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Joule heating of plasma in current-carrying coronal magnetic loops
V.V. Zaitsev1 , A.A. Stepanov
1 2 2

Institute of Applied Physics, Nizhny Novgorod, Russia Central Astronomical Observatory, Pulkovo, St.Peterburg, Russia

Introduction
Already rst X-ray observations made on "Skylab" in 1969 Vaiana, et al., 1970 have shown that solar corona consists of a lots of magnetic eld loops, lled with a hot plasma. The average temperature of plasma in these loops is 1:3 1:7 106 K, whereas plasma density inside of them at least 2 , 4 times exceeds the average coronal plasma density Golub, et al., 1974. Observations of X-ray loops indicate that their heating happens on the time intervals about the time of radiative energy losses. More rapid temperature increase in the loop however is also possible. Nolte, et al. 1979 suppose that the observed behavior of the temperature dynamics can be explained by the presence of two heating components: slowly changing or constant component, and an impulsive one, causing rapid increase of the brightness of X-ray loops. Rosner et al., 1978 assumed the presence inside a coronal magnetic loop of a heating source of inde nite nature and considered a steady-state heating of such a magnetic loop. In this work a steady-state heating of bases of current-carrying magnetic loops is considered. As a source of heating we assume the Joule energy dissipation of currents, which are generated by a convective ow of partially ionized photospheric plasma in the loop's foot-points Zaitsev and Khodachenko, 1997, Zaitsev et al., 1998. Belowwe'll show, rstly, that coronal magnetic loops with currents should be hot this follows from the equilibrium conditions, and, secondary, we'll nd a relation between a maximum temperature in bases of a loop and a velocity of photospheric convection.

Equilibrium of a magnetic current-carrying loop
Before to analyze an energy dissipation of currents, running in a magnetic loop formed bya converging convective ow of a photospheric plasma in joining points of few supergranulas, let us consider a balance of kinetic and magnetic pressures in bases of the loop. From the condition of balance of vertical magnetic tube in a dynamo region, where a radial component of a plasma convection velocity Vr 6= 0, and a current density j is perpendicular to a magnetic eld B

@P = 1 j B , j B = , Vr B 2 @r c ' z z ' c2 1+ B

2

1

F2 e2 n where = m + Coulomb conductivity, = 2 , F c2nm , n density of e ei ea i ia electrons, F relative density of neutrals, ei and ea are respectively e ective frequencies of electron-ion and electron-atom collisions, it follows that kinetic pressure in a magnetic tube increases towards its periphery. On a boundary of such a magnetic tube r = r0


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the pressure is of the order of magnitude of a magnetic pressure in the central part of the tube:

B2 P r0 , P 0 = 81 , b2 ln z 0+ b2 =b2 j ; j 1
!

2

B'r0 2 , B 0 is a vertical component of a magnetic eld on the axis of the where b = B r z z0 tube. If to take that convective motion of plasma takes place up to heights h = 500 km above the photospheric level 5000 = 1, then the standard atmosphere pressure on these heights appears to be about 1:35 103 dyne cm,2 Vernazza, et al., 1981. Therefore, a magnetic tube with the standard atmosphere pressure is in equilibrium with a convective ow only if the value of a magnetic eld in the tube is less then approximately 102 G. For more intensive magnetic tubes Bz 0 = 103 G and the twisting ratio b2 = 1 the balance with a convective ow could be reached only in the case when gas pressure in the tube is P r0 1:3 105 dyne cm,2. For usual photospheric temperatures such values of pressure are realized only in regions, situated about 25 km above the photosphere and deeper h ,25 km. Thus, for the balance to take place in a magnetic tube plasma in the tube should be hot. As far as the existence in solar corona of intensive magnetic loops with the elds of a few kilogauss is of no doubt, then it seems that photospheric bases of such loops should be lled with a hot plasma if one assumes that generation of these loops is caused by converging ows of photospheric plasma.
2

Joule dissipation of currents energy
What are the mechanisms, which cause the heating of magnetic loops in the photospheric dynamo region? One of the possible mechanisms can be Joule heating, taking place during energy dissipation of currents, generated in the magnetic tube by a convection ow. As it was already noted, the current density j is perpendicular to the self-consistent magnetic eld B in this case, and the Joule heating rate is determined by Cowling conductivity. This appears to be quite an e ective heating mechanism. The steady-state Joule energy release in the conditions of a sunspot with an Evershed's ow was studied rst by Sen and White 1972 in the approximation of a purely vertical magnetic eld B' = 0. For a magnetic tube with a convective plasma ow a value of the Joule energy release ratio depends on both components of a magnetic eld B' 6= 0; Bz 6= 0. The energy, qJ = E + 1 V Bj, released in 1s in a unit volume in the case of E = fEr ; 0; 0g, c V = fVr ; 0;Vz g and j = f0;j';jz g is
2 22 2 2 qJ = j ; j 2 = jz2 + j' = cVr 1 +B B 22 ; 2 C 22 B qJ = 1 +Vr B 2c2 :

C

or

= 1+ B 2 ;

3 4

It is interesting to note here that the equation 1 can be considered now in a sense of a heat-transfer equation, which describes a pressure increase due to the Joule heating in direction from the tube's center towards its periphery:


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22 B ,Vr @P = qJ = 1 +Vr B 2 c2 : @r

67 5

The expression for the rate of Joule energy release becomes to be the most simple in a case of intensive magnetic tubes, when B 2 1
2 2 qJ Vr2 = nmi ia VrFr2 , F : 2 c

6

As it follows from 6, if Vr increases with the r increase, then the energy release also increases in this direction. In this case one can expect that the surface of the magnetic tube will be hotter than its inner regions.

Maximum temperature
Let's consider now the temperature evolution with height in a photospheric part of a magnetic tube. By this, we'll suppose that plasma pressure is constant and is determined by a value of a magnetic eld in the tube. In this case the temperature evolution is determined by a balance between the Joule heating and energy losses due to the thermoconductivity, as well as optical radiation of plasma. Because of the rapid temperature increase plasma in the tube can be assumed to be completely ionized, and the following state equation P = 2nkB T can be used. Here P; n and T are the averaged over the tube's cross-section values of pressure, density and temperature. For the case of intensive magnetic loops B 2 1 the average over the tube cross-section Joule energy release rate can be written as qJ = n2 erg cm,3 s,1 , where = 8:3 10,33 V02. The radiative energy losses function in the temperature interval 5 logT 7 we approximate as the following Golub and Pasacho , 1998: qJ = n2 T ,1erg cm,3 s,1 , where =10,16:22. The thermo-conductivity heat ux along a magnetic eld is Fc = ,k0T 5=2 z dT , where dz 2 k0 =9:2 10,7 cgs units. For simplicity's sake assume a small twisting of the tube B' Bz2 . In this case the magnetic eld of the tube is approximately parallel to the z, axis. Within the frame of the assumptions made above let's consider the thermo-equilibrium equation

d k T 5=2 dT = P 2 , P 2 2 2 dz 0 dz 4kB T 3 4kB T

2

7

We take the z =0 level on some height under the photosphere where magnetic pressure of the tube begins to exceed a value of an unperturbed atmosphere's gas pressure, and the tube starts to be heated. On this zero-level there should be determined the values of temperature and its derivative:

T z =0 = T0;

dT dz

z

=0

= dT dz


!

0

:

8


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Let's assume that the temperature increases with height upward from the point z = 0 and reaches its maximum value on the level z = z1 , i.e.

T z = z1 = T1 ;

dT j dz

z=z

1

=0 :

9

After multiplying 7 on T 5=2 dT and integration over the coordinate, starting from z1 one dz obtains 1k T 20


5

dT dz

!

2

P 2 2 pT , T , 2 T =2 1 4kB 3
q

3=2

, T13=2 :

10

As far as T1 T0 , and the heat ux k0T05=2 dT =dzz=z0 is small in comparison with 2 2 2 P 2T11=2=4kB and 2P 2T13=2 =12kB , then from 10 one obtains an equation for the maximal temperature ,2 T11=2 + 2 T13=2 =0, which gives avalue of T1 : 3 3 2 1016 : T1 = V02 11

For the values of a horizontal component of a convection velocity V0 = 0:3 1km s,1 11 gives T1 2 106 107K. Let's determine the height on which the temperature reaches the obtained maximal values. After exclusion of from 10 with taking account of 11 and taking a square root the equation 10 transforms to

P2 T dT = k2 k dz B0
9=4


!

1=2 

T 1, T 1



1=2

:

12

Integration of 12 over z from 0 till z1, for T0=T1 1 yields the relation between z1 the height of the maximal temperature of the photospheric bases of a magnetic tube, pressure P and avalue of the maximal temperature T1 :
2 z1 P = kB k
Z

0

!1

=

2

IT

13=4

;

13

where I = t9=4 1 , t,1=2 dt 0 assume that the pressure in a a pressure of a magnetic eld, formula 11 into account, then B3 = Bz 0=103 G, and v0 = eld and horizontal convection

1

is dimensionless value of the order of a unit. If one will photospheric base of a magnetic loop is determined by Bz2 i.e. P = 161 , b2 ln j 1 + b2 =b2 j see 2, and take ,, from 13 one will obtain z1 0:8105B3 2 v0 13=2 cm, where V0 =105 cm s,1 are dimensionless values of a magnetic velocity. For Bz 0 = 103 G and V0 0:3km s,1 one


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can obtain z1 103 km. Thus the temperature reaches a maximal value on a scale of the order of the scale of a dynamo region in the loop's foot-point. This scale decreases, i.e. the temperature increases faster, if the convection velocity and magnetic eld increase. And vice versa for small velocities for example V0 0:1km s,1 the maximal temperature is not reached at all, since the scale z1 becomes to be greater then the dynamo region's scale in the photospheric bases of the loop. There is no convection in a chromospheric part of a magnetic loop Vr = 0 and Joule heating is une ective here, since the plasma conductivity increases signi cantly in comparison with the Cowling conductivity of photospheric partially ionized plasma. Therefore the temperature of plasma in the bases of a magnetic loop decreases rather quickly with height increase. This happens because of the energy losses due to the thermo-conductivity and radiation. A characteristic scale on which the temperature decreases signi cantly is also of the order of magnitude of the scale z1 , i.e. about 103 km. If however an additional heating source there exist in corona this heating source could be caused for example by dissipation of Alfven or acoustic waves, Joule dissipation of coronal currents then the plasma temperature in the tube can continue to grow with height, reaching its maximal value in the top of the loop. In this case for su ciently low magnetic loops with a height being less then the gravitational height scale H =5 103T cm the following relation between a maximal temperature in the top of a loop Tmax , its length l and plasma pressure P inside is ful lled Rosner et al., 1978; Serio et al., 1981:

T

max

1:4 103Pl1=3 K

14

The heating source qH of an uncertain nature should satisfy the following relation Rosner et al., 1978:

qH 9:8 104P 7=6 l

,5=6

erg cm,3 s,1

15

which means that the pressure, in uencing the radiative energy losses ratio, has a positive correlation with the heating rate.

Conclusion
We have shown above that current-carrying magnetic loops with a potential magnetic eld of about 103 G and convective ows of photospheric plasma should have hot bases. The source of their heating is connected with a Joule dissipation of energy of current in the photospheric foot-points. The high rate of the energy release is caused by Cowling conductivity. In particular, for horizontal component of a velocity of convection ow 0:3 1km s,1 the maximal temperature in the bases of the loop can reach values of 2 106 107 K. Such loops can be associated with bright X-ray points, which are observed by "Skylab" and other space observatories.
This research was supp orted by Russian Foundation of Basic Researches grant 99-02-18244.


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References
Vaiana G.S., Krieger A.S., Van Speybroeck L.P., Zehnpfennig T., 1970, Bull. Am. Phys. Soc., 15, 611 Golub L., Krieger A.S., Silk J.K., Timothy A.F., Vaiana G.S., 1974, Astrophys. J. Lett., 189, L93 Nolte J.T., Solodyna C.V., Gerasimenko M., 1979, Solar Phys., 63, 113 Rosner R., Tucker W.H., Vaiana G.S., 1978, Astrophys. J., 220, 643 Zaitsev V.V., Khodachenko M.L., 1997, Radiopys. and Quantum Electron., 40, 114 Zaitsev V.V., Stepanov A.V., Urpo S., Pohjalainen S., 1998, Astron. and Astrophys., 337, 887 Sen H.K., White M.L., 1972, Solar Phys., 23, 146 Golub L., Pasacho J.M., 1998, The Solar Corona, Cambridge Univ. Press, p.95 Vernazza J.E., Avrett E.H., Loeser R., 1981, Astrophys. J. Suppl., 45, 635 Serio S., Peres G., Vaiana G.S., Golub L., Rosner R., 1981, Astrophys.J., 243, 288