Annals 48, 3, 2005+app1 515 ANNALS OF GEOPHYSICS, VOL. 48, N. 3, June 2005 Key words ionosphere – electron density models – Travelling Ionospheric Disturbances – TID model 1. Introduction Travelling Ionospheric Disturbances (TIDs) are the plasma signatures of Atmospheric (acous- tic) 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 wave- lengths 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 The TID model for modulation of large scale electron density models Reinhart Leitinger and Markus Rieger Institut für Physik, Institutsbereich für Geophysik, Astrophysik und Meteorologie (IGAM), Universität Graz, Austria Abstract Various modern applications of empirical electron density models need realistic structures of the electron densi- ty distribution with smaller scales than the model background. Travelling Ionospheric Disturbances (TIDs) pro- duce three dimensional and time dependent disturbances of the background ionization. We present a TID mod- el suitable to «modulate» large scale electron density distributions by multiplication. A model TID takes into ac- count 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, COST- prof and NeUoG-plas). Mailing address: Dr. Reinhart Leitinger, Institut für Phy- sik, Institutsbereich für Geophysik, Astrophysik und Meteo- rologie (IGAM), Universität Graz, Universitaetsplatz 5, A- 8010 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 «geo- physical noise») to 30% and more for LSTIDs. There are two source regions for the AGWs be- hind the TIDs: 1) the lower atmosphere (tropo- sphere 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 ampli- tudes and are thought to have their source re- gions in the other hemisphere. In some cases the occurrence of large scale TIDs could be linked to auroral substorm activity (Kirchen- gast, 1997; see also Kishcha et al., 1993). The main purpose of the TID model present- ed here is the combination by multiplication with a suitable electron density background model like the «family» of models developed at Trieste 516 Reinhart Leitinger and Markus Rieger and Graz (NeQuick, COSTprof and NeUoG- plas) 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 resolu- tion ionosphere tomography; see e.g., Leitinger, 1999) or on Radio Occultation data (see e.g., Jakowski et al., 2004). 1.1. 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) ( ) dt d z g dz d c g z s 0 2 2 0 2 0 = +t t t∆ ∆d n (1.1) c ps 0 0= c t^ h is the velocity of sound, γ = cp/cv is the ratio of specific heats at constant pressure and at constant volume, p0 and ρ0 are pressure and mass density at the starting level for the dis- placement. 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); j 1= - , ωB 2= = ωb 2 + (g/cs2) (d(cs2)/dz) is the square of the (gen- eral) Brunt-Väisälä or buoyancy angular fre- quency, ωb 2 = [(γ −1)g2]/cs2 being the square of the isothermal Brunt-Väisälä angular frequency. Forcing of oscillations will lead to wave propagation because the air parcel in consider- ation is connected to its surroundings. Deriving wave solutions from the linearized hydrodynamic equations for an isothermal at- mosphere over a flat Earth leads to the follow- ing dispersion relation for AGWs, first pub- lished by Hines (1960) k k c z b x s a2 2 2 2 2 2 2 2 = - - - ~ ~ ~ ~ ~ (1.2) 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 dis- persion relation degenerates to the dispersion re- lation for pure acoustic waves k2= ω2/cs2; 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 kz b 2 2 2 $= ~ ~^ h k cx a s 2 2 2 $ - ~^ h. 1.2. The response of the ionosphere to AGWs The equation of continuity for electrons can be written as t N q L N ve e e e e$2 2 d= - - ^ h (1.3) where Ne is the electron density; ve, the electron velocity; qe, the electron production rate; and Le, the loss of electrons by recombination. The electron density can be split into a sta- tionary background electron density (Ne 0) and a perturbation (Ne 1), due to the passage of the Table I. TID classes according to their horizontal scales. Class Horizontal wavelength Periods Horizontal phase velocities Large Scale (LSTIDs) >1000 km (0.5 ... 3) h (300 ... 1000 ) m/s Medium Scale (MSTIDs) (100 ... 1000) km 12 min ... 1 h (100 ... 300) m/s Small Scale (SSTIDs) <100 km A few minutes < 200 m/s 517 The TID model for modulation of large scale electron density models AGW. Assuming ve1 to be the velocity induced by the AGW leads to N N Ne e e0 1= + , v v ve e e0 1= + . (1.4) Assuming no perturbation in production and loss and setting ve0 = 0 gives t N N ve e e 1 0 1$2 2 d= - ^ h. (1.5) The relation between ve1 and vn1 is given by a balance between the Lorentz force and the «ion drag» force e mv B v v 0e i e n1 1 1# + - =o^ ^h h where vn1 is the perturbation velocity of the neutral gas; B, the geomagnetic induction vec- tor; mi, the ion mass; and νi, the effective ion- neutrals collision frequency. In the F-region of the ionosphere (h > 150 km) the mobility of the ions perpendicular to the geomagnetic field vector is negligibly small and we can approximate v v b be n1 1$. ^ h (1.6) where b is the unit vector in the direction of the geomagnetic field. This leads to the electron density distur- bance in the form N j j Nv b b k be n e1 0$ $ $d0 -^ ^ ^h h h" , . (1.7) If in (1.7) the second term of the expression in { } can be neglected we approximate ( ) ( ) ( ) ( ) cos cos cos N N k v U k s k h h t v b Ψ e e n ph n x z 0 1 $ $0 = + + - ~ ~ Ξ Σ Ξ cos$ ^ ^ h h (1.8) where Un is the amplitude of the AGW; vph, the phase velocity of the AGW; s, the horizontal co- ordinate; h, the height coordinate; and Ψ(h) is a height dependent phase constant. The projection factors are ( ) ( )cos k b k$=Ξ and ( ) ( )cos v b vn n1 1$=Σ . Energy transport upwards corresponds to a downward directed wave normal, kz < 0. Then χ = atan(−kx / kz ) is the famous «forward tilt» of the planes of constant phase of the AGWs. For AGWs Ψ is a constant value. Because of the pro- jection 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. Be- low the F-region the condition (1.6) is not valid and so variations of vn perpendicular to the ge- omagnetic field cannot be excluded. 1.3. AGW signatures in Total Electron Content (TEC) The integration of electron density along slant rays from a satellite transmitter to a ground re- ceiver 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). Fig. 1. Examples for TIDs in observed electron con- tent (Leitinger, 1992). NNSS Difference Doppler ob- servations, band pass filtered data. Receiving stations: Bonn (50.5°N, 6.9°E), Lindau/Harz (51.6°N, 10.1°E), Graz (47.1°N, 15.5°E), Penc (47.8°N, 19.3°E). Mar- ked amplitude range: 1×1016 m−2. Note the differences in TID amplitudes for the closely spaced receiving sta- tions. For more examples see Putz et al. (1990), Lei- tinger and Putz (1991). 518 Reinhart Leitinger and Markus Rieger Because of the tilt of AGW fronts the TID distur- bance in TEC is smoothed out polewards of the receiving station if the AGW propagates equator- wards. The tilt of the wavefront itself does not ap- pear (figs. 1 and 2: observed TIDs; figs. 5 to 7: model TIDs). Projection from slant to vertical in a «mean ionospheric height» does not change this situatuation. 2. Some TID observations Travelling ionospheric disturbances have been observed with many different instruments. Fig. 2. Examples for TIDs in observed electron con- tent (Leitinger, 1992). SIRIO Faraday observations, band pass filtered data. Receiving stations: Graz (47.1°N, 15.5°E) and Firenze (43.0°N, 10.7°E). Ab- scissae in hours CET (top) or UT (bottom), ordinates in units of 1015 m−2. For more examples see Putz et al. (1992). Table II. Examples for the large scale «model basis». The «family» of models developed at Trieste and Graz NeQuick (Q), COSTprof (C) and NeUoG-plas (P) are profilers using the peaks of the E-layer, the F1- layer and the F2-layer as anchor points. They are identical from 100 km to the F2 peak (Epstein layer formulations). C and P have identical ionosphere topsides (height aligned O+-H+ Diffusive Equilibrium – DE), Q uses a semi-Epstein layer with a height dependent thickness parameter. Above 2000 km P uses a magnetic field aligned H+ DE («plasmasphere»). 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 ar- bitrarily 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 ob- served by means of propagation effects on satel- lite signals. Figure 1 shows TIDs observed by means of the Difference Doppler effects on sig- nals of polar orbiting satellites. The evaluation of the observations gave the latitude dependence of the vertical electron content (TEC) of the iono- sphere. The disturbance by large scale TIDs is demonstrated by means of band pass filtering of the electron content data. Figure 1 is a clear ex- ample for the effect of the tilt of the wave fronts and of the filtering effect introduced by the geo- magnetic field. The equatorward travelling TIDs appear only equatorwards of the receiving sta- tions. Furthermore the data for the closely spaced receiving stations show strong differences in TID amplitudes which indicates that the AGWs be- hind 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 519 The TID model for modulation of large scale electron density models of the geostationary satellite SIRIO. In this case evaluation of the raw data gives the time de- pendence 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 larg- er during nighttime than during daytime. This is in accordance with the idea that AGW attenua- tion is caused by «ion drag» and therefore stronger during daytime (higher ionization) than during nighttime. 3. Model requirements and «modulation» technique In general, three dimensional and time de- pendent electron density models for the iono- sphere 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) ( , , , ) ( , , , ) ( , , , ) ( , , , ) ( , , , ) h t h t h t h t h t M L T S Sn1 $ g = + + { m { m { m { m { m$6 @ where M is the resulting electron density mod- el; L, the large scale model (member of our model family, IRI, etc.); T and S, the modula- tions e.g., for the main trough and for TIDs; h, the height; ϕ, geographic latitude; λ, the geo- graphic longitude; and t, the universal time. Fig. 3. Scheme for the multiplicative «modulation» of electron density models to add smaller scale structures. Fig. 4. Horizontal radiation pattern of the TID model: radiation into a fan beam of given half width and azimuth. 520 Reinhart Leitinger and Markus Rieger F ig . 5. L at it ud e de pe nd en ce o f T ID m od ul at io n in te gr at ed a lo ng s tr ai gh t li ne s fr om a t ra ns m it te r in t he 1 5° E m er id ia n at 1 00 0 km t o a gr ou nd r ec ei ve r at 4 5° N a nd 1 5° E . P an el 1 ( to p) : sn ap sh ot ( si m ul ta ne ou s tr an sm it te r po si ti on s) . P an el s 2 an d 4 (b ot to m ): t ra ns m it te r m ov in g so ut hw ar ds w it h 3° /m in . P an - el 3 : tr an sm it te r m ov in g no rt hw ar ds w it h 3 °/ m in . T ID p ro pe rt ie s: P an el 1 : pe ri od τ = 60 m in , ho r. w av el en gt h λ x = 10 00 k m ; pa ne ls 2 a nd 3: τ = 30 m in , λ x = 50 0 km ; pa ne l 4: τ = 30 m in , λ x = 30 0 km . S ou rc e po in t of t he T ID s: 7 0° N , 3 0° E . H al fw id th o f fa n be am : 10 °, a zi m ut h of c en tr e of f an b ea m : 19 0° . O rd in at es : re la ti ve a m pl it ud es o f T ID f lu ct ua ti on s. F ig . 6. L at it ud e de pe nd en ce o f in te gr at ed T ID m od ul at io n (r el at iv e T ID -T E C ) as i n fi g. 5 . T ra ns m it te r m ov in g w it h 3 °/ m in , so ut hw ar ds i n pa ne ls 1 an d 3, n or th w ar ds i n pa ne ls 2 a nd 4 . T ID p ro pe rt ie s: p er io d τ = 3 m in i n al l ca se s; h or iz on ta l w av e le ng th λ x = 50 0 km i n pa ne ls 1 a nd 2 ; λ x = 30 0 km i n pa ne ls 3 a nd 4 . S ou rc e po in t of t he T ID s: 7 0° N , 60 °E . H al fw id th o f fa n be am : 10 °, a zi m ut h of c en tr e of f an b ea m : 21 0° . O rd in at es : re la ti ve am pl it ud es o f T ID f lu ct ua ti on s. 5 6 521 The TID model for modulation of large scale electron density models 4. The TID model The TID model makes use of the following properties of Atmospheric Gravity Waves (AG- Ws): – the horizontal component kx of the wave vector; – the AGW period τ = 2πω; – the dispersion relation to derive the verti- cal component kz of the wave vector; – the velocity of the disturbance cosv Un n= $ ( ( ) )k s k h h tΨx z$ + + - ~ (where s is the hori- zontal 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 /k2x x=m r , /k2z z=m r , /v kx x= ~ , /v kz z= ~ . The model takes into account the projection factors discussed above, geometric dilution, hori- zontal 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 re- al «source points», meaning that the vertical struc- ture 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 proper- ties 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 pro- file 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 pro- duced by (−kx / kz) in accordance with the dis- persion relation of the AGWs. – The geometric dilution of horizontally travelling AGWs and horizontal attenuation are also taken into account. Fig. 7. Latitude dependence of TID modulated electron density model NeQuick. Geometry and TID properties as in fig. 5, panels 3 and 4. Solid lines: TID modulated electron density. Dotted lines: background model. 522 Reinhart Leitinger and Markus Rieger of observations with model results show that we obtain realistic TID signatures in electron content (see figs. 1 and 5 to 7). 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 success- fully 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 sever- al TID wave trains. An example for a single TID wave train is given in figs. 8 and 9. 5. 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 pa- rameters, one of the most important being satel- lite 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. 8. Example from the TID model: TID in a fixed height of 250 km. Horizontal coordinates: geo- graphic longitude (−10°E-20°E) versus geographic latitude (0°N to 60°N). TID properties: wave period 60 min, horizontal wavelength 2000 km, source point in 70°N, 15°E, half width of fan beam: 5°, az- imuth of centre of fan beam: 195°. Fig. 9. Example from the TID model: TID in a fixed height of 250 km. Isolines of relative TID am- plitude over a geographic coordinate system (longi- tude −10°E ... 20°E versus latitude −90°N ... 90°N). Solid lines: positive levels (0.1, 0.2, ..., 0.9), dotted lines: negative levels (−0.1, −0.2, ... −0.9). TID prop- erties: wave period 60 min, horizontal wavelength 2000 km, source point in 70°N, 15°E, half width of fan beam: 5°, azimuth of centre of fan beam: 195°. The «projection effect» is cleanly seen in the latitude region around 10°N (AGW wind nearly perpendicu- lar to geomagnetic induction vector). 523 The TID model for modulation of large scale electron density models REFERENCES BILITZA, D. (2001): International Reference Ionosphere 2000, Radio Sci., 36, 261-275. FRANCIS, S.H. (1975): Global propagation of atmospheric gravity waves: a review, J. Atmos. Terr. Phys., 37, 1011- 1054. GEORGES, T.M. and W.H. HOOKE (1970): Wave-induced fluctuations in the ionospheric electron content: a mod- el indicating some observational biases, J. Geophys. Res., 75, 6295-6308. HINES, C.O. (1960): Internal atmospheric gravity waves at ionospheric heights, Canad. J. Phys., 38, 1441-1481. 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