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Boundary Layer Control of Rotating Convection Systems

TL;DR: This work forms a predictive description of the transition between the two regimes on the basis of the competition between these two boundary layers, and unifies the disparate results of an extensive array of previous experiments, and is broadly applicable to natural convection systems.
Abstract: Turbulent rotating convection is an important dynamical process occurring on nearly all planetary and stellar bodies, influencing many observed features such as magnetic fields, atmospheric jets and emitted heat flux patterns. For decades, it has been thought that the importance of rotation's influence on convection depends on the competition between the two relevant forces in the system: buoyancy (non-rotating) and Coriolis (rotating). The force balance argument does not, however, accurately predict the transition from rotationally controlled to non-rotating heat transfer behaviour. New results from laboratory and numerical experiments suggest that the transition is in fact controlled by the relative thicknesses of the thermal (non-rotating) and Ekman (rotating) boundary layers. Turbulent rotating convection controls many observed features in stars and planets, such as magnetic fields. It has been argued that the influence of rotation on turbulent convection dynamics is governed by the ratio of the relevant global-scale forces: the Coriolis force and the buoyancy force. This paper presents results from laboratory and numerical experiments which exhibit transitions between rotationally dominated and non-rotating behaviour that are not determined by this global force balance. Instead, the transition is controlled by the relative thicknesses of the thermal (non-rotating) and Ekman (rotating) boundary layers. Turbulent rotating convection controls many observed features of stars and planets, such as magnetic fields, atmospheric jets and emitted heat flux patterns1,2,3,4,5,6. It has long been argued that the influence of rotation on turbulent convection dynamics is governed by the ratio of the relevant global-scale forces: the Coriolis force and the buoyancy force7,8,9,10,11,12. Here, however, we present results from laboratory and numerical experiments which exhibit transitions between rotationally dominated and non-rotating behaviour that are not determined by this global force balance. Instead, the transition is controlled by the relative thicknesses of the thermal (non-rotating) and Ekman (rotating) boundary layers. We formulate a predictive description of the transition between the two regimes on the basis of the competition between these two boundary layers. This transition scaling theory unifies the disparate results of an extensive array of previous experiments8,9,10,11,12,13,14,15, and is broadly applicable to natural convection systems.
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Journal ArticleDOI
TL;DR: The scaling laws for planetary dynamos relate the characteristic magnetic field strength, characteristic flow velocity and other properties to primary quantities such as core size, rotation rate, electrical conductivity and heat flux as discussed by the authors.
Abstract: Scaling laws for planetary dynamos relate the characteristic magnetic field strength, characteristic flow velocity and other properties to primary quantities such as core size, rotation rate, electrical conductivity and heat flux. Many different scaling laws have been proposed, often relying on the assumption of a balance of Coriolis force and Lorentz force in the dynamo. Their theoretical foundation is reviewed. The advent of direct numerical simulations of planetary dynamos and the ability to perform them for a sufficiently wide range of control parameters allows to test the scaling laws. The results support a magnetic field scaling that is not based on a force balance, but on the energy flux available to balance ohmic dissipation. In its simplest form, it predicts a field strength that is independent of rotation rate and electrical conductivity and proportional to the cubic root of the available energy flux. However, rotation rate controls whether the magnetic field is dipolar or multipolar. Scaling laws for velocity, heat transfer and ohmic dissipation are also discussed. The predictions of the energy-based scaling law agree well with the observed field strength of Earth and Jupiter, but for other planets they are more difficult to test or special pleading is required to explain their field strength. The scaling law also explains the very high field strength of rapidly rotating low-mass stars, which supports its rather general validity.

264 citations


Cites background from "Boundary Layer Control of Rotating ..."

  • ...King et al. (2009) showed for rotating convection that the large value of β applies only when the rotational constraints on the flow are strong and the Ekman boundary layer is thinner than the thermal boundary layer....

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Journal ArticleDOI
TL;DR: The geophysical relevance of the experiments and simulations is called into question: the dynamics of Earth's core are too complex, and operate across time and length scales too broad to be captured by any single laboratory experiment, or resolved on present-day computers.
Abstract: Few areas of geophysics are today progressing as rapidly as basic geomagnetism, which seeks to understand the origin of the Earth's magnetism. Data about the present geomagnetic field pours in from orbiting satellites, and supplements the ever growing body of information about the field in the remote past, derived from the magnetism of rocks. The first of the three parts of this review summarizes the available geomagnetic data and makes significant inferences about the large scale structure of the geomagnetic field at the surface of the Earth's electrically conducting fluid core, within which the field originates. In it, we recognize the first major obstacle to progress: because of the Earth's mantle, only the broad, slowly varying features of the magnetic field within the core can be directly observed. The second (and main) part of the review commences with the geodynamo hypothesis: the geomagnetic field is induced by core flow as a self-excited dynamo. Its electrodynamics define 'kinematic dynamo theory'. Key processes involving the motion of magnetic field lines, their diffusion through the conducting fluid, and their reconnection are described in detail. Four kinematic models are presented that are basic to a later section on successful dynamo experiments. The fluid dynamics of the core is considered next, the fluid being driven into motion by buoyancy created by the cooling of the Earth from its primordial state. The resulting flow is strongly affected by the rotation of the Earth and by the Lorentz force, which alters fluid motion by the interaction of the electric current and magnetic field. A section on 'magnetohydrodynamic (MHD) dynamo theory' is devoted to this rotating magnetoconvection. Theoretical treatment of the MHD responsible for geomagnetism culminates with numerical solutions of its governing equations. These simulations help overcome the first major obstacle to progress, but quickly meet the second: the dynamics of Earth's core are too complex, and operate across time and length scales too broad to be captured by any single laboratory experiment, or resolved on present-day computers. The geophysical relevance of the experiments and simulations is therefore called into question. Speculation about what may happen when computational power is eventually able to resolve core dynamics is given considerable attention. The final part of the review is a postscript to the earlier sections. It reflects on the problems that geodynamo theory will have to solve in the future, particularly those that core turbulence presents.

228 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied rapid rotating Rayleigh-benard convection using an asymptotically reduced equation set valid in the limit of low Rossby numbers and identified four distinct dynamical regimes: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger Rayleigh numbers, followed by a breakdown of the convective columns into disordered plume regime characterized by reduced efficiency and finally by geostrophic turbulence.
Abstract: Rapidly rotating Rayleigh–Benard convection is studied using an asymptotically reduced equation set valid in the limit of low Rossby numbers. Four distinct dynamical regimes are identified: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger Rayleigh numbers, followed for yet larger Rayleigh numbers by a breakdown of the convective Taylor columns into a disordered plume regime characterized by reduced efficiency and finally by geostrophic turbulence. The transitions are quantified by examining the properties of the horizontally and temporally averaged temperature and thermal dissipation rate. The maximum of the thermal dissipation rate is used to define the width of the thermal boundary layer. In contrast to the non-rotating Rayleigh–Benard convection, the temperature drop across this layer decreases monotonically with increasing Rayleigh number and does not saturate. The breakdown of the convective Taylor column regime is attributed to the onset...

215 citations


Cites background or methods from "Boundary Layer Control of Rotating ..."

  • ...This result is also in agreement with DNS of the incompressible Navier–Stokes equations performed at moderately low Rossby number Ro¼ 0.75 for which Ra¼O(E 2)>RaT and the result Nu Ra2/7 is also found (Julien et al. 1996)....

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  • ...The location of this transition in parameter space is a subject of current debate (e.g. King et al. 2009, Liu and Ecke 2009, Schmitz and Tilgner 2009, 2010)....

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  • ...Laboratory experiments and simulation data (King et al. 2009) allude to thermal and momentum boundary layers as the principal players in this result, despite the presence of a similar transition in the presence of stress-free boundary conditions where no viscous layers exist (Schmitz and Tilgner…...

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  • ...Recent theory of rotating convection with no-slip boundaries posited by (King et al. 2009) attributes changes in the heat transport to the crossing of the thermal and Ekman boundary layers....

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  • ...It has been posited that the crossover of the boundary layers is responsible for the transition to a heat transport scaling law identical to that observed non-rotating RBC where for no-slip boundaries Nu Ra2/7 (King et al. 2009)....

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Journal ArticleDOI
TL;DR: In this paper, an exact scaling law for heat transfer by geostrophic convection, by considering the stability of the thermal boundary layers, where, and are the Nusselt, Rayleigh and Ekman numbers, respectively, and is the critical Rayleigh number for the onset of convection.
Abstract: Turbulent, rapidly rotating convection has been of interest for decades, yet there exists no generally accepted scaling law for heat transfer behaviour in this system. Here, we develop an exact scaling law for heat transfer by geostrophic convection, , by considering the stability of the thermal boundary layers, where , and are the Nusselt, Rayleigh and Ekman numbers, respectively, and is the critical Rayleigh number for the onset of convection. Furthermore, we use the scaling behaviour of the thermal and Ekman boundary layer thicknesses to quantify the necessary conditions for geostrophic convection: . Interestingly, the predictions of both heat flux and regime transition do not depend on the total height of the fluid layer. We test these scaling arguments with data from laboratory and numerical experiments. Adequate agreement is found between theory and experiment, although there is a paucity of convection data for low .

185 citations

Journal ArticleDOI
TL;DR: In this paper, a closely coupled suite of advanced asymptotically-reduced theoretical models, efficient Cartesian direct numerical simulations (DNS) and laboratory experiments are presented.

156 citations


Cites background from "Boundary Layer Control of Rotating ..."

  • ...5) necessary to identify fundamental changes in system behavior (e.g., King et al., 2009, 2012; Cheng et al., 2015)....

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References
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Book
01 Jul 1968

2,123 citations

Journal ArticleDOI
TL;DR: In this paper, an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude, were studied and their scaling laws were established.
Abstract: SUMMARY We study numerically an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude. Convection is driven by a fixed temperature contrast between rigid boundaries. There are two distinct classes of solutions with strong and weak dipole contributions to the magnetic field, respectively. Non-dipolar dynamos are found when inertia plays a significant role in the force balance. In the dipolar regime the critical magnetic Reynolds number for self-sustained dynamos is of order 50, independent of the magnetic Prandtl number Pm. However, dynamos at low Pm exist only at sufficiently low Ekman number E. For dynamos in the dipolar regime we attempt to establish scaling laws that fit our numerical results. Assuming that diffusive effects do not play a primary role, we introduce non-dimensional parameters that are independent of any diffusivity. These are a modified Rayleigh number based on heat (or buoyancy) flux Ra ∗ , the Rossby number Ro measuring the flow velocity, the Lorentz number Lo measuring magnetic field strength, and a modified Nusselt number Nu ∗ for the advected heat flow. To first approximation, all our dynamo results can be collapsed into simple power-law dependencies on the modified Rayleigh number, with approximate exponents of 2/5, 1/2 and 1/3 for the Rossby number, modified Nusselt number and Lorentz number, respectively. Residual dependencies on the parameters related to diffusion (E, Pm, Prandtl number Pr) are weak. Our scaling laws are in agreement with the assumption that the magnetic field strength is controlled by the available power and not necessarily by a force balance. The Elsasser number � , which is the conventional measure for the ratio of Lorentz force to Coriolis force, is found to vary widely. We try to assess the relative importance of the various forces by studying sources and sinks of enstrophy (squared vorticity). In general Coriolis and buoyancy forces are of the same order, inertia and viscous forces make smaller and variable contributions, and the Lorentz force is highly variable. Ignoring a possible weak dependence on the Prandtl numbers or the Ekman number, a surprising prediction is that the magnetic field strength is independent both of conductivity and of rotation rate and is basically controlled by the buoyancy flux. Estimating the buoyancy flux in the Earth’s core using our Rossby number scaling and a typical velocity inferred from geomagnetic secular variations, we predict a small growth rate and old age of the inner core and obtain a reasonable magnetic field strength of order 1 mT inside the core. From the observed heat flow in Jupiter, we predict an internal field of 8 mT, in agreement with Jupiter’s external field being 10 times stronger than that of the Earth.

719 citations

Journal ArticleDOI
TL;DR: In this article, an experimental study of the response of a thin uniformly heated rotating layer of fluid is presented, and it is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number.
Abstract: An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 104, whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 105. At all other Taylor numbers there is good agreement between linear theory and experiment.The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 104, it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.

584 citations

Journal ArticleDOI
TL;DR: In this paper, the authors take the optimistic view that present convective models are qualitatively reasonable, what can one expect of an improved theory? One desirable feature would be the prediction of convective transfer with, in addition, some reasonable estimate of the accuracy of the prediction.
Abstract: Convection occurs somewhere in most stars, yet our lack of understanding of convection has not seemed a major impediment to progress in stellar structure in recent years. In part this is true because convection often achieves the idealized adiabatic limit that is expected in convective cores of stars. I t has also been true that uncertainties in the other physical processes in stars have been reduced considerably, and this has permitted a better empirical determination of the arbitrary parameters used in stellar convec­ tion theory. Of course, there is always the possibility that things are not as satisfactory as one thinks. But if we take the optimistic view that present convective models are qualitatively reasonable, what can one expect of an improved theory? One desirable feature would be the prediction of convective transfer with, in addition, some reasonable estimate of the accuracy of the prediction. For this, a minimal but inadequate test is found in laboratory convection for which some quantitative data are available. Thus, a principal goal of stellar convection theory should be the development of a reasonable deductive theory whose reasonability can be minimally established by laboratory tests. Having obtained a theory at this level we would next be interested in finer details that characterize stellar convection. That is, we would like to be able to be quantitative about the time dependence and scales of the con­ vection motion and to compare these with solar observations; we would like to know how far convection may penetrate beyond the regions of in­ stability and by large-scale mixing remove chemical inhomogeneities; we would be interested in the precise temperature variations at the tops of convective envelopes to have better input for model atmospheres. And these are only a sample of some of the questions that one would hope to answer at this level of difficulty. There is, in addition, a series of dynamical questions which raise problems about the interaction of convection with other processes of stellar fluid dynamics. These bring in new instabilities and are probably the most in-

309 citations

Journal ArticleDOI
10 Nov 2005-Nature
TL;DR: This work presents a numerical model of three-dimensional rotating convection in a relatively thin spherical shell that generates both types of jets and implies that Jupiter's latitudinal transition in jet width corresponds to a separation between the bottom-bounded flow structures in higher latitudes and the deep equatorial flows.
Abstract: The bands of Jupiter represent a global system of powerful winds. Broad eastward equatorial jets are flanked by smaller-scale, higher-latitude jets flowing in alternating directions. Jupiter's large thermal emission suggests that the winds are powered from within, but the zonal flow depth is limited by increasing density and electrical conductivity in the molecular hydrogen-helium atmosphere towards the centre of the planet. Two types of planetary flow models have been explored: shallow-layer models reproduce multiple high-latitude jets, but not the equatorial flow system, and deep convection models only reproduce an eastward equatorial jet with two flanking neighbours. Here we present a numerical model of three-dimensional rotating convection in a relatively thin spherical shell that generates both types of jets. The simulated flow is turbulent and quasi-two-dimensional and, as observed for the jovian jets, simulated jet widths follow Rhines' scaling theory. Our findings imply that Jupiter's latitudinal transition in jet width corresponds to a separation between the bottom-bounded flow structures in higher latitudes and the deep equatorial flows.

269 citations