7. Results and discussion
In this section we first consider the case of small amplitude waves that are excited by an initial pulse with amplitude , corresponding to . Later, larger amplitude motions will be excited and the role of nonlinearities will be investigated.
7.1. Linear waves in a uniform density medium
A constant value of the unperturbed density is realized by setting (see formulae  and ). An initial impulse (shown in Fig. 2a) is given to the system and the simulation is run until . As the initial impulse is not a stationary solution of the original set of MHD equations, it has to evolve in time in accordance with these equations. At the early stages of its temporal evolution, the central part of the impulse subsides and at (Fig. 2b) the centre of the impulse reaches a negative value. Later, the wavefront takes a defined shape and the whole structure spreads nearly symmetrically in space (Figs. 2c and 2d). Once the wavefront is formed it travels in all directions with an Alfvén speed that depends on z. In particular, for the Alfvén speed decreases exponentially with height (cf. Eq. ), so the down-going wave component moves faster than its up-going counterpart; at (Fig. 2d), the down-going wave has already reached the lower boundary of the simulation region whereas the up-going wave has traversed a smaller distance.
7.2. Linear waves for various profiles of and
In this part of the paper we compare numerically obtained results for several values of defined in expression (13). In particular we discuss the following cases:
a) the equilibrium mass density is uniform and the Alfvén speed decays exponentially with altitude. This case can be realized by choosing , for which the decay rate of is the largest;
b) both the mass density and Alfvén speed decay with height (). As representative cases we take and . The decay rate of is smaller than in case a) and gets smaller with larger . See formula (14);
c) the mass density decays with height (as it does for every ) but the Alfvén speed is constant with z. This case corresponds to ;
d) the mass density decays with height and the Alfvén speed grows with height. For this case is chosen as a representative value.
The normal velocity profile at the same time is plotted for in Fig. 2c and for 0.5, 1, 2, 3 in Figs. 3a-d. A fixed value is chosen for the amplitude of the initial impulse. A first comparison of these results indicates that the shape of the propagating wave is similar for all values of , although it moves with a speed that depends on this parameter. The reason for this behaviour must be found in the fact that the velocity never reaches large amplitudes during the simulation, which implies that the propagation of disturbances takes place in the linear regime and only the fast mode exists (cf. section 4). Since the fast mode carries information along with the local Alfvén speed, one must expect the wavefront to travel faster whenever is larger. Near the position of the initial pulse ( L) the Alfvén speed increases for larger , so the wavefront spreads faster as this parameter is varied from to .
In addition, also varies with height, so propagation takes place anisotropically and the wavefront is not circular. To prove this point, cuts of along the planes and are shown in Fig. 4 ( and are the coordinates of the initial impulse). The two cuts are overplotted for each value of using a common spatial coordinate r, equal to the distance from . The Alfvén speed remains constant for horizontal propagation and so the dotted line is symmetric about in all cases. Moreover, for the Alfvén speed is uniform and propagation is isotropic, as shown by Fig. 4c. For values of between 0 and 2, decays with height and so the down-going part of the front travels faster than its up-going counterpart (Figs. 4a and b). On the other hand, for the Alfvén speed grows with height and the disturbance moves faster away from the photosphere than towards it (Fig. 4d). In summary, the wavefront is circular for , while it is oval-like for , with a degree of deformation that grows in time. The front is `squeezed' at its top for and it is `stretched' at the top for .
A further comment about Fig. 4 must be made at this point. Rather than plotting the normal component, in this figure we have multiplied it by a decreasing exponential, , to reduce the difference between the amplitude of the up-going and down-going parts of the front. The form of the multiplying factor is suggested by the analysis presented in section 7.3, which indicates that the function v (cf. Eqs.  and ) propagates isotropically for . This is confirmed by the numerical results shown in Fig. 4c.
The position of the wavefront at any time can be easily predicted. Consider, for instance, propagation in the x -direction with . At time t the position x of the front satisfies the equation
where the positive and negative signs correspond to propagation to the right and left. Similarly, for propagation in the vertical direction, the position z of the front satisfies the equation
for . A comparison between the results in Fig. 4 and the above formulae has been performed and they appear to be in excellent agreement, which confirms the good performance of the numerical code.
Fig. 5 presents time signatures which are made by measuring the normal velocity at a fixed spatial location. These time signatures correspond to impulsively generated waves with initial amplitude . The behaviour of the normal velocity at the point (, ) is displayed. As a consequence of the fact that the Alfvén speed decays (grows) with height for (), the time signatures are shifted in time for different values of . For large values of the waves propagate faster and consequently need less time to reach the detection point, whereas the opposite happens for small values of . For example, for the wave needs about 1.05 to reach the detection point, while for that time is shortened to . The arrival time of the various wavefronts in Fig. 5 are in good agreement with the analytical predictions given by Eqs. (31) and (32).
The influence of on the time signature is also felt in the time it takes to the front to pass over the detection point. As the Alfvén speed gets smaller with decreasing , the time signature becomes more extended when one moves from Fig. 5d to Fig. 5a. These time signatures together with the spatial structure of the wavefront, shown in Figs. 2, 3 and 4, indicate that waves in the present potential arcade are less complex than waves trapped in coronal loops (Murawski & Roberts 1994). This is a consequence of the fact that the waves in coronal loops are dispersive, while waves in this study are non-dispersive. The temporal scales associated with loop waves are of the order of 1 s, while the temporal scales associated with the time signatures for coronal arcades are of the order of 10 s for . These scales are larger for lower values of . For example, for , they are about 70 s.
The shape of the front is clearly shown in Figs. 3 and 4. It consists of a positive normal velocity component followed by a trailing `wake'. Wavefronts with similar properties are obtained in the propagation of disturbances in two-dimensional uniform media (such as an elastic membrane), although the wake is absent in one or three dimensions (Morse & Feshbach 1953).
7.3. Comparison with analytical results ()
A further check of the performance of the numerical code may be obtained by considering the case . The idea is just to insert the shape of the initial pulse used in the numerical simulations (Eq. ) into the analytical expressions (21)-(24). Nevertheless, the analytical solution to the problem is obtained in terms of the variable v, defined in Eq. (17), which differs from in an exponential term. For a very localised initial pulse, as the one considered here, we have
at . Hence, the functions and are given by
The Fourier transforms of and are
and Eq. (21) yields for
with defined in Eq. (24). This expression has been integrated numerically using the NAG subroutine D01DAF and the normal component has been obtained from Eq. (17). A comparison with numerical results is displayed in Fig. 6 and confirms the good performance of the numerical code. The wavefront asymmetry which can be seen on the numerically obtained profiles of Fig. 6 is a numerical artifact and is a consequence of clipping caused by numerical diffusion. This asymmetry, a typical feature of FCT schemes, is lower for higher-order (in space) numerical schemes. It is quite large for second-order schemes and much lower for a fourth-order scheme such as the one used in our studies.
7.4. Nonlinear waves
When small velocity amplitudes are excited one should expect nonlinear effects to be of little importance for the evolution of the system. This means that, after the work by ade & Ballester (1995b), there should be no conversion between normal and parallel motions in the linear regime and that the velocity component parallel to the equilibrium magnetic field lines should not propagate when excited in the coronal arcade. However, the system of equations we are solving numerically is nonlinear and therefore the linear assumption is valid for very small amplitude perturbations at the initial stages of their temporal evolution. Consider the case of an initial impulse with a low amplitude . At the beginning, the parallel component of the velocity is absent in the system as only the normal velocity, , is set in. However, as a consequence of nonlinear effects, the magnitude of grows in time, reaching at a maximum value of about (Fig. 7a), some 50 times smaller than the amplitude of (Fig. 3c). For a larger amplitude of the initial impulse, equivalent to 3000 km s-1, the parallel component grows even more (see Fig. 7b) and at it reaches a value of about 0.15, much larger than the corresponding value of for the smaller initial impulse. In the second case the amplitude of is even similar to that of .
It is noteworthy that the maximum of is located at the excitation point, . This is a consequence of the removal of the slow magnetosonic wave from the system, which is achieved by neglecting all pressure terms in the original set of MHD equations. As, if not excited initially, is driven by nonlinear terms only, the parallel velocity component in Fig. 7 can be called nonlinear remnant of the slow magnetosonic wave in a cold plasma.
Other interesting effects associated with the nonlinear nature of the MHD equations can be appreciated by looking at the normal velocity component. To realize this it is worth to compare time signatures for small and large amplitude initial pulses. These signatures are measured at the point and therefore correspond to up-going waves. Fig. 8 shows such signatures for the initial pulse (27) with amplitude (solid line) and (broken line). The differences between the two signals must be ascribed to nonlinearities, which appear to slow down the propagation of up-going positive amplitudes (the low amplitude wave arrives to the detection point before the large amplitude wave.) The retardation of the positive part of the wave front gives rise to a delay of the whole perturbation. Nevertheless, one can appreciate that the maximum and minimum perturbations occur closer together in time for than for , a consequence of the speeding of up-going negative perturbations by nonlinearities.
Upwards and downwards propagation of disturbances is also greatly influenced by nonlinearities. Fig. 9 shows time signatures which correspond to . Solid (broken) line is associated with down-going (up-going) propagating waves. Since the maximum of the solid line occurs at an earlier time than the corresponding maximum of the broken line, we conclude that down-going positive () waves are speeded up by nonlinearity, while up-going positive waves are slowed down. On the other hand, the comparison of delays between maximum and minimum for the two signals indicates that the opposite happens for negative perturbations.
To explain this peculiar behaviour of nonlinear waves we have derived a model wave equation for weakly nonlinear fast waves propagating in the vertical direction (see Appendix). For the case and we end up with the following expression for v, defined in Eq. (17),
where , , , and correspond to dimensionless quantities defined in the Appendix.
The solution to this equation is of the form
where is an arbitrary function. So, waves propagate with the speed . This expression for the propagation speed can also be obtained from the equation of characteristics
Hence, is constant along the characteristics
The dependence of the wave speed on is displayed in Fig. 10, which allows us to extract the following conclusions:
Conclusion a) is in agreement with Fig. 8 and also indicates that, since the leading part of the wave front travels slower than the trailing part, the transition from to within the front will be steepened by nonlinear effects. Moreover, conclusions a) and b) together are in full agreement with Fig. 9, where the trailing part of the wave front arrives first to the low detection point as a result of the speeding of down-going positive perturbations. The trailing, negative amplitudes also suffer from a different propagation speed and hence the overall shape of the wave front is distorted accordingly - the up-going perturbation is more localised than the down-going one, which presents a longer delay between the maximum and minimum perturbations.
Finally, it is worth mentioning that we also performed many runs for various parameters of the plasma and the initial pulse. In particular, we considered the following values of the amplitude of the driver: , , , and the ratio of the magnetic to pressure scale height: , , , , . For all these combination of parameters, the nonlinear effects mentioned above were essentially the same.
© European Southern Observatory (ESO) 1998
Online publication: January 16, 1998