4.1 In trodu ction .
Ill this chapter^ we consider the tem poral evolution of the resonant coupling be tween fast and Alfven waves. As we have discussed in the previous chapter, with the pre scription for plasma displacements given in equation (2.1), when the “azim uthal” wavenum
ber A 7^ 0 or oo, the decoupled fast and Alfven waves become inexorably coupled to each
other.
Here, we are interested in determining the time dependent behaviour of an inho mogeneous plasma which has an Alfven frequency continuum w^(æ) and which contains the eigenfrequency of a A = 0 decoupled discrete fast eigenmode. W hen A ^ O , and is 0 (1 ), compressional energy which initially resides in the displacements can resonantly drive Alfvénic oscillations on those held lines around positions dehned by = wa(>'*'*)•
The evolution of these inhomogeneous plasma systems has been subjected to much theoretical (both analytical and numerical) study in the contexts of solar coronal heating (e.g., Poedts et al. (1990)), laboratory fusion plasma heating (e.g., Hasegawa and Chen (1974) and Hasegawa and Chen (1976)) and in describing magnetic pulsations in the E arth ’s magnetosphere (e.g., Southwood (1974) and Kivelson and Southwood (1986)). Many im portant physical mechanisms are at work in these inhomogeneous systems, and we briefly
^Based on Mann, I.R., A.N. Wright, and P.S. Cally, “Coupling of Magnetospheric Cavity Modes to Field Line Resonances: A Study of Resonance Widths”, J. Geophys. Res. , (1995), in press.
C H APTE R 4. TEM PO RAL EVO LU TIO N OF R E SO N A N T WAVES. 77
discuss them here.
As we have mentioned in Chapter 3, these systems are generally governed by singu lar differential equations (Barston, 1964; Sedlacek, 1971a; Adam, 1986), having singularities which may be associated with the accumulation of energy at a resonance, and the generation of the line scales which are necessary if they are to be utilised as solar or laboratory plasma heating mechanisms. In general, the accumulation of energy at a resonance is referred to as resonant absorption. In the resonant absorption process, the plasma is driven with a monochromatic driving frequency u>d. In the laboratory, this usually takes the form of an external antenna (e.g., see Poedts et al. (1990)), whilst in the solar coronal loop case the plasma is driven at the loop photospheric footpoints by some assumed dominant part of the frequency spectrum of granular or super-granular motions. In the magnetosphere, the original work on resonant absorption proposed that Kelvin-Helmholtz vortices travelling on the magnetopause surface would couple to held lines having the same natural frequency, deep within the magnetosphere. This explained the ground magnetometer observations of pulsations having a localised structure in latitude (Southwood, 1974; Chen and Hasegawa, 1974a; Chen and Hasegawa, 1974b).
In addition to resonances being driven by evanescent (surface) waves, or by incident travelling fast waves driven from an external plasma boundary, it was subsequently realised th at in a plasma system with strongly reflecting boundaries, standing global wave modes could be set up. Kivelson et al. (1984) first proposed that this could occur in the E arth ’s magnetosphere. These global modes would be excited by the incident solar wind, and adopt a standing nature between an outer boundary (often assumed to be the magnetopause) and a turning point in the body of the magnetosphere (these are analogous to the global, low A, dominantly eigenmodes discussed in Chapter 3). If these modes have frequencies which lie inside the natural Alfven frequency continuum, then they can resonantly drive pulsations (Kivelson and Southwood, 1985). In an entirely analogous way, global modes can become trapped (ducted) inside solar coronal loops (flux tubes) and similarly drive resonances which then heat the coronal plasma (e.g., Steinolfson and Davila (1993)).
Recently, the importance of this global mode behaviour has been stressed further by Wright and Rickard (1995). In their study, they drove an inhomogeneous plasma system with a random external driver. The effect of the global (cavity) modes was to filter the incident broadband spectrum so th at the internal plasma response was dominated by the global mode eigenfrequencies u)g. These global modes could then drive resonances at the
C H APTE R 4. TEM PO RAL EVO LUTIO N OF R E SO N A N T WAVES. 78
frequency LOg in an entirely similar way to the monochromatically driven studies (having a driving frequency Wj), the only difference being that u)g need not be the dominant driving frequency. Hence the weU known results from monochromatic normal mode studies can be expected to be equally valid in the more natural randomly driven situation.
The inhomogeneous coupled system which we are considering clearly has singular eigenmodes, as discussed in Chapter 3, whilst the physical wave fields remain weU behaved in time. It is interesting to note th at the growth of energy at the resonance in time can be considered purely in terms of the independent oscillations of the individual singular eigenmodes which were used to construct the displacements corresponding to the initial conditions. Instead of viewing the system as being one whereby fast waves drive Alfven waves (as is often the case in the magnetospheric literature, using the terminology of the decoupled wave modes as the reference point), you can equally well view the evolution of the resonant interactions in a purely m athem atical way. Here, the combined oscillations of the eigenmodes “reveal” the energy at the resonant location as each of the eigenmodes oscillates with its frequency a;„.
In this chapter, we consider the tem poral evolution of an undriven inhomogeneous plasm a cavity. The results which we present could be applied to many of the inhomogeneous plasma systems which we have just mentioned, although we choose to interpret them in the context of magnetospheric pulsations.