## 5. LRC-circuit analogIn the self-consistent model of LRC-circuit both capacitance
where , and the values of , and satisfy Eqs. (5) describing the structure of a steady-state loop. The linearized equations describing region 1 (photosphere-convection zone where the loop footpoints are located) are expressed as here We assume that the eigenfrequencies of an equivalent LRC-circuit for the coronal loop are small compared with the inverse Alfvén transit time in the scale of the order of the loop thickness. In this case the circuit eigenmodes can be considered as adiabatic for the perturbations of the loop magnetic field. It means that for eigenmodes the magnetic field can be expressed as in the case of a steady-state situation: Substituting Eq. (11) into Eq. (15) and taking Eqs. (14) into account, we obtain the following relations between the components of the magnetic field and currents: Using Eqs. (16) we can write the following expression for the velocity: Excluding the velocity from Eq. (13) and by using Eq. (17) we get: We introduce also the average values for the current as well as the magnetic and electric fields: where and are the
constant and deviated where and and are the conductivity and the plasma density in the region 1. Consider now the coronal part of a loop (region 2). We will assume that the currents and magnetic fields in region 2 vary in space according to Eqs. (16). This is due to the fact that Eqs. (5), which result in Eq. (15) for adiabatic magnetic field perturbations (), are true for the coronal part of a loopwhere = 0 (fully ionized plasma). It should be taken into account that the small diffusion plasma flow runs through the loop surface in region 2, because both the plasma density and temperature inside and outside the loop are different. An additional argument in favour of the similarity of the space structure of magnetic fields in photospheric and coronal parts of a loop has been given by the observations of an almost constant loop cross-section along the entire loop length (Klimchuk et al. 1992). Keeping in mind these arguments we obtain the following equation for the coronal part of the electric circuit (region 2): where and are the conductivity and the plasma density in the region 2. Deep in the photosphere where the electric current closure between the loop footpoints take place (region 3), we use the formula Here and are the plasma conductivity and the cross-sectional area of the current channel in the region 3. To describe the global electrodynamics of the current-carrying loop we must integrate over the entire circuit keeping in mind Eqs. (20), (21), and (22), and the expression where and , , and are the lengths of the circuits' parts in the dynamo-region, corona, and the region of the current closure, respectively. Eq. (23) suggests that deviated part of the electric current is small compared the current mean value. It opens a possibility for the interpretation of quasi-periodic low amplitude modulation of mcw-emission in terms of eigen oscillations of an equivalent LRC-circuit. Therewith the oscillation period depends on the current mean value. Some peculiarities should be noted in Eq. (23). First, the
steady-state e.m.f. acting in the photospheric dynamo-region is
included in Eq. (23) by a self-consistent way via Eqs. (5). This
e.m.f. generates the steady-state component of the electric current
, persisting in Eq. (23) as a parameter which
determines the main characteristics of the equivalent LRC-circuit.
Both the circuit resistance This means that the essential steady-state Joule losses for a current-carrying magnetic loop occur mainly in the photosphere. Slow variations of the current (for the time ) in the global electric circuit are described by: The right hand side of Eq. (28) denotes e.m.f. driven by the photospheric convection in the loop footpoints: In a self-consistent approach the -component of the magnetic field depends on the current , and hence we can write approximately , where is the average value of the radial velocity inside the tube. A typical time scale of current increase is determined by the minimum value of the two time scales: Variations of the circuit inductance in time can be dealing with, for example, an emerging magnetic loop. Steady-state current value can be found from the equation , and it is easy to show that it coincides with Eq. (10) if the coronal part and the region of current closure in the total circuit resistance are omitted, and the structure coefficient when . © European Southern Observatory (ESO) 1998 Online publication: August 27, 1998 |