The Astrophysical Journal, 506:L143–L146, 1998 October 20 q 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.
CORONAL SCALE-HEIGHT ENHANCEMENT BY MAGNETOHYDRODYNAMIC WAVES C. Litwin and R. Rosner Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637;
[email protected] Received 1998 July 8; accepted 1998 August 19; published 1998 September 16
ABSTRACT We discuss the possibility that the transmission of Alfve´n waves from the photosphere into coronal loops can increase the density scale height in these loops. The mechanism involved is the ponderomotive force of transmitted waves, which opposes the force of gravity. We propose that this effect may account for cool coronal loops observed by the Solar and Heliospheric Observatory (SOHO). We also suggest that it may be an explanation of the emission measure enhancement found at the top of coronal X-ray loops observed by Yohkoh. Subject headings: MHD — waves — gravitation — Sun: corona — Sun: magnetic fields density ∼1010 cm23), the resonant frequency of the n 5 1 mode is near the peak of the power spectrum of photospheric fluctuations (cf. Parker 1994), on the order of the frequency of granular motions (∼0.02 rad s21). Inhomogeneous electromagnetic fields (such as those of standing MHD waves) exert a force on a dielectric medium (such as plasma). For an isotropic medium, with dielectric permeability e, in an inhomogeneous electric field E, this so-called ponderomotive force has the form (Landau & Lifshitz 1960)
Observations in the O v l629 line emission performed with the Coronal Diagnostic Spectrometer (CDS) on the Solar and Heliospheric Observatory (SOHO) show the existence of long, cool loops (viz., with temperature ∼240,000 K and height ∼50,000 km), at coronal altitudes significantly exceeding their density scale height and persisting over times much longer than the free-fall time. The present paper deals with a possible process that may lead to the appearance of such loops. A simple way of accounting for the lack of balance between the gravitational force and the thermal pressure in steady state is to postulate the presence of downflows, which in turn would require a constant resupply of plasma at coronal heights. The resupply could be the result of coronal condensations (Foukal 1978), but the occurrence of such condensations is uncertain. Force balance in the coronal loop can also be affected by the presence of magnetohydrodynamic (MHD) waves. Such an effect of electromagnetic waves has been the object of numerous investigations in the past, in contexts ranging from acceleration and confinement of laboratory plasmas (Motz & Watson 1967), magnetic field generation (Mora & Pellat 1979), low-frequency stability (Yasaka & Itatani 1984), to the solar wind (e.g., Hollweg 1973) and interstellar gasdynamics (McKee & Zweibel 1995), among others. In solar physics, sound wave pressure has been invoked in the context of planar models of the solar atmosphere (viz., McWhirter, Thoneman, & Wilson 1975; more recently, Woods, Holzer, & MacGregor 1990 and references therein). Such effects have also been considered in the context of dynamic loop structures, in which torsional wave packets were assumed to move material along a loop axis (e.g., Shibata & Uchida 1986). Magnetohydrodynamic waves in coronal loops can be either excited in situ (e.g., by reconnection processes in the corona) or transmitted from the photosphere. Since the Alfve´n velocity cA in the corona is much larger than the photospheric Alfve´n velocity cAph, magnetohydrodynamic waves excited in the photosphere tend to reflect in the transition region; significant energy transmission occurs only at resonant frequencies at which standing waves are excited in the coronal portion of the loop (Hollweg 1984). If the loop magnetic field is uniform, this occurs for Alfve´n waves with parallel wavenumber k res 5 np/L and frequency qres 5 k res cA, where n is an integer and L is the length of the loop (for the twisted field case, see Litwin & Rosner 1998). For the cool loops seen by CDS (see, e.g., Brekke, Kjeldseth-Moe, & Harrison 1997), with length L ∼ 1.5 # 10 10 cm, and for typical coronal Alfve´n wave velocities cA ∼ 10 8 cm s21 (e.g., for magnetic field ∼50 G and number
Fp 5
e21 =E 2. 8p
(1)
This expression is derived for static electric fields, in the absence of the background magnetic field that introduces anisotropy of the medium. In the past, the time-averaged, ponderomotive force due to electromagnetic oscillations in a magnetized medium has been calculated in a variety of approximations, including stress tensor (Pitaevskii 1961), guiding-center (Motz & Watson 1967), collisionless two-fluid (Klima 1968), and collisionless Vlasov (D’Ippolito & Myra 1985) theories. For the problem at hand, involving the ponderomotive force of Alfve´n waves in the magnetized collisional plasma of solar corona loops, the single-fluid MHD approximation is the most appropriate. The interaction between a magnetohydrodynamic wave and a weakly inhomogeneous, slowly evolving medium has been analyzed by Dewar (1970), who showed that the force exerted by Alfve´n waves takes the form of a pressure gradient. This work has been the basis of studies of magnetohydrodynamic wave effects on the solar atmosphere (e.g., Jacques 1977). However, the employed eikonal approximation makes it inapplicable to our problem because the density scale height in coronal loops is comparable to or shorter than the spatial scale of wave-associated fluctuations. It turns out, nevertheless, that the parallel component of the ponderomotive force, which determines the density scale height in coronal loops, can be derived under sufficiently general conditions. As we shall see, this force does not, in fact, have the form of a pressure gradient but rather one that is similar to equation (1). Thus we start our discussion with a simple derivation of the parallel ponderomotive force in the MHD approximation, for an arbitrary density profile; an arbitrary force-free magnetic field, relevant to the low-pressure coronal plasma; and an arbitrary spatial distribution of fluctuating fields. Subsequently, L143
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we shall exploit the derived result to investigate the effect of the aforementioned standing Alfve´n waves on the mass distribution in coronal loops, in particular, those observed by CDS and the Soft X-Ray Telescope (SXT) aboard the Yohkoh satellite. We start with the single-fluid momentum balance equation:
r
(
1 1 V · = V 5 J 3 B 2 =P 1 rg, t c
)
(2)
where V, J, r, P, and B are plasma velocity, current, density, pressure, and magnetic field, respectively, and g is the gravitational acceleration. We are interested in coronal loops that are stationary on a timescale that is long compared with the wave period. The density scale height in such loops is determined by the time-averaged force balance along the magnetic field. In order to determine it, we shall represent every dynamical quantity Q as a sum of the mean part Q 0 { AQS, where the brackets denote the time-averaging operation, and of the fluctuating part q˜ { Q 2 Q 0. In a stationary (V0 5 0 ) state, the time-averaged momentum balance equation has the form =P0 5
1 J0 3 B0 1 r 0 g 1 Fp , c
(3)
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i.e., J0 # B0 5 0, a condition that is well satisfied in coronal plasmas. Similarly, the fluctuating magnetic field is determined by the induction equation from which its time-averaged part has been subtracted. For transverse wavelengths comparable to the observed loop width (*108 cm), the frequency of photospheric motions is much higher than the rate of resistive diffusion, which can therefore be disregarded. Neglecting again resonant wave-wave coupling, the fluctuating magnetic field is thus found to obey the equation b˜ 5 = 3 v˜ 3 B0 . t
(7)
We shall now exploit equations (6) and (7) to find the parallel component of the ponderomotive force. First, we observe that, for a force-free magnetic field, equation (6) implies that v˜ · B0 5 0, so that the last term on the right-hand side of equation (4) does not contribute to the parallel ponderomotive force. Next, we shall compute the contribution of the time-averaged Lorentz force (the first term on the right-hand side of eq. [4]). With the aid of equation (6), it follows that 1 ˜ ˜ 1 ˜ 5 2r Ab˜ · v˜ S. A j 3 b · B0 S 5 2 A ˜j 3 B0 · bS 0 t c c
(8)
where Fp is the (time-averaged) ponderomotive force due to fluctuations: 1 ˜ 2 r Av˜ · =v˜ S 2 Ar ˜ tv˜ S. Fp 5 A ˜j 3 bS 0 c
(4)
1 ˜ ˜ ˜ A j 3 b · B0 S 5 r 0Av˜ · t bS. c
Thus, in the presence of MHD waves, the usual “hydrostatic” force balance is replaced by the time-averaged parallel force balance in the form ∇k P0 5 r 0 gk 1 Fpk,
(5)
where subscript k denotes the component parallel to the mean magnetic field B0 and ∇k { (B0 /B 0 ) · =. Subtracting equation (3) from equation (2), one obtains an equation governing the oscillating velocity. If, as is customary, resonant wave-wave coupling is neglected (which is justified for a peaked excitation spectrum of the kind that is expected in coronal loops; see earlier discussion), the oscillating velocity is governed by the momentum balance equation in the form r0
v˜ 1 ˜ 1 ˜ 5 j 3 B0 1 J0 3 b. t c c
Since for arbitrary periodic functions f and g, with equal periods, A ft gS 5 2Agt f S, equation (8) now implies that (9)
Eliminating t b˜ with the aid of the induction equation (7) and taking into account that v˜ · B0 5 0, equation (9) leads, after simple algebra, to 1 ˜ ˜ r A j 3 b · B0 S 5 0 B0 · =Av˜ 2 S 1 r 0Av˜ · =v˜ S · B0 . c 2
(10)
Inserting equation (10) into the parallel component of the ponderomotive force (eq. [4]), one thus finally obtains Fpk 5
1 r ∇ Av˜ 2 S. 2 0 k
(11)
(6)
Contributions of pressure and gravitational forces are neglected here as small compared with the magnetic forces, as is the case for MHD waves in low-b ({8pP0 /B 02) plasmas. Also omitted is the term due to the cubic nonlinearity, which is small for sufficiently small amplitudes and which, moreover, vanishes exactly for axisymmetric incompressible modes, discussed later. We retain, however, nonlinearity on the left-hand side of equation (6), implicit through r 0, which depends on the fluctuation amplitude through equation (5). For the sake of simplicity, we assume that the mean magnetic field is force-free,
Note that the above result is derived for an arbitrary, albeit force-free, mean magnetic field. If the latter is uniform, equation (11) can be transformed into a form analogous to equation ˜ # B /B 2 (where (1). From Ohm’s law it follows that v˜ 5 cE 0 0 ˜E is the oscillating electric field), which implies Fpk 5
e⊥ 2 1 ˜ 2 S, ∇kAE 8p
(12)
where e⊥ 5 1 1 4pr 0 c 2/B 02 is the transverse dielectric constant for low-frequency waves (see, e.g., Stix 1992). Thus the parallel momentum balance in steady state (eq. [5])
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CORONAL SCALE-HEIGHT ENHANCEMENT
can be expressed in the form n 0∇k(Fg 1 Fp ) 5 2∇k P0 ,
(13)
where Fg is the gravitational potential, Fp 5 2mu 2/2 is the ponderomotive potential, u 2 { Av˜ 2 S, n 0 is the mean particle number density, and m 5 r 0 /n 0 is the ion mass. If the equation of state is given in the form n 0 5 n 0 (P0 ), equation (13) can be integrated to yield the spatial mass distribution for an arbitrary oscillating field profile. In particular, if the plasma temperature T is constant along the magnetic field, integrating the above equation leads to
(
n 0 5 n 0 (0) exp 2
)
F , 2k B T
(14)
where the effective potential F { Fg 1 Fp, n 0 (0) is the integration constant, and kB is the Boltzmann constant. Thus if the ponderomotive potential decreases with height, this will counter the effect of the increasing gravitational potential; consequently, the density scale height will be increased. The magnitude of this increase depends on the wave amplitude. The above result is quite general and does not depend on the characteristics of any particular type of fluctuations (as long as b K 1). In the following we shall focus specifically on the effect of Alfve´n waves transmitted from the photosphere. As mentioned earlier, the wave field in the coronal portion of the loop must be approximately a standing wave for the wave transmission coefficient to be on the order of unity. Let us consider a simplified model (Hollweg 1984) in which the coronal portion of the loop is approximated by a cylinder with uniform mean plasma density and magnetic field; the excited modes are assumed to be axisymmetric and incompressible, so that only the azimuthal components of fluctuating velocity and magnetic field are nonvanishing. Expressing the oscillating field as the superposition of standing waves (with different resonant frequencies) and averaging over time leads to
O `
u 2 (s) ≈
un2 sin2 np
n51
s , L
(15)
where un are constant excitation amplitudes and s is the distance along the loop. If the coronal excitation is caused by photospheric motions and the latter have a power spectrum peaked in the vicinity of the frequency of granular motions (Parker 1994), higher n modes will have smaller amplitudes than the n 5 1 fundamental mode. Let us therefore consider only the effect of the n 5 1 eigenmode. Assuming that the loop is a semicircle, of radius R 5 L/p, the wave field variation with height z 5 R sin ps/L above the photosphere is in this case given by u(z) 5 u1
z R
(16)
(0 ≤ z ≤ R). For such a standing wave excitation, the effective potential has the form 2
()
1 z F 5 mgz 2 mu12 2 R
(17)
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and has the maximum value Fmax 5 mg 2R 2/2u12 at z 5 gR 2/u12 (which requires that u12 1 gR). Thus the effective scale height can be significantly increased if, say, Fmax ! k B T. This occurs if u rms 1 mvt ,
(18)
where u rms 5 [∫ 0 d(s/L)u 2 (s)]1/2 is the rms average of u along the loop (for the fundamental excitation considered above, u rms 5 u1 /Î2), m { mgR/2k B T is the ratio of the loop height and the density scale height (in the absence of waves), and vt 5 Îk B T/m is the ion thermal velocity. For the inequality (18) to be consistent with the condition u12 1 gR (see above), m 1 1 is required, which is satisfied, in particular, for the aforementioned cool coronal loops observed by CDS. The above estimate of the required wave amplitude was derived under the assumption that the plasma density was uniform in the loop. Actually, since F varies along the loop, so does the plasma density, as indicated by equation (14). For an excitation with frequency q, the variation of the oscillation amplitude along the loop is governed by the equation (assuming the mean magnetic field is uniform) L
d 2u 1 k 2 (s)u 5 0, ds 2
(19)
where k 2 (s) 5 q 2/cA2 (s) and cA(s) 5 B 0 /Î4pr 0 (s) is the local Alfve´n velocity; it follows from equation (14) that k 2 (s) 5 k 2 (0) exp (2F/T ). Introducing dimensionless variables y 5 u/2 vt, x 5 s/L, and l { k(0)L, equation (19) becomes d2y 1 l2 y exp (y 2 2 m sin px) 5 0. dx 2
(20)
We investigate numerically the solutions of equation (20), subject to the boundary conditions y(0) 5 y(1) 5 0 for various values of l and m. For fixed l and m, solutions are found for discrete values of y 0(0) ( 0, corresponding to fundamental and harmonic excitations. In the parameter range characteristic of cool CDS loops, mentioned earlier [cA(0) ∼ 10 8 cm s21, L ∼ 1.5 # 10 10 cm, T ∼ 2.4 # 10 5 K], and for the oscillation frequency q ∼ 0.02 rad s21 near the peak of the power spectrum, m ∼ 3.4 and l ∼ p. In Figure 1 we compare the numerically obtained fundamental solution y(x) for m 5 3.4 and l 5 2, 3.14, and 4.5 with y( 12 ) sin px: we see that the solutions of equation (20) are similar to, albeit somewhat more peaked than, the solution obtained in the simplified model (cf. eq. [16]). Unlike the simplified model, however, the amplitude is now no longer arbitrary but has a specific value (depending on l and m); in particular, for l 5 3.14, u rms ≈ 2.65 vt is somewhat smaller than the lower bound mvt indicated by the inequality (18). In Figure 2 we show density profiles for various values of m and l. We see that for cool loops (m 5 3.4) and l 5 3.14, the plasma density along the loop oscillates about its value at the footpoints; for a higher value (l 5 4.5), it is approximately constant over most of the loop and is lower than the footpoint density; for a smaller l (52), it peaks at the center at a value higher than the footpoint density. For measured temperatures of the CDS cool loops, the above implies that the mean value of the velocity fluctuations in the loop u rms ∼ 120 km s21 for l ∼ p. This value is significantly higher than the 10–25 km s21 maximum nonthermal velocity
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Fig. 1.—Comparison of sin px (solid line) with the fundamental solution of eq. (20) [normalized to y( 12 )] for m 5 3.4 and l 5 2 (long-dashed line), l 5 3.14 (medium-dashed line), and l 5 4.5 (short-dashed line).
found by Cheng, Doschek, & Feldman (1979). It should be noted, however, that the observations of Cheng et al. concerned motions in the lower corona or top of the transition region (primarily between 00 and 200 above the solar limb), while urms is the mean amplitude of wave motion, primarily in the corona proper. As seen from equation (16), the amplitude of wave motion at lower heights is lower; if one assumed that the distance from the limb can be identified with the distance from the loop footpoints, the required mean velocity amplitude in the region observed by Cheng et al. would be ∼30 km s21, much closer to the found upper bounds. Furthermore, if the filling factor of high-velocity regions observed by Cheng et al. is significantly smaller than unity, this would additionally decrease the discrepancy. Doppler shifts corresponding to higher velocities (∼50–60 km s21) have been measured by Brekke et al. (1997) in the aforementioned cool loops. As Brekke et al. pointed out, the true velocity is likely higher since Doppler shifts indicate only the velocity component along the line of sight and not its magnitude. More recently, Kjeldseth-Moe & Brekke (1998) have reported measurements of typical velocities of 50–100 km s21 and as high as 200–300 km s21. A similar analysis can be applied to the problem of emission measure (EM) enhancement at coronal looptops, reported by Kano & Tsuneta (1996) for loops imaged by SXT. Kano & Tsuneta rejected, for the lack of a physical mechanism, the natural explanation of EM enhancement as being caused by pressure imbalance, attributing it instead to an increased filling
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Fig. 2.—Comparison of mean density profile for cool loops (with m 5 3.4) in the absence of waves (solid line) and for the fundamental solution of eq. (20) with l 5 2 (long-dashed line), l 5 3.14 (medium-dashed line), and l 5 4.5 (short-dashed line); also indicated is density profile for hot loops, with m 5 0.1, for l 5 2.1 (dotted line).
factor. The ponderomotive force of Alfve´n waves, discussed in this paper, provides a mechanism that can explain pressure imbalance along a loop. In fact, as shown in Figure 2, we can obtain a factor of 3 pressure imbalance between looptop and footpoints (implied by the observed EM enhancement if the filling factor is constant) for physical parameters consistent with those found by Kano & Tsuneta [for m 5 0.1 and l 5 2.1, corresponding to, e.g., B 0 5 100 G, T 5 6 MK, n 0 (0) 5 5 # 10 9 cm23, L 5 10 10 cm, and q 5 0.068 rad s21] and for u rms ≈ 300 km s21. This paper has been in part inspired by the recently held Second SOHO-Yohkoh workshop, during which observations of active regions from different instruments were discussed. We benefited greatly from the interaction with participants of this workshop, in particular with Spiro Antiochos, Pa˚l Brekke, James Klimchuk, Piet Martens, and Giovanni Peres. Special thanks are due to Hugh Hudson and Saku Tsuneta for numerous illuminating discussions of Yohkoh observations. This research has been supported by the National Science Foundation grant ATM-9422213 and NASA grants to the University of Chicago under the aegis of the Space Physics Theory Program and the SOHO/UVCS program. C. L. acknowledges support from the European Space Agency, which made his participation in the workshop possible.
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