This study offers an explanation of a recently observed effect of destabilization of free convective flows by weak rotation. After studying several models where flows are driven by a simultaneous action of convection and rotation, it is concluded that the destabilization is observed in the cases where centrifugal force acts against main convective circulation. At relatively low Prandtl numbers this counter action can split the main vortex into two counter rotating vortices, where the interaction leads to instability. At larger Prandtl numbers, the counter action of the centrifugal force steepens an unstable thermal stratification, which triggers Rayleigh-Bénard instability mechanism. Both cases can be enhanced by advection of azimuthal velocity disturbances towards the axis, where they grow and excite perturbations of the radial velocity. The effect was studied considering a combined convective/rotating flow in a cylinder with a rotating lid and a parabolic temperature profile at the sidewall. Next, explanations of the destabilization effect for rotating magnetic field driven flow and melt flow in a Czochralski crystal growth model were derived.
The horizontal convection within a rectangle tank is numerically simulated. The flow is found to be unsteady at high Rayleigh numbers. There is a Hopf bifurcation of $Ra$ from steady solutions to periodic solutions, and the critical Rayleigh number $Ra_c$ is obtained as $Ra_c=5.5377\times 10^8$ for the middle plume forcing at $Pr=1$, which is much larger than the formerly obtained value. Besides, the unstable perturbations are always generated from the central jet, which implies that the onset of instability is due to velocity shear (shear instability) other than thermally dynamics (thermal in…
In an earlier paper we showed that the combination of azimuthal magnetic fields and super-rotation in Taylor-Couette flows of conducting fluids can be unstable against non-axisymmetric perturbations if the magnetic Prandtl number of the fluid is $Pm\neq 1$. Here we demonstrate that the addition of a weak axial field component may allow axisymmetric perturbation patterns for $Pm$ of order unity depending on the boundary conditions. The axisymmetric modes only occur for magnetic Mach numbers (of the azimuthal field) of order unity, while higher values are necessary for non-axisymmetric modes. Th…
The onset of thermal convection in a uniformly rotating and horizontally isotropic bi-disperse porous medium, taking into account the Vadasz term, is investigated. Via linear instability analysis, it has been proven that the Vadasz term allows the onset of convection via an oscillatory state but does not affect convection via a stationary motion.
We investigate by direct numerical simulation Rayleigh-Bénard convection in a rotating rectangular cell with rotation vector and gravity perpendicular to each other. The flow is two dimensional near the onset of convection with convection rolls aligned parallel to the rotation axis of the boundaries. At a sufficiently large Rayleigh number, the flow becomes unstable to three dimensional disturbances which changes the scaling of heat transport and kinetic energy with Rayleigh number. The mechanism leading to the instability is identified as an elliptical instability. At the transition, the Reyn…
We investigate the effect of weak rotation on the large-scale circulation (LSC) of turbulent Rayleigh-Bénard convection, using the theory for cessations in a low-dimensional stochastic model of the flow previously studied. We determine the cessation frequency of the LSC as a function of rotation, and calculate the statistics of the amplitude and azimuthal velocity fluctuations of the LSC as a function of the rotation rate for different Rayleigh numbers. Furthermore, we show that the tails of the reorientation probability distribution function remain unchanged for rotating systems, while the di…
We present measurements of the azimuthal orientation θ(t) and thermal amplitude δ(t) of the large-scale circulation (LSC) of turbulent rotating convection within an unprecedented large Rossby number range 1<Ro<314. Results of θ(t) reveal persistent rotation of the LSC flow in the retrograde direction over the entire Ro range. The rotation speed ratio remains a constant of 0.13(${\pm}$0.01) for 10<Ro<70, but starts to increase with increasing Ro when Ro>70. We identify the mechanism through which the mean retrograde rotation speed can be enhanced by stochastic cessations in the presence of weak…
We show the possibility of quasiperiodic waves at the onset of thermal convection in a thin horizontal layer of slowly rotating zero-Prandtl number Boussinesq fluid confined between stress-free conducting boundaries. Two independent frequencies emerge due to an interaction between a stationary instability and a self-tuned wavy instability in presence of coriolis force, if Taylor number is raised above a critical value. Constructing a dynamical system for the hydrodynamical problem, the competition between the interacting instabilities is analyzed. The forward bifurcation from the conductive st…
The frequency spectra of the entropy and kinetic energy along with the power spectrum of the thermal flux are computed from direct numerical simulations for turbulent Rayleigh-Bénard convection with uniform rotation about a vertical axis in low-Prandtl-number fluids ($\mathrm{Pr} < 0.6$). Simulations are done for convective Rossby numbers $\mathrm{Ro} \ge 0.2$. The temporal fluctuations of these global quantities show two scaling regimes: (i) $ω^{-2}$ at higher frequencies for all values of $\mathrm{Ro}$ and (ii) $ω^{-γ_1}$ at intermediate frequencies with $γ_1 \approx 4$ for $\mathrm{Ro} > 1$…
According to Rayleigh's criterion, rotating flows are linearly stable when their specific angular momentum increases radially outward. The celebrated magnetorotational instability opens a way to destabilize those flows, as long as the angular velocity is decreasing outward. Using a short-wavelength approximation we demonstrate that even flows with very steep positive shear can be destabilized by azimuthal magnetic fields which are current-free within the fluid. We illustrate the transition of this instability to a rotationally enhanced kink-type instability in case of a homogeneous current in …
We report heat transfer and temperature profile measurements in laboratory experiments of rapidly rotating convection in water under intense thermal forcing (Rayleigh number $Ra$ as high as $\sim 10^{13}$) and unprecedentedly strong rotational influence (Ekman numbers $E$ as low as $10^{-8}$). Measurements of the mid-height vertical temperature gradient connect quantitatively to predictions from numerical models of asymptotically rapidly rotating convection, separating various flow phenomenologies. Past the limit of validity of the asymptotically-reduced models, we find novel behaviors in a re…
We consider rotating Rayleigh-Bénard convection of a fluid with a Prandtl number of $Pr = 0.8$ in a cylindrical cell with an aspect ratio $Γ= 1/2$. Direct numerical simulations were performed for the Rayleigh number range $10^5 \leq Ra \leq 10^9$ and the inverse Rossby number range $0 \leq 1/Ro \leq 20$. We propose a method to capture regime transitions based on the decomposition of the velocity field into toroidal and poloidal parts. We identify four different regimes. First, a buoyancy dominated regime occurring as long as the toroidal energy $e_{tor}$ is not affected by rotation and remains…
The expansion of the rotating fluid will change the vorticity and rotational speed of the expanding region. In turbulent thermal convection, this microscopic effect is preserved. Tracking the fluid micelles shows that the average vorticity varies with density, producing vorticity transport and angular momentum transport from the low-density area to the high-density area, forming a macroscopic vorticity difference and rotational speed difference. Taking the axial thermal convection model of the solar polar region, it can generate axial differential rotation, and the centrifugal force difference…
We present a broad range of measurements of the angular orientation theta_0(t) of the large-scale circulation (LSC) of turbulent Rayleigh-Benard convection as a function of time. We used two cylindrical samples of different overall sizes, but each with its diameter nearly equal to its height. The fluid was water with a Prandtl number of 4.38. The time series theta_0(t) consisted of meanderings similar to a diffusive process, but in addition contained large and irregular spontaneous reorientation events through angles Delta theta. We found that reorientations can occur by two distinct mechanism…
We discuss recent observational, theoretical and modeling progress made in understanding the Sun's internal dynamics, including its rotation, meridional flow, convection and overshoot. Over the past few decades, substantial theoretical and observational effort has gone into appreciating these aspects of solar dynamics. A review of these observations, related helioseismic methodology and inference and computational results in relation to these problems is undertaken here.
In the present paper double-diffusive convection, taking into account Coriolis effects, in a horizontal layer of Brinkman-anisotropic bi-disperse porous medium is analysed. Via linear instability analysis, we found that convection can set in through stationary or oscillatory motions and the critical Rayleigh numbers for the onset of stationary secondary flow (steady convection) and overstability (oscillatory convection) are determined.
Weakly-rotating turbulent Rayleigh-Benard convection was studied experimentally and numerically. With increasing rotation and large enough Rayleigh number an abrupt transition from a turbulent state with nearly rotation-independent heat transport to another turbulent state with enhanced heat transfer is observed at a critical inverse Rossby number $1/Ro_c \simeq 0.4$. Whereas for $1/Ro < 1/Ro_c$ the strength of the large-scale convection-roll is either enhanced or essentially unmodified depending on parameters, its strength is increasingly diminished beyond $1/Ro_c$ where it competes with Ekma…
We study rotating homogeneous turbulent convection forced by a mean vertical temperature gradient by means of direct numerical simulations (DNS) in the Boussinesq approximation in a rotating frame. In the absence of rotationour results are in agreement with the "ultimate regime of thermal convection" for the scaling of the Nusselt and Reynolds numbers vs Rayleigh and Prandtl numbers. Rotation is found to increase both $Nu$ and $Re$ at fixed $Ra$ with a maximum enhancement for intermediate values of the Rossby numbers, qualitatively similar, but with stronger intensity, to what observed in Rayl…
We present spatial-resolved measurements of the columnar vortex structures in rotating Rayleigh-Bénard convection. The scaled radial profiles of the azimuthal velocity $u_ϕ(r)$ and vertical vorticity $ω(r)$ of the vortices are analyzed and compared with the predictions of the asymptotic theory. The results reveal that the asymptotic theory predicts accurately $u_ϕ(r)$ and $ω(r)$ in the geostrophic convection regime, but extension of the theory in the weak rotation regime is needed to interpret the rotation-dependence of the experimental data. Our measurements of the mean velocity, vorticity of…
In RB convection for fluids with Prandtl number $Pr\gtrsim 1$, rotation beyond a critical (small) rotation rate is known to cause a sudden enhancement of heat transfer which can be explained by a change in the character of the BL dynamics near the top and bottom plates of the convection cell. Namely, with increasing rotation rate, the BL signature suddenly changes from Prandtl--Blasius type to Ekman type. The transition from a constant heat transfer to an almost linearly increasing heat transfer with increasing rotation rate is known to be sharp and the critical Rossby number $Ro_{c}$ occurs t…
The heat transfer and flow structure in rotating Rayleigh-Bénard convection are strongly influenced by the Rayleigh ($Ra$), Prandtl ($Pr$), and Rossby ($Ro$) number. For $Pr\gtrsim 1$ and intermediate rotation rates, the heat transfer is increased compared to the non-rotating case. We find that the regime of increased heat transfer is subdivided into a low and a high $Ra$ number regime. For $Ra\lesssim 5\times10^8$ the heat transfer at a given $Ra$ and $Pr$ is highest at an optimal rotation rate, at which the thickness of the viscous and thermal boundary layer is about equal. From the scaling …
Here we summarize the results from our direct numerical simulations (DNS) and experimental measurements on rotating Rayleigh-Bénard (RB) convection. Our experiments and simulations are performed in cylindrical samples with an aspect ratio Γvarying from 1/2 to 2. Here Γ=D/L, where D and L are the diameter and height of the sample, respectively. When the rotation rate is increased, while a fixed temperature difference between the hot bottom and cold top plate is maintained, a sharp increase in the heat transfer is observed before the heat transfer drops drastically at stronger rotation rates. He…
We report experimental measurements of heat transport in rotating Rayleigh-B{é}nard convection in a cylindrical convection cell with aspect ratio $Γ= 1/2$. The fluid was helium gas with Prandtl number Pr = 0.7. The range of control parameters was Rayleigh number $4 \times 10^9 < {\rm Ra} < 4 \times 10^{11}$ and Ekman number $2 \times 10^{-7} < {\rm Ek} < 3 \times 10^{-5}$(corresponding to Taylor number $4 \times 10^9 < {\rm Ta} < 1 \times 10^{14}$ and convective Rossby number $0.07 < {\rm Ro} < 5$). We determine the crossover from weakly rotating turbulent convection to rotation dominated geos…
A model for the large-scale circulation (LSC) dynamics of turbulent Rayleigh-Benard convection is presented. It consists of two stochastic ordinary differential equations motivated by the Navier-Stokes equation, one each for the strength and the azimuthal orientation of the LSC. Stochastic forces represent phenomenologically the action of the turbulent fluctuations on the LSC. Without adjustable parameters, the model yields a meandering LSC with occasional rotations, and with more rare cessations that occur a few times per day -- differing only by about a factor of two from experiment. Also as…
We present measurements of the orientation theta_0(t) of the large-scale circulation (LSC) of turbulent Rayleigh-Benard convection in cylindrical cells of aspect ratio 1. theta_0(t) undergoes irregular reorientations. It contains reorientation events by rotation through angles Delta theta with a monotonically decreasing probability distribution p(Delta theta), and by cessations (where the LSC stops temporarily) with a uniform p(Delta theta). Reorientations have Poissonian statistics in time. The amplitude of the LSC and the magnitude of the azimuthal rotation rate have a negative correlation.
The effect of rotation on the boundary layers (BLs) in a Rayleigh-Benard (RB) system at a relatively low Rayleigh number, i.e. $Ra = 4\times10^7$, is studied for different Pr by direct numerical simulations and the results are compared with laminar BL theory. In this regime we find a smooth onset of the heat transfer enhancement as function of increasing rotation rate. We study this regime in detail and introduce a model based on the Grossmann-Lohse theory to describe the heat transfer enhancement as function of the rotation rate for this relatively low Ra number regime and weak background rot…
Direct numerical simulations are performed to study turbulent Rayleigh-Bénard convection in a vertical cylindrical cavity with uniform axial magnetic field. Flows at high Hartmann and Rayleigh numbers are considered. The calculations reveal that, similarly to the behavior observed in Rayleigh-Bénard convection with strong rotation, flows at strong magnetic field develop a central vortex, while the heat transfer is suppressed.
Laboratory experiments were conducted to study heat transport characteristics in a nonhomogeneously heated fluid annulus subjected to rotation along the vertical axis (z). The nonhomogeneous heating was obtained by imposing radial and vertical temperature gradient (ΔT). The parameter range for this study was Rayleigh number, Ra=2.43x10^8-3.66x10^8, and Taylor number, Ta=6.45x10^8-27x10^8. The working fluid was water with a Prandtl number, Pr=7. Heat transport was measured for varying rotation rates (Ω) for fixed values of ΔT. The Nusselt number, Nu, plotted as a function of Ta distinctly showe…
We present results of direct numerical simulations on homoclinic gluing and ungluing bifurcations in low-Prandtl-number ($ 0 \leq Pr \leq 0.025 $) Rayleigh-Bénard system rotating slowly and uniformly about a vertical axis. We have performed simulations with \textit{stress-free} top and bottom boundaries for several values of Taylor number ($5 \leq Ta \leq 50$) near the instability onset. We observe a single homoclinic ungluing bifurcation, marked by the spontaneous breaking of a larger limit cycle into two limit cycles with the variation of the reduced Rayleigh number $r$ for smaller values of…
We present results for entropy and kinetic energy spectra computed from direct numerical simulations for low-Prandtl-number ($Pr < 1$) turbulent flow in Rayleigh-Bénard convection with uniform rotation about a vertical axis. The simulations are performed in a three-dimensional periodic box for a range of Taylor number ($ 0 \leq Ta \leq 10^8$) and reduced Rayleigh number $r = Ra/Ra_{\circ} (Ta, Pr)$ ($1.0 \times 10^2 \le r \le 5.0 \times 10^3$). The Rossby number $Ro$ varies in the range $1.34 \le Ro \le 73$. The entropy spectrum $E_θ(k)$ shows bi-splitting into two branches for lower values of…
When a fluid system is subject to strong rotation, centrifugal fluid motion is expected, i.e., denser (lighter) fluid moves outward (inward) from (toward) the axis of rotation. Here we demonstrate, both experimentally and numerically, the existence of an unexpected outward motion of warm and lighter vortices in rotating turbulent convection. This anomalous vortex motion occurs under rapid rotations when the centrifugal buoyancy is sufficiently strong to induce a symmetry-breaking in the vorticity field, i.e., the vorticity of the cold anticyclones overrides that of the warm cyclones. We show t…
We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number ($Pr = 0.1$) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box ($L_x \times L_y \times 1$) and a wide range of Taylor numbers ($750 \le Ta \le 5000$) for the purpose. The horizontal aspect ratio $η= L_y/L_x$ of the box was varied from $0.5$ to $10$. The primary instability appeared in the …
This paper experimentally investigates the convection in a fast rotating Tangent Cylinder (TC), for Ekman numbers down to $E=3.36\times10^{-6}$, in a configuration relevant to the liquid core of the Earth. In the apparatus, the TC results from the Proudman-Taylor constraint incurred by rotating a hemispherical fluid vessel heated in its centre by a protruding heating element of cylindrical shape. The resulting convection that develops above the heater, i.e within the TC, is shown to set in for critical Rayleigh numbers and wavenumbers respectively scaling as $Ra_c\sim E^{4/3}$ and $a_c\sim E^{…
Influence of Aspect ratio in the Convection in Rotating Annulus In the Presence of Localized Heating
School of Artificial Intelligence, Amrita Vishwa Vidyapeetham（阿迈塔维迪亚佩塔姆大学人工智能学院）；Department of Mechanical Engineering, Indian Institute of Technology（印度理工学院机械工程系）
A mean-field theory of differential rotation in a density stratified turbulent convection has been developed. This theory is based on a combined effect of the turbulent heat flux and anisotropy of turbulent convection on the Reynolds stress. A coupled system of dynamical budget equations consisting in the equations for the Reynolds stress, the entropy fluctuations and the turbulent heat flux has been solved. To close the system of these equations, the spectral tau approach which is valid for large Reynolds and Peclet numbers, has been applied. The adopted model of the background turbulent conv…
We present a direct numerical simulation to investigate the dynamics and statistics of reorientations of large-scale circulation (LSC) in turbulent rotating Rayleigh-Bénard convection (RRBC) for air ($Pr=0.7$) contained in a cylindrical cell with unit aspect ratio. A wide range of rotation rates ($0\leq Ro^{-1}\leq 30$) is considered for two different Rayleigh numbers $Ra=2\times10^6$ and $2\times10^7$. Using the Fourier mode analysis of time series data obtained from the different probes placed in the azimuthal direction of the container at the mid-plane, the orientation and associated dynami…
We study the effects of rotation on the growth and saturation of the double-diffusive fingering (thermohaline) instability at low Prandtl number. Using direct numerical simulations, we estimate the compositional transport rates as a function of the relevant non-dimensional parameters - the Rossby number, inversely proportional to the rotation rate, and the density ratio which measures the relative thermal and compositional stratifications. Within our explored range of parameters, we generally find rotation to have little effect on vertical transport. However, we also present one exceptional ca…
We simulate three-dimensional, horizontally periodic Rayleigh-Bénard convection between free-slip horizontal plates, rotating about a distant horizontal axis. When both the temperature difference between the plates and the rotation rate are sufficiently large, a strong horizontal wind is generated that is perpendicular to both the rotation vector and the gravity vector. The wind is turbulent, large-scale, and vertically sheared. Horizontal anisotropy, engendered here by rotation, appears necessary for such wind generation. Most of the kinetic energy of the flow resides in the wind, and the ver…
Numerical MHD simulations play increasingly important role for understanding mechanisms of stellar magnetism. We present simulations of convection and dynamos in density-stratified rotating spherical fluid shells. We employ a new 3D simulation code for the solution of a physically consistent anelastic model of the process with a minimum number of parameters. The reported dynamo simulations extend into a "buoyancy-dominated" regime where the buoyancy forcing is dominant while the Coriolis force is no longer balanced by pressure gradients and strong anti-solar differential rotation develops as a…
Oscillatory double-diffusive convection (ODDC, more traditionally called semiconvection) is a form of linear double-diffusive instability that occurs in fluids that are unstably stratified in temperature (Schwarzschild unstable), but stably stratified in chemical composition (Ledoux stable). This scenario is thought to be quite common in the interiors of stars and giant planets, and understanding the transport of heat and chemical species by ODDC is of great importance to stellar and planetary evolution models. Fluids unstable to ODDC have a tendency to form convective thermo-compositional lay…
Overshooting of turbulent motions from convective regions into adjacent stably stratified zones plays a significant role in stellar interior dynamics as this process may lead to mixing of chemical species, and contribute to the transport of angular momentum and magnetic fields. We present a series of fully non-linear, three-dimensional (3D) anelastic simulations of overshooting convection in a spherical shell which are focused on the dependence of the overshooting dynamics on the density stratification and the rotation, both key ingredients in stars which however have not been studied systemat…
The dynamics of convecting fluids in rotating spherical shells is governed at Prandtl numbers of the order unity by the interaction between differential rotation and roll-like convection eddies. While the differential rotation is driven by the Reynolds stresses of the eddies, its shearing action inhibits convection and causes phenomena such as localized convection and turbulent relaxation oscillations. The response of the system is enriched in the case of dynamo action. Lorentz forces may brake either entirely or partially the geostrophic differential rotation and give rise to two rather diffe…
We present the first global 3D simulations of thermal convection in the oblate envelopes of rapidly-rotating solar-type stars. This has been achieved by exploiting the capabilities of the new Compressible High-ORder Unstructured Spectral difference (CHORUS) code. We consider rotation rates up to 85\% of the critical (breakup) rotation rate, which yields an equatorial radius that is up to 17\% larger than the polar radius. This substantial oblateness enhances the disparity between polar and equatorial modes of convection. We find that the convection redistributes the heat flux emitted from the …
The problem of the turbulent thermal convection in the Earth's liquid core is considered. Following assumptions on decreasing of the spatial scales due to the rapid rotation, we propose the subgrid model of the eddy diffusivity, which is used in the large-scale model. This approach makes it possible to model realistic regimes with small Ekman and Rossby numbers ($E\sim 10^{-14}$, $R_o \sim 10^{-8}$) and a sufficiently large Rayleigh number $R_a \sim 10^{12}$. The obtained estimate of the averaged kinetic energy is comparable with observations. The model includes rotation of the solid core due …
Numerical simulations of rotating Rayleigh-Bénard convection are presented for both no slip and free slip boundaries. The goal is to find a criterion distinguishing convective flows dominated by the Coriolis force from those nearly unaffected by rotation. If one uses heat transport as an indicator of which regime the flow is in, one finds that the transition between the flow regimes always occurs at the same value of a certain combination of Reynolds, Prandtl and Ekman numbers for both boundary conditions. If on the other hand one uses the helicity of the velocity field to identify flows nearl…
In this paper we examine the interaction of convection, rotation and mean flows in a thermal annulus. In this system mean flows are driven by correlations induced by rotation leading to non-trivial Reynolds stresses. The mean flows act back on the convective turbulence acting as a barrier to transport. For this system we demonstrate that the Generalised Quasilinear Approximation (GQL) (Marston et al 2016) may provide a much better approximation to the complicated full nonlinear dynamics than the widely used Quasilinear Approximation (QL). This result will enable the construction of more accura…
We present experimental heat transport measurements of turbulent Rayleigh-Bénard convection with rotation about a vertical axis. The fluid, water with Prandtl number ($σ$) about 6, was confined in a cell which had a square cross section of 7.3 cm$\times$7.3 cm and a height of 9.4 cm. Heat transport was measured for Rayleigh numbers $2\times 10^5 <$ Ra $ < 5\times 10^8$ and Taylor numbers $0 <$ Ta $< 5\times 10^{9}$. We show the variation of normalized heat transport, the Nusselt number, at fixed dimensional rotation rate $Ω_D$, at fixed Ra varying Ta, at fixed Ta varying Ra, and at fixed Rossb…
We study thermal convection in a rotating fluid in order to better understand the properties of convection zones in rotating stars and planets. We first derive mixing-length theory for rapidly-rotating convection, arriving at the results of Stevenson (1979) via simple physical arguments. The theory predicts the properties of convection as a function of the imposed heat flux and rotation rate, independent of microscopic diffusivities. In particular, it predicts the mean temperature gradient; the rms velocity and temperature fluctuations; and the size of the eddies that dominate heat transport. …
