried out for Rayleigh numbers between $10^{7}$ and $10^{9}$ and a Prandtl number of 5.3. The investigated Nusselt numbers include the volume-averaged $Nu_{\text{vol}}$, the wall-averaged $Nu_{\text{wall}}$, the kinetic energy dissipation based $Nu_{\text{kinetic}}$, and the thermal …
Internal gravity waves play a primary role in geophysical fluids: they contribute significantly to mixing in the ocean and they redistribute energy and momentum in the middle atmosphere. Until recently, most studies were focused on plane wave solutions. However, these solutions are not a satisfactory description of most geophysical manifestations of internal gravity waves, and it is now recognized that internal wave beams with a confined profile are ubiquitous in the geophysical context. We will discuss the reason for the ubiquity of wave beams in stratified fluids, related to the fact that th…
Here we show that there exist internal gravity waves that are inherently unstable, that is, they cannot exist in nature for a long time. The instability mechanism is a one-way (irreversible) harmonic-generation resonance that permanently transfers the energy of an internal wave to its higher harmonics. We show that, in fact, there are countably infinite number of such unstable waves. For the harmonic-generation resonance to take place, nonlinear terms in the free surface boundary condition play a pivotal role, and the instability does not obtain for a linearly-stratified fluid if a simplified …
We explore the dynamics of inclined temporal gravity currents using direct numerical simulation, and find that the current creates an environment in which the flux Richardson number $Ri_f$, gradient Richardson number $Ri_g$, and turbulent flux coefficient $Γ$ are constant across a large portion of the depth. Changing the slope angle $α$ modifies these mixing parameters, and the flow approaches a maximum Richardson number $Ri_\textrm{max}\approx 0.15$ as $α\rightarrow 0$ at which the entrainment coefficient $E \rightarrow 0$. The turbulent Prandtl number remains $O(1)$ for all slope angles, dem…
Considered in this paper is a bi-directional model for the propagation of interfacial capillary-gravity waves in a two-layer system of fluids with rigid lid condition for the upper layer and lower layer with a much larger or infinite depth. The system is derived from a reformulation of the Euler equations for internal waves with nonnegligible surface tension effects in the interface and the corresponding asymptotic model under the Benjamin-Ono regime. Another unidirectional model, so-called regularized Benjamin equation, generalizing the Benjamin equation, is also introduced. Well-posedness of…
Phytoplankton patchiness, namely the heterogeneous distribution of microalgae over multiple spatial scales, dramatically impacts marine ecology. A spectacular example of such heterogeneity occurs in thin phytoplankton layers (TPLs), where large numbers of photosynthetic microorganisms are found within a small depth interval. Some species of motile phytoplankton can form TPLs by gyrotactic trapping due to the interplay of their particular swimming style (directed motion biased against gravity) and the transport by a flow with shear along the direction of gravity. Here we consider gyrotactic swi…
