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In this paper we investigate the effectiveness of direct statistical simulation (DSS) for two low-order models of dynamo action. The first model, which is a simple model of solar and stellar dynamo action, is third-order and has cubic nonlinearities whilst the second has only quadratic nonlinearities and describes the interaction of convection and an aperiodically reversing magnetic field. We show how DSS can be utilised to solve for the statistics of these systems of equations both in the presence and the absence of stochastic terms, by truncating the cumulant hierarchy at either second or third order. We compare two different techniques for solving for the statistics, timestepping -- which is able to locate only stable solutions of the equations for the statistics and direct detection of the fixed points. We develop a complete methodology and symbolic package in Python for deriving the statistical equations governing the Low-order dynamic systems in cumulant expansions. We demonstrate that although direct detection of the fixed points is efficient and accurate for DSS truncated at second order, the addition of higher order terms leads to the inclusion of many unstable fixed points that may be found by direct detection of the fixed point by iterative methods. In those cases timestepping is a more robust protocol for finding meaninful solutions to DSS.
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Surface observations indicate that the speed of the solar meridional circulation in the photosphere varies in anti-phase with the solar cycle. The current explanation for the source of this variation is that inflows into active regions alter the global surface pattern of the meridional circulation. When these localized inflows are integrated over a full hemisphere, they contribute to the slow down of the axisymmetric poleward horizontal component. The behavior of this large scale flow deep inside the convection zone remains largely unknown. Present helioseismic techniques are not sensitive enough to capture the dynamics of this weak large scale flow. Moreover, the large time of integration needed to map the meridional circulation inside the convection zone, also masks some of the possible dynamics on shorter timescales. In this work we examine the dynamics of the meridional circulation that emerges from a 3D MHD global simulation of the solar convection zone. Our aim is to assess and quantify the behavior of meridional circulation deep inside the convection zone, where the cyclic large-scale magnetic field can reach considerable strength. Our analyses indicate that the meridional circulation morphology and amplitude are both highly influenced by the magnetic field, via the impact of magnetic torques on the global angular momentum distribution. A dynamic feature induced by these magnetic torques is the development of a prominent upward flow at mid latitudes in the lower convection zone that occurs near the equatorward edge of the toroidal bands and that peaks during cycle maximum. Globally, the dynamo-generated large-scale magnetic field drives variations in the meridional flow, in stark contrast to the conventional kinematic flux transport view of the magnetic field being advected passively by the flow.
In this study we discuss two key issues related to a small-scale dynamo instability at low magnetic Prandtl numbers and large magnetic Reynolds numbers, namely: (i) the scaling for the growth rate of small-scale dynamo instability in the vicinity of the dynamo threshold; (ii) the existence of the Golitsyn spectrum of magnetic fluctuations in small-scale dynamos. There are two different asymptotics for the small-scale dynamo growth rate: in the vicinity of the threshold of the excitation of the small-scale dynamo instability, $lambda propto ln({rm Rm}/ {rm Rm}^{rm cr})$, and when the magnetic Reynolds number is much larger than the threshold of the excitation of the small-scale dynamo instability, $lambda propto {rm Rm}^{1/2}$, where ${rm Rm}^{rm cr}$ is the small-scale dynamo instability threshold in the magnetic Reynolds number ${rm Rm}$. We demonstrated that the existence of the Golitsyn spectrum of magnetic fluctuations requires a finite correlation time of the random velocity field. On the other hand, the influence of the Golitsyn spectrum on the small-scale dynamo instability is minor. This is the reason why it is so difficult to observe this spectrum in direct numerical simulations for the small-scale dynamo with low magnetic Prandtl numbers.
We present a three--pronged numerical approach to the dynamo problem at low magnetic Prandtl numbers $P_M$. The difficulty of resolving a large range of scales is circumvented by combining Direct Numerical Simulations, a Lagrangian-averaged model, and Large-Eddy Simulations (LES). The flow is generated by the Taylor-Green forcing; it combines a well defined structure at large scales and turbulent fluctuations at small scales. Our main findings are: (i) dynamos are observed from $P_M=1$ down to $P_M=10^{-2}$; (ii) the critical magnetic Reynolds number increases sharply with $P_M^{-1}$ as turbulence sets in and then saturates; (iii) in the linear growth phase, the most unstable magnetic modes move to small scales as $P_M$ is decreased and a Kazantsev $k^{3/2}$ spectrum develops; then the dynamo grows at large scales and modifies the turbulent velocity fluctuations.
Much work on turbulent three-dimensional dynamos has been done using triply periodic domains, in which there are no magnetic helicity fluxes. Here we present simulations where the turbulent intensity is still nearly homogeneous, but now there is a perfect conductor boundary condition on one end and a vertical field or pseudo-vacuum condition on the other. This leads to migratory dynamo waves. Good agreement with a corresponding analytically solvable alpha^2 dynamo is found. Magnetic helicity fluxes are studied in both types of models. It is found that at moderate magnetic Reynolds numbers, most of the magnetic helicity losses occur at large scales. Whether this changes at even larger magnetic Reynolds numbers, as required for alleviating the catastrophic dynamo quenching problem, remains still unclear.
We present a statistical analysis of turbulent convection in stars within our Reynolds-Averaged Navier Stokes (RANS) framework in spherical geometry which we derived from first principles. The primary results reported in this document include: (1) an extensive set of mean-field equations for compressible, multi-species hydrodynamics, and (2) corresponding mean-field data computed from various simulation models. Some supplementary scale analysis data is also presented. The simulation data which is presented includes: (1) shell convection during oxygen burning in a 23 solar mass supernova progenitor, (2) envelope convection in a 5 solar mass red giant, (3) shell convection during the helium flash, and (4) a hydrogen injection flash in a 1.25 solar mass star. These simulations have been partially described previously in Meakin [2006], Meakin and Arnett [2007a,b, 2010], Arnett et al. [2009, 2010], Viallet et al. [2011, 2013a,b] and Mocak et al. [2009, 2011]. New data is also included in this document with several new domain and resolution configurations as well as some variations in the physical model such as convection zone depth and driving source term. The long term goal of this work is to aid in the development of more sophisticated models for treating hydrodynamic phenomena (e.g., turbulent convection) in the field of stellar evolution by providing a direct link between 3D simulation data and the mean fields which are modeled by 1D stellar evolution codes. As such, this data can be used to test previously proposed turbulence models found in the literature and sometimes used in stellar modeling. This data can also serve to test basic physical principles for model building and inspire new prescriptions for use in 1D evolution codes.
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