Entropy and plasma sheet transport

Transcription

1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2009ja014044, 2009 Entropy and plasma sheet transport R. A. Wolf, 1 Yifei Wan, 1 X. Xing, 1,2 J.-C. Zhang, 1,3 and S. Sazykin 1 Received 2 January 2009; revised 28 February 2009; accepted 13 March 2009; published 21 May [1] This paper presents a focused review of the role of entropy in plasma sheet transport and also describes new calculations of the implications of plasma sheet entropy conservation for the case where the plasma pressure is not isotropic. For the isotropic case, the entropy varies in proportion to log[pv 5/3 ], where P is plasma pressure and V is the volume of a tube containing one unit of magnetic flux. Theory indicates that entropy should be conserved in the ideal MHD approximation, and a generalized form of entropy conservation also holds when transport by gradient/curvature drift is included. These considerations lead to the conclusion that under the assumption of strong, elastic pitch angle scattering, PV 5/3 should be approximately conserved over large regions of the plasma sheet, though gradient/curvature drift causes major violations in the innermost region. Statistical magnetic field and plasma models lead to the conclusion that PV 5/3 increases significantly with distance downtail (pressure balance inconsistency). We investigate the possibility that the inconsistency could be removed or reduced by eliminating the assumption of strong, elastic pitch angle scattering but find that the inconsistency becomes worse if the first two adiabatic invariants are conserved as the particles drift. We consider two previously suggested mechanisms, bubbles and gradient/curvature drift, and conclude that the combination of the two is likely adequate for resolving the pressure balance inconsistency. Quantitatively accurate estimation of the efficiency of these mechanisms depends on finding a method of estimating PV 5/3 (or equivalent) from spacecraft measurements. Two present approaches to that problem are discussed. Citation: Wolf, R. A., Y. Wan, X. Xing, J.-C. Zhang, and S. Sazykin (2009), Entropy and plasma sheet transport, J. Geophys. Res., 114,, doi: /2009ja Introduction [2] In the early days of magnetospheric physics, largescale modelers often viewed transport in the plasma sheet as more or less uniform sunward convection, though with significant fluctuations. By about 1990, the discovery of bursty bulk flows [Baumjohann et al., 1990; Angelopoulos et al., 1992; Angelopoulos, 1993] had made it clear that plasma sheet transport was dominated by much more complex and interesting mechanisms, and it soon became equally clear that entropy played a crucial role in that transport. [3] The purpose of this paper is to discuss the role of entropy in plasma sheet dynamics from a large-scale theory perspective. We start in section 2 with a theoretical discussion of entropy and the entropy parameter PV 5/3, where V = R ds/b is the volume per unit magnetic flux. Section 3 summarizes evidence for the pressure balance 1 Physics and Astronomy Department, Rice University, Houston, Texas, USA. 2 Now at Atmospheric Sciences and Oceanic Sciences Department, University of California, Los Angeles, California, USA. 3 Now at Space Science Center, University of New Hampshire, Durham, New Hampshire, USA. inconsistency, specifically the evidence that on average, the entropy parameter PV 5/3, which is defined for conditions where the plasma pressure is isotropic, is not uniform in the plasma sheet. Section 4 presents a generalization of the pressure balance inconsistency argument to the case where the plasma pressure is anisotropic but the first two adiabatic invariants are conserved. Section 5 reviews mechanisms that have been suggested for resolving the inconsistency. Observational estimation of PV 5/3 remains a central challenge, and some approaches to solving that problem are discussed in Appendix A. 2. Basic Physics of Plasma Sheet Entropy for Isotropic Distribution Functions [4] The entropy S of an ideal gas is given by Z S ¼ Z d 3 x d 3 pf ln f where f is the distribution function, p is particle momentum, and the phase space integral extends over some volume V and all of momentum space. On each flux tube, f for the ideal gas in equilibrium is assumed to be a function of ð1þ Copyright 2009 by the American Geophysical Union /09/2009JA014044$ of14

2 [5] Equation (7) expresses the thermodynamic entropy, which can be evaluated from spacecraft measurements. Statistical studies [e.g., Borovsky et al., 1998] show that P/n 5/3 tends to decrease earthward in a very undramatic way. Combining (7) with conservation of particles produces an entropy expression that has much more impact on global modeling. We set n = N/V, which converts (7) to the form S N ¼ 3 2 5=3 ln PV 3 2 5=3 ln N L 0 ð8þ Figure 1. Cartooned result of tracing drift trajectories of different charged particles back from point P to the points where they cross the surface C. particle energy W and not on pitch angle or position. The number of particles in volume V is, of course, given by Z N ¼ V Z d 3 x d 3 pf If the distribution function is isotropic, it can be written in the form f ¼ n W 3=2 o gðþ x where W o is the average particle energy, and x W W o (The appropriateness of the form (3) can be verified by substituting (3) and (4) in (2).) The fact that f = f(w) along a field line implies that W o and n are constant along the line. The function g(x) expresses the shape of the distribution function (e.g., Maxwellian, kappa); it depends on particle mass but not on temperature or density. Dividing (1) by (2) and substituting (3) and (4) gives S N ¼ ln n W 3=2 o! L where L is a ratio of integrals that depend only on the shape of the distribution function, as specified by g(x), and not on n or W o. Since P/n =2W o /3, we have and (5) can be rewritten P n 5=3 ¼ 2W o 3n 2=3 S N ¼ 3 2 ln P L 0 n 5=3 where L 0 is another constant that depends on the shape of the distribution function and particle mass. ð2þ ð3þ ð4þ ð5þ ð6þ ð7þ Define V to include one unit of magnetic flux. If we assume ideal MHD (and thus isentropic flow) and we also assume that particle loss and ion upflow are negligible, then N should be conserved along a drift path because of frozen-in flux, and conservation of entropy (8) implies that the entropy parameter PV 5/3 and the specific entropy P 3/5 V are also conserved along the drift path. (If the flux tube is not in equilibrium, so that P is not constant along it, then PV 5/3 in (8) should be replaced by [ R P 3/5 ds/b] 5/3.) [6] Let the outer boundary of the plasma sheet be represented by a surface C, and assume that the fluid arriving at point x in the plasma sheet at time t crossed into the plasma sheet through C at point x C and time t C. (See Figure 1.) Conservation of S and N then implies that Pðx; t ÞV ðx; tþ 5=3 ¼ P½x C ðx; tþ; t C ðx; tþšv½x C ðx; tþ; t C ðx; tþš 5=3 ð9þ In addition to this ideal MHD demonstration, adiabatic drift theory also leads to the conclusion that nv and PV 5/3 are conserved along a drift path, assuming that cross-field drift is ExB drift and that the pitch angle distribution is isotropic [Wolf, 1983]. In addition, the adiabatic drift formalism also allows generalization of (8) and the associated proof of conservation of PV 5/3, to cover situations where gradient and curvature drifts contribute importantly to particle transport. The conclusion of that analysis is that n s V and P s V 5/3 are conserved along a drift path for particles of charge q s and given value of the isotropic invariant l s ¼ W s V 2=3 ð10þ where W s is particle energy, and n s and P s are the partial number density and partial pressure of particles characterized by q s and l s. As in the argument that led to (9), assume that each particle arriving at a given point x in the plasma sheet at time t lies on an open trajectory, so that its drift trajectory can be traced back to a point x Cs (x,t) and time t Cs (x,t) on surface C. Figure 1 illustrates in cartoon form how particles of different energy invariant and charge arrive at P from different points on the boundary surface. Then adiabatic drift theory [e.g., Wolf, 1983] implies that Pðx; t ÞV ðx; tþ 5=3 ¼ X s P s ½x Cs ðx; tþ; t Cs ðx; tþšv½x Cs ðx; tþ; t Cs ðx; tþš 5=3 ð11þ Equation (11) thus represents a generalization of (9) for the case where gradient and curvature drifts are important. Note that if the distribution function for given l is uniform along the part of the boundary surface C where particles enter the 2of14

3 Figure 2. Distribution of log 10 (PV 5/3 ) in the equatorial plasma sheet, based on a T96 magnetic field model [Tsyganenko, 1995] and a Tsyganenko and Mukai [2003] model of the plasma sheet, for average solar wind conditions (n =5cm 3, v = 400 km/s, B x = B y = 5 nt, and B z =0).PV 5/3 has units of npa(r E /nt) 5/3. From Xing and Wolf [2007]. plasma sheet and is independent of time, then (11) implies that PV 5/3 is uniform in the plasma sheet. Of course, the predicted uniformity of PV 5/3 breaks down near the inner edge of the plasma sheet, where particles of large l are on trapped orbits that circle the Earth, so that their populations are not directly related to conditions on surface C. [7] Some comments are needed concerning loss processes, which should tend to cause entropy to decrease earthward. In strong pitch angle scattering, the loss lifetime for electrons in a dipole field is (2.8 h) (L/10) 4 /W K (kev) 1/2 and for ions it is (120 h) A 1/2 (L/10) 4 /W K (kev) 1/2, where A is atomic weight [Kennel, 1969]. The electron lifetime is not enormously long compared to the convection time, so a significant fraction of electrons may be lost from the inner plasma sheet, especially since upward field-aligned electric fields can increase the loss rate above the strong-pitch angle-scattering limit. However, electrons carry only about one seventh of the plasma sheet pressure [Baumjohann et al., 1989]. The ion precipitation rate in strong pitch angle scattering, however, is much longer than the convection time, and there is no evidence that large downward fieldaligned electric fields are widespread in the plasma sheet. Therefore, we conclude that particle precipitation does not reduce PV 5/3 on plasma sheet flux tubes by anything like an order of magnitude, which would be required to resolve the pressure balance inconsistency (see Figure 2). It should be noted that Borovsky et al. [1998] concluded that ionospheric dissipation could consume much of the particle energy in the plasma sheet, in terms of overall energy balance. However, the idea that the precipitation rate is limited by the rate of strong pitch angle scattering places a much more stringent upper limit on ion precipitation. [8] Outflow of ions from the ionosphere should tend to make entropy increase earthward in the plasma sheet and thus compensate for loss by precipitation, but the effect on PV 5/3 is relatively minor, since most of the upflowing ions have energies well below the average plasma sheet energies. 3. Pressure Balance Inconsistency for Isotropic Pressure [9] Erickson and Wolf [1980] used empirical magnetic field models of that time to estimate PV 5/3 as a function of position in the equatorial plasma sheet [see also Schindler and Birn, 1982], and several similar studies have been done since then using more sophisticated models of the plasma magnetic field [e.g., Kivelson and Spence, 1988; Garner et al., 2003; Kaufmann et al., 2004; Xing and Wolf, 2007; Wang et al., 2009]. All of these have indicated that PV 5/3 decreases earthward. An example is shown in Figure 2, which shows an increase of more than an order of magnitude in PV 5/3 between X = 10 and 25. This result, combined with the theoretical expectation that PV 5/3 should be roughly uniform, was called the pressure balance inconsistency or pressure crisis, though entropy inconsistency would have been a better name. [10] In the statistical models, PV 5/3 tends to increase tailward, because pressure decreases tailward very slowly, 3of14

4 Figure 3. Two-dimensional equilibrium calculation of equatorial values of pressure, magnetic field, and entropy parameter for a case where the entropy is nearly constant for a large part of the plasma sheet. Units are arbitrary. From Hau [1991]. while the flux tube volume increases rapidly, because the field line length increases tailward and magnetic field strength decreases in most statistical average models. [11] One might be tempted to conclude from these statistics-based analyses of the average entropy distribution in the equatorial plane that it is impossible to construct a magnetotail equilibrium with uniform entropy. However, that was found not to be the case. The solution shown in Figure 3 has approximately uniform entropy over nearly the whole plasma sheet. The magnetic configuration differs from the statistical models in the existence of a clear minimum in equatorial magnetic field in the inner plasma sheet. The fact that the equatorial field strength decreases earthward through much of the plasma sheet allows the flux tube volume to decrease earthward only very slowly, which allows PV 5/3 to remain constant. [12] If one starts with a force-balanced magnetospheric configuration that resembles statistical models and forces strong convection with strict conservation of PV 5/3 tailward of the inner edge region, the inner plasma sheet becomes more and more stretched with time, with weaker and weaker equatorial B z, gradually approaching a configuration like that shown in Figure 3 [Erickson, 1992]. Several other groups have come to the same conclusion in different ways. Sergeev et al. [1994] combined data from steady convection events with their isotropic boundary algorithm to modify a Tsyganenko [1989] model and found a deep minimum in B z. Pritchett and Coroniti [1995] found the formation of a deep B z minimum based on fully kinetic 2-D simulations. Zaharia and Cheng [2003] found a magnetic field minimum in their computation of a 3-D force-balanced configuration with a pressure profile that was designed to represent a substorm growth phase. [13] The same kind of result has been obtained from more elaborate calculations with the RCM-E code, which couples the Rice Convection Model to a friction code equilibrium solver [Toffoletto et al., 2003]. Figure 4 shows a sample result, obtained after a 45-min run with 120 kv cross-tail potential drop. The computed magnetic field strength has dropped to 1 nt at 12 R E. Note that PV 5/3 is nearly uniform across a wide area of the plasma sheet, and the magnetic field configuration is so stretched that the equilibrium solver is having problems computing equilibria, as evidenced by the fluctuations in B z. [14] The conclusion from Figures 3 and 4 is that assuming a large potential drop across the tail leads to magnetic field configurations that are much more highly stretched than statistical models. Although such highly stretched configurations may sometimes exist in the magnetosphere, the fact that they do not show up in statistical averages indicates that they are either rare or of short duration. This is the essence of the pressure balance inconsistency. It is an inconsistency between statistical magnetic field and plasma models and the assumption that PV 5/3 is conserved. [15] It must be acknowledged that there is less-thancomplete agreement with the conclusions summarized in the last paragraph. Simulations of long-duration strong convection carried out by Wang et al. [2004], do not exhibit severe stretching like that shown in Figures 3 and 4. The Wang et al. [2004] simulations are force-balanced only in the xz plane, and they utilize the Magnetospheric Specification Model potential electric fields, rather than computing them self-consistently as in the RCM and RCM-E. The potential and boundary conditions of the simulations are different. The Wang et al. [2004] simulations are set up so that the LLBL is an important source of the plasma sheet in these times of strong convection, whereas the RCM-E simulations assume that for strong-convection conditions, the modeled part of the plasma sheet comes from the distant Figure 4. Equatorial plot of PV 5/3 (units of npa(r E /nt) 5/3 ) and a log plot of B z along the midnight line for an RCM-E run after 45 min with assumed 120 kv cross-tail potential drop. Adapted from Lemon [2005]. 4of14

5 tail. It is not clear which of the computational assumptions cause the difference in conclusions. 4. Pressure Balance Inconsistency for the Case Where First and Second Adiabatic Invariants Are Conserved [16] A theoretical loophole has long existed in the pressure balance inconsistency argument, because it has always been framed in terms of the assumption of strong pitch angle scattering and thus isotropic pressure. That assumption is generally realistic on tail-like field lines, where ion motion is chaotic near the current sheet, but it is not necessarily realistic for field lines that are dipolar in shape. That raises the question of whether calculations based on the assumption of isotropy may substantially overestimate the severity of the pressure balance inconsistency in regions where ion motion is not chaotic. To assess this possibility, we calculate particle energies and pressures for the assumption that there is no pitch angle scattering, so that the first and second adiabatic invariants are conserved in sunward convection through the plasma sheet. In this section, we perform these calculations for Tsyganenko [1989] models for Kp = 0, 3, and 6. [17] Suppose that the distribution function on flux tube 1, which extends to an equatorial crossing distance of r e1,is given by f ¼ A dðw W 1 Þ ð12þ where W is particle kinetic energy and A and W 1 are constants. (Of course, we are assuming here that the gyro and bounce motion is much faster than the convective motion.) The second adiabatic invariant J is written Z sm pffiffiffiffiffiffiffiffiffi J ¼ 2 p k ds ¼ 2 2mm K ð13þ s m where s m is the distance along the field line from the equatorial plane to the mirror point, m is the first adiabatic invariant, and K ¼ Z sm s m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B m Bs ðþds ð14þ is the geometrical invariant. Equation (12) can be rewritten f ¼ A dmb ½ m ðk; r e1 Þ W 1 Š ð15þ where B m (K, r e1 ) is the mirror field corresponding to geometric invariant K on the flux tube that crosses the equatorial plane at r e1. [18] Since f should be constant along a drift path, and m and K are also conserved along that path, equation (15) also specifies the value of the distribution function anywhere on that path. The average energy of particles arriving at flux tube 2 closer to Earth is given by W 2;inv W 1 R R dmm 3=2 dk dmb ½ m ðk; r e1 Þ W 1 ŠB m ðk; r e2 Þ ¼ R W R 1 dmm 1=2 dk dmb ½ m ðk; r e1 Þ W 1 Š ð16þ Figure 5. Log 10 (E crit (kev)) versus x for the three T89 models under consideration, computed from (19) for protons assuming k crit =2. In the approximation of strong, elastic pitch angle scattering, we have, simply, W 2;iso ¼ V 2=3 1 ð17þ W 1 V 2 [Wolf, 1983]. The occurrence of chaos in plasma sheet particle orbits is often characterized by the parameter sffiffiffiffiffiffiffiffiffiffiffi R c;min k ¼ a c;max ð18þ where R c,min is the minimum radius of curvature on the field line, and a c,max is the maximum Larmor radius for a particle of given energy E [Büchner and Zelenyi, 1989]. We will be considering field lines in the midnight meridian plane, and, for those field lines, the minimum radius of curvature and maximum Larmor radius occur on the x axis. Suppose we define a critical value of k below which the assumption of conservation of the first two adiabatic invariants fails, and the pitch angle distribution is best approximated as being isotropic. The corresponding critical energy can be derived from (18) by substituting the usual expression for the Larmor radius. The result is E crit ¼ R 2 c;mineb min 2mk 4 crit ð19þ Several estimates have been made of k crit, from the viewpoint of isotropy of ions near the loss cone. For their observational study, Sergeev et al. [1983] chose k crit =8. Detailed theoretical calculations of Delcourt et al. [1996] suggested values k crit =2 4. Figure 5 shows a plot of E crit for the lowest value k crit = 2. Figure 5 suggests that the vast majority of ions that contribute importantly to plasma sheet pressure should be isotropic beyond a critical distance of 8 12 R E. We perform calculations for a critical distance of 12 R E ; assuming a smaller critical distance would reduce the region where the adiabatic approximation is imposed and would tend to reduce the estimated difference from the isotropic approximation. Our calculations will therefore tend to overestimate the effect. 5of14

6 [21] Choosing r e1 < 12 would obviously lead to smaller departures from isotropic conditions. However, it is illuminating to consider r e1 =18R E, even though plasma sheet ions are almost certain to be isotropic beyond 12 R E. Results are shown in Figure 7. In the Kp = 3 and 6 cases, note that the difference between results of the two calculations is smaller for r e1 = 18 than for r e1 = 12. The explanation for this lies in the competition between Fermi-type acceleration, which primarily increases P k by shortening the field line and moving mirror points closer together, and betatron acceleration, which primarily increases P? as the drifting particles encounter stronger and stronger fields. For a dipole field, betatron acceleration wins for all geocentric distances, and the same is true for all distances considered in the Kp = 0 case, where the field lines are always quasi-dipolar. For the more stretched models representing more active times, Fermi acceleration Figure 6. Energization factors for particles drifting from 12 R E to r e2. The plots compare log 10 (W 2,inv /W 1 ) from equation (16) and log 10 (W 2,iso /W 1 ) from equation (17) for T89 models with three different Kp values. [19] Figure 6 compares the results from (16) and (17) for three T89 magnetic field models (Kp =0,3,6)andforr e1 = 12. It is clear that the assumption that m and J are conserved leads to greater average energization than the assumption of strong pitch angle scattering. In both the isotropic case and the fully adiabatic case, entropy is conserved as the particles drift. However, for given energy, the entropy is greatest when the distribution is isotropic. Therefore to conserve entropy, particles drifting fully adiabatically must gain more energy than in the isotropic case. [20] Note that the Kp dependence is not very dramatic. There is less acceleration in the high-kp case, because the inner magnetospheric magnetic field is highly inflated in that case and therefore not so much different from the tail fields. Figure 7. Same as Figure 6 but with the particles starting from 18 R E with an isotropic distribution. 6of14

7 is stronger than betatron in much of the region beyond 12 R E, though betatron acceleration becomes more efficient close to Earth; thus the pitch angle effects of the two regions partly cancel each other. For a detailed discussion of the competing effects of Fermi and betatron acceleration in the near tail based on electron pitch angle distributions, see Sergeev et al. [2001]. [22] It also turns out to be interesting to compare equatorial pressure ratios computed from the full-adiabatic approximation to those computed assuming isotropic pressure. We find that the ratio of the perpendicular pressure at r e2 to the isotropic pressure at r e1 is given by P?2 ¼ 3B2 2 P 1 4 BZ m2;max B 2 db m2 B m1 ðb m2 Þ 5=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B m2 B 2 ð20þ where B 2 is the magnetic field at the point on the line through r e2 where P?2 is to be evaluated, B m2,max is the mirror magnetic field at the ionosphere on that field line. The function B m1 (B m2 ) is the mirror magnetic field at field line 1 that corresponds to the K that corresponds to mirror field B m2 on field line 2. The corresponding ratio for the average pressure P 2 ¼ 2P?2 þ P k2 ð21þ 3 at the same point on line 2 is given by P 2 ¼ B 2 P 1 2 BZ m2;max B 2 B m2 db m2 B m1 ðb m2 Þ 5=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B m2 B 2 ð22þ The parallel pressure on the field line through r e2 can easily be computed from (20) (22). [23] Figure 8 compares P?2, P k2, and P 2 computed at the equatorial plane assuming conservation of the adiabatic invariants to the pressure value at point 2 computed from (V 1 /V 2 ) 5/3. Note that for r e2 much less than 12, P?2 and P 2 are significantly larger than either P k2 or the pressure computed from the isotropic approximation. Although Figure 6 indicates that the difference between the adiabatic and isotropic calculations fairly minor, when the comparison was done in terms of average energy density on the flux tube. However, Figure 8 shows that the difference is large in terms of equatorial pressure for all Kp, simply because the fully adiabatic calculation concentrates the particle energy near the equatorial plane. This means that for meaningful comparisons of entropies between different flux tubes earthward of the isotropic region, it is important to integrate properly over each flux tube, considering the pitch angle distribution. [24] The overall conclusion of this section is that fully adiabatic drift makes the pressure balance inconsistency worse. Specifically, we have shown that conservation of the first two invariants in the innermost plasma sheet should theoretically cause PV 5/3 to increase earthward, though, of course, it is observed to decrease, on average. The theoretically expected earthward increase can be relatively mild if P is estimated from a flux tube average, stronger when it is taken to be pitch angle-averaged equatorial pressure. It must be noted that the analysis of this section applies to plasma sheet particles that have drifted from r e1 to r e2, which can happen only if the effective Alfvén layer lies earthward of Figure 8. Log 10 of ratio of equatorial pressure at r e2 to pressure at r e1 =12R E, computed several different ways. Isotropic means that the pressure ratio was computed from (V 1 /V 2 ) 5/3. Perp adiabatic and average adiabatic were computed from (20) and (22), respectively. The parallel adiabatic curve was then computed from (21). r e2. The effective Alfvén layer will lie slightly further from the Earth for most particles that conserve their adiabatic invariants than for particles of the same initial energy that are subject to strong pitch angle scattering. That more efficient energization makes the gradient/curvature drift effect, to be discussed in the next section, slightly stronger. 5. Mechanisms That Help Resolve the Pressure Balance Inconsistency [25] The analysis of the last section indicates that departures from isotropy are unlikely to resolve the pressure 7of14

8 inconsistency, namely, the fact that statistical models indicate that the entropy parameter systematically decreases earthward in the plasma sheet, which seems inconsistent with the theoretical expectation that entropy should be roughly conserved in adiabatic drift. However, two other physical mechanisms, discussed in the following two subsections, do together seem strong enough to resolve the problem Gradient/Curvature Drift [26] One much-discussed mechanism is gradient/curvature drift [Tsyganenko, 1982; Kivelson and Spence, 1988; Spence and Kivelson, 1990, 1993; Wang et al., 2001, 2004, 2009]. The trajectories sketched in Figure 1 suggest that energetic ions observed in the inner plasma sheet may come from the near-earth low-latitude boundary layer (LLBL) and not from the distant tail. The distribution function f(l) and partial entropy parameter P s V 5/3 may be smaller in the LLBL than in the distant tail for high l s, and we might expect this to reduce the total PV 5/3 in the inner plasma sheet (equation (11)). The effectiveness of this mechanism depends on the ratio of ExB to gradient/curvature drift. Calculations indicate that the gradient/curvature drift mechanism may well be strong enough to resolve the pressure balance inconsistency for times of weak convection [Kivelson and Spence, 1988; Spence and Kivelson, 1990, 1993; Wang et al., 2001]. Wang et al. [2004, 2009] claim that it can resolve much or all of the problem even in times of strong convection, though this remains a controversial point Plasma Sheet Bubbles [27] The topic of plasma sheet bubbles deserves a longer discussion, because it relates to other phenomena, including flow channels, bursty bulk flows (BBFs), poleward boundary intensification, and substorm expansions. It has been clear for many years that flows in the plasma sheet are highly variable and nonuniform. For example, Sergeev and Lennartsson [1988] reported steady earthward streaming motion that was apparently confined to flow channels. Then Baumjohann et al. [1990], Angelopoulos et al. [1992], and Angelopoulos [1993] defined the BBF phenomenon, which involves plasma near the center of the current sheet moving earthward at speeds far greater than average convection. They showed that these sporadic flows carry a large fraction of the total earthward transport in the plasma sheet. In the original definition of BBFs, which required peak velocities greater than 400 km/s, the events nearly always occurred beyond 12 R E geocentric distance. More recent analyses have, however, demonstrated the existence of slightly slower fast flows near the Earth [Zhang et al., 2009; Hori et al., 2005]. Walsh et al. [2009] recently reported observation of the same bubble near 15 R E and also near 7 R E. [28] On the theoretical side, Pontius and Wolf [1990] and Chen and Wolf [1993, 1999] pointed out that a bubble, defined as a set of flux tubes with significantly lower entropy PV 5/3 than their neighbors, would move systematically earthward through the plasma sheet. Because of its reduced entropy, the bubble carries less cross-tail gradient/ curvature current than its neighbors. Current continuity is assured partly by westward acceleration drift current, which accelerates the bubble earthward in the equatorial plane. An Alfvén wave propagates to the Earth, eventually creating a Figure 9. Cartoon demonstration that the Lui et al. [1992] tail current interruption model can create a bubble. The highly stretched growth-phase magnetic field is subjected to a region of anomalous resistivity, where there is a strong current out of the page and a corresponding electric field in the same direction (in the plasma rest frame). Magnetic field line 2 consequently slips earthward relative to the plasma. The volume of the flux tube between field lines 2 and 3 is reduced, which reduces its PV 5/3, making it into a bubble. The volume of the tube between field lines 1 and 2 is correspondingly increased, making it into a blob (a region with higher PV 5/3 than its neighbors). current wedge with downward field-aligned current on the east side of the bubble, westward ionospheric current across the bubble, and upward current on the west side of the bubble. While these field-aligned currents are flowing, the bubble has a more dipolar shape than its neighbors. Eventually, the bubble decelerates and comes to rest where its entropy and shape match those of its neighbors. Many observers have interpreted BBF events as bubbles [e.g., Sergeev et al., 1996; Kauristie et al., 2000; Nakamura et al., 2001; Walsh et al., 2009]. Several mechanisms have been suggested for creating them: [29] 1. Uneven filling of flux tubes in the distant tail has been invoked in the interpretation of poleward boundary intensifications [Zesta et al., 2006]. [30] 2. A patch of reconnection in the plasma sheet in the absence of a significant B y produces a plasmoid and a shortened closed field line; the entropy of the original stretched closed flux tube is divided between the two, so the daughter closed tube has lower entropy than its parent. Birn et al. [2006] and Sitnov et al. [2005] have modeled this process computationally. A small reconnection event might cause a flow burst, while a large one might produce a substorm expansion [e.g., Hones, 1977]. [31] 3. A tail current interruption that violates the perfectconductivity condition [Lui et al., 1992] could produce a 8of14

9 Figure 10. MHD simulation of a thin-filament bubble that initially had the shape of a background field line that crossed the equatorial plane at 40 R E. (top) Field line shapes for times 0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 18, 23, 28, 33, 38, and 43 min. Enlarged views of the (middle) tailward and (bottom) earthward portions of Figure 10 (top). The near-equatorial part of the filament exhibits overshoot and oscillation, while motion along the left boundary, representing the ionosphere, is always equatorward. The final position is marked in Figure 10 (bottom). Adapted from Chen and Wolf [1999]. bubble, if it allows field lines to slip on the plasma and produces some flux tubes with reduced PV 5/3. Figure 9 shows the mechanism in cartoon approximation theory. [32] Turning now to the theory of how bubbles evolve once they are created, we start with the paper of Pontius and Wolf [1990]. That paper visualized a bubble flux tube as having the same shape as its neighbors, used the Vasyliunas equation and standard ionosphere-magnetosphere coupling theory to compute the bubble velocity, concluding that the bubble moves earthward with a speed that is proportional to DJ GC /S P, where DJ GC is the difference between the gradient/curvature drift current that can be carried by the bubble s neighbors and the bubble itself, and S P is ionospheric Pedersen conductance. This analysis indicated that highly depleted bubbles connected to regions of low conductance could reach speeds 1000 km/s. The bubble finally comes to rest when its PV 5/3 value equals that of its neighbors. [33] Chen and Wolf [1993] calculated the equilibrium shape of a thin-filament bubble in a quasi-static approximation that was consistent with force balance but neglected inertial terms. They also explicitly defined the bubble as a flux tube with reduced PV 5/3. [34] Chen and Wolf [1999] continued to picture the bubble as a thin filament but treated it with full ideal MHD, so that inertial effects were properly included. The time evolution of a thin filament is shown in Figure 10. A depleted filament, representing a small bubble, was given the same shape as its neighbors and started at rest. Its nearequatorial region accelerated earthward, sending both an Alfvén wave and a slow wave propagating earthward. The bubble quickly reached a terminal velocity through the plasma sheet, that velocity being limited mainly by inertial effects. Both waves reflected from the near-earth region, which was represented by a hard, finitely conducting wall 5R E behind the Earth. When the reflected slow compressional wave arrived back at the equatorial plane, that part of the filament braked sharply. The filament executed a damped oscillation about its final equilibrium, where its entropy matched the surrounding medium. The net result of the bubble s motion is rapid earthward transport of a lowentropy flux tube from the distant plasma sheet; this process supplies low-entropy flux tubes to the inner plasma sheet. [35] Birn et al. [2004] studied the motions of bubbles in full 3-D MHD, for much less idealized conditions than the thin-filament simulations. They showed that the equatorial cross section of the bubble developed an elongated tail stretching in the antisunward direction. Overall their results were consistent with the main features of the Chen and Wolf [1999], even though the initial and boundary conditions were somewhat different. The earthward boundary of these simulations was a perfectly conducting wall (rather than a finitely conducting wall used in the filament calculations), so Birn et al. [2004] did not make predictions of the motion of the ionospheric footprint of the bubble, and the bubble could not approach a final state where its entropy matched that of its neighbors. [36] The ideal MHD simulations of plasma sheet bubbles (both filaments and full 3-D MHD) were for only a few cases. Many more simulations are needed to establish the sensitivity of the conclusions to the initial and boundary conditions. For example, there is still uncertainty about the maximum velocity that a bubble can reach. The example 9of14

10 Figure 11. Bubble observations by Geotail on 22 July The first and second panels show the northward normal component of the magnetic field and the x component of velocity. The third panel shows the y component of electric field, derived from E = v B, and the fourth panel shows R E y dt. The fifth, sixth, and seventh panels give ion number density, temperature, and pressure, while the eighth and ninth panels show values of V and PV 5/3, based on estimated values from a magnetic field model that has been tailored to this event. Dashed red curves are RCM values. The green and blue dashed curves in the second and third panels represent the components of drift associated with potential and induction electric fields, respectively. 10 of 14

11 Figure 12. Equatorial view of an RCM simulation of the injection of a substorm-associated bubble into the inner magnetosphere. The Sun is to the left. Colors show the entropy parameter PV 5/3 (npa(r E /nt) 5/3 ). Black curves are contours of constant F + hl i i V 2/3 /e, with contour spacing 5 kv. Dashed equipotentials are negative. The pink circle has radius 6.6 R E. The bubble was launched from the tailward boundary at 0655 UT. shown by Chen and Wolf [1999] reached a velocity of about 500 km/s, consistent with bursty bulk flows, but the 3-D Birn et al. [2004] simulations showed a peak at a somewhat lower velocity. More numerical work is needed to determine how the peak velocity depends, for example, on the location of an X line that generates the bubble, the plasma sheet conditions involved, or (in the 3-D case) on grid size. [37] Near the Earth, gradient and curvature drifts become comparable to ExB drift and ideal MHD becomes inadequate. On the other hand, total flow velocities become smaller near the Earth, and configurations come closer to equilibrium. For these conditions, the Rice Convection Model becomes a useful tool. On theoretical grounds, we concluded several years ago that relatively low PV 5/3 plasma sheet flux tubes are injected into the inner magnetosphere during the substorm expansion phase [Toffoletto et al., 2000; Wolf et al., 2002], and observations reported by Lyons et al. [2003] seem consistent with that view. Some years ago, we published a first-try simulation that represented the substorm expansion phase as a bubble [Toffoletto et al., 2000]. We are now pursuing a much more extensive series of simulations of the bubble injection process using both the Rice Convection Model (RCM) and RCM-Equilibrium model (RCM-E). A simulation of an idealized substorm was published by Zhang et al. [2008]. Figures 11 and 12 show some results of a more recent study of a real substorm that occurred 22 July That event was chosen because of the availability of Geotail observations of a well-defined bubble, as indicated by an increase in the component of B normal to the estimated current sheet (labeled B n in the first panel of Figure 11) and rapid spiky earthward flow for 5 min (second panel of Figure 11). The eighth and ninth panels of Figure 11 show estimated flux tube volume V and entropy parameter PV 5/3. (Methods of estimating those key parameters are discussed in Appendix A.) The dashed curves represent RCM values and indicate our attempt to tailor RCM inputs for approximate agreement with a smoothed version of Geotail data. Figure 12 shows sample RCM results during the injection. The blue area of the plasma sheet is the bubble. Note its rapid earthward motion, helped by the strong westward potential electric field near the earthward tip of the bubble. That electric field results from Birkeland currents (not shown), which flow down on the east side of the bubble, up on the west side Some Concluding Comments on the Pressure Balance Inconsistency [38] It should be emphasized that the pressure balance inconsistency is a conflict between statistical magnetic field and pressure models and the assumption of steady, largescale adiabatic convection. Theoretical models can be constructed that are consistent with steady or long-term strong convection, as illustrated in Figures 3 and 4, but the resulting magnetic field configurations that the Rice group has been able to construct exhibit much stronger field stretching in the inner plasma sheet than the statistical models do. Our position is that the magnetosphere probably sometimes attains such highly stretched configurations, but only in events that are short enough or rare enough that they are not clearly represented in statistical models. In our view, they probably occur briefly near the end of a substorm growth phase and possibly in steady magnetospheric convection events. [39] We emphasize that the two PV 5/3 -reducing mechanisms discussed in this section, gradient/curvature drift and bubbles, are not mutually exclusive. It is clear that both operate. In fact, transport by bubbles tends to enhance the effectiveness of the gradient/curvature mechanism, the efficiency of which is determined by the relative magnitudes of ExB and gradient/curvature drifts. If half of the cross-tail potential occurs across bubbles, for example, then only half appears across the much larger nonbubble region. That makes gradient/curvature drift a more effective loss mechanism for that region. [40] It seems likely that a combination of the gradient/ curvature drift and bubble effects described above are adequate to explain the observed systematic increase of average entropy with distance down the tail. The main lesson from the pressure balance inconsistency idea is that adiabatic sunward convection in the plasma sheet naturally produces a highly stretched growth phase-type magnetic field in the plasma sheet, if convection is strong enough to overwhelm the effects of gradient/curvature drift, and small bubbles are not being generated fast enough to relieve the 11 of 14

12 Figure A1. Comparison of optimized models with satellite measurements of magnetic fields during the 18 April 2002 sawtooth event. Adapted from Kubyshkina et al. [2008]. stress. These conditions seem likely to result in a substorm expansion. [41] Studies of plasma sheet transport are now properly focused more on event studies than on statistics, because the THEMIS mission was designed for event studies. That focus also makes sense from a convection theory point of view, since bubbles seem to be mainly transient processes that are best studied as specific events. However, a major problem with using event studies to understand plasma sheet transport is that the key parameter for understanding bubble transport is PV 5/3, in which the flux tube volume V cannot be directly measured by spacecraft at present. Two approaches to making observational estimates of these quantities are discussed in Appendix A. Appendix A: Observational Estimation of Plasma Sheet Flux Tube Volumes [42] As discussed in section 2, statistical modeling indicates that the basic entropy function P/n 5/3 varies only rather weakly through the plasma sheet, and the assumption that it is conserved does not seem to be systematically violated in a dramatic manner. It is when we add the assumption of frozen-in flux (in the form of conservation of N = nv) to the assumption of entropy conservation that we come to the conclusion that PV 5/3 should be conserved, and that is the theoretical conclusion that is strongly violated in the plasma sheet. Thus the interesting physics is included in the parameter PV 5/3,which cannot be measured directly by a spacecraft, while the parameter that is fairly easy to measure does not exhibit that interesting physics. [43] In recent years, the need for a way to estimate the flux tube volume V from spacecraft measurements for specific events has become increasingly apparent. Statistical models cannot describe local dynamic changes in flux tube volume, e.g., inside the bubble shown in Figure 11, but that is exactly the kind of dynamic situation for which estimation of entropy is most needed. Two approaches have been considered. [44] One approach centers on constructing event-optimized magnetic field models. The most sophisticated approach to that kind of magnetic field modeling has been developed by Kubyshkina et al. [1999, 2002, 2008]. In addition to utilizing magnetic field data from spacecraft measurements during the event, the procedure incorporates pressure measurements (assuming force equilibrium) as well as the location of the isotropy boundary. Internal parameters in Tsyganenko models are adjusted to optimize agreement with observations, including the intensities of ring and tail currents, position of the tail current inner edge, and the position and amplitude of localized thinning of the plasma sheet. The dipole tilt angle is also adjusted to represent shifts in the z coordinate of the current sheet peak. Kubyshkina et al. [2008] compared several versions of this type of model, based on T89 [Tsyganenko, 1989], T01 [Tsyganenko, 2002a, 2002b] and T96 [Tsyganenko, 1995] magnetic fields and slightly different adjustable parameters. Figure A1 compares fits of the three models to measurements by several different spacecraft. Flux tube volumes could easily be calculated from these models. [45] A vastly simpler and less ambitious approach to estimating flux tube volume near a single spacecraft was proposed by Wolf et al. [2006]. They derived an analytic formula for the flux tube volume in a simple analytic model of pressure and magnetic field in a tail-like equilibrium configuration, then altered that flux tube volume formula to include more adjustable parameters, which were best fit to a suite of Tsyganenko models. This Wolf et al. [2006] formula estimates the flux tube volume and pressure for a flux tube that crosses the center of the current sheet at the point nearest the spacecraft. The inputq parameters ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for the formula are measured values of B z, B r = B 2 x þ B2 y, particle pressure P. Here B x, B y and B z are in a coordinate system where z is locally normal to the current sheet. In practice, our group estimates the tilt of the current sheet by the method of Tsyganenko and Fairfield [2004]. The formula of Wolf et al. [2006] is easy to apply but is very limited. It provides only values of V and PV 5/3 near the measuring spacecraft and 12 of 14

13 does not give information about the global magnetic field configuration. Wolf et al. [2006] provide tables that show average errors in the formula s values when it was tested against a suite of Tsyganenko models as well as some other model results. When given measurements in the center of the current sheet, the model s RMS error in the flux tube volume is about 24%. Accuracy decreases with increasing distance from the center of the sheet, and our group normally does not use the formula when B r /B n > 3. In one test against an MHD thin-filament calculation, the formula, which is based on the assumption of quasi-static equilibrium, overestimates PV 5/3 by a factor of 2 3 in cases where the Mach number exceeds 0.2, suggesting that the formula only provides an upper limit on PV 5/3 in conditions of fast earthward flow. [46] Acknowledgments. This work benefited from useful conversations with Gary Erickson and Frank Toffoletto. 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