In this section, a summary of the assumptions made in setting up the model and the basic equations which result is given. In order to investigate the magnetic field structure in the region of the prominence, a section across the tube, perpendicular to the axis is considered. Following Priest et al. (1989) we assume a small inverse
aspect ratio for the tube. The large-scale curvature can then be
neglected as a first approximation and we define a cylindrical polar coordinate system (r,0,z) for the tube as illustrated in Figure 3.1. The prominence is considered to be an infinitesimally thin, vertical sheet of mass and current lying on the line 0 = dm: with the upper end point of the prominence located at r = 0.
Within the tube, in the neighbourhood surrounding the prominence, the magnetic field is assumed to be z-invariant and force- free. Following section 2.3, the field may then be expressed in terms of a flux function A(r,0) as
B = (Bn Be.B^) = A ^ ( r , 6), - % y(r,e), B ,(r,e )] (2.28) where A satisfies
^ a , = V2A + F(A) = 0 (2.27)
In Priest et al. (1989), a solution for A(r,0) of the above
Fig. 3.1 The notation used for a flux tube In which a prominence sheet Is supported along the line 6
considering this field to be a perturbation A^(r,9) about a cylindrically
symmetric field Ao(r) in which no current sheet is present. Thus
A(r,0) = Ao(r) + A^(r,0) (3.1)
where Ag(r) satisfies
V^Aq + Fo(Aq) « 0 (3.2)
However, the function Fq(A) for the cylindrical field is not necessarily the same as the function F(A) for the perturbed system since the function 82(A) in the presence of a prominence sheet may not be the same as in the absence of the sheet before the prominence forms. Let us denote by F^(A) the difference between these functions so that
F(A) « Fo(A) + Fi(A) (3.3)
A Taylor expansion to first order then gives us
F(A) « Fo(Ao) + Fi(Ao) + F*o(Ao)Ai (3.4) and subtracting equation (3.2) from equation (2.27) then yields
V2Ai + F‘o(Ao)Ai = -Fi(Ao) (3.5)
The linearised equation (4.13) in Priest et al. (1989) neglected the term on the right-hand side of equation (3.5), making the assumption that
F(A) and hence 82(A) does not change as the model evolves from the original symmetric state in which no prominence exists to the perturbed state containing the prominence sheet.
Now the amount of plasma per unit length in a volume element with section r dr de is
M * p r dr de (3.6)
and the corresponding axial magnetic flux threading the element is
f * 82 r dr de (3.7)
These may be integrated over 0 to give the mass and flux between neighbouring flux surfaces. If one starts with a cylindrical flux tube and then supposes that the prominence forms to give a flux tube with a
current sheet, the question is: how does 82(A) change? This is a
nontrivial question, which is outside the scope of this chapter to answer. 82 will depend on several factors, such as the end conditions where the flux tube comes down and concentrates towards sources in the photosphere or the details of the prominence formation process. Furthermore, initially the density is spread fairly uniformly over azimuth (0), but after formation it is concentrated in the current sheet. The total flux (f integrated over r and 0) will be unlikely to change during formation but its distribution in r and 0 may well alter. If it does not do so and the density and mass outside the sheet remain constant (due to a mass inflow by evaporation or injection to provide the prominence mass) then we can see from the above two equations
that 82(A) will be unaltered. If on the other hand the density p
decreases (because of, for instance, prominence condensation from the
corona) then, if M is conserved, r dr d0 would increase and 82 would
decrease.
In this chapter we will not incorporate such complications. We will, however, consider much larger deviations away from cylindrical symmetry than Priest et al. (1989). In doing so we can choose between two possibilities. We could consider our model as
describing the evolution from a cylindrically symmetric state Ag(r) to
a perturbed state A(r,0) during which the function Fq(A) changes by Fi(A ). Assuming Fq(A) and F^(A) then to be known, we may solve equation (3.2) to obtain Ao(r) and then equation (3.5) to obtain Ai(r,0) and hence obtain the final state A(r,0) from equation (3.1). This approach is difficult to implement in practise because the function Fi(A) cannot easily be determined.
Alternatively, we shall consider our model to describe only the final state of the system without reference to the initial state. Thus F(A) corresponds now to the functional form of Bz(A) in the presence of a particular prominence sheet and the solution of equation (2.27) for A(r,0) describes the field everywhere outside this sheet. This greatly simplifies the mathematical analysis of the model as we no longer need to introduce the arbitrary function F-j(A). We are however, unable to determine the original cylindrically symmetric field from which the final system was derived (unless of course we assume that Bz(A) does not alter during the evolution). This is the course which we will pursue in this chapter.
The field is assumed to be symmetrical about the vertical axis and so above the prominence sheet we have
Br = 0 on 0 = 0 (3.8)
It should be noted that a change in the direction of B@ or B^ within the tube is physically unrealistic and so, as r is increased from zero for a
particular value of 0, we have the condition that Be and B^ remain of
constant sign within the extent of the tube. Furthermore, for a physically reasonable model it is necessary that the field components are bounded at all points within the tube.
The mass in the prominence sheet is assumed to be in static equilibrium between the action of a downward uniform gravitational field and a necessarily upward magnetic tension force provided by B. For this force to be in the required direction there must be a field line dip at 0 « ±7c for all r so that (c.f. section 2.5)
de
If this is indeed the case then the local force balance equation gives the mass density per unit length m as
m(r) « [BJ(r) Be(r,jc) / p gc (3.10)
where [BJ is the jump in B^ across the prominence from 0 = -7c to 0 = +%.
The total mass per unit length M may then be determined from rtc
M » J m dr (3.11)
0
where r, is the radius of the outermost field line at 0 « jc.
in the following two sections, we seek solutions to equation (2.27) for A(r,0), the field around the prominence when it has already formed, after prescribing one of two functional forms of 8%(A). The
field B must satisfy the four properties outlined above, namely
a) The deformation of the tube is symmetric about the vertical axis (equation (3.8)).
b) There is no sign change in either or Bq within the extent
of the tube.
c)There are no singularities of the field components in the neighbourhood of the prominence.
d) There exists a field line dip along the line 0 ~ tt in which a prominence may be supported (equation (3.9)).