Convective heat transfer in planetary dynamo models

Abstract

[1] The magnetic fields of planets and stars are generated by the motions of electrically conducting fluids within them. These fluid motions are thought to be driven by convective processes, as internal heat is transported outward. The efficiency with which heat is transferred by convection is integral in understanding dynamo processes. Several heat transfer scaling laws have been proposed, but the range of parameter space to which they apply has not been firmly established. ) to show that heat transfer (Nu) in spherical dynamo simulations occurs in two distinct regimes. We argue that heat transfer scales as Nu ∼ Ra 6/5 in the rapidly rotating regime and Nu ∼ Ra 2/7 in the weakly rotating regime. The transition between these two regimes is controlled by the competition between the thermal and viscous boundary layers. Boundary layer scaling theory allows us to predict that the transition between the regimes occurs at a transitional Rayleigh number, Ra t = E −7/4 . Furthermore, boundary layer control of heat transfer is shown to relate to the interior temperature profiles of the models. In the weakly rotating regime, the interior fluid is nearly adiabatic. In the rapidly rotating regime, adverse mean temperature gradients abide, irrespective of the Reynolds number (Re). Extrapolating our results to Earth's core, we estimate that core convection resides in the rapidly rotating regime, with Ra ≈ 2 × 10 24 (Ra/Ra t ≈ 0.02), corresponding to a superadiabatic density variation of Dr/r o ≈ 10 −7 , which is significantly below the sensitivity of present seismic observations.

Laboratory-numerical models of rapidly rotating convection in planetary cores

“…However, this effect is strong only at relatively high E. In fact, our data shows that the overshoot becomes small for E 10 −5 . The slope of the steep scaling regime changes as a function of E. At the highest E values (E 10 −3 ), the data conform to a β 6/5 law (Christensen 2002;Aurnou 2007;King et al 2009King et al , 2010Schmitz & Tilgner 2009). At lower E values in the vicinity of 10 −5 , the data fit a steeper, roughly cubic scaling law in agreement with .…”

“…The best-fitting heat transfer trend of Nu (Ra/Ra C ) 3.6 is plotted for E ∼ 10 −7 . For comparison, Nu = (Ra/Ra C ) 3 ) is plotted for E ∼ 10 −5 and Nu = (Ra/Ra C ) 6/5 (King et al 2009(King et al , 2010 for E ∼ 10 −3 . Note that with each study at lower E, the scaling exponent becomes larger.…”

Turbulent convection in liquid metal with and without rotation

“…Numerical simulations of planetary dynamo action by rotating convection also concentrate on fluids with unit order Pr (16). Scaling analysis of heat transfer in such simulations agrees quantitatively with rotating convection experiments in water (17). Although it is often assumed that planetary cores can be modeled as Pr = Oð1Þ fluids (e.g., ref.…”

Stochastic generation of MAC waves and implications for convection in Earth’s core

“…The resulting proportionalities from (55) to (57) are then used to extrapolate to Earth-like conditions. A representative state for the core is defined by settingV rms = 0.38 mm s −1 (Holme 2015), B rms = 2.5 to 4 mT (Buffett 2010;Gillet et al 2010) and T = 10 mK (King et al 2010). These values correspond to dimensionless parameters Ro = 2 × 10 −6 , = 10 to 20 and Ra * = 8 × 10 −5 .…”