Self-consistent single mode investigations of the quasi-geostrophic convection-driven dynamo model

Abstract

The quasi-geostrophic dynamo model (QGDM) is a multiscale, fully nonlinear Cartesian dynamo model that is valid in the asymptotic limit of low Rossby number. In the additional limit of small magnetic Prandtl number investigated here, the QGDM is a self-consistent, asymptotically exact form of an $\unicode[STIX]{x1D6FC}^{2}$ large-scale dynamo. This article explores methods for simulating the multiscale QGDM and investigates how convection is altered by the magnetic field in the planetary regime of small Rossby number and small magnetic Prandtl number. At present, this combination is beyond the reach of direct numerical simulations. We use a simplified class of solutions whose horizontal structure is restricted to a periodic hexagonal lattice characterized by a single horizontal wavenumber (single mode). In contrast with previous kinematic investigations of the QGDM, the Lorentz force is included to study saturated, self-consistent dynamos. Two methodologies are used to assess handling of the multiple time scales of the QGDM: a stiff, common-in-time approach where all time scales are converted to a single time variable and a heterogeneous multiscale modelling approach employing fast time averaging on the Reynolds, magnetic and buoyancy eddy fluxes that feed back onto the slow scales. These strategies produce consistent results and each illustrates self-similar dynamics as the time-averaging window is increased. The properties of the convection are significantly altered by the dynamo-generated magnetic field. All solutions show a decrease in the overall heat transfer efficiency as compared to non-magnetic convection, suggesting that a change in length scale or flow planform plays a critical role in the enhanced heat transfer efficiency observed in previous dynamo studies. All dynamo solutions show a trend of increasing ohmic dissipation relative to viscous dissipation as the buoyancy forcing is increased.

On the inverse cascade and flow speed scaling behaviour in rapidly rotating Rayleigh–Bénard convection

“…The barotropic, time-dependent kinetic energy density is defined as follows: where indicates an average over the small, horizontal spatial scales, consistent with the notation employed in Plumley et al. (2018). In Fourier space, the barotropic kinetic energy equation is derived by multiplying the Fourier representation of (2.9) by the complex conjugate of , the spectral representation of , and integrating over physical space to obtain where the box-normalised horizontal wavenumber vector is , and .…”

“…The mean temperatureΘ evolves on the slow time scale τ = Ek 2/3 t associated with the vertical diffusion time H 2 /ν. However, Julien, Knobloch & Werne (1998) and Plumley et al (2018) found that spatial averaging over a sufficient number of convective elements on the small scales is sufficiently accurate to (i) omit fast-time averaging and (ii) assume a statistically stationary state where the slow evolution term ∂ τΘ that would appear in (2.4) is omitted.…”

The turbulent dynamo

“…2015; Calkins, Julien & Tobias 2017; Calkins 2018; Plumley et al. 2018 and the references therein) the asymptotic theory for modelling of quasi-geostropic dynamo models (QGDM) was developed.…”

“…These models have recently been extended into the nonlinear regime by Plumley et al. (2018).

Figure 10.Volumetric renderings of the small-scale vertical vorticity ( a , b ) and current density ( c , d ) illustrating two flow regimes observed in the reduced convection simulations, namely the plume regime , ( a , c ) and the geostrophic turbulence regime ( b , d ) , .

…”

Dynamo saturation down to vanishing viscosity: strong-field and inertial scaling regimes

“…(a) Some studies retain the full complexity of the convective dynamo problem, and aim at reproducing the multiple-branch picture conjectured by Roberts. These studies either consider the full set of convective MHD equations, or focus on precise asymptotic limits to derive reduced sets of equations that can be simulated at lower computational cost (Calkins et al 2015(Calkins et al , 2016Plumley et al 2018). In this quest for numerically tractable asymptotic regimes, another line of work focused on the rapid-rotation limit in otherwise viscous flows (Hughes & Cattaneo 2016;Cattaneo & Hughes 2017;Dormy 2016).…”