ISSN 0016-7932, Geomagnetism and Aeronomy, 2017, Vol. 57, No. 7, pp. 841–843. © Pleiades Publishing, Ltd., 2017.

On the Constancy of the Width of Coronal Magnetic Loops V. V. Zaitsev* and P. V. Kronshtadtov Federal Research Center Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, 603950 Russia *e-mail: [email protected] Received April 11, 2017

Abstract—The variation of the width of coronal current-carrying magnetic loops with height is considered. Two invariants are taken into consideration: conservation of the longitudinal magnetic field flow through the tube cross-section, and conservation of the electric current through the cross-section. It is shown that, if gas kinetic pressure inside the tube is low in comparison with the longitudinal magnetic field pressure, the tube width does not vary with height in the corona, which is typical for most magnetic loops observed in the corona. Otherwise, when the gas kinetic pressure exceeds the longitudinal magnetic field pressure, the tube width increases within the double scale of height of the inhomogeneous atmosphere. The increase in width with height is typical for certain postflare loops that feature higher parameters of temperature and plasma concentration. DOI: 10.1134/S001679321707026X

1. INTRODUCTION A quite complete statistical study of “warm” magnetic loops observed by TRACE satellite (171, 195 and 284 Å) is presented in the literature (Aschwanden and Nightingale, 2005). These loops are thermally homogeneous, with an average temperature of T = 10 6 K and concentration n ≈ 10 9 cm −3, which considerably exceeds the plasma concentration in the ambient corona. This follows from the good visibility of loops in ultraviolet radiation against the background of the ambient corona. The loop overall length averages l = 2L ≈ 6.5 × 10 9 cm, where L is the length of the loop from base to vertex. A distinctive feature of these loops is their width, which varies little with height and averages w ≈ 1.4 × 10 8 cm. This is quite a remarkable property, since the loop height is comparable with the inhomogeneous corona size H ≈ 3 × 109 cm or exceeds it, i.e. the coronal plasma pressure varies considerably over the scale of the loop. A recent study aimed to discover the lower limit of magnetic loop width based on data from the AIA/Solar Dynamics Observatory (Aschwanden and Hardi, 2017). The discovered lower limit of width was wmin ≈ 10 7 cm, and the most probable average w ≈ 3 × 10 7 cm appeared to be of the order of a granule size. An example of a magnetic loop with an approximately twofold increase in width over the distance from the loop base to its apex is also considered in the paper. Similar arch structures with widths increasing with height were observed in hot flare loops by the Yohkoh satellite (Kano and Tsuneta,

1995). In Fig. 1, examples of “warm” magnetic loops with approximately constant widths are presented, and Fig. 2 displays an example of a “hot” loop with the width increasing from its base to vertex. In this paper, the conditions at which the coronal magnetic loop width does not vary with height have been found, and the conditions at which the loop width is not constant have been formulated. 2. MAGNETIC TUBE MODEL We consider for simplicity a vertical magnetic tube, which can be used to roughly approximate the “legs” of coronal magnetic loops, though the results obtained below are easily expanded to the case in which the axial magnetic field of a loop makes a nonzero angle with the vertical line. We designate by z the coordinate axis set along the magnetic tube axis, for which we introduce a local cylindrical coordinate system ( r, ϕ, z) and assume axial symmetry, i.e., we consider all values independent of coordinate ϕ. Then, from the equilibrium equation    (1) −∇ p + 1 j × B + ρ g = 0, c we obtain the barometric law of pressure variation with height inside the tube

p = p0 exp(−z H ), H = k BT mi g .

(2)

Here p0 is the pressure at the tube base, g is the acceleration of gravity, Т is the temperature, kB is the Boltzmann constant. Moreover, in Eq. (1), j = j z is the electric current flowing along the tube axis, and, for

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does not result in electrical charge accumulation in the tube. Consequently, the longitudinal electric current flow through the tube cross-section is constant: a( z )

∫ 2πrj (r, z)dr = I z

= const.

z

(4)

0

Proceeding further, we obtain from Eq. (1) the condition of tube equilibrium along the radial coordinate

B2 dp(r, z) 1 d (5) + Bϕ2 + B z2 + 1 ϕ = 0. dr 8π dr 4π r Hereinafter, we neglect magnetic field B z at r > a(z), since that magnetic field decreases outside the tubes generated by photospheric convection as B z (r = a)(a r ) −RM (Zaitsev and Khodachenko, 1997), where RM @ 1 is the magnetic Reynolds number. We also neglect the external gas kinetic pressure, since the plasma density inside the tubes considerably exceeds the density of the ambient corona. These approximations mean that the Ampere force induced by current Iz flowing in the magnetic tube should balance the plasma pressure gradient and magnetic field B z inside the tube.

(

Fig. 1. Example of coronal magnetic loops with constant width observed by the TRACE 2005 satellite, 1142 UT, September 8 (http:/trace.lmsal.com/POD/TRACEpodarchive24.html).

)

SXT: 0310–0320 UT February 21, 1992

3. DEPENDENCE OF THE TUBE WIDTH ON HEIGHT Let us rewrite Eq. (5) in the form of 2 ⎛ dp dB z2 ⎞ 1 d(rBϕ ) (6) r2 ⎜ + 1 + =0 ⎟ ⎝ dr 8π dr ⎠ 8π dr and integrate it over r from 0 to a(z) + 0 with regards to the assumptions p(a + 0) = 0, B z (a + 0) = 0. As a result, we obtain the generalized Bennett’s criterion as the condition of equilibrium of the current-carrying plasma magnetic tube:

a( z )

∫ 0

(7)

which takes into consideration that, at an even distribution of the current density over the cross-section, the total current is

Fig. 2. Example of a hot magnetic loop with increasing width observed by the Yohkoh satellite, 0310–0320 UT, February 21, 1992 (solar. physics. Montana.edu/nuggets/ 2001/010914/010914.html).

simplicity, we assume even distribution of the current over the tube cross-section of radius а(z). We designate the magnetic field components inside the tube as   B{Bϕ, B z }. The condition div B = 0 then implies conservation of the longitudinal magnetic field flow through the tube cross-section:

B z (z)a 2(z) = B z (0)a 2(0) = Φ z ,

⎛ B z2 ⎞ I z2 ⎜ p + ⎟2πrdr = 2 , 8π ⎠ 2c ⎝

(3)

where a(z) is the tube radius at height z . In addition, we assume that the current through the cross-section

caBϕ(a) (8) . 2 Condition (7) is independent of the distributions of pressure p and field B z along coordinate r ; therefore, assuming these distributions to be homogeneous over the tube cross-section and with allowance for invariants (3) and (4), we obtain the following equation for radius a(z): Iz =

( )

Φ2 I2 2 π a (z) p0 exp − z + 2 z = z2 . H 8a (z ) 2c

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ON THE CONSTANCY OF THE WIDTH OF CORONAL MAGNETIC LOOPS

Let us consider two extreme cases following from Eq. (9). (1) Plasma pressure in the magnetic tube is low. If the following condition is satisfied 2

B z (0) (10) , 8π which corresponds to small values of plasma beta, then from (9) we obtain a 4 ( z ) p0e

−z H

! a 4 (0)

Φ zc (11) . 2I z In this case, conservation of the longitudinal magnetic field flow and the electric current through the tube cross section implies constancy of the width of the magnetic tube. (2) Plasma pressure in the magnetic tube is high. If the condition inverse to inequality (10) is satisfied a(z) = const =

B 2(0) (12) @ a (0) z , a ( z ) p0e 8π the longitudinal magnetic field does not play a significant role in the balance of forces ensuring the tube equilibrium state. In this case, the tube width increases with height, 4

−z H

4

( )

Iz (13) exp z ; 2H c 2πp(0) however, the typical scale of tube width variation is twice the barometric scale of gas pressure variation, i.e., the tube widens more slowly than the gas pressure varies. a(z) =

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corona, which is typical for most “warm” magnetic loops observed in the corona. In this case, Eq. (11) makes it possible to evaluate the electric current in the magnetic loops. Assuming the magnetic field at the loop base to be B z (0) ≈ (1.2 − 1.3) × 10 3 G (Katsukawa and Tsuneta, 2005)) and the radius of photospheric bases of current-carrying loops to be a(0) = 5 × 10 6 cm (Sharykin and Kosovichev, 2015)), we obtain from (11) electric current I z ≈ 1.5 × 10 9 A. Otherwise, when gas kinetic pressure exceeds the longitudinal magnetic field pressure, the tube width increases with height within the double scale of the height of the inhomogeneous atmosphere. In this case, as follows from (13), the equilibrium is maintained in the widening tube at lower values of electric current I z ≈ 2 × 10 8 A for the most probable average radius of the “warm” loop a ≈ 3 × 10 7 cm. Arch structures with widths that increase with height were sometimes observed by the Yohkoh satellite in hot X-ray loops (Kano and Tsuneta, 1995). ACKNOWLEDGMENTS This work was supported by the Russian Scientific Foundation, project nos. 16-12-10528 (section 2) and 16-12-10448 (section 3), and the Russian Foundation for Basic Research, project no. 17-02-00091 and contract 14Z5D.31.0007 (sections 1 and 4). REFERENCES

4. DISCUSSIONS The remarkable constancy of the width of “warm” quasi-stationary magnetic loops observed by the TRACE satellite (Aschwanden and Nightingale, 2005) indicates the presence of electric current in them, which leads to the emergence of the Ampere force and related transverse “contraction” of the magnetic tube (pinch effect). We consider the loop width variation for the case of current-carrying magnetic loops and plasma pressure variation with height according to the barometric law. In this paper, the important role of the conservation of axial magnetic field flow Φ z and longitudinal electric current intensity I z through the magnetic loop cross-section (see Eqs. (3) and (4)) is considered. These two invariants, together with Bennett’s generalized criterion (7), provide Eq. (9) for radius a(z) variation with height. We have shown that, if the gas kinetic pressure inside the tube is low in comparison with the longitudinal magnetic field pressure, the tube width does not vary with its height in the

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Aschwanden, M.J. and Hardi, P., The width distribution of loops and strands in the solar corona—are we hitting rock bottom?, Jan 4, 2017. https://arxiv.org/abs/1701.01177v1. Aschwanden, M.J. and Nightingale, R.W., Elementary loop structures in the solar corona analyzed from trace triplefilter images, Astrophys. J., 2005, vol. 633, pp. 499–517. Kano, R. and Tsuneta, S., Scaling law of solar coronal loops obtained with YOHKOH, Astrophys. J., 1995, vol. 454, pp. 934–944. Katsukawa, J. and Tsuneta, S., Magnetic properties at footpoints of hot and cool loops, Astrophys. J., 2005, vol. 621, pp. 498–511. Sharykin, I.N. and Kosovichev, A.G., Dynamics of electric currents, magnetic field topology, and helioseismic response of solar flare, Astrophys. J., 2015, vol. 808, no. 1, id 72. Zaitsev, V.V. and Khodachenko, M.L., Energy release in corona magnetic loops, Radiophys. Quantum Electron., 1997, vol. 40, nos. 1–2, pp. 114–138.

Translated by N. Semenova

2017