Measuring and Modeling Magnetic Flux Transport on the Sun Dr. David Hathaway NASA Marshall Space Flight Center 2013 August 22 – Ames Research Center

Three Major Solar Mysteries What causes the sunspot cycle? What causes flares (and other explosive events)? How is the Sun’s Corona heated to million degree temperatures (that drive the solar wind)?

Solar Magnetism – The Key

The Sun’s Magnetic Dynamo Key characteristics of the sunspot cycleKey characteristics of the sunspot cycle The role of magnetic flux transportThe role of magnetic flux transport Characterizing the axisymmetric flowsCharacterizing the axisymmetric flows Characterizing the non-axisymmetric flowsCharacterizing the non-axisymmetric flows Modeling the magnetic flux transportModeling the magnetic flux transport Future directionsFuture directions

Highly Variable Amplitudes The average sunspot cycle has a peak sunspot number of ~100 but with variations from 50 to nearly 200 and periods of no sunspots like the Maunder minimum (the magnetic cycle continued through the Maunder minimum but did not produce fields strong enough to make sunspots). There appears to be a cycle (the Gleissberg Cycle) of ~100 years in the amplitudes.

Sunspot Latitudes Sunspots appear in two bands on either side of the equator. These bands drift toward the equator as the cycle progresses. Big cycles have wider bands that extend to higher latitudes. Cycles overlap by 2-3 years.

Hale’s Magnetic Polarity Law Hale’s Magnetic Polarity Law “…the preceding and following spots … are of opposite polarity, and that the corresponding spots of such groups in the Northern and Southern hemispheres are also opposite in sign. Furthermore, the spots of the present cycle are opposite in polarity to those of the last cycle” Hale et al. (1919).

Active Region Tilt- Joy’s Law In that same 1919 paper Joy noted that sunspot groups are tilted with the leading spots closer to the equator than the following spots. This tilt increases with latitude.

Babcock (1961) a) Dipolar field at cycle minimum threads through a shallow layer below the surface. b) Latitudinal differential rotation shears out this poloidal field to produce a strong toroidal field (first at the mid-latitudes then progressively lower latitudes). c) Buoyant fields erupt through the photosphere giving Hale’s polarity law and Joy’s Tilt. d) Meridional transport gives reconnection at the poles and equator. (son Horace, not father Harold)

Surface Field over 3 Cycles One map is constructed for each 27-day rotation of the Sun using data from near the central meridian. These maps show that the field: 1) Emerges in bi-polar active regions (sunspot groups) 2) Diffuses (undergoes a random walk in longitude and latitude) 3) Drifts in longitude (Differential Rotation) 4) Drifts poleward from the equator (Meridional Flow)

The Magnetic Butterfly Diagram Models for the solar dynamo must reproduce this diagram – active latitudes that drift equatorward, Joy’s Law tilt giving different polarity to low and high latitudes, poleward transport of high latitude (following polarity) flux that reverses the field at about the time of cycle maximum.

Polar Fields – Seeds for Cycles The Sun’s polar fields are the seeds of the next solar cycle in most dynamo models. We have direct observations for the last three cycles and now a proxy (polar faculae) for the last 10 cycles. Muñoz-Jaramillo et al. (2013)

Polar Fields as Predictors Muñoz-Jaramillo et al. (2013) find a strong correlation between polar fields and the amplitude of the next solar cycle (hemisphere by hemisphere in terms of peak sunspot area). North – Blue squares South – Red circles Cycles are from direct magnetic measurements. Earlier cycles are from counting polar faculae. This was noted years ago by Schatten (1978) and was used by Svalgaard, Cliver and Kamide (2005) to successfully predict Cycle 24.

What Makes the Polar Fields? Tilted active regions and flux transport.

Surface Flux Transport Surface magnetic flux transport models were developed in the early 1980s by the Naval Research Laboratory (NRL) group including Neil Sheeley, Yi-Ming Wang, Rick DeVore, and Jay Boris. They found that they could reproduce the evolution of the Sun’s surface magnetic field using active region flux emergence as the only source of magnetic flux – that flux is then transported across the Sun’s surface by: 1.Differential Rotation, U(θ) 2.Meridional Flow, V(θ) 3.Supergranule Diffusion,  ∂B/∂t + 1/(R sinθ) ∂(BV sinθ)/∂θ + 1/(R sinθ) ∂(BU)/∂  =   2 B + S(θ,  ) Neither the meridional flow nor the supergranule diffusion were well constrained at that time – so they used what worked.

Flux Transport Details Four days of HMI data drive home the fact that flux transport is dominated by the cellular flows. The extent to which this can be represented by a diffusion coefficient and a Laplacian operator is to be determined. 300 Mm 500 Mm

Supergranules and the Magnetic Network Tracking the motions of granules (Local Correlation Tracking with 6- minute time lags from HMI Intensity data) reveals the flow pattern within supergranules and the relationship to the magnetic pattern – the magnetic network forms at the supergranule boundaries (convergence zones).

My Goal Produce a surface flux transport model based on observationsProduce a surface flux transport model based on observations –Characterize the axisymmetric flows –Characterize the non-axisymmetric flows –Produce synchronic maps of the entire surface for comparison with observations Determine any missing processesDetermine any missing processes –Is 2D sufficient or are there 3D processes? –Are active regions the only sources? –Are there sinks? Use these synchronic maps for space weather applicationsUse these synchronic maps for space weather applications –Coronal and Interplanetary field configuration

The Axisymmetric Flows We (Hathaway & Rightmire 2010, Science, 327, 1350) measured the axisymmetric transport of magnetic flux by cross-correlating 11x600 pixel strips at 860 latitude positions between ±75˚ from magnetic images acquired at 96-minute intervals by MDI on SOHO.

Average Flow Profiles Average ( ) differential rotation profile with 2σ error limits. Average ( ) meridional flow profile with 2σ error limits. Our MDI data included corrections for CCD misalignment, image offset, and a 150 year old error in the inclination of the ecliptic to the Sun’s equator. We extracted differential rotation and meridional flow profiles from over 60,000 image pairs from May of 1996 to September of 2010.

Meridional Flow Comparisons The Meridional Flow we measure is very unlike that still used by the NRL group (Wang et al.). It is similar at low latitudes to that used by other groups but differs by not vanishing at high latitudes.

High-Latitude Meridional Flow In Rightmire-Upton, Hathaway, & Kosak (2012) we repeated the analysis with the higher resolution SDO/HMI data and confirmed a poleward flow all the way to 85° north and south.

Solar Cycle Variations in the Axisymmetric Flows While the differential rotation does vary slightly over the solar cycle, it is the meridional flow that shows the most significant variation. The Meridional Flow slowed from 1996 to 2001 but then increased in speed again after maximum. The slowing of the meridional flow at maximum seems to be a regular solar cycle occurrence (Komm, Howard, & Harvey, 1993). The greater speed up after maximum is specific to Cycle 23. Differential rotation variations Meridional flow variations

Solar Cycle Variations in Flow Structure Differential rotation profiles Meridional flow profiles The differential rotation and meridional flow profiles for each solar rotation also show that the differential rotation changes very little while the meridional flow changes substantially. The change in the meridional flow is primarily a weakening of the poleward flow poleward of the active latitudes that was strong in Cycle 23 and weak in Cycle 24.

Characterizing Supergranules We (Hathaway et al. 2010, ApJ 725, 1082) analyzed and simulated Doppler velocity data from MDI to determine the characteristics of supergranulation. These cellular flows have a broad spectrum characterized by a peak in power at wavelengths of about 35 Mm. MDI SIM

Measuring their motions The axisymmetric flows can be measured using the Doppler velocity pattern using the same method used with the magnetic pattern. A key difference is the use of several different time lags between images.

Reproducing the Lifetimes The cellular structures are given finite lifetimes by adding random perturbations to the phases of the complex spectral coefficients. The amplitude of the perturbation was inversely proportional to a lifetime given by the size of the cell (from its wavenumber) divided by its flow velocity (from the amplitude of the spectral coefficient). This process can largely reproduce the strength of the cross- correlation as a function of both time and latitude.

Reproducing their Rotation The motions of the cellular patterns in longitude can be reproduced by making systematic changes to the complex spectral coefficient phases (Hathaway et al. 2010).

Reproducing their Meridional Flow The motions of the cellular patterns in latitude can be reproduced by making systematic changes to the complex spectral coefficient amplitudes (Hathaway et al. 2010).

Surface Shear Layer Variations in the rotation rate of the supergranules indicate that they are advected by the flows at depths equal to their widths (Hathaway, ApJ 749, L13, 2012). The differential rotation of the supergranules using different time-lags indicates that the shear layer extends to high latitudes. Equatorial rotation rate of the longitudinal velocity pattern as a function of wavelength from an FFT analysis. Equatorial rotation rate of the line-of-sight velocity pattern as a function of wavelength from an FFT analysis compared with rotation rate from cross-correlation analysis.

Meridional Return Flow The reversal of the meridional flow direction at time-lags greater than 24- hours indicates a shallow return (equatorward) flow.

Magnetic Element Motions Comparing the differential rotation and meridional flow of the magnetic elements to that of the supergranules indicates that the magnetic elements are anchored at a depth of ~20Mm. Magnetic Elements Supergranules

Lagrangian Flux Transport We did a Lagrangian simulation of flux transport using supergranules that have the differential rotation and meridional flow of the magnetic elements. The magnetic field from a synoptic map was represented by ~100,000 magnetic elements with field strengths of 1 kG. We calculated the position of each element as they were advected by the evolving supergranule flow pattern.

Supergranules Rule! If we add differential rotation and meridional flow on top of supergranules that don’t move with those flows we get magnetic element motion with no differential rotation or meridional flow. Magnetic elements move to the boundaries of supergranules where they follow the differential rotation and meridional flow of the supergranules themselves.

Eulerian Flux Transport We now do the flux transport on a 1024x512 grid in longitude and latitude. This is faster than the Lagrangian formulation and easily allows for the assimilation of data from full-disk magnetograms. We solve the advection equation using supergranule flows calculated at 15-minute intervals with meridional flow and differential rotation velocities appropriate for the given date. ∂B/∂t + 1/(R sinθ) ∂(BV sinθ)/∂θ + 1/(R sinθ) ∂(BU)/∂  =   2 B The 15-minute time step is required by the Courant condition given the fast flows and relatively high spatial resolution. The diffusion term is purely for numerical stability – to minimize ringing around features.

Data Assimilation Data from the entire visible hemisphere should be assimilated – but with weights inversely proportional to the noise level. Data Assimilation. A) The data simulated with the flux transport model. B) The data observed with a magnetograph. C) The weights for the simulated data. D) The weights for the observed data.

Synchronic Maps for 2001 Updated every 15 minutes (shown every 8 hours)

Polar Field Reversal 2001

Ongoing/Future Work Polar field production experimentsPolar field production experiments –Polar fields with data assimilation from active latitudes only With observed DR and MF – does it work?With observed DR and MF – does it work? With average DR and MF – are variations importantWith average DR and MF – are variations important Farside dataFarside data –Helioseismology? –STEREO? –Kalman Smoothing? ApplicationsApplications –Coronal/Interplanetary field configuration for CME evolution (UMich) –Coronal heating by footpoint motions (St. Andrews)

Conclusions 1.To produce synchronic magnetic maps we need to properly transport magnetic flux and assimilate up-to-date observations. 2.To do the transport we need to measure and monitor the structure and variations in the flow components – differential rotation and meridional flow. 3.To do the non-axisymmetric transport we need to characterize the supergranule flow properties. 4.We still need to explore methods to accommodate the emergence of active regions on the far side of the Sun.