Wave Properties

We wish to investigate the wave behaviour across the β ≈ 1 layer but it is also important to know the properties of waves away from this region. Table 3.1 shows the typical speed and direction of propagation for slow and fast magnetoacoustic waves in high- and low-β plasma. From this table it is clear that the high-β slow wave shares its properties with the low-β fast wave, and similarly the low-βslow wave and high-β fast wave have common properties. So an uncoupled slow magnetoacoustic wave (kx = 0limit) propagating through low-β plasma will change its behaviour to that of a fast magnetoacoustic wave as it passes into high-β plasma. Similarly an uncoupled fast wave will change its behaviour to that of a slow wave as it travels from low- to high-β plasma. Despite this change in terminology the wave mode is the same - no mode conversion has occurred. Thus, when we discuss mode conversion the slow wave driven on the upper boundary retains the properties of a slow wave as it propagates down into high-βplasma. We do not see any evidence of upward-propagating fast waves from the mode-conversion region. The transmitted component of the incident slow wave will continue into the high-βplasma as a fast wave.

In the numerical simulation it is clear that something is happening to the wave as it crosses the region wherecs=vA(Figure 3.2) especially in the horizontal velocity, and the horizontal and vertical magnetic field. This change displays itself as a change in the phase and the behaviour of the amplitude. It is not easy to pick out what is happening, however, as all of the plots display a strong exponential nature which is disguising other underlying effects. We can uncover these by making a simple transformation: vx →

˜

vxe−z/2_{,vz → vze˜} z/2_{, Bx → Bxe˜} −z/2_{,Bz → Bze˜} −z/2_{, p → pe˜} −z/2_{, and ρ → ρe˜} −z/2_{. The data}

resulting from this transformation is shown in Figure 3.3. In the low-β plasma to the right of the dashed red line only one wave mode is present - this is the slow mode which we are driving. To the left of the red dashed line both the fast and slow modes are present. The converted slow mode is clearly visible in the plots of the horizontal velocity and the horizontal and vertical magnetic field, where we can see that the wavefront has slowed right down. The transmitted fast mode is apparent in the plots of the vertical velocity, pressure and density, where we can see it has almost reached the edge of the computational domain. The slow mode is also present in these plots and can be seen as interference with the fast mode just to the left of the red dashed line.

It is possible to predict the position of these different modes at any given time. The position of the acoustic mode (slow in lowβ, fast in highβ) may be found from

dz

dt =−cs, (3.42)

3.3 Numerical Simulations 62

Figure 3.2: Results of the numerical simulation withω = 2π√6andkx =πatt = 13.5Alfv´en times. The plots show the horizontal and vertical velocity, the horizontal and vertical magnetic field, pressure and density respectively from top left to bottom right. The red dashed line indicates wherecs=vA.

Figure 3.3: Results of the numerical simulation withω = 2π√6andkx =πatt = 13.5Alfv´en times. The plots show a transformation of the horizontal and vertical velocity, the horizontal and vertical magnetic field, pressure and density respectively from top left to bottom right. The red dashed line indicates where cs=vA.

3.3 Numerical Simulations 63

Figure 3.4: Surface plot of the horizontal velocity forω= 2π√6andkx=π. The red dashed line shows the position of the acoustic mode, the green dashed line the position of the magnetic mode, and the blue dashed line the position of the slow mode.

which tells us that att = 13.5Alfv´en times the fast wave should have reachedz ≈ −7.5. Similarly the position of the magnetic mode (the slow wave in highβ) may be found from

dz dt =−vA, (3.44) z=−2 ln t 2 + 1− 3 cs , (3.45)

so the slow mode will have reachedz ≈ −3.1att = 13.5Alfv´en times. This is in agreement with the simulations shown in Figure 3.3. We may also use the equation

dz

dt =−cT, (3.46)

wherecT =csvA/pc2

s+vA2 is the tube speed. This is easier to solve in terms oft t= 2 ₁ cT − 1 cT(6) + 1 csln (cs−cT) (cs+cT(6)) (cs+cT) (cs−cT(6)) . (3.47)

This equation models the behaviour of the slow mode throughout the computational domain, following the incident slow wave in the low-βplasma and the converted slow wave in the high-βplasma. Figure 3.4 shows the horizontal velocity viewed from above, overplotted on this are the paths predicted by Equations (3.43), (3.45) and (3.47). The path of the acoustic mode is modelled well by Equation (3.43) and the path of the magnetic mode (given by Equation (3.45)) also agrees afterz = 0, which is unsurprising as the magnetic mode is not present before the conversion point. Equation (3.47) does not seem to agree as well with the simulations as the others; but as the magnitude ofkxis increased in comparison toω, it turns out that this prediction improves and that given by Equation (3.45) actually worsens.

3.4 Analytical Approximations 64