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Astron. Astrophys. 318, 947-956 (1997)

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4. Effects of the currents in the photosphere

Without radial and vertical currents, the dependence of the radial velocity would be linear in the photosphere. However, the electromagnetic forces generated by the currents modify the [FORMULA] law, and at a height of 350 km the radial velocity of neutral changes of sign and stops the influx of angular momentum. Consequently, the layers at 350 km and above cannot act as current generator and the radial current density [FORMULA] is null at [FORMULA]  km.

The radial currents are generated near the boundary of the flux tube in low [FORMULA] field regions as it can be seen by comparison of Figs. 2a,b,c with Fig. 1. The vertical current density [FORMULA] profile shows a negative and a positive peak indicating the presence of two cylindrical shells of currents flowing in opposite directions. In each shell, the vertical current density [FORMULA] changes of sign at about 200 km, as shown on Fig. 2c. As it can be seen in Fig. 2d, since the two cylindrical shells of current flowing in opposite directions are generating repulsive forces they create a depression between them. On the other hand, the most internal current shell of current produces a pinch effect and increases the gas pressure inside the flux tube.

4.1. Photospheric upflows inside the internal current shell

In the flux tube the pressure enhancement due to the internal current shell reaches 10 [FORMULA] and it can generate upward flows and modify consequently the pressure of the upper atmospheric layers. The amplitude of the upflow can be estimated by solving the system of fluid dynamics equations for vertical motions in presence of some local overpressure [FORMULA]

[EQUATION]

Here z is the altitude in the solar atmosphere, and [FORMULA] and [FORMULA] are the pressure and density corresponding to hydrostatic equilibrium. Then we assume steady state ([FORMULA]) and no density changes, i.e. [FORMULA].

The equations (13) and (14) lead to

[EQUATION]

This equation is independent of density. Using the numerical expression of its coefficients at an altitude of about 50 km we obtain

[EQUATION]

At the height of 50 km, the relative pressure increase is [FORMULA], and in the photospheric layers the vertical gradient [FORMULA] of the relative pressure increase is negative or null. A lower limit of the vertical velocity gradient is found for a null value of this gradient. This limit is

[EQUATION]

Consequently, assuming a constant value of [FORMULA], up to the height where the electric current goes through zero and changes of sign, i.e. over a vertical distance of 250 km, the vertical velocity can reach 3.5 km s-1.

4.2. Upflows between the two current shells

Between the two current shells, the dominant force is the vertical [FORMULA] force. This force is directed upwards in the low photosphere and downwards in the upper photosphere where the sign of [FORMULA] reverses. However, the dominant contribution comes from the denser lower photospheric layers. The velocity gradient is such that

[EQUATION]

where [FORMULA] is the density. Assuming that the density [FORMULA] is equal to [FORMULA] the density in the hydrostatic atmosphere, equation (17) simplifies as

[EQUATION]

At [FORMULA] km, [FORMULA] reaches [FORMULA]  Newton. This force is nearly equal to the gravity force. Consequently, a significant velocity gradient is present and

[EQUATION]

In the hypothesis of constant density, a vertical velocity of about 7 km s-1 is reached over a vertical distance of 100 km. In order to explain the acceleration of spicules, the electromagnetic force [FORMULA] must lead to velocities of about 60 km s-1 at their base in order to rise matter to a height of about 7000 km. These velocities can be achieved since velocities in the 40 to 60 km s-1 range can be reached at the top of the photosphere since the density decreases with height by nearly two orders of magnitude.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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