Thermal Generation of Alfvén Waves in Oscillatory Magnetoconvection: Diffusively Modified Modes

Thermal instabilities in the form of oscillatory magnetoconvection representing diffusively modified Alfvén waves in an electrically-conducting Bénard fluid layer of rigid walls in the presence of a vertical magnetic field are investigated. Emphasis of the article is on the transition from a nearly undamped Alfvén wave to diffusively modified Alfvén waves, and on the effect of physically realisable magnetic field boundary conditions on magnetoconvection. It is found that the extra magnetic dissipation in the magnetic Hartmann boundary layers can enhance oscillatory magnetoconvection in the form of strongly modified Alfvén waves. Oscillatory magnetoconvection produced solely by the Alfvén wave mechanism can be the most unstable mode even in the presence of a strong viscous effect. This article also represents the first study on the effect of an electrically conducting wall on magnetoconvection which is associated with a nonlinear eigenvalue problem. We find that the electrically perfectly conducting condition does not yield a good approximation for magnetoconvection with an electrically highly conducting wall. The size of oscillation frequency with an electrically highly conducting wall can be more than a factor of 2 larger than that obtained using the perfectly conducting condition.     

Sunspots

Abstract

Some examples of research on structure and formation of sunspots are briefly recollected in historical sequence. They relate to many facets of sunspots, first: magnetic inhibition of convection, the conjecture of a fiat penumbra, the stratification beneath the umbra, the observable magnetic profile, the Evershed effect as syphon flow, the concept of a magnetopause; next: cooling by Alfven waves, evolution and stability, the “bright ring”, the observed change of umbra brightness with the phase of the sunspot cycle, the hypothetical cluster of separate flux strands underneath the umbra, the profile of the magnetopause, the structure of the penumbra and the inclination of its field and finally: the concept of a deep penumbra with volume currents, exchange convection and the concept of a second current sheet separating umbra and penumbra.

Of course, the rigorous theoretical modeling of local magnetoconvection is an essential tool for our understanding of all these processes. I do not deal with it here, but the reader has a fascinating review of magnetoconvection already in his hands (Weiss, 1991).     

Effect of a uniform magnetic field on nonlinear magnetocenvection in a rotating fluid spherical shell

Abstract

Dynamic interaction between magnetic field and fluid motion is studied through a numerical experiment of nonlinear three-dimensional magnetoconvection in a rapidly rotating spherical fluid shell to which a uniform magnetic field parallel to its spin axis is applied. The fluid shell is heated by internal heat sources to maintain thermal convection. The mean value of the magnetic Reynolds number in the fluid shell is 22.4 and 10 pairs of axially aligned vortex rolls are stably developed. We found that confinement of magnetic flux into anti-cyclonic vortex rolls was crucial on an abrupt change of the mode of magnetoconvection which occurred at Δ = 1 ～ 2, where A is the Elsasser number. After the mode change, the fluid shell can store a large amount of magnetic flux in itself by changing its convection style, and the magnetostrophic balance among the Coriolis, Lorentz and pressure forces is established. Furthermore, the toroidal/poloidal ratio of the induced magnetic energy becomes less than unity, and the magnetized anti-cyclones are enlarged due to the effect of the magnetic force. Using these key ideas, we investigated the causes of the mode change of magnetoconvection. Considering relatively large magnetic Reynolds number and a rapid rotation rate of this model, we believe that these basic ideas used to interpret the present numerical experiment can be applied to the dynamics in the Earth's and other planetary cores.     

Nonlinear waves on a coupled density front

Abstract

It is shown that the inclusion of the nonlinear terms in the equations of motion of a coupled density front of zero potential vorticity results in wave solutions which merely propagate with time. The linear theory, on the other hand, predicts an exponential temporal growth. The nonlinear equation admits steady solutions representing standing waves whereas if the nonlinear terms are omitted no steady solutions exist. The general initial value problem is difficult to solve numerically since the linear problem is ill posed.

In addition we prove that the general similarity solution of the nonlinear equation tends to zero for large times, at any point in space, regardless of the initial condition.     

Magneto-acoustic-gravity waves in the horizontal magnetic field

Abstract

In this paper we consider the propagation of magneto-acoustic-gravity waves in a compressible, conducting isothermal atmosphere permeated by a uniform horizontal magnetic field. The singular levels, arising in a horizontal magnetic field, are considered in their most general form. Exact analytical solutions for a number of particular cases of wave propagation are obtained. The wave transformation is analyzed for all these cases using the solutions obtained.

Based on the theory of wave propagation across a magnetic field, low-frequency wave trapping in a chromospheric resonator is explained, and some properties of running penumbral waves are discussed.     

Symmetries of time-dependent magnetoconvection

Abstract

In the presence of a magnetic field, convection may set in at a stationary or an oscillatory bifurcation, giving rise to branches of steady, standing wave and travelling wave solutions. Numerical experiments provide examples of nonlinear solutions with a variety of different spatiotemporal symmetries, which can be classified by establishing an appropriate group structure. For the idealized problem of two-dimensional convection in a stratified layer the system has left-right spatial symmetry and a continuous symmetry with respect to translations in time. For solutions of period P the latter can be reduced to Z ₂ symmetry by sampling solutions at intervals of ½P. Then the fundamental steady solution has the spatiotemporal symmetry D ₂ = Z ₂ ? Z ₂ and symmetry-breaking yields solutions with Z ₂ symmetry corresponding to travelling waves, standing waves and pulsating waves. A further loss of symmetry leads to modulated waves. Interactions between the fundamental and its first harmonic are described by the group D _2h = D ₂ ? Z ₂ and its invariant subgroups, which describe solutions that are either steady or periodic in a uniformly moving frame. For a Boussinesq fluid in a layer with identical top and bottom boundary conditions there is also an up-down symmetry. With fixed lateral boundaries the spatiotemporal symmetries, again described by D _2h and its invariant subgroups, can be related to results obtained in numerical experiments and analysed by Nagata et al. (1990). With periodic boundary conditions, the full symmetry group, D _2h ?Z ₂, is of order 16. Its invariant subgroups describe pure and mixed-mode solutions, which may be steady states, standing waves, travelling waves, pulsating waves or modulated waves.     

Topology of Rayleigh–Bénard convection and magnetoconvection in plane layer

ABSTRACT

The present study aims to link the dynamics of geophysical fluid flows with their vortical structures in physical space and to study the transition of these structures due to the control parameters. The simulations are carried in a rectangular box filled with liquid gallium for three different cases, namely, Rayleigh–Bénard convection (RBC), magnetoconvection (MC) and rotating magnetoconvection (RMC). The physical setup and material properties are similar to those considered by Aurnou and Olson in their experimental work. The simulated results are validated with theoretical results of Chandrasekhar and experimental results of Aurnou and Olson. The results are also topologically verified with the help of Euler number given by Ma and Wang. For RBC, the onset is obtained at Ra greater than 1708 and at this Ra, the symmetric rolls are orientated in/along a horizontal axis. As the value of Ra increases further, the width of the horizontal rolls starts to amplify. It is observed that these two-dimensional rolls are nothing but the cross-sections of three-dimensional (3D) cylindrical rolls with wave structures. When the vertically imposed magnetic field is added to RBC, the onset of convection is delayed due to the effect of Lorentz force on the thermal buoyancy force. The presence of 3D rectangular structures is highlighted and analysed. When the magnetically influenced rectangular box rotates about vertical axis at low rotation rates in magnetoconvection model, the onset of convection gets further delayed by magnetic field, which is in general agreement with the theoretical predictions. The critical Ra increases linearly with magnetic field intensity. Coherent thermal oscillations are detected near the onset of convection, at moderate rotation rates.     

Periodic and aperiodic dynamo waves

Abstract

In order to show that aperiodic magnetic cycles, with Maunder minima, can occur naturally in nonlinear hydromagnetic dynamos, we have investigated a simple nonlinear model of an oscillatory stellar dynamo. The parametrized mean field equations in plane geometry have a Hopf bifurcation when the dynamo number D=1, leading to Parker's dynamo waves. Including the nonlinear interaction between the magnetic field and the velocity shear results in a system of seven coupled nonlinear differential equations. For D>1 there is an exact nonlinear solution, corresponding to periodic dynamo waves. In the regime described by a fifth order system of equations this solution remains stable for all D and the velocity shear is progressively reduced by the Lorentz force. In a regime described by a sixth order system, the solution becomes unstable and successive transitions lead to chaotic behaviour. Oscillations are aperiodic and modulated to give episodes of reduced activity.     

Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo

Mean-field theory describes magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In this study magnetoconvection and dynamo processes in a rotating spherical shell are considered. Mean fields are defined by azimuthal averaging. In the framework of mean-field theory, the coefficients which determine the traditional representation of the mean electromotive force, including derivatives of the mean magnetic field up to the first order, are crucial for analyzing and simulating dynamo action. Two methods are developed to extract mean-field coefficients from direct numerical simulations of the mentioned processes. While the first method does not use intrinsic approximations, the second one is based on the second-order correlation approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. Both methods are applied to simulations of rotating magnetoconvection and a quasi-stationary geodynamo. The mean-field induction effects described by these coefficients, e.g., the α-effect, are highly anisotropic in both examples. An α²-mechanism is suggested along with a strong γ-effect operating outside the inner core tangent cylinder. The turbulent diffusivity exceeds the molecular one by at least one order of magnitude in the geodynamo example. With the aim to compare mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all previously determined mean-field coefficients was constructed. Various tests with different sets of mean-field coefficients reveal their action and significance. In the magnetoconvection and geodynamo examples considered here, the match between direct numerical simulations and mean-field simulations is only satisfying if a large number of mean-field coefficients are involved. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from the numerical simulation is in good agreement with its counterpart in the mean-field model. However, this match is not completely satisfactory in the geodynamo case anymore. Here the traditional representation of the mean electromotive force ignoring higher than first-order spatial derivatives of the mean magnetic field is no longer a good approximation.     

Interpretations and observations of ocean wave spectra

The paper starts with a discussion of the linear stochastic theory of ocean waves and its various nonlinear extensions. The directional spectrum, with its unique dispersion relation connecting frequency (ω) and wavenumber (k), is no longer valid for nonlinear waves, and examples of $\left( \mathbf{k},\omega\right) The paper starts with a discussion of the linear stochastic theory of ocean waves and its various nonlinear extensions. The directional spectrum, with its unique dispersion relation connecting frequency (ω) and wavenumber (k), is no longer valid for nonlinear waves, and examples of ( k,w)\left( \mathbf{k},\omega\right) -spectra based on analytical expressions and computer simulations of nonlinear waves are presented. Simulations of the dynamic nonlinear evolution of unidirectional free waves using the nonlinear Schr?dinger equation and its generalizations show that components above the spectral peak have larger phase and group velocities than anticipated by linear theory. Moreover, the spectrum does not maintain a thin well-defined dispersion surface, but rather develops into a continuous distribution in ( k,w)\left( \mathbf{k,}\omega\right) -space. The majority of existing measurement systems rely on linear theory for the interpretation of their data, and no measurement systems are currently able to measure the full spectrum in the open ocean with high accuracy. Nevertheless, there exist a few low-resolution systems where data may be interpreted within a minimal assumption of a non-restricted ( k,w)\left( \mathbf{k,}\omega\right) -spectrum. The theory is reviewed, and analyses based on conventional spectral analysis as well as a directional wavelet analysis are carried out on data from a compact laser array at the Ekofisk field in the North Sea. The investigation confirms the strong impact of the second order spectrum below the spectral peak, but is non-conclusive about the off-set in the support of the first order spectrum seen in the dynamical simulations.     

Planetons: An example of large amplitude solitary waves

Abstract

The term ‘‘solitary wave'’ is usually used to denote a steadily propagating permanent form solution of a nonlinear wave equation, with the permanency arising from a balance between steepening and dispersive tendencies. It is known that large-scale thermal anomalies in the ocean are subject to a steepening mechanism driven by the beta effect, while at the smaller deformation scale, such phenomena are highly dispersive. It is shown here that the evolution of a physical system subject to both effects is governed by the ‘‘frontal semi-geostrophic equation'’ (FSGE), which is valid for large amplitude thermocline disturbances. Solitary wave solutions of the FSGE (here named planetons) are calculated and their properties are described with a view towards examining the behavior of finite amplitude solitary waves. In contrast, most known solitary wave solutions belong to weakly nonlinear wave equations (e.g., the Korteweg—deVries (KdV) equation).

The FSGE is shown to reduce to the KdV equation at small amplitudes. Classical sech² solitons thus represent a limiting class of solutions to the FSGE. The primary new effect on planetons at finite amplitudes is nonlinear dispersion. It is argued that due to this effect the propagation rates of finite amplitude planetons differ significantly from the ‘‘weak planeton'', or KdV, dispersion relation. Planeton structure is found to be simple and reminiscent of KdV solitons. Numerical evidence is presented which suggests that collisions between finite amplitude solitary waves are weakly inelastic, indicating the loss of true soliton behavior of the FSGE at moderate amplitudes. Lastly, the sensitivity of solitary waves to the existence of a nontrivial far field is demonstrated and the role of this analysis in the interpretation of lab experiments and the evolution of the thermocline is discussed.     

The galactic dynamo: Axisymmetric and non-axisymmetric modes

Abstract

We discuss recent developments in the theory of large-scale magnetic structures in spiral galaxies. In addition to a review of galactic dynamo models developed for axisymmetric disks of variable thickness, we consider the possibility of dominance of non-axisymmetric magnetic modes in disks with weak deviations from axial symmetry. Difficulties of straightforward numerical simulation of galactic dynamos are discussed and asymptotic solutions of the dynamo equations relevant for galactic conditions are considered. Theoretical results are compared with observational data.     

Large wavenumber convection in the rotating annulus

Abstract

The annulus model considers convection between concentric cylinders with sloping endwalls. It is used as a simplified model of convection in a rapidly rotating sphere. Large azimuthal wavenumbers are preferred in this problem, and this has been exploited to develop an asymptotic approach to nonlinear convection in the annulus. The problem is further reduced because the Taylor-Proudman constraint simplifies the dependence in the direction of the rotation vector, so that a nonlinear system dependent only on the radial variable and time results. As Rayleigh number is increased a sequence of bifurcations is found, from steady solutions to periodic solutions and 2-tori, typically ending in chaotic behaviour. Both the magnetic (MHD convection) and non-magnetic problem has been considered, and in the non-magnetic case our bifurcation sequence can be compared with those found by previous two-dimensional numerical simulations.     

Onset of an energy cascade and nonperiodic behaviour in the nonlinear propagation of MHD waves in the solar atmosphere

Abstract

We study the nonlinear stability of MHD waves propagating in a two-dimensional, compressible, highly magnetized, viscous plasma. These waves are driven by a weak, shear body force which could be imposed by large scale internal fluctuations present in the solar atmosphere.

The effects of anisotropic viscosity (leading to a cubic damping) and of the nonlinear coupling of the Alfven and the magnetoacoustic waves are analysed using Galerkin and multiple-scale analysis: the MHD equations are reduced to a set of nonlinear ordinary differential equations which is then suitably truncated to give a model dynamical system, representing the interaction of two complex Galerkin modes.

For propagation oblique to the background magnetic field, analytical integration shows that the low-wavenumber mode is physically unstable. For propagation parallel to the background magnetic field the high-wavenumber wave can undergo saddlenode bifurcations, in way that is similar to the van der Pol oscillator; these bifurcations lead to the appearance of a hysteresis cycle.

A numerical integration of the dynamical system shows that a sequence of Hopf bifurcations takes place as the Reynolds number is increased, up to the onset of nonperiodic behaviour. It also shows that energy can be transferred from the low- wavenumber to the high-wavenumber mode.     

Properties of nonlinear dynamo waves

Abstract

Dynamo theory offers the most promising explanation of the generation of the sun's magnetic cycle. Mean field electrodynamics has provided the platform for linear and nonlinear models of solar dynamos. However the nonlinearities included arc (necessarily) arbitrarily imposed in these models. This paper conducts a systematic survey of the role of nonlinearities in the dynamo process, by cousidering the behaviour of dynamo waves in the nonlinear regime. It is demonstrated that only by considering realistic nonlinearities that are non-local in space and time can modulation of the basic dynamo wave be achieved. Moreover this modulation is greatest when there is a large separation of timescales provided by including a low magnetic Prandtl number in the equation for the velocity perturbations.     

Convective dynamos with intermediate and strong fields

Abstract

The weak-field Benard-type dynamo treated by Soward is considered here at higher levels of the induced magnetic field. Two sources of instability are found to occur in the intermediate field regime M ～ T ^1/12, where M and T are the Hartmann and Taylor numbers. On the time scale of magnetic diffusion, solutions may blow up in finite time owing to destabilization of the convection by the magnetic field. On a faster time scale a dynamic instability related to MAC-wave instability can also occur. It is therefore concluded that the asymptotic structure of this dynamo is unstable to virtual increases in the magnetic field energy.

In an attempt to model stabilization of the dynamo in a strong-field regime we consider two approximations. In the first, a truncated expansion in three-dimensional plane waves is studied numerically. A second approach utilizes an ad hoc set of ordinary differential equations which contains many of the features of convection dynamos at all field energies. Both of these models exhibit temporal intermittency of the dynamo effect.     

The weak taylor state in an αω-dynamo

Abstract

A spherical αω-dynamo is studied for small values of the viscous coupling parameter ε ～ v^1/2, paying attention particularly to large dynamo numbers. The present study is a follow-up of the work by Hollerbach et al. (1992) with their choice of α-effect and Archimedean wind including also the constraint of magnetic field symmetry (or antisymmetry) due to equatorial plane. The magnetic field scaled by ε^1/2 is independent of ε in the solutions for dynamo numbers smaller than a certain value of D _b (the Ekman state) which are represented by dynamo waves running from pole to equator or vice-versa. However, for dynamo numbers larger than D _b the solution bifurcates and subsequently becomes dependent on ε. The bifurcation is a consequence of a crucial role of the meridional convection in the mechanism of magnetic field generation. Calculations suggest that the bifurcation appears near dynamo number about 33500 and the solutions for larger dynamo numbers and ε = 0 become unstable and fail, while the solutions for small but non-zero ε are characterized by cylindrical layers of local maximum of magnetic field and sharp changes of geostrophic velocity. Our theoretical analysis allows us to conclude that our solution does not take the form of the usual Taylor state, where the Taylor constraint should be satisfied due to the special structure of magnetic field. We rather obtained the solution in the form of a “weak” Taylor state, where the Taylor constraint is satisfied partly due to the amplitude of the magnetic field and partly due to its structure. Calculations suggest that the roles of amplitude and structure are roughly fifty-fifty in our “weak” Taylor state solution and thus they can be called a Semi-Taylor state. Simple estimates show that also Ekman state solutions can be applicable in the geodynamo context.     

Nonlinear Convection in a Rotating Annulus with a Finite Gap

Thermal convection in a fluid-filled gap between the two corotating, concentric cylindrical sidewalls with sloping curved ends driven by radial buoyancy was first studied by Busse (Busse, F.H., "Thermal instabilities in rapidly rotating systems", J. Fluid Mech . 44 , 441-460 (1970)). The annulus model captures the key features of rotating convection in full spherical geometry and has been widely employed to study convection, magnetoconvection and dynamos in planetary systems, usually in connection with the small-gap approximation neglecting the effect of azimuthal curvature of the annulus. This article investigates nonlinear thermal convection in a rotating annulus with a finite gap through numerical simulations of the full set of nonlinear convection equations. Three representative cases are investigated in detail: a large-gap annulus with the ratio of the radii ( s i and s o ) of the sidewalls ξ = s i / o s = 0.1, a medium-gap annulus with ξ = 0.35 and a small-gap annulus with ξ = 0.8. Near the onset of convection, the effect of rapid rotation through the sloping ends forces the first (Hopf) bifurcation in the form of small-scale, steadily drifting rolls (thermal Rossby waves). At moderately large Rayleigh numbers, a variety of different convection patterns are found, including mixed-mode steadily drifting, quasi-periodic (vacillating) and temporally chaotic convection in association with various temporal and spatial symmetry-breaking bifurcations. Our extensive simulations suggest that competition between nonlinear and rotational effects with increasing Rayleigh number leads to an unusual sequence of bifurcation characterized by enlarging the spatial scale of convection.     

Subharmonic oscillations of a forced hydromagnetic cavity

Abstract

We study the propagation of nonlinear MHD waves in a highly magnetized plasma cavity. The cavity's moving boundaries generate Alfvén waves, which in turn drive and interact with slow magnetosonic waves. The interacting wave system is analyzed by a Galerkin and multiple-scale analyses leading to simple dynamical equations. When the frequency of the forcing provided by the moving boundaries and that of the fundamental Alfvén eigenmode are close, the cavity behaves like a Duffing oscillator. Application of the Melnikov function theory shows that the Alfvén wave's amplitude undergoes both flip and saddle-node bifurcations as the amplitude and the phase of the boundary forcing vary. Direct numerical integration confirms these results and provides an estimate of the amount of energy dissipated in the bifurcations.     

Linear Magnetoconvection in Rotating Fluid Spheres Permeated by a Uniform Axial Magnetic Field

A linear analysis of thermally driven magnetoconvection is carried out with emphasis on its application to convection in the Earth's core. We consider a rotating and self-gravitating fluid sphere (or spherical shell) permeated by a uniform magnetic field parallel to the spin axis. In rapidly rotating cases, we find that five different convective modes appear as the uniform field is increased; namely, geostrophic, polar convective, magneto-geostrophic, fast magnetostrophic and slow magnetostrophic modes. The polar convective (P) and magneto-geostrophic (E) modes seem to be of geophysical interest. The P mode is characterized by such an axisymmetric meridional circulation that the fluid penetrates the equatorial plane, suggesting that generation of quadrapole from dipole fields could be explained by a linear process. The E mode is characterized by a few axially aligned columnar rolls which are almost two-dimensional due to a modified Proudman-Taylor theorem.