Dynamo action in rapidly rotating Rayleigh–Bénard convection at infinite Prandtl number

In order better to understand the processes that lead to the generation of magnetic fields of finite amplitude, we study dynamo action driven by turbulent Boussinesq convection in a rapidly rotating system. In the limit of infinite Prandtl number (the ratio of viscous to thermal diffusion) the inertia term drops out of the momentum equation, which becomes linear in the velocity. This simplification allows a decomposition of the velocity into a thermal part driven by buoyancy, and a magnetic part driven by the Lorentz force. While the former velocity defines the kinematic dynamo problem responsible for the exponential growth of the magnetic field, the latter encodes the magnetic back reaction that leads to the eventual nonlinear saturation of the dynamo. We argue that two different types of solution should exist: weak solutions in which the saturated velocity remains close to the kinematic one, and strong solutions in which magnetic forces drive the system into a new strongly magnetised state that is radically different from the kinematic one. Indeed, we find both types of solutions numerically. Interestingly, we also find that, in our inertialess system, both types of solutions exist on the same subcritical branch of solutions bifurcating from the non-magnetic convective state, in contrast with the more traditional situation for systems with finite inertia in which weak and strong solutions are thought to exist on different branches. We find that for weak solutions, the force balance is the same as in the non-magnetic case, with the horizontal size of the convection varying as the one-third power of the Ekman number (the ratio of viscous to Coriolis forces), which gives rise to very small cells at small Ekman numbers (i.e. high rotation rates). In the strong solutions, magnetic forces become important and the convection develops on much larger horizontal scales. However, we note that even in the strong cases the solutions never properly satisfy Taylor’s constraint, and that viscous stresses continue to play a role. Finally, we discuss the relevance of our findings to the study of planetary dynamos in rapidly rotating systems such as the Earth

Jets and large-scale vortices in rotating Rayleigh-Bénard convection

One of the most prominent dynamical features of turbulent, rapidly rotating convection is the formation of large-scale coherent structures, driven by Reynolds stresses resulting from the small-scale convective flows. In spherical geometry, such structures consist of intense zonal flows that are invariant along the rotation axis. In planar geometry, long-lived, depth-invariant structures also form at large scales, but, in the absence of horizontal anisotropy, they consist of vortices that grow to the domain size. In this work, through the introduction of horizontal anisotropy into a numerical model of planar rotating convection by the adoption of unequal horizontal box sizes (i.e., Lx≤Ly, where the xy plane is horizontal), we investigate whether unidirectional flows and large-scale vortices can coexist. We find that only a small degree of anisotropy is required to bring about a transition from dynamics dominated by persistent large-scale vortices to dynamics dominated by persistent unidirectional flows parallel to the shortest horizontal direction. When the anisotropy is sufficiently large, the unidirectional flow consists of multiple jets, generated on a time scale smaller than a global viscous time scale, thus signifying that the upscale energy transfer does not spontaneously feed the largest available mode in the system. That said, the multiple jets merge on much longer time scales. Large-scale vortices of size comparable with Lx systematically form in the flanks of the jets and can be persistent or intermittent. This indicates that large-scale vortices, either coexisting with jets or not, are a robust dynamical feature of planar rotating convection

The convective instability of a Maxwell-Cattaneo fluid in the presence of a vertical magnetic field

We study the instability of a Bénard layer subject to a vertical uniform magnetic field, in which the fluid obeys the Maxwell–Cattaneo (MC) heat flux–temperature relation. We extend the work of Bissell (Proc. R. Soc. A 472, 20160649 (doi:10.1098/rspa.2016.0649)) to non-zero values of the magnetic Prandtl number pm. With non-zero pm, the order of the dispersion relation is increased, leading to considerably richer behaviour. An asymptotic analysis at large values of the Chandrasekhar number Q confirms that the MC effect becomes important when C Q1/2 is O(1), where C is the MC number. In this regime, we derive a scaled system that is independent of Q. When CQ1/2 is large, the results are consistent with those derived from the governing equations in the limit of Prandtl number p → ∞ with pm finite; here we identify a new mode of instability, which is due neither to inertial nor induction effects. In the large pm regime, we show how a transition can occur between oscillatory modes of different horizontal scale. For Q ≫ 1 and small values of p, we show that the critical Rayleigh number is non-monotonic in p provided that C > 1/6. While the analysis of this paper is performed for stress-free boundaries, it can be shown that other types of mechanical boundary conditions give the same leading-order results

Onset of interchange instability in a coupled core–SOL plasma

The dynamics at the edge of fusion confinement devices is driven by interchange instabilities and involves the motion of plasma across two regions—the “core region” and the scrape-off layer (SOL)—distinguished by whether field lines are, respectively, closed or connected to the wall. Motivated by this phenomenon, we present an extensive linear stability analysis of a two-layer plasma model encompassing the coupled interactions between the region with closed field lines and the SOL. We focus on the effect of varying the particle diffusivity and ion viscosity, revealing the significant variation in the spatial structure of the critical modes. In addition, we have investigated the dependence of the stability threshold on the ratio of the width of the region with closed field lines to that of the SOL; this dependence is strong when the ratio is sufficiently small, but becomes insignificant once the ratio is of order unity

Scale Selection in the Stratified Convection of the Solar Photosphere

We examine the role of stratification in determining the length scales of turbulent anelastic convection. Motivated by the range of scales observed in convection at the solar photosphere, we perform local numerical simulations of convection for a range of density contrasts in large domains, analyzing both the Eulerian and Lagrangian statistics of the flow. We consider the two cases of constant dynamic viscosity and constant kinematic viscosity. We discuss the implications of our results to the issue of solar mesogranulation

Stability of two-layer miscible convection

The dynamics at the edge of fusion confinement devices involves the motion of plasma across two regions—the core and the scrape-off layer—that may have very different properties. Motivated by this problem, and exploiting the analogy between the equations governing plasma interchange dynamics and those of classical Rayleigh-Bénard convection, we consider the linear stability of two-dimensional, two-layer miscible convection. We focus specifically on the influence of three particular parameters: the ratio of the viscosities in the two layers, the ratio of the thermal diffusivities, and the ratio of the depths of the two layers. The key result is that, depending on the parameters of the problem, the most unstable mode can take one of three quite distinct forms: whole layer solutions, in which the eigenfunctions of the stream function and temperature extend over both layers of fluid; localized solutions, with the velocity cells or the temperature perturbation (or both) confined to just one of the layers; and segregated solutions, in which the fluid motion and temperature perturbation are confined to different fluid layers

How was the Earth–Moon system formed? New insights from the geodynamo

The most widely accepted scenario for the formation of the Earth–Moon system involves a dramatic impact between the proto-Earth and some other cosmic body. Many features of the present-day Earth–Moon system provide constraints on the nature of this impact. Any model of the history of the Earth must account for the physical, geochemical, petrological, and dynamical evidence. These constraints notwithstanding, there are several radically different impact models that could in principle account for all the evidence. Thus, in the absence of further constraints, we may never know for sure how the Earth–Moon system was formed. Here, we put forward the idea that additional constraints are indeed provided by the fact that the Earth is strongly magnetized. It is universally accepted that the Earth’s magnetic field is maintained by a dynamo operating in the outer liquid core. However, because of the rapid rotation of the Earth, this dynamo has the peculiar property that it can maintain a strong field but cannot amplify a weak one. Therefore, the Earth must have been magnetized at a very early epoch, either preimpact or as a result of the impact itself. Either way, any realistic model of the formation of the Earth–Moon system must include magnetic field evolution. This requirement may ultimately constrain the models sufficiently to discriminate between the various candidates

Rapidly rotating Maxwell-Cattaneo convection

Motivated by astrophysical and geophysical applications, the classical problem of rotating Rayleigh-Bénard convection has been widely studied. Assuming a classical Fourier heat law, in which the heat flux is directly proportional to the temperature gradient, the evolution of temperature is governed by a parabolic advection-diffusion equation; this, in turn, implies an infinite speed of propagation of information. In reality, the system is rendered hyperbolic by extending the Fourier law to include an advective derivative of the flux—the Maxwell-Cattaneo (M-C) effect. Although the correction (measured by the parameter Γ , a nondimensional representation of the relaxation time) is nominally small, it represents a singular perturbation and hence can lead to significant effects when the rotation rate (measured by the Taylor number T ) is sufficiently high. In this paper, we investigate the linear stability of rotating convection, incorporating the M-C effect, concentrating on the regime of T ≫ 1 , Γ ≪ 1 . On increasing Γ for a fixed T ≫ 1 , the M-C effect first comes into play when Γ = O ( T − 1 / 3 ) . Here, as in the classical problem, the preferred mode can be either steady or oscillatory, depending on the value of the Prandtl number σ . For Γ > O ( T − 1 / 3 ) , the influence of the M-C effect is sufficiently strong that the onset of instability is always oscillatory, regardless of the value of σ . Within this regime, the dependence on σ of the critical Rayleigh number and of the scale of the preferred mode are explored through the analysis of specific distinguished limits

Incorporating velocity shear into the magneto-Boussinesq approximation

Motivated by consideration of the solar tachocline, we derive, via an asymptotic procedure, a new set of equations incorporating velocity shear and magnetic buoyancy into the Boussinesq approximation. We demonstrate, by increasing the magnetic field scale height, how these equations are linked to the magneto-Boussinesq equations of Spiegel and Weiss (Magnetic buoyancy and the Boussinesq approximation. Geophys. Astrophys. Fluid Dyn. 1982, 22, 219-234)