A Fixed-point Scheme for the Numerical Construction of Magnetohydrostatic Atmospheres in Three Dimensions.

Magnetohydrostatic models of the solar atmosphere are often based on idealized analytic solutions because the underlying equations are too difficult to solve in full generality. Numerical approaches, too, are often limited in scope and have tended to focus on the two-dimensional problem. In this article we develop a numerical method for solving the nonlinear magnetohydrostatic equations in three dimensions. Our method is a fixed-point iteration scheme that extends the method of Grad and Rubin (Proc. 2nd Int. Conf. on Peaceful Uses of Atomic Energy31, 190, 1958) to include a finite gravity force. We apply the method to a test case to demonstrate the method in general and our implementation in code in particular.

Probing the Solar Meridional Circulation Using Fourier Legendre Decomposition.

We apply the helioseismic methodology of Fourier Legendre decomposition to 88 months of Dopplergrams obtained by the Helioseismic and Magnetic Imager (HMI) as the basis of inferring the depth variation of the mean meridional flow, as averaged between 20° and 60° latitude and in time, in both the northern and southern hemispheres. We develop and apply control procedures designed to assess and remove center-to-limb artifacts using measurements obtained by performing the analysis with respect to artificial poles at the east and west limbs. Forward modeling is carried out using sensitivity functions proportional to the mode kinetic energy density to evaluate the consistency of the corrected frequency shifts with models of the depth variation of the meridional circulation in the top half of the convection zone. The results, taken at face value, imply substantial differences between the meridional circulation in the northern and southern hemispheres. The inferred presence of a return (equatorward propagating) flow at a depth of approximately 40 Mm below the photosphere in the northern hemisphere is surprising and appears to be inconsistent with many other helioseismic analyses. This discrepancy may be the result of the inadequacy of our methodology to remove systematic errors in HMI data. Our results appear to be at least qualitatively similar to those by Gizon et al., which point to an anomaly in HMI data that is not present in MDI or GONG data.

Average motion of emerging solar active region polarities: II. Joy's law.

CONTEXT: The tilt of solar active regions described by Joy's law is essential for converting a toroidal field to a poloidal field in Babcock-Leighton dynamo models. In thin flux tube models the Coriolis force causes what we observe as Joy's law, acting on east-west flows as they rise towards the surface. AIMS: Our goal is to measure the evolution of the average tilt angle of hundreds of active regions as they emerge, so that we can constrain the origins of Joy's law. METHODS: We measured the tilt angle of the primary bipoles in 153 emerging active regions (EARs) in the Solar Dynamics Observatory Helioseismic Emerging Active Region survey. We used line-of-sight magnetic field measurements averaged over 6 h to define the polarities and measure the tilt angle up to four days after emergence. RESULTS: We find that at the time of emergence the polarities are on average aligned east-west, and that neither the separation nor the tilt depends on latitude. We do find, however, that EARs at higher latitudes have a faster north-south separation speed than those closer to the equator at the emergence time. After emergence, the tilt angle increases and Joy's law is evident about two days later. The scatter in the tilt angle is independent of flux until about one day after emergence, when we find that higher-flux regions have a smaller scatter in tilt angle than lower-flux regions. CONCLUSIONS: Our finding that active regions emerge with an east-west alignment is consistent with earlier observations, but is still surprising since thin flux tube models predict that tilt angles of rising flux tubes are generated below the surface. Previously reported tilt angle relaxation of deeply anchored flux tubes can be largely explained by the change in east-west separation. We conclude that Joy's law is caused by an inherent north-south separation speed present when the flux first reaches the surface, and that the scatter in the tilt angle is consistent with buffeting of the polarities by supergranulation.

Flows around Averaged Solar Active Regions.

We explore the general properties of near-surface flows around solar active regions. Helioseismic holography is applied to HMI Dopplergrams yielding nearly 5000 flow measurements of 336 unique active regions observed by the Solar Dynamics Observatory between 2010 and 2014. Ensemble averages of the flows, over subsets of regions sorted on the basis of magnetic flux, are performed. These averages show that converging flows, with speeds of about 10 m s-1 and extending up to 10° from the active-region centers, are prevalent and have similar properties for all regions with magnetic flux above 1021 Mx. Retrograde flows are also detected, with amplitudes around 10 m s-1, which predominantly, but not exclusively, flank the polar side of the active regions. We estimate the expected contribution of these active-region flows to longitudinal averages of zonal and meridional flows and demonstrate the plausibility that they are responsible for at least some component of the time-varying global-scale flows. The reliability of our flow determination is tested using publicly available MHD simulations of both quiet-Sun convection and of a sunspot. While validating the overall methodology in general, the sunspot simulation demonstrates the presence of artifacts that may compromise quantitative flow inferences from some helioseismic measurements.

Average surface flows before the formation of solar active regions and their relationship to the supergranulation pattern.

CONTEXT: The emergence of solar active regions is an important but poorly understood aspect of the solar dynamo. AIMS: Knowledge of the flows associated with the rise of active-region-forming magnetic concentrations through the near-surface layers will help determine the mechanisms of active region formation. METHODS: We used helioseismic holography and granulation tracking to measure the horizontal flows at the surface that precede the emergence of active regions. We then averaged these flows over about sixty emerging active regions to reduce the noise, selecting active regions that emerge into relatively quiet Sun. To help interpret the results, we constructed a simple model flow field by generating synthetic "emergence locations" that are probabilistically related to the locations of supergranulation-scale convergence regions in the quiet Sun. RESULTS: The flow maps obtained from helioseismology and granulation tracking are very similar (correlation coefficients for single maps around 0.96). We find that active region emergence is, on average, preceded by converging horizontal flows of amplitude about 40 ms-1. The convergence region extends over about 40 Mm in the east-west direction and about 20 Mm in the north-south direction and is centered in the retrograde direction relative to the emergence location. This flow pattern is largely reproduced by a model in which active region emergence occurs preferentially in the prograde direction relative to supergranulation inflows. CONCLUSIONS: Averaging over many active regions reveals a statistically significant pattern of near-surface flows prior to emergence. The qualitative success of our simple model suggests that rising flux concentrations and supergranule-scale flows interact during the emergence process.

VALIDATING FORWARD MODELING AND INVERSIONS OF HELIOSEISMIC HOLOGRAPHY MEASUREMENTS.

Here we use synthetic data to explore the performance of forward models and inverse methods for helioseismic holography. Specifically, this work presents the first comprehensive test of inverse modeling for flows using lateral-vantage (deep-focus) holography. We derive sensitivity functions in the Born approximation. We then use these sensitivity functions in a series of forward models and inversions of flows from a publicly available magnetohydrodynamic quiet-Sun simulation. The forward travel times computed using the kernels generally compare favorably with measurements obtained by applying holography, in a lateral-vantage configuration, on a 15-hour time series of artificial Dopplergrams extracted from the simulation. Inversions for the horizontal flow components are able to reproduce the flows in the upper 3Mm of the domain, but are compromised by noise at greater depths.

Helioseismic Holography of Simulated Sunspots: dependence of the travel time on magnetic field strength and Wilson depression.

Improving methods for determining the subsurface structure of sunspots from their seismic signature requires a better understanding of the interaction of waves with magnetic field concentrations. We aim to quantify the impact of changes in the internal structure of sunspots on local helioseismic signals. We have numerically simulated the propagation of a stochastic wave field through sunspot models with different properties, accounting for changes in the Wilson depression between 250 and 550 km and in the photospheric umbral magnetic field between 1500 and 3500 G. The results show that travel-time shifts at frequencies above approximately 3.50 mHz (depending on the phase-speed filter) are insensitive to the magnetic field strength. The travel time of these waves is determined exclusively by the Wilson depression and sound-speed perturbation. The travel time of waves with lower frequencies is affected by the direct effect of the magnetic field, although photospheric field strengths below 1500 G do not leave a significant trace on the travel-time measurements. These results could potentially be used to develop simplified travel-time inversion methods.

HELIOSEISMIC HOLOGRAPHY OF SIMULATED SUNSPOTS: MAGNETIC AND THERMAL CONTRIBUTIONS TO TRAVEL TIMES.

Wave propagation through sunspots involves conversion between waves of acoustic and magnetic character. In addition, the thermal structure of sunspots is very different than that of the quiet Sun. As a consequence, the interpretation of local helioseismic measurements of sunspots has long been a challenge. With the aim of understanding these measurements, we carry out numerical simulations of wave propagation through sunspots. Helioseismic holography measurements made from the resulting simulated wavefields show qualitative agreement with observations of real sunspots. We use additional numerical experiments to determine, separately, the influence of the thermal structure of the sunspot and the direct effect of the sunspot magnetic field. We use the ray approximation to show that the travel-time shifts in the thermal (non-magnetic) sunspot model are primarily produced by changes in the wave path due to the Wilson depression rather than variations in the wave speed. This shows that inversions for the subsurface structure of sunspots must account for local changes in the density. In some ranges of horizontal phase speed and frequency there is agreement (within the noise level in the simulations) between the travel times measured in the full magnetic sunspot model and the thermal model. If this conclusion proves to be robust for a wide range of models, it would suggest a path toward inversions for sunspot structure.