Small Scale Kinematic Dynamos: Beyond the α-Effect

In geodynamo simulations which simulate the generation of an axial dipolar magnetic field, the generation mechanism appears to be adequately described as an α²-dynamo with an anisotropic α-effect. The anisotropy in the α-effect favors an equatorial dipole field, however, which calls into question the interpretation in terms of an α²-dynamo. It is shown in this article with kinematic dynamo calculations and exemplary velocity fields with an anisotropic α-effect that both types of dipolar magnetic field can be generated. Two examples of working dynamos in a sphere with flows with zero α-effect are also provided.     

Buoyancy-driven perturbations in a rapidly rotating,electrically conducting fluid: part II – dynamo action

The helicity, electromotive force and α-effect produced in a homogeneous, rapidly rotating, electrically conducting fluid by an isolated source of buoyancy at small Elsasser number are calculated, visualized and analyzed. Due to physical symmetries of the system, the integrals of helicity and electromotive force over all space are zero. However, each has a significant non-zero value when integrated over the cross section of the Taylor column. The local α-effect is found to be significantly anisotropic; it is strongest when the applied magnetic field is toroidal and the resulting EMF is parallel to the applied field.     

Geomagnetic polarity reversals in a turbulent core

The behavior of the main magnetic field components during a polarity transition is investigated using the α²-dynamo model for magnetic field generation in a turbulent core. It is shown that rapid reversals of the dipole field occur when the helicity, a measure of correlation between turbulent velocity and vorticity, changes sign. Two classes of polarity transitions are possible. Within the first class, termed component reversals, the dipole field reverses but the toroidal field does not. Within the second class, termed full reversals, both dipole and toroidal fields reverse. Component reversals result from long term fluctuations in core helicity; full reversals result from short term fluctuations. A set of time-evolution equations are derived which govern the dipole field behavior during an idealized transition. Solutions to these equations exhibit transitions in which the dipole remains axial while its intensity decays rapidly toward zero, and is regenerated with reversed polarity. Assuming an electrical conductivity of 3 × 10⁵ mho m^?1 for the fluid core, the time interval required to complete the reversal process can be as short as 7500 years. This time scale is consistent with paleomagnetic observations of the duration of reversals. A possible explanation of the cause of reversals is proposed, in which the core's net helicity fluctuates in response to fluctuations in the level of turbulence produced by two competing energy sources—thermal convection and segregation of the inner core. Symmetry considerations indicate that, in each hemisphere, helicity generated by heat loss at the core-mantle boundary may have the opposite sign of helicity generated by energy release at the inner core boundary. Random variations in rates of energy release can cause the net helicity and the α-effect to change sign occasionally, provoking a field reversal. In this model, energy release by inner core formation tends to destabilize stationary dynamo action, causing polarity reversals.     

Calculation of magnetic field turbulent diffusivities and α-effect

Abstract

First the exact numerical solutions of DIA system of equations describing the transportation of magnetic field in an infinite medium are presented. It is assumed that the turbulence is stationary, homogeneous, isotropic and incompressible. The spectra of turbulence of δ-type and Kolmogorov's type were used. The steady state values of magnetic field diffusivity D_T and the α-effect coefficient α _T were calculated for various values of space-scale and lifetimes of these spectra and the spectra of helicity. Also investigated is the dependence of D_T and α _T on the degree of helicity. The corrections to the α _T -coefficient due to the contribution of four-order velocity correlators are given. The results are compared with those due to the self-consistent technique.     

Scale-to-scale stabilization of the magnetic field in the dynamo models

The three-dimensional dynamo model in the fast-rotating plane layer heated from below is considered. The transition from the linear generation of the magnetic field to the nonlinear generation is studied. With the use of the wavelet analysis, it is demonstrated how the spatial spectra of the kinetic and magnetic energies, as well as the hydrodynamic, magnetic, cross-, and current helicity, vary in time. The scenarios of the suppression of α-effect (α-quenching) by the magnetic field are suggested.     

α2-Dynamos and taylor's constraint

Abstract

An idealised α²ω-dynamo is considered in which the α-effect is prescribed. The additional ω-effect results from a geostrophic motion whose magnitude is determined indirectly by the Lorentz forces and Ekman suction at the boundary. As the strength of the α-effect is increased, a critical value α? _c is reached at which dynamo activity sets in; α? _c is determined by the solution of the kinematic α²-dynamo problem. In the neighbourhood of the critical value of α? the magnetic field is weak of order E ^1/4(μηρω)^½ due to the control of Ekman suction; E(?1) is the Ekman number. At certain values of α?, viscosity independent solutions are found satisfying Taylor's constraint. They are identified by the bifurcation of a nonlinear eigenvalue problem. Dimensional arguments indicate that following this second bifurcation the magnetic field is strong of order (μηρω)^½. The nature of the transition between the kinematic linear theory and the Taylor state is investigated for various distributions of the α-effect. The character of the transition is found to be strongly model dependent.     

Tuning of the mean-field geodynamo model

Parker’s two-dimensional (2D) dynamo model with an algebraic form of nonlinearity for the α-effect is considered. The model uses geostrophic distributions for the α-effect and differential rotation, which are derived from the three-dimensional (3D) convection models. The resulting configurations of the magnetic field in the liquid core are close to the solutions in Braginsky’s Z-model. The implications of the degree of geostrophy observed in the 3D dynamo models for the behavior of the mean magnetic field are explored. It is shown that the reduction in geostrophy leads to magnetic field reversals accompanied by the relative growth of the nondipole component of the field on the surface of the liquid core. The simulations with a random α-effect which causes turbulent pulsations are carried out. The approach is capable of producing realistic sequences of magnetic reversals.     

Book Review of Accretion Power in Astrophysics (3rd Edition) by Juhan Frank,Andrew King,Derek Raine Cambridge University Press, 2002, XIV+384 pp., £27.95, hardback (ISBN 0 521 62053 8), paperback (ISBN 0 521 62957 8).

The velocity, pressure, perturbation magnetic field, helicity and electromotive force driven by an isolated buoyant parcel in an unbounded, rapidly rotating, electrically conducting fluid in the limit of small Elsasser number and very small Ekman number are calculated, visualized and analyzed. On the scale of the parcel, the solution is identical to that obtained in the limit of small Ekman number and zero Elsasser number. On the scale of the Taylor-column, it is elongated in the direction of the applied magnetic field and compressed in the direction perpendicular to it. The α-effect calculated by averaging the electromotive force on planes normal to rotation is strongly anisotropic: near the parcel and in the inner part of the Taylor-column it is strongest when the applied magnetic field is perpendicular to rotation and gravity; in the outer part of the Taylor-column it is strongest when the applied magnetic field is in the same plane as rotation and gravity.     

Dynamos with a flat α-effect distribution

Abstract

In order to obtain a better insight into the excitation conditions of magnetic fields in flat objects, such as galaxies, we have calculated critical dynamo numbers of different magnetic field modes for spherical dynamos with a flat α-effect distribution. A simple but realistic approximation formula for the rotation curve is employed. In most cases investigated a stationary quadrupole-type solution is preferred. This is a consequence of the flat distribution of the α-effect. Non-axisymmetric fields are in all cases harder to excite than axisymmetric ones. This seems to be the case particularly for flat objects in combination with a realistic rotation curve for galaxies. The question of whether non-axisymmetric (bisymmetric) fields, which are observed in some galaxies, can be explained as dynamos generated by an axisymmetric αω-effect is therefore still open.     

α-effect due to wave reflection from the core mantle boundary

Abstract

Latitude behavior of the boundary α-effect based on reflection of hydromagnetic waves from the Core-Mantle Boundary (CMB) studied earlier by Anufriev (1991), is discussed. Its form given in Figure 2 is in good agreement with that used in model-Z of Braginsky. We want to emphasize the following main features of our α-effect which resemble those of Braginsky: concentration near the boundary, sign-changing of α in the layer and geometrical behavior. The last includes the vanishing of α near the poles and the equator which is characteristic for Braginsky's α-effect. It is also shown that for Magnetic Reynolds Number of order 100 the amplitude of the α-effect and the thickness of the α-layer is of the order of those used in model-Z.     

赤道附近局地激发LS-TAD的事例观测

大尺度行进式大气扰动(LS-TAD)通常被认为起源于极区高纬,本文报道了一起激发于低纬赤道地区、在北半球向高纬传播的LS-TAD观测事例.2002年8月10日07:30 UT,CHAMP卫星在30°W磁赤道附近观测到显著的大气密度增强,在随后的2个连续轨道上,该密度增强依次出现在北半球更高纬度上,表现出明显的极向位移,...     

The flux ejection dynamo effect

Abstract

The mean-field effects of cyclonic convection become increasingly complex when the cyclonic rotation exceeds ½-π. Net helicity is not required, with negative turbulent diffusion, for instance, appearing in mirror symmetric turbulence. This paper points out a new dynamo effect arising in convective cells with strong asymmetry in the rotation of updrafts as against downdrafts. The creation of new magnetic flux arises from the ejection of reserve flux through the open boundary of the dynamo region. It is unlike the familiar α-effect in that individual components of the field may be amplified independently. Several formal examples are provided to illustrate the effect. Occurrence in nature depends upon the existence of fluid rotations of the order of π in the convective updrafts. The flux ejection dynamo may possibly contribute to the generation of field in the convective core of Earth and in the convective zone of the sun and other stars.     

Self-consistent dynamo models driven by hydromagnetic instabilities

The dynamics of the Earth's core are dominated by a balance between Lorentz and Coriolis forces. Previous studies of possible (magnetostrophic) hydromagnetic instabilities in this regime have been confined to geophysically unrealistic flows and fields. In recent papers we have treated rather general fields and flows in a spherical geometry and in a computationally simple plane-layer model. These studies have highlighted the importance of differential rotation in determining the spatial structure of the instability. Here we have proceeded to use these results to construct a self-consistent dynamo model of the geomagnetic field. An iterative procedure is employed in which an α-effect is calculated from the form of the instability and is then used in a mean field dynamo model. The mean zonal field calculated there is then input back into the hydromagnetic stability problem and a new α-effect calculated. The whole procedure is repeated until the input and output zonal fields are the same to some tolerance.     

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.     

Mean-field equations for weakly nonlinear two-scale perturbations of forced hydromagnetic convection in a rotating layer

We consider stability of regimes of hydromagnetic thermal convection in a rotating horizontal layer with free electrically-conducting boundaries, to perturbations involving large spatial and temporal scales. Equations governing the evolution of weakly nonlinear mean perturbations are derived under the assumption that the α-effect is insignificant in the leading-order (e.g. due to a symmetry of the system). The mean-field equations generalise the standard equations of hydromagnetic convection: New terms emerge – a second-order linear operator representing the combined eddy diffusivity and quadratic terms associated with the eddy advection. If the perturbed CHM regime is nonsteady and insignificance of the α-effect in the system does not rely on the presence of a spatial symmetry, the combined eddy diffusivity operator also involves a nonlocal pseudodifferential operator. If the perturbed CHM state is almost symmetric, α-effect terms appear in the mean-field equations as well. Near a point of a symmetry-breaking bifurcation, cubic nonlinearity emerges in the equations. All the new terms are in general anisotropic. A method for evaluation of their coefficients is presented; it requires solution of a significantly smaller number of auxiliary problems than in a straightforward approach.     

The α-Effect and Current Helicity for Fast Sheared Rotators

We explore the f -effect and the small-scale current helicity, , for the case of weakly compressible magnetically driven turbulence that is subjected to the differential rotation. No restriction is applied to the amplitude of angular velocity, i.e., the derivations presented are valid for an arbitrary Coriolis number, z * = 2 z cor , though the differential rotation itself is assumed to be weak. The expressions obtained are used to explore the possible distributions of f -effect and h c in convection zones (CZ) of the solar-type stars. Generally, our theory gives f { { > 0 in the northern hemisphere of the Sun and the opposite case in the southern hemisphere. In most cases the h c has the opposite sign to f { { . However, we show that in the depth of CZ where the influence of rotation upon turbulence (associated with z *) and the radial shear of angular velocity are strong, the distribution of f { { might be drastically different from a classical cos è -dependence, where è is colatitude. It is shown that f { { has a negative sign at the bottom and below of CZ at mid latitudes. There, the distribution of h c is also different from cos è , but it does not change its sign with the depth. Further, we briefly consider these quantities in the disk geometry. The application of the developed theory to dynamos in the accretion disk is more restrictive because they usually have a strong differential rotation, | ‘ log z / ‘ log r | > 1.     

Axially asymmetric velocities in the boundary layer of the nearly symmetric hydromagnetic dynamo

Abstract

An attempt has been made to include the axially asymmetric velocities into the calculation of Braginsky's Z-model of the nearly symmetric hydromagnetic dynamo. In this axisymmetric non-linear model dominated by Lorentz and Coriolis forces and maintained by a specified convection, the α-effect is prescribed. An example is shown of the axially asymmetric Archimedean buoyancy, which can imply an arbitrary alpha effect in the model with viscous core-mantle coupling. The formalisms of Tough and Roberts (1968) is also discussed and a modified α-effect in the Z-model is suggested.     

Deep two-dimensional turbulence: An idealized model for atmospheric jets of the giant outer planets

The present paper proposes a new type of two–dimensional turbulence with a deep configuration as an idealization of the dynamics of the global circulations of the Jovian atmospheres under the constraint of the Taylor–Proudman theorem. A major difference of the deep two-dimensional turbulence from a standard shallow version is that the former reverses the sign of the planetary β-effect, leading to an opposite sign for the equatorial jet under the homogenization tendency of the potential vorticity. As a result, a westerly (prograde) equatorial jet is generated, which contrasts with the easterly (retrograde) equatorial jet obtained using the shallow configuration. The results suggest that the zonal flows of Jupiter and Saturn are probably of deep origin, whereas those of Uranus and Neptune are likely to be of shallow origin.     

The ionospheric F2-region at low geomagnetic latitudes during the geomagnetic storms of 22–26 April 1990: Comparison of observed and modeled response

A comparison between the modeled NmF2 and hmF2 and NmF2 and hmF2, which were observed by the Kokubunji, Okinawa, Manila, Vanimo, and Darwin ionospheric sounders and by the middle and upper (MU) atmosphere radar, have been used to study the time-dependent response of the low-latitude ionosphere to geomagnetic forcing during a time series of geomagnetic storms from 22 to 26 April 1990. The reasonable agreement between the model results and data requires the modified equatorial meridional E×B plasma drift, the modified HWM90 wind, and the modified NRLMSISE-00 neutral densities. We found that changes in a flux of plasma into the nighttime equatorial F2-region from higher L-shells to lower L-shells caused by the meridional component of the E×B plasma drift lead to enhancements in NmF2 close to the geomagnetic equator. The equatorward wind-induced plasma drift along magnetic field lines, which cross the Earth equatorward of about 20° geomagnetic latitude in the northern hemisphere and about −19° geomagnetic latitude in the southern hemisphere, contributes to the maintenance of the F2-layer close to the geomagnetic equator. The nighttime weakening of the equatorial zonal electric field (in comparison with that produced by the empirical model of Fejer and Scherliess [Fejer, B.G., Scherliess, L., 1997. Empirical models of storm time equatorial zonal electric fields. J. Geophys. Res. 102, 24047–24056] or Scherliess and Fejer [Scherliess, L., Fejer, B.G., 1999. Radar and satellite global equatorial F region vertical drift model. J. Geophys. Res. 104, 6829–6842) in combination with corrected equatorward nighttime wind-induced plasma drift along magnetic field lines in the both geomagnetic hemispheres are found to be the physical mechanism of the nighttime NmF2 enhancement formation close to the geomagnetic equator over Manila during 22–26 April 1990. The model crest-to-trough ratios of the equatorial anomaly are used to study the relative role of the main mechanisms of the equatorial anomaly suppression for the 22–26 April 1990 geomagnetic storms. During the most part of the studied time period, a total contribution from geomagnetic storm disturbances in the neutral temperature and densities to the equatorial anomaly changes is less than that from meridional neutral winds and variations in the E×B plasma drift. It is shown that the latitudinal positions of the crests are determined by the E×B drift velocity and the neutral wind velocity.     

The rotational quenching of the rotation-induced kinetic alpha-effect

Abstract

A theory of the non-diffusive anisotropic kinetic alpha-effect (“Γ-effect”) for densitystratified rotating turbulent fluids is developed. No limitations on the rotation rate are imposed and the fully nonlinear dependence of the Γ-effect on the angular velocity is studied. When the Coriolis number, ω? = 2τ ω, is small the dimensionless “dynamo number”, Cτ, characterising the power of the Γ-effect, grows with ω?. The dependence, however, reaches a maximum for ω? ～ 2. For still higher rotation rates C_Λ decreases as 1/ω?. In opposition, the corresponding number, C_x, of the hydromagnetic α² -dynamo problems remains finite for very large ω?. Hence, for fast rotation the hydrodynamic Γ-effect is small while the hydromagnetic α-effect remains large. In consequence, the large-scale magnetic and velocity structures are expected to be generated with roughly equal power in slowly rotating objects. In the rapid rotators, however, generation of the large-scale flows is problematic.