We present numerical investigations into three principal properties of the small-scale dynamo in stably stratified turbulence: the onset criterion, the growth rate, and the nature of the magnetic field anisotropy in the kinematic regime. The results suggest that all three dynamo properties are controlled by the scale separation between the Ozmidov scale and the viscous or resistive scale. In addition to the critical magnetic Reynolds number, this allows for the definition of critical buoyancy and magnetic buoyancy Reynolds numbers for stratified small-scale dynamo onset in the high and low magnetic Prandtl number regimes, respectively. The presence of a small-scale dynamo in stellar radiative zones could affect dynamics through resulting Maxwell stresses and/or influence large-scale dynamo mechanisms in regions of differential rotation. Taking the solar radiative zone as a representative example and applying the onset criterion, we find that the stratification is strong enough to make the small-scale dynamo marginally active in the stably stratified turbulence of the solar tachocline.
Small-scale dynamo action is often held responsible for the generation of quiet-Sun magnetic fields. We aim to determine the excitation conditions and saturation level of small-scale dynamos in non-rotating turbulent convection at low magnetic Prandtl numbers. We use high resolution direct numerical simulations of weakly stratified turbulent convection. We find that the critical magnetic Reynolds number for dynamo excitation increases as the magnetic Prandtl number is decreased, which might suggest that small-scale dynamo action is not automatically evident in bodies with small magnetic Prandt…
Stably stratified layers are present in stellar interiors (radiative zones) as well as planetary interiors - recent observations and theoretical studies of the Earth's magnetic field seem to indicate the presence of a thin, stably stratified layer at the top of the liquid outer core. We present direct numerical simulations of this region, which is modelled as an axisymmetric spherical Couette flow for a stably stratified fluid embedded in a dipolar magnetic field. For strong magnetic fields, a super-rotating shear layer, rotating nearly 30% faster than the imposed rotation rate difference betw…
We report the first successful simulation of spontaneous formation of surface magnetic structures from a large-scale dynamo by strongly-stratified thermal convection in Cartesian geometry. The large-scale dynamo observed in our strongly-stratified model has physical properties similar to those in earlier weakly-stratified convective dynamo simulations, indicating that the $α^2$-type mechanism is responsible for it. Additionally to the large-scale dynamo, we find that large-scale structures of the vertical magnetic field are spontaneously formed in the convection zone surface only for the case …
We use a simplified model to study wave reflection and transmission at interface of convective region and stably stratified region (e.g. radiative zone in star or stratification layer in gaseous planet). Inertial wave in convective region and gravito-inertial wave in stably stratified region are considered. We begin with polar area and then extend to any latitude. Six cases are discussed (see Table 1), and in Case 2 both waves co-exist in both regions. Four configurations are further discussed for Case 2. The angles and energy ratios of wave reflection and transmission are calculated. It is fo…
It is necessary to introduce an external forcing to induce turbulence in a stably stratified fluid. The Heisenberg eddy viscosity technique should in this case suffice to calculate a space-time averaged quantity like the global anisotropy parameter as a function of the Richardson number. We find analytically that the anisotropy increases linearly with the Richardson number, with a small quadratic correction. A numerical simulation of the complete equations shows the linear behaviour.
We use a database of direct numerical simulations to evaluate parametrizations for energy dissipation rate in stably stratified flows. We show that shear-based formulations are more appropriate for stable boundary layers than commonly used buoyancy-based formulations. As part of the derivations, we explore several length scales of turbulence and investigate their dependence on local stability.
The topic of turbulent transport in a stellar radiative zone is vast and poorly understood. Many physical processes can potentially drive turbulence in stellar radiative zone but the limited observational constraints and the uncertainties in modelling turbulent transport make it difficult to identify the most relevant one. Here, we focus on the effect of stable stratification on the radial turbulent transport of chemicals and more particularly on the case where the turbulence is driven by a radial shear. Results of numerical simulations designed to test phenomenological models of turbulent tra…
In this paper, the properties of stably stratified turbulence (SST) and turbulent thermal convection are contrasted. A key difference between these flows is the sign of the kinetic energy feed by buoyancy, $\mathcal{F}_B$. For SST, $\mathcal{F}_B < 0$ due to its stable nature; consequently, the kinetic energy flux $Π_u(k)$ decreases with wavenumber $k$ that leads to a steep kinetic energy spectrum, $E_u(k) \sim k^{-11/5}$. Turbulent convection is unstable, hence $\mathcal{F}_B > 0$ that leads to an increase of $Π_u(k)$ with $k$; this increase however is marginal due to relatively weak buoyancy…
We present models for single-particle dispersion in vertical and horizontal directions of stably stratified flows. The model in the vertical direction is based on the observed Lagrangian spectrum of the vertical velocity, while the model in the horizontal direction is a combination of a continuous-time eddy-constrained random walk process with a contribution to transport from horizontal winds. Transport at times larger than the Lagrangian turnover time is not universal and dependent on these winds. The models yield results in good agreement with direct numerical simulations of stratified turbu…
Linear stability of inviscid, parallel, and stably stratified shear flow is studied under the assumption of smooth strictly monotonic profiles of shear flow and density, so that the local Richardson number is positive everywhere. The marginally unstable modes are systematically found by solving a one-parameter family of regular Sturm-Liouville problems, which can determine the stability boundaries more efficiently than solving the Taylor-Goldstein equation directly. By arguing for the non-existence of a marginally unstable mode, we derive new sufficient conditions for stability, which generali…
The universal velocity log law first proposed by von Kármán in the near-wall region of turbulent shear flows is one of the cornerstones of turbulence theory. When buoyancy effects are important, the universal velocity log law is typically believed to break down according to Monin-Obukhov similarity theory (MOST), which has been used in almost all global weather and climate models to describe the dependence of the mean velocity profiles on buoyancy in the atmospheric boundary layer. In contrast to MOST, we propose new logarithmic profiles of near-wall mean velocity in the stably stratified atmo…
The dynamics of stably stratified stellar radiative zones is of considerable interest due to the availability of increasingly detailed observations of Solar and stellar interiors. This article reports the first non-axisymmetric and time-dependent simulations of flows of anelastic fluids driven by baroclinic torques in stably stratified rotating spherical shells -- a system serving as an elemental model of a stellar radiative zone. With increasing baroclinicity a sequence of bifurcations from simpler to more complex flows is found in which some of the available symmetries of the problem are bro…
We study the effect of different forcing functions and of the local gradient Richardson number Rig on the vertical mixing of Lagrangian tracers in stably stratified turbulence under the Boussinesq approximation, and present a wave and continuous-time random walk model for single- and two- particle vertical dispersion. The model consists of a random superposition of linear waves with their amplitude based on the observed Lagrangian spectrum of vertical velocity, and a random walk process to capture overturning that depends on the statistics of Rig among other Eulerian quantities. The model is i…
In this study, we explore several integral and outer length scales of turbulence which can be formulated by using the dissipation of temperature fluctuations ($χ$) and other relevant variables. Our analyses directly lead to simple yet non-trivial parameterizations for both $χ$ and the structure parameter of temperature ($C_T^2$). For our purposes, we make use of high-fidelity data from direct numerical simulations of stratified channel flows.
We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In the case of the Couette flow $U(y)=y$, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in $L^2$ for the vorticity. For the previously…
Nonlinear interaction between dynamo-generated magnetic fields, mean flows and internal gravity waves in stellar stably-stratified layers: From 3D to 1D
Laboratoire de Physique de l’Ecole normale superieure ENS Universite PSL CNRS Sorbonne Universite Universite de Paris（巴黎高等师范学院物理实验室）；Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) Northwestern University（西北大学天体物理学跨学科探索与研究中心）；Laboratoire d’Instrumentation et de Recherche en Astrophysique (LIRA) Observatoire de Paris Université PSL Sorbonne Université Université Paris Cité CY Cergy Paris Université CNRS（巴黎天文台天体物理仪器与研究实验室）；Université Paris-Saclay CNRS Institut d’astrophysique spatiale（巴黎萨克雷大学空间天体物理研究所）；Departament d’Astronomia i Astrofisica Universitat de Valencia（瓦伦西亚大学天文学与天体物理学系）；Institut Universitaire de France (IUF)（法国高等研究院）
Magnetic reversals in a geodynamo model with a stably-stratified layer
Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，ENS，巴黎文理研究大学，法国国家科学研究中心，索邦大学，巴黎西岱大学）；Institute Universitaire de France (IUF)（法国大学研究所）
According to the celebrated Bolgiano--Obukhov \citep{Bolgiano_1959,Obukhov_1959} phenomenology for moderately stably stratified turbulence, the energy spectrum in the inertial range shows a dual scaling; the kinetic energy follows (i) $\sim k^{-11/5}$ for $k < k_B$, and (ii) $\sim k^{-5/3}$ for $k > k_B$, where $k_B$ is Bolgiano wavenumber. The $k^{-5/3}$ scaling akin to passive scalar turbulence is a direct consequence of the assumption that buoyancy is insignificant for $k>k_B$. We revisit this assumption, and using constancy of kinetic and potential energy fluxes and simple theoretical anal…
Inertial particles in stably stratified flows play a fundamental role in geophysics, from the dynamics of nutrients in the ocean to the dispersion of pollutants in the atmosphere. We consider the Maxey-Riley equation for small neutrally buoyant inertial particles in the Boussinesq approximation, and discuss its limits of validity. We show that particles behave as forced damped oscillators, with different regimes depending on the particles Stokes number and the fluid Brunt-Väisälä frequency. Using direct numerical simulations we study the particles dynamics and we show that small neutrally buoy…
Observations of low-mass stars reveal a variety of magnetic field topologies ranging from large-scale, axial dipoles to more complex magnetic fields. At the same time, three-dimensional spherical simulations of convectively driven dynamos reproduce a similar diversity, which is commonly obtained either with Boussinesq models or with more realistic models based on the anelastic approximation, which take into account the variation of the density with depth throughout the convection zone. Nevertheless, a conclusion from different anelastic studies is that dipolar solutions seem more difficult to …
Marangoni instabilities can emerge when a liquid interface is subjected to a concentration or temperature gradient. It is generally believed that for these instabilities bulk effects like buoyancy are negligible as compared to interfacial forces, especially on small scales. Consequently, the effect of a stable stratification on the Marangoni instability has hitherto been ignored. Here we report, for an immiscible drop immersed in a stably stratified ethanol-water mixture, a new type of oscillatory solutal Marangoni instability which is triggered once the stratification has reached a critical v…
On dissipation time scales of the basic second-order moments: the effect on the Energy and Flux-Budget (EFB) turbulence closure for stably stratified turbulence
The evolution of a localized vortex in stably stratified flow, within the Boussinesq approximation, is analyzed using the fluid impulse concept. The set of equations describing the temporal development of the fluid impulse has an integro-differential character where the terms representing the effect of stratification appear as convolution integral of the component of the fluid impulse and time-depended 'memory' functions. These functions are calculated for the case where the external parallel shear flow varies only in the direction gravitational force and is subjected to localized two- and thr…
We extend the scaling relations of stably stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $PrRb<1$, which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of t…
We observe the emergence of strong vertical drafts in direct numerical simulations of the Boussinesq equations in a range of parameters of geophysical interest. These structures, which appear intermittently in space and time, generate turbulence and enhance kinetic and potential energy dissipation, providing a possible explanation for the observed variability of the local energy dissipation in the bulk of oceanic flows, and the modulation of its probability distribution function. We show how, due to the extreme drafts, in runs with Froude numbers observable in geophysical scenarios, roughly 10…
We study a new type of large-scale instability, which arises in obliquely rotating stratified electroconductive fluid with an external uniform magnetic field and a small-scale external force having zero helicity. This force gives rise to small-scale oscillations of the velocity with a small Reynolds number. Using the method of multi-scale asymptotic expansions there are obtained nonlinear equations for vortex and magnetic perturbations in the third order in Reynolds number. Studied is the linear stage of magneto-vortex dynamo caused by instabilities of $α$-effect type. Stationary solutions for…
Global intermittency is observed in the stably stratified Atmospheric Boundary Layer (ABL) and corresponds to having large nonturbulent flow regions to develop in an otherwise turbulent flow. In this paper, the differences between continuous and intermittent turbulence are quantified with the help of coherent structures. Eight classes of coherent structures are identified from literature, most of which are indicated by scalar criteria derived from velocity fields. A method is developed to geometrically classify structures into three categories: blob-like, tube-like or sheet-like. An alternate …
We examine how known unstable equilibria of the Navier-Stokes equations in plane Couette flow adapt to the presence of an imposed stable density difference between the two boundaries for varying values of the Prandtl number $Pr$, the ratio of viscosity to density diffusivity, and fixed moderate Reynolds number, $Re=400$. In the two asymptotic limits $Pr \to 0$ and $Pr \to \infty$, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the $Pr \to 0$ limit, density variations away from a constant stable density gradient become vanish…
In this study, the stability dependence of turbulent Prandtl number ($Pr_t$) is quantified via a novel and simple analytical approach. Based on the variance and flux budget equations, a hybrid length scale formulation is first proposed and its functional relationships to well-known length scales are established. Next, the ratios of these length scales are utilized to derive an explicit relationship between $Pr_t$ and gradient Richardson number. In addition, theoretical predictions are made for several key turbulence variables (e.g., dissipation rates, normalized fluxes). The results from our p…
Turbulent fluxes in the atmospheric surface layer are key input for the prediction of weather, hydrology, and carbon dioxide concentration. In numerical modelling of turbulent fluxes, a -7/3 power-law scaling in turbulence cospectra is usually assumed at high wavenumbers. In eddy-covariance (EC) measurements of turbulent fluxes, an assumed shape of turbulence cospectra is typically required for high-frequency spectral corrections, typically assuming a -7/3 power law. The derivation of -7/3 power-law scaling is based primarily on dimensional analysis, and other cospectral scaling has also been …
We investigate exact and near resonant triad interactions (RTI) in a two-dimensional stably stratified uniform shear flow confined between two infinite parallel walls in the absence of viscous and diffusive effects. RTI occur when three interacting waves satisfy the resonance conditions of the form $k_1 \pm k_2 = k_3$ and $ω_1 \pm ω_2 = ω_3$ with $k_i$ and $ω_i$ being the wavenumber and frequency of the ith wave ($i \in {1,2,3}$), respectively. The linear stability problem is solved analytically, which gives the eigenfunctions in the form of the modified Bessel functions. It is identified that…
The effects of turbulent dynamic range on scalar mixing in stably stratified turbulence are investigated by an adaptation of the theoretical passive scalar modelling arguments of Beguier et al. (1978) and demonstrated statistically using direct numerical simulations of statistically stationary homogeneous stratified and sheared turbulence (SHSST). By analysis of inertial and inertial-convective subrange scaling, we show that the relationship between active scalar and turbulence time scales is predicted by the ratio of the Kolmogorov and Oboukhov-Corrsin constants provided there is sufficient s…
We investigated experimentally stably stratified turbulent flows in a lid driven cavity with a non-zero vertical mean temperature gradient in order to identify the parameters governing the mean and turbulent flows and to understand their effects on the momentum and heat transfer. We found that the mean velocity patterns (e.g., the form and the sizes of the large-scale circulations) depend strongly on the degree of the temperature stratification. In the case of strong stable stratification, the strong turbulence region is located in the vicinity of the main large-scale circulation. We detected …
We have advanced the energy and flux budget (EFB) turbulence closure theory that takes into account a two-way coupling between internal gravity waves (IGW) and the shear-free stably stratified turbulence. This theory is based on the budget equation for the total (kinetic plus potential) energy of IGW, the budget equations for the kinetic and potential energies of fluid turbulence, and turbulent fluxes of potential temperature for waves and fluid flow. The waves emitted at a certain level, propagate upward, and the losses of wave energy cause the production of turbulence energy. We demonstrate …
In this paper we characterize the scaling of energy spectra, and the interscale transfer of energy and enstrophy, for strongly, moderately and weakly stably stratified two-dimensional (2D) turbulence under large-scale random forcing. In the strongly stratified case, a large-scale vertically sheared horizontal flow (VSHF) co-exists with small scale turbulence. The VSHF consists of internal gravity waves and the turbulent flow has a kinetic energy (KE) spectrum that follows an approximate $k^{-3}$ scaling with zero KE flux and a robust positive enstrophy flux. The spectrum of the turbulent poten…
In this paper we advance physical background of the energy- and flux-budget turbulence closure based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time-scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth rotation. In accordance to modern experimental evidence, the closure implies maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the tw…
