Astron. Astrophys. 356, 301-307 (2000)

3. Perturbation method

3.1. Initial potential field

Let us write the total magnetic field in the half-space above the photosphere as the sum of the potential field satisfying the observed component at the photosphere , and a small deviation

where denotes the unit vector on the line of sight. The total current density splits into

with

and is assumed to be potential

Now, we are interested in currents yielding magnetic forces counterbalanced by pressure gradients, like in (4)

Therefore, writing and as the sum of their components along and perpendicular to it

and inserting (15a)(16) and (20) into (19) with use of (18), one gets the Lorentz force

after neglect of the quantity assumed one order smaller than . Likewise, from (15a) and (20), the expressions and can be approximated as

after neglect of the quantities and assumed one order smaller than and respectively. Since by definition, and , we eventually get the equalities

which provide the other two equations we shall use. Actually, relation (23a) means that the first term in (21) disappears, and there simply remains the system

This system can be now solved directly for , without any iteration

with

wherever . Then by Biot-Savart's law, the magnetic field vector is derived (h denotes here the thickness of the photosphere)

and the contribution due to (second integral) can be computed. Note that the contribution due to (first integral) is arbitrary and can be dropped since it does not produce any magnetic force

Moreover, it should be kept in mind that, in general, there is no simple relation between the quantities , , and (see Appendix). For any vector (in particular or ), its components and along and perpendicular to it must be calculated by means of the formulas

wherever .

Let us now demonstrate the well-posedness of this boundary value problem restricted to the -contribution. It is defined by the three properties of existence, unicity of the solution and its continuous dependence on the boundary conditions (Amari et al. 1997). In our problem, the existence and unicity are guaranteed by the existence and unicity of the potential field , which is known to be solution of a well-posed boundary value problem, and by the fact that the explicit relations (25) and (28) determine uniquely the current and induction corrections to . Moreover, since the potential field is solution of a well-posed boundary value problem, it depends continuously on the boundary condition . Relation (25) shows that depends continuously on and provided , and relation (28) shows that depends continuously on . It results that depends continuously on and .

3.2. Initial force-free field

The basic method described above can be extended to the situation where the initial non-perturbed field , instead of being potential, is linear force-free. Then (18) is replaced by

with constant in space. Therefore the magnetic force becomes, after inserting (15a)(16) and (20) into (19) with use of (30)

and neglect of the quantity , like in (21). Inserting the expression of (30b) into (31) yields the magnetic force

Using the same arguments as previously, one derives (23) which holds here again, and moreover the expressions and can be approximated as

after neglect of the quantities and assumed one order smaller than and respectively. Since by definition, and , we eventually get the equalities

The relations (23a) and (34a) imply that the first term in (32) disappears, and there simply results the system

Now this system can be solved directly for the following linear combination of and

corresponding to (25), with and given by (26), wherever . Then, as previously, the magnetic field vector produced by the -contribution is derived by Biot-Savart's law, Eq. (28).

Unlike the potential case yielding an explicit solution (25), the present situation needs solving an integro-differential system (36) and (28). In a way similar to that one used for force-free fields (Grad & Rubin 1958; Bineau 1972; Aly 1989), we may imagine an iterative solving scheme, based on the sequence

for any integer n, with the initial seed

The well-posedness of this problem can be demonstrated by means of Bineau's arguments (Bineau 1972). From (37a), using the same norm as Bineau, we derive the sequence

and after Bineau's second lemma, we know there exists a scalar such that

Therefore, provided is not too large , the difference converges to zero, and eventually and converge toward a limit. This proves the existence of and solutions of (36) and (28).

Next, in order to establish the unicity of solutions, let and be two different solutions of Eq. (36), i.e. such that

If we subtract Eqs. (41) to each other, and take the norm of both sides, there results the equality

and again using Bineau's second lemma, we know there exists a scalar such that

or

Provided is not too large , the inequality (44) leads to , which means unicity of the -solution and therefore of the -solution from (36).

As for the third necessary condition of well-posedness, we know that the linear force-free field is solution of a well-posed boundary value problem (Bineau 1972), which implies that it depends continuously on the boundary condition . Then, inverting (35a) for yields

wherever or equivalently . This latter relation (45) shows the continuous dependence of the field upon and . We can say that if did not depend continuously on and , then would not as well, and this would contradict the assumptions. Finally, by contraposition of this argument we derive the desired property.

© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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