The connection between the heat transfer and characteristic flow velocities of planetary core-style convection remains poorly understood. To address this, we present novel laboratory models of rotating Rayleigh–Bénard convection in which heat and momentum transfer are simultaneously measured. Using water (Prandtl number, Pr≃6) and cylindrical containers of diameter-to-height aspect ratios of Γ≃3,1.5,0.75, the non-dimensional rotation period (Ekman number, E) is varied between 10−7≲E≲3×10−5 and the non-dimensional convective forcing (Rayleigh number, Ra) ranges from 107≲Ra≲1012. Our heat transfer data agree with those of previous studies and are largely controlled by boundary layer dynamics. We utilize laser Doppler velocimetry (LDV) to obtain experimental point measurements of bulk axial velocities, resulting in estimates of the non-dimensional momentum transfer (Reynolds number, Re) with values between 4×102≲Re≲5×104. Behavioral transitions in the velocity data do not exist where transitions in heat transfer behaviors occur, indicating that bulk dynamics are not controlled by the boundary layers of the system. Instead, the LDV data agree well with the diffusion-free Coriolis–Inertia–Archimedian (CIA) scaling over the range of Ra explored. Furthermore, the CIA scaling approximately co-scales with the Viscous–Archimedian–Coriolis (VAC) scaling over the parameter space studied. We explain this observation by demonstrating that the VAC and CIA relations will co-scale when the local Reynolds number in the fluid bulk is of order unity. We conclude that in our experiments and similar laboratory and numerical investigations with E≳10−7, Ra≲1012, Pr≃7, heat transfer is controlled by boundary layer physics while quasi-geostrophically turbulent dynamics relevant to core flows robustly exist in the fluid bulk.
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The effect of rotation on double diffusive convection: perspectives from linear stability analysis
Abstract Diffusive convection can occur when two constituents of a stratified fluid have opposing effects on its stratification and different molecular diffusivities. This form of convection arises for the particular temperature and salinity stratification in the Arctic Ocean and is relevant to heat fluxes. Previous studies have suggested that planetary rotation may influence diffusive-convective heat fluxes, although the precise physical mechanisms and regime of rotational influence are not well understood. A linear stability analysis of a temperature and salinity interface bounded by two mixed layers is performed here to understand the stability properties of a diffusive-convective system, and in particular the transition from non-rotating to rotationally-controlled heat transfer. Rotation is shown to stabilize diffusive convection by increasing the critical Rayleigh number to initiate instability. In the rotationally-controlled regime, a −4/3 power law is found between the critical Rayleigh number and the Ekman number, similar to the scaling for rotating thermal convection. The transition from non-rotating to rotationally-controlled convection, and associated drop in heat fluxes, is predicted to occur when the thermal interfacial thickness exceeds about 4 times the Ekman layer thickness. A vorticity budget analysis indicates how baroclinic vorticity production is counteracted by the tilting of planetary vorticity by vertical shear, which accounts for the stabilization effect of rotation. Finally, direct numerical simulations yield generally good agreement with the linear stability analysis. This study, therefore, provides a theoretical framework for classifying regimes of rotationally-controlled diffusive-convective heat fluxes, such as may arise in some regions of the Arctic Ocean.
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- Award ID(s):
- 1950077
- NSF-PAR ID:
- 10327115
- Date Published:
- Journal Name:
- Journal of Physical Oceanography
- ISSN:
- 0022-3670
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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SUMMARY We present investigations of rapidly rotating convection in a thick spherical shell geometry relevant to planetary cores, comparing results from quasi-geostrophic (QG), 3-D and hybrid QG-3D models. The 170 reported calculations span Ekman numbers, Ek, between 10−4 and 10−10, Rayleigh numbers, Ra, between 2 and 150 times supercritical and Prandtl numbers, Pr, between 10 and 10−2. The default boundary conditions are no-slip at both the ICB and the CMB for the velocity field, with fixed temperatures at the ICB and the CMB. Cases driven by both homogeneous and inhomogeneous CMB heat flux patterns are also explored, the latter including lateral variations, as measured by Q*, the peak-to-peak amplitude of the pattern divided by its mean, taking values up to 5. The QG model is based on the open-source pizza code. We extend this in a hybrid approach to include the temperature field on a 3-D grid. In general, we find convection is dominated by zonal jets at mid-depths in the shell, with thermal Rossby waves prominent close to the outer boundary when the driving is weaker. For the thick spherical shell geometry studied here the hybrid method is best suited for studying convection at modest forcing, $Ra \le 10 \, Ra_c$ when Pr = 1, and departs from the 3-D model results at higher Ra, displaying systematically lower heat transport characterized by lower Nusselt and Reynolds numbers. We find that the lack of equatorially-antisymmetric motions and z-correlations between temperature and velocity in the buoyancy force contributes to the weaker flows in the hybrid formulation. On the other hand, the QG models yield broadly similar results to the 3-D models, for the specific aspect ratio and range of Rayleigh numbers explored here. We cannot point to major disagreements between these two data sets at Pr ≥ 0.1, with the QG model effectively more strongly driven than the hybrid case due to its cylindrically averaged thermal boundary conditions. When Pr is decreased, the range of agreement between the hybrid and 3-D models expands, for example up to $Ra \le 15 \, Ra_c$ at Pr = 0.1, indicating the hybrid method may be better suited to study convection in the low Pr regime. We thus observe a transition between two regimes: (i) at Pr ≥ 0.1 the QG and 3-D models agree in the studied range of Ra/Rac while the hybrid model fails when $Ra\gt 15\, Ra_c$ and (ii) at Pr = 0.01 the QG and 3-D models disagree for $Ra\gt 10\, Ra_c$ while the hybrid and 3-D models agree fairly well up to $Ra \sim 20\, Ra_c$. Models that include laterally varying heat flux at the outer boundary reproduce regional convection patterns that compare well with those found in similarly forced 3-D models. Previously proposed scaling laws for rapidly rotating convection are tested; our simulations are overall well described by a triple balance between Coriolis, inertia and Archimedean forces with the length-scale of the convection following the diffusion-free Rhines-scaling. The magnitude of Pr affects the number and the size of the jets with larger structures obtained at lower Pr. Higher velocities and lower heat transport are seen on decreasing Pr with the scaling behaviour of the convective velocity displaying a strong dependence on Pr. This study is an intermediate step towards a hybrid model of core convection also including 3-D magnetic effects.
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Abstract Double‐diffusive stratification in the ocean is characterized by staircase structures consisting of mixed layers separated by high‐gradient interfaces in temperature and salinity. These double‐diffusive layers, which flux heat vertically, are observed over a vast region of the Arctic Ocean at the top boundary of the relatively warm and salty Atlantic water layer. In one formalism for the origin of double‐diffusive layers, staircase formation arises when a heat source is applied at the base of water that is stably stratified in salinity. This framework is extended to consider the effect of intermittent shear‐driven turbulence on diffusive‐convective staircase formation. One‐dimensional numerical model results indicate that there is a critical level of intermittent turbulence above which a staircase cannot form. This is framed in terms of a critical diffusivity ratio (ratio of effective salinity diffusivity to effective thermal diffusivity) that cannot be exceeded for a staircase to persist. This critical ratio is not a universal constant but rather differs for each staircase. Model results further indicate that layer thicknesses decrease with height in a staircase, with the variation in thickness over a staircase being more pronounced in the presence of intermittent turbulence. Finally, results suggest that increased diffusivity ratios lead to decreased heat fluxes across interfaces; if a staircase is subject to intermittent turbulence levels (below the critical level), vertical heat fluxes will be smaller than in the absence of shear‐driven turbulence. Findings are related to double‐diffusive staircases, and associated heat fluxes, in the weakly turbulent Arctic Ocean.
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Abstract The mechanisms that control the export of freshwater from the East Greenland Current, in both liquid and solid form, are explored using an idealized numerical model and scaling theory. A regional, coupled ocean–sea ice model is applied to a series of calculations in which key parameters are varied and the scaling theory is used to interpret the model results. The offshore ice flux, occurring in late winter, is driven primarily by internal stresses and is most sensitive to the thickness of sea ice on the shelf coming out of Fram Strait and the strength of alongshore winds over the shelf. The offshore liquid freshwater flux is achieved by eddy fluxes in late summer while there is an onshore liquid freshwater flux in winter due to the ice–ocean stress, resulting in only weak annual mean flux. The scaling theory identifies the key nondimensional parameters that control the behavior and reproduces the general parameter dependence found in the numerical model. Climate models predict that winds will increase and ice export from the Arctic will decrease in the future, both of which will lead to a decrease in the offshore flux of sea ice, while the influence on liquid freshwater may increase or decrease, depending on the relative changes in the onshore Ekman transport and offshore eddy fluxes. Additional processes that have not been considered here, such as more complex topography and synoptic wind events, may also contribute to cross-shelf exchange.
Significance Statement The purpose of this study is to provide a basic understanding of what controls the flux of sea ice and low-salinity water from the East Greenland shelf into the interior of the Greenland and Iceland Seas. This is a potentially important process since it has been shown that sufficient freshening of the surface waters in the interior of the Nordic seas can inhibit deep convection and the associated air–sea heat flux and water mass transformation. A combination of idealized computer models and basic theory indicates that the fluxes of liquid and solid freshwater are controlled by different mechanisms and occur at different times of the year. Accurate representation in climate models will require representation of small-scale processes such as mesoscale eddies and gradients of ice thickness across the shelf.
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Dynamos driven by rotating convection in the plane layer geometry are investigated numerically for a range of Ekman number ( $E$ ), magnetic Prandtl number ( $Pm$ ) and Rayleigh number ( $Ra$ ). The primary purpose of the investigation is to compare results of the simulations with previously developed asymptotic theory that is applicable in the limit of rapid rotation. We find that all of the simulations are in the quasi-geostrophic regime in which the Coriolis and pressure gradient forces are approximately balanced at leading order, whereas all other forces, including the Lorentz force, act as perturbations. Agreement between simulation output and asymptotic scalings for the energetics, flow speeds, magnetic field amplitude and length scales is found. The transition from large-scale dynamos to small-scale dynamos is well described by the magnetic Reynolds number based on the small convective length scale, $\widetilde {Rm}$ , with large-scale dynamos preferred when $\widetilde {Rm} \lesssim O(1)$ . The magnitude of the large-scale magnetic field is observed to saturate and become approximately constant with increasing Rayleigh number. Energy spectra show that all length scales present in the flow field and the small-scale magnetic field are consistent with a scaling of $E^{1/3}$ , even in the turbulent regime. For a fixed value of $E$ , we find that the viscous dissipation length scale is approximately constant over a broad range of $Ra$ ; the ohmic dissipation length scale is approximately constant within the large-scale dynamo regime, but transitions to a $\widetilde {Rm}^{-1/2}$ scaling in the small-scale dynamo regime.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1175/JPO-D-21-0060.1
