SISTEMA DE CONTROL AUTOMÁTICO DE LA SEMBRADORA

At the end of the numerical experiment, the kinetic energy drops to zero and the system has reached an equilibrium. In two dimensions, a double check can be done by evaluating the plasma pressure, p, as a function of the flux function,Az. At equilibrium, the plasma pressure is constant along field lines, and in 2D, that translates to plasma

pressure being a unique function of the flux function, as stated in equation (1.3.8). This is equivalent toB_·∇_{p= 0}. For this set of experiments, this is satisfied straight forwardly, and no further considerations have to be made.

We now look at the distributions of plasma pressure, density, current density and magnetic field, in the final equilibrium. Figures 3.3, 3.4 and 3.5 show two-dimensional maps of plasma pressure, density and perpendicular current density, respectively, with magnetic field lines overplotted in both the initial and final states. At first sight, we can get the main characteristics of the final equilibrium. There are horizontal gradients on the plasma pressure, and at the same locations, we find current density accumulations. That is, there is an equilibrium and it is non- force-free. Plasma density is not constant along field lines. A deficit in density occurs at the location of the initial pressure perturbation, which is mainly balanced by an increase in the direction of the magnetic field (Figure 3.4b). This is in agreement with the adiabatic condition,p/ργ _{= constant.}

Hence, from a qualitative point of view, the numerical results seem to agree with the predictions of the linear analysis made at the beginning of the chapter. Now, the question is how accurate are these predictions, and how far are the numerical results from the linear solutions.

Figures 3.6 and 3.7 show vertical cuts of plasma pressure and density, and horizontal cuts of plasma density, plasma pressure, magnetic field and total pressure, respectively, in the final equilibrium. These are compared with the linear analysis predictions given by equations (3.2.54), (3.2.55), (3.2.56) and (3.3.6), and, in case of the plasma density, we also compare with the solution given by the approximation of adiabaticity, equation (3.2.57), which we have already discussed is probably a better approximation.

We show that the match is almost perfect for the plasma pressure, total pressure and magnetic field, but does not work well for the plasma density. This is not surprising, as the vertical evolution of the plasma pressure is accurate, i.e. it is not constrained by the linear analysis, as shown in equation (3.2.51). Hence, the magnitude of the initial perturbation for the plasma pressure that must be taken into account when checking the accuracy of the linear analysis, is the one after the vertical non-magnetic redistribution, which is, of course, much smaller than the original one.

The calculation of the plasma density are determined by the linear approximation all way through, as seen in equation (3.2.52), and hence, the prediction for the density in the final equilibrium cannot be expected to be good. However, if the process is adiabatic, the density can be obtained directly from the final plasma pressure distribution, using equation (3.2.57). In contrast with the linear analysis, this adiabatic approximation does a very good job, as shown in Figures 3.6b and 3.7b. In Figure 3.8 we plot the quantityp/ργ _{in the final state compared}

to the initial state, for both the vertical an horizontal cuts. Since the numerical experiments have been performed using a full MHD code that solves the non-linear equations, the process is not entirely adiabatic, but has a finite amount of viscous heating that will become important as the initial perturbation is increased.

3.4.3

Overview

We have been able to predict the distributions of the final equilibrium quantities after a two-dimensional hydromag- netic perturbation over a background homogeneous magnetic field embedded in a plasmas. The linear calculations

Figure 3.3: Two-dimensional contour plots of plasma pressure in (a) the initial state and (b) the final equilibrium, for the same experiment as in Figure 3.1. White lines are magnetic field lines.

Figure 3.4: As Figure 3.3, with plasma density in (a) the initial state and (b) the final equilibrium.

Figure 3.6: Vertical cuts of (a) plasma pressure and (b) plasma density, for the same experiment as in Figure 3.1. Initial perturbed state (dashed) is compared with the final equilibrium, as found by the full MHD numerical simulations (solid) and predicted by the linear analysis (red crosses). For the density predictions, the blue crosses represent the prediction from the adiabatic condition given by equation (3.2.57).

Figure 3.7: Horizontal cuts for (a) plasma pressure, (b) plasma density, (c) total pressure and (d) magnetic field strength, for the same experiment as in Figure 3.1.

Figure 3.8: Adiabaticity condition for the same numerical experiment as in Figure 3.1. Plots ofp/ργ_{in the initial}

(dashed) and final (solid) state, for (a) horizontal cut, across the field lines, and (b) vertical cut, along the field lines.

are well behaved for the experiment presented in this section, and are based in the one-dimensional propagations by fast magnetoacoustic waves in the direction across the field, and slow sound waves in the direction along the field lines. Although in reality, the initial disturbance evolves into the final relaxed state through different families of magnetoacoustic waves. There exist an extra contribution of slow magnetoacoustic waves propagates along the magnetic field lines, which introduce a magnetic tension term during the relaxation (i.e. curve the magnetic field as they propagate up and downwards). Nevertheless, these dissipate the magnetic tension in such a way that it is totally unimportant when determining the final equilibrium distributions. The vertical redistribution of the plasma pressure to a homogeneous value demands the magnetic tension to disappear completely, so both the plasma pressure and total pressure are one-dimensional at the end of the relaxation.

Within the linear regime, the final distributions are completely independent of the viscosity, even though it is required to permit the relaxation to occur, as it is the only damping mechanism of the waves. An increase in the viscosity enhances the diffusive term in the wave equation, and so, accelerates the process, but the final distribution is not modified. Second order terms, however, might be dependent on the kinematic viscosity, since the heating term is proportional to it. Within the linear regime, in the final equilibrium, all the quantities are simply determined by the behavior of the final equilibrium total pressure, involving plasma and magnetic effects. Hence, the final equilibrium states for plasma pressure and magnetic field do not differ if the initial perturbation is of the density or internal energy.

Now, we compare the results with experiments in which the initial perturbations are increased systematically, evaluating the validity of the analytical calculations for the total pressure and plasma density in the final equilibrium state, and their departure from the linear and adiabatic regime.