Axel Brandenburg

Astrophysical magnetic fields and nonlinear dynamo theory

The invention concerns a new process for the production of 2-alkoxy-4-propen-1-yl phenols of the formula (I), in which R represents the methyl or ethyl radical wherein 2-alkoxy phenols of the formula (II), in which R is as defined above, are initially condensed with propionaldehyde in the presence of acid catalysts, and the resulting condensation product is subsequently split by heating in the presence of a basic catalyst to form the phenols of formula I.

Dynamo-generated Turbulence and Large-Scale Magnetic Fields in a Keplerian Shear Flow

The nonlinear evolution of magnetized Keplerian shear ows is simulated in a local, three-dimensional model, including the eeects of compressibility and stratiication. Supersonic ows are initially generated by the Balbus-Hawley magnetic shear instability. The resulting ows regenerate a turbulent magnetic eld which, in turn, reinforces the turbulence. Thus, the system acts like a dynamo that generates its own turbulence. However, unlike usual dynamos, the magnetic energy exceeds the kinetic energy of the turbulence by a factor of 3{10. By assuming the eld to be vertical on the outer (upper and lower) surfaces we do not constrain the horizontal magnetic ux. Indeed, a large scale toroidal magnetic eld is generated, mostly in the form of toroidal ux tubes with lengths comparable to the toroidal extent of the box. This large scale eld is mainly of 1 even (i.e. quadrupolar) parity with respect to the midplane and changes direction on a timescale of about 30 orbits, in a possibly cyclic manner. The eeective Shakura-Sunyaev alpha viscosity parameter is between 0.001 and 0.005, and the contribution from the Maxwell stress is about 3-7 times larger than the contribution from the Reynolds stress.

The Inverse Cascade and Nonlinear Alpha-Effect in Simulations of Isotropic Helical Hydromagnetic Turbulence

A numerical model of isotropic homogeneous turbulence with helical forcing is investigated. The resulting flow, which is essentially the prototype of the α2 dynamo of mean field dynamo theory, produces strong dynamo action with an additional large-scale field on the scale of the box (at wavenumber k = 1; forcing is at k = 5). This large-scale field is nearly force free and exceeds the equipartition value. As the magnetic Reynolds number Rm increases, the saturation field strength and the growth rate of the dynamo increase. However, the time it takes to build up the large-scale field from equipartition to its final superequipartition value increases with magnetic Reynolds number. The large-scale field generation can be identified as being due to nonlocal interactions originating from the forcing scale, which is characteristic of the α-effect. Both α and turbulent magnetic diffusivity ηt are determined simultaneously using numerical experiments where the mean field is modified artificially. Both quantities are quenched in an Rm-dependent fashion. The evolution of the energy of the mean field matches that predicted by an α2 dynamo model with similar α and ηt quenchings. For this model an analytic solution is given that matches the results of the simulations. The simulations are numerically robust in that the shape of the spectrum at large scales is unchanged when changing the resolution from 303 to 1203 mesh points, or when increasing the magnetic Prandtl number (viscosity/magnetic diffusivity) from 1 to 100. Increasing the forcing wavenumber to 30 (i.e., increasing the scale separation) makes the inverse cascade effect more pronounced, although it remains otherwise qualitatively unchanged.

The Case for a distributed solar dynamo shaped by near-surface shear

Arguments for and against the widely accepted picture of a solar dynamo being seated in the tachocline are reviewed, and alternative ideas concerning dynamos operating in the bulk of the convection zone, or perhaps even in the near-surface shear layer, are discussed. Based on the angular velocities of magnetic tracers, it is argued that the observations are compatible with a distributed dynamo that may be strongly shaped by the near-surface shear layer. Direct simulations of dynamo action in a slab with turbulence and shear are presented to discuss filling factor and tilt angles of bipolar regions in such a model.

Time Evolution of the Magnetic Activity Cycle Period. II. Results for an Expanded Stellar Sample

We further explore nondimensional relationships between the magnetic dynamo cycle period Pcyc, the rotational period Prot, the activity level (as observed in Ca II HK), and other stellar properties by expanding the stellar sample studied in the first paper in this series. We do this by adding photometric and other cycles seen in active stars and the secondaries of CV systems and by selectively adding less certain cycles from the Mount Wilson HK survey; evolved stars, long-term HK trends and secondary Pcyc are also considered. We confirm that most stars with age t 0.1 Gyr occupy two roughly parallel branches, separated by a factor of ~6 in Pcyc, with the ratio of cycle and rotational frequencies ωcyc/Ω Ro-0.5, where Ro is the Rossby number. Using the model of the first paper in this series, this result implies that the α effect increases with mean magnetic field (contrary to the traditional α-quenching concept) and that α and ωcyc decrease with t. Stars are not strictly segregated onto one or the other branch by activity level, though the high-ωcyc/Ω branch is primarily composed of inactive stars. The expanded data set suggests that for t 1 Gyr, stars can have cycles on one or both branches, though among older stars, those with higher (lower) mass tend to have their primary Pcyc on the lower (upper) ωcyc/Ω branch. The Suns ~80 yr Gleissberg cycle agrees with this scenario, suggesting that long-term activity trends in many stars may be segments of long (Pcyc ~ 50-100 yr) cycles not yet resolved by the data. Most very active stars (Prot < 3 days) appear to occupy a new, third branch with ωcyc/Ω Ro0.4. Many RS CVn variables lie in a transition region between the two most active branches. We compare our results with various models, discuss their implications for dynamo theory and evolution, and use them to predict Pcyc for three groups: stars with long-term HK trends, stars in young open clusters, and stars that may be in Maunder-like magnetic minima.

Simulations of nonhelical hydromagnetic turbulence

Nonhelical hydromagnetic forced turbulence is investigated using large scale simulations on up to 256 processors and 1024(3) mesh points. The magnetic Prandtl number is varied between 1/8 and 30, although in most cases it is unity. When the magnetic Reynolds number is based on the inverse forcing wave number, the critical value for dynamo action is shown to be around 35 for magnetic Prandtl number of unity. For small magnetic Prandtl numbers we find the critical magnetic Reynolds number to increase with decreasing magnetic Prandtl number. The Kazantsev k(3/2) spectrum for magnetic energy is confirmed for the kinematic regime, i.e., when nonlinear effects are still unimportant and when the magnetic Prandtl number is unity. In the nonlinear regime, the energy budget converges for large Reynolds numbers (around 1000) such that for our parameters about 70% is in kinetic energy and about 30% is in magnetic energy. The energy dissipation rates are converged to 30% viscous dissipation and 70% resistive dissipation. Second-order structure functions of the Elsasser variables give evidence for a k(-5/3) spectrum. Nevertheless, the three-dimensional spectrum is close to k(-3/2), but we argue that this is due to the bottleneck effect. The bottleneck effect is shown to be equally strong both for magnetic and nonmagnetic turbulence, but it is far weaker in one-dimensional spectra that are normally studied in laboratory turbulence. Structure function exponents for other orders are well described by the She-Leveque formula, but the velocity field is significantly less intermittent and the magnetic field is more intermittent than the Elsasser variables.

Dynamo action in stratified convection with overshoot

Results are presented from direct simulations of turbulent compressible hydromagnetic convection above a stable overshoot layer. Spontaneous dynamo action occurs followed by saturation, with most of the generated magnetic field appearing as coherent flux tubes in the vicinity of strong downdrafts, where both the generation and destruction of magnetic field is most vigorous. Whether or not this field is amplified depends on the sizes of the magnetic Reynolds and magnetic Prandtl numbers. Joule dissipation is balanced mainly by the work done against the magnetic curvature force. It is this curvature force which is also responsible for the saturation of the dynamo.

Magnetic Field Generation in Fully Convective Rotating Spheres

Magnetohydrodynamic simulations of fully convective, rotating spheres with volume heating near the center and cooling at the surface are presented. The dynamo-generated magnetic field saturates at equipartition field strength near the surface. In the interior, the field is dominated by small-scale structures, but outside the sphere, by the global scale. Azimuthal averages of the field reveal a large-scale field of smaller amplitude also inside the star. The internal angular velocity shows some tendency to be constant along cylinders and is antisolar (fastest at the poles and slowest at the equator).

Magnetic structures in a dynamo simulation

We use three-dimensional simulations to study compressible convection in a rotating frame with magnetic fields and overshoot into surrounding stable layers. The, initially weak, magnetic field is amplified and maintained by dynamo action and becomes organized into flux tubes that are wrapped around vortex tubes. We also observe vortex buoyancy which causes upward flows in the cores of extended downdraughts. An analysis of the angles between various vector fields shows that there is a tendency for the magnetic field to be parallel or antiparallel to the vorticity vector, especially when the magnetic field is strong. The magnetic energy spectrum has a short inertial range with a slope compatible with k +1/3 during the early growth phase of the dynamo. During the saturated state the slope is compatible with k −1 . A simple analysis based on various characteristic timescales and energy transfer rates highlights important qualitative ideas regarding the energy budget of hydromagnetic dynamos.

Dynamic Nonlinearity in Large-Scale Dynamos with Shear

We supplement the mean field dynamo growth equation with the total magnetic helicity evolution equation. This provides an explicitly time-dependent model for � -quenching in dynamo theory. For dynamos without shear, this approach accounts for the observed large-scale field growth and saturation in numerical simulations. After a significant kinematic phase, the dynamo is resistively quenched, i.e., the saturation time depends on the microscopic resistivity. This is independent of whether or not the turbulent diffusivity is resistively quenched. We find that the approach is also successful for dynamos that include shear and exhibit migratory waves (cycles). In this case, however, whether or not the cycle period remains of the order of the dynamical timescale at large magnetic Reynolds numbers does depend on how the turbulent magnetic diffusivity quenches. Since this is unconstrained by magnetic helicity conservation, the diffusivity is currently an input parameter. Comparison with current numerical experiments suggests a turbulent diffusivity that depends only weakly on the magnetic Reynolds number, but higher resolution simulations are needed. Subject headings: magnetic fields — MHD — turbulence