An initial zonally symmetric quasigeostrophic potential-vorticity (PV) distribution q_i(y) is subjected to complete or partial mixing within some finite zone |y| < L, where y is latitude. The change in M, the total absolute angular momentum, between the initial and any later time is considered. For standard quasigeostrophic shallow-water beta-channel dynamics it is proved that, for any q_i(y) such that dq_i/dy > 0 throughout |y| < L, the change in M is always negative. This theorem holds even when "mixing" is understood in the most general possible sense. Arbitrary stirring or advective rearrangement is included, combined to an arbitrary extent with spatially inhomogeneous diffusion. The theorem holds whether or not the PV distribution is zonally symmetric at the later time. The same theorem governs Boussinesq potential-energy changes due to buoyancy mixing in the vertical. For the standard quasigeostrophic beta-channel dynamics to be valid the Rossby deformation length L_D >> εL where εis the Rossby number; when L_D = \infty the theorem applies not only to the beta-channel, but also to a single barotropic layer on the full sphere, as considered in the recent work of Dunkerton and Scott on "PV staircases". It follows that the M-conserving PV reconfigurations studied by those authors must involve processes describable as PV unmixing, or anti-diffusion, in the sense of time-reversed diffusion. Ordinary jet self-sharpening and jet-core acceleration do not, by contrast, require unmixing, as is shown here by detailed analysis. Mixing in the jet flanks suffices. The theorem extends to multiple layers and continuous stratification. A corollary is a new nonlinear stability theorem for shear flows.
The velocity-vorticity correlation integral (Chkhetiani invariant) is an invariant of a viscous Karman-Howarth equation. In the tree-dimensional space this invariant can naturally determine a turbulent diffusivity (viscosity). It is shown, using results of direct numerical simulations, that distributed chaos dominated by this integral can provide an adequate description of the turbulent Rayleigh-Bénard convection, stably stratified turbulence and Rayleigh-Taylor mixing at Prandtl number $Pr \sim 1$.
The theoretical analysis of the energetics of mechanically-stirred horizontal convection for a Boussinesq fluid yields the formula: G(APE) = γ_{mixing} G(KE) + (1+γ_{mixing}) W_{r,laminar} where G(APE) and G(KE) are the work rate done by the buoyancy and mechanical forcing respectively, γ_{mixing} is the mixing efficiency, and W_{r,laminar} is the background rate of increase in gravitational potential energy due to molecular diffusion. The formula shows that mechanical stirring can easily induce a very strong buoyancy-driven overturning cell (meaning a large G(APE)) even for a relatively low m…
In rotating Rayleigh-Bénard convection, columnar vortices advect horizontally in a stochastic manner. When the centrifugal buoyancy is present the vortices exhibit radial motions that can be explained through a Langevin-type stochastic model. Surprisingly, anomalous outward motion of cyclones is observed in a centrifugation-dominant flow regime, which is contrary to the well-known centrifugal effect. We interpret this phenomenon as a symmetry-breaking of both the population and vorticity magnitude of the vortices brought about by the centrifugal buoyancy. Consequently, the cyclones submit to t…
We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size $\varepsilon$. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t \approx \varepsilon^{-2}$. Notice that the density behav…
Turbulent mixing layers in nature are often characterized by the presence of a mean shear and an unstable buoyancy gradient between two streams of different velocity. Depending on the relative strength of shear versus buoyancy, either the former or the latter may dominate the turbulence and mixing between the two streams. In this paper, we present a phenomenological theory that leads to the identification of two distinct turbulent regimes: an early regime, dominated by the mean shear, and a later regime dominated by the buoyancy. The main theoretical result consists of the identification of a …
Bridging the local and the global: a physically constrained buoyancy--drag model for unified prediction of Rayleigh--Taylor and Richtmyer--Meshkov mixing widths across density ratios
College of Mechanical and Electrical Engineering, Beijing University of Chemical Technology（北京化工大学机电工程学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；National Key Laboratory of Computational Physics（计算物理学国家重点实验室）；Center for Applied Physics and Technology, HEDPS, College of Engineering, Peking University（北京大学工程与应用物理与技术中心）
We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_B^2$ imposed in the third, vertical direction in a fluid with kinematic viscosity $ν$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this sy…
This study derives a complete set of equatorially confined wave solutions from an anelastic equation set with the complete Coriolis terms, which include both the vertical and meridional planetary vorticity. The propagation mechanism can change with the effective static stability. When the effective static stability reduces to neutral, buoyancy ceases, but the role of buoyancy as an eastward-propagation mechanism is replaced by the compressional beta effect, i.e., vertical density-weighted advection of the meridional planetary vorticity. For example, the Kelvin mode becomes a compressional Ross…
We consider the time-dependence of a hierarchy of scaled $L^{2m}$-norms $D_{m,ω}$ and $D_{m,θ}$ of the vorticity $\boldsymbol ω = \boldsymbol{\nabla} \times {\mathbf u}$ and the density gradient $\boldsymbol{\nabla} θ$, where $θ=\log (ρ^*/ρ^*_0)$, in a buoyancy-driven turbulent flow as simulated by \cite{LR2007}. $ρ^*({\mathbf x},\,t) $ is the composition density of a mixture of two incompressible miscible fluids with fluid densities $ρ^*_2 > ρ^*_1$ and $ρ^*_{0}$ is a reference normalisation density. Using data from the publicly available Johns Hopkins Turbulence Database we present evidence t…
Rayleigh--Taylor fluid turbulence through a bed of rigid, finite-size, spheres is investigated by means of high-resolution Direct Numerical Simulations (DNS), fully coupling the fluid and the solid phase via a state-of-the art Immersed Boundary Method (IBM). The porous character of the medium reveals a totally different physics for the mixing process when compared to the well-known phenomenology of classical RT mixing. For sufficiently small porosity, the growth-rate of the mixing layer is linear in time (instead of quadratical) and the velocity fluctuations tend to saturate to a constant valu…
We report quantitative evidence of mixing-layer elastic instability in a viscoelastic fluid flow between two widely spaced obstacles hindering a channel flow at $Re\ll1$ and $Wi\gg1$. Two mixing layers with nonuniform shear velocity profiles are formed in the region between the obstacles. The mixing-layer instability arises in the vicinity of an inflection point on the shear velocity profile with a steep variation in the elastic stress. The instability results in an intermittent appearance of small vortices in the mixing layers and an amplification of spatio-temporal averaged vorticity in the …
To investigate the intrinsic mechanism for mixing enhancement by variable density behaviour, a canonical variable density (VD) mixing extracted from a supersonic streamwise vortex protocol, shock bubble interaction (SBI), is numerically studied and compared with a counterpart of passive scalar (PS) mixing. It is meaningful to observe that the maximum concentration decays much faster in VD SBI than in PS SBI regardless of the shock Mach number (Ma = 1.22 ~ 4). The quasi-Lamb-Oseen type velocity distribution in the PS SBI is found by analyzing the azimuthal velocity that stretches the bubble. Me…
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…
The discrete baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the…
Turbulent mixing exerts a significant influence on many physical processes in the ocean. In a stably stratified Boussinesq fluid, this irreversible mixing describes the conversion of available potential energy (APE) to background potential energy (BPE). In some settings the APE framework is difficult to apply and approximate measures are used to estimate irreversible mixing. For example, numerical simulations of stratified turbulence often use triply periodic domains to increase computational efficiency. In this setup however, BPE is not uniquely defined and the method of Winters et al. (1995,…
This paper is the second part of a two-fold study of mixing, i.e. the formation of layers and upwelling of buoyancy, in axially stratified Taylor--Couette flow, with fixed outer cylinder. In a first paper, we showed that the dynamics of the flow was dominated by coherent structures made of a superposition of nonlinear waves. (Mixed)-ribbons and (mixed)-cross-spirals are generated by interactions between a pair of linearly unstable helical modes of opposite `handedness', and appear to be responsible for the formation of well-mixed layers and sharp density interfaces. In this paper, we show that…
The presence of stratified layer in atmosphere and ocean leads to buoyant vertical motions, commonly referred to as plumes. It is important to study the mixing dynamics of a plume at a local scale in order to model their evolution and growth. Such a characterization requires measuring the velocity and density of the mixing fluids simultaneously. Here, we present the results of a buoyant plume propagating in a linearly stratified medium with a density difference of 0.5%, thus yielding a buoyancy frequency of N=0.15 s^{-1}. To understand the plume behaviour, statistics such as centerline and axi…
A new mixing layer can be generated if the rotation of either of the two cylinders in a Taylor--Couette apparatus varies discontinuously along the symmetry axis. The mixing zone between the two resulting co-current streams gives rise to radial vorticity in addition to the primary axial vorticity. An analytic solution for the azimuthal velocity has been derived from which we show that the width of the mixing zone varies only with radial position.
We investigate the non-linear equilibration of a two-layer quasi-geostrophic flow in a channel forced by an imposed unstable zonal mean flow, paying particular attention to the role of bottom friction. In the limit of low bottom friction, classical theory of geostrophic turbulence predicts an inverse cascade of kinetic energy in the horizontal with condensation at the domain scale and barotropization on the vertical. By contrast, in the limit of large bottom friction, the flow is dominated by ribbons of high kinetic energy in the upper layer. These ribbons correspond to meandering jets separat…
We perform the analytic study of the the buoyancy-drag equation with a time-dependent acceleration $γ(t)$ by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of $γ(t)$ define a time-dependent Hamiltonian from which the buoyancy-drag equation can be derived. We then determine the Lie point symmetries of the buoyancy-drag equation, which only exist for values of $γ(t)$ including the previous ones, plus additional classes of accelerations for which the equation is reducible to an Abel equation. This allows us to exh…
We introduce a dynamical description based on a probability density $ϕ(σ,x,y,t)$ of the vorticity $σ$ in two-dimensional viscous flows such that the average vorticity evolves according to the Navier-Stokes equations. A time-dependent mixing index is defined and the class of probability densities that maximizes this index is studied. The time dependence of the Lagrange multipliers can be chosen in such a way that the masses $m(σ,t):=\intdxdy ϕ(σ,x,y,t)$ associated with each vorticity value $σ$ are conserved. When the masses $m(σ,t)$ are conserved then 1) the mixing index satisfies an H-theorem …
A theory of potential vorticity (PV) mixing in a disordered (tangled) magnetic field is presented. The analysis is in the context of $β$-plane MHD, with a special focus on the physics of momentum transport in the stably stratified, quasi-2D solar tachocline. A physical picture of mean PV evolution by vorticity advection and tilting of magnetic fields is proposed. In the case of weak-field perturbations, quasi-linear theory predicts that the Reynolds and magnetic stresses balance as turbulence Alfvénizes for a larger mean magnetic field. Jet formation is explored quantitatively in the mean fiel…
Experimental observations of turbulent buoyant plumes, produced by a constant source of buoyancy, have been described with great success using a horizontally averaged model for the conservation of mass, momentum and buoyancy flux. However, experimental observations of plumes with time-dependent buoyancy fluxes has proved more challenging for quantitative models. At each level in the plume, the horizontal variation in velocity leads to an along-axis shear and hence dispersive transport relative to the mean. With a time-dependent source, axial dispersion of the dynamic properties of the plume ha…
An intrinsic feature of turbulent flows is an enhanced rate of mixing and kinetic energy dissipation due to the rapid generation of small-scale motions from large-scale excitation. The transfer of kinetic energy from large to small scales is commonly attributed to the stretching of vorticity by the strain-rate, but strain self-amplification also plays a role. Previous treatments of this connection are phenomenological or inexact, or cannot distinguish the contribution of vorticity stretching from that of strain self-amplification. In this paper, an exact relationship is derived which quantitat…
There exist two central measures of turbulent mixing in turbulent stratified fluids, both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy (APE); 2) the turbulent rate of change Wr,turbulent of background potential energy GPEr. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPEr, owing to D(APE)=Wr,turbulent holding exactly for a Boussinesq fluid with a linear equation of state. It was recently pointed out, however, that this equality no longer holds for a thermally-s…
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 monochro-matic 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…
A series of laboratory experiments on energy conserving gravity currents in a lock-exchange facility are conducted for a range of Reynolds numbers, $Re= \frac{U_Fh}ν =$ 485-12270. The velocity and density fields are captured simultaneously using a PIV-PLIF system. A moving average method is employed to compute the mean field and a host of turbulence statistics, namely, turbulent kinetic energy ($K$), shear production ($P$), buoyancy flux ($B$), and energy dissipation ($ε$) during the slumping phase of the current. The subsequent findings are used to ascertain the quantitative values of mixing …
In the context of two-dimensional (2D) turbulence, we apply the maximum entropy production principle (MEPP) by enforcing a local conservation of energy. This leads to an equation for the vorticity distribution that conserves all the Casimirs, the energy, and that increases monotonically the mixing entropy ($H$-theorem). Furthermore, the equation for the coarse-grained vorticity dissipates monotonically all the generalized enstrophies. These equations may provide a parametrization of 2D turbulence. They do not generally relax towards the maximum entropy state. The vorticity current vanishes for…
Small density differences, large Richardson numbers: buoyancy decides where gypsum deposits in a centifluidic confluence cell
Norwegian University of Science and Technology (NTNU)（挪威科技大学）；Technical University of Denmark（丹麦技术大学）；Universidad Autónoma de Madrid (UAM)（马德里自治大学）；Eindhoven University of Technology（埃因霍温理工大学）
It has hitherto been widely considered that a mixing layer is unstable at all Reynolds numbers. However this is untenable from energy considerations, which demand that there must exist a non-zero Reynolds number below which disturbances cannot extract energy from the mean flow. It is shown here that a linear stability analysis of similarity solutions of the plane mixing layer, including the effects of flow non-parallelism, using the minimal composite theory and the properties of adjoints following Govindarajan & Narasimha (2005), resolves the issue by yielding non-zero critical Reynolds number…
Statistical equilibria of two-dimensional turbulent flows for generic initial vorticity fields on a sphere, calculated on the basis of the original Miller-Robert-Sommeria theory
Research Institute for Applied Mechanics, Kyushu University（九州大学应用力学研究所）；Graduate School of Science, Kyoto University（京都大学理学研究科）
In this Part 2 we study further experimental properties of two-layer exchange flows in a stratified inclined duct (SID), which are turbulent, strongly-stratified, shear-driven, and continuously-forced. We analyse the same state-of-the-art data sets using the same core shear layer methodology as in Part 1, but we focus here on turbulent energetics and mixing statistics. The detailed analysis of kinetic and scalar energy budgets reveals the specificity and scalings of SID turbulence, while energy spectra provide insight into the current strengths and limitations of our experimental data. The ani…
We present a simulation for the interactions of shockwaves with light spherical density inhomogeneities. Euler equations for two-phase compressible flows are solved in a 3D uniform resolution finite volume based solver using 5th order WENO reconstructions of the primitive quantities, HLL-type numerical fluxes and 3rd order TVD time stepping scheme. In this study, a normal Mach 3 shockwave in air is directed at a helium bubble with an interface Atwood number of -0.76. We employ 4 billion cells on a supercomputing cluster and demonstrate the development of this flow until relatively late times. …
A two-field model of potential vorticity (PV) staircase structure and dynamics relevant to both beta-plane and drift-wave plasma turbulence is studied numerically and analytically. The model evolves averaged PV whose flux is both driven by and regulates, a potential enstrophy field, $\varepsilon$. The model employs a closure using a mixing length model. Its link to bistability, vital to staircase generation, is analyzed and verified by integrating the equations numerically. Long-time staircase evolution consistently manifests a pattern of meta-stable quasi-periodic configurations, lasting for …
In Hamiltonian systems characterized by a degenerate Poisson algebra, we show how to construct a local energy-like quantity that can be used to study diabatic effects on the evolution of the Available Energy of the system, the latter concept formalizing the original idea of Margules'. We calculate the local diagnostic energy for geophysically relevant flows. For the particular case of stratified Boussinesq flows, we show that under moderately general conditions, in inertial frames where the initial distribution of potential vorticity is even around the origin, our framework recovers the Availa…
