An Introduction to Magnetohydrodynamics

We review the properties of solar convection that are directly observable at the solar surface, and discuss the relevant underlying physics, concentrating mostly on a range of depths from the temperature minimum down to about 20 Mm below the visible solar surface. The properties of convection at the main energy carrying (granular) scales are tightly constrained by observations, in particular by the detailed shapes of photospheric spectral lines and the topology (time- and length-scales, flow velocities, etc.) of the up- and downflows. Current supercomputer models match these constraints very closely, which lends credence to the models, and allows robust conclusions to be drawn from analysis of the model properties. At larger scales the properties of the convective velocity field at the solar surface are strongly influenced by constraints from mass conservation, with amplitudes of larger scale horizontal motions decreasing roughly in inverse proportion to the scale of the motion. To a large extent, the apparent presence of distinct (meso- and supergranulation) scales is a result of the folding of this spectrum with the effective “filters” corresponding to various observational techniques. Convective motions on successively larger scales advect patterns created by convection on smaller scales; this includes patterns of magnetic field, which thus have an approximately self-similar structure at scales larger than granulation. Radiative-hydrodynamical simulations of solar surface convection can be used as 2D/3D time-dependent models of the solar atmosphere to predict the emergent spectrum. In general, the resulting detailed spectral line profiles agree spectacularly well with observations without invoking any micro- and macroturbulence parameters due to the presence of convective velocities and atmosphere inhomogeneities. One of the most noteworthy results has been a significant reduction in recent years in the derived solar C, N, and O abundances with far-reaching consequences, not the least for helioseismology. Convection in the solar surface layers is also of great importance for helioseismology in other ways; excitation of the wave spectrum occurs primarily in these layers, and convection influences the size of global wave cavity and, hence, the mode frequencies. On local scales convection modulates wave propagation, and supercomputer convection simulations may thus be used to test and calibrate local helioseismic methods. We also discuss the importance of near solar surface convection for the structure and evolution of magnetic patterns: faculae, pores, and sunspots, and briefly address the question of the importance or not of local dynamo action near the solar surface. Finally, we discuss the importance of near solar surface convection as a driver for chromospheric and coronal heating.

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Solar Surface Convection

Hydrodynamic and hydromagnetic stability

The Sun and other stars are magnetic: magnetism pervades their interiors and affects their evolution in a variety of ways In the Sun, both the fields themselves and their influence on other phenomena can be uncovered in exquisite detail, but these observations sample only a moment in a single star's life By turning to observations of other stars, and to theory and simulation, we may infer other aspects of the magnetism-eg, its dependence on stellar age, mass, or rotation rate-that would be invisible from close study of the Sun alone Here, we review observations and theory of magnetism in the Sun and other stars, with a partial focus on the "Solar-stellar connection": ie, ways in which studies of other stars have influenced our understanding of the Sun and vice versa We briefly review techniques by which magnetic fields can be measured (or their presence otherwise inferred) in stars, and then highlight some key observational findings uncovered by such measurements, focusing (in many cases) on those that offer particularly direct constraints on theories of how the fields are built and maintained We turn then to a discussion of how the fields arise in different objects: first, we summarize some essential elements of convection and dynamo theory, including a very brief discussion of mean-field theory and related concepts Next we turn to simulations of convection and magnetism in stellar interiors, highlighting both some peculiarities of field generation in different types of stars and some unifying physical processes that likely influence dynamo action in general We conclude with a brief summary of what we have learned, and a sampling of issues that remain uncertain or unsolved

Magnetism, dynamo action and the solar-stellar connection

Young solar-type stars rotate rapidly and many are magnetically active. Some appear to undergo magnetic cycles similar to the 22 yr solar activity cycle. We conduct simulations of dynamo action in rapidly rotating suns with the three-dimensional magnetohydrodynamic anelastic spherical harmonic (ASH) code to explore dynamo action achieved in the convective envelope of a solar-type star rotating at five times the current solar rotation rate. We find that dynamo action builds substantial organized global-scale magnetic fields in the midst of the convection zone. Striking magnetic wreaths span the convection zone and coexist with the turbulent convection. A surprising feature of this wreath-building dynamo is its rich time dependence. The dynamo exhibits cyclic activity and undergoes quasi-periodic polarity reversals where both the global-scale poloidal and toroidal fields change in sense on a roughly 1500 day timescale. These magnetic activity patterns emerge spontaneously from the turbulent flow and are more organized temporally and spatially than those realized in our previous simulations of the solar dynamo. We assess in detail the competing processes of magnetic field creation and destruction within our simulations that contribute to the global-scale reversals. We find that the mean toroidal fields are built primarily through an Ω-effect, while the mean poloidal fields are built by turbulent correlations which are not well represented by a simple α-effect. During a reversal the magnetic wreaths propagate toward the polar regions, and this appears to arise from a poleward propagating dynamo wave. As the magnetic fields wax and wane in strength and flip in polarity, the primary response in the convective flows involves the axisymmetric differential rotation which varies on similar timescales. Bands of relatively fast and slow fluid propagate toward the poles on timescales of roughly 500 days and are associated with the magnetic structures that propagate in the same fashion. In the Sun, similar patterns are observed in the poleward branch of the torsional oscillations, and these may represent poleward propagating magnetic fields deep below the solar surface.

/pdf/magnetic-cycles-in-a-convective-dynamo-simulation-of-a-young-1lrntjc5rv.pdf

Magnetic cycles in a convective dynamo simulation of a young solar-type star

The vigorous magnetic dynamo action achieved within the convective cores of A-type stars may be influenced by fossil magnetic fields within their radiative envelopes. We study such effects through three-dimensional simulations that model the inner 30% by radius of a 2 M ☉ A-type star, capturing the convective core and a portion of the overlying radiative envelope within our computational domain. We employ the three-dimensional anelastic spherical harmonic code to model turbulent dynamics within a deep rotating spherical shell. The interaction between a fossil field and the core dynamo is examined by introducing a large-scale magnetic field into the radiative envelope of a mature A star dynamo simulation. We find that the inclusion of a twisted toroidal fossil field can lead to a remarkable transition in the core dynamo behavior. Namely, a super-equipartition state can be realized in which the magnetic energy built by dynamo action is 10-fold greater than the kinetic energy of the convection itself. Such strong-field states may suggest that the resulting Lorentz forces should seek to quench the flows, yet we have achieved super-equipartition dynamo action that persists for multiple diffusion times. This is achieved by the relative co-alignment of the flows and magnetic fields in much of the domain, along with some lateral displacements of the fastest flows from the strongest fields. Convection in the presence of such strong magnetic fields typically manifests as 4-6 cylindrical rolls aligned with the rotation axis, each possessing central axial flows that imbue the rolls with a helical nature. The roll system also possesses core-crossing flows that couple distant regions of the core. We find that the magnetic fields exhibit a comparable global topology with broad, continuous swathes of magnetic field linking opposite sides of the convective core. We have explored several poloidal and toroidal fossil field geometries, finding that a poloidal component is essential for a transition to super-equipartition to occur.

http://iopscience.iop.org/article/10.1088/0004-637X/705/1/1000/pdf

Effects of fossil magnetic fields on convective core dynamos in a-type stars

Velocity amplitudes in global convection simulations: The role of the Prandtl number and near-surface driving

Morphological Classification of the Convective Regimes in Rotating Stars

The observational absence of giant convection cells near the Sun’s outer surface is a long-standing conundrum for solar modelers. We herein propose an explanation. Rotation strongly influences the internal dynamics, leading to suppressed convective velocities, enhanced thermal-transport efficiency, and (most significantly) relatively smaller dominant length scales. We specifically predict a characteristic convection length scale of roughly 30-Mm throughout much of the convection zone, implying weak flow amplitudes at 100- to 200-Mm giant cells scales, representative of the total envelope depth. Our reasoning is such that Coriolis forces primarily balance pressure gradients (geostrophy). Background vortex stretching balances baroclinic torques. Both together balance nonlinear advection. Turbulent fluxes convey the excess part of the solar luminosity that radiative diffusion cannot. We show that these four relations determine estimates for the dominant length scales and dynamical amplitudes strictly in terms of known physical quantities. We predict that the dynamical Rossby number for convection is less than unity below the near-surface shear layer, indicating rotational constraint.

https://www.pnas.org/content/pnas/118/31/e2022518118.full.pdf

Rotation suppresses giant-scale solar convection

We present a local helioseismological analysis of the convective flows around sunspots. The flow deductions were obtained from MDI Dopplergrams using ring-analysis techniques to measure Doppler shifts induced in solar acoustic oscillations. A novel multi-scale 3-D inversion procedure was used to self-consistently combine all ring-analysis data taken from a mosaic of analysis tiles spanning the solar disk. The inversion is multi-scale because in addition to folding together information from tiles located at different positions on the solar surface, it is capable of incorporating tiles of different sizes, thus, enabling fine control of the horizontal resolution and the probing depth. The inversion is based on sensitivity kernels computed with the Born approximation. Our inversion results indicate that outflow observed at the surface surrounding sunspots persists to unexpectedly deep depths (~ 7 Mm). These outflows appear to have two components, being comprised of a superficial moat flow and a deeper outflow whose strength peaks around 5 Mm.

https://iopscience.iop.org/article/10.1088/1742-6596/271/1/012002/pdf

Nicholas A. Featherstone

Papers

Effects of fossil magnetic fields on convective core dynamos in a-type stars

Velocity amplitudes in global convection simulations: The role of the Prandtl number and near-surface driving

Morphological Classification of the Convective Regimes in Rotating Stars

Rotation suppresses giant-scale solar convection

Ring-analysis flow measurements of sunspot outflows