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.