Application of the method for a particular choice of boundary conditions

In this section we demonstrate the construction of magnetic field configurations for particular forms of the arbitrary functions

f(y) and g'(x) and particular values of the parameters c, y^and yg. The

procedure is to select f(y) to be of the sign required for the chosen type of configuration (negative or positive for N-type or l-type respectively), then select g’(x) that satisfies the conditions described in section 5.4.5. Values must also be given to c, y^ and yg. From equations (5.37) and (5.38) the field components in Q can now be calculated everywhere and they automatically satisfy the constraints given by equations (5.34) and (5.35) which ensure that the field is bounded at the endpoints of the sheet r. The field line projections in Q may then be plotted by the method described below. Finally we must check that the equilibrium condition given by equation (5.54) is satisfied everywhere along the sheet to ensure that we have constructed a physically valid model in which support of the entire sheet against gravitational forces is possible.

5.5.1 Method used to obtain plots of the magnetic field

Since A(x,y) is constant along field lines the easiest method of plotting field lines is to produce a contour plot in Q of equally spaced values of A(x,y). Unfortunately, we do not have an explicit expression for A(x,y) and so must resort to a numerical approach.

Along the photosphere, since we have imposed By(x,0) we obviously know (to within a constant) A(x,0) from equation (5.13d). For a particular point (Xp,yp) in Q we can find A(Xp,yp) from

A(Xp,yp) = A(Xp,0) + J dA (5.57)

0

which from equation (2.25) becomes j:p

A(Xp,yp) = A(Xp,0) + J B ,(X p ,y) dy (5 .5 8 )

0

Thus for each value of Xp we can calculate from equation (5,37) B%(Xp,y)

at several values of y and so evaluate numerically the integral in equation (5.58) to obtain A(Xp,yp) for various values of yp. This enables us to obtain values of A on a grid of points in Q and hence produce a

5.5.2 A particular choice of boundary conditions

We now construct particular configurations of the magnetic field for a simple form of the imposed normal field components. The most straightforward choice for the field across the prominence is

Bx(0,y) = f(y) = a on {yi < y < yg} (5.59) where a is a real constant, positive or negative for an l-type or N-type model, respectively.

For the vertical field along the photosphere we choose

By(x,0) = -g'(x) = bx/(x2 + d^) + c^x on {-oo < x < oo} (5.60)

where b and d are real constants. Note that g’(x) -> -c^x as x oo, as required by equation (5.47), is nowhere singular and satisfies the conditions required for a bipolar field given by equation (5.47) and

(5.48) provided

b>-c2d2 (5.61)

Values of y^ and yg are then selected and from equation (5.54) we see that to ensure equilibrium the condition simply becomes

By(0+,y) < 0 on {yi < y ^ yg) (5.62)

for an N-type configuration

By(0+,y) > 0 on {yi ^ y < yg} (5.63)

for an l-type configuration where By(0+,y) is obtained from equation (5.44).

Examples of both types of configuration are seen in Figures 5.2 and 5.3 along with the corresponding variation of By(0+,y) along the prominence sheet in order to verify that the above condition holds. Note that for both types of model the field strength increases without bound as we move to infinite distances from the sheet but that in the localised area about the sheet (and in particular at the extreme points) the field is everywhere finite. We may construct models in which the prominence is either attached to the photosphere (yi = 0) or is detached (yi > 0). The 0-type neutral point above the sheet in the case of the I- type model is to be expected from the arguments in section 5.4.4

concerning the direction of Bx(0,y). Note especially that for the first

time we have been able to produce an l-type model in equilibrium everywhere in which the field is locally finite with closed field lines above the prominence and an X-type neutral point below. This is particularly important since the majority of mature quiescent

0.0 0.25 0.5 0.75 1.0 1.25 1.25 1.25- 0.75 -0.75 0.5- :-o.5 0.25- -0,25 0.0 0.0 0.0 0.25 0.5 0.75 1.0 1.25 0.5 0.75 1.0 0.0 0.0 —0.1- - 0.1 — 0.2 0.2- — 0 .3 -0.3- 1.0 0.75 0.5 (a) (b)

Fig. 5.2 (a) The projection in the x-y plane of the field lines for a normal configuration constructed using the particular functional form of the boundary conditions detailed in section 5.5. y-j = 0.4, y2

= 1.0, a = -1.0, b = -1.0, c^ s 1.0, d^ = 1.0. (b) The vertical field component By(0+,y) plotted as a function of height along the vertical extent of the prominence sheet.

0.75 1.0 0.075- -0.075 0.05- -0.05 0.025- -0.025 0.0 0.0 0.75 1.0 (a) (b)

Fig. 5.3 (a) The projection in the x-y plane of the field lines for an inverse configuration, y-j = 0.6, V2 = 1.0, a = 1.0, b * 1.1, c2 = 1.0, d2 = 0.01. (b) The vertical field component By(0+,y) plotted

as a function of height along the vertical extent of the prominence sheet.

? prominences are known observationally to be of I-type.

The mass density per unit length of the prominence m(y) which will be supported by the field is given by equation (5.4) as

m(y) = 2 |a| |By(0+,y)| / \i g^ (5.64)

In this simple case we see that m(y) is directly proportional to By(0+,y) and the shapes of the two graphs against y will be identical.

The total prominence mass per unit length M may be obtained by numerically integrating equation (5.64) to get

Y2

M

= ^ I