Standard report - UCLA IGPP

2/21/09

Entropy and Plasma Sheet Transport R. A. Wolf1, Yifei Wan1, X. Xing1,2, J.-C. Zhang1,3, and S. Sazykin1 1

Physics and Astronomy Dept., Rice University, Houston, Texas Now at Atmospheric Sciences and Oceanic Sciences Dept., University of California, Los Angeles, California 3 Now at Space Science Center, University of New Hampshire, Durham, New Hampshire 2

2 Abstract. This paper presents a viewpointed 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[PV5/3], 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 holds also when transport by gradient/curvature drift is included. These considerations lead to the conclusion that, under the assumption of strong, elastic pitch-angle scattering, PV5/3 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 PV5/3 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 pressurebalance inconsistency. Quantitatively accurate estimation of the efficiency of these mechanisms depends on finding a method of estimating PV5/3 (or equivalent) from spacecraft measurements. Two present approaches to that problem are discussed.

3

I. Introduction In the early days of magnetospheric physics, large-scale 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, 1990; Angelopoulos et al., 1992, 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. 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 II with a theoretical discussion of entropy and the entropy parameter PV5/3, where V=∫ds/B is the volume per unit magnetic flux. Section III summarizes evidence for the pressure-balance inconsistency, specifically the evidence that, on average, the entropy parameter PV5/3, which is defined for conditions where the plasma pressure is isotropic, is not uniform in the plasma sheet. Section IV 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 V reviews mechanisms that have been suggested for resolving the inconsistency. Observational estimation of PV5/3 remains a central challenge, and some approaches to solving that problem are discussed in the appendix.

4

II. Basic Physics of Plasma-Sheet Entropy for Isotropic Distribution Functions The entropy S of an ideal gas is given by

S = − ∫ d 3 x ∫ d 3 p f ln f

(1)

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 particle energy W and not on pitch angle or position. The number of particles in volume V is, of course, given by N = ∫ d 3x ∫ d 3 p f

(2)

V

If the distribution function is isotropic, it can be written in the form f =

n g (ξ ) Wo3/2

(3)

where Wo is the average particle energy, and W (4) Wo (The appropriateness of the form (3) can be verified by substituting (3) and (4) in (2).) The fact

ξ≡

that f=f(W) along a field line implies that Wo and n are constant along the line. The function

g (ξ ) 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 ⎛ n ⎞ S (5) = − ln ⎜ 3/2 ⎟ − Λ N ⎝ Wo ⎠ where Λ is a ratio of integrals that depend only on the shape of the distribution function, as

specified by g (ξ ) , and not on n or Wo. Since P / n = 2Wo / 3 , we have

5 2W P = 2/3o 5/3 n 3n

(6)

and (5) can be rewritten S 3 ⎛ P ⎞ = ln ⎜ (7) ⎟−Λ' N 2 ⎝ n5/3 ⎠ where Λ ' is another constant that depends on the shape of the distribution function and particle mass. Equation (7) expresses the thermodynamic entropy, which can be evaluated from spacecraft

measurements. Statistical studies [e.g., Borovsky, 1998] show that P/n5/3 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 3 3 (8) = ln PV 5/3 − ln( N 5/3 ) − Λ ' 2 N 2 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 PV5/3 and the specific entropy P3/5V are also conserved along the drift path. (If the flux tube is not in equilibrium, so that P is not constant along it, then PV5/3 in (8) should be replaced by ⎡ ∫ P 3/5 ds / B ⎤ ⎣ ⎦

5/3

.)

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 xC and time tC. (See Figure 1.) Conservation of S and N then implies that P (x, t )V (x, t )5/3 = P [ xC (x, t ), tC (x, t ) ]V [ xC (x, t ), tC (x, t ) ]

5/3

(9)

6 In addition to this ideal-MHD demonstration, adiabatic drift theory also leads to the conclusion that NV and PV5/3 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 PV5/3, to cover situations where gradient and curvature drifts contribute importantly to particle transport. The conclusion of that analysis is that NsV and PsV5/3 are conserved along a drift path for particles of charge qs and given value of the isotropic invariant

λs = WsV 2/3

(10) where Ws is particle energy, and Ns and Ps are the number and partial pressure of particles characterized by qs and λs . 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 xCs (x, t ) and time tCs (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 P (x, t )V (x, t )5/3 = ∑ Ps [ xCs (x, t ), tCs (x, t ) ]V [ xCs (x, t ), tCs (x, t ) ]

5/3

(11)

s

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 λ is uniform along the part of the boundary surface C where particles enter the plasma sheet and is independent of time, then (11) implies that PV5/3 is uniform in the plasma sheet. Of course, the predicted uniformity of PV5/3 breaks down near the inner edge of the plasma sheet, where particles of large λ 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 hr) (L/10)4/WK(keV)1/2 and for ions it is ~ (120 hr) A1/2 (L/10)4/WK(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 strongpitch-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 field-aligned electric fields are widespread in the plasma sheet. Therefore, we conclude that particle precipitation does not reduce PV5/3 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. 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 PV5/3is relatively minor, since most of the upflowing ions have energies well below the average plasma-sheet energies.

8

III. Pressure-Balance Inconsistency for Isotropic Pressure Erickson and Wolf [1980] used empirical magnetic field models of that time to estimate PV5/3

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 PV5/3 decreases

earthward. An example is shown in Figure 2, which shows an increase of more than an order of magnitude in PV5/3 between X=-10 and -25. This result, combined with the theoretical expectation that PV5/3 should be roughly uniform, was called the "pressure balance inconsistency" or "pressure crisis", though "entropy inconsistency" would have been a better name. In the statistical models, PV5/3 tends to increase tailward, because pressure decreases tailward very slowly, while the flux tube volume increases rapidly, because the field line length increases tailward and magnetic field strength decreases in most statistical-average models. 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 PV5/3 to remain constant.

9 If one starts with a force-balanced magnetospheric configuration that resembles statistical models and forces strong convection with strict conservation of PV5/3 tailward of the inner-edge region, the inner plasma sheet becomes more and more stretched with time, with weaker and weaker equatorial Bz, 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 Bz. Pritchett and Coroniti [1995] found the formation of a deep Bz minimum based on fully kinetic 2D

simulations. Zaharia and Cheng [2004] found a magnetic-field minimum in their computation of a 3D force-balanced configuration with a pressure profile that was designed to represent a substorm growth phase. 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-minute run with 120 kV cross-

tail potential drop. The computed magnetic field strength has dropped to ~1 nT at 12 RE. Note that PV5/3 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 Bz. 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

10 inconsistency between statistical magnetic field and plasma models and the assumption that PV5/3 is conserved.

It must be acknowledged that there is less-than-complete 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 selfconsistently 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 tail. It is not clear which of the computational assumptions cause the difference in conclusions.

11

IV. Pressure Balance Inconsistency for the Case Where First and Second Adiabatic Invariants are Conserved 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. Suppose that the distribution function on flux tube 1, which extends to an equatorial crossing distance of re1, is given by (12) f = A δ (W − W1 ) where W is particle kinetic energy and A and W1 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 J =2

sm

∫

− sm

p|| ds = 2 2mμ K

(13)

where sm is the distance along the field line from the equatorial plane to the mirror point, μ is the first adiabatic invariant, and

12

K=

sm

∫

Bm − B( s ) ds

(14)

− sm

is the geometrical invariant. Equation (12) can be rewritten f = A δ [ μ Bm ( K , re1 ) − W1 ] (15) where Bm(K, re1) is the mirror field corresponding to geometric invariant K on the flux tube that

crosses the equatorial plane at re1. Since f should be constant along a drift path, and μ 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 W2,inv W1

∫ d μ μ ∫ dK δ ⎡⎣ μ B ( K , r ) − W ⎤⎦ B ( K , r ) = W ∫ d μ μ ∫ dK δ ⎡⎣ μ B ( K , r ) − W ⎤⎦ 3/2

m

e1

1

m

e1

e2

m

1/2

1

(16)

1

In the approximation of strong, elastic pitch-angle scattering, we have, simply, W2,iso W1

⎛V ⎞ =⎜ 1 ⎟ ⎝ V2 ⎠

2/3

(17)

[Wolf, 1983]. The occurrence of chaos in plasma sheet particle orbits is often characterized by the parameter

κ=

Rc ,min ac ,max

(18)

where Rc,min is the minimum radius of curvature on the field line, and ac,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 κ 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

13 corresponding critical energy can be derived from (18) by substituting the usual expression for the Larmor radius. The result is Ecrit =

( Rc ,min eBmin ) 2

(19) 4 2mκ crit Several estimates have been made of κ crit , from the viewpoint of isotropy of ions near the loss cone. For their observational study, Sergeev et al. [1983] chose κ crit = 8 . Detailed theoretical calculations of Delcourt et al. [1996] suggested values κ crit = 2 − 4 . Figure 5 shows a plot of Ecrit for the lowest value κ crit = 2 . The figure suggests that the vast majority of ions that contribute importantly to plasma sheet pressure should be isotropic beyond a critical distance of 8-12 RE. We perform calculations for a critical distance of 12 RE; 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. Figure 6 compares the results from (16) and (17) for three T89 magnetic field models (Kp=0, 3, 6) and for re1=12. It is clear that the assumption that μ 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. 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.

14 Choosing re13. In one test against an MHD thin-filament calculation, the formula, which is based on the assumption of quasi-static equilibrium, overestimates PV5/3 by a

26 factor of 2-3 in cases where the Mach number exceeds 0.2, suggesting that the formula only provides an upper limit on PV5/3 in conditions of fast earthward flow.

Acknowledgements. This work benefitted from useful conversations with Gary Erickson and Frank Toffoletto. The paper was supported in part by the NASA Heliospheric Physics Program under grants NNX08AI55G and NNG05GH93G and also in part by NASA grant NNX07AF44G.

27

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Figure Captions 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. Figure 2. Distribution of log10(PV5/3) in the equatorial plasma sheet, based on a T96 magnetic field model [Tsyganenko, 1995] and a Tsyganenko-Mukai [2003] model of the plasma sheet, for average solar wind conditions (n=5 cm-3, v=400 km/s, Bx=By=5 nT, Bz=0). PV5/3 has units of nPa(RE/nT)5/3. From Xing and Wolf [2007]. Figure 3. 2D-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]. Figure 4. Equatorial plot of PV5/3 (units of nPa(RE/nT)5/3) and a log plot of Bz along the midnight line, for an RCM-E run, after 45 minutes with assumed 120 kV cross-tail potential drop. Adapted from Lemon [2005]. Figure 5. Log10(Ecrit(keV)) vs. x for the three T89 models under consideration, computed from (19) for protons assuming κ crit = 2 . Figure 6. Energization factors for particles drifting from 12 RE to re2. The plots compare log10(W2,inv/W1) from equation (16) and log10(W2,iso/W1) from equation (17), for T89 models with three different Kp values. Figure 7. Same as Figure 6, but with the particles starting from 18 RE with an isotropic distribution. Figure 8. Log10 of ratio of equatorial pressure at re2 to pressure at re1 =12 RE, computed several different ways. "Isotropic" means that the pressure ratio was computed from (V1/V2)5/3. "Perp

34 adiabatic" and "Average adiabatic" were computed from (20) and (22), respectively. The "Parallel adiabatic" curve was then computed from (21). Figure 9. Cartoon demonstration that the Lui [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 PV5/3, 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 PV5/3 than its neighbors). 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 RE. Field line shapes are shown in the top panel for times 0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 18, 23, 28, 33, 38, and 43 minutes. The middle and lower panels are enlarged views of the tailward and earthward portions of the top panel. The nearequatorial 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 the bottom panel. Adapted from Chen and Wolf [1999]. Figure 11. Bubble observations by Geotail on July 22, 1998. The top two panels show the northward normal component of the magnetic field and the x-component of velocity. The middle panel shows ion pressure, and the bottom two panels show values of V and PV5/3 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 panel represent the components of drift associated with potential and induction electric fields, respectively.

35 Figure 12. Equatorial view ofan 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 PV5/3 (nPa(RE/nT)5/3). Black curves are contours of constant Φ + λi V −2/3 / e , with contour spacing 5 kV. Dashed equipotentials are negative. The pink circle has radius 6.6 RE. The bubble was launched from the tailward boundary at 0655 UT. Figure 13. Comparison of optimized models with satellite measurements of magnetic fields during the 18 April 2002 sawtooth event. Adapted from Kubyshkina et al. [2008].