EN CONCLUSIÓN; QUÉ RELACIÓN EXISTE ENTRE LA REVISORÍA FISCAL Y LA TEORÍA DE LA

[+ve] I

_[+ve]

\IR

T

Schematic representation of energy balance for the current sheet. The two equilibrium solutions are shown at points (1) and (2) where the radiation curve (IR) intersects the Joule heating and wave

heating (J+IH) curve. As described in the text, only the low temperature equilibrium (1) is stable.

solution (2) is traversed F(T) changes sign from negative to positive. Thus at solution (1)

0

3T

and so the lower temperature solution is stable while the upper temperature solution (2) is unstable.

Although we have found a useful method of predicting the central, temperature and density we still require to solve the full system of

equations. Here we must vary two eigenvalues and D (effectively the width and length of the sheet) to obtain a solution satisfying all the required boundary conditions. It is only from the full solution that we can obtain the dimensions of the sheet and also be sure of the existence of solutions with the predicted central values. The problem of finding a solution by varying two eigenvalues is not simple and we would like to find some way of simplifying the problem. To this end it is useful to consider the reduced system of equations

^ - y7(1 + vB)(l + (1 - ¥^)/6)/FP , ... (4.65) dp

^ = R DfF(l + (1 - B^)/6)(p(l - B^)) Vlp^g^ - ¥/p ... (4.66)

dp “

obtained by eliminating the x dependence from equations (4.42)-(4.44). In the above we have put

-2, a-3/2 (1 + vi)2 - rp^ ( ■L ± . ( V -,,r iZê )

+ r„p , _ , . g

We note with interest that in the above system the parameters and D appear only in the combination R^D and so we only have one parameter to vary. This aids our computation considerably. In the reduced system of equations (4.65), (4.66) we have p as the independent variable, and we know

the necessary central value of p without solving the equations. Thus we can vary the combination R D unt_m il the extra conditions of B and v

vanishing at the centre are satisfied.

Finally we note that the full system (4.42)-(4,44) may be scaled to generate a class of solutions from one solution. That is if we define X = Xx R = XR* an_{m m} d D = D'/X we obtain the same set of equations for the

dashed and undashed variables while the outer conditions are imposed at

— ^ —1 — ,

X = X and x = 1 respectively.

Thus to determine the eigenvalues R^ and D, and thus the full solution, we need only consider the simpler single-eigenvalue problem presented by the reduced equations (4.65), (4.66). The full solution for the profiles B(x), p(x), v(x) then follows by solving the system of

equations (4.42)-(4,44) for any values of R^ and D which conserve the

product R^D found previously and then deducing the required values of R^ and D by a suitable scaling which makes the outer conditions satisfied at

X = 1,

Another feature readily apparent from the reduced system is that

difficulties may arise if F changes sign during the integration. If we consider the phase space (B,p,v) this eventuality would imply B decreasing towards zero while p and v reach maxima and subsequently decrease. Should this maximum value of p be less than the necessary central value (as given by the method described previously) we see that no trajectory in the phase space will pass through both the outer and central values required to satisfy the imposed conditions. Since the behaviour of the term F in the phase space is determined by the parameter 3 and r we expect to find ranges of this parameter for which no solution exists. This phase plane behaviour is illustrated in Figure (4.7). A solution which satisfies all the conditions must pass through both point (1) and point (2). The curves marked (a) represent the skeletal structure for a surface which obviously

m

o

in

in

Ô

o

IÛÛ

in

CD

>

Skeletal structures for two possible phase surfaces of the reduced system of equations. A solution satisfying the boundary conditions must pass through both points (1) and (2). A solution lying on the

surface (a) cannot satisfy the boundary conditions while one on surface (b) can do so.

will not allow a trajectory passing through the points (1) and (2). Any values of 3 and r which give such a surface do not permit a solution. Alternatively, the curves marked (b) give the skeletal structure for a surface which obviously can permit a trajectory passing through both points and hence the corresponding values of 3 and r do permit a solution*

Thus, armed with the above information about the properties of the equations we may proceed to solve the system numerically. We have

outlined a simple method for obtaining the central values of the temperature and pressure but the phase plane analysis indicates that there may not

always be a solution for which these values are attained. 4.3.3 Results

In the previous Section we noted that solutions do not exist for

certain values of the parameters r and 3. In Figure 4.8 we show the ■

regions in the parameter space where solutions are possible. We see that J if 3 is too small (corresponding to a low external pressure, or high

external magnetic field strength) no solution will exist. Similarly if r || ■s is too small (corresponding to convection dominating radiative cooling) no

solution exists, A simple-minded justification of the above results may J be obtained by considering the full system of equations with the limiting

cases of 3 ® 0 or r = 0. Firstly with 3 identically zero we have that the

magnetic field must be constant across the sheet, which obviously cannot satisfy the condition that B be zero at the centre, unless, B = 0 everywhere. Secondly, with r identically zero (and thus r^ = 0 also) our energy

equation is a balance between convection and Joule heating. The condition that the velocity vanish at the centre would thus imply from (4.40) that

0 #

dx §

N at the centre. Ohms’ Law, however, shows that the field gradient cannot

vanish when v = 0 and thus no solution is possible.