Measuring and Modeling Magnetic Flux Measuring and Modeling Magnetic Flux Transport on the SunTransport on the Sun

Dr. David HathawayDr. David Hathaway

NASA Marshall Space Flight CenterNASA Marshall Space Flight Center

2013 August 22 – Ames Research Center2013 August 22 – Ames Research Center

Three Major Solar MysteriesThree Major Solar Mysteries

1.1. 1844 - What causes the 1844 - What causes the sunspot cycle?sunspot cycle?

2.2. 1859 - What causes flares 1859 - What causes flares (and other explosive events)?(and other explosive events)?

3.3. 1939 - How is the Sun’s 1939 - How is the Sun’s Corona heated to million Corona heated to million degree temperatures (that degree temperatures (that drive the solar wind)?drive the solar wind)?

Solar Magnetism – The KeySolar Magnetism – The Key

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

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

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

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

Babcock (1961)Babcock (1961)

a) Dipolar field at cycle minimum a) Dipolar field at cycle minimum threads through a shallow layer threads through a shallow layer below the surface.below the surface.

b) Latitudinal differential rotation b) Latitudinal differential rotation shears out this poloidal field to shears out this poloidal field to produce a strong toroidal field produce a strong toroidal field (first at the mid-latitudes then (first at the mid-latitudes then progressively lower latitudes).progressively lower latitudes).

c) Buoyant fields erupt through c) Buoyant fields erupt through the photosphere giving Hale’s the photosphere giving Hale’s polarity law and Joy’s Tilt.polarity law and Joy’s Tilt.

d) Meridional transport gives d) Meridional transport gives reconnection at the poles and reconnection at the poles and equator.equator.

(son Horace , not father Harold)(son Horace , not father Harold)

Surface Field over 3 CyclesSurface Field over 3 CyclesOne map is constructed for each 27-day rotation of the Sun using data 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:from near the central meridian. These maps show that the field:

1) Emerges in bi-polar active regions (sunspot groups)1) Emerges in bi-polar active regions (sunspot groups)2) Diffuses (undergoes a random walk in longitude and latitude)2) Diffuses (undergoes a random walk in longitude and latitude)3) Drifts in longitude (Differential Rotation)3) Drifts in longitude (Differential Rotation)4) Drifts poleward from the equator (Meridional Flow)4) Drifts poleward from the equator (Meridional Flow)

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

Polar Fields – Seeds for CyclesPolar Fields – Seeds for Cycles

The Sun’s polar fields are the The Sun’s polar fields are the seeds of the next solar cycle in seeds of the next solar cycle in most dynamo models. We have most dynamo models. We have direct observations for the last direct observations for the last three cycles and now a proxy three cycles and now a proxy (polar faculae) for the last 10 (polar faculae) for the last 10 cycles. cycles.

Muñoz-Jaramillo et al. (2013)Muñoz-Jaramillo et al. (2013)

Polar Fields as PredictorsPolar Fields as PredictorsMuñoz-Jaramillo et al. (2013) find a strong correlation between polar Muñoz-Jaramillo et al. (2013) find a strong correlation between polar fields and the amplitude of the next solar cycle (hemisphere by fields and the amplitude of the next solar cycle (hemisphere by hemisphere in terms of peak sunspot area).hemisphere in terms of peak sunspot area).

North – Blue squaresNorth – Blue squaresSouth – Red circlesSouth – Red circles

Cycles 21-23 are from Cycles 21-23 are from direct magnetic direct magnetic measurements.measurements.

Earlier cycles are from Earlier cycles are from counting polar faculae.counting polar faculae.

This was noted years ago This was noted years ago by Schatten (1978) and by Schatten (1978) and was used by Svalgaard, was used by Svalgaard, Cliver and Kamide (2005) Cliver and Kamide (2005) to successfully predict to successfully predict Cycle 24.Cycle 24.

What Makes the Polar Fields?What Makes the Polar Fields?

Tilted active regions and flux transport.Tilted active regions and flux transport.

Surface Flux TransportSurface Flux Transport

Surface magnetic flux transport models were developed in the early Surface magnetic flux transport models were developed in the early 1980s by the Naval Research Laboratory (NRL) group including Neil 1980s by the Naval Research Laboratory (NRL) group including Neil Sheeley, Yi-Ming Wang, Rick DeVore, and Jay Boris. They found that Sheeley, Yi-Ming Wang, Rick DeVore, and Jay Boris. They found that they could reproduce the evolution of the Sun’s surface magnetic field they could reproduce the evolution of the Sun’s surface magnetic field using active region flux emergence as the only source of magnetic flux using active region flux emergence as the only source of magnetic flux – that flux is then transported across the Sun’s surface by:– that flux is then transported across the Sun’s surface by:

1.1. Differential Rotation, U(Differential Rotation, U(θθ))2.2. Meridional Flow, V(Meridional Flow, V(θθ))3.3. Supergranule Diffusion, Supergranule Diffusion,

∂∂B/∂t + 1/(R sinB/∂t + 1/(R sinθθ) ∂(BV sin) ∂(BV sinθθ)/∂)/∂θθ + 1/(R sin + 1/(R sinθθ) ∂(BU)/∂) ∂(BU)/∂ = = 22B + S(B + S(θθ,,))

Neither the meridional flow nor the supergranule diffusion were well Neither the meridional flow nor the supergranule diffusion were well constrained at that time – so they used what worked.constrained at that time – so they used what worked.

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

300 Mm300 Mm

500 Mm500 Mm

Supergranules and the Supergranules and the Magnetic NetworkMagnetic Network

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

My GoalMy Goal

• Produce a surface flux transport model based on observationsProduce a surface flux transport model based on observations– Characterize the axisymmetric flowsCharacterize the axisymmetric flows– Characterize the non-axisymmetric flowsCharacterize the non-axisymmetric flows– Produce synchronic maps of the entire surface for Produce synchronic maps of the entire surface for

comparison with observationscomparison with observations

• Determine any missing processesDetermine any missing processes– Is 2D sufficient or are there 3D processes?Is 2D sufficient or are there 3D processes?– Are active regions the only sources?Are active regions the only sources?– Are there sinks?Are there sinks?

• Use these synchronic maps for space weather applicationsUse these synchronic maps for space weather applications– Coronal and Interplanetary field configurationCoronal and Interplanetary field configuration

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

Average Flow ProfilesAverage Flow Profiles

Average (1996-2010) differential Average (1996-2010) differential rotation profile with 2rotation profile with 2σσ error error limits.limits.

Average (1996-2010) meridional Average (1996-2010) meridional flow profile with 2flow profile with 2σσ error limits. error limits.

Our MDI data included corrections for CCD misalignment, image offset, 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 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 equator. We extracted differential rotation and meridional flow profiles from over 60,000 image pairs from May of 1996 to September of 2010.from over 60,000 image pairs from May of 1996 to September of 2010.

Meridional Flow ComparisonsMeridional Flow Comparisons

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

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

Solar Cycle Variations in the Solar Cycle Variations in the Axisymmetric FlowsAxisymmetric Flows

While the differential rotation does vary slightly over the solar cycle, it is While the differential rotation does vary slightly over the solar cycle, it is the meridional flow that shows the most significant variation. The the meridional flow that shows the most significant variation. The Meridional Flow slowed from 1996 to 2001 but then increased in speed Meridional Flow slowed from 1996 to 2001 but then increased in speed again after maximum. The slowing of the meridional flow at maximum again after maximum. The slowing of the meridional flow at maximum seems to be a regular solar cycle occurrence (Komm, Howard, & Harvey, seems to be a regular solar cycle occurrence (Komm, Howard, & Harvey, 1993). The greater speed up after maximum is specific to Cycle 23.1993). The greater speed up after maximum is specific to Cycle 23.

Differential rotation variationsDifferential rotation variations Meridional flow variationsMeridional flow variations

Solar Cycle Variations in Flow Solar Cycle Variations in Flow StructureStructure

Differential rotation profilesDifferential rotation profiles Meridional flow profilesMeridional flow profiles

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

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

MDIMDI

SIMSIM

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

Reproducing the LifetimesReproducing the LifetimesThe cellular structures are given finite lifetimes by adding random The cellular structures are given finite lifetimes by adding random perturbations to the phases of the complex spectral coefficients. The perturbations to the phases of the complex spectral coefficients. The amplitude of the perturbation was inversely proportional to a lifetime amplitude of the perturbation was inversely proportional to a lifetime given by the size of the cell (from its wavenumber) divided by its flow given by the size of the cell (from its wavenumber) divided by its flow velocity (from the amplitude of the spectral coefficient).velocity (from the amplitude of the spectral coefficient).

This process can largely reproduce the strength of the cross-This process can largely reproduce the strength of the cross-correlation as a function of both time and latitude.correlation as a function of both time and latitude.

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

Reproducing their Meridional FlowReproducing their Meridional Flow

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

Surface Shear LayerSurface Shear LayerVariations in the rotation rate of the supergranules indicate that they are Variations in the rotation rate of the supergranules indicate that they are advected by the flows at depths equal to their widths (Hathaway, advected by the flows at depths equal to their widths (Hathaway, ApJApJ 749, 749, L13, 2012). The differential rotation of the supergranules using different L13, 2012). The differential rotation of the supergranules using different time-lags indicates that the shear layer extends to high latitudes.time-lags indicates that the shear layer extends to high latitudes.

Equatorial rotation rate of the Equatorial rotation rate of the longitudinallongitudinal velocity pattern as a function of wavelength velocity pattern as a function of wavelength from an FFT analysis.from an FFT analysis.

Equatorial rotation rate of the Equatorial rotation rate of the line-of-sightline-of-sight velocity pattern as a function of wavelength velocity pattern as a function of wavelength from an FFT analysis compared with from an FFT analysis compared with rotation rate from cross-correlation rotation rate from cross-correlation analysis.analysis.

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

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

Magnetic ElementsMagnetic Elements SupergranulesSupergranules

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

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

Eulerian Flux TransportEulerian Flux Transport

We now do the flux transport on a 1024x512 grid in longitude and We now do the flux transport on a 1024x512 grid in longitude and latitude. This is faster than the Lagrangian formulation and easily latitude. This is faster than the Lagrangian formulation and easily allows for the assimilation of data from full-disk magnetograms.allows for the assimilation of data from full-disk magnetograms.

We solve the advection equation using supergranule flows calculated We solve the advection equation using supergranule flows calculated at 15-minute intervals with meridional flow and differential rotation at 15-minute intervals with meridional flow and differential rotation velocities appropriate for the given date.velocities appropriate for the given date.

∂∂B/∂t + 1/(R sinB/∂t + 1/(R sinθθ) ∂(BV sin) ∂(BV sinθθ)/∂)/∂θθ + 1/(R sin + 1/(R sinθθ) ∂(BU)/∂) ∂(BU)/∂ = = 22BB

The 15-minute time step is required by the Courant condition given the The 15-minute time step is required by the Courant condition given the fast flows and relatively high spatial resolution. The diffusion term is fast flows and relatively high spatial resolution. The diffusion term is purely for numerical stability – to minimize ringing around features.purely for numerical stability – to minimize ringing around features.

Data AssimilationData AssimilationData from the entire visible hemisphere should be assimilated – but with Data from the entire visible hemisphere should be assimilated – but with weights inversely proportional to the noise level.weights inversely proportional to the noise level.

Data Assimilation. A) The data simulated with the flux transport model. Data Assimilation. A) The data simulated with the flux transport model. B) The data observed with a magnetograph. C) The weights for the B) The data observed with a magnetograph. C) The weights for the simulated data. D) The weights for the observed data.simulated data. D) The weights for the observed data.

Synchronic Maps for 2001Synchronic Maps for 2001

Updated every 15 minutes (shown every 8 hours)Updated every 15 minutes (shown every 8 hours)

Polar Field Reversal 2001Polar Field Reversal 2001

Ongoing/Future WorkOngoing/Future Work

• Polar field production experimentsPolar field production experiments

– Polar fields with data assimilation from active latitudes onlyPolar 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?Helioseismology?

– STEREO?STEREO?

– Kalman Smoothing?Kalman Smoothing?

• ApplicationsApplications

– Coronal/Interplanetary field configuration for CME evolution Coronal/Interplanetary field configuration for CME evolution (UMich)(UMich)

– Coronal heating by footpoint motions (St. Andrews)Coronal heating by footpoint motions (St. Andrews)

ConclusionsConclusions1.1. To produce synchronic magnetic maps we need to properly To produce synchronic magnetic maps we need to properly

transport magnetic flux and assimilate up-to-date transport magnetic flux and assimilate up-to-date observations.observations.

2.2. To do the transport we need to measure and monitor the To do the transport we need to measure and monitor the structure and variations in the flow components – differential structure and variations in the flow components – differential rotation and meridional flow.rotation and meridional flow.

3.3. To do the non-axisymmetric transport we need to characterize To do the non-axisymmetric transport we need to characterize the supergranule flow properties.the supergranule flow properties.

4.4. We still need to explore methods to accommodate the We still need to explore methods to accommodate the emergence of active regions on the far side of the Sun.emergence of active regions on the far side of the Sun.