in "Mathematical Aspects of Natural Dynamos", E. Dormy, A.M. Soward (eds.), Grenoble Sciences and CRC Press, Boca Raton, pp. 119-198, ISBN-13:978-1-58488-954-0, 2007
The chapter begins with a description of the annulus model of rotationg convection. The basic equations for the spherical problem are then introduced and the onset of columnar convection in spherical shells is discussed. The onset of inertial mode convection is described which prevails at very low Prandtl numbers. The properties of finite amplitude convection are outlined for a wide range of Prandtl numbers. Equatorially attached convection is considered which evolves from inertial mode convection.
Numerical simulations of dynamos in rotating Rayleigh-Bénard convection in plane layers are presented. Two different types of dynamos exist which obey different scaling laws for the amplitude of the magnetic field. The transition between the two occurs within a hydrodynamically uniform regime which can be classified as rapidly rotating convection.
Numerical simulation of rotating convection in plane layers with free slip boundaries show that the convective flows can be classified according to a quantity constructed from the Reynolds, Prandtl and Ekman numbers. Three different flow regimes appear: Laminar flow close to the onset of convection, turbulent flow in which the heat flow approaches the heat flow of non-rotating convection, and an intermediate regime in which the heat flow scales according to a power law independent of thermal diffusivity and kinematic viscosity.
The onset of convection in the form of inertial waves in a rotating fluid sphere is studied through a perturbation analysis in an extension of earlier work by Zhang (1994). Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived and new results for the case of a nearly thermally insulating boundary are obtained.
Patterns of convection in internally heated, self-gravitating rotating spherical fluid shells are investigated through numerical simulations. While turbulent states are of primary interest in planetary and stellar applications the present paper emphasizes more regular dynamical features at Rayleigh numbers not far above threshold which are similar to those which might be observed in laboratory or space experiments. Amplitude vacillations and spatial modulations of convection columns are common features at moderate and large Prandtl numbers. In the low Prandtl number regime equatorially attache…
Double-diffusive convection driven by both thermal and compositional buoyancy in a rotating cylindrical annulus with conical caps is considered with the aim to establish whether a small fraction of compositional buoyancy added to the thermal buoyancy (or vice versa) can significantly reduce the critical Rayleigh number and amplify convection in planetary cores. It is shown that the neutral surface describing the onset of convection in the double-buoyancy case is essentially different from that of the well-studied purely thermal case, and does indeed allow the possibility of low-Rayleigh number…
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…
Pattern forming with externally imposed symmetry is ubiquitous in nature but lightly studied.We present experimental studies of pattern formation and selection by spatial periodic forcing in rapidly rotating convection. We observe symmetric convection patterns in form of regular vortex lattice near the instability onset, when the periodicity of the external forcing is set close to the intrinsic vortex spacing. We show that the new patterns arise as a dynamical process of imperfect bifurcation which can be well described by a Ginzburg-Landau-like model. With increasing buoyancy strength the eff…
The range of existence and the properties of two essentially different chaotic attractors found in a model of nonlinear convection-driven dynamos in rotating spherical shells are investigated. A hysteretic transition between these attractors is established as a function of the rotation parameter \ttau. The width of the basins of attraction is also estimated.
Rapidly rotating Rayleigh-Bénard convection is studied by combining results from direct numerical simulations (DNS), laboratory experiments and asymptotic modeling. The asymptotic theory is shown to provide a good description of the bulk dynamics at low, but finite Rossby number. However, large deviations from the asymptotically predicted heat transfer scaling are found, with laboratory experiments and DNS consistently yielding much larger Nusselt numbers than expected. These deviations are traced down to dynamically active Ekman boundary layers, which are shown to play an integral part in con…
Direct numerical simulations are employed to reveal three distinctly different flow regions in rotating spherical Rayleigh-Bénard convection. In the low-latitude region $\mathrm{I}$ vertical (parallel to the axis of rotation) convective columns are generated between the hot inner and the cold outer sphere. The mid-latitude region $\mathrm{II}$ is dominated by vertically aligned convective columns formed between the Northern and Southern hemispheres of the outer sphere. The diffusion-free scaling, which indicates bulk-dominated convection, originates from this mid-latitude region. In the equato…
We present high-precision experimental and numerical studies of the Nusselt number $Nu$ as functions of the Rayleigh number $Ra$ in geostrophic rotating convection with domain aspect ratio $Γ$ varying from 0.4 to 3.8 and the Ekman number Ek from $2.0{\times}10^{-7}$ to $2.7{\times}10^{-5}$. The heat-transport data $Nu(Ra)$ reveal a gradual transition from buoyancy-dominated to geostrophic convection at large $Ek$, whereas the transition becomes sharp with decreasing $Ek$. We determine the power-law scaling of $Nu{\sim}Ra^γ$, and show that the boundary flows give rise to pronounced enhancement …
Natural dynamos such as planets and stars generate global scale magnetic field despite the inferred presence of small scale turbulence. Such systems are known as large scale dynamos and are typically driven by convection and influenced by rotation. Previous numerical studies of rotating dynamos generally find that the large scale magnetic field becomes weaker as the flow becomes more turbulent. The underlying physical processes necessary for sustaining so-called large scale dynamos is therefore still debated. Here we use a suite of numerical simulations to show that strong large scale magnetic…
The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically-relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e., incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilise the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convecti…
Planetary cores consist of liquid metals (low Prandtl number $Pr$) that convect as the core cools. Here we study nonlinear convection in a rotating (low Ekman number $Ek$) planetary core using a fully 3D direct numerical simulation. Near the critical thermal forcing (Rayleigh number $Ra$), convection onsets as thermal Rossby waves, but as the $Ra$ increases, this state is superceded by one dominated by advection. At moderate rotation, these states (here called the weak branch and strong branch, respectively) are smoothly connected. As the planetary core rotates faster, the smooth transition is…
In this paper we present numerical simulations of rapidly-rotating Rayleigh-Bénard convection in the Boussinesq approximation with stress-free boundary conditions. At moderately low Rossby number and large Rayleigh number, we show that a large-scale depth-invariant flow is formed, reminiscent of the condensate state observed in two-dimensional flows. We show that the large-scale circulation shares many similarities with the so-called vortex, or slow-mode, of forced rotating turbulence. Our investigations show that at a fixed rotation rate the large-scale vortex is only observed for a finite ra…
Rotating Rayleigh-Bénard convection is investigated numerically with the use of an asymptotic model that captures the rapidly rotating, small Ekman number limit, $Ek \rightarrow 0$. The Prandtl number ($Pr$) and the asymptotically scaled Rayleigh number ($\widetilde{Ra} = Ra Ek^{4/3}$, where $Ra$ is the typical Rayleigh number) are varied systematically. For sufficiently vigorous convection, an inverse kinetic energy cascade leads to the formation of a depth-invariant large-scale vortex (LSV). With respect to the kinetic energy, we find a transition from convection dominated states to LSV domi…
We show, using direct numerical simulations with experimentally realizable boundary conditions, that wall modes in Rayleigh-Bénard convection in a rapidly rotating cylinder persist even very far from their linear onset. These nonlinear wall states survive in the presence of turbulence in the bulk and are robust with respect to changes in the shape of the boundary of the container. In this sense these modes behave much like the topologically protected states present in two-dimensional chiral systems even though rotating convection is a three-dimensional nonlinear driven dissipative system. We s…
Bistability and hysteresis of magnetohydrodynamic dipolar dynamos generated by turbulent convection in rotating spherical fluid shells is demonstrated. Hysteresis appears as a transition between two distinct regimes of dipolar dynamos with rather different properties including a pronounced difference in the amplitude of the axisymmetric poloidal field component and in the form of the differential rotation. The bistability occurs from the onset of dynamo action up to about 9 times the critical value of the Rayleigh number for onset of convection and over a wide range of values of the ordinary a…
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…
Motivated by the recent discovery of subsurface oceans on planetary moons and the interest they have generated, we explore convective flows in shallow spherical shells of dimensionless gap width $\varepsilon^2\ll 1$ in the rapid rotation limit $\mathrm{E}\ll1$, where $\mathrm{E}$ is the Ekman number. We employ direct numerical simulations (DNS) of the Boussinesq equations to compute the local heat flux $\mathrm{Nu}(λ)$ as a function of the latitude $λ$ and use the results to characterize the trapping of convection at low latitudes, around the equator. We show that these results are quantitativ…
We study a three-dimensional fluid model describing rapidly rotating convection that takes place in tall columnar structures. The purpose of this model is to investigate the cyclonic and anticyclonic coherent structures. Global existence, uniqueness, continuous dependence on initial data, and large-time behavior of strong solutions are shown provided the model is regularized by a weak dissipation term.
Double-diffusive convection driven by both thermal and compositional buoyancy in a rotating spherical shell can exhibit a rather large number of behaviours often distinct from that of the single diffusive system. In order to understand how the differences in thermal and compositional molecular diffusivities determine the dynamics of thermo-compositional convection we investigate numerically the linear onset of convective instability in a double-diffusive setup. We construct an alternative equivalent formulation of the non-dimensional equations where the linearised double-diffusive problem is d…
The value of the Prandtl number $P$ exerts a strong influence on convection-driven dynamos in rotating spherical shells filled with electrically conducting fluids. Low Prandtl numbers promote dynamo action through the shear provided by differential rotation, while the generation of magnetic fields is more difficult to sustain in high-Prandtl-number fluids where higher values of the magnetic Prandtl number $P_m$ are required. The magnetostrophic approximation often used in dynamo theory appears to be valid only for relatively high values of $P$ and $P_m$. Dynamos with a minimum value of $P_m$ s…
Convection and magnetic field generation in the Earth and planetary interiors are driven by both thermal and compositional gradients. In this work numerical simulations of finite-amplitude double-diffusive convection and dynamo action in rapidly rotating spherical shells full of incompressible two-component electrically-conducting fluid are reported. Four distinct regimes of rotating double-diffusive convection identified in a recent linear analysis (Silva et al., 2019, Geophys. Astrophys. Fluid Dyn., doi:10.1080/03091929.2019.1640875) are found to persist significantly beyond the onset of ins…
The competition between turbulent convection and global rotation in planetary and stellar interiors governs the transport of heat and tracers, as well as magnetic-field generation. These objects operate in dynamical regimes ranging from weakly rotating convection to the `geostrophic turbulence' regime of rapidly rotating convection. However, the latter regime has remained elusive in the laboratory, despite a worldwide effort to design ever-taller rotating convection cells over the last decade. Building on a recent experimental approach where convection is driven radiatively, we report heat tra…
Using direct numerical simulations, we study rotating Rayleigh-Bénard convection in a cylindrical cell for a broad range of Rayleigh, Ekman, and Prandtl numbers from the onset of wall modes to the geostrophic regime, an extremely important one in geophysical and astrophysical contexts. We connect linear wall-mode states that occur prior to the onset of bulk convection with the boundary zonal flow that coexists with turbulent bulk convection in the geostrophic regime through the continuity of length and time scales and of convective heat transport. We quantitatively collapse drift frequency, bo…
The possibility of subcritical behaviour in the geostrophic turbulence regime of rapidly rotating thermally driven convection is explored. In this regime a non-local inverse energy transfer may compete with the more traditional and local direct cascade. We show that, even for control parameters for which no inverse cascade has previously been observed, a subcritical transition towards a large-scale vortex state can occur when the system is initialized with a vortex dipole of finite amplitude. This new example of bistability in a turbulent flow, which may not be specific to rotating convection,…
Rayleigh-Bénard convection in rotating spherical shells can be considered as a simplified analogue of many astrophysical and geophysical fluid flows. Here, we use three-dimensional direct numerical simulations to study this physical process. We construct a dataset of more than 200 numerical models that cover a broad parameter range with Ekman numbers spanning $3\times 10^{-7} \leq E \leq 10^{-1}$, Rayleigh numbers within the range $10^3 < Ra < 2\times 10^{10}$ and a Prandtl number unity. We investigate the scaling behaviours of both local (length scales, boundary layers) and global (Nusselt an…
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…
The role of the moments of helicity distribution in rotating turbulence has been studied using the notion of helical distributed chaos. Results of the direct numerical simulations, laboratory experiments and geophysical observations have been used in this investigation. It is shown, in particular, that even for the cases when the global helicity is equal to zero at least even moments are usually non-zero (due to the appearance of the local spatial regions with strong negative and positive helicity) and can play a significant role in the rotating turbulence as adiabatic invariants. Rotating buo…
The heat transfer behavior of convection-driven dynamos in a rotating plane layer between two parallel plates, heated from below and cooled from the top, is investigated. At a fixed rotation rate (Ekman Number, $E=10^{-6}$) and fluid properties (thermal and magnetic Prandtl numbers, $Pr=Pr_m=1$), both dynamo convection (DC) and non-magnetic rotating convection (RC) simulations are performed to demarcate the effect of magnetic field on heat transport at different thermal forcings (Rayleigh Number, $Ra=3.83\times10^{9}-3.83\times10^{10}$). In this range, our turbulence resolving simulations demo…
We investigate the processes that lead to the generation of mean flows in two-dimensional anelastic convection. The simple model consists of a plane layer that is rotating about an axis inclined to gravity. The results are two-fold: firstly we numerically investigate the onset of convection in three-dimensions, paying particular attention to the role of stratification and highlight a curious symmetry. Secondly, we investigate the mechanisms that drive both zonal and meridional flows in two dimensions. We find that, in general, non-trivial Reynolds stresses can lead to systematic flows and, usi…
When the Rayleigh number is low, Rayleigh-Bénard convection in a nonrotating spherical shell with central gravity has symmetric solutions in terms of three-dimensional discrete rotation. All the known patterns with the regular polyhedral symmetries accompany reflection symmetry. We found a new type of steady convection in a nonrotating spherical shell by computer simulations. The pattern has the discrete rotational symmetry of a regular tetrahedron with no reflection symmetry. The convection consists of six pairs of spiral rolls placed on 12 faces of a spherical dodecahedron. Depending on the …
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 consider triadic nonlinear interaction in the Navier-Stokes equation for quasi-geostrophic convection in a spherical shell. This approach helps understanding the origin of kinetic energy transport in the system and the particular scheme of mode interaction, as well as the locality of the energy transfer. The peculiarity of convection in the sphere, concerned with excitation of Rossby waves, is considered. The obtained results are compared with our previous study in Cartesian geometry.
For rapidly rotating turbulent Rayleigh--Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like $Ra^{1/4}Ek^{2/3}$ where the…
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…
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…
Rotating Rayleigh--Bénard convection is an oft-employed model system to evaluate the interplay of buoyant forcing and Coriolis forces due to rotation, an eminently relevant interaction of dynamical effects found in many geophysical and astrophysical flows. These flows display extreme values of the governing parameters: large Rayleigh numbers $Ra$, quantifying the strength of thermal forcing, and small Ekman numbers $E$, a parameter inversely proportional to the rotation rate. This leads to the dominant geostrophic balance of forces in the flow between pressure gradient and Coriolis force. The …
We present the first detailed thermal and velocity field characterization of convection in a rotating cylindrical tank of liquid gallium, which has thermophysical properties similar to those of planetary core fluids. Our laboratory experiments, and a closely associated direct numerical simulation, are all carried out in the regime prior to the onset of steady convective modes. This allows us to study the oscillatory convective modes, sidewall modes and broadband turbulent flow that develop in liquid metals before the advent of steady columnar modes. Our thermo-velocimetric measurements show th…
The surface winds of Jupiter and Saturn are primarily zonal. Each planet exhibits strong prograde equatorial flow flanked by multiple alternating zonal winds at higher latitudes. The depth to which these flows penetrate has long been debated and is still an unsolved problem. Previous rotating convection models that obtained multiple high latitude zonal jets comparable to those on the giant planets assumed an incompressible (Boussinesq) fluid, which is unrealistic for gas giant planets. Later models of compressible rotating convection obtained only few high latitude jets which were not amenable…
We perform direct numerical simulations of rotating Rayleigh--Bénard convection of fluids with low ($Pr=0.1$) and high ($Pr=5$) Prandtl numbers in a horizontally periodic layer with no-slip top and bottom boundaries. At both Prandtl numbers, we demonstrate the presence of an upscale transfer of kinetic energy that leads to the development of domain-filling vortical structures. Sufficiently strong buoyant forcing and rotation foster the quasi-two-dimensional turbulent state of the flow, despite the formation of plume-like vertical disturbances promoted by so-called Ekman pumping from the viscou…
Experimental and numerical data for the heat transfer as a function of the Rayleigh-, Prandtl-, and Rossby numbers in turbulent rotating Rayleigh-Benard convection are presented. For relatively small $Ra ~ 10^8$ and large Pr modest rotation can enhance the heat transfer by up to 30%. At larger Ra there is less heat-transfer enhancement, and at small Pr = 0.7. there is no heat-transfer enhancement at all. We suggest that the small-Pr behavior is due to the breakdown of the heat-transfer-enhancing Ekman pumping because of larger thermal diffusion.
We propose a new self-consistent dynamo mechanism for the generation of large-scale magnetic fields in natural objects. Recent computational studies have described the formation of large-scale vortices (LSVs) in rotating turbulent convection. Here we demonstrate that for magnetic Reynolds numbers below the threshold for small-scale dynamo action, such turbulent flows can sustain large-scale magnetic fields --- i.e. fields with a significant component on the scale of the system.
Spherical shell dynamo models based on rotating convection show that the flow within the tangent cylinder is dominated by an off-axis plume that extends from the inner core boundary to high latitudes and drifts westward. Earlier studies explained the formation of such a plume in terms of the effect of a uniform axial magnetic field that significantly increases the lengthscale of convection in a rotating plane layer. However, rapidly rotating dynamo simulations show that the magnetic field within the tangent cylinder has severe lateral inhomogeneities that may influence the onset of an isolated…
Mixed convection above a horizontal disk rotating in a semi-infinite fluid is examined when the disk is heated so that its temperature varies quadratically with distance away from its centre. Steady similarity solutions are presented for a range of values of the two dimensionless parameters, a Prandtl number and a Grashof number. The results are corroborated by asymptotic analyses undertaken in the limits of small and large Prandtl number. For finite Prandtl number, the existence of multiple solutions at fixed Grashof number is revealed. The similarity structure for steady solutions fails beyo…
We study nonlinear convection in a rapidly rotating sphere with internal heating for values of the Prandtl number relevant for liquid metals ($Pr\in[10^{-2},10^{-1}]$). We use a numerical model based on the quasi-geostrophic approximation, in which variations of the axial vorticity along the rotation axis are neglected, whereas the temperature field is fully three-dimensional. We identify two separate branches of convection close to onset: (i) a well-known weak branch for Ekman numbers greater than $10^{-6}$, which is continuous at the onset (supercritical bifurcation) and consists of thermal …
The non-hydrostatic, quasigeostrophic approximation for rapidly rotating Rayleigh-Bénard convection admits a class of exact `single mode' solutions. These solutions correspond to steady laminar convection with a separable structure consisting of a horizontal planform characterized by a single wavenumber multiplied by a vertical amplitude profile, with the latter given as the solution of a nonlinear boundary value problem. The heat transport associated with these solutions is studied in the regime of strong thermal forcing (large reduced Rayleigh number $\widetilde{Ra}$). It is shown that the N…
Zonal flows are often found in rotating convective systems. Not only are these jet-flows driven by the convection, they can also have a profound effect on the nature of the convection. In this work the cylindrical annulus geometry is exploited in order to perform nonlinear simulations seeking to produce strong zonal flows and multiple jets. The parameter regime is extended to Prandtl numbers that are not unity. Multiple jets are found to be spaced according to a Rhines scaling based on the zonal flow speed, not the convective velocity speed. Under certain conditions the nonlinear convection ap…
