Diferencias entre la firma electrónica y el certificado digital

For low frequencies (periods longer than 20 s) the approximation of the thin layer ionosphere may be used for the description of MHD wave interactions with ionosphere. However, this theory cannot be used for waves in a higher frequency range, whose field-aligned wavelengths are comparable to the vertical scale of the ionospheric inho- mogeneity.

A characteristic feature of the upper ionosphere is the occurrence of the IAR and the fast magnetoacoustic waveguide (Sec. 3.8.1), which can trap MHD wave energy in the range from fractions of a Hz to a few Hz. The occurrence of the IAR and fast magnetoacoustic waveguide ensures the strong dependence of the ionospheric transmis- sion/reflective properties of incident Alfv´en waves on the frequency (Ostapenko and Poliakov 1990). The ionosphere can act as a multi-band band-pass filter for magneto- spheric MHD waves, and determines, together with the magnetospheric wave source, the spectral structure observed on the ground. Numerical models accounting for the Alfv´en wave interaction with a realistic ionosphere, predicted that at nighttime the transmission coefficient T(f, k) of Alfv´en waves to the ground has an oscillatory de- pendence on frequency, with many narrow “transmission windows” at the IAR eigen- frequencies. For a broadband magnetospheric emission the occurrence of the “trans- parency windows” may result in the formation of narrow-band emissions observed on the ground after transmission through the ionosphere.

A typical model describing the propagation of MHD waves through the magnetosphere– ionosphere–atmosphere system to the ground is a set of multiple layers with different electrodynamic parameters, permeated by the geomagnetic field B0 with inclination I. An incidence of Alfv´en waves with various frequencies and transverse scales, az- imuthally extended waves withk2k1, is considered, wherek1andk2are the effective

radial (latitudinal) and azimuthal (longitudinal) wave numbers. The wave structure is determined from the solution to Maxwell’s equations with the complex tensor permit- tivity ˆε⊥(ω, z), comprising the local Pedersen and Hall conductivities, σP(z, ω) and

σH(z, ω). In an oblique-angled coordinate system under the assumptionsσk→ ∞and

k2= 0, withk≡k1, Maxwell’s equations are reduced to the following set of equations:

∂3b1=ikcotIb1−µ0σH(sinI)−1e1+iω−1

k2−iµ0ωσP

e2, (60)

∂3b2=−µ0σP(sinI)−2e1−µ0σH(sinI)−1e2, (61)

∂3e1=iωb2, ∂3e2=−iωb1+ikcotIe2,

where e1,2 andb1,2 are the perturbations of the electric and magnetic fields, respec-

tively. If the Hall conductivityσHcan be neglected, then the set of equations (61) splits

into two uncoupled sub-sets, for Alfv´en waves (with the non-vanishing componentse1

andb2), and for fast magnetoacoustic waves (with componentse2andb1).

For terrestrial applications, the altitude profile of the ionospheric parameters given by the International Reference Ionosphere (IRI) model is used. Using the altitude profiles of the electron and ion collision frequencies, νe and νi, Alfv´en speed VA(z),

the plasma conductivitiesσP(z, ω) andσH(z, ω), one can calculate the refraction index nA(z) = Re

√

εP,εP=iσP/(ε0ω).

During the interaction with the anisotropically-conducting ionosphere, Alfv´en and fast magnetoacoustic modes are linearly coupled. Therefore, the reflection and trans- mission matrices are needed for the description of the coupling of the horizontal com- ponents of the magnetic fields in the incident(i)and reflected(r)waves. Further, we present the results of the incidence and reflection of Alfv´en mode only, characterised by the reflection coefficient R(f) (element of the reflection matrix), the ratio of the horizontal magnetic components of reflected and incident Alfv´en wave R=b(₂r)/b(₂i), and the transmission coefficient, which is defined as the ratio between the amplitudes of magnetic field disturbance on the ground and magnetic component of an incident Alfv´en waveT =b(₁g)/b(₂i)

As an example, we show results of the numerical modelling for typical mid-latitude location during nighttime (Fig. 41). To reconstruct the altitude profile of the plasma conductivities σP(z, ω) and σH(z, ω) and refraction index nA(z), the IRI and Mass–

Spectrometer–Incoherent–Scatter (MSIS) models3 were used. During nighttime the height-integrated (60–400 km) conductances areΣP= 0.26 S andΣH= 0.17 S, whereas

during daytime the conductances are ΣP = 10.0 S and ΣH = 10.56 S. We consider

the features of reflection and transmission through a realistic IRI-derived ionosphere of Alfv´en waves with various transverse scales, fromk= 10−3km−1 tok= 10−1km−1. The effect of the IAR and fast magnetoacoustic waveguides on the ionospheric reflection coefficientR(f) as a function of the frequency is clearly visible in Fig. 41, upper panel. An important issue is the modification of the spectral content of the mag- netospheric noise and waves upon transmission through the ionosphere to the ground, characterised by the transmission coefficientT(f, k). The coefficientT(f) has a strong oscillatory dependence on the frequency, with maxima at the IAR eigenfrequencies

withf < 4 Hz (Figure 41, bottom panel). Atf > 3 Hz the behaviour is drastically

different for different scales: fork⊥= 10−2km−1T(f) it is oscillatory till rather high

f '10 Hz, whereas fork⊥ = 10−3 km−1 broadband “transmission windows” appear

at higher frequencies, namely 4–6 Hz. This “transmission window” may result in an enhanced magnetospheric noise leakage to the ground in this frequency band.

Fig. 41 Reflection R(f) and transmissionT(f) coefficients for Alfv´en waves with the wave numbersk⊥= 10−3km−1 (dark line) andk⊥= 10−2 km−1 (grey line), interacting with the nightside ionosphere at mid-latitude (I= 58.6◦).

For very small scales, a collisional field-aligned conductivity σk (neglected in the

above model) should be taken into account. The collisional damping in the ionosphere leads to absorption of Alfv´enic waves with wave numbers larger thank∗_⊥'µoσk/τA,

whereτAis the Alfv´enic time. According to Lessard and Knudsen (2001), Alfv´en waves

with spatial scales less than a few km are strongly diffused and damped upon the reflection from the realistic ionosphere. This effect limits the transverse scale of IAR from below. The possible occurrence in the ionosphere of small-scale Alfv´enic structures with the transverse scales comparable to the dispersive scale (typically, about few hundred m), commonly invoked to interpret the mechanism for the auroral electron acceleration, is still disputable.