The formation of magnetic depletions and flux annihilation due to reconnection in the heliosheath
aa r X i v : . [ phy s i c s . s p ace - ph ] F e b The formation of magnetic depletions and flux annihilation due toreconnection in the heliosheath
J. F. Drake , , M. Swisdak , M. Opher , and J. D. Richardson ABSTRACT
The misalignment of the solar rotation axis and the magnetic axis of the Sunproduces a periodic reversal of the Parker spiral magnetic field and the sectoredsolar wind. The compression of the sectors is expected to lead to reconnectionin the heliosheath (HS). We present particle-in-cell simulations of the sectoredHS that reflect the plasma environment along the Voyager 1 and 2 trajectories,specifically including unequal positive and negative azimuthal magnetic flux asseen in the Voyager data (Burlaga et al. 2003). Reconnection proceeds on in-dividual current sheets until islands on adjacent current layers merge. At latetime bands of the dominant flux survive, separated by bands of deep magneticfield depletion. The ambient plasma pressure supports the strong magnetic pres-sure variation so that pressure is anti-correlated with magnetic field strength.There is little variation in the magnetic field direction across the boundaries ofthe magnetic depressions. At irregular intervals within the magnetic depressionsare long-lived pairs of magnetic islands where the magnetic field direction re-verses so that spacecraft data would reveal sharp magnetic field depressions withonly occasional crossings with jumps in magnetic field direction. This is typicalof the magnetic field data from the Voyager spacecraft (Burlaga & Ness 2011;Burlaga et al. 2016). Voyager 2 data reveals that fluctuations in the density andmagnetic field strength are anti-correlated in the sector zone as expected fromreconnection but not in unipolar regions. The consequence of the annihilation ofsubdominant flux is a sharp reduction in the “number of sectors” and a loss inmagnetic flux as documented from the Voyager 1 magnetic field and flow data(Richardson et al. 2013). Department of Physics, the Institute for Physical Science and Technology and the Joint Space Institute,University of Maryland, College Park, MD 20742; [email protected] Institute for Research in Electronics and Applied Physics,, University of Maryland, College Park, MD20742 Astronomy Department, Boston University, MA 02215 Kavli Center for Astrophysics and Space Science, Massachusetts Institute of Technology, Cambridge,MA 02139
Subject headings:
1. INTRODUCTION
The rotation of the Sun twists the solar dipole field into the dominant azimuthal Parkerspiral magnetic field B T separated by the heliospheric current sheet. Because of the mis-alignment of the solar rotation axis and magnetic axis the current sheet flaps in the verticaldirection as it propagates outward from the Sun, producing the sectored magnetic field(Wilcox & Ness 1965) in which azimuthal magnetic field B T reverses sign around every 13days (the RTN coordinate system is defined with R in the radial direction, T in the azimuthaldirection with positive T in the direction of the Sun’s rotation and N, which points Northin the equatorial plane, completes the triad). The sector-zone occupies a latitudinal extentthat varies during the solar cycle, reaching nearly the poles when the fields from the Sun area maximum (Smith 2001).An important question is whether the sectored magnetic field can reconnect to releasemagnetic energy and accelerate particles. In the solar wind around 1 AU the heliospheric cur-rent only occasionally undergoes reconnection (Gosling 2007), probably because the currentsheet is far wider than the characteristic ion inertial scale d i = c/ω pi (where collisionless re-connection onsets) (Cassak et al. 2005). As a result, the sector stucture of the solar magneticfield survives out to the termination shock (TS) even though the periodicity of the currentsheet becomes increasingly irregular with distance from the Sun (Burlaga et al. 2003, 2005,2006). It has been suggested that the drop in the solar wind density with distance and there-fore the increase in d i , combined with the compression of the current sheets downstream ofthe TS, leads to the onset of reconnection in the sectored heliosheath (HS). Such recon-nection has been proposed as a source of free energy to drive the production of anomalouscosmic rays (ACRs) (Drake et al. 2010; Opher et al. 2011) and as a source of turbulence inthe heliosheath that might control the transport of energetic particles (Burgess et al. 2016).The reduction of the plasma flow in the HS on its approach to the heliospause is expectedto further compress the sectored magnetic field, inevitably leading to the onset of reconnec-tion in the HS (Czechowski et al. 2010; Drake et al. 2010; Opher et al. 2011; Borovikov et al.2011). However, determining the thickness of the heliospheric current sheets downstreamof the TS to confirm that collisionless reconnection should onset in the HS is a challengebecause of the low and variable speed of plasma flows in the HS and because weak magneticfields there are difficult to measure. It has been suggested that current sheets in the HS arethicker than the ion inertial scale (Burlaga & Ness 2011) but post reconnection final statesare characterized by magnetic islands in which current layers are comparable in width to 3 –the island width. Thick current sheets might therefore suggest that reconnection in HS hasalready taken place.In any case, smoking gun evidence from the Voyager observations that reconnection inthe sectored HS has taken place has not been identified. The challenge is that the Voyagermagnetometers were not designed to measure the weak magnetic fields in the outer helio-sphere ( ∼ . ∼ . β environment of the HS remains relativelypoorly understood (Schoeffler et al. 2011; Schoeffler et al. 2013) compared with the typically β ∼ AU . The multiple sharp dropouts of energetic particles of heliosphericorigin measured by Voyager 1 at the HP boundary (Stone et al. 2013) do suggest the exis-tance of magnetic islands and therefore magnetic reconnection at and in the vicinity of theHP(Swisdak et al. 2013; Strumik et al. 2013, 2014).There are a variety of indirect indicators that reconnection is active in the HS, includingthe loss in magnetic flux documented by Voyager 1 (Richardson et al. 2013) and the dropoutsin the “low energy” electrons at Voyager 2 (Opher et al. 2011; Hill et al. 2014). In the ab-sence of reconnection the azimuthal magnetic flux V R RB T is preserved in the heliosphere inregions where the flow is dominantly radial. While Voyager 1 was in the HS, the radial flow V R dropped essentially to zero (Krimigis et al. 2011) while there was no significant increasein B T . A significant flow in the N direction (to the North in the case of Voyager 1) mightconvect the flux away and therefore prevent the pileup of B T . However, the Voyager 1 datasuggested that V N was also very small (Decker et al. 2012; Stone & Cummings 2012). Thus,it seems likely that reconnection must be playing a role in the Voyager 1 flux loss measure-ments. On the other hand, reconnection actually does not normally reduce the magneticflux in a reconnecting current layer. Rather, the flux that reconnects at an x-line is con-vected into the adjacent magnetic island such that the integrated magnetic flux is preserved(Fermo et al. 2010). The flux loss issue can therefore not be simply be resolved by invokingreconnection without a more careful analysis. The dropouts in low energy electrons at Voy-ager 2 were attributed to Voyager 2 leaving the sector zone (Opher et al. 2011; Hill et al.2014). The argument was that electrons can rapidly escape from a region of the heliosheathwith laminar (unreconnected) magnetic fields while reconnected magnetic fields and associ-ated magnetic islands would be more effective in suppressing electron transport along theambient magnetic field. Thus, higher electron fluxes in the sectored heliosheath are evidencethat the heliosheath magnetic field had reconnected – in the absence of reconnection thereis no reason that the transport properties of the sectored and non-sectored HS should differ.The Voyager observations in the HS have uncovered other issues related to the re-connection or not of the sectored magnetic field. The first concerns the polarity of the 4 –HS magnetic field. The nominal polarity of the magnetic field in the Northern hemi-sphere during the time period 2000-2011 is negative (corresponding to the azimuthal angle λ = arctan( B T /B R ) = 90 ◦ ) (Burlaga & Ness 2012; Richardson et al. 2016). When Voyager1 is in the sector zone, λ typically flips back and forth between 90 ◦ and 270 ◦ and a longperiod of 90 ◦ , which occurred in 2011, would normally be interpreted as an excursion outof the sector zone and into the unipolar zone. On the other hand, the MHD models of theglobal heliosphere predict that the sector zone should be convected to the North as the HSplasma approaches the HP so that Voyager 1 should remain within that sector zone priorto its crossing of the HP (Opher et al. 2011; Borovikov et al. 2011). Similarly, in 2011-2012Voyager 2 saw significantly fewer excursions into negative (North latitude polarity) thanexpected based on the Wilcox Solar Observatory predictions (Richardson et al. 2016). Whythe Voyagers are seeing fewer excursions into magnetic field polarities that are opposite totheir heliolatitude is a mystery.A second mystery concerns the distinct magnetic structures seen in the Voyager 1 and2 data. At “proton boundary layers (PBLs)” the magnetic field strength either rises ordrops by factors of up to three with no measureable change in the azimuthal angle λ or theelevation angle δ = arcsin( B N /B ) (the angle of the magnetic field with respect to the R-Tplane) (Burlaga & Ness 2011; Burlaga & Ness 2012). Since these regions of magnetic fieldenhancement or depletion can last for days, it seems unlikely that they are associated withkinetic scale instabilities such as mirror modes, which typically produce localized “humps”in the magnetic field rather than depletions in high β plasma (Baumg¨artel et al. 2003).Here we address the dynamics of reconnection in the sectored HS with the goal of under-standing the signatures and consequences of the reconnection and the resultant structure ofthe HS magnetic field. The simulations extend earlier models by considering more realisticinitial conditions that account for unequal positive and negative azimuthal magnetic flux.That the sector zone does not carry equal postive and negative flux at the latitudes of theVoyager spacecraft trajectories is evident from the Voyager observations in the solar wind inthe outer heliosphere but upstream of the termination shock (Burlaga et al. 2003). In thisregion because the solar wind velocity greatly exceeds the spacecraft velocity, time in a regionof given polarity is linked to the integrated magnetic flux in a given sector. The Voyager datain the high-speed solar wind of the outer heliosphere reveals that sector spacing is highlyerratic and therefore positive and negative fluxes are unequal. For the time period 2000-2011one might expect that for Voyager 1 the negative polarity dominates because the spacecraftis closer to the Northern boundary of the sector zone, which has negative polarity, while forVoyager 2 positive polarity dominates. Unequal magnetic fluxes were required to reproduceVoyager 1 magnetic data in MHD simulations of reconnection at the HP (Strumik et al.2014). 5 –The consequence of unequal fluxes is profound. At late time when reconnection is nearlycomplete, bands of single polarity flux survive, which tends to organize the sector structuremore than in earlier simulations in which magnetic islands dominated the magnetic structureat late time (Drake et al. 2010; Opher et al. 2011). The simulations are also carried out withhigh initial β and with initial force-free current layers rather than Harris type current layers,which are typically not seen even at 1AU (Smith 2001). Because of the high β the magneticfield at late time exhibits large-scale depletions in which the magnetic field strength drops byaround a factor of three over a narrow boundary layer with little variation in the direction ofthe magnetic field. The plasma density and temperature rise slightly within the depletionsto maintain pressure balance. The radial width of these magnetic depletions is aroundthree times the width of the initial sector with the subdominant magnetic flux. The depthof these depletions and their widths (when normalized to the initial separation of currentlayers) are universal values that are linked to the intrinsic properties of collisionless magneticreconnection. The boundaries of these magnetic depletions exhibit a striking resemblanceto the “proton boundary layers” seen in the Voyager data. We show that reconnectionon adjacent current layers conspires to completely annihilate the subdominant magneticflux, leading to regions of unipolar flux. Pairs of magnetic islands do survive at late timealthough the volume of plasma associated with these remnant islands is small compared withthe regions of magnetic depletion. The island structures exhibit magnetic dips and rotationsin the magnetic field direction that might be interpreted as sector crossings in satellite data.The simulations therefore offer a possible explanation of the predominance of unipolar fluxand the loss of magnetic flux seen in the Voyager 1 magnetic field data. Finally, analysisof the Voyager 2 magnetic field and plasma data reveals that fluctuations in magnetic fieldstrength and density are anti-correlated in the sectored HS, as expected from reconnection,but not in the unipolar HS. As a whole, the consistency of the Voyager data with uniquereconnection signatures establishes with high likelihood that reconnection in the sectored HShas taken place.
2. PIC model and initial conditions
We carry out 2-D particle-in-cell (PIC) simulations of the sector structure in the x − y plane of the simulation, which maps to the heliospheric R − T plane. The simulations areperformed with the PIC code p3d (Zeiler et al. 2002) using a periodic equilibrium magnetic 6 –field B y B = tanh (cid:18) x − . L x w (cid:19) − tanh (cid:18) x − . L x w (cid:19) + tanh (cid:18) x − . L x w (cid:19) − tanh (cid:18) x − . L x w (cid:19) − . (1)For this magnetic configuration there are four current layers centered at x/d i = 20 .
48, 30 . .
68 and 81 .
92 on a computational domain that is L y × L x = 409 . × . d i = c A / Ω i the proton inertial length. The total magnetic flux in thepositive y direction is four times that in the negative y direction. The initial plasma density n and temperatures T e and T i are constants and the out-of-plane magnetic field B z = B − B y is chosen to produce force balance. This initial state is not a rigorous kinetic equilibrium,especially for ions, but does not display unusual behavior at early time. The results arepresented in normalized units: the magnetic field to the asymptotic value of the reversedfield B , the density to n , velocities to the proton Alfv´en speed c A = B / √ πm i n , timesto the inverse proton cyclotron frequency, Ω − i = m i c/eB y , and temperatures to m i c A . Wedefine some additional scale lengths as follows: w = 0 . d i is the half-width of an individualcurrent sheet and ∆ x = ∆ y = 0 . d i are the grid scales. To maximize the separation betweenthe macroscales L x and L y and the kinetic scales, we choose a modest ion to electron massratio of 25 and velocity of light c of 15 c A . The particle temperatures are initially uniformwith T i = 5 . m i c A and T e = 2 . m i c A so β = 8 πn ( T e + T i ) /B = 14 is large, as expected forthe heliosheath. The average number of particles per cell is 100. Reconnection begins fromparticle noise.The overall scale sizes of our simulations are much smaller than those of the heliosphericsectored field. The widths of the sectors upstream of the TS are around 1 . × km ,which at a density of 0 . /cm , is around 2 × d i . Compression across the shock andthe approach to the heliopause reduces the sector width somewhat but the sector spacingcontinues to be far larger in units of d i than the values we can implement in our simulations.However, we have shown earlier that the rate of growth of islands is insensitive to the kineticscale d i (Schoeffler et al. 2012) and the same conclusion applies to the simulations presentedhere. Thus, the reconnection rates and associated bulk ion flows can be translated to theheliosheath by normalizing to the Alfv´en speed and the Alfv´en transit time L x /c A .
3. Simulation results
In Fig. 1 we show 2D plots of the magnetic field strength B and the magnetic field linesaround the two lower current layers in the system at times Ω i t = 50, 200 and 350. The 7 –initial magnetic field points to the left above and below the two current layers and to theright between the two layers. In (a) and (b) a large number of very small islands grow onthe two current layers at early time. These islands coalesce and continue to grow until in (c)and (d) they span the entire region between the adjacent current layers. At this point allof the magnetic field lines spanning the simulation domain in the positive y direction havereconnected. On the other hand, positive magnetic flux still exists. Along a single cut in x in(d) B y has positive and negative values. However, island merging and reconnection continuesand at later time in (e) and (f) much of the positive magnetic flux has been annihilated.How this happens is important first because of the flux loss documented by the Voyager1 observations and second because magnetic reconnection normally preserves the magneticflux. At the magnetic x-line field lines reconnect but the integrated unsigned flux througha magnetic island is unchanged as reconnection proceeds. Namely, there is no flux loss atthe center of the island and magnetic flux is preserved elsewhere so the total magnetic fluxcontained in an island is preserved. On the other hand, it is evident from Fig. 1(f) that ona cut along x at y ∼ B x will only have positive values – there is no surviving negativeflux in this region.The pressure anisotropy that develops during magnetic reconnection in a high β systemsuch as the HS weakens the tension force exerted by the magnetic field on the plasma andtherefore slows reconnection and allows islands take elongated forms (Drake et al. 2010;Opher et al. 2011; Schoeffler et al. 2011). A video animation of the magnetic field B for thefull duration of the simulation domain (up until Ω ci t = 440 . ) is available online. The moviereveals that the evolution of the magnetic field slows dramatically at late time, a consequenceof the pressure anisotropy, and allows isolated elongated magnetic islands to survive late intime.In Fig. 2 we show a blowup of B and associated magnetic field lines at four timesto illustrate how flux anniliation in the geometry of the sectored heliosphere takes place.Again, the magnetic field lines above and below the two current layers initially point to theleft and between the two current layers point to the right. In (a) magnetic islands on the twocurrent layers do not yet cross-connect with the adjacent current layer and there is positivemagnetic flux that spans the domain in y . In (b) the magnetic separatrix of the large islandon the lower current layer connects with the upper current layer. At this point there is stillsubstantial negative magnetic flux even though there are no positive B y field lines that spanthe system along the y direction. However, the positive magnetic field B y of the island onthe lower current layer begins reconnecting with the negative flux above the upper currentlayer, which annihilates the positive flux. In (d) almost all of the magnetic flux in the islandfrom the lower current layer has been annihilated, eliminating the surviving positive flux. 8 –The late-time (Ω p t = 440) structure of the magnetic fields, density, and temperatureof the full simulation domain are shown in Figs. 3 and 4. In Fig. 3(a) and (b) are B andfield lines of the entire x − y computational domain. Surviving at late time are two pairsof islands embedded in a lower and upper band of depleted magnetic energy. Elsewherethe islands have reconnected away, leaving only negative magnetic flux. The structure ofthe magnetic depletions is futher illustrated in cuts of B in (c), n in (d), T i in (e) and theazimuthal angle λ = arctan( B y /B x ) in (f), where λ defines the direction of the magneticfield in the x − y plane with respect to the x direction. The location of the cut is marked bythe white line in Fig. 3a. The cut in Fig. 3a is chosen specifically because it represents whata satellite might typically see in a heliosheath where reconnection has already annihilatedsignificant magnetic flux. The strong depletions of the magnetic field seen in (c) span theregion around each pair of adjacent current layers in the initial system and are a consequenceof growth and merger of magnetic islands on adjacent current layers. The boundaries of themagnetic depletions mark the maximum spatial extent of reconnection of adjacent currentlayers. Pressure balance is maintained within the magnetic depletions by small increases inthe density and ion temperature. The increase in electron temperature (not shown) is evensmaller. In the cut shown here the azimuthal angle λ is nearly constant at 270 ◦ . In thisregion all of the initial positive flux has been annihilated. The scale length of the boundariesof the magnetic depletions are expected to scale with the proton Larmor radius ρ i becausethe protons carry most of the pressure and they are able to decouple from the magneticfield on scale lengths of the order of several ρ i (Drake et al. 2009). For the simulation ofFig. 3, the proton Larmor radius is around 2 . d i so the scale lengths of the boundaries ofthe magnetic depletions in Fig. 3(c) are 3 − ρ i . We emphasize that the scale lengths ofthese boundary layers are insensitive to the initial width of the initial current sheets becausethe boundaries are associated with the upstream edges of the reconnection exhaust, which iscontrolled by local physics – ions move from upstream into the exhaust and are acceleratedup to the Alfv´en speed across a narrow boundary layer (Drake et al. 2009).Figure 4 is similar to Fig. 3 but the cuts are now taken through the pair of magneticislands on the top band of depleted magnetic field. Again the white line shows the locationof the cut in Fig. 4(a). The centers of the two islands can be seen in Fig. 4(b) but are mostevident in the cut of λ , which jumps sharply from 270 ◦ to 90 ◦ and back across the centers ofthe islands. The island centers are locations of minima of B and small peaks in the densityand ion temperature. Distinguishing the crossing of such an island from the crossing ofthe heliospheric current sheet in the heliosheath would be difficult because the traditionalsignatures of reconnection, such as high-speed flow, have died away – the system has evolvedto a quasistatic state. We emphasize that the probability of crossing a region where thesubdominant magnetic flux survives at late time in the system is small. In Fig. 5 the spatial 9 –distribution of λ at late time is shown in (a) and in (b) is the probability distribution of λ .The probability of finding λ ∼ ◦ in this simulation at late time is finite but small.We can give estimates for the widths and size of the magnetic depletions based on howreconnection developed to produce the final states shown in Figs. 3 and 4. The total width ofthe magnetic depletion can be calculated by determining the spatial extent of reconnectionleading to the final state. Consider two adjacent current layers separated by a distance δL with a total magnetic flux between the two layers δψ . The separatrix magnetic fieldline that connects to the two x-lines in Fig. 2(b) was originally at the center of the regionbetween the two nearby current layers at x/d i = 20 and 30. Thus, during the reconnectionthat led to this state half of the magnetic flux in the region between the two current δψ layers reconnected with the flux above the current layer at 30 and half reconnected with theflux below the current layer at 20. Thus, the spatial extent in x of the entire region insideof the reconnected field lines is 2 δL ( δL/ δL/ δL between the two current layers). During the time fromFig. 2(b) to late time the remaining positive flux δψ/ x ∼
20 and y ∼
185 reconnects with the negative flux above the upper current layer. Atthe end of this reconnection process the island is gone. The same happens as the positiveflux in the island centered at x ∼
30 and y ∼
145 reconnects with the negative flux belowthe lower current layer. The reconnection of these two islands with flux above and belowextends the reconnection zone another distance δL/ δL . The total surviving magnetic flux over this domain is δψ ( δψ eachfrom above and below the initial current layers and − δψ from between the current layers).Since this flux is spread out over a distance that is three times the initial separation of thecurrent layers, the magnetic field strength is B /
3, independent of the size of the simulationdomain, current layer separation or plasma parameters. The cuts of B in Fig. 3(c) displaydepletions with widths that are around 30, consistent with this estimate, and with magneticfield minima that are close to 0 . B . A Probability Distribution Function (pdf) of themagnitude of the in-plane magnetic field B plane is presented in Fig. 6. There are distinctpeaks in the pdf at the initial magnetic field strength B and at B /
3. Of course, if thedisparity between positive and negative flux is insufficient, the magnetic islands growingon all current layers will ultimately overlap and the organized magnetic depletions shownin Figs. 3 and 4 will not characterize the final state. The disparity between positive andnegative flux needs to exceed two for the isolated depletions to survive at late time. 10 –
4. Observational Results from Voyager 2
Our simulations of the sectored HS suggest that large depletions in the magnetic fieldresult from magnetic reconnection. The “proton boundary layers” (Burlaga & Ness 2011;Burlaga & Ness 2012) that have been identified in the Voyager 1 and 2 data look very muchlike the boundaries of the strong magnetic depletions that we see in our simulations. Be-cause reconnection-driven flows are Alfv´enic and therefore fall below the local magnetosonicvelocity in the high- β heliosheath, reconnection dynamics is to lowest order incompressible,which means that depletions in the magnetic field pressure correspond to enhancements inthe plasma density and pressure. This can be seen in the cuts across the magnetic depletionsin the simulation data presented in Figs. 3 and 4. Thus, if the magnetic field disturbancesin the sectored heliosheath are driven by reconnection, they should correspond to perturba-tions in the density such that deviations of the magnetic field and density from the ambientbackground are anti-correlated – reductions (increases) of the magnetic strength should cor-respond to local increases (decreases) in the plasma density. In the non-sectored heliosheaththis anti-correlation should not be present unless there are mechanisms for generating non-compressible turbulence other than reconnection.Thus, we have explored the correlation between magnetic field and density fluctuationsin the Voyager 2 datasets. The plasma instrument on Voyager 1 failed many years agoso correlation studies with the Voyager 1 data are not possible. We have compared thefluctuations of the density ( dn ) and magnetic field magnitude ( dB ) in a heliosheath regionwhere the sector structure is observed (2008.2 - 2009.15, 344 days of data) with those in aunipolar region (2009.15 -2010.5, 455 days of data) (Burlaga & Ness 2011; Richardson et al.2016). We use daily averages of the magnetic field magnitudes from the SPDF website andof the density from the MIT Voyager web site. Figure 7 shows histograms of < dBdn >/ √ < dB >< dn > where dB = B − < B > and dn = n − < n > . The average < A > of any quantity A is defined over a 25 day averaging window. The histograms are clearlydifferent for the two regions, with dB and dn usually having opposite signs (anti-correlated)in the sector region but not in the unipolar region. Thus, the Voyager 2 data suggests thatfluctuations in the sectored heliosheath are driven by reconnection. Alternative explanationsfor the anti-correlation between the fluctuations in the magnetic field intensity and densitywould need to explain why the fluctuations are anti-correlated in the sectored zone but notin the unipolar region.To compare this correlation data with that from our simulation, we evaluate δnδB plane q < δn >< δB plane >
11 –with δf = f − < f > for any function f and where < f > is an average over the simulationdomain. The pdf of this correlation is shown in Fig. 8. As in the observational data thedensity and B plane are anti-correlated and there is a distinct tail on the negative side of thedistribution.
5. Discussion
We have carried out kinetic simulations of magnetic reconnection in the sectored he-liosheath that include the asymmetry in the magnetic flux in the sectors that is expectednear the Northern and Southern latitude boundaries of the heliospheric sector zone. We showthat when the magnetic flux asymmetry is more than a factor of two, bands of unreconnectedmagnetic flux B T survive and sandwich bands that have strongly depleted magnetic field.In the final state the widths of the depletion regions are around three times the width of theinitial current layer separations with magnetic field strengths that are around one third ofthe pre-reconnection intensity. The boundaries of the depletion regions are sharp – severaltimes the proton Larmor radius. The reconnection of magnetic islands on adjacent currentsheets leads to the nearly complete annihilation of flux in the subdominant direction. Suchflux annihilation does not typically take place during magnetic island growth on a singlecurrent layer because magnetic islands conserve the integrated magnetic flux. The finalstate has scattered pairs of remnant magnetic islands in which the subdominant magneticflux survives. Cuts across these islands would appear like crossings of an undisturbed he-liospheric current sheet – the azimuthal angle λ jumps sharply from 90 ◦ to 270 ◦ and thenreverses across the cores of these islands (Fig. 4). The probability of crossing such an island,however, is low compared to that of crossing a pristine region of magnetic depletion (Fig. 3).In the spherically expanding solar wind the conservation of magnetic flux implies that V R B T R is a constant. For a constant solar velocity V R this expression yields the usualfalloff of B T as 1 /R , which has been documented in the solar wind. In the heliosheaththe magnetic field is more complex because of the development of latitudinal flows V N .However, it was a major surprise when the estimated radial plasma flows V R at the Voyager1 spacecraft dropped to essentially zero in 2010 (Krimigis et al. 2011) and yet the magneticfield strength did not increase to compensate (Burlaga & Ness 2012). Since the velocity V N was also close to zero, the loss of magnetic flux through a flow to high latitude wasinsufficient to explain the apparent loss of flux. The conclusion therefore was that theVoyager 1 observations documented a flux loss in the heliosheath (Richardson et al. 2013).In contrast, the magnetic field and flow measurements at Voyager 2 suggested that flux wasconserved along its trajectory. 12 –The model presented here might explain how magnetic flux could be lost in the sectoredheliosheath. Of course, if flux annihilation as discussed here did take place in the heliosheathat the location of Voyager 1, one would expect to see fewer sector crossings than expectedin the magnetic field data. From 2010 to the heliopause crossing in mid 2012 the Voyager 1spacecraft did see less Southern polarity flux than expected from the WSO data. However,during the long period during which the measured value of V R decreased at Voyager 1, thedata does not suggest a reduced probability of Southern polarity magnetic flux. Indeed,there is an unexplained period during 2008-2010 when the probability of seeing Southernpolarity flux is much higher than expected (Richardson et al. 2016). Magnetic reconnectionnevertheless seems to be the only viable mechanism that can explain the flux loss along theVoyager 1 trajectory.The challenge is to identify a more direct method of establishing whether reconnectionis taking place in the heliosheath. This is not easy because crossing a current sheet wherereconnection is actually taking place is highly improbable. There have now been severalhundred identifications of active reconnection in the solar wind at 1AU and yet there is nota single documented observation of the crossing of a magnetic x-line where active recon-nection is ongoing – how does one distinguish a static current layer from a current layerwhere reconnection is active? This requires the accurate measurement of the intense Hallelectric field that bounds the current layer on either side of the x-line (Drake et al. 2008),which has not been measured in the solar wind. The documented reconnection observations(Gosling et al. 2005; Gosling 2007) are crossings of the reconnection exhaust, where themeasured exhaust velocity has been cross-checked with the predictions based on the Wal´encondition (Hudson 1970). What are the corresponding direct signatures of reconnection inthe heliosheath? We have argued previously that if reconnection onsets just downstreamof the termination shock, where the heliospheric current sheet should be compressed belowthe ion inertial scale d i , the growth time for islands to reach the characterisic sector spacingshould be around 60 days, which translates to a distance around 2-3 AU downstream of thetermination shock (Schoeffler et al. 2012). Deep within the heliosheath the sectors shouldtherefore have reconnected and the signatures of reconnection should reflected in the late-time structure of the magnetic remnants of reconnection rather than an active reconnectionsite. That reconnection is taking place in the heliosheath is also supported by the ACR spec-tra, which peak well downstream of the termination shock (Stone et al. 2005; Decker et al.2005; Decker et al. 2008). Reconnection downstream of the termination shock is one pos-sible explanation of these observations (Drake et al. 2010; Opher et al. 2011). Some morerecent theoretical models suggest that reconnection dynamics downstream of the shock is anintrinsic component of the termination shock structure and associated particle acceleration(Zank et al. 2015). 13 –In the core of the sector zone, where the positive and negative polarity fluxes are nearlyequal, the late time state consists of elongated magnetic islands (Opher et al. 2011) whilecloser to the latitudinal boundaries of the sector zone where positive and negative polar-ity fluxes are not equal, the spacecraft observations should resemble the cuts shown inFigs. 3 and 4. The magnetic depletions across rather narrow boundary layers, which canextend to several AU in width, are the clearest direct signatures of the post reconnection he-liosheath. The “proton boundary layers (PBLs)” that have been documented in the Voyager1 (Burlaga & Ness 2011; Burlaga & Ness 2012) and 2 (Burlaga et al. 2016) data are verysimilar to the magnetic depletions that appear in our simulations. A surprise is that manyof the clearest examples of PBLs from the Voyager 1 data seem to have jumps in magneticfield strength that are around three, which is the value which follows from our analysis ofthe reconnection dynamics. The measured scale lengths of the measured PBLs are 5 − ρ i ,which are modestly wider than those seen in our simulations. Other suggestions are that thePBLs result from the growth of mirror modes (Burlaga & Ness 2011; Burlaga & Ness 2012).However, mirror modes tend to take the form of humps rather than depletions in high β systems (Baumg¨artel et al. 2003) and the overall spatial scale of the depletions from mirrormodes are far smaller than the typical depletions measured in the heliosheath. In contrastthe size of the depletions from reconnection are not linked to any kinetic scale but to theradial scale length of the sectors, which can be of the order of an AU. The depletions frommirror modes would also require a significant magnetic field component in the N directionso that the elevation angle δ would be substantial. Large values of δ that extend over theregions of magnetic field depletion are not typically seen in the spacecraft data.Finally, intrinsic to reconnection in a high β system such as the HS, where the Alfv´enspeed is well below the magnetosonic speed, is that the dynamics is nearly incompressibleso that the magnetic depletions seen in the simulations are supported by correspondingincreases in the plasma pressure (and density). Thus, the fluctuations in the magnetic fieldstrength and density from reconnection should be anti-correlated. The data from Voyager2 confirms the anti-correlation of fluctuations in magnetic field strength and density in thesectored HS but not in the unipolar HS as expected if reconnection is taking place in thesectored HS.This work has been supported by NASA Grand Challenge NNX14AIB0G, NASA awardsNNX14AF42G, NNX13AE04G and NNX13AE04G, and NASA contract 959203 from JPLto MIT. The simulations were performed at the National Energy Research Scientific Com-puting Center. We acknowledge fruitful discussions with Dr. Len Burlaga on the Voyagerobservations and with Dr. Obioma Ohia on outer heliosphere reconnection. This researchbenefited greatly from discussions held at the meetings of the Heliopause International Team 14 –Facing the Most Pressing Challenges to Our Understanding of the Heliosheath and its OuterBoundaries” at the International Space Science Institute in Bern, Switzerland. REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
17 –Fig. 1.— (Color online) The magnetic structure in the lower half of the simulation domainat Ω i t = 50, 200, and 350 during the simulation. The magnetic field B is shown in (a), (c)and (e) and the in-plane field lines are shown in (b), (d) and (f). The magnetic field B fromthe simulation is available online as a video. The video runs from time zero to Ω i t = 440 . and shows the actual aspect ratio of the lower half of the computational domain. 18 –Fig. 2.— (Color online) A blowup view of the time evolution of a pair of magnetic islandsat Ω i t = 100, 150, 200 and 250 showing how the subdominant flux is annihilated. Themagnetic field B is shown in color and the overlaid white lines are magnetic field lines. Thetimes shown are before the islands overlap the adjacent current layer in (a), when they firstintersect the adjacent current layer in (b) and when the magnetic flux in the islands is erodedin (c) and (d). 19 –Fig. 3.— (Color online) The late-time (Ω i t = 440) structure of the magnetic field B in (a)and magnetic field lines in (b) over the entire simulation domain. Cuts across the magneticdepletions along the white line in (a) showing B in (c), of the density n in (d), the iontemperature T i in (e) and the azimuthal angle λ in (f). 20 –Fig. 4.— (Color online) The late-time (Ω i t = 440) structure of the magnetic field B in (a)and magnetic field lines in (b) over the entire simulation domain. Cuts through a remnantpair of magnetic islands along the white line in (a) showing B in (c), of the density n in (d),the ion temperature T i in (e) and the azimuthal angle λ in (f). 21 –Fig. 5.— (Color online) At late time the spatial distribution of the azimuthal angle λ in thefull simulation domain (a) and its probability distribution in (b). Note the prominence of thedominant magnetic polarity at late time. The ratio probability of dominant to subdominantpolarity in the initial state was four. 22 –Fig. 6.— From the simulation at late time the Probability Distribution Function (pdf) ofthe strength of the in-plane magnetic field B plane = p B x + B y . Note the distinct peak at B /
3. 23 –Fig. 7.— Histograms of < dBdn > / √ < dB >< dn > where dB = B − < B > and dn = n − < n > from Voyager 2 in the sectored region (top) and the unipolar region(bottom). We use 25 day averaging windows. 24 –Fig. 8.— From the simulation a histogram of δnδB plane / q < δB plane >< δn > where δn = n − < n > , δB plane = B plane − < B plane > with n the density, B plane the magnitude of thein-plane magnetic field and < n > and < B plane >>