Astron. Astrophys. 319, 699-719 (1997)

Appendix A: existence of a neutral point of on the z-axis

A.1. Two elementary lemmas

We first give two simple lemmas which are repeatidly used in this paper.

Lemma A: Let satisfy the conditions (8), and and be two positive functions (possibly vanishing for some isolated values of y), with being strictly increasing. Then

If is strictly decreasing, the reversed inequality is satisfied.

To prove Eq. (A1), we just need to note that: (i) ; (ii) and on ; (iii) and on . The case where is decreasing is handled in the same manner.

Lemma B:Let be a continuously differentiable function defined on () and satisfying

with a constant. Then the equation has a solution in if and only if

and this solution is unique.

Here, we first note that Eq. (A2) implies that if . Then the equation has at most one solution. On the other hand, the same relation implies that Eq. (A3) is necessarily satisfied if such a solution does exist and belongs to . Reciprocally, equation has obviously a solution if Eq. (A3) holds true.

A.2. Existence of a neutral point of on the z -axis

As noted in Sect. 3, has a neutral point on the z -axis if the equation

has a solution . We have:

where use has been made of Lemma A to get the first relation. Using Lemma B (with and some large number), we thus have:

• If , , and there is no neutral point.
• If , , and we have a V-point at the origin.
• If and , the equation has a unique positive solution , and then there is one neutral point on the z -axis, located at height . Note for later use that for and for .
• If and , , and there is no neutral point (for , the neutral point of the previous case reaches infinity).

A.3. An example of quadrupolar potential configuration

Let us choose

where , and are constants. Then we obtain after some algebra

Then the field is quadrupolar if

in which case

On the other hand, its topology is of type

In the X-case,

A.4. A second example of quadrupolar potential configuration

A field which has been used quite often before (e.g., Priest & Raadu 1975, Low 1987, Sneyd 1993) is the one produced by two dipoles of moment located at , respectively (h, and ). For this field, we have

Then the field is quadrupolar if

and dipolar otherwise. In the former case, we have

and the field has a topology of type

In the X-case

Appendix B: computation of the field of a QPSE on the y - and z -axis

B.1. Field on the y -axis

The value of the field on the y -axis is obtained by computing the limit of (Eq. (76)) when . This can be done by using the relation

which is an immediate consequence of Plemelj's formula (Muskhelishvili 1953)

In these relations, stands for the distribution principal part and for the Dirac measure.

Then, owing to the antisymmetry of , we obtain after separating the real and imaginary parts of

where . No particular problem arises at since , with .

B.2. Field on the z -axis

To compute the value of the field on the z -axis, we take the limit of when . Here, the only point that needs to be treated properly is the computation of the square root in Eq. (76), which requires the choice of a particular determination in the complex plane.

being some arbitrary number, we take , with . Then:

If and :

If and :

If and :

This implies:

If and :

If and :

If and :

Therefore:

If :

If :

If :

Appendix C: conditions of appearance of a vertical CS

We discuss here the existence of a solution to the equations determining the position at time t of a vertical CS in a QPSE: Eqs (77) and (81) when the initial field is of the X-type, Eqs (110) and (84), respectively, when is of the U- or I-type. We first discuss Eqs (77) and (81) from a general point of view.

C.1. Analysis of Eq. (77)

In the domain of the plane , we set

Our aim is to determine the set of couples for which

Fixing the value of in , we obtain an equation for the unknown , to which we shall apply Lemma B:

a. Differentiating F with respect to and , and applying Lemma A, we have

where , ... Then we can use Lemma B.

b. To apply the latter, we first determine the sign of F at the lower bound of the interval . Making , we obtain

where f is the function already defined by Eq. (A4). As shown in Appendix A2:

• If has a U-topology, when , while and when .
• If has a X-topology, for , and for .
• If has a I-topology, .

c. Next we determine the sign of f for large values of . When , we have

h satisfies

where Lemma A has been applied to get the two first relations. Then

• If has a X-topology and , by Eq. (C9), whence by Lemma B.
• If has a X-topology and , then by Eq. (C9) while by Eq. (C8) (in which we take ). Hence (Lemma B) there does exist a number such that for , , and for . This result also holds true when , in which case .
• If has a I-topology, , whence by Eq. (C8).

Putting together the results of (b) and (c) and applying Lemma B (which implies that Eq. (C2) has a solution if and only if and ), we can conclude that, for a given value of :

• If has a U-topology, Eq. (C2) has no solution if . If , it has the solution if (O is a V-point).
• If has a X-topology and , Eq. (C2) has a solution (and a unique one) if and only if . For , we denote the solution by (). Of course, we have .
• If has a X-topology and , Eq. (C2) has a solution (and a unique one) if and only if . For , , while we still have .
• If has a I-topology, Eq. (C2) has no solution.

Then, when has a X-topology, a solution exists on an interval. An important point is that is a decreasing function. We have indeed

because of Eqs (C3)-(C4), and then

This implies in particular that (remember that )

C.2. Analysis of Eq. (81)

Let us set

where and . Then

where and are defined by Eqs (90) and (91), respectively.

We now discuss the system of equations

We thus have to consider the equation on the curve of on which . From Sect. C.1, exists if and only if has a X-topology, and it can be parametrized by , with the latter variable belonging to some interval.

a. On , we have

where use has been made of Eqs (C14), (C15), (C11) and (D4). Then G vanishes at most once on .

b. At the right endpoint of , we have

c. At the left endpoint of , we have

if , and

if . Then, G vanishes once (and only once) at a point of if and only if

(we convene here that is an admissible solution when ). Of course, it vanishes at if .

C.3. Conditions for the existence of a vertical CS

The previous results give at once the conditions which are necessary and sufficient for a vertical CS to be possibly present in the evolving field .

a. If is of the X-type, a vertical CS can exist when the system (C16) has a solution , with . This is the case if and only if is of the X-type and condition (C21) is satisfied. The solution is unique, and because of Eqs (C12) and (16) ()), it satisfies

b. If is of the U-type, a vertical CS can exist if Eq. (C2) has a solution with . This is the case if and only if is of the X-type. This solution (which coincides with ) is unique, and it satisfies

because of Eq. (C12). In fact, we could have also looked for a solution to Eq. (C16), without setting a priori . As Eqs (14) and (16) imposes , this system has a solution if and only if is of the X-type, and it is unique and of the form indeed. This can be considered as a formal proof of the fact that the CS starts from the origin.

c. If is of the I-type, a vertical CS can exist if the equation has a solution - the CS extending between and . As , the results of Sect. C.1.c implies that this is the case indeed if and only if is of the X-type. This solution (coinciding with ) is unique, and it satisfies

because of Eq. (C12). Here too, we could have looked for a solution of the system (C16) (F and G being defined for by taking appropriate limits) without imposing a priori . As we have by Eqs (17) and (16), this system has a solution if and only if is of the X-type, and it is unique and of the form - which provides a formal proof of the fact that the CS has to extend to infinity.

Two remarks are worth here. First, the equations for and have a solution only when the topological constraints are violated. Second, they have only one solution, and therefore, owing to remark (c) at the end of Sect. 5.1, there is at most one QPSE containing a vertical CS and satisfying the constraints imposed at some time t. These statements are in accordance with the uniqueness result of Paper I (quoted at the end of Sect. 4.2). But they do not provide a complete independant proof of uniqueness, as they have been obtained by assuming a priori a y -symmetric field and a vertical CS.

Appendix D: reduction formulas for , and I

We express here the three quantities , and I defined by Eqs (90)-(92), respectively, in terms of the complete elliptic integrals of first, second and third kind , and , respectively. For that, we set

and we use formulas 8.111.2-4 and 8.112.1-2 of Gradshteyn and Ryzhik (1965).

As for the two first quantities, we have:

Whence in particular

As for the third quantity, we have

Then, using the decomposition

we obtain eventually

This expression allows us to reduce the double integral in the LHS of Eq. (81) to a single integral with respect to y.

Appendix E: computation of a QPSE when

Some authors (e.g., Tur & Priest 1976, Low 1987, Sneyd 1993) have considered the problem of the formation of a CS in the case where the field approaches an horizontal constant value at infinity. Then the asymptotic condition (64) is replaced by

where is a given nonzero constant.

In that case, the initial potential field has a complex topology even if the boundary condition on is of the dipolar type, which will be assumed hereafter. Rather than presenting a detailed study of the topology of , let us consider a simple example. Consider the complex field

where and h are real positive constants. Clearly, has all the required virtues. The topology of its lines in depends on the parameter and changes at the critical value :

• If , has a neutral point in , located at .
• If , has no neutral point in , but its lines admit a separatrix of the U-type tangent to the boundary at the origin. The topology is as shown in Fig. 4a.

Then, when a velocity is applied to the footpoints on the boundary along the y -axis, a CS will appear either at once (first topology) or after an X-point has formed in the potential field (second topology). The second situation has been considered by Low (1987), whose results are represented in Fig. 4b.

Fig. 4. a Topology of the field lines of given by Eq. (D2) when . has no X-point above the photosphere, but has a U-separatrix tangent to the boundary at the origin. tends towards a uniform field at infinity. b Structure of the field lines of the QPSE resulting from a compressive velocity field being applied to the footpoints of the potential configuration shown in (a). The resulting CS starts from the origin.

To compute the QPSE which may be produced in situations of this type (the CS being still assumed to be vertical), we thus need to solve BVP (59)-(64) whith the asymptotic condition (64) being replaced by Eq. (E1). It is readily shown that, if we give up for a while with the latter condition and merely assume that the field is bounded in , then the solution to Eqs (59)-(63) is given by

Taking the limit of when and applying condition (E1), we thus obtain

This relation replaces Eq. (77) as the condition of existence of a solution to the BVP. Clearly, it reduces to Eq. (77) when . Hence Eq. (77) may be interpreted as implying either an asymptotic condition or a boundedness condition.

When has a U-separatrix and a CS has formed, and can be determined from Eq. (E4). If the initial separatrix is of the X-type, we need one more equation to compute both and . It is easy to see that Eq. (81) provides the seeked relation. Eqs (E4) and (81) can be analyzed by merely adapting the arguments of Appendix C. For avoiding this paper becoming too long, we leave to the interested reader the care of completing the discussion.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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