4. Current-carrying magnetic flux tubes in the photosphere
The formation of intensive axially symmetric flux tubes with (), j () in the steady-state situation in the case of axially symmetric ), converging, , flux of partially ionized photospheric plasma, supposing the tubes to be vertical inside the convective zone, was considered by Zaitsev (1996, 1997), and Zaitsev & Khodachenko (1997).
Supposing , , , where , , and are Alfvén, sound and free-fall velocities, respectively we obtain from Eqs. (2) and (3) the following expressions for the magnetic field components of a steady-state flux tube (Zaitsev 1996):
Let us suppose the following approximations for the plasma convection velocity field near a steady-state magnetic flux tube:
where is the radius of a magnetic flux tube.
For such a model of the convective flow in the photosphere, where , the solution for Eq. (5) can be expressed as (Zaitsev & Khodachenko 1997):
where , and = .
It follows from Eq. (6) that for magnetic field has a maximum on the axis of the tube and decreases with increasing . Assuming that at the tube boundary the magnetic field value appears to be much less than , we can estimate from Eq. (6) the radius of flux tube (Zaitsev & Khodachenko 1997):
In particular, if on the tube boundary , then for the height km upon the level , where cm-3, cm-3, and K, and the magnetic field on the axis of the flux tube G, we obtain from Eq. (8) the tube radius . This yields for the convection velocity m/s the radius of the magnetic flux tube cm.
Keeping in mind that on the boundary of the tube, in the region , the magnetic field becomes rather small and consequently the photospheric parameters, and Eq. (5), give the spatial behavior of the magnetic field:
is the magnetic Reynolds number. Thus, we have very rapid power-law decrease of the magnetic field vs. distance for , because in the photosphere.
Taking Eq. (7) into account we obtain the total longitudinal current in the magnetic flux tube driven by the photospheric convection (Zaitsev 1997):
Under the physical conditions of the upper photosphere in height h = 500 km upon the level , for the plasma flow velocity m/s, and the magnetic field G, Eq. (10) yields for the longitudinal current value A.
Thus, thin magnetic flux tubes with strong magnetic field and longitudinal currents of about 1011-1012 A can be generated in the photosphere-convection zone. We assume that these currents flow from one footpoint to another through the coronal part of a loop, and close deep down in the photosphere forming an electric circuit.
© European Southern Observatory (ESO) 1998
Online publication: August 27, 1998