Rainer Hollerbach

A spectral solution of the magneto-convection equations in spherical geometry

A fully three-dimensional solution of the magneto-convection equations-the nonlinearly coupled momentum, induction and temperature equations-is presented in spherical geometry. Two very different methods for solving the momentum equation are presented, corresponding to the limits of slow and rapid rotation, and their relative advantages and disadvantages are discussed. The possibility of including a freely rotating, finitely conducting inner core in the solution of the momentum and induction equations is also discussed.

On the theory of the geodynamo

Abstract I trace the development of geodynamo theory leading from Larmors original hypothesis (Larmor, 1919, Rep. Br. Assoc. Adv. Sci., A, 159–160) to the present. I consider a number of kinematic results, from Cowlings proof (Cowling, 1934, Mon. Not. R. Astron. Soc., 94: 39–48) that two-dimensional dynamo action is not possible, to the proofs by Backus (1958, Ann. Phys., 4: 372–447) and Herzenberg (1958, Philos. Trans. R. Soc. London, Ser. A, 250: 543–585) that three-dimensional dynamo action is possible. I next turn to various mean-field and convective models in which the fluid flow is no longer kinematically prescribed, but is itself dynamically determined. In these dynamical models, I describe the distinction between weak and strong field regimes that comes about owing to the effect of the field on the pattern of convection in a rapidly rotating system. I consider the dynamics of Taylors constraint (Taylor, 1963, Proc. R. Soc. London, Ser. A, 274: 274–283), and demonstrate how it makes the analysis of the geophysically appropriate strong field regime particularly difficult.

Experimental evidence for magnetorotational instability in a Taylor-Couette flow under the influence of a helical magnetic field.

A recent paper [R. Hollerbach and G. Rudiger, Phys. Rev. Lett. 95, 124501 (2005)] has shown that the threshold for the onset of the magnetorotational instability (MRI) in a Taylor-Couette flow is dramatically reduced if both axial and azimuthal magnetic fields are imposed. In agreement with this prediction, we present results of a Taylor-Couette experiment with the liquid metal alloy GaInSn, showing evidence for the existence of the MRI at Reynolds numbers of order 1000 and Hartmann numbers of order 10.

On the magnetically stabilizing role of the Earth's inner core

Abstract We consider the effect that a finitely conducting inner core may have on the dynamo processes in the outer core. Because a finitely conducting inner core has a diffusive timescale of its own of a few thousand years, which is long compared with the most rapid advective timescales possible in the outer core, the field in the inner core must necessarily average over these very rapid timescales. This averaging-out may then have a stabilizing influence, preventing the very rapid timescales from dominating the dynamo processes in the outer core. In this work we present a solution to the mean-field geodynamo equations which seems to exhibit these features. In this way it may be possible to reconcile the complicated, time-dependent nature of the field and flow in the dynamically active portion of the outer core with the simple, relatively stable nature of the externally observed dipole component of the Earths magnetic field.

New type of magnetorotational instability in cylindrical taylor-couette flow

We study the stability of cylindrical Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields, and show that adding an azimuthal field profoundly alters the previous results for purely axial fields. For small magnetic Prandtl numbers Pm, the critical Reynolds number Re(c) for the onset of the magnetorotational instability becomes independent of Pm, whereas for purely axial fields it scales as Pm-1. For typical liquid metals, Re(c) is then reduced by several orders of magnitude, enough that this new design should succeed in realizing this instability in the laboratory.

A self-consistent convection driven geodynamo model, using a mean field approximation

The magnetic fields generated by thermal convection in a rapidly rotating fluid spherical shell are studied. The shell is sandwiched between a finitely conducting solid inner core and a non-conducting mantle. As the Rayleigh number is increased, the convective motion becomes stronger; when the magnetic Reynolds number becomes larger than a few hundred, dynamo action onsets, and a magnetic field with both axisymmetric and nonaxisymmetric components develops. The magnetic fields generated are generally of the same order of magnitude as the geomagnetic field, and the outer core fluid velocity is consistent with the values deduced from secular variation observations. A mean field approximation is used in which the dynamics of one non-axisymmetric convective mode (the rn = 2 mode being most frequently used) and the associated axisymmetric components are followed. This scheme involves significantly less computation than a fully three-dimensional code, but does not require an arbitrary a-effect to be imposed. Although the Roberts number, q, the ratio of thermal to magnetic diffusion, is small in the Earth, we find that dynamo action is most easily obtained at larger values of q. The Ekman number in our calculations has been taken in the range O(10-3)-O(10-4), which, although small, is larger than the appropriate value for the Earths core. At q = 10 we find solutions at Rayleigh numbers close to critical; two such runs are presented, one corresponding to a weak field dynamo, another to a strong field dynamo; the solution found depends on the initial conditions. At q = 1, the solutions have a complex spatial and temporal structure, with few persistent large-scale features, and our solutions reverse more frequently than the geodynamo. The final run presented has an imposed stable region near the core-mantle boundary. This solution has a weaker non-axisymmetric field, which fits better with the observed geomagnetic field than the solution without the stable layer.

A geodynamo model incorporating a finitely conducting inner core

Hollerbach, R. and Jones, C.A., 1993. A geodynamo model incorporating a finitely conducting inner core. Phys. Earth Planet. Inter., 75:317-327 A direct spectral solution is presented for the mean-field geodynamo equations. The momentum equation, with inertia neglected but viscosity retained, is solved in a spherical shell. The induction equation is solved in a similar fashion, and includes Ohmic dissipation in the finitely conducting inner core. The existence of a second Taylors constraint is pointed out, requiring the integrated Lorentz torque on the inner core to vanish in the limit of vanishing viscosity. In the slightly supercritical regime this constraint is shown to be violated, resulting in a differential rotation of the inner core actively driving a portion of the outer core flow. In the more strongly supercritical regime this constraint is shown to be satisfied, the process of adjustment involving a minimization of the Ohmic dissipation in the inner core.

Magnetohydrodynamic Ekman and Stewartson Layers in a Rotating Spherical Shell

I investigate numerically the flow of an electrically conducting fluid in a differentially rotating spherical shell, in the presence of an imposed magnetic field. For a very weak field the flow is seen to consist of an Ekman layer on the inner and outer spherical boundaries, and a Stewartson layer on the cylinder circumscribing the inner sphere and parallel to the axis of rotation, in agreement with the classical non-magnetic analysis. As the field strength is increased, the non-magnetic Ekman layers merge smoothly into magnetic Ekman-Hartmann layers, and the Stewartson layer is suppressed. In the fully magnetic régime the interior flow consists essentially of a solid-body rotation, with the precise rate determined by a torque balance between the inner and outer Ekman-Hartmann boundary layers.

The influence of Hall drift on the magnetic fields of neutron stars

We consider the evolution of magnetic fields under the influence of Hall drift and Ohmic decay. The governing equation is solved numerically, in a spherical shell with ri/ro = 0.75. Starting with simple free-decay modes as initial conditions, we then consider the subsequent evolution. The Hall effect induces so-called helicoidal oscillations, in which energy is redistributed among the different modes. We find that the amplitude of these oscillations can be quite substantial, with some of the higher harmonics becoming comparable with the original field. Nevertheless, this transfer of energy to the higher harmonics is not sufficient to accelerate significantly the decay of the original field, at least not at the RB = O(100) parameter values accessible to us, where this Hall parameter RB measures the ratio of the Ohmic time-scale to the Hall time-scale. We do find clear evidence though of increasingly fine structures developing for increasingly large RB, suggesting that perhaps this Hall-induced cascade to ever-shorter lengthscales is eventually sufficiently vigorous to enhance the decay of the original field. Finally, the implications for the evolution of neutron star magnetic fields are discussed.

Magnetic processes in astrophysics : theory, simulations, experiments

1. Differential Rotation of Stars 2. Radiation Zones: Magnetic Stability and Rotation 3. Quasilinear Theory of Driven Turbulence 4. The Galactic Dynamo 5. The Magneto-Rotational Instability (MRI) 6. The Tayler Instability (TI) 7. Magnetic Spherical Couette Flow