Heat transport in rotating convection without Ekman layers

Heat transport in rotating convection without Ekman layers

Abstract

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.

pacs:
47.27.te, 44.25.+f, 47.32.-y, 91.25.Za
1

It is a central problem for many areas of geo- and astrophysics to determine the heat flux through a rotating and convecting fluid layer. For example, the heat flux through the atmosphere governs weather and climate, the heat flux through stellar atmospheres determines stellar evolution, and the heat flux through planetary cores is essential for the generation of the magnetic fields of these bodies. A correspondingly large effort has already been spent on the problem. Buoyancy is driving the flow and can be balanced by either viscous or Coriolis forces, or the nonlinear terms in the equations of motion, or any combination of these. If the Coriolis force dominates the dynamics, a special type of boundary layer appears near solid boundaries, the Ekman layers, in which the viscous force is balanced by the Coriolis term. In addition, the flow in the bulk is organized into columnar vortices with their axes aligned with the rotation axis. If on the contrary nonlinear advection supersedes the Coriolis term, these columns are broken up and the style of flow known from non-rotating convection is approached Julien et al. (1996); Stellmach and Hansen (2004); King et al. (2009). There is an ongoing debate concerning the parameters at which the transition between these two flow regimes occurs Canuto and Dubovikov (1998); Ecke (1999); King et al. (2009), and we are still lacking reliable relations between the heat flux and the control parameters of the flow that would allow us to extrapolate data from laboratory experiments and numerical simulation to astrophysical objects.

Some recent work on rotating convection has focused on the Ekman layers. For instance, ref. King et al. (2009) relates the Ekman layers to the transition mentioned above. Despite the inhibiting effect of rotation on turbulence, the heat flux in a rotating flow can exceed that of a non-rotating flow at equal Rayleigh number Zhong et al. (1993). In ref. Zhong et al. (2009), this phenomenon is attributed to so called Ekman pumps, a term reserved for a certain flow pattern associated with Ekman boundary layers Cushman-Roisin (1994). Here we investigate convection with free slip boundary conditions. This eliminates Ekman layers and one can discern which effect really depends on their presence. Free slip boundaries are realized to a good approximation in Nature, for example at the surface of the oceans or at the top of atmospheric layers.

Consider a plane layer of thickness in the direction and of infinite extent in the plane. Let the layer be filled with fluid of kinematic viscosity , thermal diffusivity , and thermal expansion coefficient . Gravitational acceleration is pointing in the negative direction and the layer is rotating with angular velocity about the axis. The temperatures of the top and bottom boundaries are fixed at and , respectively. These two boundaries are assumed to be free slip, whereas periodic boundary conditions are applied in the and directions. The equations of evolution are made non-dimensional by using , and for units of time, length, and temperature, respectively. These equations then become within the Boussinesq approximation for the dimensionless velocity and temperature :

(1)
(2)
(3)

is the unit vector in -direction and collects the pressure and the centrifugal acceleration. The boundary conditions require that , , and that at both and . Three independent dimensionless control parameters appear: The Rayleigh number , the Ekman number , and the Prandtl number . They are defined by:

(4)

The Reynolds number and the Nusselt number are an output of the simulations:

(5)

The angular brackets denote average over time and the integrals extend over the computational volume for and over the surface of either the top or the bottom boundary for .

The equations of motion were solved with the same spectral method as used in Hartlep et al. (2003), except that free slip boundaries were implemented and that the Coriolis term was added and treated implicitly together with the diffusion terms. Resolutions reached up to 129 Chebychev polynomials for the discretization of the coordinate and Fourier modes in the plane. The periodicity lengths along the and directions were always chosen to be identical. The aspect ratio, defined as the ratio of the periodicity length in the plane and the layer height, was fixed at 10 for simulations without rotation. In rotating convection, the typical size of flow structures varies considerably as a function of the control parameters, so that it is not useful to use a single aspect ratio. Instead, the aspect ratio was adjusted for each to fit at least 8 columnar vortices along both the and directions at the onset of convection, and kept constant as and were varied.

Figure 1: as a function of for (red symbols and continuous line) and (blue symbols and dot dashed line), and (diamonds), (squares), (triangles) and (stars). Zero rotation is indicated by circles and the power law fits have an exponent of .

Fig. 1 shows as a function of for various and two different . The case of zero rotation is included for comparison. The basic features visible in this figure are known from previous experiments and simulations Rossby (1969); Zhong et al. (1993); Julien et al. (1996); King et al. (2009). The onset of convection is delayed by rotation. After onset, rises more steeply as a function of than in the non-rotating case. does not follow any simple power in this range of . For large enough , the dependence asymptotes towards the dependence valid for zero rotation, which is well approximated by a power law in the investigated range of .

Figure 2: as a function of for the same data and with the same symbols as in fig. 1. The dashed lines are power laws with exponents 2 and 2/3. The two vertical lines indicate the interval outside of which the fit to one of the two power laws is considered satisfactory.

All the different curves in fig. 1 collapse to a single curve in most of the parameter range when is plotted as a function of as shown in fig. 2. For large values of one finds or . This law is independent of as it should be: At any fixed and , the limit of large corresponds to the situation in which the nonlinear term dominates the Coriolis term, so that one has to recover the behavior of non-rotating convection, which is of course independent of . The data for zero rotation cannot be included in fig. 2 because has no finite value in this case, but is also found for strictly zero rotation.

Low values of on the other hand correspond to laminar flows near the onset of convection. Forming the dot product of eq. (1) and , integrating over the whole volume and averaging over time, one finds

(6)

where is the adimensional average dissipation rate of kinetic energy. In a laminar flow, one expects , where is a characteristic length scale of the flow. For , convection starts at a critical Rayleigh number obeying and forms stationary cells of size with Chandradekhar (1961). Eq. (6) becomes . Close to onset, and therefore . This corresponds to the left asymptote in fig. 2. Both the left asymptote and become straight lines in a logarithmic plot of vs. , which explains the simple appearance of fig. 2.

Fig. 2 in summary identifies three regimes of rotating convection. Rotating laminar flow characterizes one of them, and heat transport behaves the same as in non-rotating convection in another. The transition occurs where the two asymptotes in fig. 2 cross, i.e. at . There is a transition interval around this point of about one decade in width in which is close to neither asymptote. This third regime will receive detailed attention below.

Even though behaves as if there was no rotation for in fig. 2, visualizations of the flow still reveal differences. In the rotating case, the flow forms columnar vortices extending from one boundary to the other, whereas for zero rotation, plumes advected by a large scale circulation are observed. Enough visualizations of vortices in rotating convection have already appeared Julien et al. (1996); Stellmach and Hansen (2004); King et al. (2009) so that there is no need to reproduce any here. The size of the vortices can be quantified by the method already used in Hartlep et al. (2003): Compute the time averaged advective heat transport through the plane , , with , compute the Fourier transform of , and plot the spectrum of as a function of wavelength (see Hartlep et al. (2003) for detailed formulas). The median wavelength is extracted from the spectra, such that the heat advected at wavelengths smaller than equals the heat advected at larger wavelengths. The value of matches the diameter of the columnar vortices identified visually in the flow field. Fig. 3 shows as a function of . It is seen that stays at the onset wavelength well into the transition interval and decreases at high . This decrease follows at fixed , which is compatible with experimental data in Vorobieff and Ecke (1998).

Figure 3: as a function of . The symbols have the same meaning as in fig. 1. The dashed lines indicate power laws with exponents 0 and -1/2.

Near the onset of convection, the heat transport is determined by a balance between buoyancy, Coriolis and diffusive terms. For high , the Coriolis term is overwhelmed by the nonlinear term in eq. (1) so that is the same as in turbulent, non-rotating convection. Diffusive processes play a role because all heat has to cross the thermal boundary layers diffusively. Let us assume as a working hypothesis that the heat flow in the intermediate regime is governed by a competition between the nonlinear and Coriolis terms, and that the constraints imposed by rotation on the flow structure control the heat flux, not diffusion in the boundary layers. The dimensional heat flow must then be given by an expression independent of and . In order to check this hypothesis, it is convenient to use a control parameter independent of and . The only combination of , and meeting this requirement is . An appropriate measure of heat flux independent of and is , in which stands for the density and for the heat capacity.

It is useful to replace by the flux Rayleigh number given by . This combination is strictly speaking a control parameter only when Neumann conditions are imposed on the temperature field, which was not the case in our simulations. However, a parameter based on instead of is preferable in astrophysical applications because heat fluxes are better constrained by observations than vertical temperature differences. We will therefore seek a relation between and . Furthermore, in a flow dominated by rotation, which is necessarily nearly two dimensional, it seems plausible that heat flow through a plane should be determined solely by the dynamics in that plane. would then be independent of the layer height . If our working hypothesis is correct that is independent of and , and assuming is given by a power law, one has to find a scaling of the form . If in addition is independent of , one has to find .

Figure 4: as a function of . The symbols have the same meaning as in fig. 1. This figure contains only those data points which lie in the interval marked by vertical lines in fig. 2.

Fig. 4 shows as a function of . The figure contains only those points for which . This transition interval is small and does not corroborate any power law at fixed and . However, the data for different and collectively define an envelope which we regard to be the genuine scaling obeyed by the Nusselt number in the transition regime. The best fit to the data in fig. 4 yields

(7)

The exponent is measurably different from . There is some scatter in the points in fig. 4 around the power law (7). This scatter can be reduced by retaining data from a smaller interval of , so that the data are less affected by scalings valid in the neighboring intervals.

Ref. Christensen (2002) investigates thermal convection in a rotating spherical shell. In this geometry, convection occurs mostly outside a cylinder tangent to the inner core and coaxial with the rotation axis, whereas the flow velocities are much smaller inside the tangent cylinder. Gravitational acceleration varies radially in the simulations in ref. Christensen (2002) and there is a zonal flow along circles of constant latitude which has no analog in our simulations. Despite all these differences, the heat flux in the spherical geometry obeys according to ref. Christensen (2002) and the best fit to a compilation of data in ref. Aurnou (2007) yields . The exponent in (7) appears to be very robust.

It is also interesting to draw a parallel with dimensional arguments for non-rotating convection Spiegel (1971). If the heat transfer is independent of the layer thickness because it is determined by boundary layer dynamics, has to behave like . This exponent is generally not observed experimentally because of the presence of a large scale circulation. The assumption that heat transport is independent of thermal diffusivity and kinematic viscosity leads without rotation to . While this scaling has been found in simulations avoiding boundary layers Lohse and Toschi (2003) it remains elusive in any bounded geometry. In rotating convection, fig. 4 shows that a power law independent of diffusivities is a useful fit to the data, but the heat flow still depends on the layer depth.

In summary, three different regimes of convection could be identified as a function of . For small and large values of , one approaches asymptotically the scalings valid for rotating convection near onset and non-rotating convection, respectively. The cross-over occurs in a transition interval around . This contradicts the naive expectation that the transition should occur when the Rossby number equals 1. Even though is not a control parameter, the transition criterion is useful when observations yield some information about the flow velocities in a celestial body. A case in point is the Earth’s core, for which magnetic secular variations provide us with estimates of typical flow velocities around . Together with and the generally accepted material properties inside the core of and Schubert (2007, volume 8, tables on pp. 140 and 191), one finds , which places the Earth’s core inside the transition interval. If on the other hand the Earth’s core is driven by compositional convection, a diffusivity of should be used Schubert (2007, volume 8, tables on pp. 140 and 191), leading to .

Acknowledgements.
This work was supported by the Deutsche Forschungsgemeinschaft (DFG).

Footnotes

  1. preprint: APS/123-QED

References

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  11. S. Chandradekhar, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford, 1961).
  12. P. Vorobieff and R. E. Ecke, Physica D 123, 153 (1998).
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