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.     

Dissipative structures in a nonlinear dynamo

This article considers magnetic field generation by a fluid flow in a system referred to as the Archontis dynamo: a steady nonlinear MHD state is driven by a prescribed body force. The field and flow become almost equal and dissipation is concentrated in cigar-like structures centred on straight-line separatrices. Numerical scaling laws for energy and dissipation are given that extend previous calculations to smaller diffusivities. The symmetries of the dynamo are set out, together with their implications for the structure of field and flow along the separatrices. The scaling of the cigar-like dissipative regions, as the square root of the diffusivities, is explained by approximations near the separatrices. Rigorous results on the existence and smoothness of solutions to the steady, forced MHD equations are given.     

Understanding the solar dynamo

Paul Bushby and Joanne Mason take a look at the workings of the Sun's dynamo, from the development of physical theory through current ideas and methods, to the possibilities for future understanding of this enigmatic magnetic engine.     

Higher helicity invariants and solar dynamo

Modern models of nonlinear dynamo saturation in celestial bodies (specifically, on the Sun) are largely based on the consideration of the balance of magnetic helicity. This physical variable has also a topological meaning: it is associated with the linking coefficient of magnetic tubes. In addition to magnetic helicity, magnetohydrodynamics has a number of topological integrals of motion (the so-called higher helicity moments). We have compared these invariants with magnetic helicity properties and concluded that they can hardly serve as nonlinear constraints on dynamo action.     

An overshoot solar dynamo with a strong return meridional flow

The meridional circulation plays an essential role in determining the basic mechanism of the dynamo action in the case of a low eddy diffusivity. Flux-transport dynamos with strong return flow and a deep stagnation point are discussed in the case of a positive α-effect located in the overshoot layer and a rotation law consistent with helioseismology. By means of a linear dynamo model, it will be shown that the migration of the toroidal belts at lower latitudes and the periods of the activity cycles are consistent with the observations. Moreover, at variance with previous investigations, the typical critical dynamo numbers of dipolar solutions are significantly smaller than those of quadrupolar solutions even in the regime of strong flow.     

The solar magnetic flux mid-term periodicities and the solar dynamo

We investigate here the fluctuations in the total, open and closed solar magnetic flux (SMF) for the period 1971–1999 by means of the maximum entropy method in the frequency range 5×10⁻⁹–10⁻⁷ Hz (6 yr to 120 days). We use monthly data for the total, open and closed magnetic solar fluxes. Periodicities found in the series are similar showing that there is some relationship between the fluxes. The most important finding of this work is the existence of fluctuations at around 1.3 and 1.7 yr in the SMF with alternating importance during consecutive even and odd solar cycles. These fluctuations are directly related with variations present in cosmic rays, solar wind parameters and geomagnetic activity indexes. A quasi-triennial periodicity previously found in sunspots and other solar phenomena is also of importance. The SMF is generated by the action of the solar dynamo; therefore, it is through the magnetic flux that the solar dynamo influences several heliospheric phenomena.     

The dynamo basis of solar cycle precursor schemes

We investigate the dynamo underpinning of solar cycle precursor schemes based on direct or indirect measures of the solar surface magnetic field. We do so for various types of mean-field-like kinematic axisymmetric dynamo models, where amplitude fluctuations are driven by zero-mean stochastic forcing of the dynamo number controlling the strength of the poloidal source term. In all stochastically forced models considered, the surface poloidal magnetic field is found to have precursor value only if it feeds back into the dynamo loop, which suggests that accurate determination of the magnetic flux budget of the solar polar fields may hold the key to dynamo model-based cycle forecasting.     

A shell model for a large-scale turbulent dynamo

This article addresses the interesting and important problem of large-scale magnetic field generation in turbulent flows, using a self-consistent dynamo model recently developed. The main idea of this model is to consider the induction equation for the large-scale magnetic field, integrated consistently with the turbulent dynamics at smaller scales described by a magnetohydrodynamic shell model. The questions of dynamo action threshold, magnetic field saturation, magnetic field reversals, nature of the dynamo transition and the changes of small-scale turbulence as a consequence of the dynamo onset are discussed. In particular, the stability curve obtained by the model integration is shown in a very wide range of values of the magnetic Prandtl number not yet accessible by direct numerical simulation but more realistic for natural dynamos. Moreover, from our analysis it is shown that the large-scale dynamo transition displays a hysteretic behaviour and therefore a subcritical nature. The model successfully reproduces magnetic polarity reversals, showing the capability to generate persistence times which are increasing for decreasing magnetic diffusivity. Moreover, when the system reaches a statistically stationary dynamo state, where the large-scale magnetic field can abruptly reverse its polarity (magnetic reversal state) or not, keeping the same polarity (steady state), it shows an unmistakable tendency towards the energy equipartition for the turbulence at small scale.     

A nonlinear dynamo driven by rapidly rotating convection

In Kim et al. (Kim, E., Hughes, D.W. and Soward, A.M., “An investigation into high conductivity dynamo action driven by rotating convection”, Geophys. Astrophys. Fluid Dynam. 91, 303–332 ().) we investigated kinematic dynamo action driven by rapidly rotating convection in a cylindrical annulus. Here we extend this work to consider self-consistent nonlinear dynamo action in which the back-reaction of the Lorentz force on the flow is taken into account. In particular, we investigate, as a function of magnetic Prandtl number, the evolution of an initially weak magnetic field in two different types of convective flow – one chaotic and the other integrable. On saturation, the latter shows a systematic dependence on the magnetic Prandtl number whereas the former appears not to. In addition, we show how, in keeping with the findings of Cattaneo et al. (Cattaneo, F., Hughes, D.W. and Kim, E., “Suppression of chaos in a simplified nonlinear dynamo model”, Phys. Rev. Lett. 76, 2057–2060 ().), saturation of the growth of the magnetic field is brought about, for the originally chaotic flow, by a strong suppression of chaos.     

A generalized two-disk dynamo model

Abstract

A generalized two-disk dynamo model is considered that includes mechanical friction; this model is intended to simulate in its broad character the behavior of the geodynamo. Fixed points, limit cycles and chaotic attractors are located for different input parameters of the model. The chaotic regimes are of several kinds as are the “routes to chaos”. Several approximate models, helpful for studying the dynamo are discussed. A number of essential differences from the well-known Rikitake dynamo are demonstrated.     

Hydromagnetic dynamo model with spiral convection

Summary The present paper deals with a hydromagnetic dynamo model of the generation mechanism of the Earth's magnetic field. An attempt has been made at selecting a flow-velocity field in the Earth's core which would satisfy the condition 0 for regenerating the field according to [2], and which would yield a velocity field pattern on the core surface as given in the papers by Kahle et al. [9]. These conditions are satisfied by the velocityv=V ₁+U ₂ ^c –V ₂ ^c and, geometrically, this velocity field is represented in space by a spiral convective motion. On the core surface two downflows and two upflows with the corresponding rotating cells may then be found. Only the axisymmetric harmonic component regeneration of the magnetic field has been considered. Adequate regeneration equations have been obtained by means of Braginski's method of quantity estimates in order of magnitude.     

Note on the scalar dynamo model

Abstract

Bayly (1993) introduced and investigated the equation (? _t + v·▽-η ▽²)S = RS as a scalar analogue of the magnetic induction equation. Here, S(r, t) is a scalar function and the flow field v(r, t) and “stretching” function R(r, t) are given independently. This equation is much easier to handle than the corresponding vector equation and, although not of much relevance to the (vector) kinematic dynamo problem, it helps to study some features of the fast dynamo problem. In this note the scalar equation is considered for linear flow and a harmonic potential as stretching function. The steady equation separates into one-dimensional equations, which can be completely solved and therefore allow one to monitor the behaviour of the spectrum in the limit of vanishing diffusivity. For more general homogeneous flows a scaling argument is given which ensures fast dynamo action for certain powers of the harmonic potential. Our results stress the singular behaviour of eigenfunctions in the limit of vanishing diffusivity and the importance of stagnation points in the flow for fast dynamo action.     

On the inverse dynamo problem in a simplified mean-field model

Inverse dynamo theory seeks to gain information about the motion of a liquid conductor from measurements of the magnetic field in the surrounding vacuum. We consider here a highly simplified model problem, namely a steady α²-dynamo in plane geometry with an α-field varying only in the z-direction normal to the conductor–vacuum interface. Based on perturbation theory about constant-α solutions, we find as many integral conditions on α(z) as modes are present in the vacuum field. This result is corroborated by the complete solution of a special case.     

Numerical treatment of the kinematic dynamo model

Summary Since the initial equations are complicated, the treatment of the kinematic dynamo model requires the use of numerical methods. In applying them to the given problem difficulties are encountered, which are not easy to overcome. This paper deals with the analysis of the experience acquired in treating the model of a nearly symmetric dynamo. Three different methods were employed (stationary, oscillatory and general non-stationary), because a combination of several solutions will yield more comprehensive information about the model being studied. Although the results are based on the study of a single particular model, similar problems also occur in other excercises and, therefore, the conclusions have a more general validity.     

Non-steady kinematic dynamo model with spiral convection

Summary The stability of steady states, the evolution of the magnetic field and possible changes of the magnetic field under small changes of velocity are studied on a non-stationary solution of a kinematic dynamo model.     

Model of intermittency of grand minimums and maximums in the solar dynamo

The αΩ-dynamo model with casual fluctuations of parameter α reproduces all main indications of solar grand minimums and maximums. If we take the dependence of turbulent diffusivity on the magnetic field into account, we obtain the phenomenon of hysteresis, when two solutions are possible in a certain interval of dynamo number values: decaying oscillations of weak fields and magnetic cycles with a constant and a large amplitude, which are formed depending on initial conditions. Fluctuations in parameter α result in transitions between these regimes, and the computations indicate that magnetic cycles with a relative large amplitude alternate with epochs of weak magnetic fields. Such behavior can be used as a model of grand minimums and maximums of solar activity.     

A fully implicit model for simulating dynamo action in a Cartesian domain

We present a fully implicit numerical method to solve the incompressible MHD equations in a strongly rotating Cartesian domain. The equations are solved in a primitive variable formulation using a finite volume discretization. In order to use massively parallel computers, we applied a domain decomposition approach in space. The performance of this model is compared with an earlier model, which treated the convective terms of the equations in an explicit manner. Our results indicate that although the fully implicit method needs about three times the memory of the implicit–explicit method, it is superior in terms of computational efficiency. As an application of this model, we investigated the influence of the Prandtl number in the range of 0.01–1000 on the dynamics of the dynamo.     

Nonlinear dynamical modeling of solar cycles using dynamo formulation with turbulent magnetic helicity

Abstract

We describe a sequence of two-dimensional numerical simulations of inflection point instability in a stably stratified shear flow near the ground. The fastest growing Kelvin-Helmholtz modes are studied in detail; in particular we investigate the growth inhibiting effect of the ground which is predicted by linear theory and the Reynolds number dependence of the process of growth to finite amplitude. We consider flows which are both above and below the critical Reynolds number (Re = 300) which has been reported by Woods (1969) to mark the boundary between flows which have turbulent final states and those which do not. A global energy budget reveals a fundamental difference in character of the finite amplitude billows in these two Reynolds number regimes. However, for relatively high Reynolds numbers (Re = 10³) we do not find any explicit evidence for secondary instability. Above the transition Reynolds number the modified mean flow induced by wave growth is characterized by a splitting of the original shear layer and of the in version in which it is embedded.