Category: SHINE 2002

Presenter: LEN FISK

UNIVERSITY OF MICHIGAN

OVER THE PAST SIX YEARS NOW, WE HAVE BEEN WRITING A NUMBER OF PAPERS ABOUT THE BEHAVIOR OF THE SO-CALLED OPEN MAGNETIC FLUX OF THE SUN.  THAT IS, THE PORTION OF THE MAGNETIC FIELD OF THE SUN THAT CREATES THE HELIOSPHERIC MAGNETIC FIELD; IT OPENS INTO THE HELIOSPHERE.  [VU-GRAPH 1] AT THE HEART OF ALL OF THESE DISCUSSIONS IS A MECHANISM THAT WE ARGUE IS IMPORTANT TO UNDERSTAND THE BEHAVIOR OF THE OPEN FLUX.  THAT IS, OPEN FIELD LINES CAN RECONNECT WITH CLOSED MAGNETIC LOOPS, AND THE FOOTPOINT OF THE OPEN FIELD LINE AT THE SUN IS DISPLACED, BY THE SEPARATION OF THE FOOTPOINTS OF THE LOOPS.

CLEARLY, IF THE LOOPS, WITH WHICH THE OPEN FIELD LINES ARE  RECONNECTING, ARE RANDOMLY ORIENTED, THE OPEN FIELD LINES ARE TRANSPORTED IN A DIFFUSIVE MANNER; WHICH CAN BE MORE RAPID  THAN FOR EXAMPLE, THE STANDARD LEIGHTON DIFFUSION OF FIELD LINES, IN WHICH THE FIELD LINES ARE CONVECTED WITH THE RANDOM MOTIONS OF SUPERGRANULAR FLOWS.  THE JUMP DISTANCE CAN BE RELATIVELY LARGE, SINCE THE SIZE OF CORONAL LOOPS GENERALLY EXCEED THE SIZE OF SUPERGRANULES.

WHAT I WANT TO TALK ABOUT TODAY IS THREE THINGS:  FIRST, WHAT IS THE EVIDENCE THAT A PROCESS SUCH AS THIS IS IMPORTANT AT THE SUN. SECOND, GIVEN THAT IT OCCURS, WHAT IS THE BEST WAY TO DESCRIBE – MATHEMATICALLY – SO THAT ITS IMPLICATIONS CAN BE UNDERSTOOD.  AND FINALLY WHAT RELEVANCE DOES THIS HAVE FOR UNDERSTANDING THE FIELD REVERSAL OF THE SUN.

[VU GRAPH 2] NOW THAT SUCH A PROCESS COULD OCCUR ON THE SUN IS NOT REALLY IN DEBATE.  CONSIDER, FOR EXAMPLE, THE COMMON UNDERSTANDING OF THE FORMATION OF CORONAL LOOPS ON THE QUIET SUN.  FROM THE EXTENSIVE OBSERVATIONS OF MDI ON SOHO, AND THE EXTENSIVE ANALYSIS BY THE LOCKHEED GROUP AND COLLABORATORS, WHAT YOU SEE IS SMALL BI-POLAR LOOPS EMERGING EVERYWHERE ON THE SUN, USUALLY INTERIOR TO THE EDGES OF SUPERGRANULES.  THE FLUX CONCENTRATIONS, WHICH REPRESENT THE INTERSECTIONS OF THESE LOOPS WITH THE SUN – THE FOOTPOINTS OF THE LOOPS, EXPAND WITH THE SUPERGRANULAR AND GRANULAR FLOW, AND ENTER THE LANES SURROUNDING THE SUPERGRANULES.  WHAT YOU SEE ARE FLUX CONCENTRATIONS – THE FOOTPOINTS  –RUNNING INTO FLUX CONCENTRATIONS OF OPPOSITE POLARITY, AND DISAPPEARING.  WHAT IS HAPPENING, IN THREE DIMENSIONS, IS THAT TWO LOOPS HAVE RECONNECTED, AND FORMED A LARGER LOOP – THE LOOPS HAVE COALESCED.  THERE MUST BE A SMALLER SECONDARY LOOP FORMED, WHICH SUBDUCTS BACK INTO THE PHOTOSPHERE, SINCE THE FLUX CONCENTRATION AT THE POINT OF RECONNECTION IS GENERALLY OBSERVED TO DISAPPEAR. INDEED, IN A NICE PAPER BY HANDY AND SCHRIJVER A COUPLE OF YEARS AGO, THIS MECHANISM – THE COALESCENCE OF SMALL LOOPS TO MAKE BIGGER LOOPS, IS ARGUED TO BE THE PRINCIPAL MECHANISM BY WHICH LOOPS FORM ON THE QUIET SUN.

NOW ALL YOU HAVE TO DO IS IMAGINE THAT SOME OF THESE FLUX CONCENTRATIONS REPRESENT OPEN FIELD LINES, MIXED AMONG THE CLOSED LOOPS.  THE IDENTICAL PROCESS, THEN, CAUSES A DISPLACEMENT OF THE OPEN FIELD LINES, AND EFFECTIVELY ELIMINATES THE ORIGINAL LOOP, EXCEPT FOR THE FORMATION OF THE SMALL SECONDARY LOOP WHICH WE ASSUME SUBDUCTS AWAY.

THE EVIDENCE THAT A PROCESS SUCH AS THIS IS IMPORTANT COMES MAINLY FROM HELIOSPHERIC OBSERVATIONS, SINCE THE ONLY DIRECT OBSERVATIONS THAT WE HAVE OF THE OPEN MAGNETIC FLUX OF THE SUN IS IN THE HELIOSPHERE.  THE MAGNETIC FIELD CAN BE OBSERVED AT THE SOLAR SURFACE, BUT WHAT IS OPEN AND CLOSED IS A MATTER FOR MODEL CALCULATIONS. 

LET ME CITE, THEN A FEW DIFFERENT HELIOSPHERIC OBSERVATIONS, WHICH ARGUE FOR THIS ENHANCED DIFFUSIVE TRANPORT OF OPEN FLUX, BY RECONNECTING WITH CLOSED LOOPS.  TIME DOES NOT PERMIT A DETAILED DISCUSSION OF ANY OF THESE OBSERVATIONS –HOWEVER, THIS IS ALL PUBLISHED WORK, WHICH CAN BE DISCUSSED IN MORE DETAIL IN THE WORKSHOP.

THE FIRST IS THE ORIGINAL ONE [VU GRAPH 3] THAT MOTIVATED THIS WORK.  ULYSSES AT HIGH LATITUDES AT SOLAR MINIMUM SAW ENERGETIC PARTICLES, WHICH WERE KNOW TO BE PRODUCED IN CO-ROTATING INTERACTION REGIONS NEAR THE SOLAR EQUATOR.  THESE ENERGETIC PARTICLES FOLLOW THE HELIOSPHERIC MAGNETIC FIELD LINES. [VU GRAPH 4] IN A NORMAL PARKER SPIRAL MODEL, THE HELIOSPHERIC MAGNETIC FIELD LIES ON CONES OF CONSTANT LATITUDE.  TRANSPORT ALONG MAGNETIC FIELD LINES FROM LOW LATITUDES, WHERE THESE PARTICLES ARE PRODUCED TO HIGH LATITUDES WHERE THEY ARE OBSERVED IS CERTAINLY NOT EASY – HENCE A PUZZLE.

I PROPOSED A SIMPLE MECHANISM [VU GRAPH 5].  CONSIDER THE OPEN MAGNETIC FLUX, SHOWN IN RED, FROM THE POLAR CORONAL HOLE AT SOLAR MINIMUM.  IT EXPANDS SUPER-RADIALLY.  IT DIFFERENTIALLY ROTATES, WHEREAS THE CORONAL HOLE IS OBSERVED TO ROTATE MORE RIGIDLY.  [VU GRAPH 6] PUT ALL THAT TOGETHER, AND THE OPEN FIELD LINES ARE FORCED TO MOVE SUBSTANTIALLY IN LATITUDE, WHICH RESULTS IN A HELIOSPHERIC MAGNETIC FIELD WITH A POLAR COMPONENT, WHICH PROVIDES A PATH FOR ENERGETIC PARTICLES FROM LOW TO HIGH LATITUDES. 

BUT THE THEORY HAS A PROBLEM UNLESS YOU INVOKE A RAPID DIFFUSIVE TRANSPORT MECHANISM AT LOW LATITUDES.  NOTE ALL THE OPEN FLUX MOVES SYSTEMATICALLY DOWNWARD IN LATITUDE IN THIS FIGURE.  BUT IT CAN’T CROSS THE EQUATORIAL CURRENT SHEET – IT HAS A DEFINED POLARITY – AND HENCE IT MUST TURN AND TRANSPORTED AROUND THE EQUATOR BY A RELATIVELY RAPID MECHANISM, SUCH AS RECONNECTING WITH THE LARGE CORONAL LOOPS AT LOW LATITUDES NEAR SOLAR MINIMUM.

THIS RAPID TRANSPORT WILL EITHER BE IN THE DIRECTION OF, OR OPPOSITE TO THAT OF THE SOLAR ROTATION.  WHEN IT IS IN THE OPPOSITE DIRECTION OF THE SOLAR ROTATION, IT WILL CREATE AN UNDERWOUND – MORE RADIAL — HELIOSPHERIC MAGNETIC FIELD.  THIS IS SEEN, EXACTLY WHERE IT SHOULD OCCUR, IN THE ULYSSES OBSERVATIONS OF MURPHEY ET AL.  INDEED, IT IS HARD TO IMAGINE HOW YOU COULD CREATE A NEAR RADIAL HELIOSPHERIC MAGNETIC FIELD OVER SEVERAL DAYS, UNLESS YOU HAVE SYSTEMATIC MOTIONS OF THE FOOTPOINTS OF THE HELIOSPHERIC FIELD AT A SPEED COMPARABLE TO THE SOLAR ROTATION RATE, WHICH REQUIRES A QUITE FAST TRANSPORT, INVOLVING RECONNECTIONS WITH RELATIVELY LARGE LOOPS.

THE HELIOSPHERIC MAGNETIC FIELD IS ALSO OBSERVED TO BE RELATIVELY CONSTANT OVER THE SOLAR CYCLE [VU GRAPH 7].  IT DOES VARY BY 50% OR SO OVER THE SOLAR CYCLE, BUT COMPARED WITH THE 500% VARIATION IN THE TOTAL MAGNETIC FIELD OF THE SUN, THE OPEN FLUX IS RELATIVELY CONSTANT.  THE HELIOSPHERIC MAGNETIC FIELD IS ALSO WELL ORGANIZED – AND THIS IS ONE OF THE MOST IMPORTANT OBSERVATIONS FROM ULYSSES.  [VU GRAPH 8] AT SOLAR MINIMUM THERE IS A SINGLE CURRENT SHEET NEAR THE SOLAR EQUATOR, SEPARATING TWO HEMISPHERES EACH WITH UNIFORM BUT OPPOSITE POLARITY OPEN MAGNETIC FLUX. AS FAR AS CAN BE INFERRED, THERE IS A SINGLE CURRENT SHEET, AT ALL TIMES DURING THE SOLAR CYCLE.  [VU GRAPH 9]THIS FIGURE IS FROM A PAPER BY SANDERSON, ET AL.; THERE IS SIMILAR WORK BY SMITH, SHOWING THE LOCATION OF THE CURRENT SHEET.  AT SOLAR MINIMUM IT IS NEAR THE EQUATORIAL PLANE; AT SOLAR MAXIMUM IT IS NEARLY VERTICAL.  THE CURRENT SHEET APPEARS TO SIMPLY ROTATE, AS THE SOLAR CYLE PROGRESSES; INDEED THIS APPARENT ROTATION CAN BE THOUGHT OF AS ACCOMPLISHING THE FIELD REVERSAL OF THE OPEN MAGNETIC FLUX.

THE RELATIVE CONSTANCY OF THE OPEN FLUX, AND A WELL-DEFINED CURRENT SHEET ARE RELATED.  [VU GRAPH 10] IF YOU ASK, HOW CAN I ELIMINATE OPEN MAGNETIC FLUX FROM THE SUN, I HAVE TO TAKE AN OPEN FIELD LINE OF ONE POLARITY AND RECONNECT WITH AN OPEN FIELD LINE OF THE OPPOSITE POLARITY.  BUT I CAN ONLY DO THIS AT THE CURRENT SHEET, WHICH SHOWS NO SIGN OF EXTENSIVE RECONNECTION OCCURRING.

IT IS NOT SURPRISING THEN THAT THE SUN HAS A RELATIVELY CONSTANT AMOUNT OF OPEN FLUX.  INDEED, THE CHALLENGE IS IN THE REVERSE.  CORONAL MASS EJECTIONS HAVE THE POTENTIAL TO BRING NEW OPEN FLUX INTO THE HELIOSPHERE.  IN SEEMS UNLIKELY THAT THE ADDITION OF OPEN FLUX FROM CME’S WOULD BE EXACTLY BALANCED BY A SEPARATE PROCESS OF RECONNECTION AT THE CURRENT SHEET, IN ORDER TO MAINTAIN A RELATIVELY CONSTANT OPEN FLUX IN THE HELIOSPHERE.  [VU GRAPH 11] WHICH HAS LED, NANCY CROOKER AND CO-WORKERS TO ARGUE FOR WHAT THEY CALL INTERCHANGE RECONNECTION, IN WHICH A CME FIELD LINE, COMING OUT FROM THE SUN AS A VERY LARGE LOOP, RECONNECTS WITH AN OPEN FIELD LINE, AND THEREBY DOES NOT ADD TO THE OPEN FLUX IN THE HELIOSPHERE.  THIS IS OF COURSE JUST AN EXAGGERATED FORM OF THE BASIC PROCESS OF OPEN FIELD LINES RECONNECTING WITH CORONAL LOOPS, IN THIS CASE VERY LARGE LOOPS.

IF THE AMOUNT OF OPEN FLUX IS RELATIVELY CONSTANT OVER THE SOLAR CYCLE, CERTAINLY THE DISTRIBUTION IS NOT.  THERE ARE LARGE CONCENTRATIONS OF OPEN FLUX IN THE POLAR CORONAL HOLES AT SOLAR MINIMUM.  THE POLAR CORONAL HOLES DISAPPEAR AS THE CYCLE DEVELOPS, AND TRANSIENT CORONAL HOLES DEVELOP AT LOW LATITUDES.  AND AGAIN, IF YOU NEITHER ADD NOR SUBTRACT SUBSTANTIAL OPEN FLUX OVER THE SOLAR CYCLE, TO CREATE THESE CHANGES IN THE SPATIAL DISTRIBUTION, YOU HAVE TO MOVE THE OPEN FLUX AROUND, SOMETIMES FAIRLY RAPIDLY, PERHAPS DUE TO OUR BASIC MECHANISM.

FINALLY LET ME MENTION A VERY RECENT PAPER BY SCHRIJVER, DE ROSA AND TITLE, WHO DID A VERY DETAILED ANALYSIS OF THE BEHAVIOR OF THE HIGH LATITUDE MAGNETIC FIELD OF THE SUN, OVER CENTURIES IN FACT, AND CONCLUDED THAT THERE WAS A NEED FOR AN ENHANCED TRANSPORT OF OPEN MAGNETIC FLUX TO THE POLES, THAT SIMPLE LEIGHTON-LIKE DIFFUSION IN SUPERGRANULE FLOW WAS NOT SUFFICIENT, AND INDEED THEY SUGGESTED A MECHANISM FOR THIS ENHANCED TRANSPORT, WHICH IS EFFECTIVELY THE SAME AS THE ONE WE HAVE BEEN DISCUSSING.

LET’S SUPPOSE THEN THAT WE NEED ENHANCED TRANSPORT OF OPEN MAGNETIC FLUX, DUE TO RECONNECTION WITH LOOP  — HOW WOULD WE CALCULATE ITS EFFECTS.  [VU GRAPH 12] THE GOOD NEWS, IS WE ARE DEALING PRIMARILY WITH THE QUIET SUN, OPEN FIELD LINES DON’T COME FROM ACTIVE REGIONS, AND THE LOOPS WE WANT TO RECONNECT WITH ARE OBSERVED, BY FOR EXAMPLE HANDY AND SCHRIJVER, TO HAVE NO PREFERRED ORIENTATION.  AND IN THAT SENSE THE TRANSPORT OF OPEN FIELD LINES THAT RESULTS FROM THEIR RECONNECTION WITH LOOPS CAN BE DESCRIBED BY A DIFFUSION COEFFICIENT.  HERE H-BAR SQUARED IS THE MEAN SEPARATION OF THE FOOTPOINTS OF THE LOOP, WHERE THE RECONNECTION IS ASSUMED TO OCCUR ON ONE END – IT IS THE MEAN SQUARED DISPLACEMENT OF THE OPEN FIELD LINE, AND TAU IS THE CHARACTERISTIC TIME FOR THE RECONNECTION.  THE FACTOR OF 4 ENTERS, FIRST OF ALL BECAUSE THE DIFFUSION COEFFICIENT IS THE COMPONENT OF A TENSOR, WHEREAS H-BAR SQUARED IS A RANDOM DIRECTION, THAT GIVES YOU A FACTOR OF 2; AND THE OTHER FACTOR OF 2 IS BECAUSE THE JUMPS CAN BE IN EITHER DIRECTION.  THIS PROCESS, BY THE WAY WOULD BE HARD TO DESCRIBE IN SOME OF THE MONTE-CARLO CALCULATIONS THAT SCHRIJVER AND OTHERS DO, SINCE YOU ARE DESCRIBING A THREE-DIMENSINAL PROCESS, YOU HAVE TO KEEP TRACK NOT ONLY OF WHERE YOU RECONNECT, BUT ALSO WHERE YOU GO.  DIFFUSION CALCULATIONS ARE EASIER IN THIS REGARD.

[VU GRAPH 13] THINK OF THE OPEN FIELD STRENGTH OF THE SUN AS A DENSITY, NUMBER OF FIELD LINES PER AREA. THEN THIS IS THE APPROPRIATE DIFFUSION EQUATION.  IT IS, IN FACT, ACTUALLY A SPECIAL FORM.  I HAVE SPENT MOST OF MY CAREER DOING DIFFUSION PROBLEMS AND FOUND THAT IT IS NOT ALWAYS RECOGNIZED THAT THERE ARE DIFFERENT FORMS OF THE DIFFUSION EQUATION.  THIS IS THE FORM THAT APPLIES WHEN THE SUBSTANCE THAT IS DIFFUSING, IS NOT MOVING ON ITS OWN, BUT IS CARRIED AROUND BY SOME EXTERNAL PROCESS, IN THIS CASE THE RECONNECTION.  YOU CAN SHOW THAT THIS IS THE EXACT SAME FORM AS YOU WOULD GET IN EDDY DIFFUSION, WHERE SOME TRACE GAS IS BEING CONVECTED AROUND BY TURBULENT MOTIONS OF THE ATMOSPHERE.

ONE OF THE REAL DISTINCTIONS OF THIS FORM, IS YOU WILL NOTE THAT IT DOES NOT TEND TO B-OPEN CONSTANT, BUT RATHER KAPPA-B-OPEN CONSTANT. 

[VU GRAPH 14] THERE ARE PROBABLY THREE TYPES OF DIFFUSION THAT YOU NEED TO WORRY ABOUT.  THE FIRST IS THE STANDARD LEIGHTON DIFFUSION.  IF DELTA H IS A STEP SIZE ALONG A SUPERGRANULE LANE, AND DELTA T THE CHARACTERISTIC TIME TO TRANSPORT ALONG THE LANE, THEN THE APPROPRIATE DIFFUSION COEFFICIENT IS DELTA H SQUARED OVER 2 DELTA T.  IT IS TWO NOT 4 SINCE THE CONVECTION ALONG THE LANE IS ALL IN ONE DIRECTION.

THE SECOND DIFFUSION COEFFICIENT IS THE ONE RESULTING FROM THE RECONNECTION WITH LOOPS.  IN THE RECENT PAPER BY SCHRIJVER, WHERE THEY REQUIRE ENHANCED TRANSPORT BEYOND LEIGHTON DIFFUSION, THIS WOULD BE THE TERM.

NOW INTERESTINGLY ENOUGH, YOU CAN SHOW ON VERY GENERAL GROUNDS, THAT H-BAR SQUARED TIMES THE OPEN FIELD STRENGTH, OVER 4 TAU, MUST ALWAYS EQUAL DELTA H – THE LANE SIZE –SQUARED, TIMES THE AVERAGE FIELD STRENGTH OF THE LOOPS WITH WHICH THE OPEN FIELD LINES RECONNECT, OVER DELTA T.  THE ARGUMENT IS AN ELECTRO-MAGNETIC ONE.  THE MOTIONS OF THE FIELD LINES IN THE CONVECTIVE MOTIONS OF THE PHOTOSPHERE CREATES A DIFFUSIVITY OF THE OPEN FIELD LINES, AND THE LOOP FIELD STRENGTH PLAYS THE SAME ROLE AS A MAGNETIZATION.  BUT WHAT THIS SAYS IS THAT THIS TERM IN OUR DIFFUSION EQUATION IS DIRECTLY MEASURABLE ON THE SUN – JUST MEASURE THE LOOP STRENGTH.

THERE IS ANOTHER FORM OF DIFFUSION WHICH I WON’T GO INTO, WHAT CAN ARISE IF OPEN FIELD LINES BECOME CONCENTRATED IN CORONAL HOLES, EXPAND AND OVERLIE THE CORONAL LOOPS, AND RECONNECT NOT AT THE BASE, BUT IN THE CANOPY.

A COUPLE OF FINAL POINTS.  THIS DIFFUSION EQUATION HAS A COUPLE OF INTERESTING SOLUTIONS THAT WERE POINTED OUT IN A RECENT AP-J PAPER WITH SCHWADRON.  YOU CAN SOLVE THIS EQUATION QUITE GENERALLY IN A STEADY STATE.  AT SOLAR MINIMUM, WHEN THERE IS A WELL DEVELOPED POLAR CORONAL HOLE, THERE IS A SOLUTION WITH A SINGLE CURRENT SHEET, STATIONARY NEAR THE SOLAR EQUATOR.  AND MOST INTERESTINGLY, WHEN THERE IS NO POLAR CORONAL HOLE, AS IN SOLAR MAXIMUM, THE ONLY TIME STATIONARY SOLUTION, IS A ROTATING CURRENT SHEET, AS OBSERVED.

FINALLY THERE IS THE ISSUE OF THE FIELD REVERSAL ON THE SUN.  THE STANDARD THEORY IS THAT LARGE BI-POLAR ACTIVE REGIONS EMERGE, WITH AN ORIENTATION SUCH THAT THE POLARITY OF THE EMERGING FIELD, SAY, TOWARDS THE NORTH, IS OPPOSITE TO THE POLARITY AT THE NORTH POLE IN THE PREVIOUS CYCLE.  THE NEWLY EMERGED FLUX THEN MIGRATES TO THE NORTH POLE, BY CONVECTIVE, LEIGHTON TYPE DIFFUSION, CHANGING THE POLARITY.

YOU CAN SEE THE IMMEDIATE PROBLEMS WITH THIS.  THE EMERGING FLUX IS CLOSED MAGNETIC FLUX- IT’S LOOPS.  WHEN IT GETS TO THE POLES IT IS STILL GOING TO BE CLOSED FLUX.  BUT WHAT YOU WANT TO CREATE AT THE POLE IS OPEN FLUX.  A LARGE CORONAL HOLE WILL DEVELOP AT THE POLES AS THE CYCLE PROGRESSES, WHICH IS PRIMARILY OPEN FLUX.

NOW YOU CAN IMAGINE THAT YOU TURN YOUR EMERGING CLOSED FLUX INTO OPEN FLUX, BY HAVING IT EXPAND INTO THE HELIOSPHERE, EFFECTIVELY AS A CME.  YOU DON’T SEE THAT, BUT MORE IMPORTANT WHAT DO YOU DO, THEN, WITH THE OPEN FLUX IN THE HELIOSPHERE, WHICH YOU ALREADY HAD, AND WHICH YOU CAN ELIMINATE ONLY BY RECONNECTING WITH ITSELF. AND OF COURSE NO BUILD UP OF FLUX IS OBSERVED.

YOU COULD, OF COURSE, DO INTERCHANGE RECONNECTION.  YOUR NEW EMERGING FLUX, CANCELS AGAINST OPEN FLUX, AND THEN NO BUILD UP RESULTS.  BUT THEN YOU HAVE AN INCONSISTENCY IN YOUR MODEL, BECAUSE THE RECONNECTION OF OPEN FIELD LINES WITH THE EMERGING LOOPS, IS A TRANSPORT PROCESS – THE OPEN FIELD LINES MOVE, AND YOU DIDN’T INCLUDE THAT IN YOUR MODELS, YOU HAD ONLY LEIGHTON-TYPE CONVECTIVE DIFFUSION.

YOU COULD ARGUE I SUPPOSE THAT THE ONLY WAY YOU GET LOOPS NEAR THE POLES IS TO HAVE ACTIVE REGIONS DISPERSE THERE.  THEN TO TRANSPORT THE OPEN FLUX THERE I WOULD HAVE TO GET THE LOOP THERE FIRST, AND I WOULD GET THE SAME ANSWER AS IF I JUST USED LEIGHTON-TYPE DIFFUSION ALONE.  BUT SUCH A PROCESS WOULD DEFY THE FACT THAT PERHAPS AS MUCH AS 10 TIMES THE AMOUNT OF MAGNETIC FLUX REACHES THE SOLAR SURFACE OUTSIDE OF ACTIVE REGIONS AS THROUGH ACTIVE REGIONS, AND THERE ARE MANY WAYS TO MAKE LOOPS BESIDES THE DISPERSAL OF ACTIVE REGIONS.

 THANK YOU VERY MUCH.

Presenter: N. R. Sheeley

1.Introduction: As Yi-Ming described yesterday, we use a model in which new flux  erupts in active regions and is transported on the photosphere by supergranular diffusion, differential rotation, and meridional flow.  The field extends upward in a current-free configuration and open flux is simulated by requiring the non-radial components to vanish at a spherical source surface located at 2.5 R. Today, I am going to consider how the open flux changes in response to variations of the surface field.  The general answer is that the open flux increases when opposite polarities are separated and the open flux decreases when the opposite polarities are mixed together.

2.Four Examples: It is easy to verify this answer for the source surface model because the open flux is precisely defined.  Let’s consider four examples: a horizontal dipole, a tilted dipole, an equatorial configuration, and the eruption of a new doublet.

     [a] HORIZONTAL DIPOLE:  Differential rotation winds the horizontal dipole into stripes of alternating magnetic polarity.  (FIG. 1)  The source surface does not see such mixtures and its field is determined by the dwindling remnant of unwound flux at the equator.  In the language of spherical harmonics, power is transferred from the dipole to the higher order harmonics (l = 3, 5, 7, …) which have little contribution at 2.5 R.  The open flux mimics the dipole amplitude, hovering unchanged for a brief moment and then falling rapidly to half its initial value in the time it takes to wind the meridional boundary through 360 degrees (4.8 rotations). (FIG. 2)  Thereafter, the amplitude decays inversely as the square root of the elapsed time.  Because differential rotation does not change the total amount of flux on the Sun, open flux must have been converted to closed flux by reconnecting  at the source surface and collapsing downward over the growing photospheric neutral line.  Adding supergranular diffusion to the problem has little effect during the first windup time because the diffusion time scale is 15 yrs for a dipole. (FIG3)  It’s main contribution is to decrease the total flux by removing the narrow-spaced stripes. (FIG. 4)  Meridional flow also does not contribute during the first windup time because its time scale is about 1 yr.  Thus, in general, we expect reconnection and collapse during the first windup time as open flux is converted to closed flux.

     [b] TILTED DIPOLE:  Now, let’s add a vertical component to make a tilted dipole.  The photospheric neutral line starts as a sinusoid, but is quickly wound up into stripes as before.  (FIG. 5)  However, in this case, the dwindling remnant of unwound flux at the equator is accompanied by unwound flux at the poles, and the amplitude of the tilted dipole decays more slowly than the amplitude of its horizontal component.  (FIG. 6)  In fact, the  net change of open flux is reduced by the factor tan(tilt_angle/2), which approaches zero for very strong vertical fields.  Thus, we would expect the Sun to shed less open flux during the declining years of the sunspot cycle when the polar field is strengthening than in the years around sunspot maximum when the polar field is weak.  As before, the addition of diffusion and flow do not change this behavior appreciably during the first windup time (when half of the change occurs).

     [c] A LATITUDINALLY COMPRESSED DISTRIBUTION: If the non-axisymmetric flux were concentrated near the equator, then it would be unaffected by differential rotation, and the field would not change with time. (FIG. 7)  The source-surface field would maintain a tilted dipole configuration with no change of open flux, and the meridional lobe of the coronal hole would not be sheared by differential rotation.  The same thing would happen if the equatorial flux were compressed in longitude to simulate a low-latitude emerging flux region.  (FIG. 8)  In this case, one would obtain a large meridional coronal hole like Skylab CH#1 in 1973, which resisted the shearing effect of differential rotation for several months. 

     [d] FLUX ERUPTIONS: Because most newly erupting doublets are tilted relative to the parallels of latitude, they contribute to both the vertical and the horizontal components of the large-scale field.  The vertical contribution is small, but systematically opposite the polar field at the start of the sunspot cycle, and therefore will weaken it.  The horizontal contribution is larger, but may be out of phase with the background field or in phase, depending on the longitude and north-south hemisphere of the eruption.  If the eruption is in phase with the horizontal dipole, it will strengthen it and cause the total dipole to tip further toward the equator.  If it is out of phase, the eruption will weaken the horizontal dipole and tilt the total dipole back toward the pole.  This is why some large active regions increase the warp of the coronal streamer belt, while other equally large regions do not.  In the source-surface model, these changes occur in step with the growing active regions and do not depend on the rates of photospheric flux transport.

     It is instructive to consider what would happen during a sunspot cycle if we were to neglect flux transport and simply add together the dipole contributions of all of the erupting active regions.  There would be a systematic erosion and reversal of the polar fields.  But the amplitude of the horizontal dipole would increase without limit in a random walk produced by the longitudinal fluctuations of the sources.  The flux transport provides an effective dissipation that limits this growth and allows the evolving field to come into equilibrium with its sources.  It would strengthen the horizontal dipole around sunspot maximum and weaken it again at minimum, as observed.

     But this concept of a global dipole tipping over has two more problems.  First, it leaves out quadrupole terms, which result from a variety of active region configurations (FIG. 10) and have a major contribution to the open flux in the years around sunspot maximum.  (FIG. 9) Second, it ignores interactions between multipoles, such as occur when differential rotation increases the pole separation of a newly erupted doublet.  In desperation, one might hope to salvage this idea of  `something’  tipping over during the sunspot cycle by considering a more general concept like `open flux’ or `streamer belt topology’, and then averaging over transients and interactions between multipoles.  But in doing so, he would be throwing away the opportunity for new discoveries that the flux-transport/source-surface model provides.

     So where do we stand?  We have a model that is useful for discovering new things about the Sun.  We found that open flux decreases when the horizontal dipole is wound up by differential rotation, as often occurs in the years around sunspot maximum.  And we learned that much less open flux is shed when the non-axisymmetric field is accompanied by a strong axisymmetric component, as occurs during the declining phase of the cycle.

     The accompanying movies show coronal inflows at sector boundaries where opposite-polarity field lines are being pushed together by shearing motions of their footpoints.(http://lasco-www.nrl.navy.mil/public_movies.html )  Most inflows occur in long-lived streams where low-latitude coronal holes are being sheared by solar differential rotation and their oppositely directed coronal field lines are being pushed together (FIG. 11).  This suggest that they indicate magnetic field reconnection associated with the decreasing open flux of shearing coronal holes. 

Most of these inflows lack visible outward components, perhaps due to their extreme faintness.  However, some inflows do have outward components.  It is possible that these in/out pairs involve reconnection between open field lines and closed loops.  If so, we might expect to see more of them during the declining phase of the sunspot cycle when the polar fields are stronger and coronal holes begin to rotate rigidly again.

References

“Simulations of the Mean Solar Magnetic Field During Sunspot Cycle 21”, Sheeley, Jr., N.R., DeVore, C.R., and Boris, J.P.: 1985, Solar Phys. 98, 219.

“Mechanisms For the Rigid Rotation of Coronal Holes”, Nash, A.G., Sheeley, Jr., N.R., and Wang, Y.-M.: 1988, Solar Phys. 117, 359.

“Magnetic Flux Transport On the Sun”, Wang, Y.-M., Nash, A.G., and Sheeley, Jr., N.R.: 1989, Science 245, 712.

“Understanding the Rotation of Coronal Holes”, Wang, Y.-M. and Sheeley,  Jr., N.R.: 1993, Astrophys. J. 414, 916.

“The Magnetic Nature of Coronal Holes”, Wang, Y.-M., Hawley, S.H., and Sheeley, N.R., Jr.: 1996, Science 271, 464.

“Streamer Disconnection Events Observed With the LASCO Coronagraph”, Wang, Y.-M., Sheeley, N.R., Jr., Howard, R.A., and Lamy, P.L.: 1999, Geophys. Res. Letters 26, 1349.

“The Long-Term Variation of the Sun’s Open Magnetic Flux”, Wang, Y.-M., Lean, J., & Sheeley, N.R., Jr.: 2000, Geophys. Res. Letters 27, 505.

“Understanding the Evolution of the Sun’s Open Magnetic Flux”, Wang, Y.-M., Sheeley, N.R., Jr., & Lean, J.: 2000, Geophys. Res. Letters 27, 621.

“Coronal Inflows and the Sun’s Nonaxisymmetric Open Flux”, Sheeley, N.R. Jr., Knudson, T.N., & Wang, Y.-M.: 2001, Astrophys. J. Letters 546, L131.

“Coronal Inflows and Sector Magnetism”, Sheeley, N.R., Jr. & Wang, Y.-M.: 2001, Astrophys. J. Letters  562, L107.

“Meridional Flow and the Solar Cycle Variation of the Sun’s Open Magnetic Flux”, Wang, Y.-M., Sheeley, N.R., Jr., & Lean, J.: 2002, Astrophys. J. (in press).