Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit

Dritschel, David G.; Tobias, Steven M.

doi:10.1017/jfm.2012.195

Cited by 14 publications

(16 citation statements)

References 33 publications

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“…Finally, it would be valuable to study dynamical flux expulsion numerically by means of full simulations in unbounded geometry and explore the limitations of the quasi-linear approximation as set out here. Simulations with R e ≫ R m ≫ 1 can be undertaken efficiently using contourspectral methods (Dritschel & Tobias 2012) while regimes with R e ∼ R m ≫ 1 would be best simulated with standard pseudo-spectral codes.…”

Section: Discussionmentioning

confidence: 99%

Flux expulsion with dynamics

2016

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In the process of flux expulsion, a magnetic field is expelled from a region of closed streamlines on a T R 1/3 m time scale, for magnetic Reynolds number R m ≫ 1 (T being the turnover time of the flow). This classic result applies in the kinematic regime where the flow field is specified independently of the magnetic field. A weak magnetic 'core' is left at the centre of a closed region of streamlines, and this decays exponentially on the T R 1/2 m time scale.The present paper extends these results to the dynamical regime, where there is competition between the process of flux expulsion and the Lorentz force, which suppresses the differential rotation. This competition is studied using a quasi-linear model in which the flow is constrained to be axisymmetric. The magnetic Prandtl number R m /R e is taken to be small, R m large, and a range of initial field strengths b 0 is considered. ), below which the field follows the kinematic evolution and decays rapidly. Between these two thresholds the magnetic field is strong enough to suppress differential rotation leaving a magnetically controlled core spinning in solid body motion, which then decays slowly on a time scale of order T R m .

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Section: Discussionmentioning

confidence: 99%

Flux expulsion with dynamics

2016

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“…All other aspects of the algorithm rely on fast Fourier transforms, particularly in the dealiased pseudospectral evolution of the ageostrophic horizontal vorticity A h . While the grid resolution may seem modest, the effective resolution of the CASL algorithm is much higher, more than 10 times in each direction, as demonstrated in ; see also Dritschel & Scott (2009) and Dritschel & Tobias (2012).…”

Section: Initialisation and Parameter Settingsmentioning

confidence: 99%

Effect of Prandtl’s ratio on balance in geophysical turbulence

Dritschel

McKiver

2015

J. Fluid Mech.

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The fluid dynamics of the atmosphere and oceans is to a large extent controlled by the slow evolution of a scalar field called ‘potential vorticity’ (PV), with relatively fast motions such as inertia-gravity waves playing only a minor role. This state of affairs is commonly referred to as ‘balance’. Potential vorticity is a special scalar field which is materially conserved in the absence of diabatic effects and dissipation, effects that are generally weak in the atmosphere and oceans. Moreover, in a balanced flow, PV induces the entire fluid motion and its thermodynamic structure (Hoskins et al., Q. J. R. Meteorol. Soc., vol. 111, 1985, pp. 877–946). While exact balance is generally not achievable, it is now well established that balance holds to a high degree of accuracy in rapidly rotating and strongly stratified flows. Such flows are characterised by both a small Rossby number, $\mathit{Ro}\equiv |{\it\zeta}|_{max}/f$, and a small Froude number, $\mathit{Fr}\equiv |{\bf\omega}_{h}|_{max}/N$, where ${\it\zeta}$ and ${\bf\omega}_{h}$ are the relative vertical and horizontal vorticity components, while $f$ and $N$ are the Coriolis and buoyancy frequencies. In fact, balance can even be a good approximation when $\mathit{Fr}\lesssim \mathit{Ro}\sim \mathit{O}(1)$. In this study, we examine how balance depends specifically on Prandtl’s ratio, $f/N$, in unforced freely evolving turbulence. We examine a wide variety of turbulent flows, at a mature and complex stage of their evolution, making use of the fully non-hydrostatic equations under the Boussinesq and incompressible approximations. We perform numerical simulations at exceptionally high resolution in order to carefully assess the degree to which balance holds, and to determine when it breaks down. For this purpose, it proves most useful to employ an invariant PV-based Rossby number ${\it\varepsilon}$, together with $f/N$. For a given ${\it\varepsilon}$, our key finding is that – for at least tens of characteristic vortex rotation periods – the flow is insensitive to $f/N$ for all values for which the flow remains statically stable (typically $f/N\lesssim 1$). Only the vertical velocity varies in proportion to $f/N$, in line with quasi-geostrophic (QG) scaling for which $\mathit{Fr}^{2}\ll \mathit{Ro}\ll 1$. We also find that as ${\it\varepsilon}$ increases towards unity, the maximum $f/N$ attainable decreases towards 0. No statically stable flows occur for ${\it\varepsilon}\gtrsim 1$. For all stable flows, balance is found to hold to a remarkably high degree: as measured by an energy norm, imbalance never exceeds more than a few per cent of the balance, even in flows where $\mathit{Ro}>1$. The vertical velocity $w$ remains a tiny fraction of the horizontal velocity $\boldsymbol{u}_{h}$, even when $w$ is dominantly balanced. Finally, typical vertical to horizontal scale ratios $H/L$ remain close to $f/N$, as found previously in QG turbulence for which $\mathit{Fr}\sim \mathit{Ro}\ll 1$.

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“…Note, however, that for the present case, this peak decreases toward a positive limit (which is the maximum of µ j 2 ) because µ is fixed. The vanishing of kinetic energy dissipation in the limit Pm → 0 partly justifies the numerical approach of Dritschel & Tobias (2012), who simulated 2D MHD turbulence at low Pm using a conservative numerical scheme for the vorticity. Given strong support for solution regularity discussed below, this justification could be considered complete.…”

Section: Numerical Resultsmentioning

confidence: 86%

“…The former case is of practical interest as plasmas in nature and liquid metals in laboratories are relatively low in viscosity but high in resistivity. In fact, Pm can be as low as 10 −5 − 10 −3 in stellar interiors and 10 −5 in liquid metals (Iskakov et al 2007;Dritschel & Tobias 2012). 3.1.…”

Section: Theoretical Considerationsmentioning

confidence: 99%

Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number

Tran

Blackbourn

2013

J. Fluid Mech.

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We study both theoretically and numerically two-dimensional magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl number $\mathit{Pm}$ (and the limits thereof), with an emphasis on solution regularity. For $\mathit{Pm}= 0$, both $\Vert \omega \Vert ^{2} $ and $\Vert j\Vert ^{2} $, where $\omega $ and $j$ are, respectively, the vorticity and current, are uniformly bounded. Furthermore, $\Vert \boldsymbol{\nabla} j\Vert ^{2} $ is integrable over $[0, \infty )$. The uniform boundedness of $\Vert \omega \Vert ^{2} $ implies that in the presence of vanishingly small viscosity $\nu $ (i.e. in the limit $\mathit{Pm}\rightarrow 0$), the kinetic energy dissipation rate $\nu \Vert \omega \Vert ^{2} $ vanishes for all times $t$, including $t= \infty $. Furthermore, for sufficiently small $\mathit{Pm}$, this rate decreases linearly with $\mathit{Pm}$. This linear behaviour of $\nu \Vert \omega \Vert ^{2} $ is investigated and confirmed by high-resolution simulations with $\mathit{Pm}$ in the range $[1/ 64, 1] $. Several criteria for solution regularity are established and numerically tested. As $\mathit{Pm}$ is decreased from unity, the ratio $\Vert \omega \Vert _{\infty } / \Vert \omega \Vert $ is observed to increase relatively slowly. This, together with the integrability of $\Vert \boldsymbol{\nabla} j\Vert ^{2} $, suggests global regularity for $\mathit{Pm}= 0$. When $\mathit{Pm}= \infty $, global regularity is secured when either $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $, where $\boldsymbol{u}$ is the fluid velocity, or $\Vert j\Vert _{\infty } / \Vert j\Vert $ is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range $\mathit{Pm}\in [1, 64] $ show that $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $ varies slightly (with similar behaviour for $\Vert j\Vert _{\infty } / \Vert j\Vert $), thereby lending strong support for the possibility $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert \lt \infty $ in the limit $\mathit{Pm}\rightarrow \infty $. The peak of the magnetic energy dissipation rate $\mu \Vert j\Vert ^{2} $ is observed to decrease rapidly as $\mathit{Pm}$ is increased. This result suggests the possibility $\Vert j\Vert ^{2} \lt \infty $ in the limit $\mathit{Pm}\rightarrow \infty $. We discuss further evidence for the boundedness of the ratios $\Vert \omega \Vert _{\infty } / \Vert \omega \Vert $, $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $ and $\Vert j\Vert _{\infty } / \Vert j\Vert $ in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields.

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Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit

Cited by 14 publications

References 33 publications

Flux expulsion with dynamics

Flux expulsion with dynamics

Effect of Prandtl’s ratio on balance in geophysical turbulence

Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number

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