scholarly journals Magnetic Helicity and the Geodynamo

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 99
Author(s):  
John V. Shebalin
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Magnetic Energy ◽  
Liquid Core ◽  
Dipole Field ◽  
Magnetic Helicity ◽  
Quasi Equilibrium ◽  
Mhd Turbulence ◽  
Poloidal Magnetic Field ◽  
Dipole Alignment

We present theoretical and computational results in magnetohydrodynamic turbulence that we feel are essential to understanding the geodynamo. These results are based on a mathematical model that focuses on magnetohydrodynamic (MHD) turbulence, but ignores compressibility and thermal effects, as well as imposing model-dependent boundary conditions. A principal finding is that when a turbulent magnetofluid is in quasi-equilibrium, the magnetic energy in the internal dipole component is equal to the magnetic helicity multiplied by the dipole wavenumber. In the case of the Earth, measurement of the exterior magnetic field gives us, through boundary conditions, the internal poloidal magnetic field. The connection between magnetic helicity and dipole field in the liquid core then gives us the toroidal part of the internal dipole field and a model value of 3 mT for the average core dipole magnetic field. Here, we present the theoretical analysis and numerical simulations that lead to these conclusions. We also test an earlier assertion that differential oblateness may be related to dipole alignment, and while there is an effect, rotation appears to be far more important. In addition, the relationship between dipole quasi-stationarity, broken ergodicity and broken symmetry is clarified. Lastly, we discuss how inertial waves in a rotating magnetofluid can affect dipole alignment.

scholarly journals Nonlinear equilibrium structure of thin currents sheets: influence of electron pressure anisotropy

2004 ◽  
Vol 11 (5/6) ◽  
pp. 579-587 ◽  
Author(s):  
L. M. Zelenyi ◽  
H. V. Malova ◽  
V. Yu. Popov ◽  
D. Delcourt ◽  
A. S. Sharma
Keyword(s):  
Magnetic Field ◽  
Current Sheet ◽  
Magnetic Energy ◽  
Plasma Pressure ◽  
Quasi Equilibrium ◽  
Electrostatic Effects ◽  
Current Sheets ◽  
Thin Current Sheet ◽  
Field Lines ◽  
The Magnetic Field

Abstract. Thin current sheets represent important and puzzling sites of magnetic energy storage and subsequent fast release. Such structures are observed in planetary magnetospheres, solar atmosphere and are expected to be widespread in nature. The thin current sheet structure resembles a collapsing MHD solution with a plane singularity. Being potential sites of effective energy accumulation, these structures have received a good deal of attention during the last decade, especially after the launch of the multiprobe CLUSTER mission which is capable of resolving their 3D features. Many theoretical models of thin current sheet dynamics, including the well-known current sheet bifurcation, have been developed recently. A self-consistent 1D analytical model of thin current sheets in which the tension of the magnetic field lines is balanced by the ion inertia rather than by the plasma pressure gradients was developed earlier. The influence of the anisotropic electron population and of the corresponding electrostatic field that acts to restore quasi-neutrality of the plasma is taken into account. It is assumed that the electron motion is fluid-like in the direction perpendicular to the magnetic field and fast enough to support quasi-equilibrium Boltzmann distribution along the field lines. Electrostatic effects lead to an interesting feature of the current density profile inside the current sheet, i.e. a narrow sharp peak of electron current in the very center of the sheet due to fast curvature drift of the particles in this region. The corresponding magnetic field profile becomes much steeper near the neutral plane although the total cross-tail current is in all cases dominated by the ion contribution. The dependence of electrostatic effects on the ion to electron temperature ratio, the curvature of the magnetic field lines, and the average electron magnetic moment is also analyzed. The implications of these effects on the fine structure of thin current sheets and their potential impact on substorm dynamics are presented.

Magnetic Field And Chemical Anomalies of Ap Stars

1976 ◽  
Vol 32 ◽  
pp. 43-48
Author(s):  
A.Z. Dolginov
Keyword(s):  
Magnetic Field ◽  
Liquid Core ◽  
Close Binaries ◽  
Poloidal Magnetic Field ◽  
Stellar Matter ◽  
The Earth ◽  
Chemical Anomalies ◽  
Ap Stars ◽  
The Magnetic Field ◽  
Chemical Anomaly

SummaryIt is shown that owing to the “battery” effect the spatial nonuniform helium distribution in Ap stars causes a toroidal magnetic field of about 103- 104gauss. Possible circulation of stellar matter in this field produce a poloidal magnetic field of the same order as the toroidal one even if the velocities of circulation are about 10-4-10-5cm/s. The chemical anomaly in turn is supported and amplified by the magnetic field. It is pointed out that the “battery” effect is likely to take place in the liquid core of the earth and in components of close binaries if they are nonuniformly heated by their companions.

Parametric investigation of self-similar decay laws in MHD turbulent flows

1999 ◽  
Vol 61 (3) ◽  
pp. 507-541 ◽  
Author(s):  
S. GALTIER ◽  
E. ZIENICKE ◽  
H. POLITANO ◽  
A. POUQUET
Keyword(s):  
Magnetic Field ◽  
Turbulent Flows ◽  
Magnetic Energy ◽  
Nauk Sssr ◽  
Mhd Turbulence ◽  
Integral Scale ◽  
Self Similar ◽  
Grid Points ◽  
The One ◽  
The Magnetic Field

An investigation of the decay laws of energy and of higher moments of the Elsässer fields z±=v±b in the self-similar regime of magnetohydrodynamic (MHD) turbulence is presented, using phenomenological models as well as two-dimensional numerical simulations with periodic boundary conditions and up to 20482 grid points. The results are compared with the generalization of the parameter-free model derived by Galtier et al. [Phys. Rev. Lett.79, 2807 (1997)], which takes into account the slowing down of the dynamics due to the propagation of Alfvén waves. The new model developed here allows for a study in terms of one parameter governing the wavenumber dependence of the energy spectrum at scales of the order of (and larger than) the integral scale of the flow. The one-dimensional compressible case is also dealt with in two of its simplest configurations. Computations are performed for a standard Laplacian diffusion as well as with a hyperdiffusive algorithm. The results are sensitive to the amount of correlation between the velocity and the magnetic field, but rather insensitive to all other parameters such as the initial ratio of kinetic to magnetic energy or the presence or absence of a uniform component of the magnetic field. In all cases, the decay is significantly slower than for neutral fluids in a way that favours for MHD flows the phenomenology of Iroshnikov [Soviet Astron.7, 566 (1963)] and Kraichnan [Phys. Fluids8, 1385 (1965)] as opposed to that of Kolmogorov [Dokl. Akad. Nauk. SSSR31, 538 (1941)]. The temporal evolution of q-moments of the generalized vorticities 〈[mid ]ω±[mid ]q〉 =〈[mid ]ω±j[mid ]q〉 up to order q=10 is also given, and is compared with the prediction of the model. Less agreement obtains as q grows – a fact probably due to intermittency and the development of coherent structures in the form of eddies, and of vorticity and current sheets.

Relaxed states of an ideal MHD plasma with external magnetic field

1995 ◽  
Vol 53 (3) ◽  
pp. 373-385 ◽  
Author(s):  
G. Knorr ◽  
M. Mond ◽  
C. Grabbe
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Vector Fields ◽  
Magnetic Energy ◽  
Adjoint Operator ◽  
Magnetic Helicity ◽  
Turbulent Transition ◽  
Ideal Mhd ◽  
Field Fluctuations ◽  
Base Vector

We study an ideal MHD plasma with the non-vanishing invariants energy, crosshelicity and magnetic helicity, confined in a cylinder with infinitely conducting walls and an externally applied magnetic field B0. The magnetic and velocity fields are expanded in base vector fields, satisfying Δ × Bλ = Bλ. Boundary conditions are imposed to make the curl a self-adjoint operator. The three invariants depend on the time-dependent coefficients of the base vector fields, and are used to construct the partition function to gather statistical information about the equlibrium thermodynamic state to which the plasma relaxes after a turbulent transition. For zero external magnetic field but large magnetic helicity, the energy resides preferentially in magnetic field fluctuations. A sizeable fraction of the kinetic energy initially present is transformed into magnetic energy. The energy condenses via an inverse cascade predominantly to the lowest energy eigenstate, in agreement with results obtained by Taylor. However, since we consider the whole spectrum of eigenstates, the energy does not exclusively occupy the lowest eigenstate. If the eigenvalues are densely spaced (as in a thin torus), the higher eigenmodes also contain appreciable amounts of energy, resulting in a finite pressure of the plasma. For constant and finite external magnetic field, the average induced magnetic field exactly cancels the external field. This indicates that, on a statistical average, the plasma is diamagnetic or superconducting. Superimposed on the average statistical state are fluctuations that may become large if the magnetic helicity is large.

Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds number

1998 ◽  
Vol 358 ◽  
pp. 299-333 ◽  
Author(s):  
OLEG ZIKANOV ◽  
ANDRE THESS
Keyword(s):  
Magnetic Field ◽  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
Boundary Conditions ◽  
Interaction Parameter ◽  
Three Dimensional ◽  
Magnetic Interaction ◽  
Two Dimensional ◽  
Mhd Turbulence ◽  
Magnetic Interaction Parameter

The transformation of initially isotropic turbulent flow of electrically conducting incompressible viscous fluid under the influence of an imposed homogeneous magnetic field is investigated using direct numerical simulation. Under the assumption of large kinetic and small magnetic Reynolds numbers (magnetic Prandtl number Pm[Lt ]1) the quasi-static approximation is applied for the computation of the magnetic field fluctuations. The flow is assumed to be homogeneous and contained in a three-dimensional cubic box with periodic boundary conditions. Large-scale forcing is applied to maintain a statistically steady level of the flow energy. It is found that the pathway traversed by the flow transformation depends decisively on the magnetic interaction parameter (Stuart number). If the magnetic interaction number is small the flow remains three-dimensional and turbulent and no detectable deviation from isotropy is observed. In the case of a strong magnetic field (large magnetic interaction parameter) a rapid transformation to a purely two-dimensional steady state is obtained in agreement with earlier analytical and numerical results for decaying MHD turbulence. At intermediate values of the magnetic interaction parameter the system exhibits intermittent behaviour, characterized by organized quasi-two-dimensional evolution lasting several eddy-turnover times, which is interrupted by strong three-dimensional turbulent bursts. This result implies that the conventional picture of steady angular energy transfer in MHD turbulence must be refined. The spatial structure of the steady two-dimensional final flow obtained in the case of large magnetic interaction parameter is examined. It is found that due to the type of forcing and boundary conditions applied, this state always occurs in the form of a square periodic lattice of alternating vortices occupying the largest possible scale. The stability of this flow to three-dimensional perturbations is analysed using the energy stability method.

scholarly journals On uniqueness of transfer rates in magnetohydrodynamic turbulence

2019 ◽  
Vol 85 (5) ◽  
Author(s):  
Franck Plunian ◽  
Rodion Stepanov ◽  
Mahendra Kumar Verma
Keyword(s):  
Kinetic Energy ◽  
Transfer Rate ◽  
Stokes Equations ◽  
Magnetic Energy ◽  
A Priori ◽  
Magnetic Helicity ◽  
Mhd Turbulence ◽  
Magnetohydrodynamic Turbulence ◽  
Transfer Rates ◽  
Induction Equation

In hydrodynamic and MHD (magnetohydrodynamic) turbulence, formal expressions for the transfer rates rely on integrals over wavenumber triads $(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})$ satisfying $\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=0$ . As an example $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$ denotes the kinetic energy transfer rate to the mode $\boldsymbol{k}$ , from the two other modes in the triad, $\boldsymbol{p}$ and $\boldsymbol{q}$ . However as noted by Kraichnan (Phys. Rev., vol. 111, 1958, pp. 1747–1747), in $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$ , what fraction of the energy transferred to the mode $\boldsymbol{k}$ originated from $\boldsymbol{p}$ and which from $\boldsymbol{q}$ is unknown. Such an expression is thus incongruent with the customary description of turbulence in terms of two-scale energy exchange. Notwithstanding this issue, Dar et al. (Physica D, vol. 157 (3), 2001, pp. 207–225) further decomposed these transfers into separate contributions from $\boldsymbol{p}$ -to- $\boldsymbol{k}$ and $\boldsymbol{q}$ -to- $\boldsymbol{k}$ , thus introducing the concept of mode-to-mode transfers that they applied to MHD turbulence. Doing so, they had to set aside additional transfers circulating within each triad, but failed to calculate them. In the present paper we explain how to derive the complete expressions of the mode-to-mode transfers, including the circulating transfers. We do it for kinetic energy and kinetic helicity in hydrodynamic turbulence, for kinetic energy, magnetic energy and magnetic helicity in MHD turbulence. We find that the degree of non-uniqueness of the energy transfers derived from the induction equation is a priori higher than the one derived from the Navier–Stokes equations. However, separating the contribution of magnetic advection from magnetic stretching, the energy mode-to-mode transfer rates involving the magnetic field become uniquely defined, in striking contrast to the hydrodynamic case. The magnetic helicity mode-to-mode transfer rate is also found to be uniquely defined, contrary to kinetic helicity in hydrodynamics. We find that shell-to-shell transfer rates have the same properties as mode-to-mode transfer rates. Finally calculating the fluxes, we show how the circulating transfers cancel in accordance with conservation laws.

scholarly journals The nature of mean-field generation in three classes of optimal dynamos

2020 ◽  
Vol 86 (1) ◽  
Author(s):  
Axel Brandenburg ◽  
Long Chen
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Boundary Conditions ◽  
Magnetic Energy ◽  
Mean Field ◽  
Dynamo Action ◽  
Magnetic Diffusivity ◽  
The Mean ◽  
The Magnetic Field ◽  
Mean Field Dynamo

In recent years, several optimal dynamos have been discovered. They minimize the magnetic energy dissipation or, equivalently, maximize the growth rate at a fixed magnetic Reynolds number. In the optimal dynamo of Willis (Phys. Rev. Lett., vol. 109, 2012, 251101), we find mean-field dynamo action for planar averages. One component of the magnetic field grows exponentially while the other decays in an oscillatory fashion near onset. This behaviour is different from that of an $\unicode[STIX]{x1D6FC}^{2}$ dynamo, where the two non-vanishing components of the planar averages are coupled and have the same growth rate. For the Willis dynamo, we find that the mean field is excited by a negative turbulent magnetic diffusivity, which has a non-uniform spatial profile near onset. The temporal oscillations in the decaying component are caused by the corresponding component of the diffusivity tensor being complex when the mean field is decaying and, in this way, time dependent. The growing mean field can be modelled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chen et al. (J. Fluid Mech., vol. 783, 2015, pp. 23–45), we find, to some extent, similar mean-field dynamo actions. When the magnetic boundary conditions are mixed, the two components of the planar averaged field grow at different rates when the dynamo is 15 % supercritical. When the mean magnetic field satisfies homogeneous boundary conditions (where the magnetic field is tangential to the boundary), mean-field dynamo action is found for one-dimensional averages, but not for planar averages. Despite having different spatial profiles, both dynamos show negative turbulent magnetic diffusivities. Our finding suggests that negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos.

Axisymmetric global gravitational equilibrium for magnetized, rotating hot plasma

2015 ◽  
Vol 81 (6) ◽  
Author(s):  
Peter J. Catto ◽  
Istvan Pusztai ◽  
Sergei I. Krasheninnikov
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Plasma Density ◽  
Equatorial Plane ◽  
Magnetic Energy ◽  
Plasma Pressure ◽  
Gravitational Energy ◽  
Equilibrium Solutions ◽  
Poloidal Magnetic Field ◽  
Hot Plasma

We present analytic solutions for three-dimensional magnetized axisymmetric equilibria confining rotating hot plasma in a gravitational field. Our up–down symmetric solution to the full Grad–Shafranov equation can exhibit equatorial plane localization of the plasma density and current, resulting in disk equilibria for the plasma density. For very weak magnetic fields and high plasma pressure, we find strongly rotating thin plasma disk gravitational equilibria that satisfy strict Keplerian motion provided the gravitational energy is much larger than the plasma pressure, which must be large compared to the magnetic energy of the poloidal magnetic field. When the rotational energy exceeds the gravitational energy and it is larger than the plasma pressure, diffuse disk equilibrium solutions continue to exist provided the poloidal magnetic energy remains small. For stronger magnetic fields and lower plasma pressure and rotation, we can also find gravitational equilibria with strong localization to the equatorial plane. However, a toroidal magnetic field is almost always necessary to numerically verify these equilibria are valid solutions in the presence of gravity for the cases considered in Catto & Krasheninnikov (J. Plasma Phys., vol. 81, 2015, 105810301). In all cases both analytic and numerical results are presented.

Kinematic dynamos and the Earth’s magnetic field

1973 ◽  
Vol 275 (1251) ◽  
pp. 425-461 ◽  
Keyword(s):  
Magnetic Field ◽  
Inner Core ◽  
Magnetic Energy ◽  
Liquid Core ◽  
Outer Part ◽  
Ohmic Dissipation ◽  
Tidal Dissipation ◽  
Magnetic Dipoles ◽  
Magnetic Energy Density ◽  
The Magnetic Field

The Bullard—Gellman formalism is applied to investigate the existence of convergent solutions for steady kinematic dynamos. It is found that the solutions for the Bullard—Gellman dynamo, as well as for Lilley’s modification of it, do not converge. Convergent solutions have been found for a class of spherical convective cells which would be stationary in a perfect fluid in the absence of rotation and of the magnetic field. By calibrating the theoretical magnetic dipole so as to fit the observed value at the Earth’s surface, one can find a dynamo in the above class which also matches the observed equatorial magnetic dipoles. There is a dynamo which has a rate of total ohmic dissipation of only 1.8 x 1016 erg s-1 for an assumed electrical conductivity of 3 x 10~6 e.m.u.'f This is one thousandth the rate of tidal dissipation, and one hundred thousandth the rate of heat outflow from the surface of the Earth. The required velocities are of the order of 10~3 cm s_1, and the average magnetic energy density is 4 erg cm-3. The internal structure of the magnetic field in this model shows a dynamo mechanism situated in the outer part of the liquid core and is thus insensitive to possible rigidity of the material in the * inner core.

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