arXiv subjects

D. W. Hughes

Publications and source records attributed to D. W. Hughes.

14 recordsLinked to original sources

arXiv2025

Self-organisation through layering of $β$-plane like turbulence in plasmas and geophysical fluids

Staircase formation and layering is studied in simplified, potential vorticity conserving models of plasmas and geophysical fluids, by investigating turbulent self-organisation and nonlinear saturation with different mechanisms of free energy production -- forcing or linear instability -- and with standard or modified zonal flow responses. To this end, staircase formation in both the standard and modified Charney-Hasegawa-Mima equations with stochastic forcing, along with two different simple instability driven models -- one from a plasma and from a geophysical context -- are studied and compared. In these studies, it is observed that $β$-plane turbulence that does not distinguish between zonal and non-zonal perturbations (i.e., standard zonal response) gradually forms large-scale, elliptic zonal structures that merge progressively, regardless of whether it is driven by forcing (though it should be slow enough to allow wave couplings) or by the baroclinic instability, using for example a two-layer model. Conversely, the plasma system, with its modified zonal response, can rapidly form straight, stationary jets of well-defined size, again regardless of the way it is driven: by stochastic forcing or by the dissipative drift instability. Furthermore, the instability-driven plasma system exhibits a phase transition between a zonal flow dominated state and an eddy dominated state. In both states, saturation is possible without large-scale friction.

physics.plasm-ph↗

arXiv2020

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, 2016) to non-zero values of the magnetic Prandtl number $p_m$. With non-zero $p_m$, 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 $C Q^{1/2}$ is $O(1)$, where $C$ is the Maxwell-Cattaneo number. In this regime, we derive a scaled system that is independent of $Q$. When $CQ^{1/2}$ is large, the results are consistent with those derived from the governing equations in the limit of Prandtl number $p\to \infty$ with $p_m$ finite; here we identify a new mode of instability, which is due neither to inertial nor induction effects. In the large $p_m$ regime, we show how a transition can occur between oscillatory modes of different horizontal scale. For $Q \gg 1$ 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.

physics.flu-dyn↗

arXiv2018

Scale Selection in the Stratified Convection of the Solar Photosphere

We examine the role of stratification in determining the scale for 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. We analyse both the Eulerian and Lagrangian statistics of the convection and demonstrate that increasing the stratification shifts the scale of the most energetic structures in the flow to smaller scales; furthermore, the relative amplitude of vertical to horizontal flows in the convection decreases with increasing stratification. We discuss the implications of our results to the issue of solar mesogranulation.

astro-ph.SR↗

arXiv2016

Vortex disruption by magnetohydrodynamic feedback

In an electrically conducting fluid, vortices stretch out a weak, large-scale magnetic field to form strong current sheets on their edges. Associated with these current sheets are magnetic stresses, which are subsequently released through reconnection, leading to vortex disruption, and possibly even destruction. This disruption phenomenon is investigated here in the context of two-dimensional, homogeneous, incompressible magnetohydrodynamics. We derive a simple order of magnitude estimate for the magnetic stresses --- and thus the degree of disruption --- that depends on the strength of the background magnetic field (measured by the parameter $M$, a ratio between the Alfvén speed and a typical flow speed) and on the magnetic diffusivity (measured by the magnetic Reynolds number $\mbox{Rm}$). The resulting estimate suggests that significant disruption occurs when $M^{2}\mbox{Rm} = O(1)$. To test our prediction, we analyse direct numerical simulations of vortices generated by the breakup of unstable shear flows with an initially weak background magnetic field. Using the Okubo--Weiss vortex coherence criterion, we introduce a vortex disruption measure, and show that it is consistent with our predicted scaling, for vortices generated by instabilities of both a shear layer and a jet.

physics.flu-dyn↗

arXiv2015

Shear instabilities in shallow-water magnetohydrodynamics

Within the framework of shallow-water magnetohydrodynamics, we investigate the linear instability of horizontal shear flows, influenced by an aligned magnetic field and stratification. Various classical instability results, such as Høiland's growth rate bound and Howard's semi-circle theorem, are extended to this shallow-water system for quite general profiles. Two specific piecewise-constant velocity profiles, the vortex sheet and the rectangular jet, are studied analytically and asymptotically; it is found that the magnetic field and stratification (as measured by the Froude number) are generally both stabilising, but weak instabilities can be found at arbitrarily large Froude number. Numerical solutions are computed for corresponding smooth velocity profiles, the hyperbolic-tangent shear layer and the Bickley jet, for a uniform background field. A generalisation of the long-wave asymptotic analysis of Drazin & Howard (1962) is employed in order to understand the instability characteristics for both profiles. For the shear layer, the mechanism underlying the primary instability is interpreted in terms of counter-propagating Rossby waves, thereby allowing an explication of the stabilising effects of the magnetic field and stratification.

physics.flu-dyn↗

arXiv2013

Short wavelength magnetic buoyancy instability

Magnetic buoyancy instability plays an important role in the evolution of astrophysical magnetic fields. Here we revisit the problem introduced by \citet{Gilman_1970} of the short wavelength linear stability of a plane layer of compressible isothermal fluid permeated by a horizontal magnetic field of strength decreasing with height. Dissipation of momentum and magnetic field is neglected. By the use of a Rayleigh-Schrödinger perturbation analysis, we explain in detail the limit in which the transverse horizontal wavenumber of the perturbation, denoted by $k$, is large (i.e.\ short horizontal wavelength) and show that the fastest growing perturbations become localized in the vertical direction as $k$ is increased. The growth rates are determined by a function of the vertical coordinate $z$ since, in the large $k$ limit, the eigenmodes are strongly localized in the vertical direction. We consider in detail the case of two-dimensional perturbations varying in the directions perpendicular to the magnetic field, which, for sufficiently strong field gradients, are the most unstable. The results of our analysis are backed up by comparison with a series of initial value problems. Finally we extend the analysis to three-dimensional perturbations.

astro-ph.SR↗

arXiv2012

The Effect of Velocity Shear on Dynamo Action Due to Rotating Convection

Recent numerical simulations of dynamo action resulting from rotating convection have revealed some serious problems in applying the standard picture of mean field electrodynamics at high values of the magnetic Reynolds number, and have thereby underlined the difficulties in large-scale magnetic field generation in this regime. Here we consider kinematic dynamo processes in a rotating convective layer of Boussinesq fluid with the additional influence of a large-scale horizontal velocity shear. Incorporating the shear flow enhances the dynamo growth rate and also leads to the generation of significant magnetic fields on large scales. By the technique of spectral filtering, we analyse the modes in the velocity that are principally responsible for dynamo action, and show that the magnetic field resulting from the full flow relies crucially on a range of scales in the velocity field. Filtering the flow to provide a true separation of scales between the shear and the convective flow also leads to dynamo action; however, the magnetic field in this case has a very different structure from that generated by the full velocity field. We also show that the nature of the dynamo action is broadly similar irrespective of whether the flow in the absence of shear can support dynamo action.

astro-ph.SR↗

arXiv2011

The alpha-effect in rotating convection: a comparison of numerical simulations

Numerical simulations are an important tool in furthering our understanding of turbulent dynamo action, a process that occurs in a vast range of astrophysical bodies. It is important in all computational work that comparisons are made between different codes and, if non-trivial differences arise, that these are explained. Kapyla et al (2010: MNRAS 402, 1458) describe an attempt to reproduce the results of Hughes & Proctor (2009: PRL 102, 044501) and, by employing a different methodology, they arrive at very different conclusions concerning the mean electromotive force and the generation of large-scale fields. Here we describe why the simulations of Kapyla et al (2010) are simply not suitable for a meaningful comparison, since they solve different equations, at different parameter values and with different boundary conditions. Furthermore we describe why the interpretation of Kapyla et al (2010) of the calculation of the alpha-effect is inappropriate and argue that the generation of large-scale magnetic fields by turbulent convection remains a problematic issue.

astro-ph.SR↗

arXiv2010

The mean electromotive force resulting from magnetic buoyancy instability

Motivated both by considerations of the generation of large-scale astrophysical magnetic fields and by potential problems with mean magnetic field generation by turbulent convection, we investigate the mean electromotive force (emf) resulting from the magnetic buoyancy instability of a rotating layer of stratified magnetic field, considering both unidirectional and sheared fields. We discuss why the traditional decomposition into $α$ and $β$ effects is inappropriate in this case, and that it is only consideration of the entire mean emf that is meaningful. By considering a weighted average of the unstable linear eigenmodes, and averaging over the horizontal plane, we obtain depth-dependent emfs. For the simplified case of isothermal, ideal MHD we are able to obtain an analytic expression for the emf; more generally the emf has to be determined numerically. We calculate how the emf depends on the various parameters of the problem, particularly the rotation rate and the latitude of the magnetic layer.

astro-ph.SR↗

arXiv2010

A Self-Consistent Treatment of the Electromotive Force in Magnetohydrodynamics for Large Diffusivities

The coupled equations that describe the effect of large-scale magnetic and velocity fields on forced high-diffusivity magnetohydrodynamic flows are investigated through an extension of mean field electrodynamics. Our results generalise those of Rädler & Brandenburg (2010), who consider a similar situation but assume that the effect of the Lorentz force on the momentum equation can be neglected. New mean coupling terms are shown to appear, which can lead to large-scale growth of magnetic and velocity fields even when the usual $α$-effects are absent.

astro-ph.SR↗

arXiv2009

Self-consistent mean field MHD

We consider the linear stability of two-dimensional nonlinear magnetohydrodynamic basic states to long-wavelength three-dimensional perturbations. Following Hughes & Proctor (2009a), the 2D basic states are obtained from a specific forcing function in the presence of an initially uniform mean field of strength $\mathcal{B}$. By extending to the nonlinear regime the kinematic analysis of Roberts (1970), we show that it is possible to predict the growth rate of these perturbations by applying mean field theory to \textit{both} the momentum and the induction equations. If $\mathcal{B}=0$, these equations decouple and large-scale magnetic and velocity perturbations may grow via the kinematic $α$-effect and the AKA instability respectively. However, if $\mathcal{B} \neq 0$, the momentum and induction equations are coupled by the Lorentz force; in this case, we show that four transport tensors are now necessary to determine the growth rate of the perturbations. We illustrate these situations by numerical examples; in particular, we show that a mean field description of the nonlinear regime based solely on a quenched $α$ coefficient is incorrect.

astro-ph.SR↗

arXiv2009

Dynamo Action in the Presence of an Imposed Magnetic Field

We consider the linear stability to three-dimensional perturbations of two-dimensional nonlinear magnetohydrodynamic basic states obtained from a specified forcing function in the presence of an imposed initially uniform magnetic field of strength $B_0$. The forcing is chosen such that it drives the CP flow of Galloway & Proctor (1992) when $B_0=0$. We first examine the properties of these basic states and their dependence on $B_0$ and on the magnetic Reynolds number $Rm$. The linear stability of these states is then investigated. It is found that at a given $Rm$ the presence of a background field is stabilising. The results also allow us to speculate that at a fixed value of $B_0$ the growth of the unstable perturbations is `fast', in the sense that the growth rate becomes independent of $Rm$ as $Rm \to \infty$.

physics.flu-dyn↗

arXiv2008

The effects of flux transport on interface dynamos

The operation of an interface dynamo (as has been suggested for the Sun and other stars with convective envelopes) relies crucially upon the effective transport of magnetic flux between two spatially disjoint generation regions. In the simplest models communication between the two regions is achieved solely by diffusion. Here we incorporate a highly simplified anisotropic transport mechanism in order to model the net effect of flux conveyance by magnetic pumping and by magnetic buoyancy. We investigate the influence of this mechanism on the efficiency of kinematic dynamo action. It is found that the effect of flux transport on the efficiency of the dynamo is dependent upon the spatial profile of the transport. Typically, transport hinders the onset of dynamo action and increases the frequency of the dynamo waves. However, in certain cases there exists a preferred magnitude of transport for which dynamo action is most efficient. Furthermore, we demonstrate the importance of the imposition of boundary conditions in drawing conclusions on the role of transport.

astro-ph↗

arXiv2008

Problems with kinematic mean field electrodynamics at high magnetic Reynolds numbers

We discuss the applicability of the kinematic $α$-effect formalism at high magnetic Reynolds numbers. In this regime the underlying flow is likely to be a small-scale dynamo, leading to the exponential growth of fluctuations. Difficulties arise with both the actual calculation of the $α$ coefficients and with its interpretation. We argue that although the former may be circumvented -- and we outline several procedures by which the the $α$ coefficients can be computed in principle -- the interpretation of these quantities in terms of the evolution of the large-scale field may be fundamentally flawed.

astro-ph↗