Statistical dynamo theory: Mode excitation
📝 Abstract
We compute statistical properties of the lowest-order multipole coefficients of the magnetic field generated by a dynamo of arbitrary shape. To this end we expand the field in a complete biorthogonal set of base functions, viz. B = sum_k a^k(t) b^k(r). We consider a linear problem and the statistical properties of the fluid flow are supposed to be given. The turbulent convection may have an arbitrary distribution of spatial scales. The time evolution of the expansion coefficients a^k(t) is governed by a stochastic differential equation from which we infer their averages <a^k>, autocorrelation functions <a^k(t) a^{k*}(t+tau)>, and an equation for the cross correlations <a^k a^l*>. The eigenfunctions of the dynamo equation (with eigenvalues lambda_k) turn out to be a preferred set in terms of which our results assume their simplest form. The magnetic field of the dynamo is shown to consist of transiently excited eigenmodes whose frequency and coherence time is given by Im(lambda_k) and -1/(Re lambda_k), respectively. The relative r.m.s. excitation level of the eigenmodes, and hence the distribution of magnetic energy over spatial scales, is determined by linear theory. An expression is derived for <|a^k|^2> / <|a^0|^2> in case the fundamental mode b^0 has a dominant amplitude, and we outline how this expression may be evaluated. It is estimated that <|a^k|^2>/<|a^0|^2> ~ 1/N where N is the number of convective cells in the dynamo. We show that the old problem of a short correlation time (or FOSA) has been partially eliminated. Finally we prove that for a simple statistically steady dynamo with finite resistivity all eigenvalues obey Re(lambda_k) < 0.
💡 Analysis
We compute statistical properties of the lowest-order multipole coefficients of the magnetic field generated by a dynamo of arbitrary shape. To this end we expand the field in a complete biorthogonal set of base functions, viz. B = sum_k a^k(t) b^k(r). We consider a linear problem and the statistical properties of the fluid flow are supposed to be given. The turbulent convection may have an arbitrary distribution of spatial scales. The time evolution of the expansion coefficients a^k(t) is governed by a stochastic differential equation from which we infer their averages <a^k>, autocorrelation functions <a^k(t) a^{k*}(t+tau)>, and an equation for the cross correlations <a^k a^l*>. The eigenfunctions of the dynamo equation (with eigenvalues lambda_k) turn out to be a preferred set in terms of which our results assume their simplest form. The magnetic field of the dynamo is shown to consist of transiently excited eigenmodes whose frequency and coherence time is given by Im(lambda_k) and -1/(Re lambda_k), respectively. The relative r.m.s. excitation level of the eigenmodes, and hence the distribution of magnetic energy over spatial scales, is determined by linear theory. An expression is derived for <|a^k|^2> / <|a^0|^2> in case the fundamental mode b^0 has a dominant amplitude, and we outline how this expression may be evaluated. It is estimated that <|a^k|^2>/<|a^0|^2> ~ 1/N where N is the number of convective cells in the dynamo. We show that the old problem of a short correlation time (or FOSA) has been partially eliminated. Finally we prove that for a simple statistically steady dynamo with finite resistivity all eigenvalues obey Re(lambda_k) < 0.
📄 Content
The origin of the magnetic field of the Earth and the Sun is well understood at the qualitative level. Helical convection acting on the toroidal component of the field generates new poloidal field, while the toroidal field is regenerated either by shear flows acting on the poloidal field or by the same helical convection now acting on the poloidal field. These dynamo processes are in principle able to balance resistive decay and to maintain a magnetic field for very long times. The large scale magnetic fields observed in galactic discs are likewise believed to be due to similar dynamo processes [1]. Self-consistent hydromagnetic simulations that became available since 1995 have confirmed this dynamo picture for the geomagnetic field [2]. Many groups have since then published numerical geodynamo models. Even though the available computational means do not permit the parameters of the models to be ’earth-like’, the magnetic field of the simulations has many properties in common with the observed geomagnetic field [2,3,4,5,6]. For the Sun with its much higher magnetic Reynolds number such simulations are not yet feasible.
The availability of numerical geodynamo models has opened up the possibility for a detailed diagnostics of dynamo action. Kageyama and Sato [7] and Olson et al. [8] found that the regeneration of poloidal and toroidal field resembles the α 2 -dynamo scenario of mean field the- * Electronic address: p.hoyng@sron.nl ory [9]. Wicht and Olson [10] analysed the sequence of events leading to a reversal in a simple numerical dynamo model. Schrinner et al. [11] have inferred mean field tensors α ik and β ikℓ and the mean field B from simulations. The mean field is then compared with the mean field predicted by the dynamo equation. Reasonable agreement was found for a simple magnetoconvection model and for simple dynamo models. For a recent review we refer to Wicht et al. [12].
The evolution of the magnetic field B in the conducting fluid of a dynamo is governed by the induction equation:
The dynamo is located in a volume V with exterior vacuum E, see Fig. 1. The shape of the dynamo need not be spherical. The flow V consists of a stationary component v and the turbulent convection u, which may have an arbitrary distribution of spatial scales:
A popular line of attack is to average over the turbulent convection u and to derive an equation for the mean field B (the so-called mean field dynamo equation or briefly dynamo equation [9,13]). We follow a different path, and we expand the field B in a complete set of functions b i (r). We then determine the statistical properties of the expansion coefficients. The mean field B will appear only occasionally, as a mathematical concept without much physical meaning attached to it.
The idea to study dynamo-generated magnetic fields by an expansion in multipoles goes back to Elsasser [14]. We follow the same technique and obtain a set of equations for the expansion or mode coefficients. These equations contain a random element as the fluid motion V in the induction equation consists of a steady part with a superposed turbulent convective component. The new aspect is that we use the theory of stochastic differential equations [15] to infer the statistical properties of the mode coefficients. We consider a statistically steady, saturated dynamo with a selfconsistent mean flow u and turbulent flow v.
We treat a linear problem and consider u and v as given. Until recently it had been tacidly assumed that the solution of the induction equation (1) represents the selfconsistent field B obtained from a nonlinear solution of the MHD equation (provided one uses the exact selfconsistent flow u + v). But we now know that this is not correct. There are statistically steady, saturated dynamos whose velocity field, taken as a given input flow in the induction equation, acts as a kinematic dynamo with exponentially growing solutions [16,17]. In those cases the induction equation on its own is obviously unable to reproduce the selfconsistent field B. Many questions regarding this unexpected phenomenon remain to be answered. For example, it is not known to what extent it is a universal feature. Schrinner and coworkers (to be submitted) have found several counterexamples, and their results suggest that the flow fields of fast rotating geodynamo models are also kinematically stable. The solution of the induction equation with the (selfconsistent) flow taken from these dynamos is (after a transitory period) up to a constant factor equal to the selfconsistent field B, independent of the initial condition. In the absence of a generally agreed-upon terminology we shall in this paper refer to these dynamos as kinematically stable dynamos.
Here we restrict ourselves to kinematically stable dynamos, so that the solution of (1) faithfully represents the actual field B. Otherwise, the dynamo model is general and may be of the geodynamo or solar type. We take the existence of a linear d
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