density structure of a preeruption coronal flux rope - IOPscience

31 downloads 0 Views 504KB Size Report
ABSTRACT. A density model is developed for a coronal flux rope with a specified magnetic field configuration in the non- interacting limit, for which transverse ...
The Astrophysical Journal, 628:1046–1055, 2005 August 1 Copyright is not claimed for this article. Printed in U.S.A.

DENSITY STRUCTURE OF A PREERUPTION CORONAL FLUX ROPE J. Krall and J. Chen Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375-5346 Received 2004 August 19; accepted 2005 April 7

ABSTRACT A density model is developed for a coronal flux rope with a specified magnetic field configuration in the noninteracting limit, for which transverse pressure forces can be neglected and flows are rapid enough (>1 km s1) to invalidate a hydrostatic solution. In the coronal portion of the flux rope, we find that the plasma density is depleted on-axis, where the magnetic field lines are least twisted. Subject headinggs: Sun: coronal mass ejections (CMEs) — Sun: magnetic fields — Sun: prominences

1. INTRODUCTION

In our flux rope density calculation below, the flux rope plasma will be characterized as having a uniform density f at the footpoints and a nonuniform density  in the corona. We will show that this characterization is valid at the footpoints, where the plasma is highly collisional, i 3 i ( Kleimann & Hornig 2001). Here i is the thermal ion collision rate for protons and i is the ion gyrofrequency. In the coronal portion of the flux rope, the plasma is highly magnetized, i T i , and magnetic forces dominate. In solar MHD, the situation in which the magnetic forces are balanced is commonly referred to as the force-free limit; here pressure forces can be neglected relative to magnetic forces. In our calculation of  we will indeed neglect pressure forces transverse to the magnetic field, but we will depart from MHD and instead consider the ballistic propagation of fluid along coronal flux rope field lines, in the limit that these flows do not interact with those on neighboring field lines. Table 1 shows parameters for an isolated solar flux tube based on representative solar atmospheric density and temperature parameters ( Vernazza et al. 1981; Hansteen et al. 1997; Sa´nchez Almeida & Lites 2000; Aschwanden & Acton 2001), on an assumption of equipartition, B2  p, and an assumed 5 kG field in the photosphere. Consistent with initial conditions used for model flux rope CME events (Chen et al. 1997, 2000; Krall et al. 2001), the representative flux rope field in the corona is of the order of 1 G. In Table 1, i ½s1  ’ 4:8 ; 108 ni ½cm3 kT ½eV3/2 and

i ¼ eB/mc are also given, where k ’ 23  log ½(2ni T 3 )1/2  is the Coulomb logarithm ( Trubnikov 1965). Note that, while an active region coronal loop may be expected to have higher field values than those given in Table 1, the same scalings apply. Based on Table 1, we identify a significant transition from the highly collisional chromosphere (i 3 i ) to the highly collisionless transition region and corona (i T i ). Thus, as plasma flows through the corona from footpoint to footpoint, the protons making up that plasma suffer very few collisions. That is, plasma particles experience negligible collisional diffusion across flux surfaces during a single transit along the length of the coronal flux loop. Specifically, let us consider a loop of length L and width a (see Fig. 1) and a fluid element moving along this loop with velocity component vz . Ions within this fluid element will suffer Nc ¼ i L/vz collisions in a single transit. In each collision the ion ‘‘jumps,’’ in the transverse direction, a distance equal to the cyclotron radius rc ¼ vth / i . Here we assume that the thermal velocity vth ¼ (kT/mp )1/2 > vz. In one transit, ions within the fluid element randomly walk, in the transverse direction, a distance r ¼ rc Nc1/2 .

The typical rim-cavity-prominence coronal mass ejection (CME ) morphology ( Illing & Hundhausen 1986) has been hypothesized to be the result of an underlying flux rope geometry (Chen 1996; Low 1996). Papers supporting this hypothesis have focused on new observational results (Dere et al. 1999; Plunkett et al. 2000), on model development (Gibson & Low1998, 2000; Roussev et al. 2003), or on model-data comparisons for specific events (Chen et al. 1997, 2000; Wood et al. 1999; Wu et al. 1999; Krall et al. 2001; Chen & Krall 2003). In this paper we provide a new analytical description of a stationary coronal flux rope through which we determine the physical relationship between the density structure of this model preeruption flux rope and its underlying magnetic field. We show that, for a physically realistic field configuration, the density in the coronal portion of the flux rope is somewhat hollow, with a density minimum along the flux rope axis. In so doing, we consider the idea that, if the plasma in a coronal flux rope flows quickly enough that the transit time along the flux rope is shorter than the relevant collisional interaction time, then the pressure forces transverse to the magnetic field, which are associated with those interactions, can be neglected. In this limit the resulting density structure is determined by the magnetic field structure (which is, in turn, determined by the magnetic forces that dominate the system) and by the boundary conditions of the plasma, which flows ballistically along the flux rope field. Mathematically, this is a departure from the usual magnetohydrodynamic ( MHD) approach. Physically, this approximation is similar to the ‘‘force-free’’ approximation, which is commonly used when magnetic forces dominate, as in the corona. The remainder of this paper proceeds as follows. The ‘‘noninteracting fluid approximation’’ is described in x 2, the calculation of the coronal density is presented in x 3, examples are given in x 4, and x 5 concludes this paper with a brief discussion. 2. AN IDEALIZED MODEL FLUX ROPE Consider a quiescent coronal flux rope. In obtaining the flux rope density, let us assume that plasma flows along the length of the flux rope, which has footpoints embedded in the photosphere, and that plasma motion across field lines is inhibited. If we ‘‘stretch out’’ the model flux rope into a straight cylindrical geometry as shown in Figure 1, and if we specify the magnetic field, its density can be computed in a simple way. Before we proceed, however, let us consider the physical properties of the flux rope field and its associated plasma. 1046

1047

DENSITY STRUCTURE OF A CORONAL FLUX ROPE

is flowing with a velocity component vz > i (rc L)1/2 . Because plasma flowing along a given field line does not interact with that of neighboring field lines, the net effect of these flows is not expected, or required by the analysis, to be coherent. That is, bulk flow velocities on neighboring field lines can differ greatly from each other in magnitude or direction without affecting the validity of the results, so long as the flow at the footpoints, averaged over any localized ensemble of neighboring field lines, is uniform, as dictated by the boundary conditions at the footpoints. 3. CORONAL FLUX ROPE DENSITY IN THE NONINTERACTING LIMIT

Fig. 1.—Model flux rope schematic in cylindrical geometry.

We can see that, for any likely flow velocity vz, r/aT1. For example, with L ¼ 1010 cm, a ¼ 109 cm, vz ¼ 1 km s1, and coronal parameters from Table 1, r/a ’ 3 ; 105 T1. Thus, our plasma indeed follows the direction of the field lines as it flows. Further, this has minimal interaction with plasma on neighboring field lines. Based on a simple geometrical argument, ions within a flowing plasma have a characteristic transverse velocity as a result of collisions: v? ’ vz rc /kc , where kc ¼ vz /i is the mean free path between collisions. The approximate number of collisions that occur between a given ion and ions associated with plasma on neighboring field lines is N? ’ Nc v? / vz ¼ Lrc (i /vz ) 2 . Thus, if vz is large enough, N? < 1 and plasma flowing along any given field line does not interact with plasma associated with neighboring field lines. With L ¼ 1010 cm, and again using coronal parameters from Table 1, these flows are noninteracting if vz > i (rc L)1/2 ’ 0:8 km s1. Given that bulk plasma flows are often observed in coronal loops, a new conception emerges, which we refer to as the ‘‘noninteracting fluid approximation.’’ Here a coronal magnetic loop represents a conduit through which plasma can flow, between a high-density, low-temperature, highly collisional reservoir at one footpoint and a similar plasma reservoir at the other footpoint, without interacting with plasma on neighboring flux surfaces. In practical terms, the noninteracting fluid approximation is one in which thermal pressure forces transverse to flux surfaces are neglected. Physically, this approximation is similar to the force-free approximation, which is commonly used when magnetic forces dominate, as in the corona. It is possible, of course, that the bulk of the plasma within the loop resides in a static equilibrium state with vz < i (rc L)1/2 , in which case this approximation would not hold. However, given that high-speed flows in the corona are commonly observed ( Winebarger et al. 2002), this is not likely to be the case. In x 3, we make use of the noninteracting fluid approximation in order to determine the density structure within the coronal portion of a preeruption coronal loop. In this approximation, the bulk of the plasma in the coronal portion of the loop is assumed to consist of individual field lines, each containing plasma that

We now present a simple density calculation in the noninteracting fluid approximation, in which fluid flowing along a given field line does not interact with that of neighboring field lines (see x 2). In this regime, transverse thermal pressure forces do not enter the analysis. In fact, we will set aside the forcebalance equation, for now, and determine the density for a given magnetic field configuration based on the continuity equation. This result will be valid only in the corona. In producing our result, we will account for the effect of the gravitational potential but will not impose gravitational stratification. Finally, we will account for magnetic forces by choosing field parameters for which transverse magnetic forces are approximately balanced. Because heat-transfer processes in a coronal flux rope are not well understood, these processes will be neglected and a temperature profile will be specified. For a flux rope in a stationary state, the continuity equation is : = (v) ¼ 0. If we ‘‘stretch out’’ the model flux rope into a straight cylindrical geometry as shown in Figure 1, we can compute its density under the simplifying assumption of axisymmetry. Because our key approximation is that plasma follows field lines without interaction with plasma flowing along neighboring field lines, and because field lines lie within flux surfaces, the problem becomes more transparent when formulated in terms of a coordinate system that follows flux surfaces. With  as an ignorable coordinate, the magnetic field can be written as ( Bateman 1978) 1 @ z @Az ˆ 1 @ z rˆ  fþ zˆ ; 2r @z 2r @r @r ˆ þ Az zˆ is the vector potential, where A ¼ z /2rf B¼

I z (r; z) ¼

B = zˆ ds ¼

Z

Z

2

r

d 0

Bz r dr

is the magnetic flux threading a constant-z surface within a circle of radius r centered on the axis, and (r; ; z) are the usual cylindrical coordinates. Note that currents flowing within our

Region

n (cm3)

T (eV )

p (dyne cm2)

B2 (G 2)

i (s1)

Photosphere...................... Chromosphere .................. Transition Region............. Corona..............................

2 ; 1017 2 ; 1014 1010 108

0.5 0.5 20 200

1.6 ; 105a 160a 0.64b 0.064b

2.5 ; 107 2.5 ; 104 100 10

4.6 ; 1010 1.4 ; 108 84 0.036

b

Here p ¼ nkT . Here p ¼ 2nkT .

ð2Þ

0

TABLE 1 Solar Flux Tube Parameters Assuming Equipartition, B2  p

a

ð1Þ

i (s1) 4.8 1.5 9.6 3.0

; ; ; ;

107 106 104 104

1048

KRALL & CHEN

model flux rope, which are related to z and Az above, will be confined to a current channel of radius a(z) (see Fig. 1). We now define a nonorthogonal coordinate system that conforms to flux surfaces: ¼ z =(2a0 B¯ t0 ); # ¼ ;  ¼ z;

e ¼ : ¼ r (Bz rˆ  Br zˆ )=(a0 B¯ t0 ); ˆ e# ¼ :# ¼ f=r; e ¼ : ¼ zˆ ;

ð4Þ

such that any vector v may be expressed in terms of contravariant components v i  v = ei (see Appendix A). In this formalism, the continuity equation has the form 

@ : = (v) ¼ D @



v D



    @ v# @ v þ þ ¼ 0; @# D @ D ð5Þ

where D ¼ e = (e# < e ) ¼ Bz /a0 B¯ t0 is the reciprocal of the Jacobian. We now invoke our assumption that plasma flows follow field lines in order to write the fluid velocity as v ¼ v B=B;

ð6Þ

where v ¼ jvj and B ¼ jBj. Thus, in our nonorthogonal coordinate system, v ¼ v = e ¼ 0 and v ¼ v = e ¼ vz ¼ vBz /B everywhere. Substituting v , v, and D into equation (5) and applying axisymmetry (@/@# ¼ 0), we obtain @(vz =Bz )=@ ¼ @(v=B)=@ ¼ 0:

ð7Þ

Thus, along any given flux surface,  ¼ f

where Bf; z is the z-component of Bf . Here we assume that all flux rope field lines emanating from one footpoint remain within the flux rope and connect to the other footpoint. Similarly, continuity of axial current (assuming that the axial current, if present, remains within the flux rope as it flows from footpoint to footpoint) gives Bf; ¼ B a/af . Substituting Bf; z and Bf; into equation (8), the coronal flux rope density is obtained in terms of f , Bf; r , a/af , v/vf , and B:

ð3Þ

where a0 is the value of a(z) at the apex of the flux rope in the corona (the apex is the midpoint of the flux rope in Fig. 1) and B¯ t 0 is the axial field at the apex, averaged over r  a. The covariant basis vectors associated with these coordinates are ( Bateman 1978)

vf B : v Bf

ð8Þ

Through equation (8), the continuity of the flow allows us to relate , v, and B in the corona to their values f , vf , and Bf at some reference point, which we take to be one of the footpoints of our coronal loop. We specifically consider a case in which the plasma at the footpoints has a uniform density f (due to collisionality). In the model flux rope, electric currents are confined to a current channel of radius a(z) (see Fig. 1), which expands from a small size af at the footpoints to larger values a(z) in the corona. We now assume that expansion of the flux rope cross section is self-similar so that the magnetic field profile versus radius can be specified as a function of r/a(z). Thus, for any given flux surface (r/a), conservation of flux z gives Bf;z ¼ Bz a2 /a2f ,

Vol. 628

( ;  ) ¼ f

vf  af 2 B h    2 i1=2 : v a 4 B2z þ B2f; r af =a þ B2 af =a ð9Þ

We write equation (9) in this form to separate out the ½af /a(z)2 factor, which accounts for variations in density resulting from the self-similar spreading or converging of flux surfaces versus distance along the flux rope, from the density enhancement 1/2 factor B/½B2z þ B2f; r (af /a)4 þ B2 (af /a)2  , which increases from unity on-axis, where B ¼ Bz zˆ , to a/af > 1 at the edge of the ˆ This off-axis density enhancement flux rope, where B ’ B f. represents the accumulation of plasma in the highly twisted field lines at the edge of the flux rope, where the field lines are longest and the plasma velocity component along the length of the flux rope, vz , is slowest (see eq. [6]). Equation (9) provides the radial flux rope density structure in the corona for a specified magnetic field, but it includes unknown factors f , af /a, and vf /v. While af /a can be estimated using Table 1 and the constant f can be estimated based on solar observations, the significance of vf /v is less clear. We argue that, while vf /v has a value that most likely varies from field line to field line, there is no expectation of any systematic variation of vf /v as a function of radius (flux surface) within the flux rope. Let us first consider vf . Due to collisionality at the footpoints, it is reasonable to assume that the velocity distribution is independent of r at the footpoints. Thus, while vf presumably varies from field line to field line, the average value of vf over any localized ensemble of field lines within the flux rope is expected to be constant versus radius. Let us next consider the effect of the gravitational potential on the value of v in the coronal segment of our solar flux rope. Consistent with the noninteracting fluid approximation (see x 2) we assume that the coronal flux rope is populated by plasma flowing independently along individual field lines from the highdensity, low-temperature, highly collisional reservoir at one footpoint to a similar plasma reservoir at the other footpoint. These flows may be driven by any of a number of mechanisms, including pressure forces parallel to the field lines. We assume only that the driving mechanism does not introduce a variation of v versus r within the flux rope. Because previous studies (Chen et al. 1997, 2000; Krall et al. 2001) suggest that the preeruption flux rope dimension a is typically smaller than the coronal gravitational scale height 2kT /gmp ’ 1:4 ; 1010 cm, we can assume that all such fluid flows overcome the same average gravitational potential, independent of flux surface, as they spiral along the length of our twisted flux rope. Thus, at any fixed z, the average value of v over any localized ensemble of field lines within the flux rope is expected to be constant versus radius. The velocity distribution of plasma fluid elements that flow through our model coronal flux rope from end to end therefore produces a density profile that is given by equation (9), with vf /v being replaced by an overall constant factor, which we absorb, along with the constant factors f and af , into a specified constant

No. 2, 2005

1049

DENSITY STRUCTURE OF A CORONAL FLUX ROPE TABLE 2 Model Solar Flux Tube Parametersa

Region

z ( Mm)

a/af

Bz (G)

B (G)

Photosphere....................... Chromosphere ................... Transition Region.............. Corona...............................

0 1 3 20

0.18 1.0 4.0 7.1

5000 160 10 3.2

67 12 3.0 1.7

a /2 Assuming self-similarity B ¼ B½z; r/a(z), axial flux continuity, a  B1 , z and axial current continuity B  1/a.

0. The density profile in the coronal portion of our model flux rope can be written as  a 2 B 0  ( ;  ) ¼ 0 h    2 i1=2 : ð10Þ a 4 B2z þ B2f; r af =a þ B af =a Here 0 and a0 are the on-axis density and the current-channel radius, respectively, of the flux rope at its apex, and a is the local current-channel radius in the corona. If we assume B2f; rTB2f; z at the footpoints, we obtain the density profile in the corona in terms of the coronal flux rope field, the local current-channel radius a, and a/af :  a 2 B 0  (r; z) ’ 0 ð11Þ h  2 i1=2 : a B2z þ B af =a The validity of this assumption (B2f; rTB2f; z ) for a twodimensional model coronal flux rope is addressed in Appendix B. For given values of 0, a0, and af , and a specified magnetic field profile versus r, equation (11) provides the density structure of the coronal portion of a stationary flux rope. We now revisit the flux rope parameters of Table 1. Taking the representative field values B from Table 1 to be Bz values in our model flux rope, relative values of a/af in the photosphere, chromosphere, transition region, and corona are obtained from and are listed in Table 2. Assuming flux conservation a  B1/2 z that the peak value of B is comparable to the peak value of Bz in the corona, B values for our model flux rope are also given in Table 2. Identifying the chromosphere (i 3 i ) as the location of our footpoints and the transition region and corona (i T i ) as regions through which the magnetized plasma flows consistent with equation (11), we find acorona /af ’ 7. It is clear from either equation (10) or equation (11) that at the apex of the flux rope,  ! 0 a0 /af at the edge of the flux rope current channel, where Bz /B ! 0. This limiting value is obtained at the point at the edge of the flux rope at which the flux rope current drops to zero. At this point, Bz ¼ 0 while B 6¼ 0. This is illustrated in Figure 2 in which, based on equation (11) with a/af taken from Table 2, /0 is plotted versus Bz /B . Thus, in the corona, (r ¼ a)=(r ¼ 0) ’ a(z)=af ’ 7:

ð12Þ

Equation (12) represents a rough approximation based on the values given in Tables 1 and 2. We note that an active region flux rope might have larger field values. Relative to the calculation above, an active region result might feature increased gyrofrequencies, a shift in the footpoint location downward toward the photosphere (a footpoint being where the plasma becomes

Fig. 2.—Ratio of peak density to on-axis density 0 for a coronal flux rope plotted vs. the value of Bz /B at the edge of the flux rope (see eq. [11]) for a/af ¼ 7.

collisional ), a smaller footpoint radius, and a greater center-toedge density enhancement in the corona. We note again that the result of equation (12) is valid only in the noninteracting fluid approximation, for which the bulk of the plasma in the coronal flux rope is assumed to be flowing with velocity v > 1 km s1. We note also that additional factors that would affect (z) in the legs of the flux rope have been neglected, such as the details of the process(es), not determined here, that drive the observed flows along the flux rope. In obtaining equation (10) from equation (9), a key approximation is that the forces that drive flows along the flux rope do not introduce changes in the ensemble-averaged fluid speed v as a function of radius. That is, on average, v ¼ jvj is constant versus r at any fixed z. While v is constant, the direction of v ¼ vB/B changes systematically versus r in the corona, giving rise to a density enhancement at large r. For example, pressure gradients parallel to B may have a role in driving these flows, but this role is expected to be most significant near the footpoints, where Bf; z 3 Bf; r ; Bf; (see Table 2 and Appendix B). Thus, the leading-order effect of any force parallel to B at the footpoints is on vz and is not expected to introduce a significant variation in v (r), so long as the force itself is independent of r. In contrast to what is presented above, the static case [v < = i (rc L)1 2 ’ 1 km s1] can be addressed by solving the MHD momentum equation. This approach is discussed in Appendix C, in which we show that, in this limit, a coronal loop with a nearly force-free magnetic field can support either an on-axis density depletion, qualitatively similar to the result from equation (11), or an on-axis density enhancement. It is further shown that neither of these solutions (the density depletion or the density enhancement) are clearly favored. One of the conclusions of Appendix C is that the naive view, wherein the ‘‘magnetic pressure’’ of a flux rope magnetic field naturally leads to reduced plasma pressure in the region in which the magnitude of the flux rope field is largest, is not correct. 4. EXAMPLES For examples, we first consider the flux rope CME model of Chen (Chen & Garren 1993; Chen 1996; Krall et al. 2000). Here the coronal portion of the initial equilibrium flux rope has a current channel with an initially constant radius a(z) ¼ a0 . In the model, the magnetic field profile versus r is defined in a locally cylindrical fashion with a previously used field profile being (Chen 1996; Chen et al. 1997, 2000; Wood et al. 1999; Krall et al. 2000, 2001) 8   r 2 r4 < ¯ 3Bt 1  2 2 þ 4 ; Bz ¼ a a : 0;

r  a; r > a;

ð13Þ

1050

KRALL & CHEN

Vol. 628

Fig. 3.—(a) Field profiles Bz (solid line) and B (dashed line) from eqs. (13) and (14), with B¯ t ¼ 1:08 G and Bp ¼ 1:5 G. (b) Corresponding density profile  (solid line) from eq. (11) with a/af ¼ 7. The density profile from eq. (10) is also shown as a dashed line (see Appendix B).

8   r r2 r4 > > < 3Bp 1 2 þ 4 ; a a 3a B ¼ > a > : Bp ; r

r  a; ð14Þ r > a;

where B¯ t is the z-component of the field averaged over r  a and Bp is the -component of the field at r ¼ a. This field profile is shown in Figure 3a. Setting a ¼ a0 and a0 /af ¼ 7, the resulting coronal flux rope density profile is shown in Figure 3b. Note, however, that for r > a, Bz ¼ 0 and the density is undefined in equation (11), because field lines in this region do not connect to the flux rope footpoints. The shaded portion of the density profile shown in Figure 3b therefore represents an assumption that the B field outside of the current channel (r > a) contains plasma. The sharp cutoff at r ¼ 2a represents an assumed cutoff at which the flux rope fields are too weak to contain this plasma. This specific correspondence between the model flux rope field and the flux rope density, which was until now an ad hoc assumption, has been used in numerous successful comparisons of model results to observed CME dynamics (Chen et al. 1997, 2000; Wood et al. 1999; Krall et al. 2001). Note that, consistent with the noninteracting fluid approximation, field parameters in Figure 3 are chosen such that the two radial magnetic force components, J Bz /c and Jz B /c, are balanced as in the force-free Lundquist solution ( Lundquist 1950). This is shown in Figure 4a in which the two magnetic force components (dashed lines) and the net force (solid line) are plotted versus r for this case. We see that these forces are approximately balanced, such that the deviation in the net force fr (solid line) from equilibrium is everywhere small in comparison to the individual contributions (dashed lines). For comparison, the forces associated with a Lundquist field, in which the force balance is exact instead of approximate, is shown in Figure 4c. In Figure 4c, Bpeak ¼ 1:6 G. In all cases the gravitational force, which is vertical rather than radial, has been neglected in Figure 4. This force is 2 orders of magnitude smaller than magnetic forces for these parameters. Our results suggest that, while density profiles obtained via equation (11) do not necessarily represent exact equilibria, equilibrium states are possible, consistent with this analysis. In the field profile of Figure 3, a significant portion of the flux rope field energy lies in a region r > a in which Bz ¼ 0 and field lines are purely azimuthal. While this field profile seems idealized, observed magnetic clouds often share these properties, with the full rotation of the field direction occurring well within the region of enhanced field ( Zhang & Burlaga 1988).

Fig. 4.—Transverse magnetic forces (1010 dyne cm3) at the apex of the model coronal flux rope: ðJ < B/cÞr (solid line), J Bz /c (short-dashed line), and Jz B /c (long-dashed line). The cases correspond to (a) Fig. 3, (b) Fig. 5, (c) a Lundquist field with Bpeak ¼ 1:6 G, and (d ) Fig. 6. Here a ¼ 109 cm in (a) and (c) and a ¼ 2 ; 109 cm in (b) and (d ).

A field profile that is similar to that of Figure 3, but with a nonzero axial field throughout, is shown in Figure 5a. Here the azimuthal current does not vanish completely in the highdensity region, as it does in Figure 3, and the ‘‘edge’’ of the flux rope is now assumed to lie at r ¼ a. This field profile is qualitatively similar to that of Figure 3 in that the current in both cases is somewhat localized so that the width of the central peak in Bz is narrow compared to the overall width of the flux rope and its associated density profile. The fields in Figure 5 are ( Bz ¼

 1 B¯ t exp ( 1 r 2 =a 2 )=(1  e 1 ); 0; 8   r > < Bp exp  2   2 r 2 =a 2 ; a B ¼ a > : Bp ; r

r  a; r > a;

ð15Þ

r  a; ð16Þ r > a;

where the transition in Bz at r ¼ a is assumed to be smooth and, for Figure 5,  1 ¼ 8 and  2 ¼ 4:6. The corresponding density profile from equation (11) is also shown. As in Figure 3, field

Fig. 5.—(a) Field profiles Bz (solid line) and B (dashed line) from eqs. (15) and (16), with B¯ t ¼ 0:4 G, Bp ¼ 0:093 G,  1 ¼ 8, and  2 ¼ 4:6. (b) Corresponding density profile  (solid line) from eq. (11) with a/af ¼ 7. The density profile from eq. (10) is also shown as a dashed line (see Appendix B).

No. 2, 2005

DENSITY STRUCTURE OF A CORONAL FLUX ROPE

Fig. 6.—(a) Field profiles Bz (solid line) and B (dashed line) from eqs. (15) and (16), with B¯ t ¼ 0:4 G, Bp ¼ 0:33 G,  1 ¼ 3, and  2 ¼ 2. (b) Corresponding density profile  (solid line) from eq. (11) with a/af ¼ 7. The density profile from eq. (10) is also shown as a dashed line (see Appendix B).

parameters in Figure 5 are chosen such that the magnetic forces are approximately balanced. This is shown in Figure 4b. Finally, we consider a broader field profile, relative to Figures 3 and 5. This is shown in Figure 6 and is given by equations (15) and (16) with  1 ¼ 3 and  2 ¼ 2. The resulting density profile is also shown. As in Figures 3 and 5 the magnetic forces are approximately balanced in this case; see Figure 4d. In this case, Bz /B ’ 0:21 at the edge of the flux rope and the largest possible density enhancement from equation (11) is not obtained (see Fig. 2). This might correspond to a situation in which our twisted coronal flux rope abuts an external current system and its associated fields such that the flux rope current drops abruptly to zero at r ¼ a. Figures 3, 5, and 6 show that the density profile is sensitive to the field profile, with a more narrowly peaked current channel corresponding to a much stronger off-axis density enhancement. We reiterate that the density profiles shown in these figures are computed using equation (11), in which the Bf; r term, which appears in equation (10), has been neglected under the assumption that B2f; r TB2f; z . It is straightforward, however, to estimate the value of this term by constructing a two-dimensional model flux rope consistent with Tables 1 and 2. This is done in Appendix B, with the resulting density profiles being shown as dashed lines in Figures 3, 5, and 6. Clearly, it is reasonable to neglect Bf; r in this analysis. 5. DISCUSSION We have analyzed the radial density structure of a coronal flux rope, which is a function of its magnetic field configuration. We find that a density depletion can occur along the axis of a coronal flux rope, that the bulk of the current flowing along a coronal flux rope lies within this reduced-density region, and that a more narrowly peaked current channel results in a greater center-to-edge density ratio. Our results, particularly equations (7) and (11), can be interpreted in terms of the time of flight of particles or fluid elements moving along the length of a magnetic flux rope. Where the time of flight for a fluid element moving along a field line is large, as with a highly twisted field line, the density is also high. This approach to the continuity equation has been used to directly calculate densities in collisionless current sheets (Holland & Chen 1993). Assuming that a flux rope CME results from the eruption of a coronal flux rope, this result provides an explanation for the commonly observed ‘‘rim’’ and ‘‘cavity’’ of the well-established

1051

three-part CME morphology ( Illing & Hundhausen 1986; Hundhausen 1999). This conception, in which the preeruption flux rope density structure is preserved in the posteruption CME, is supported by model calculations ( Krall et al. 2000), which show that a typical erupting flux rope accelerates rapidly beyond the point at which the CME velocity exceeds the sound speed within the flux rope. That is, typical subsonic flows are too slow to affect the density structure during the eruption. While this simple conception does not explain the role of the helmet streamer (other than to say that the streamer density that is associated with open field lines is not part of the underlying flux rope and therefore does not contribute to the posteruption bright rim), it is nevertheless attractive. In particular, the density profiles shown in Figures 3 and 5 are qualitatively similar to the ad hoc density profile used by Chen et al. (2000) in order to produce model synthetic coronagraphs corresponding to a specific CME event. In that case, a model flux rope that was both highly tilted and shifted away from the limb resulted in a synthetic CME image that was strikingly similar to the observed asymmetric CME event that was the subject of the study. We note, however, that a slowly evolving, large (a 31010 cm ¼ 100 Mm) flux rope, as might be embedded within a large helmet streamer, might not be well described by this analysis, because the width of the streamer would exceed the local gravitational scale height. For a streamer-associated flux rope with a length of the order of L  350 Mm ’ R /2 and a plasma flow velocity v  10 km s1, ‘‘slowly evolving’’ implies a timescale longer than L/v  10 hr. An alternative viewpoint is that of Gibson & Low (1998), who specify a quasi-static magnetic field (a distorted spheromak field) to which they selectively add and subtract mass and/or pressure in order to balance net magnetic forces with gravitational and pressure forces. For example, the curved flux rope in the core of their specified field structure tends to expand outward, as does any three-dimensional, curved, twisted flux rope (see, e.g., Fig. 1 of Chen 1996). To balance this force, Gibson & Low (1998) add cold mass to their solution in the region of the twisted flux rope, which provides a downward gravitational force ( by contrast, Chen [1996] adds a background magnetic field for this purpose). Gibson & Low (1998) suggest that this additional cold mass could correspond to the observed prominence. Similarly, they are able to reproduce the cavity by subtracting mass in a region of the solution in which an additional net upward force is needed, and finally, similar to Figure 8b of Appendix C, they add pressure near the edge of their specified magnetic field to provide radial force balance. The Gibson & Low (1998) model is attractive in that it seems to reproduce all features of the threepart CME structure, but it is valid only for a quasi-static solar atmosphere. In contrast, we have shown that, if a typical preeruption solar flux rope manifests itself in a dynamic, rather than static, solar atmosphere, with typical coronal flows >1 km s1 (see x 2), then the flux rope structure will be determined by a combination of boundary conditions and ballistic flows rather than by hydrostatic forces in the corona.

This work was supported by NASA ( DPR W-10106, LWS TRT program) and the Office of Naval Research. The authors wish to thank Peter W. Schuck of NRL for assistance with the formulation of the problem in terms of z and Steven Slinker of NRL for assistance with the numerics.

1052

KRALL & CHEN

Vol. 628

APPENDIX A FLUX COORDINATES For completeness, we include here a brief discussion of nonorthogonal coordinates and the ( ; #;  ) coordinate system introduced in x 3. As an example, the covariant and contravariant coordinates of the magnetic field are determined. Consider a curvilinear coordinate system defined by coordinates (1, 2, 3). In these coordinates, ei ¼ : i

ðA1Þ

are called covariant basis vectors, where i ¼ 1; 2; 3, and e1 ¼ :2 < :3 =D

(and permutations)

ðA2Þ

are called contravariant basis vectors, where D ¼ :1 = :2 < :3

ðA3Þ

is the reciprocal of the Jacobian ( Bateman 1978). Keeping in mind that neither ei nor ei are unit vectors, any vector can be expressed in terms of covariant basis vectors a ¼ ai e i ; ai  a = e i ;

ðA4Þ

where ai are called the covariant components and the repeated index on the right-hand side of equation (A4) indicates summation. Similarly, a can be expressed as a ¼ ai ei , where ai  a = ei are the contravariant components. As an example, we consider the flux-coordinate system of x 3: ¼ z /(2a0 B¯ t0 ), # ¼ ,  ¼ z. For these coordinates, the covariant basis vectors are given in equation (4) of x 3, D ¼ Bz /a0 B¯ t0 , and the contravariant basis vectors are a0 B¯ t 0 rˆ ; rBz ˆ e# ¼ rf; Br e ¼ rˆ þ zˆ : Bz e ¼

ðA5Þ

It is straightforward to write B in terms of either covariant components and basis vectors, B¼

a0 B¯ t 0 Br B2 þ B2z  e þ rB e# þ r e; rBz Bz

ðA6Þ

B e# þ Bz e : r

ðA7Þ

a = B ¼ a i bi ¼ ai bi ;

ðA8Þ

or contravariant components and basis vectors, B¼

We close with a few useful identities:

ða < BÞ1 ¼ ða2 b3  a3 b2 ÞD

(and permutations);

:f ¼ @f =@i ei ;   : = a ¼ D@ ai =D =@i ;   (and permutations): ð: < aÞ1 ¼ @a3 =@2 þ @a2 =@3 D

ðA9Þ ðA10Þ ðA11Þ ðA12Þ

No. 2, 2005

DENSITY STRUCTURE OF A CORONAL FLUX ROPE

1053

Fig. 7.—Model a(z) profile (top) and contours of z (bottom) for a two-dimensional solar flux rope. Atmospheric temperature profile log ½T (K) is also shown.

APPENDIX B A TWO-DIMENSIONAL MODEL FLUX ROPE A simple two-dimensional model flux rope can be constructed as outlined in xx 2 and 3 (see Tables 1 and 2, eq. [1], and the discussion preceding eq. [9]) if an appropriate function a(z) is specified. Specifically, recall that we assume self-similar expansion of the flux rope cross section so that the radial field profile can be expressed as B½r/a(z). For example, let us consider Bz and B as specified in equations (15)–(16) and Figure 5. Recalling that conservation of flux z gives B¯ t ¼ B¯ t0 a20 /a2 , conservation of current within r  a gives Bp ¼ B¯ p0 a0 /a, we have  

z ¼ B¯ t 0 a02 1  exp  1 r 2 =a 2 =ð1  e 1 Þ

ðB1Þ

  B¯ t 0 a02  1 r da exp  1 r 2 =a 2 =ð1  e 1 Þ: 3 a dz

ðB2Þ

and, from equation (1), Br ¼

Returning to Tables 1 and 2, we now specify a function a(z) consistent with these values. The resulting two-dimensional model flux rope is shown in Figure 7 as a(z) (top) and z contours (bottom). To clarify the orientation of this model flux rope relative to the stratified solar atmosphere, a solar atmospheric temperature function log ½T (z), consistent with Tables 1 and 2, is also shown. This model flux rope, while far from exact, provides a useful estimate of the Br; f term in equation (10). Note that, with Bz and B specified as in Figure 5 and a(z) as specified in Figure 7, the field values given in Table 2 approximate peak model Bz and B values. Applying this simple formalism to the examples given in Figures 3, 5, and 6, the density profile from equation (10) is evaluated and plotted in each figure as a dashed line. The results justify the neglect of the Br; f term in equation (11). APPENDIX C MODEL FLUX ROPE EQUATIONS IN THE STATIC LIMIT As an alternative to the noninteracting limit addressed above, we describe the limit in which plasma fluid elements within a coronal flux rope are coupled via the pressure term. We note again that, if the plasma lying along any given coronal flux rope field line flows quickly enough, then the plasma conditions along that field line will be dictated by boundary conditions (the highly collisional plasma at the footpoints) rather than by collisional coupling with plasma on neighboring field lines. Further, it was shown in x 2 that ‘‘quickly enough’’ corresponds to vz > i (rc L)1/2 ’ 0:8 km s1. Thus, the limit in which plasma fluid elements within a coronal flux rope are coupled via the pressure term is essentially the static limit: (: < B) < B=4  :p  g ˆz ¼ 0:

ðC1Þ

1054

KRALL & CHEN

Vol. 628

Fig. 8.—(a) Transverse magnetic forces: (J < B/c)r (solid line), J Bz /c (short-dashed line), and Jz B /c (long-dashed line); this case corresponds to Fig. 4a, but with the radially outward force increased by 15%; (b) density for a static transverse equilibrium for case (a); (c) transverse magnetic forces corresponding to Fig. 4a, but with the radially outward force decreased by 15%; and (d ) density for case (c).

Employing the flux coordinates of Appendix A, we find :p ¼

@p @p  e þ e @ @

ðC2Þ

and   @B# # @B#   @B  @B # @B# B B e  B# e; (: < B) < B ¼ B e þ B @ @ @ @ @

ðC3Þ

where we have used @/@# ¼ 0 and B ¼ 0. Substituting equations (C2), (C3), and zˆ ¼ e into equation (C1) and considering the result component by component, we find   @p 1  @B  @B # @B# ¼ B B B ; ðC4Þ @ 4 @ @ @ 0 ¼ B @B# =@;

ðC5Þ

@p ¼ g ; @

ðC6Þ

and

where we have employed equation (C5) to obtain equation (C6). Note that equation (C5) is easily verified for a self-similarly expanding flux rope [B (r; z) ¼ Bp0 a0 /a(z)f (r/a)]. Equation (C6) implies that the spatially uniform (vs. r) plasma density that we have assumed at each footpoint carries over to a spatially uniform density in the corona unless transverse forces, equation (C4), are significant. One solution to these equations, corresponding to a (r) density profile that is everywhere flat, is the Lundquist solution ( Lundquist 1950). Forces associated with the Lundquist field are shown in Figure 4c. We can consider static cases in which the transverse forces in the corona are significant by specifying a field with mismatched forces and using equation (C4) to compute the resulting density profile, assuming a fully ionized plasma with T ¼ 200 eV, as indicated in Table 1, and a transverse dimension a ¼ 109 cm. If, for example, the radially outward force in Figure 4a were increased by 15%, then an inward pressure force, corresponding to reduced on-axis density, would be required to satisfy equation (C4). This is shown in Figure 8a, in which the mismatched magnetic forces (dashed curves) and net magnetic force (solid curve) are shown, and Figure 8b, in which the corresponding density profile is shown. Similarly, by specifying a magnetic field in which the net magnetic force is inward, Figure 8c, an increased on-axis density is obtained, Figure 8d. In fact, there is nothing in the analysis to indicate that either result (onaxis density depletion or on-axis peak) is preferred. We note here that the naive view, wherein the ‘‘magnetic pressure’’ of a flux rope magnetic field naturally leads to reduced plasma pressure in the region in which the magnitude of the flux rope field is largest, is not always correct. We conclude that the hollow density profile, which is inherent for a twisted flux rope field in the noninteracting limit, is not necessarily obtained in the collisionally coupled ‘‘static’’ limit.

No. 2, 2005

DENSITY STRUCTURE OF A CORONAL FLUX ROPE

1055

REFERENCES Aschwanden, M. J., & Acton, L. W. 2001, ApJ, 550, 475 Krall, J., Chen, J., & Santoro, R. 2000, ApJ, 539, 964 Bateman, G. 1978, MHD Instabilities (Cambridge: MIT Press) Low, B. C. 1996, Sol. Phys., 167, 217 Chen, J. 1996, J. Geophys. Res., 101, 27499 Lundquist, S. 1950, Ark. Fys., 2, 361 Chen, J., & Garren, D. A. 1993, Geophys. Res. Lett., 20, 2319 Plunkett, S. P., et al. 2000, Sol. Phys., 194, 371 Chen, J., & Krall, J. 2003, J. Geophys. Res., 108, doi:10.1029/2003JA009849 Roussev, I. I., Forbes, T. G., Gombosi, T. I., Sokolov, I. V., DeZeeuw, D. L., & Chen, J., et al. 1997, ApJ, 490, L191 Birn, J. 2003, ApJ, 588, L45 ———. 2000, ApJ, 533, 481 Sa´nchez Almeida, J., & Lites, B. W. 2000, ApJ, 532, 1215 Dere, K. P., Brueckner, G. E., Howard, R. A., & Michels, D. J. 1999, ApJ, 516, Trubnikov, B. A. 1965, Rev. Plasma Phys., 1, 105 465 Vernazza, J. E., Avrett, E. H., & Loeser, R. 1981, ApJS, 45, 635 Gibson, S. E., & Low, B. C. 1998, ApJ, 493, 460 Winebarger, A. R., Warren, H., van Ballegooijen, A., DeLuca, E. E., & Golub, L. ———. 2000, J. Geophys. Res., 105, 18187 2002, ApJ, 567, L89 Hansteen, V. H., Leer, E., & Holzer, T. E. 1997, ApJ, 482, 498 Wood, B. E., Karovska, M., Chen, J., Bruekner, G. E., Cook, J. W., & Howard, Holland, D. L., & Chen, J. 1993, Geophys. Res. Lett., 20, 1775 R. A. 1999, ApJ, 512, 484 Hundhausen, A. J. 1999, in The Many Faces of the Sun, ed. K. T. Strong et al. Wu, S. T., Guo, W. P., Michels, D. J., & Burlaga, L. F. 1999, J. Geophys. Res., ( New York: Springer), 143 104, 14789 Illing, R. M. E., & Hundhausen, A. J. 1986, J. Geophys. Res., 90, 10951 Zhang, G., & Burlaga, L. F. 1988, J. Geophys. Res., 93, 2511 Kleimann, J., & Hornig, G. 2001, Sol. Phys., 200, 47 Krall, J., Chen, J., Duffin, R. T., Howard, R. A., & Thompson, B. J. 2001, ApJ, 562, 1045