A global study of hot flow anomalies using Cluster multi-spacecraft measurements
Gabor Facsko, Zoltan Nemeth, Geza Erdos, Arpad Kis, Iannis Dandouras
aa r X i v : . [ phy s i c s . s p ace - ph ] J u l Manuscript prepared for Ann. Geophys.with version 1.3 of the L A TEX class copernicus.cls.Date: 20 July 2018
A global study of hot flow anomalies using Cluster multi-spacecraftmeasurements
G. Facsk´o *1 , Z. N´emeth , G. Erd˝os , A. Kis , and I. Dandouras KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary Geodetic and Geophysical Research Institute, Sopron, Hungary CERS, CNRS, Toulouse, France * Now at LPCE, CNRS, Orl´eans, France
Abstract.
Hot flow anomalies (HFAs) are studied using ob-servations of the magnetometer and the plasma instrumentaboard the four Cluster spacecraft. We study several specificfeatures of tangential discontinuities on the basis of Clustermeasurements from the time periods of February-April 2003,December 2005-April 2006 and January-April 2007, whenthe separation distance of spacecraft was large. The previ-ously discovered condition (Facsk´o et al., 2008) for form-ing HFAs is confirmed, i.e. that the solar wind speed andfast magnetosonic Mach number values are higher than av-erage. Furthermore, this constraint is independent of theSchwartz et al. (2000)’s condition for HFA formation. Theexistence of this new condition is confirmed by simultane-ous ACE magnetic field and solar wind plasma observationsat the L1 point, at 1.4 million km distance from the Earth.The temperature, particle density and pressure parametersobserved at the time of HFA formation are also studied andcompared to average values of the solar wind plasma. Thesize of the region affected by the HFA was estimated by us-ing two different methods. We found that the size is mainlyinfluenced by the magnetic shear and the angle between thediscontinuity normal and the Sun-Earth direction. The sizegrows with the shear and (up to a certain point) with the an-gle as well. After that point it starts decreasing. The resultsare compared with the outcome of recent hybrid simulations.
Keywords.
Hot flow anomaly, solar wind, tangential discon-tinuity, bow-shock, hybrid simulation
Although hot flow anomalies (HFAs), explosive events nearthe Earth’s bow shock have been known more than 20 years(Schwartz et al., 1985; Thomsen et al., 1986), their theoreti-cal explanation needs further studies (Burgess and Schwartz,
Correspondence to:
G. Facsk´o ([email protected]) 1988; Thomas et al., 1991; Lin, 2002). The most reliable de-scription of HFAs is so far based on hybrid plasma simula-tions where electrons are considered as a massless and neu-tralizing fluid. The original motivation of this work was toverify several predictions presented in Lin (2002), but thisstudy led us much further than we expected. In order to dothis we determined the size-angle plot (described in the fol-lowing section). We calculated the related angles and esti-mated the size in two different ways. Lin’s hybrid simula-tion (Lin, 2002) uses a larger simulation box than in otherstudies mentioned above, and inserts a zero-resistivity sur-face (magnetopause) to the super-Alfv´enic plasma flow whenthe simulation is initialized. This plasma flow moves parallelto the x axis of the box and a shock is formed. A tangen-tial discontinuity is created ahead of the shock, and then theangle between flow direction and normal vector ( γ ) can bechanged. The simulations were run using different anglesand their results suggested that average radius of HFAs is ap-proximately 1-3 R Earth . A prediction of her theory is thatthe size of HFAs increases monotonically with γ until 80 ◦ and then begins to decrease. Another prediction is that thesize of HFAs is a monotonically increasing function of themagnetic field vector direction change angle ( ∆Φ ) across thediscontinuity (Lin, 2002). The goal of this study was to checkthe validity of these predictions based on simulation results.The four spacecraft Cluster mission provides an ex-cellent opportunity to study HFAs (Lucek et al., 2004;Kecskem´ety et al., 2006). We have identified 124 HFAs inthe Cluster dataset, which enables a statistical survey. Thisexpands the database of known events since previous analy-sis was based on significantly fewer events (Schwartz et al.,2000). Our results confirm the results of Lin (2002) that thesize depends on the shear and on the angle between the dis-continuity normal and Sun-Earth direction as well; further-more these results strongly support the recently suggestednew condition of HFA formation namely that during HFAformation the typical value of the solar wind speed is higher G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurementsthan the average (Facsk´o et al., 2008). We have used part ofSchwartz et al. (2000)’s calculations so we have checked hisformula (Eq. 2) too. Finally the original purpose led us toconfirm the findings of three different previous theories andto discover several new independent condition of HFA for-mation.The structure of this paper is as follows: we first describethe observational methods and the observed events in Sec-tion 2 and 3, discuss and present our analysis methods inSection 4, and explain and summarize the result of our studyin Section 5. For our study we used 1 s and (22 . Hz ) − temporal resolu-tion Cluster FGM (Fluxgate Magnetometer) magnetic fielddata (Balogh et al., 2001) and spin averaged time resolu-tion CIS (Cluster Ion Spectrometry) HIA (Hot Ion Analyzer)plasma measurement data (R`eme et al., 2001). We oftenfound the magnetic signatures of the TD – which interactswith the bow shock and generates the HFA later – in ACE(Advanced Composition Explorer) MAG (Magnetometer In-strument) 16 s temporal resolution magnetic field data se-ries (Smith et al., 1998). Alfv´en Mach numbers were calcu-lated and solar wind velocity was determined based on ACESWEPAM (Solar Wind Electron, Proton, and Alpha Mon-itor) 16 s temporal resolution data (McComas et al., 1998).ACE SWEPAM data series were used instead of Cluster CISHIA prime parameter data because in the case of very coldplasmas, as in the solar wind, where thermal velocities arevery small compared to the plasma bulk velocity and to theinstrument intrinsic energy (and thus velocity) resolution, therelative error in temperature can be large (R`eme et al., 2001;CIS Team, 1997-present); furthermore not all the necessaryCIS HIA data has been uploaded onto the Cluster ActiveArchive yet.We set a series of criteria for the selection of HFA eventsbased on Thomsen et al. (1986, 1993); Sibeck et al. (1999,2002) that were:1. The rim of the cavity must be visible as a sudden in-crease of magnetic field magnitude compared to the un-perturbed solar wind region’s value. Inside the cavitythe magnetic field strength drops and its direction turnsaround.2. The solar wind speed drops and its direction alwaysturns away from the Sun-Earth direction.3. The solar wind temperature increases and its valuereaches up to several ten million Kelvin degrees.4. The solar wind particle density also increases on the rimof the cavity and drops inside the HFA. Fig. 1.
HFA locations (a) in XY GSE and (b) XZ GSE plane pro-jections and the average bow shock and magnetopause positions.The coordinates were plotted in units of R Earth . The shapes ofthe magnetopause and the bow shock were calculated with the aver-age solar wind pressure (Sibeck et al., 1991; Tsyganenko, 1995) andAlfv´en-Mach number during HFA formation (Peredo et al., 1995).The black, red and blue points show Cluster positions when HFAswere observed in 2003, 2006 and 2007, respectively.
Using these criteria we identified 124 events in the 2003,2006 and 2007 data. Two of these events were studied byKecskem´ety et al. (2006), and a statistical study of 33 eventsin the 2003 data was analyzed by Facsk´o et al. (2008). Thepositions of the events are given in Tab. 1 and Fig. 1. All ofthem were observed beyond the bow shock in the February-April, 2003, December 2005-April 2006 and January-April,2007 time intervals. A fraction of these events was locatedvery far from the bow shock and the Earth ( ≥ R Earth ),occurring mainly in 2007. Only the position of tetrahedroncenter of the Cluster SC is plotted in Fig. 1, 2 because thelength of the orbital section is comparable with the thicknessof the lines drawn. The bow shock position was calculatedusing the average Alfv´en Mach number during formation of. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 3
Table 1.
The list of studied HFA events and spacecraft positions where HFA was observed in GSE system, in R Earth units. An empty cellindicates that the satellite in question did not observe the magnetic signature of a HFA.date time s/c positions(yymmdd) (UT) C1 C2 C3 C4030216 10:04 10.57, -1.19, -9.57 11.25, -0.55, -9.43 11.95, -0.67, -9.58 12.14, -0.47, -9.00030216 10:48 9.82, -1.45, -9.66 10.53, -0.78, -9.54 11.27, -0.90, -9.69 11.46, -0.73, -9.12030216 11:00 9.58, -1.53, 9.67 11.06, -0.98, -9.72030217 09:59 10.32, 5.78, 6.88 9.59, 10.77, 6.93 10.91, 5.70, 6.41030217 10:05 10.43, 5.79, 6.84 10.90, 5.10, 6.88 11.03, 5.71, 6.36030217 10:07 10.47, 5.79, 6.82030221 04:18 10.43, -2.08, -9.60 11.17, -1.49, -9.46 11.85, -1.67, -9.62 12.06, -1.50, -9.03030307 09:12 11.29, -4.56, -9.35 12.08, -4.21, -9.16 12.62, -4.51, -9.33 12.89, -4.37, -8.73030307 09:19 11.18, -4.57, -9.38 11.98, -4.22, -9.19 12.52, -4.52, -9.36 12.78, -4.38, -8.76030307 10:15 11.13, -4.23, -9.41 11.71, -4.56, -9.56 11.97, -4.44, -8.98030308 12:07 12.89, 1.71, 6.23 13.07, 0.92, 6.30 12.90, 1.21, 5.69 13.40, 1.46, 5.80030317 23:57 12.51, -0.22, 6.42 12.55, -1.03, 6.48 12.41, -0.70, 5.88 12.95, -0.55, 5.97030318 00:41 13.14, -0.51, 6.11 13.18, -1.32, 6.18 13.07, -1.00, 5.57 13.58, -0.83, 5.68030319 06:20 10.47, -6.86, -9.31 11.30, -6.68, -9.11 11.74, -7.07, -9.28 12.03, -6.98, -8.68030319 06:52 9.96, -6.78, -9.44030319 07:01 9.83, -6.76, -9.47030321 15:15 10.33, -7.30, -9.28030321 15:48 9.84, -7.21, -9.41 10.70, -7.06, -9.22 11.14, -7.49, -9.38 11.44, -7.42, -8.79030321 16:57 8.76, -6.99, -9.64 9.67, -6.82, -9.49 10.17, -7.29, -9.64 10.45, -7.24, -9.06030321 17:12 8.52, -6.93, -9.68 9.44, -6.76, -9.54 9.96, -7.25, -9.68 10.22, -7.19, -9.10030321 17:56 7.79, -6.75, -9.78 8.75, -6.58, -9.65 9.30, -7.09, -9.80 9.55, -7.04, -9.23030322 19:58 13.84, -2.92, 5.67 13.79, -2.62, 5.05 14.28, -2.47, 5.19030323 23:22 10.86, -7.87, -9.01 11.66, -7.79, -8.77 12.02, -8.17, -8.96 12.34, -8.10, -8.36030324 00:25 9.96, -7.70, -9.30 10.80, -7.59, -9.10 11.21, -8.02, -9.27 11.51, -7.95, -8.67030324 00:57 9.50, -7.59, -9.43 10.36, -7.48, -9.24 10.79, -7.93, -9.40 11.08, -7.87, -8.81030324 01:08 10.63, -7.89, -9.45030412 01:38 7.76,-11.04, -9.44030412 01:42 7.73,-11.01, -9.45 8.02,-11.05, -8.87030416 16:07 8.32,-12.45, -8.35 8.96,-12.69, -8.09 9.12,-13.10, -8.31 9.47,-13.13, -7.71030416 16:23 8.16,-12.36, -8.44 8.81,-12.60, -8.19 8.98,-13.02, -8.40 9.32,-13.05, -7.80030416 18:18 6.90,-11.58, -9.09 7.65,-11.82, -8.87 7.85,-12.33, -9.04 8.17,-12.37, -8.45051228 11:17 6.15, 17.33, -3.18 7.28, 17.15, -2.32 7.39, 16.48, -4.19 6.97, 16.49, -3.29051228 12:10 6.42, 17.31, -3.69 7.52, 17.13, -2.85051228 21:51 8.41, 14.50, -8.60 8.95, 14.42, -7.98 9.14, 13.38, -9.69 9.08, 14.07, -9.09051228 22:09 8.43, 14.34, -8.72 8.96, 14.28, -8.10 9.14, 13.22, -9.81 9.10, 13.93, -9.22051228 22:34 8.47, 14.12, -8.89 8.96, 14.05, -8.29 9.15, 12.99, -9.97 9.12, 13.77, -9.35051228 22:39 8.47, 14.08, -8.92 8.96, 14.00, -8.32 9.15, 12.94,-10.00 9.13, 13.68, -9.43051229 00:01 8.55, 13.30, -9.41 8.94, 13.24, -8.87051229 01:20 8.57, 12.47, -9.85051229 01:54 8.56, 12.09,-10.01 8.82, 12.04, -9.54 9.02, 10.96,-11.04 9.13, 11.95,-10.59051229 02:28 8.54, 11.70,-10.17 8.76, 11.66, -9.71060117 04:50 11.50, 7.12,-10.56 11.60, 7.09,-10.16 11.42, 5.96,-11.53 11.93, 6.83,-11.26060126 21:22 10.25, 2.38,-11.00 10.09, 2.49,-10.76 9.77, 1.44,-11.83 10.69, 2.28,-11.84060128 05:56 12.97, 12.00, -1.09 13.92, 11.18, -0.17 13.78, 10.71, -2.05 13.19, 10.76, -1.16060128 06:12 13.09, 12.00, -1.26 14.03, 11.18, -0.34 13.88, 10.69, -2.22 13.32, 10.75, -1.34060128 07:24 13.56, 11.91, -1.98060128 08:24 13.91, 11.79, -2.59 14.80, 11.02, -1.68 14.62, 10.42, -3.63 14.20, 10.60, -2.77060128 13:23 15.00, 10.67, -5.43 15.73, 10.05, -4.59 15.44, 9.23, -6.56 15.33, 9.61, -5.80060214 22:33 10.23, -1.38,-11.02 10.18, -1.15,-10.82 9.46, -2.05,-11.84 10.58, -1.59,-11.88
G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements
Table 1.
The list of studied HFA events and spacecraft positions where HFA was observed in GSE system, in R Earth units. An empty cellindicates that the satellite in question did not observe the magnetic signature of a HFA.date time s/c positions(yymmdd) (UT) C1 C2 C3 C4060215 23:29 11.55, 7.51, 3.19 12.20, 6.40, 4.04 12.17, 6.30, 2.53 10.93, 6.21, 3.34060221 01:50 17.04, 5.47, -1.77 17.53, 4.36, -0.81 17.23, 3.93, -2.82 16.81, 4.22, -1.96060222 01:09 9.82, -3.03,-11.89060223 04:14 13.62, 5.91, 2.30 14.12, 4.71, 3.20 14.03, 4.56, 1.58 13.02, 4.69, 2.39060310 15:31 10.84, -5.30,-11.01 10.97, -5.22,-10.72 9.83, -5.76,-11.91 10.84, -5.72,-11.82060320 04:16 9.58, -7.01,-11.07 9.75, -6.93,-10.80 8.51, -7.27,-11.95 9.49, -7.41,-11.89060322 08:00 13.35, -7.40,-10.08 13.47, -7.71, -9.55 12.23, -8.02,-11.14 12.96, -8.01,-10.80060410 04:37 12.65,-11.59, -8.68 12.54,-12.17, -7.96 11.34,-12.11, -9.83 11.85,-12.15, -9.37060410 05:27 12.23,-11.59, -9.01 12.16,-12.12, -8.32 10.93,-12.05,-10.14 11.48,-12.13, -9.71060410 07:53 10.89,-11.46, -9.85 10.90,-11.85, -9.27 9.61,-11.75,-10.94 10.23,-11.92,-10.59060410 08:28 10.54,-11.40,-10.03 10.57,-11.75, -9.47 9.26,-11.64,-11.10 9.91,-11.84,-10.78060410 12:52 7.54,-10.49,-10.98 7.72,-10.59,-10.64060416 12:39 13.07, -7.25, 1.87 12.31, -8.49, 1.06060416 12:45 13.11, -7.33, 1.81 12.40, -8.37, 2.81 12.34, -8.57, 0.99 11.92, -7.82, 1.71060416 13:22 13.34, -7.72, 1.44 14.04, -9.21, -0.12 13.07,-10.38, -1.08 12.88, -9.82, -0.36060416 16:30 14.14, -9.51, -0.48 13.46,-10.54, 0.57 13.14,-10.68, -1.47 12.99,-10.13, -0.74060416 16:39 14.16, -9.58, -0.57 13.48,-10.59, 0.50 13.16,-10.73, -1.54 13.01,-10.19, -0.82060416 18:32 14.34,-10.46, -1.71 13.70,-11.46, -0.66 13.25,-11.56, -2.77 13.21,-11.11, -2.06060416 20:01 14.35,-11.06, -2.60 13.75,-12.04, -1.56 13.19,-12.10, -3.70 13.23,-11.72, -3.01070104 03:53 10.15, 12.49,-10.41 10.59, 12.74, -9.78 10.69, 11.67,-11.24070104 04:36 10.13, 12.06,-10.57 10.53, 12.34, -9.99 10.64, 11.25,-11.40 10.64, 11.31,-11.38070104 06:20 10.01, 10.96,-10.91 10.34, 11.30,-10.44 10.44, 10.20,-11.72 10.45, 10.26,-11.71070104 05:08 10.10, 11.74,-10.69 10.48, 12.03,-10.14 10.59, 10.95,-11.51 10.59, 11.00,-11.49070106 16:07 10.39, 10.06,-11.01 10.69, 10.41,-10.60070108 11:25 10.17, 15.81, -6.43 11.03, 15.52, -5.28 11.08, 14.82, -7.27 11.06, 14.84, -7.22070116 09:40 10.08, 4.54,-11.18 10.19, 5.02,-11.14 10.11, 3.97,-11.93 10.15, 4.03,-11.93070116 10:00 9.91, 4.27,-11.15 10.00, 4.77,-11.13 9.93, 3.73,-11.89 9.97, 3.79,-11.89070116 10:49 9.48, 3.63,-11.04 9.54, 4.15,-11.08 9.48, 3.14,-11.77 9.53, 3.21,-11.78070117 16:38 10.35, 14.55, -2.64 11.24, 13.79, -1.34 11.28, 13.50, -3.37 11.24, 13.50,-3.317070118 07:49 13.27, 10.98, -9.65 13.84, 10.90, -8.91 13.69, 9.91,-10.53 13.69, 9.95,-10.50070118 09:42 13.09, 10.01,-10.20 13.59, 10.03, -9.56 13.43, 9.00,-11.06 13.44, 9.05,-11.04070118 12:13 12.65, 8.58,-10.77 13.05, 8.73,-10.28 12.89, 7.66,-11.61 12.91, 7.71,-11.59070118 14:35 12.01, 7.08,-11.12 12.31, 7.34,-10.79 12.15, 6.26,-11.93 12.18, 6.32,-11.91070118 19:34 9.83, 3.48,-11.08 9.91, 3.98,-11.10 9.83, 2.98,-11.82 9.87, 3.04,-11.82070120 18:18 13.58, 9.71,-10.06 13.90, 8.66,-10.94 13.92, 8.71,-10.90070130 16:44 10.72, 1.63,-11.13 10.85, 2.01,-11.13070201 06:48 15.78, 10.20, -6.63 16.48, 9.54, -5.61 16.23, 8.88, -7.56 16.22, 8.91, -7.51070201 22:07 13.05, 3.36, -11.2 13.32, 3.52,-10.95 12.92, 2.53,-12.00 12.97, 2.59,-11.99070201 22:16 12.97, 3.27,-11.21 13.23, 3.44,-10.97 12.83, 2.45,-12.01 12.88, 2.51,-12.00070202 01:31 11.02, 1.36,-11.17 11.17, 1.71,-11.14 10.82, 0.75,-11.90 10.88, 0.80,-11.90070209 02:14 12.68, 0.53,-12.04070215 01:35 13.61, 8.15, -0.21 14.10, 6.94, 0.93 14.16, 6.88, -0.96 14.10, 6.89, -0.89070215 02:29 14.21, 8.13, -0.74 14.74, 6.82, -1.53 14.68, 6.83, -1.46070215 02:49 14.41, 8.11, -0.94 14.93, 6.79, -1.73 14.88, 6.80, -1.67070215 03:13 14.65, 8.08, -1.17070215 03:55 15.05, 8.02, -1.58 15.56, 6.84, -0.43 15.53, 6.67, -2.41 15.49, 6.68, -2.35070215 04:01 15.10, 8.01, -1.64 15.62, 6.83, -0.48 15.58, 6.65, -2.47 15.54, 6.67, -2.41070215 08:44 17.02, 7.22, -4.29 17.57, 6.18, -3.20 17.33, 5.79, -5.23 17.31, 5.81, -5.17070215 15:16 17.86, 5.35, -7.49 18.36, 4.60, -6.59 17.89, 3.97, -8.47 17.90, 4.01, -8.42 . Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 5
Table 1.
The list of studied HFA events and spacecraft positions where HFA was observed in GSE system, in R Earth units. An empty cellindicates that the satellite in question did not observe the magnetic signature of a HFA.date time s/c positions(yymmdd) (UT) C1 C2 C3 C4070301 04:56 12.47, 5.00, 1.65 12.64, 3.71, 2.67 12.88, 3.76, 0.98 12.80, 3.77, 1.04070301 07:10 14.38, 4.72, 0.34 14.61, 3.40, 1.40 14.72, 3.37, -0.44 14.65, 3.39, -0.37070301 09:43 16.06, 4.24, -1.16 16.32, 2.93, -0.08 16.29, 2.81, -2.03 16.24, 2.83, -1.96070301 10:30 16.49, 4.07, -1.63070302 02:03 17.87, -0.58, -9.15 18.12, -1.23, -8.49 17.40, -1.80,-10.15 17.44, -1.77,-10.10070313 05:36 15.61, 1.36, -0.24 15.65, -0.02, 0.77 15.68, -0.09, -1.11 15.63, -0.05, -1.04070314 07:53 13.15, -5.91,-11.14 13.21, -6.06,-10.90 12.31, -6.54,-11.98 12.38, -6.53,-11.97070314 08:36 12.64, -6.00,-11.21 12.70, -6.10,-11.00 11.81, -6.58,-12.03 11.88, -6.57,-12.02070314 12:51 9.18, -6.27,-11.17 9.21, -6.06,-11.17 8.40, -6.56,-11.85 8.49, -6.56,-11.86070314 15:52 6.20, -6.10,-10.53 6.20, -5.68,-10.67 5.52, -6.18,-11.09 5.63, -6.19,-11.12070315 12:14 13.84, 1.56, 1.21 14.01, 0.18, 0.43 13.93, 0.22, 0.50070316 18:13 12.02, -6.57,-11.25 11.16, -7.09,-12.06070316 19:56 10.69, -6.67,-11.30 10.72, -6.61,-11.20 9.85, -7.07,-12.05 9.93, -7.07,-12.04070319 03:39 11.55, -7.10,-11.28 11.57, -7.14,-11.12 10.64, -7.55,-12.08 10.72, -7.55,-12.07070319 04:27 10.93, -7.12,-11.31 10.95, -7.10,-11.19070328 13:41 12.01, -9.01,-11.06 11.94, -9.23,-10.79 10.97, -9.47,-11.93 11.05, -9.48,-11.91070328 15:22 10.84, -8.95,-11.24 10.78, -9.04,-11.06 9.82, -9.30,-12.06 9.90, -9.31,-12.05070328 16:07 10.29, -8.89,-11.29 10.23, -8.92,-11.14 9.28, -9.19,-12.08 9.36, -9.20,-12.07070328 16:50 9.74, -8.82,-11.31 9.70, -8.80,-11.19 8.75, -9.07,-12.08 8.84, -9.09,-12.07070429 20:40 11.79,-11.03, -1.00 10.88,-12.27, -0.21 10.79,-12.27, -2.12 10.79,-12.21, -2.05070429 21:00 11.84,-11.24, -1.20 10.92,-12.47, -0.40 10.82,-12.47, -2.32 10.82,-12.40, -2.24070429 22:05 11.97,-11.89, -1.83 11.04,-13.01, -1.04 10.88,-13.07, -2.99 10.89,-13.01, -2.91070429 23:02 12.04,-12.41, -2.37 11.10,-13.58, -1.57 10.89,-13.53, -3.54 10.89,-13.47, -3.46070430 02:01 11.98,-13.79, -4.03 the events ( M A = 11 . , Sec. 3.2) according to the modeldescribed in Peredo et al. (1995). The position of the mag-netopause was calculated using the same average solar windpressure ( . ± . nPa, Sec. 3.3) as that in Sibeck et al.(1991) and Tsyganenko (1995).The cylindrical projection of the center of the Cluster SCpositions is also plotted to more easily determine whether theobservations were performed beyond or inside the averagebow shock (Fig. 2). Fig. 1 seems to indicate that the HFAsare mostly located within the magnetosheath, with some in-side the magnetosphere. However, this is only a feature ofthe applied projection. The position of the bow shock wascalculated using the average solar wind pressure during, theHFA event. All HFAs were beyond the actual bow shockwhen we observed them. However the bow shock positionchanges quickly, presenting explanation for why some of theevents seem to be located in the magnetosheath. γ , ∆Φ ) play in controllingHFA size. In the next two sections we therefore calculate theangles associated with each HFA and its size.3.1.1 Determination of anglesThe two angles (the γ and the ∆Φ ) mentioned before are con-sidered to be very important in the formation of HFA events.We are able to measure these angles and thus to compare theresults of measurements with the predictions of earlier sim-ulations. Unfortunately, triangulation techniques can not beused to determine these angles because of strong magneticfield fluctuations. Thus the direction of the TD normal vectorwas determined by the cross-product method and minimum-variance techniques using Cluster FGM (Balogh et al., 2001)and ACE MAG measurements data (Smith et al., 1998). Thetemporal resolution of FGM data series were 1 s and MAG’sresolution was 16 s. We accepted the result of minimum vari-ance method if the cross product method did not differ by G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements Table 2.
Parameters of TD normal vectors: λ /λ is the ratio of 2nd and 3rd eigenvalues, B min is the smallest magnetic field component inminimum variance system, ∆ n is the error cone of minimum variance method, γ is the angle between the Sun direction and TD normal, ∆Φ is the direction change across the discontinuity and θ the angle between the bow shock normal and the B magnetic field vector. Boldfaceletter shows quasi-perpendicular conditions; the angles were calculated by scaling a model BS to the location of Cluster-1 and 3 spacecraft.date time s/c n B u × B d n minvar λ λ B min ∆ n γ ∆Φ θ C1 , C3 (yymmdd) (UT) ( nT ) ( o ) ( o ) ( o ) ( o ) , , , , C2 -0.53,-0.43,-0.73 0.22, 0.75,-0.62 1.8 0.61 15.55C3 0.43, 0.50, 0.76 1.8 0.01 19.85C4 0.17, 0.79,-0.60 1.5 0.12 26.17030308 12:07 ACE 0.56, 0.38, 0.73 1.7 0.00 34.32 , C4 -0.36,-0.35,-0.87 0.54, 0.30, 0.78 1.8 0.68 17.86 111 87 30, 27030317 23:57 C4 0.81, 0.33,-0.48 0.89, 0.25,-0.38 4.3 -1.13 10.93 61 37030318 00:41 ACE 0.62, 0.75, 0.23 0.51, 0.80, 0.32 2.3 1.09 25.83 67 40 26, 29030319 06:20 ACE 0.27,-0.73, 0.63 0.18,-0.71, 0.67 1.4 0.38 44.64 79 44 8, 16030319 06:52 ACE -0.29,-0.37,-0.88 0.38, 0.30, 0.87 1.3 -0.24 53.73 95 19 34, , , . Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 7 Table 2.
Parameters of TD normal vectors: λ /λ is the ratio of 2nd and 3rd eigenvalues, B min is the smallest magnetic field component inminimum variance system, ∆ n is the error cone of minimum variance method, γ is the angle between the Sun direction and TD normal, ∆Φ is the direction change across the discontinuity and θ the angle between the bow shock normal and the B magnetic field vector. Boldfaceletter shows quasi-perpendicular conditions; the angles were calculated by scaling a model BS to the location of Cluster-1 and 3 spacecraft.date time s/c n B u × B d n minvar λ λ B min ∆ n γ ∆Φ θ C1 , C3 (yymmdd) (UT) ( nT ) ( o ) ( o ) ( o ) ( o ) , , , 5051229 00:00 ACE 0.59,-0.17,-0.79 0.72, -0.12, -0.69 13.4 0.56 7.28 54 98 , , , , , , , 2906:12 C4 -0.23, 0.73,-0.64 0.29, -0.71, 0.64 1.9 0.02 19.34 103 113060128 07:24 ACE 0.60,-0.78, 0.17 -0.40, 0.91, -0.07 1.5 0.74 32.51 53 30 34, 35060128 08:25 C1 -0.61, -0.19, 0.77 3.1 -0.71 12.54 , , 3323:29 C2 0.358,0.201,0.912 1.6 -0.26 16.3223:29 C4 0.37, 0.25, 0.89 0.07, 0.15, 0.99 4.0 1.00 9.63 68 130060221 01:47 ACE -0.36,-0.26,-0.89 0.22, 0.36, 0.91 5.90 -0.54 8.89 111 89 34, 3001:47 C1 -0.16, 0.31, 0.93 1.4 0.48 26.59060222 01:10 C3 0.39, 0.76, 0.52 0.48, 0.72, 0.50 2.30 0.24 23.05 66 107 , , , , G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements
Table 2.
Parameters of TD normal vectors: λ /λ is the ratio of 2nd and 3rd eigenvalues, B min is the smallest magnetic field component inminimum variance system, ∆ n is the error cone of minimum variance method, γ is the angle between the Sun direction and TD normal, ∆Φ is the direction change across the discontinuity and θ the angle between the bow shock normal and the B magnetic field vector. Boldfaceletter shows quasi-perpendicular conditions; the angles were calculated by scaling a model BS to the location of Cluster-1 and 3 spacecraft.date time s/c n B u × B d n minvar λ λ B min ∆ n γ ∆Φ θ C ,C (yymmdd) (UT) ( nT ) ( o ) ( o ) ( o ) ( o ) , , , , . Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 9 Table 2.
Parameters of TD normal vectors: λ /λ is the ratio of 2nd and 3rd eigenvalues, B min is the smallest magnetic field component inminimum variance system, ∆ n is the error cone of minimum variance method, γ is the angle between the Sun direction and TD normal, ∆Φ is the direction change across the discontinuity and θ the angle between the bow shock normal and the B magnetic field vector. Boldfaceletter shows quasi-perpendicular conditions; the angles were calculated by scaling a model BS to the location of Cluster-1 and 3 spacecraft.date time s/c n B u × B d n minvar λ λ B min ∆ n γ ∆Φ θ C ,C (yymmdd) (UT) ( nT ) ( o ) ( o ) ( o ) ( o ) , , , 4322:17 C3 -0.23, 0.84,-0.48 2.1 0.13 21.55070202 01:31 ACE -0.42, 0.91, 0.02 2.5 0.08 15.51 , ,0 G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements
Parameters of TD normal vectors: λ /λ is the ratio of 2nd and 3rd eigenvalues, B min is the smallest magnetic field component inminimum variance system, ∆ n is the error cone of minimum variance method, γ is the angle between the Sun direction and TD normal, ∆Φ is the direction change across the discontinuity and θ the angle between the bow shock normal and the B magnetic field vector. Boldfaceletter shows quasi-perpendicular conditions; the angles were calculated by scaling a model BS to the location of Cluster-1 and 3 spacecraft.date time s/c n B u × B d n minvar λ λ B min ∆ n γ ∆Φ θ C ,C (yymmdd) (UT) ( nT ) ( o ) ( o ) ( o ) ( o ) , , , 4322:17 C3 -0.23, 0.84,-0.48 2.1 0.13 21.55070202 01:31 ACE -0.42, 0.91, 0.02 2.5 0.08 15.51 , ,0 G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements Table 2.
Parameters of TD normal vectors: λ /λ is the ratio of 2nd and 3rd eigenvalues, B min is the smallest magnetic field component inminimum variance system, ∆ n is the error cone of minimum variance method, γ is the angle between the Sun direction and TD normal, ∆Φ is the direction change across the discontinuity and θ the angle between the bow shock normal and the B magnetic field vector. Boldfaceletter shows quasi-perpendicular conditions; the angles were calculated by scaling a model BS to the location of Cluster-1 and 3 spacecraft.date time s/c n B u × B d n minvar λ λ B min ∆ n γ ∆Φ θ C ,C (yymmdd) (UT) ( nT ) ( o ) ( o ) ( o ) ( o ) , , , , , , 30070319 03:47 C4 0.56, 0.14, 0.82 0.20, 0.19, 0.96 2.1 0.30 14.07 56 144070319 04:28 C1 0.50, 0.42, 0.76 2.1 -0.14 13.22 16, 13070319 04:28 C2 0.32, 0.40, 0.86 1.9 0.02 15.32070319 04:28 C3 -0.24,-0.39,-0.89 0.20, 0.38, 0.90 2.3 0.06 12.42 104 96070328 13:41 C3 0.90, 0.31, 0.29 0.90, 0.30, 0.30 1.5 -0.01 20.23 25 47 9, , , . Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 11 Fig. 2.
Cylindrical projection of Cluster SC center positions dur-ing HFA observation and the average bow shock and magnetopausepositions in GSE system. The shape of the magnetopause and thebow shock were calculated using the average solar wind pressure(Sibeck et al., 1991; Tsyganenko, 1995; Peredo et al., 1995). Theblack, red and blue points show the Cluster SC positions when HFAwas observed in 2003, 2006 and 2007, respectively. The coordinateswere plotted in R Earth units. more than 15 ◦ and the ratio of second and third eigenval-ues were equal to or larger than 2.0 (Tab. 2) (For a moredetailed description of the method see Facsk´o et al. (2008);Facsk´o et al.). It turns out that the minimum variance methodcan mostly be used at low magnetic field variation. Thismethod is very difficult and almost impossible to use in theHFA cavity and in SLAMS (Short Large Amplitude Mag-netic Structures) mostly coupled to quasi-parallel regions(Schwartz and Burgess, 1991). Many HFAs were embeddedinto SLAMS and so we were able to use the minimum vari-ance method with good accuracy only in a few cases. Besideof this feature of the method we have found more HFAs atthe quasi-parallel region ( ∼
66 %). (See Tab. 2) The localbow shock normals were calculated by scaling a model bowshock to the spacecraft location as in Schwartz et al. (2000)and we used the upstream magnetic field upstream of theHFA to calculate the angle of the shock-normal and the mag-netic field vector. This might confirm previous results: theconditions were quasi-parallel at least on one side of the TDpreviously (See Onsager et al., 1991; Thomsen et al., 1993;Kecskem´ety et al., 2006) and current simulations expect theHFAs to appear where the quasi-parallel condition turns toquasi-perpendicular (Omidi and Sibeck, 2007). We used thesame conditions for HFA observation and determination in2003 (Kecskem´ety et al., 2006; Facsk´o et al., 2008), 2006and 2007 (Facsk´o et al.), however this effect was very strongin 2007 and it was also noticeable in 2003 and 2006.
Fig. 3.
Distribution of cos (∆Φ) where ∆Φ is the angle of magneticfield directional change at the discontinuity. The ∆Φ and γ distributions differ from the typical dis-tributions associated with discontinuities in the solar wind.The ∆Φ distribution associated with HFAs (Fig. 3) peaks atsmaller values ( o − o ) when compared to the distribu-tion of solar wind distribution rotation angles, which peaksat larger values ( o − , Knetter et al., 2004, Fig. 2). The γ distribution associated with HFAs (Fig. 4) shows a wide, Fig. 4.
Polar plot of the direction of the normal vectors of TDs. Theazimuthal angle is measured between the GSE y direction and theprojection of the normal vector onto the GSE yz plane. The distancefrom the center is the γ angle as determined by the cross-productmethod. The TD normal vector is in a special polar coordinate sys-tem in which we measure the γ angle from the center, and where theazimuth is the angle of GSE y and the projection of normal vector toGSE yz plane. The regions surrounded by dashed lines are the pro-jection of error cones around the average normal vector marked by“X”. Circles and squares symbolize ACE and Cluster data, respec-tively. The black, red and blue symbols present events observed in2003, 2006 and 2007, respectively. γ angles (Knetter et al., 2004, Fig. 11).We found only one normal vector within this cone in 2003and a few others in 2006 and 2007. We observed this featurein the distribution of γ (Fig. 4). This finding strongly sup-ports the earlier theoretical and simulation results that HFAscan only be formed if ◦ ≤ γ ≤ ◦ (Lin, 2002; N´emeth,2007; Facsk´o et al., 2008). The distribution of ∆Φ showsthat HFAs can be formed if the magnetic field vector direc-tional change is sufficiently large across the TD (Tab. 2).Actually smaller values of ∆Φ were also observed, whichsupports the theoretical results by (Lin, 2002; Facsk´o et al.,2008; Facsk´o et al.). The distribution of TD normals for γ > ◦ is evenly distributed. We most often used ACEMAG measurements to determine TD normals in 2003, buthad to mainly use Cluster FGM magnetic field data in 2007because it was impossible to couple ACE and Cluster obser-vations. The simulation was a better description of the eventsof 2006 than that of 2007. This turns out to be an advantagebecause the accuracy of γ and ∆Φ increased in 2006 and2007.3.1.2 Estimations of HFA sizeCluster satellites cross HFAs but the time length of the eventholds no information about the real size of the phenomenabecause the boundaries of the cavity rim are not in pres-sure balance (Thomsen et al., 1986; Lucek et al., 2004) andthe HFA also moves in the frame of the solar wind plasma.On the other hand, we have other valuable information: thetime that the spacecraft spends inside the cavity gives a lowerlimit for the time of the existence of the HFA. One can cal-culate the error based on the measurements of four (or less)satellites. The size of the HFA must be estimated in anotherway.1. HFAs, hot diamagnetic cavities, are created by parti-cle beams accelerated by the supercritical bow shock.The beam shares its energy through electromagneticion-ion beam instability. In fact, this beam createsAlfv´en waves and these waves carry away a larger partof the energy; only 2/3 of the energy heats the plasma(Thomas and Brecht, 1988; Thomas, 1989). The prop-agation velocity of these waves does not exceed theAlfv´en velocity so that twice the Alfv´en speed multi-plied by time of existence may give a rough estimatefor the lower limit of the HFA size. Schwartz et al.(1985) determined the expansion speed of the cavityusing ISEE-1 and ISEE-2 measurements, and the mea-sured expansion speed was approximately the same asthe estimated velocity.2. HFAs are formed by the interaction of the bowshock and a tangential discontinuity. In manynumerical simulations (Burgess and Schwartz, 1988; Lin, 2002; Omidi and Sibeck, 2007) and observations(Lucek et al., 2004) one can see that the HFA appearswhen the TD reaches the quasi-parallel region and re-main while the TD sweeps the surface of the bow-show. We calculated the transit velocity of the tangen-tial discontinuity on the surface of the bow shock usingSchwartz et al. (2000)’s formula: V tr = V sw n cs sin θ cs : bs ( n cs − cos θ cs : bs n bs ) , (1)where V tr is the transient velocity, V sw is the solarwind speed, n cs is the normal of the tangential discon-tinuity (current sheet), n bs is the normal of the bowshock, and θ cs : bs is the angle between the two previ-ously mentioned normals. The bow shock shape, po-sition and normal were calculated by the model de-scribed in Peredo et al. (1995) as in the original paperwhich used ACE SWEPAM measurements. The so-lar wind vectors were determined by using Cluster CISHIA measurements. This instrument operates only onCluster SC1 and SC3. We obtained two estimates onthe size of HFA. The obtained sizes are very similar af-ter multiplying the velocity by the transition time of thespacecraft. Fig. 5.
The size distributions of HFAs estimated by Alfv´en velocityand (solid line) the speed of the TD and bow shock intersectioncalculated by the solar wind measurements of Cluster-1 and -3 CISHIA (red and blue line, scale drawn on top). The average sizes are (1 . ± . R Earth , (7 . ± . R Earth and (6 . ± . R Earth ,respectively.
We estimated the size of HFAs and the errors based on themethods above. Each of them gives four results by foursatellites. We took the average over the four points to be thesize, with the standard deviation as the error. Unfortunatelythe CIS HIA aboard Cluster-1 and Cluster-3 provided un-usually high temperatures close to the bow shock and so weused only the measurements of the ACE SWEPAM plasmainstrument and the ACE MAG magnetometer to determine. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 13the properties of the plasma. For this reason only one sizedistribution from using the first method is given (Fig. 5).The average sizes and their errors are (1 . ± . R Earth , (7 . ± . R Earth and (6 . ± . R Earth , respectively.The first result confirms the predictions of the Lin’s theoryhowever the second result seems to be much higher. Most ofthe distribution functions of the second estimation shows avalue of approximately R Earth . The reason for this higheraverage is the “tail” of the distribution at larger sizes. Unfor-tunately this size estimation is very sensitive to the errors ofthe different normals and velocity vectors (See: Eq. 1) andoften gives a very large size. After comparing the size dis-tributions of two methods on Fig. 5 one can see that most oftheir values do not differ by more then a factor of two. Theyare thus suitable for estimating the size of the phenomena.All side distributions are found to be very similar and thesize-angle functions support the simulation results.3.1.3 Size-angle and size-speed scatter plotsSize-angle relations were reported in Lin (2002). Further-more we were informed about size-speed predictions (Lin,2007, personal communication).Fig. 6 show the size- γ correlations. The error of the sizewas calculated by the method described by Sec. 3.1.2 andthe error of the angles was estimated by the cross-productmethod: we calculated the direction for every single space-craft, the average of these directions, and finally the errorcone. The error of direction was not calculated where onlyone direction was obtained. It is very important to remarkthat the size depends not on one but three parameters. Thesize was plotted as a function of one parameter ( γ ) while thespeed and ∆Φ values were fixed. In fact, fixing a parame-ter means fixed angle intervals because these were real mea-surements and not theoretical models. We fixed the speedin Alf´en-Mach number in the simulation as well. We chosethese ∆Φ intervals because these contains those points whichwere simulated by Lin with M A = 5 and ∆Φ = 80 ◦ . We ob-tained a maximum of the size- γ scattered plot but not exactlyat γ = 80 o in both cases as predicted (Lin, 2002). The otherpanels also support the theory since a maximum is visibleon every panel. When we plotted all points we obtained a“cloud” of points with a maximum value.Fig. 7 presents the size- ∆Φ functions where ∆Φ is thechange angle of magnetic field direction across the TD. Theerror of the size and angle were calculated the same way asat size- γ functions. Here γ and the solar wind speed werefixed and we used Alfv´en Mach numbers. Here the bot-tom panels show the case studied in the simulation of Lin(2002). All panels show monotonically increasing size- ∆Φ functions, confirming simulation results. We obtain a set ofpoints a dense region that increases to the larger sizes.In Fig. 8 the dependence of HFA size on velocity is visiblein several fixed angle intervals. Solar wind speed was mea-sured in Alfv´en Mach number value. The size was estimated based on the Alfv´en speed method (black) and by calculat-ing the velocity of the intersection line of the TD and the bowshock (red). The angular dependence of size was studied ina fixed intervals around γ = 80 ◦ and ∆Φ = 40 ◦ angles andthe size is the monotonically growing function of the Alfv´enMach number.3.2 Speed distributionsWe observed in our previous work (Kecskem´ety et al., 2006)that the value of the solar wind speed is close to the aver-age ∼
400 km/s but it is higher before HFAs are observed( ∼
600 km/s). We have studied this point in more detail here.The speed distributions were calculated here we used ClusterSC1 and SC3 CIS HIA; complemented by ACE SWEPAMdata measured in longer time intervals to obtain better statis-tics. We recorded these solar wind speed values again whenwe used 5-10 minute or even 30 minute long intervals beforethe bow shock. We calculated the average, its scatter andplotted the distribution (Tab. 3, Fig. 9). We determined thetime when the TD (which caused the HFA) crossed the po-sition of ACE satellite and we determined the average solarwind parameters from ACE SWEPAM measurements. Theseresults are in good agreement with earlier Cluster observa-tions (Facsk´o et al., 2008; Facsk´o et al.).These speeds are obviously higher than the long-term av-eraged solar wind speed (Fig. 9a, b, d), and a peak appearson the distribution between 400 km/s and 800 km/s measuredinstead of the expected 400 km/s or 800 km/s peaks mea-sured by Ulysses (McComas et al., 2003), but it is in ques-tion whether this difference is really significant. The averagespeed for the full-studied time period using ACE SWEPAM(Fig. 9e, black line) was (546 ± km/s in 2003. Actu-ally, the solar wind speed was higher throughout the stud-ied period in 2003 (Fig. 10). Measurements of ACE from1998 to 2008 (Fig. 9e, green line) yielded (498 ± km/s suggesting that during HFA formation the typical solar windspeed is higher than the average value by almost 200 km/sthan the average value . It seems that the presence of afast solar wind is a necessary condition of the formationof HFAs . This is obvious when one looks at the bottompanel of Fig. 10, where we plotted the studied interval us-ing 1 hour averaged solar wind speed. The HFAs marked byvertical lines and their positions all appear in fast solar windregimes. In fact almost all HFA events appeared in the sameco-rotating region (Facsk´o et al., 2008; Facsk´o et al.). Thefrequency of fast solar wind beams in the Ecliptic dependson the solar cycle. The frequency of HFAs is thus expectedto depend on solar cycle. After processing the measurementsin 2006 and 2007 this cannot be confirmed because the aver-age number of HFAs is about 2 HFAs/day with large scatter( . ± . , . ± . and . ± . in 2003, 2006 and 2007,respectively) so there is no significant difference during thedifferent seasons. There were several longer HFA series in2006 and 2007 but not in is 2003. The difference between4 G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements Fig. 6.
The size- γ functions based on the size estimation by Alfv´en Mach velocity on the left and the transition speed on the right. The fixedsolar wind speed was shown in Alfv´en Mach number. (a) ∆Φ = 60 ◦ ± ◦ and M A = 10 ± , (b) ∆Φ = 60 ◦ ± ◦ and M A = 10 ± ,(c) ∆Φ = 100 ◦ ± ◦ and M A = 10 ± , (d) ∆Φ = 100 ◦ ± ◦ and M A = 10 ± . All Alfv´en Mach numbers were calculated from theactual Alfv´en velocity. solar wind speed ( km/s ) ±
86 614 ±
84 613 ±
80 9aby C3 671 ±
92 614 ±
82 613 ±
78 9bby ACE 666 ±
84 626 ±
85 634 ±
71 9d M f numbers by ACE 8.2 ± ± ± ±
97 477 ±
97 512 ±
102 9ebetween 1998-2003/2008 by ACE 492 ±
102 498 ±
101 9e M f numbers by ACE 5.5 ± ± ∆ M f Table 3.
Solar wind speed, fast magnetosonic Mach number mean values, and their deviations measured by Cluster CIS and ACE SWEPAM.The last column gives the figure numbers shown on Fig. 9. the solar wind speeds were high –
130 km/s – but not as highas in 2003. Based on three years of measurements of we canconclude that the higher solar wind speed might be an im-portant requirement for the HFA formation mechanism. Wefound only a few HFAs out of the fast solar wind co-rotatingregions.Fig. 9c shows a more unexpected result. The figure showsthe distribution of the fast-magnetosonic Mach numbers dur-ing HFA formation. The Mach numbers are very high, with M f ≥ in 2003, this can also be observed in 2006 and 2007where the difference between them is even greater. This is made more obvious if we compare this distribution to the dis-tribution calculated by ACE SWEPAM and MAG measure-ments for the studied interval and all measurements of ACE(Fig. 9f). Both longer periods show that these high Machnumbers are very rare (Facsk´o et al., 2008). The HFAs arenot only Earth-specific features (Øieroset et al., 2001). TheMach numbers are in general much larger in the outer So-lar System, since the propagation speed of fast magnetosonicwaves is lower due to the weaker magnetic field. This factsuggests that HFA events might be even more frequent at Sat-urn, for instance the other giant planets in the Solar System.. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 15 Fig. 7.
The size- ∆Φ functions based on the size estimation by Alfv´en Mach velocity on the left and the transition speed on the right. Thefixed solar wind speed was shown in Alfv´en Mach number. (a) γ = 60 ◦ ± ◦ and M A = 10 ± , (b) γ = 60 ◦ ± ◦ and M A = 13 ± ,(c) γ = 80 ◦ ± ◦ and M A = 16 ± . , (d) γ = 80 ◦ ± ◦ and M A = 12 . ± . . All Alfv´en Mach numbers were calculated from theactual Alfv´en velocity. Fig. 8.
The size-velocity functions with Alfv´en velocity calculatedusing ACE and crossing time measured by Cluster. The sizes werecalculated using the method based on Alfv´en speed (black) and thetransition speed (red). The fixed solar wind speed was measured inunits of Alfv´en Mach number. γ = 80 ◦ ± ◦ and ∆Φ = 40 ◦ ± ◦ .All Alfv´en Mach numbers were calculated from the actual Alfv´envelocity. . ± . cm − instead of the long-term average value of . ± . cm − (based on the ACE SWEPAM 1 hour averagedata series measured between 1998 and 2008). This observa-tion is not surprising since the solar wind pressure is approx-imately constant. Thus, if the solar wind velocity is higher,the density is expected to be lower.The other studied parameter was the solar wind pres-sure. We also calculated distribution function, which sug-gested lower pressure during HFA formation than the aver-age of all measurements of ACE from 1998 to 2008. It was . ± . nP instead of the . ± . nP a (Fig. 11b). In ouropinion this difference is not significant. Unfortunately thehigh solar wind pressure does not seem to be a condition ofHFA formation in the case of those few events when the solarwind speed is not too large.6 G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements Fig. 9.
Solar wind speed distribution measured by Cluster and ACE spacecraft. Black, red, blue and green refers to measurements in 2003,2006, 2007 and 1998-2008, respectively. The figure shows the solar wind speed distribution measured by (a) Cluster-1 CIS HIA during HFAformation. (b) by Cluster-3 CIS HIA, and (d) by ACE SWEPAM; it also shows. Fast magnetosonic Mach number distribution calculatedusing ACE MAG and SWEPAM data during HFA formation (c), solar wind speed distribution measured by ACE SWEPAM from Februaryto April, 2003, December 2005-April, 2006 and January-April, 2007 and 1998-2008 (e), and fast-magnetosonic Mach-number distribution(f). (cid:12)(cid:12)(cid:12)(cid:12) V tr V g (cid:12)(cid:12)(cid:12)(cid:12) = cos θ cs : sw θ bs : sw sin θ B n sin θ cs : bs < , (2)where V tr is the transit velocity of the current sheet alongthe bow shock, V g is the gyration speed, θ cs : sw , θ bs : sw and θ cs : bs are the angles between the discontinuity normal, solarwind velocity and the bow shock, and finally θ B n is the anglebetween the magnetic field and bow shock normal. The nec-essary vectors were calculated using Cluster SC1 and SC3 CIS HIA measurements (Fig. 12). We found that the transi-tion speed is most often as low as expected by the formulaof Schwartz et al. (2000). This formula usually gives a valueof less than 1 one during HFA formation. Here the formulaoften gives a greater value than one; however, this study alsoconfirms that HFA formation also depends on the geometryof the shock, the discontinuity, and the solar wind velocity. Our resulting value of size estimation, the shape of size-angle and size-velocity distributions, as well as the func-tion of ∆Φ and γ , confirm previous predictions of numeri-cal simulations. The large number of events, as well as thehigher solar wind speed and Mach number are new results al-though Koval et al. (2005) had made similar observations us-ing INTERBALL-1 and MAGION-4 spacecraft. (That studywas performed using magnetosheath observation instead ofupstream measurements.) All our observations agree wellwith current theories and simulations.. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 17 Fig. 10.
The high solar wind velocity as an essential condition islogical and acceptable because particles of the beam whichform the HFA are accelerated at the supercritical bow shock.Here, the particles are forced to return to the foreshock regionapproximately with solar wind speed, but antiparalel to so-lar wind velocity (Gosling and Robson, 1985; Kennel et al.,1985; Scholer et al., 1993; Tanaka et al., 1983; Quest, 1989). This process causes the heating of the region and the en-ergy dissipation of the flow, and forms the beam which cre-ates the HFA. The higher the speed of the solar wind, thehigher the energy of the reflected beams. Moreover, ana-lytical calculations by N´emeth (2007) (which study the pos-sible particle trajectories of trapped ions in the vicinity ofshock-discontinuity crossings) suggest high solar wind speed8 G. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements
Fig. 11. (a) Solar wind particle density distribution during HFA events (dash-dotted line) using ACE SWEPAM measurements from 1998 to2008 (solid line). (b) Solar wind pressure in the same time intervals. as a favorable condition of particle reflection. Unfortunatelyno numerical simulation thus for can predict this condition,probably because these simulations are constrained into 2spatial dimensions. 3D hybrid simulations may be able topredict the high solar wind speed condition.The γ distribution and the size maxima of size- γ functions(Lucek et al., 2004; Schwartz et al., 2000) are explained asfollows: acceleration needs time and the TD must approachthe bow shock. If the angle is large then it approaches slowerand there is more time for acceleration. Beyond at given an-gle particles do not bounce back and nothing forms. Thesituation is different in the case of growing size- ∆Φ func-tions. Lin (2007) suggests that the electric field depends onthis angle, so larger ∆Φ generates larger electric field whichfocuses particles to the TD. It is well known that the acceler-ation happens between the TD and the quasi-parallel shock.When the TD reaches the quasi-parallel region of the bowshock or when the TD changes the magnetic field direction, Fig. 12.
The distribution of the rate given by Eq. 2. We use bothCluster SC1 and SC3 CIS HIA measurements to determine the nec-essary vectors in the formula. The red and blue lines show the dis-tribution based on Cluster-1 and -3 measurements. the particles – which form the beam – can escape from thetrap, which gives rise to the phenomenon. Larger ∆Φ causeslonger acceleration time, which can explain the growing size- ∆Φ functions.The reason of the growing size-speed function can be thefollowing: the beam that creates the HFAs is accelerated atthe supercritical bow shock. This result is not surprising be-cause their acceleration depends on the bow shock structure.A small amount of particles turns back and enters the regionin front of the bow shock, the foreshock region or the re-gion between the bow shock and the TD. TD occurs whenthe HFA is formed. The higher the velocity of the solar wind,the higher the speed of particles and size of the phenomenon.This trend can be seen on the Fig. 8, however it is not veryobvious. Earlier we showed that HFAs are not as rare a phenomenonas it was a thought prior to Cluster (Kecskem´ety et al., 2006).If a TD appears and the spacecraft are in the right positionthen the event can be observed with high probability if sev-eral special conditions are fulfilled. The numerous new HFAobservations also confirm this opinion.1. The most important condition is the larger solar windvelocity, which is typically much higher than the av-erage speed. The differences were approximately160 km/s in 2003, and approximately 130 km/s in 2006and 2007.2. The high fast magnetosonic Mach number is also apreferable condition for HFA formation. No eventswere found below M f = 6 in 2003, and this limit in-creased in 2006 and 2007.3. The pressure is irrelevant with respect to HFA forma-tion. The solar wind particle density before the HFA. Facsk´o et al.: A global study of HFAs using Cluster multi-spacecraft measurements 19events is lower than the average value of the solar winddensity.4. The angle between the TD normal ( γ ) and Earth-Sun di-rection must be greater than o . Very few events wereobserved with γ < o .5. The directional change of magnetic field within the TD( ∆Φ ) must be large. The average value was approxi-mately ◦ based on 124 events.6. Our size estimations do not contradict previous simula-tion results. We estimated − R Earth size using onemethod; the other method gave larger sizes in the rangeof R Earth . The differences can be explained with thehigh sensitivity of the methods to the accuracy of themeasurements.7. The size-angle and size-speed plots of Lin (2002) werereproduced in good agreement with the predictions.8. The conditions were mostly quasi-parallel during HFAformation, which is unexpected because the HFA deter-mination decreases the number of quasi-parallel cases.So our HFA observations confirm the previous simula-tion result of Omidi and Sibeck (2007) and showed thatHFAs appear where the quasi-perpendicular conditionturns to quasi-parallel. Furthermore, the particles of thebeam escape in the quasi-parallel part of the bow shock.9. We also confirmed the suggestion of Schwartz et al.(2000), namely that the transition velocity of the HFAat the bow shock must be slow. Furthermore, our newresult does not contradict to the formula presented inthat paper (Eq. 2).We have determined the typical size of HFAs in two differentways. The number of HFAs does not depend on solar activ-ity, only on the time of periods when the solar wind velocityis high. We compared within the theoretical predictions andproved that they are correct in 2003, 2006 and 2007 whenthe Cluster fleet separation was large. All observations agreewell with current theories and confirm the simulation results.We also publish here the detected events and their parame-ters. We hope they will be used to further studies, for exam-ple, THEMIS-Cluster multi-multispacecraft observations orfurther statistical investigations beyond and inside the bowshock.The reason why the high solar wind velocity is necessaryfor HFA formation was not explained in detail. Further –probably 3D hybrid – simulations are necessary to clarify thetheoretical background of this behavior.
Acknowledgements.
The authors thank the ACE MAG andSWEPAM working teams for the magnetic field and plasmadata; furthermore the authors are also very grateful to MariellaT´atraallyay for providing high resolution Cluster FGM data files.The present work was supported by the OTKA grant K75640 of the Hungarian Scientific Research Fund. G´abor Facsk´o thanks Pier-rette Decreau and Robert Ferdman for their help in improving theEnglish of this paper.
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