The TID model for modulation of large scale electron density models

Various modern applications of empirical electron density models need realistic structures of the electron density distribution with smaller scales than the model background. Travelling Ionospheric Disturbances (TIDs) produce three dimensional and time dependent disturbances of the background ionization. We present a TID model suitable to «modulate» large scale electron density distributions by multiplication. A model TID takes into account the forward tilt of the disturbance wave front, a distinct vertical structure, a fan type horizontal radiation characteristic, geometric dilution and attenuation. More complicated radiation patterns can be constructed by means of superposition. The model TIDs originate from source regions which can be chosen arbitrarily. We show examples for TID modulations of the background model family developed at Trieste and Graz (NeQuick, COSTprof and NeUoG-plas). Mailing address: Dr. Reinhart Leitinger, Institut für Physik, Institutsbereich für Geophysik, Astrophysik und Meteorologie (IGAM), Universität Graz, Universitaetsplatz 5, A8010 Graz, Austria; e-mail: reinhart.leitinger@uni-graz.at of TIDs increase with increasing horizontal wavelengths. Relative peak to peak amplitudes range from 0.001 to a few percent for SSTIDs (which can be considered to be part of «geophysical noise») to 30% and more for LSTIDs. There are two source regions for the AGWs behind the TIDs: 1) the lower atmosphere (troposphere and lower stratosphere) where it is thought that AGWs behind most of the MSTIDs are generated; 2) the lower thermosphere in the auroral oval (E-region heights) where it is thought that most of the LSTIDs (and some of the MSTIDs) are generated (see e.g., Leitinger, 1992; Kirchengast, 1996). In mid-latitudes most LSTIDs propagate equatorwards, those which travel polewards have smaller amplitudes and are thought to have their source regions in the other hemisphere. In some cases the occurrence of large scale TIDs could be linked to auroral substorm activity (Kirchengast, 1997; see also Kishcha et al., 1993). The main purpose of the TID model presented here is the combination by multiplication with a suitable electron density background model like the «family» of models developed at Trieste


Introduction
Travelling Ionospheric Disturbances (TIDs) are the plasma signatures of Atmospheric (acoustic) Gravity Waves (AGWs).They change the plasma distribution of the F-layer, which results in fluctuations of electron density as well as of the electron content.The theoretical understanding of these wavelike disturbances was first given by Hines (1960).According to their horizontal wavelengths we distinguish between three classes of TIDs (table I; see e.g., Van Velthoven, 1990).
With these definitions the Small Scale TIDs belong to the «acoustic branch», the others to the «gravity branch».In general, the amplitudes of TIDs increase with increasing horizontal wavelengths.Relative peak to peak amplitudes range from 0.001 to a few percent for SSTIDs (which can be considered to be part of «geophysical noise») to 30% and more for LSTIDs.There are two source regions for the AGWs behind the TIDs: 1) the lower atmosphere (troposphere and lower stratosphere) where it is thought that AGWs behind most of the MSTIDs are generated; 2) the lower thermosphere in the auroral oval (E-region heights) where it is thought that most of the LSTIDs (and some of the MSTIDs) are generated (see e.g., Leitinger, 1992;Kirchengast, 1996).In mid-latitudes most LSTIDs propagate equatorwards, those which travel polewards have smaller amplitudes and are thought to have their source regions in the other hemisphere.In some cases the occurrence of large scale TIDs could be linked to auroral substorm activity (Kirchengast, 1997; see also Kishcha et al., 1993).
The main purpose of the TID model presented here is the combination by multiplication with a suitable electron density background model like the «family» of models developed at Trieste and Graz (NeQuick, COSTprof and NeUoGplas) or the International Reference Ionosphere (IRI).In this way we are able to demonstrate the influence of TIDs on application systems like satellite assisted positioning as used in surveying or on the use of polar orbiting satellite beacons as novel ionospheric data sources (high resolution ionosphere tomography; see e.g., Leitinger, 1999) or on Radio Occultation data (see e.g., Jakowski et al., 2004).

Atmospheric Gravity Waves (AGWs) -A very short overview
Consider a stable homosphere with a given temperature profile T=T(z) (z: height).With knowledge of the acceleration of gravity g =g(z) and with one pressure and mass density value we also know the pressure profile p =p(z) and the mass density profile ρ=ρ(z).Starting with displacing air parcels adiabatically in the vertical by ∆z one can show that the buoyancy force acts as a restoring force (see Van Velthoven, 1990, and references therein;Prölss, 2001) = c t ^his the velocity of sound, γ =cp/cv is the ratio of specific heats at constant pressure and at constant volume, p 0 and ρ0 are pressure and mass density at the starting level for the displacement.This equation (Newton's second law) can be considered as a differential equation for an oscillation with the general solution (for small amplitudes A) ∆z=Aexp(jωBt); ) is the square of the (general) Brunt-Väisälä or buoyancy angular frequency, ω b 2 =[(γ −1)g 2 ]/cs 2 being the square of the isothermal Brunt-Väisälä angular frequency.
Forcing of oscillations will lead to wave propagation because the air parcel in consideration is connected to its surroundings.
Deriving wave solutions from the linearized hydrodynamic equations for an isothermal atmosphere over a flat Earth leads to the following dispersion relation for AGWs, first published by Hines (1960) with the «acoustic cut-off angular frequency» ωa= (γ g)/(2cs).kz and kx are the vertical and horizontal components of the wave vector k.
The waves described by this relation can be categorised as follows (see e.g., Van Velthoven, 1990): a) ω>ωa, acoustic branch; if ω >>ωa the dispersion relation degenerates to the dispersion relation for pure acoustic waves k 2 =ω 2 /cs 2 ; b) ω b < ω < ωa, evanescent waves; wave equation becomes diffusion equation; c) ω <ωb, gravity wave branch; if ω << ωb the dispersion relation degenerates to the dispersion relation for pure gravity waves k z b

The response of the ionosphere to AGWs
The equation of continuity for electrons can be written as where N e is the electron density; ve, the electron velocity; qe, the electron production rate; and L e, the loss of electrons by recombination.
The electron density can be split into a stationary background electron density (Ne0) and a perturbation (Ne1), due to the passage of the  (1.5) The relation between v e1 and vn1 is given by a balance between the Lorentz force and the «ion drag» force where vn1 is the perturbation velocity of the neutral gas; B, the geomagnetic induction vector; m i, the ion mass; and νi, the effective ionneutrals collision frequency.
In the F-region of the ionosphere (h >150 km) the mobility of the ions perpendicular to the geomagnetic field vector is negligibly small and we can approximate where b is the unit vector in the direction of the geomagnetic field.This leads to the electron density disturbance in the form If in (1.7) the second term of the expression in { } can be neglected we approximate where U n is the amplitude of the AGW; vph, the phase velocity of the AGW; s, the horizontal coordinate; h, the height coordinate; and Ψ(h) is a height dependent phase constant.
The projection factors are . Energy transport upwards corresponds to a downward directed wave normal, kz< 0. Then χ=atan(−kx/ k z ) is the famous «forward tilt» of the planes of constant phase of the AGWs.For AGWs Ψ is a constant value.Because of the projection unto the geomagnetic field it is (slightly) height dependent in the TID case (compare e.g., Francis, 1975;Morgan and Calderón, 1978).
The approximation (1.6) implies, that there is no ionospheric response to AGWs with wave vectors perpendicular to the magnetic field.Below the F-region the condition (1.6) is not valid and so variations of vn perpendicular to the geomagnetic field cannot be excluded.

AGW signatures in Total Electron
Content (TEC) The integration of electron density along slant rays from a satellite transmitter to a ground receiver gives (slant) Total Electron Content.AGW signatures are partly smoothed out.Perpendicular to the geomagnetic field vector we see no TIDs at all.It is necessary to have a geomagnetically meridional component to observe TIDs in TEC, but even with this geometry we have a distinct «aspect sensitivity» (Georges and Hooke, 1970).

Some TID observations
Travelling ionospheric disturbances have been observed with many different instruments.The models allow -to be updated with actual data from various sources; -to be adjusted to disturbed conditions; -to be combined with smaller scale models; -to calculate a variety of propagation effects along arbitrarily chosen raypaths.Average ratio of calculation times Q:C:P=1:2:4.
For TIDs in GPS data see Rieger and Leitinger (2002).Here we show two examples for TIDs observed by means of propagation effects on satellite signals.Figure 1 shows TIDs observed by means of the Difference Doppler effects on signals of polar orbiting satellites.The evaluation of the observations gave the latitude dependence of the vertical electron content (TEC) of the ionosphere.The disturbance by large scale TIDs is demonstrated by means of band pass filtering of the electron content data.Figure 1 is a clear example for the effect of the tilt of the wave fronts and of the filtering effect introduced by the geomagnetic field.The equatorward travelling TIDs appear only equatorwards of the receiving stations.Furthermore the data for the closely spaced receiving stations show strong differences in TID amplitudes which indicates that the AGWs behind the Travelling Ionospheric Disturbances are not radiated from an isotropic source but have a distinct «radiation pattern».
The second example (fig.2) shows TIDs in Faraday effect observations on the VHF signal of the geostationary satellite SIRIO.In this case evaluation of the raw data gives the time dependence of vertical electron content.Band pass filtering reveals LSTIDs of comparatively high amplitudes.Surprisingly there is not much amplitude variation over the day.This means that the relative TID amplitudes have been larger during nighttime than during daytime.This is in accordance with the idea that AGW attenuation is caused by «ion drag» and therefore stronger during daytime (higher ionization) than during nighttime.

Model requirements and «modulation» technique
In general, three dimensional and time dependent electron density models for the ionosphere of the Earth, like the «family» of models developed at Trieste and Graz, NeQuick, COSTprof, NeUoG-plas (table II) (Hochegger et al., 2000) or the International Reference Ionosphere (IRI; Bilitza, 2001) provide the large scale «background» only.If the models give electron density for fixed universal time as a function of geographic coordinates latitude, longitude and height, smaller scale structures can be added by a multiplicative «modulation» technique (fig.3) (Leitinger et al., 2002) ( , , , ) ( , , , ) ( , , , ) ( , , , ) ( , , , ) where M is the resulting electron density model; L, the large scale model (member of our model family, IRI, etc.); T and S, the modulations e.g., for the main trough and for TIDs; h, the height; ϕ, geographic latitude; λ, the geographic longitude; and t, the universal time.

The TID model
The TID model makes use of the following properties of Atmospheric Gravity Waves (AG-Ws): -the horizontal component k x of the wave vector; -the AGW period τ = 2πω; -the dispersion relation to derive the vertical component k z of the wave vector; -the velocity of the disturbance where s is the horizontal coordinate; h, theheight; t, the time; and Ψ is a height dependent phase constant).
Derived quantities are among others the horizontal and vertical wave lengths and phase velocities /k 2 The model takes into account the projection factors discussed above, geometric dilution, horizontal and vertical attenuation.Since the radiation patterns are far from isotropic the model assumes radiation into «fan beams» (fig.4) and allows the combination of several beams originating in one or in several different source «points».Presently we are using vertical «source lines» instead of real «source points», meaning that the vertical structure does not change with propagation.Because of a modification of the «near field» formulation it is not necessary to exclude the source points.
Some details: -The TIDs originate in chosen «source points».
-A TID travels in a «fan beam» defined by its azimuth and its half width; the wave properties are given by the wave period τ and by the horizontal wave length λx, the vertical wave length follows from the dispersion relation.
-For the vertical structure a Chapman profile was adopted defined by a scale height HT and by a peak height hmT.Values chosen for the examples shown: HT = 100 km, hmT = 250 km.
-The forward tilt of the wave fronts is produced by (−kx/ k z ) in accordance with the dispersion relation of the AGWs.
-The geometric dilution of horizontally travelling AGWs and horizontal attenuation are also taken into account.The TID model should not be used for tests of AGW sources and propagation paths.One should not expect realistic «near field» results.
The TID formulation has been used successfully to model interfering smaller scale TIDs as observed by means of a dense network of dual frequency GPS receivers (Rieger and Leitinger, 2005).
-The model allows superposition of several TID wave trains.An example for a single TID wave train is given in figs.8 and 9.

Discussion and conclusions
The main purpose of our TID model is to provide a «modulation» for large scale electron density models used primarily for the numerical calculation of transionospheric propagation parameters, one of the most important being satellite to ground electron content.The TIDs have a simplified vertical structure (Chapman layer) and therefore are not well suited for E-region and lower F-region applications.Comparisons

Fig. 3 .
Fig. 3. Scheme for the multiplicative «modulation» of electron density models to add smaller scale structures.

Fig. 4 .Fig. 5 .Fig. 6 .
Fig. 4. Horizontal radiation pattern of the TID model: radiation into a fan beam of given half width and azimuth.

Table I .
TID classes according to their horizontal scales.