We present a series of radiative MHD simulations addressing the origin and distribution of mixed polarity magnetic field in the solar photosphere. To this end we consider numerical simulations that cover the uppermost 2 − 6 Mm of the solar convection zone and we explore scales ranging from 2 km to 25 Mm. We study how the strength and distribution of magnetic field in the photosphere and subsurface layers depend on resolution, domain size and boundary conditions. We find that 50% of the magnetic energy at the τ = 1 level comes from field with the less than 500 G strength and that 50% of the energy resides on scales smaller than about 100 km. While probability distribution functions are essentially independent of resolution, properly describing the spectral energy distribution requires grid spacings of 8 km or smaller. The formation of flux concentrations in the photosphere exceeding 1 kG requires a mean vertical field strength greater than 30 − 40 G at τ = 1. The filling factor of kG flux concentrations increases with overall domain size as magnetic field becomes organized by larger, longer lived flow structures. A solution with a mean vertical field strength of around 85 G at τ = 1 requires a subsurface RMS field strength increasing with depth at the same rate as the equipartition field strength. We consider this an upper limit for the quiet Sun field strength, which implies that most of the convection zone is magnetized close to equipartition. We discuss these findings in view of recent high-resolution spectropolarimetric observations of quiet Sun magnetism.

Results of a 3D MHD simulation of a sunspot with a photospheric size of about 20 Mm are presented. The simulation has been carried out with the MURaM code, which includes a realistic equation of state with partial ionization and radiative transfer along many ray directions. The largely relaxed state of the sunspot shows a division in a central dark umbral region with bright dots and a penumbra showing bright filaments of about 2 to 3 Mm length with central dark lanes. By a process similar to the formation of umbral dots, the penumbral filaments result from magneto-convection in the form of upflow plumes, which become elongated by the presence of an inclined magnetic field: the upflow is deflected in the outward direction while the magnetic field is weakened and becomes almost horizontal in the upper part of the plume near the level of optical depth unity. A dark lane forms owing to the piling up of matter near the cusp-shaped top of the rising plume that leads to an upward bulging of the surfaces of constant optical depth. The simulated penumbral structure corresponds well to the observationally inferred interlocking-comb structure of the magnetic field with Evershed outflows along dark-laned filaments with nearly horizontal magnetic field and overturning perpendicular ('twisting') motion, which are embedded in a background of stronger and less inclined field. Photospheric spectral lines are formed at the very top and somewhat above the upflow plumes, so that they do not fully sense the strong flow as well as the large field inclination and significant field strength reduction in the upper part of the plume structures.

We present a radiative magnetohydrodynamics simulation of the formation of an active region (AR) on the solar surface. The simulation models the rise of a buoyant magnetic flux bundle from a depth of 7.5 Mm in the convection zone up into the solar photosphere. The rise of the magnetic plasma in the convection zone is accompanied by predominantly horizontal expansion. Such an expansion leads to a scaling relation between the plasma density and the magnetic field strength such that B ∝ 1/2 . The emergence of magnetic flux into the photosphere appears as a complex magnetic pattern, which results from the interaction of the rising magnetic field with the turbulent convective flows. Small-scale magnetic elements at the surface first appear, followed by their gradual coalescence into larger magnetic concentrations, which eventually results in the formation of a pair of opposite polarity spots. Although the mean flow pattern in the vicinity of the developing spots is directed radially outward, correlations between the magnetic field and velocity field fluctuations allow the spots to accumulate flux. Such correlations result from the Lorentz-force-driven, counterstreaming motion of opposite polarity fragments. The formation of the simulated AR is accompanied by transient light bridges between umbrae and umbral dots. Together with recent sunspot modeling, this work highlights the common magnetoconvective origin of umbral dots, light bridges, and penumbral filaments.

We present a simple model for the solar differential rotation and meridional circulation based on a mean field parameterization of the Reynolds stresses that drive the differential rotation. We include the subadiabatic part of the tachocline and show that this, in conjunction with turbulent heat conductivity within the convection zone and overshoot region, provides the key physics to break the Taylor-Proudman constraint, which dictates differential rotation with contour lines parallel to the axis of rotation in case of an isentropic stratification. We show that differential rotation with contour lines inclined by 10 • − 30 • with respect to the axis of rotation is a robust result of the model, which does not depend on the details of the Reynolds stress and the assumed viscosity, as long as the Reynolds stress transports angular momentum toward the equator. The meridional flow is more sensitive with respect to the details of the assumed Reynolds stress, but a flow cell, equatorward at the base of the convection zone and poleward in the upper half of the convection zone, is the preferred flow pattern.

We present a new version of the MURaM radiative MHD code that allows for simulations spanning from the upper convection zone into the solar corona. We implemented the relevant coronal physics in terms of optically thin radiative loss, field aligned heat conduction and an equilibrium ionization equation of state. We artificially limit the coronal Alfvén and heat conduction speeds to computationally manageable values using an approximation to semi-relativistic MHD with an artificially reduced speed of light (Boris correction). We present example solutions ranging from quiet to active Sun in order to verify the validity of our approach. We quantify the role of numerical diffusivity for the effective coronal heating. We find that the (numerical) magnetic Prandtl number determines the ratio of resistive to viscous heating and that owing to the very large magnetic Prandtl number of the solar corona, heating is expected to happen predominantly through viscous dissipation. We find that reasonable solutions can be obtained with values of the reduced speed of light just marginally larger than the maximum sound speed. Overall this leads to a fully explicit code that can compute the time evolution of the solar corona in response to photospheric driving using numerical time steps not much smaller than 0.1 seconds. Numerical simulations of the coronal response to flux emergence covering a time span of a few days are well within reach using this approach.

Sunspots are concentrations of magnetic field on the visible solar surface that strongly affect the convective energy transport in their interior and surroundings. The filamentary outer regions (penumbrae) of sunspots show systematic radial outward flows along channels of nearly horizontal magnetic field. These flows were discovered 100 years ago and are present in all fully developed sunspots. By using a comprehensive numerical simulation of a sunspot pair, we show that penumbral structures with such outflows form when the average magnetic field inclination to the vertical exceeds about 45 degrees. The systematic outflows are a component of the convective flows that provide the upward energy transport and result from anisotropy introduced by the presence of the inclined magnetic field.

In this paper we discuss a dynamic flux-transport dynamo model that includes the feedback of the induced magnetic field on differential rotation and meridional flow. We consider two different approaches for the feedback: mean field Lorentz force and quenching of transport coefficients such as turbulent viscosity and heat conductivity. We find that even strong feedback on the meridional flow does not change the character of the flux-transport dynamo significantly; however it leads to a significant reduction of differential rotation. To a large degree independent from the dynamo parameters, the saturation takes place when the toroidal field at the base of the convection zone reaches between 1.2 an 1.5 T, the energy converted into magnetic energy corresponds to about 0.1% to 0.2% of the solar luminosity. The torsional oscillations produced through Lorentz force feedback on differential rotation show a dominant poleward propagating branch with the correct phase relation to the magnetic cycle. We show that incorporating enhanced surface cooling of the active region belt (as proposed by Spruit) leads to an equatorward propagating branch in good agreement with observations.

We present numerical simulations of active region scale flux emergence covering a time span of up to 6 days. Flux emergence is driven by a bottom boundary condition that advects a semi-torus of magnetic field with 1.7 × 10 22 Mx flux into the computational domain. The simulations show that, even in the absense of twist, the magnetic flux is able the rise through the upper 15.5 Mm of the convection zone and emerge into the photosphere to form spots. We find that spot formation is sensitive to the persistence of upflows at the bottom boundary footpoints, i.e. a continuing upflow would prevent spot formation. In addition, the presence of a torus-aligned flow (such flow into the retrograde direction is expected from angular momentum conservation during the rise of flux ropes through the convection zone) leads to a significant asymmetry between the pair of spots, with the spot corresponding to the leading spot on the Sun being more axisymmetric and coherent, but also forming with a delay relative to the following spot. The spot formation phase transitions directly into a decay phase. Subsurface flows fragment the magnetic field and lead to intrusions of almost field free plasma underneath the photosphere. When such intrusions reach photospheric layers, the spot fragments. The time scale for spot decay is comparable to the longest convective time scales present in the simulation domain. We find that the dispersal of flux from a simulated spot in the first two days of the decay phase is consistent with self-similar decay by turbulent diffusion.