Resultados de la validación de la Estrategia

MHD mode conversion opens up a number of interesting perspectives for diagnostics of the magnetospheric plasma. There is a principal distinction between monitoring the Earth’s interior with seismic waves and the magnetospheric plasma with ULF waves. In contrast to geoseismology, the properties of ULF wave sources (location, spectral content, etc.) are known only approximately. Nonetheless, we can still use ground based observations of ULF waves for MHD diagnostics (“seismology of the magnetosphere”

or “magnetoseismology”). The physical basis of this method of plasma diagnostics is quantifying the effect of resonant conversion of MHD waves in the inhomogeneous magnetosphere. The resonant frequencyfA(Φ) for a given field line is determined by the

local field-aligned distribution of plasma, whereas a quality factorQis determined by the dissipative properties of the ionosphere and magnetosphere. In particular, reliable identification of the resonant frequency opens up the possibility of monitoring the magnetospheric plasma density via ground-based data.

The principal problem of determining the magnetospheric Alfv´en resonator param- eters is linked to the fact that in most events the contribution to the ULF spectral content of the resonant magnetospheric response and of the one from the “source”are comparable. So in most cases, a spectral peak does not necessarily correspond to a local resonant frequency, and the width of a spectral peak cannot be directly used to determine the quality factor of the magnetospheric resonator.

The most effective way the influence of the source spectrum can be discarded and local resonant effects can be retrieved is with the use ofgradient methods, based on measurements of ULF field at two latitudinally separated stations with a small base. The resonant features can be also highlighted bypolarisation methods, based on asym- metry between the resonant response of various ULF components at the observational site. Another method,travel time seismologyof the magnetospheric plasma, makes use of the propagation properties of different magnetospheric modes.

The final step in each of the techniques is inversion of the observed properties (resonant frequencies, arrival times) to obtain plasma mass densities. It requires the solution of the governing wave equation along the field line. This step requires some description of the topology and strength of the magnetic field. At mid latitudes dur- ing quiet conditions the dipole approximation of the magnetic field can be sufficient, however at lower and higher latitudes a more realistic representation of the field is needed. Field lines starting from low latitudes are so close to the Earth that higher order harmonics of the terrestrial main field has to be taken into account, e.g. by using some terrestrial geomagnetic field model, while at higher latitudes the contribution of external magnetospheric sources needs to be included by using some empirical magne- tospheric model. Ignoring the deviation of the magnetic field topology from the dipole case may lead to under- or overestimation of the mass density by at least 30% at low latitudes under quiet conditions, whereas at high latitudes under disturbed conditions the error can probably exceed 100%.

In some circumstances, if multiple harmonics are observed, the plasma mass density at off-equator points can also be estimated. Price et al. (1999) developed theharmonic derived density method for estimating the plasma mass density without any a priori assumption on the functional form of the density distribution. However, the applicabil- ity of the method is highly constrained by the insensitivity of the resonant frequency of the supposed distribution function. Information on plasma mass densities beyond

∼20◦ from the equatorial plane is unavailable, at least, using the upper harmonics. Some assumption on the functional form of the plasma mass density distribution along field lines is also necessary. Typically, the field aligned distribution is approx- imated by a power law distribution, which is sufficient in most cases, except for the lowest latitudes, where contribution of heavier ionospheric ions makes the distribution more complicated. The resonant frequency is determined by the section of the field line, where the wave spends most of the time, i.e. where the Alfv´en speed has a minimum: near the equator. Hence, the inversion is rather insensitive to the choice of the plasma distribution away from the magnetospheric equator.

5.4.1 Gradient method of the magnetospheric plasma diagnostics

The amplitude and phase characteristics of the north magnetic field componentH at the ground can be modelled by the function

H(x, f) = hA(f)

1−iζ, (35)

whereζ= (x−xA)/δdenotes the normalised distance from the resonant pointxA(f), hA(f) is the amplitude of the pulsation at the resonant point, andxis the coordinate

of the magnetic shell, as measured along the geomagnetic meridian. Upon transmission through the ionosphere, the widthδof the resonance peak, as observed at the ground, is increased compared with that above the ionosphere,δi, namelyδ=δi+h, whereh

is the height of the ionospheric E-layer.

Despite the apparent simplicity of the resonance model, the theoretically predicted amplitude and phase meridional structure given by Eq. (35) corresponds well to the observed local structure of ULF waves. Precise measurements of the gradients of the spectral amplitude and phase along a small baseline allow one to exclude the influence of the source spectrum form and to reveal even relatively weak resonant effects. The following simple relationships, stemming from the properties of the function (35), de- scribe the specific features of the ratioG(f) between the amplitude spectra and the difference of phase spectra ∆ψ(f) of the north components of the magnetic field H, recorded at pointsx1 andx2 (∆x=x1−x2>0):

G(f) = |H(x1, f)| |H(x2, f)| = 1 +ζ₂2 1 +ζ2 1 1/2 , ∆ψ(f) = arctan ζ2−ζ1 1 +ζ1ζ2 . (36)

The typical features of functions (36) are as follows:

(a) G(fA) = 1 for the frequency f = fA(xc), where the point xc = (x1+x2)/2 is

located at the midpoint between the stations;

(b)G(f) reaches the extreme valuesG+andG− at the frequenciesf+ andf− which

correspond to the pointsx±=xc±[δ2+ (∆x/2)2]1/2;

(c)G+×G−= 1 andG+−G−=∆x/δ;

(d)∆ψ(f) reaches an extreme value∆ψ∗= 2 arctan(∆x/2δ) at the frequencyfA.

The properties (a) and (d) of the functionsG(f) and∆ψ(f) enable one to estimate the resonant frequency (fA) of the field line between the stations (i.e. atxc) as the

key parameter for plasma diagnostics. The width of the resonant peak can be directly reconstructed from the extreme values of the phase difference as

δ=∆x 2 cotan ∆ψ∗ 2 . (37)

An example of results of the gradient method applied to stations from EMMA magnetometer array is shown in Fig. 21. Characteristic features, predicted by the resonant theory, are evident: the transfer of the amplitude ratio (transfer function) across 1, and the extreme value of the phase delay at the resonant frequencyfA.

The gradient method with some modification can also be applied to the obser- vations at lateral geoelectrically inhomogeneous Earth’s crust. The standard gradient method uses only the amplitude ratio or phase difference between signals for the deter- mination of the resonant frequency of the field line mid-way between the observational stations. The advanced amplitude-phase gradient method (APGM) which employs both

Tihany-Lonjsko Polje (L=1.8), 10 Nov, 2014 0 3 6 9 12 15 18 21 24 mHz 0 50 100 0 0.5 1 1.5 UT 0 3 6 9 12 15 18 21 24 mHz 0 50 100 -60 -30 0 30 60

Fig. 21 The dynamic gradient method technique: the spectral transfer functionG(f) (upper panel) and phase delay∆ψ(f) (bottom panel).

amplitude and phase information at the same time allows not only to determine the resonant frequencies of some particular field lines, but also to restore their latitudinal profile. The continuousfA(x) distribution can be obtained from the dependence of the

resonance positionxA(f) on frequency, as xA(f)−x1

∆x =

1−Gcos(∆ψ)

G2_{−2Gcos(∆ψ) + 1}. (38)

Then, inverting the dependencexA(f), the radial profilefA(x) can be restored.

Another method, suggested to restore key parameters of the resonant structure and continuous latitudinal distribution of resonant frequency, is based on the fractional- linear transform of the gradient data into the complex plane, whereas the resulting image is the hodograph. The hodograph method is a powerful and convenient tool for data analysis. The availability of several mutually controlling methods enables one to perform a reliable monitoring of the resonant frequency variations.

5.4.2 Travel time magnetoseismology

This plasma diagnostics technique is based on the detection of the arrival times of Alfv´en waves driven by the same source. The idea is based on the notion of MHD wave transmission in the magnetosphere as propagation of isolated wave packets. During the propagation through the non-uniform magnetosphere, a conversion of fast magne- toacoustic waves into Alfv´en waves occurs, as illustrated in Fig. 22. The travel time technique assumes a spatially localised impulsive source (such as the initial pulse of a geomagnetic storm at the nose of the magnetopause or substorm onset in the magne- totail) which launches fast magnetoacoustic waves that can propagate isotropically in the magnetosphere at the radially varying Alfv´en speed. The fast waves propagating across magnetic field lines couple to Alfv´en modes, which propagate further along the curved field lines. The waves arrive at the ionosphere at different times along different field lines (e.g. the travel time from the apex of the field line to its footpoint is the quarter of the Alfv´enic eigenperiod of the considered field line). The inversion starts from the arrival time differences between different spatial locations of the observations,

Fast mode

Alfven mode

Fig. 22 Schematic illustration of the MHD wave packet propagation in the magnetosphere. Upon such propagation, a conversion of fast magnetoacoustic into Alfv´en wave packets occurs.

and infers the equatorial plasma mass density for the field lines the observations are available for.

Realising that real processes are much more complicated than the above described ideal case, several attempts have been made to improve the travel time technique, by taking into account wave refraction, ionospheric effects, etc., however, further correc- tions are foreseen (Menk and Waters 2013).

It is worth noting the physical similarity of the model for travel time seismology (Fig. 22) and the possible interpretation of the global coronal wave (Fig. 6). However, the measured wave signal is in both cases different: for the global coronal wave, it is a fast magnetoacoustic wave, while for travel time seismology, the arrival time of Alfv´en waves is measured. Still, the development of travel time seismology in the solar corona may yield interesting new results.