Diagnóstico y determinación de necesidades

Alfv´en waves are very distinct from common wave processes in optics, acoustics, or geoseismology. Their spectrum is continuous even in a bounded system, and the eigen- functions are singular (see also the discussion of torsional Alfv´en waves in Sec. 4). These features result in interesting peculiarities of for Alfv´en waves: one-dimensional propa- gation patterns and absence of geometrical attenuation, the possibility of transporting non-steady field-aligned currents to large distances along field lines, the absence of steady oscillations in non-dissipative systems, and the irreversible conversion of fast magnetoacoustic disturbances into Alfv´en waves (see discussion in Sec. 5.1).

According to magnetospheric resonant theory, when MHD disturbances from the outer magnetosphere become incident upon it they are transformed into field line Alfv´en oscillations. This mechanism for resonant conversion of fast magnetoacoustic waves into Alfv´en waves is used to interpret the occurrence of narrow-band ULF signals in the Earth’s magnetosphere.

The basic notions about magnetospheric Alfv´en resonators were originally formu- lated in the frameworks of simplified 1D models (see Sec. 5.1). For a more general, 3D geometry, when the variables are not separable, the asymptotic singular solution in the vicinity of the resonant shell can be obtained by implementing the Frobenius method (e.g. Goossens et al. 1985; Kivelson and Southwood 1986):

bz∼[ky(x−xA)]2ln [ky(x−xA)], bx, ey ∼ln [ky(x−xA)],

by, ex∼[ky(x−xA)]−1. (29)

Herexis the coordinate along the meridian; xA(f) is the coordinate of the resonant

magnetic shell that corresponds to the frequency f. The singular growth of an elec- tromagnetic field amplitude at a resonant shell, where x =xA(f), is terminated by

dissipative effects, e.g., Joule damping in the conductive ionospheres. To account for this effect, the termx−xA(f) must be replaced withx−xA(f) +iδ, where the semi-

width of the resonant spatial peakδ is produced by a dissipation. Thus, a qualitative pattern of MHD wave conversion and singular wave structure in the vicinity of Alfv´en resonance remain valid even in multi-dimensional inhomogeneous systems. Thebxcom-

ponent has a weaker logarithmic singularity near the resonance in the magnetosphere, so the resonant behaviour of this component would hardly be noticeable.

The spatial structure of ULF waves can be qualitatively imagined as a superposi- tion of a source field and the resonant response to it from the magnetospheric Alfv´en resonator. This resonator is formed by geomagnetic quasi-dipole field lines terminated by the conjugate ionospheres. The source field is due to the large-scale fast magnetoa- coustic mode. The resonant response of the magnetosphere, caused by Alfv´en field line excitation, is strongly localised and produces a steep enhancement of amplitude and strong phase gradient upon the transition across the resonant shell. This transformation is most effective at the geomagnetic latitudeΦ, where the source frequencyf matches the local frequencyfAof field line Alfv´en oscillations, namelyf'fA(Φ) (Southwood

ionosphere their essential spatial structure mainly survives, but the polarisation plane rotates byπ/2 and some broadening of the resonant peak occurs.

MHD disturbances from remote parts of the magnetosphere propagate inside the magnetosphere and, through mode transformation, they excite standing Alfv´en oscil- lations on the Earth’s magnetic field lines. Alfv´en waves incident on the ionosphere are, in most cases, the sources of ULF geomagnetic pulsations (Pc3–5) observed on the ground. The leading term in the expansion, which describes the resonant singularity ofby(x, f) near the resonant shell, takes the form

by(x, f) =b0(f) iδi

x0_+iδi, (30)

wherex0=x−xA(f) is the distance from the resonant shell,xis the coordinate of a

magnetic shell,xA(f) is the point wheref=fA(x),δiis the full-width half-maximum

of the resonance region above the ionosphere, andb0 is the amplitude of the pertur-

bation. The resonance widthδi is related to the damping rateγ by the relationship

δi =−γ(2π∂fA/∂x)−1 (whereasγ > 0). The gradient sign of the Alfv´en frequency,

and correspondinglyδi, determines the direction of an apparent phase velocity in the

radial direction of the ULF waves in the resonant region. Throughout the magneto- sphere (except at the plasmapause)∂fA/∂x <0 and the meridional component of the

phase velocity is directed towards higher latitudes.

Pumping of wave energy into the resonant region causes both growth and narrowing of the spatial resonant peak. In the steady state this growth is saturated at some level depending on the rate of the dominant dissipation mechanism. Commonly, Joule dissipation is taken to be a damping mechanism in the ionosphere. The damping rate

γn of the n-th mode in the magnetospheric resonator is related to the width of the

resonant regionδn by the simple relationship

|γn/ωAn(x)|=|δn/a|=

1

πnln|R|, (31)

where a=|ωAn(x)(dωAn(x)/d x)−1 |is the typical scale of Alfv´en frequency spatial

variations, andRis the Alfv´en wave reflection coefficient from the ionosphere. In general, the magnetosphere behaves as a giant maser for MHD waves. The so- lar wind flow pumps energy into this maser, then the seed disturbances are band- filtered in the magnetospheric Alfv´en resonator, and are transmitted through the semi-transparent windows (the ionosphere). Besides the solar wind input, the injec- tion of non-equilibrium energetic particles into the magnetosphere, also can generate monochromatic waves in such a maser, similar to the inverse population in lasers. The maser mechanism results in the appearance in a space plasma of various narrow-band quasi-monochromatic oscillations, despite the turbulent character of the solar wind– magnetosphere interaction. In a greater detail, the excitation of the magnetospheric Alfv´en resonator is considered in Sec. 9.

Despite a significant progress in our understanding of MHD wave coupling in the magnetosphere, there still remain important open questions:

– Though there have been a few reports of the signatures of the field-line resonant conversion recorded by spaceborne magnetometers, the evidence of the ULF wave resonant conversion in situ is almost absent. Is it just because we had not have enough time to find them, or there are some deep physical reasons for that?

– More realistic theory of the resonant conversion, taking into account the 3-D inho- mogeneity of the magnetosphere and time-limited character of the external driver must be developed and compared with observations.