UK Magnetohydrodynamics Meeting

University of Leeds, Thursday 20 - Friday 21 May, 2010

Organiser: Dr. Rainer Hollerbach (rh_at_maths.leeds.ac.uk)

Programme

Abstracts

K. Chan, K. Zhang, X. Liao (Hong Kong and Exeter)

On librationally driven flow in planetary ellipsoidal cores

Many planets have an equatorially asymmetric distribution of mass and, consequently, gravitational interactions between the planets and star force longitudinal libration by exerting the axial torque on the planets. We present both the analytical and numerical studies of librationally driven flows in rotating ellipsoids in the mantle frame of reference, showing that the longitudinal libration causes the resonance of non-axisymmetric inertial modes and that the librationally driven flow in ellipsoidal planetary cores may be sufficiently complicated for dynamo action.

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D. Kong, K. Zhang, G. Schubert (Exeter and UCLA)

The shape of two-layered rapidly rotating planets

The interior of many planetary bodies, to first approximation, comprises a core and a mantle with significantly different densities and the shape of the interface between the core and the mantle represents an important parameter in describing the dynamics of planetary fluid cores. We present the first theory of a two-layered, rotating fluid-like planet which determines the shape of both the interface and the outer free surface without treating departure from sphericity as a small perturbation. Since the interface and the outer free surface, in general, have different shapes, two different spheroidal coordinates are required in the mathematical analysis and the transformation between them is at the heart of the complexity of the theory. We show that, in comparison to the classical Maclaurin solution which is explicitly analytical, the relevant multiple integrals for the equilibrium solution of a two-layered Maclaurin spheroid have to be evaluated numerically. We also show that the shape of a two-layer rotating planet is characterized by three dimensionless parameters which are explored systematically.

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Phil Livermore, Rainer Hollerbach (Leeds)

MHD models of the Earth's core at extremely low viscosity

Models of the MHD processes in Earth's core are hampered by numerical instabilities at low viscosity. Despite success in reproducing some features of the geomagnetic field using viscosities many orders of magnitude too large, these models are probably unrealistic. In a novel approach, we solve a similar MHD system which is forced in such a manner as to match surface observations, in which we are able to access an unprecedentedly low viscosity. In this regime, the magnetic field approaches a Taylor state and we explore the manner in which this occurs. We find a strong magnetically-driven westward jet beneath the equator that apparently arises due to a geometrical effect; this jet could be linked to equatorial waves in the observed field.

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Andrew Baggaley (Newcastle)

Magnetic structures produced by the kinematic and saturated small-scale dynamo

We present the details of numerical simulations of the fluctuation dynamo, driven by an incompressible flow. By applying morphological tools, including Minkowski functionals, we are able to quantify the structure of the resulting magnetic field in both the kinematic and saturated states, with varying values for the Magnetic Reynolds number. Our aim is to further understand the nature of the saturation mechanism of the fluctuation dynamo.

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Katy Richardson, Mike Proctor (Cambridge)

Effects of alpha-effect fluctuations on simple nonlinear dynamo models

We investigate the interaction of a fluctuating alpha-effect with large-scale shear in a simple nonlinear 1-dimensional dynamo wave model. We firstly extend the calculations of {Proctor07} to include spatial variation of the fluctuations, and find that there can be a mechanism for magnetic field generation, even when the mean alpha is zero, provided the spatiotemporal spectrum of the fluctuations has an appropriate form. We investigate mean-field dynamo action when the new term arising from the fluctuations is non-zero, and present results concerning the stability and frequency of the solutions and parity selection in the nonlinear regime. The asymptotic theory is tested by comparing the results with those of a traditional mean-field model in which the alpha-effect term is rapidly varying in space and time.

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Binod Sreenivasan, Chris Jones (Leeds)

Dipole generation and subcritical behaviour in the geodynamo

Geodynamo models based on convection-driven flow in a rapidly rotating spherical shell frequently give rise to strong stable dipolar magnetic fields. This is in sharp contrast to convection-driven flows with no rotation or slow rotation, where the large scale field is often rather weak compared to the small scale field. In particular, parameter regimes in spherical dynamos where the inertial terms play a limited role are often strongly dipolar. Kinematic dynamo theory makes a distinction between the onset of dipolar and quadrupolar modes, but for the types of flow arising in rotating convection-driven dynamos, the onset of dynamo action for quadrupolar modes often occurs close to the onset of dipolar modes, and indeed in some reasonable models occurs before the onset of dipolar modes. Here we explore nonlinear mechanisms due to the action of Lorentz force which may give rise to a strong preference for dipolar modes and also leads to subcritical behaviour.

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Piotr Boronski, Chris Jones (Leeds)

Strongly stratified dynamos in spherical shells: a numerical study

The magnetohydrodynamics equations in anelastic approximation are numerically simulated in spherical geometry using a spherical harmonics/finite differences code. Magnetic field generation via dynamo effect is investigated in strongly stratified flows. The observed dynamo states differ qualitatively from those obtained using Boussinesq approximation. In particular, a transverse Roberts-like dynamo has been observed.

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Nicolas Leprovost (Sheffield)

Stellar dynamo: no need for rotation?

In this contribution, we show that the alpha effect can exist even without rotation by considering a turbulence driven by an inhomogeneous forcing in the presence of a background shear flow. We compute both the magnetic pumping (gamma effect) and the alpha effect, which corresponds to transport of magnetic flux by turbulence and generation of magnetic field by helical turbulence, respectively. We then show that a large-scale dynamo can be possible when the inhomogenity is perpendicular to the plane of the shear flow. This has interesting implications on the structure of magnetic fields in star with slow rotation.

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Laurene Jouve, Mike Proctor, Geoffroy Lesur (Cambridge)

Buoyancy-induced time delays in Babcock-Leighton flux-transport dynamo models

We investigate the role of the magnetic buoyancy process in 2D Babcock-Leighton dynamo models, by modelling more accurately the surface source term for poloidal field. To do so, we reintroduce in mean-field models the results of full 3D MHD calculations of the non-linear evolution of a rising flux tube in a convective shell. More specifically, the Babcock-Leighton source term is modified to take into account the delay introduced by the rise time of the toroidal structures from the base of the convection zone to the solar surface. Results. We find that the time delays introduced in the equations produce large temporal modulation of the cycle amplitude even when strong and thus rapidly rising flux tubes are considered. Aperiodic modulations of the solar cycle appear after a sequence of period doubling bifurcations typical of non-linear systems. The strong effects introduced even by small delays is found to be due to the dependence of the delays on the magnetic field strength at the base of the convection zone, the modulation being much less when time delays remain constant. We do not find any significant influence on the cycle period except when the delays are made artificially strong.

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Krzysztof Mizerski, David Hughes (Leeds)

Compressible Ekman-Hartmann boundary layers

We consider the effect of compressibility on mixed Ekman-Hartmann boundary layers on an infinite plane (z=0), in the presence of an external magnetic field oblique to the boundary. The aim is to investigate the influence of the magnetic pressure on the fluid density, and hence, via mass conservation, on the mass flow into or out of the boundary layer. We find that if the z-component of vorticity in the main flow, immediately above the boundary layer, is negative, then there is a competition between Ekman suction and the magnetic pressure effect. Indeed, as the magnetic field strength is increased, the magnetic pumping may overcome the Ekman suction produced by anti-cyclonic main flow vortices.

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Jamie Douglas, Eun-jin Kim (Sheffield)

Simulations of resistive tearing modes

Resistive tearing modes (RTMs), first treated by Furth, Killeen and Rosenbluth, Phys. Fluids 6, 459 (1963), are plasma instabilities which result from the global properties of the equilibrium current density profile. This original work, which is cited in most literature, was based on RTMs in slab geometry, and is useful for elucidating the important physics. More recent work by Militello et al, Phys. Plasmas 11, 125 (2003) investigated RTMs in cylindrical geometry, a common approximation used with tokamak plasmas. We present electromagnetic simulations in cylindrical geometry using the CUTIE plasma turbulence code which confirm the results of Militello et al.

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Andrew Newton, Eun-jin Kim (Sheffield)

Chaos and transport in 2D MHD turbulence

We investigate the effect of uniform large-scale flows and magnetic fields on the chaotic properties of 2D forced MHD turbulence. Specifically, we measure exponential stretching rates of infinitesimal line elements whilst varying the strength of uniform large-scale fields and the correlation times of the forcing, and compute the probability distribution functions of instantaneous stretching rate and finite-time Lypunov exponents, spatial distribution of finite-time Lyapunov exponents, and mean Lyapunov exponent. By comparing these results with previous findings on the turbulent transport of magnetic fields [A. Newton and E. Kim, Phys. Rev. Lett. 102, 165002 (2009)], we demonstrate that there is no direct link between chaos and transport of magnetic fields. In particular, these Lyapunov exponents are found to rather sensitively depend on the complexity of the flow which is not directly reflected in the transport of magnetic fields. We also elucidate the effect of resonances on chaotic properties.

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Joanne Mason, Stanislav Boldyrev, Fausto Cattaneo (Chicago)

Dynamic alignment in driven MHD turbulence

A new theory for driven MHD turbulence in the presence of a strong background magnetic field will be summarised. The key prediction is that the velocity and magnetic fields become aligned within a small scale-dependent angle in the field-perpendicular plane. I will also describe the results of a series of high-resolution numerical simulations that have been specially designed in order to test the theoretical predictions, including the scaling of the alignment angle and its consequences on the field-perpendicular energy spectrum.

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Fred Gent (Newcastle)

Supernovae-driven turbulence in the ISM

We simulate the interstellar medium (ISM) by solving numerically the 3-dimensional non-ideal magnetohydrodynamic (MHD) set of equations. The model includes the stellar gravity field in the solar neighbourhood, a stratified gas density, shearing due to differential rotation, radiative cooling and uv-heating. We consider simulations of purely hydrodynamic properties and with the inclusion of a small seed magnetic field. The primary source of turbulence and heat is provided by supernovae (SNe). We include type I and type 2 SNe, which are exploded at rates similar to the observed rate in the solar neighbourhood and with a reasonably realistic randomised distribution by position. The aim of this project is to produce a physically motivated dynamical structure for the ISM, which saturates to a quasi-steady state of turbulence, from which the typical temperature and density filling factors and turbulent velocities and vorticity can be calculated with respect to anisotropy and height of the ISM. It is of interest to investigate the relationship between these variables and the rate of SNe. In addition the factors which determine the outward vertical velocity flows towards the galactic halo are of interest to understanding whether this constitutes a galactic wind or fountain. In consideration of the magnetic field we investigate how these factors combine to influence the galactic dynamo. In particular we investigate how the mean and random components compare and their dependence on the rate of shear and SNe rate and distribution.

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Graeme Sarson (Newcastle)

Deciphering solar turbulence from sunspots records

It is generally believed that sunspots are the emergent part of magnetic flux tubes in the solar interior. These tubes are created at the base of the convection zone and rise to the surface due to their magnetic buoyancy. The motion of plasma in the convection zone being highly turbulent, the surface manifestation of sunspots may retain the signature of this turbulence, including its intermittency. From direct observations of sunspots, and indirect observations of the concentration of cosmogenic isotopes C-14 in tree rings or Be-10 in polar ice, power spectral densities in frequency are plotted. Two different frequency scalings emerge, depending on whether the Sun is quiescent or active. From direct observations we can also calculate scaling exponents. These testify to a strong intermittency, comparable with that observed in the solar wind.

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Anthony Yeates, Gunnar Hornig, A. Wilmot-Smith, D. Pontin (Dundee)

The topology of magnetic braids

To understand the effect of turbulent reconnection on 3-d magnetic fields we need to know how to characterise their topology. While this is an unsolved problem for a general magnetic field, we show how it can be achieved for a so-called "magnetic braid": a flux tube that has everywhere non-zero field strength. We characterise the topology of such a magnetic field using a "topological flux function" defined on the cross-section of the flux tube. This function has some interesting properties, and its utility is demonstrated in the analysis of a numerical MHD simulation of turbulent relaxation.

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Gunnar Hornig (Dundee)

Turbulent magnetic reconnection: Estimation of reconnected flux

In plasmas with very low dissipation magnetic reconnection often occurs in form of a turbulent cascade of individual reconnection events. These reconnection events can overlap or cancel each other and hence the magnetic flux reconnected can significantly exceed the magnetic flux present in the initial field. We discuss how to quantify the magnetic flux reconnected in a cascade of 3D magnetic reconnection processes and present an estimation for a relaxation of a braided magnetic field. 

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Jorge Fuentes-Fernandez, Clare Parnell, Alan Hood (St Andrews)

MHD dynamical relaxation of 2D magnetic X-points

Magnetic neutral points are important locations for energy conversion in the Solar Corona, but at present, there are few studies of X-points that consider the effects of a finite beta plasma. We have run a series of experiments to study the non-force free equilibria that are reached when a hyperbolic X-point is perturbed from its potential equilibrium. We present in this talk the recent results of this work which is still in progress. One interesting feature for discussion is the appearance of currents out of the central current sheet, along the four separatrices, which had been speculated in the past.

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Andrew Haynes, Clare Parnell (St Andrews)

Separators and separator reconnection

In many astrophysical plasmas the magnetic fields have complex structures which continually evolve and change over time. One mechanism which evolves these 3D magnetic fields is that of magnetic reconnection, which by breaking and rejoining fieldlines changes their connectivities and releases magnetic energy. In this talk, we will consider the importance of 3D reconnection at a separator, a special fieldline that links a pair of null points. We will discuss using a number of examples, the locations at which separators are found, the nature of separator reconnection and the implications for energy release and the evolution of the plasma.

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Jingnan Guo (Glasgow)

Is the 3-D magnetic null point with a convective electric field an efficient particle accelerator?

We study the particle acceleration at a magnetic null point in the solar corona, considering self-consistent magnetic fields, plasma flows and the corresponding convective electric fields.

We calculate the electromagnetic fields by 3-D magnetohydrodynamic (MHD) simulations and expose charged particles to these fields within a full-orbit relativistic test-particle approach. In the 3-D MHD simulation part, the initial magnetic field configuration is set to be a potential field obtained by extrapolation from an analytic quadrupolar photospheric magnetic field with a typically observed magnitude. The configuration is chosen so that the resulting coronal magnetic field contains a null. Driven by photospheric plasma motion, the MHD simulation reveals the coronal plasma motion and the self-consistent electric and magnetic fields. In a subsequent test particle experiment the particle energies and orbits (determined by the forces exerted by the convective electric field and the magnetic field around the null) are calculated in time.

Through its convective electric field and due to magnetic nonuniform drifts and de-magnetization process, the 3-D null can act as an effective accelerator for protons but not for electrons. Protons are more easily de-magnetized and accelerated than electrons because of their larger Larmor radii. Notice that macroscopic MHD simulations are blind to microscopic magnetic structures where more non-adiabatic processes might be taking place. In the real solar corona, we expect that particles could have a higher probability to experience a de-magnetization process and get accelerated. To trigger a significant acceleration of electrons and even higher energetic protons, however, the existence of a resistive electric field mainly parallel to the magnetic field is required. A physically reasonable resistivity model included in resistive MHD simulations is direly needed for the further investigations of electron acceleration by parallel electric fields.

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Khalil Al-Ghafri, R.J. Morton, R. Erdelyi (Sheffield)

Damping of magneto-acoustic oscillations in hot and dynamic coronal plasma

In this paper we investigate the propagation of MHD waves in a homogeneous magnetized plasma in a weakly stratified atmosphere, representing hot coronal loops, where the background plasma cools during the propagation of the magneto-acoustic wave. In this model the background pressure is allowed to change as a function of time due to thermal conduction causing the cooling. The ubiquitous magnetic field is assumed to be uniform and pointing in the vertical (z) direction. The background plasma is assumed to be cooling on a time-scale comparable to the characteristic period of the perturbations. Our aim is to investigate the influence of the cooling of the background plasma on slow waves. We argue that the plasma cooling may be accountable for the damping of the MHD waves. The dispersion relation which describes the properties of the magneto-acoustic MHD waves is derived by using the WKB theory. The amplitude of waves are found by taking first order equation and solved analytically. The method of characteristics is used to find an approximate solution. Numerical calculations are applied to obtain insight into the behavior of the MHD waves in a system with variable background. The result shows that there is a heavy damping of MHD waves that can be linked to the widely observed damping of hot coronal loop oscillations.

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Rekha Jain (Sheffield)

Axisymmetric absorption and scattering of p modes by thin magnetic flux tubes

The buffeting action of the solar acoustic waves (p-modes) excites MHD tube waves. The propagation of these tube waves along the length of the tube creates a back reaction on the field-free fluid surrounding the tube, generating outgoing scattered wave field. We will present a calculation of absorption and the far-field scattering matrix for the special case of axisymmetric, vertically oriented, thin, magnetic flux tube.

The ultimate goal is to model the absorption and scattering of acoustic waves by magnetic plages. However, the first step in this future line of inquiry is the calculation of the scattering matrices (both near-and far-field) for a stratified single magnetic flux tube. The work presented will be one piece of many required to accomplish this goal.

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Nicky Chorley (Warwick)

Persistency of long period oscillations in sunspots

Long period oscillations of the gyroresonant emission from sunspot atmospheres are studied. Time series data generated from the sequences of images obtained by the Nobeyama Radioheliograph operating at a frequency of 17 GHz for three sunspots have been analysed and are found to contain significant periods in the range of several tens of minutes. Wavelet analysis shows that these periods are persistent throughout the observation periods. Spatial analysis using the techniques of period, power, correlation and time lag mapping reveals regions of enhanced oscillatory power in the umbral regions. Also seen are two regions of coherent oscillation of about 25 pixels in size, that oscillate in anti-phase with each other. Possible interpretation of the observed periodicities is discussed, in terms of the shallow sunspot model and the leakage of the solar g-modes. We also present the analysis of following one sunspot over the course of 9 days, showing the stability of the long period oscillations, in both the brightness temperature and polarisation data. To model the persistency of such oscillations, we use a nonlinear oscillator and present here preliminary results.

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Gert Botha, Tony Arber, Valery Nakariakov (Warwick)

Chromospheric resonances above sunspots

The 3 minute oscillations observed above sunspot umbrae are modelled as upward propagating slow magnetoacoustic waves guided by a vertical magnetic field. The temperature profile above the sunspot photosphere creates a cavity at the temperature minimum in the chromosphere where slow magnetoacoustic waves resonate. The acoustic resonator is not ideal and allows waves to leak into the higher regions of the solar atmosphere. Modelling the response of the chromospheric resonator to broadband impulsive excitations, we find that the leaky waves are periodic with a period of approximately 3 minutes, consistent with optical, radio, EUV and soft X-ray observations. Different temperature profiles for the sunspot atmosphere give slightly different resonance frequencies for the chromospheric resonator. Initial results show that the resonance frequency is inversely proportional to the distance between the photosphere and the transition region. We conclude that the chromospheric resonator model is fully consistent with present day observations. It explains the origin of the 3 minute oscillations in the chromosphere above sunspot umbrae and provide a criteria to choose between the various atmospheric models to fit each individual sunspot uniquely. 

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Kuan Tam, Alan Hood (St Andrews)

Coronal heating by Taylor relaxation

The problem of heating the solar corona requires the conversion of magnetic energy into thermal energy. Presently, there are two promising mechanisms for heating the solar corona: wave heating and/or nanoflares. During the talk, I will review Taylor's relaxation theory (Taylor 1974, 1986) and explain how it leads to Parker's (1988) idea of nanoflare heating. I will then show a series of simulations showing how the relaxation process in a single loop may trigger energy release in nearby loops.

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Anvar Shukurov (Newcastle)

Turbulent magnetic fields and cosmic rays in galactic spiral arms

The behaviour of frozen-in magnetic field under compression is one of the simplest problems in MHD: in 1D case, the field strength is proportional to the gas density. There is little doubt that this should apply to galactic magnetic fields when interstellar gas is compressed to form spiral arms. It is therefore quite surprising that radio observations of synchrotron emission from spiral galaxies do not show signs of the expected strong enhancement of magnetic fields in the arms despite clear signs of strong gas compression. We suggest a resolution of this long-standing paradox.

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Alban Potherat, R. Klein, V. Dymkou (Coventry)

Transition between 2D and 3D forced MHD flows

Channel flows under a strong homogeneous, transverse magnetic field are known to be quasi-two dimensional in the sense that they are invariant along the field lines, except in the vicinity of the walls where boundary layers develop. In the abundant literature on these flows, quasi-two dimensional MHD turbulence is most of the time studied as an established state, or as an asymptotic state, reached after a magnetic field is suddenly imposed on an initiall isotropic state.

Here, we are rather interested in the forced, established turbulent state, either two- or three-dimensional, and the mechanisms by which three-dimensionality appears in an initially quasi two-dimensional flow, when the forcing is increased step-wise. We present our latest experimental results on a flow electrically driven vortices in a cubic container under homogeneous magnetic field. At low magnetic field the flow exhibits some strong three-dimensionality in a nearly steady regime, as remarkable Y-shaped vortices appear. Under high magnetic fields, on the other hand, the flow remains close to quasi two-dimensionality when it becomes unsteady and three-dimensionality appears when the forcing is further increased. We also show that the electrodes where the current is injected are themselves responsible for local three-dimensionality. Finally, we present some direct numerical simulations of a similar problem where the boundaries of the domain are periodic; we show that in spite of this important difference, which underlines the role played by walls in the experiment, some of the mechanisms present in the experiment are recovered.

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Vincent Dousset (Coventry)

Vortex shedding in the MHD flow past a truncated cylinder

We consider the MHD flow past a truncated square cylinder in a duct under an externally applied magnetic field imposed along the cylinder axis. We have performed 3D direct numerical simulations within the low magnetic Reynolds number approximation by increasing the Reynolds number at a fixed Hartmann number. In comparison with the non-MHD case where the flow dynamics are greatly influenced by the spanwise confinement, the magnetic field induces a set of modifications on the flow and in particular, influences the vortex shedding mechanism as the Hartmann number is increased.