Core flows and heat transfer induced by inhomogeneous cooling with sub- and supercritical convection

Abstract

a b s t r a c tThe amount and spatial pattern of heat extracted from cores of terrestrial planets is ultimately controlled by the thermal structure of the lower rocky mantle. Using the most common model to tackle this problem, a rapidly rotating and differentially cooled spherical shell containing an incompressible and viscous liquid is numerically investigated. To gain the physical basics, we consider a simple, equatorial symmetric perturbation of the CMB heat flux shaped as a spherical harmonic Y 11 . The thermodynamic properties of the induced flows mainly depend on the degree of nonlinearity parametrised by a horizontal Rayleigh number Ra h ¼ q à Ra, where q à is the relative CMB heat flux anomaly amplitude and Ra is the Rayleigh number which controls radial buoyancy-driven convection. Depending on Ra h we identify and characterise three distinctive flow regimes through their spatial patterns, heat transport and flow speed scalings: in the linear conductive regime the radial inward flow is found to be phase shifted 90°eastwards from the maximal heat flux as predicted by a linear quasi-geostrophic model for rapidly rotating spherical systems. The advective regime is characterised by an increased Ra h where nonlinearities become significant, but is still subcritical to radial convection. There the upwelling is dispersed and the downwelling is compressed by the thermal advection into a spiralling jet-like structure. As Ra h becomes large enough for the radial convection to set in, the jet remains identifiable on time-average and significantly alters the global heat budget in the convective regime. Our results suggest, that the boundary forcing not only introduces a net horizontal heat transport but also suppresses the convection locally to such an extent, that the net Nusselt number is reduced by up to 50%, even though the mean CMB heat flux is conserved. This also implies that a planetary core will remain hotter under a non-homogeneous CMB heat flux and is less well mixed. A broad numerical parameter investigation regarding Rayleigh number and the relative heat flux anomaly further fosters these results. Crown

Regional stratification at the top of Earth's core due to core–mantle boundary heat flux variations

“…The top of the core will also be strongly influenced by thermal heterogeneity in the lowermost mantle, which is much stronger than in the core (o{10 2 K}) and evolves much more slowly, such that the mantle imposes a laterally varying pattern of heat flux across the core-mantle boundary (CMB) 24 . Estimates of the lateral variations in CMB heat flux [25][26][27] are sufficiently large that significant regional variations in core dynamics are expected 16,[28][29][30][31] . Previous models 16,[32][33][34] have considered the interaction between CMB heterogeneity and stratification at the top of the core and the extent to which such heterogeneity can drive flows that penetrate and possibly disrupt a global stratified layer 24,35 .…”

Heat transfer in rapidly rotating convection with heterogeneous thermal boundary conditions

“…The previous numerical work, conducted predominantly at Ra a few times critical, has shown that imposed boundary heterogeneity can have a substantial influence on the spatial patterns and time-dependence of convection in the fluid shell, particularly when the wavelength of the imposed pattern is large. However, to our knowledge, the only previous study that compared heat transfer behaviour in rotating spherical shells with and without heterogeneous outer boundary forcing was that of Dietrich et al (2016) who carried out simulations of internally heated rotating spherical shell convection with a Y 1 1 outer boundary heat flux pattern at Pr = 1 and E = 2.5, 5 × 10 −5 (using the definition of E in §2.1) and flux-based Rayleigh numbers up to 400 times critical for E = 5 × 10 −5 . Estimating ∆T based on the maximum and minimum values of T in the domain Dietrich et al (2016) found that Nu is reduced by up to 50% from the homogeneous case as the amplitude of heterogeneity is increased up to q ⋆ L = 2 (using our definition 1.1).…”

“…However, to our knowledge, the only previous study that compared heat transfer behaviour in rotating spherical shells with and without heterogeneous outer boundary forcing was that of Dietrich et al (2016) who carried out simulations of internally heated rotating spherical shell convection with a Y 1 1 outer boundary heat flux pattern at Pr = 1 and E = 2.5, 5 × 10 −5 (using the definition of E in §2.1) and flux-based Rayleigh numbers up to 400 times critical for E = 5 × 10 −5 . Estimating ∆T based on the maximum and minimum values of T in the domain Dietrich et al (2016) found that Nu is reduced by up to 50% from the homogeneous case as the amplitude of heterogeneity is increased up to q ⋆ L = 2 (using our definition 1.1). Here we present results from 106 numerical simulations of bottom-heated, rotating convection in a spherical shell with values of E as low as 10 −6 .…”

“…We find that addition of outer boundary heterogeneity tends to increase Nu relative to equivalent q ⋆ L = 0 cases, whereas Dietrich et al (2016) found a nearly linear reduction in Nu with increasing Hq ⋆ L for simulations with Ra/Ra C = 25, E = 10 −4 and a range of 0 Hq ⋆ L 2.0 (see figures 11 and 13 of their paper and note that their definition of q ⋆ is equal to one half our Hq ⋆ L ). This difference arises from the fact that we use ∆T in determining Nu T , whereas Dietrich et al (2016) estimate ∆T = T max −T min , where T max and T min are the maximum and minimum values of T in the domain (Dietrich, personal communication). When there are large lateral variations in boundary heat flux, and hence fluid temperature, the point-wise maximum approach can significantly overestimate the average temperature drop across the system and thus underestimate Nu T .…”

Thermal convection in rotating spherical shells: Temperature-dependent internal heat generation using the example of triple- α burning in neutron stars

“…Boussinesq and anelastic thermal convection in rotating spherical shells with differential heating mechanisms have been studied in considerable detail over the past decades [31][32][33][34] (among many others), but in the context of planetary cores and dynamos. The secular cooling of planetary cores is modelled by internal buoyancy sources and the heat flux is assumed to be nonuniform in the lateral direction of the outer boundary, to mimic the thermal structure of the lower rocky mantle.…”