Case studies of the propagation characteristics of auroral TIDS withEISCAT CP2 data using maximum entropy cross-spectral analysis

S. Y. Ma1*, K. Schlegel1, J. S. Xu2

1*Max-Planck-Institut fuÈ r Aeronomie, Germany. On leave from Department of Space Physics, Wuhan University, China1 Max-Plank-Institut fuÈ r Aeronomie, D-37189 Katlenburg-Lindau, Germany2 Dept. of Space Physics, Wuhan University, Wuhan 430072, China

Received: 15 January 1997 /Revised: 28 August 1997 /Accepted: 2 September 1997

Abstract. In this paper case studies of propagationcharacteristics of two TIDs are presented which areinduced by atmospheric gravity waves in the auroralF-region on a magnetic quiet day. By means ofmaximum entropy cross-spectral analysis of EISCATCP2 data, apparent full wave-number vectors of theTIDs are obtained as a function of height. The analysisresults show that the two events considered can beclassi®ed as moderately large-scale TID and medium-scale TID, respectively. One exhibits a dominant periodof about 72 min, a mean horizontal phase speed ofabout 180m/s (corresponding to a horizontal wave-length of about 780 km) directed south-eastwards and avertical phase speed of 55m/s for a height of about300 km. The other example shows a dominant period of44min, a mean horizontal phase velocity of about160m/s (corresponding to a horizontal wavelength ofabout 420 km� directed southwestwards, and a verticalphase velocity of about 50m/s at 250 km altitude.

Key words Ionosphere á auroral ionosphere áIonosphere-atmosphere interactions á wavepropagation)

1 Introduction

Considerable progress has been gained in recent years inthe investigation of atmospheric gravity waves (AGWs)and their ionospheric manifestations, the travellingionospheric disturbances (TIDs), (e.g., recent review byHocke and Schlegel, 1996). Particularly the incoherentscatter technique provides a versatile tool for studyingnot only the wave frequencies and vertical wave

numbers (e.g., Hearn and Yeh, 1977; Bertin et al.,1983; Schlegel, 1986; Sheen and Liu, 1988; Rice et al.,1988; Natorf et al., 1992; Ma and Schlegel, 1993;Lanchester et al., 1993; Wuttke, 1994), but also detailsof the fractional variations of various ionosphericplasma parameters such as the electron density, theelectron temperature, the ion drift and the ion temper-ature which constitute the TID (Ma and Schlegel, 1992;Hocke, 1994; Hocke et al., 1996). The inter-relationsbetween the measured TIDs and the AGWs causingthem have been explored in detail with suitable models(e.g., the GIFTS Model, Kirchengast 1996a, b; Kirc-hengast et al., 1992, 1995, 1996).

The propagation of TIDs over large distances cannotbe studied with a single instrument, but need experi-mental facilities which are separated by a suitabledistance. This technique has been successfully appliedusing EISCAT and additional facilities at mid and lowlatitudes (Morgan, 1983; Crowley and McCrea, 1988;Rice et al., 1988; Williams et al., 1988, 1993). Animportant quantity for the propagation of TIDs/AGWsis the complete wave vector of the wave-like ¯uctua-tions. Whereas its vertical component (or ®eld-alignedcomponent) can be easily derived from data of a radarwith a ®xed antenna beam, it is more di�cult to obtainreliable estimates of the horizontal wave number. TheEISCAT CP2 mode was speci®cally designed to pursuesuch studies by cycling through four antenna positions(Folkestad et al., 1983). The Japanese middle and upperatmosphere (MU) radar is also capable of observing theionospheric F-region simultaneously in multiple beamsto study horizontal propagation characteristics of TIDs(Oliver et al., 1988). In practice the relatively shortdistance between these positions in the F-region (of theorder of 100 km) requires a very accurate determinationof the phase front of the TID. Since this front is usuallycalculated from cross-spectral techniques (e.g., Mau-elshagen, 1990; Oliver et al., 1995), a high resolutionrequires in turn su�ciently long time series. Since TIDscan be traced in the data in general only over two orthree wave periods, a proper algorithm for the spectralCorrespondence to: K. Schlegel

Ann. Geophysicae 16, 161±167 (1998) Ó EGS ± Springer-Verlag 1998

estimates has to be selected. Tests have shown that themaximum entropy method (MEM, Ulrych and Jensen1974) yields more reliable spectral estimates than theusual FFT algorithm for time series of a length of 2±3times the wave frequency (Hocke, 1994). We thereforeadopted this technique in our study. Within thehorizontal wavelength range between about 500 and1000 km regarded in our examples the accuracy of thephase front determination proved to be su�cient toobtain reliable estimates of the horizontal wave number.By combining these horizontal components with simul-taneously derived vertical wave-number components,the complete vector of the wave propagation can beobtained. A di�erent approach to estimate wave num-bers has been taken by Lanchaster et al., (1993): theyinserted EISCAT-derived values of wave frequency,®eld-aligned wavelength and mean neutral wind intoHines' (1960) AGW dispersion relation in order toderive horizontal and vertical wave numbers.

2 Data and analysis method

2.1 EISCAT CP2 data

In the CP2 mode the EISCAT radar is capable ofmaking measurements during a rapid antenna scan. Thescan consists of four positions and is completed in 6min.The ®rst three positions form a triangle with vertical,south and southeast directions; while the fourth ismagnetic ®eld aligned. Thus, the EISCAT CP2 modecan provide some information on the anisotropy of theionospheric plasma parameter and the horizontal prop-agation characteristics for wave-like phenomena withcertain temporal and spatial scales.

Figure 1 illustrates the geometry of the EISCAT CP2probing. The azimuth (angle with respect to north,taking clockwise as positive) and elevation of theantenna beam for the four probing positions are:

aA � 180:0� aB � 166:5� aC � 133:3� aD � 182:5�

bA � 89:9� bB � 62:9� bC � 60:4� bD � 77:5

With simple geometric calculations, distances betweenany two probing positions and the azimuth of a lineconnecting the two positions for each height gate can beobtained. At a height of 300 km, the distances betweenthe positions vary between 65 and 170 km. For mediumscale TIDs these distances correspond to a few tenths ofthe horizontal wavelength.

2.2 Maximum entropy cross-spectral analysis method

By means of Fourier cross-spectral analysis the phasedi�erence of a TID at di�erent heights can be calculated.From the height pro®le of the relative phase, the verticalor ®eld-aligned apparent wave-number of the TID canthen be derived, as shown by Ma and Schlegel (1992,1993), and Hocke et al. (1996). In principle, thehorizontal wave-number can also be estimated byFourier cross-spectral analysis. However, the horizontalwave-number of an AGW or TID is usually muchsmaller than the vertical one. This results in small phasedi�erences of the wave-like ¯uctuations of ionosphericparameters at di�erent CP2 positions and requires highresolution cross-spectral analysis. With the help of thecomplex maximum entropy spectral analysis (Ulrychand Jensen 1974) it proved possible to obtain phasedi�erences of two time series with su�cient resolutionfor a data series limited to about 30 data points,corresponding to about 3 h in the CP2 mode. In ouranalysis we actually used 40 points (4 h). So far onlyShibata (1987) has applied multi-channel maximumentropy spectral analysis to HF Doppler data ofmedium-scale TIDs.

Suppose that x�tk� and y�tk� are two real stationaryrandom series, their cross-spectral density (CSD) is acomplex random series which can be expressed as

Pxy�fi� � Cxy�fi� � jQxy�fi� �1�with

Cxy�fi� � �Pbb�fi� ÿ Pxx�fi� ÿ Pyy�fi��=2 �2a�

Fig. 1a, b. Illustration of the EISCAT CP2 mode: a the four probing directions of the antenna beam during CP2; b geometric con®guration of thefour probing positions in a plane at a ®xed altitude

162 S. Y. Ma et al.: Case studies of the propagation characteristics of auroral TIDS with EISCAT CP2 data

Qxy�fi� � �Paa�fi� ÿ Pxx�fi� ÿ Pyy�fi��=2 �2b�Where Pxx�fi� and Pyy�fi� are auto-power spectraldensities of x�tk� and y�tk�, respectively; Pbb�fi� is theauto-power spectral density of b�tk� � x�tk� � y�tk�;Paa�fi� is the auto-power spectral density ofa�tk� � x�tk� � jy�tk�. Paa, Pbb, Pxx and Pyy can beestimated by single channel maximum entropy spectralanalysis. The arguments of the CSD yield the spectrumof phase di�erences between the two real random seriesand can be calculated from the equation

dU�f � � arctan �Im�Pxy�f ��=Re�Pxy�f ���� arctan �Qxy�f �=Cxy�f �� � 2np �3�

In writing the computer programs for the maximumentropy cross-spectral analysis, contributions by Strand(1977) and Chen (1982) have been used.

Figure 2a shows as an example the cross-spectralanalysis results for a simulated Gaussian random seriessuperimposed on a sinusoidal wave with a wavefrequency of 72min. The signal-to-noise ratio (SNR)was about 9 in this case. Figure 2b shows a similarsimulation but here a second sinusoidal wave with awave frequency of 42min has been added. The SNR wasabout 5 in this case and the phases of the two waveswere �90� and ÿ90�, respectively. Based on the centrallimit theorem the Gaussian series are created fromcomputer-produced independent random numbers fol-lowing the same uniform distribution. The number ofdata points of both series was 40, corresponding to a 4-hperiod of CP2 data. It can be realized from Fig. 2 thatthe CSD estimated by the MEM reproduced well the

peaks at 72 and 42min and also the phases of bothwaves. This result which simulates roughly our two TIDevents con®rms the reliability of our algorithm.

2.3 Estimation of the apparent wave-number vectorof the TID

We assume that the TID has the form of a monochro-matic planar wave as:

A�t; r� � Ao cosU�t; r�� Ao cos�xt ÿ k � r� �4�

At two ®xed observing points p1 and p2, the samplingstarts at the times t1o and t2o, respectively while samplingat a rate of 1=dt. The recorded discrete time series of thewave-®eld at the two positions are

A1��t1o � ndt�; r1�and A2��t2o � ndt�; r2� n � 0; 1; 2; . . . ;m �5�Their phase di�erence is

dU12 � x�t1o ÿ t2o� ÿ k � �r1 ÿ r2�� xdto � k21dr �6�

where dto is the time di�erence of the sampling start inp1 and p2, k21 is projection of the wave vector k on thedirection from p2 to p1, and dr is the distance betweenp1 and p2. From Eq. (6) we obtain

k21 � �dU12 ÿ xdto�=dr �7�

Fig. 2. a Amplitude (top) and argument (bottom) of the cross-spectraldensity estimated with the maximum entropy method for two timeseries, composed of a sinusoidal wave with a period of 72min and aGaussian random series. b The same as a but composed of two

sinusoidal waves with periods of 42min and 72min superimposed ona Gaussian random series. The phases of the two sinusoidal wavepairs are 90� and )90�, respectively

S. Y. Ma et al.: Case studies of the propagation characteristics of auroral TIDS with EISCAT CP2 data 163

For CP2 data, dr and dto of any two probing positionsare known. The angular frequency x and the phasedi�erence dU as a function of x can be estimated bymaximum entropy cross-spectral analysis, see Eq. (3).Thus, if p1 and p2 are placed in a horizontal plane, acomponent of the horizontal wave vector along thedirection from p1 to p2 can be obtained. Applied to two(non-parallel) directions in the same plane, a value ofthe horizontal wave vector kh can be determined,representing an estimate at the center of the probedregion. In practice, we omitted those position pairs forwhich the estimated phase di�erences turned out to betoo small. This situation indicates that the correspond-ing position pairs happened to fall in nearly the samephase front. If p1 and p2 are aligned along the vertical,the vertical component of the wave-number vector, kz,can then be obtained. From the vertical and thehorizontal wave-numbers, kz and kh, the complete vectorof the TID, k, can be reconstructed.

3 Case analyses: results and discussion

For these analyses we selected EISCAT CP2 dataobserved on the magnetically quiet day 14 November,1990. The Ap index was 1 and Kp not larger than 1.During the time interval of the observed TIDs the ionvelocity in the F region perpendicular to the geomag-netic ®eld was smaller than 50m/s, implying a very weekconvection. Therefore we neglected the e�ects of theconvection electric ®eld.

Figure 3 displays contours of the height dependentpower spectral density (PSD) of the Ne-TID during0730-1130 UT for position A, i.e. the vertical direction.It shows that maximum values of the estimated PSD aredistributed over a period range of 60±90 min and analtitude range of 150±350 km. Above about 300 km, thePSD over the period range in question fades out.Figure 4 shows contours of fractional ¯uctuations of theelectron density within a period band of 60±90 minversus time and height. The titled phase front is clearlyvisible. The phase propagates downward and is thus atypical feature of the TID. Above about 300 km altitudethe TID is fading away.

Figure 5 shows the height pro®les of the apparenthorizontal wave-number of the Ne-TID (top) and theTe-TID (bottom) in magnitude (left) and azimuth(right). The pro®les plotted as dashed lines are derivedfrom 5 pairs of directions namely (AB, BC), (AB, AC),(BC, BD), (BD, CD), and (AC, BC). The shortest (AD)from the 6 possible directions AB, AC, AD, BC, BD,DC was not used (see Fig. 1b). The close agreement ofall pro®les suggests that the change of the apparenthorizontal wave-number over the area surrounded bythe four probing points is only small. It also proves thereliability of our analysis method. The pro®les plottedby thick solid lines are the averaged kh. It should also benoted that the gross features of the TID in Ne and in Te

below 350 km are quite similar, as should be expected.Above 350 km the TID is not very well developed asalready mentioned. Figure 6 shows the height pro®le of

the complete apparent wave vector of the Ne-TID interms of magnitude, azimuth and elevation of k.

We can see from Figs. 5 and 6 that the magnitude ofthe horizontal wave-number varies slightly with heightbetween 200 km and 350 km. It shows a mean value of0:008 kmÿ1 in this height range, corresponding to ahorizontal wavelength of about 780 km and a horizontalphase speed of about 180m/s. At 300 km altitude themagnitude of the apparent vertical wave-number isabout 0:027 kmÿ1 and the corresponding vertical wave-

Fig. 3. Contours of the power spectral density of the Ne-TIDobserved with EISCAT (CP2, position A) on 14 November 1990,from 0730-1130 UT

Fig. 4. Electron density ¯uctuations within a period band of60±90min as a function of time and height for the same TID eventas shown in Fig. 3 (position A)

164 S. Y. Ma et al.: Case studies of the propagation characteristics of auroral TIDS with EISCAT CP2 data

length and phase speed about 230 km and 55m/s,respectively. The apparent vertical wave-number isnegative, which means a downward propagation of thephase. These propagation characteristics suggest amoderately large-scale TID. The ratio of vertical tohorizontal wave-number magnitude decreases fromabout 4.3 at 200 km height to about 2.0 at 350 km,indicating that the propagation of the TID tends to

become more horizontal with increasing height. Theazimuth changes strongly in the altitude range belowabout 250 km from south to east, and stays approxi-mately constant at eastsoutheast above 300 km altitude(see Fig. 6b). This suggests that the source which excitedthe AGW is located northwest of the observer, probablyin the auroral oval region before dawn.

The other TID studied occurred in the afternoon onthe same day. It exhibits a dominant period of about44min. Figure 7 shows contours of the fractional¯uctuations of the electron density between 12 and16 UT as a function of altitude. Again, above about300 km the TID disappears. The height pro®les of themagnitude (left) and the azimuth (right) of the apparenthorizontal wave vector are plotted in Fig. 8 (top panels).The pro®les are derived using position pairs from thedirections A, B, and C. The horizontal wave-numbervaries between 0:01 kmÿ1 and 0:02 kmÿ1 correspondingto a mean horizontal wavelength of 420 km and ahorizontal phase speed of about 160m/s. The verticalwave-number at 250 km altitude is about 0:05 kmÿ1,corresponding to a vertical wavelength of about 125 kmand a vertical phase speed of about 50m/s. These datacharacterize a typical medium-scale TID. It can also beseen from Fig. 8 that this afternoon TID propagatessouthwestwards (azimuth about 230�), suggesting asource located probably in the auroral oval after dusk.The lower two panels of Fig. 8 show the height pro®lesof magnitude and elevation of the apparent full wavevector of this TID. In comparison with the earlier72-min TID the elevation angle is practically constantwith altitude.

The estimation of the apparent wave vector of theTID contains some uncertainties. One cause of uncer-tainty is the assumption of a planar wave on whichEq. (7) is based. In fact, this assumption is valid for the®eld far from its source. For TIDs observed in theauroral ionosphere during daytime on a magnetic quietday the excitation source is usually located far away inan auroral sector before dawn or after dusk. This is thecase in our examples and therefore this assumption isvalid. The other uncertainty stems from the fact that thehorizontal wave vector estimated from CP2 data by

Fig. 5. Magnitude (left column) and azimuth (right column) of theapparent horizontal wave number, kh, versus altitude of the Ne-TID(top row) and the Te-TID (bottom row) with a period of 72min. Thedashed lines denote the kh pro®les derived from 5 combinations of the4 di�erent probing positions. The thick solid lines represent theaveraged kh

Fig. 6a±c. Height pro®le of theapparent wave vector, k, of theNe-TID for a period of about72min. a Magnitude of k; bazimuth of k with north and eastdirection as positive; c elevationangle of k with upward directionas positive

S. Y. Ma et al.: Case studies of the propagation characteristics of auroral TIDS with EISCAT CP2 data 165

cross-spectral analysis is valid not exactly for a commonprobing point, but approximated by a common regionaround its center. This approximation would also causeerrors in the estimation of the horizontal wave vector. Inaddition, our analysis method of EISCAT CP2 data isvalid only for TIDs with a horizontal wavelength belowabout 1000 km. For example, the distance betweenposition A and position C is about 162 km at a heightof 285 km. A horizontal wave vector withkh � 0:0074 kmÿ1 (actual value of the ®rst TID at285 km) and an azimuth of 120� would have a compo-nent with a magnitude of 0:0072 kmÿ1 (corresponding toan apparent wavelength of about 872 km) in thedirection from position A to position C. For a waveperiod of 72min, this would mean a phase di�erence ofdUCA � 33:4� (corresponding to a time di�erence about6:7min) for the phase fronts observed at position C andposition A. Such a phase di�erence (or time di�erence)can be easily resolved by our analysis method. In case ofa large-scale TID with a horizontal wavelength of2500 km (3±6 times the wavelength of our examples), thecorresponding time (or phase) di�erence would become0:7min (or 3:3�). Such a small value is very di�cult toresolve by the present probing mode and analyzingmethod. That means, the propagation of large-scaleTIDs over large distances cannot be studied with a singleinstrument, but need experimental facilities which areseparated by a suitable distance as mentioned above.

4 Summary

EISCAT CP2 data observed on a magnetic quiet day areanalyzed to investigate propagation characteristics of aTID. By means of a maximum entropy cross-spectralmethod, both vertical and horizontal apparent wave-number of the TID in the auroral F region are obtainedas a function of height. The main results can besummarized as follows.

1. For the ®rst time the complete wave propagationvector of TIDs as function of height was derived byusing the maximum entropy method in a cross-spectralanalysis of EISCAT CP2 data.2. The resultant apparent wave-number vectors showthat their vertical components decrease smoothly withaltitude, while their horizontal components are moreconstant. In our ®rst case the azimuth changes stronglyat heights below 250 km. The mechanism of thisvariation requires further study.3. The main periods of the considered TIDs are about72min and 44min, respectively. For the ®rst event themean horizontal wave-number is 0:008 kmÿ1 between200 and 350 km altitude, corresponding to a horizontalwavelength of about 780 km and a phase speed of180m/s. At about 300 km altitude the vertical wave-number is ÿ0:027 kmÿ1, corresponding to a verticalwavelength of 230 km and a phase speed of 55m/s. Thevertical and horizontal wave vectors are directeddownwards and southeastwards. These features indicatethat the considered event is a moderately large-scale

Fig. 7. Contours of electron density fractional ¯uctuations within aperiod band of 40±50min versus time and height for 14 November1990 (position A)

Fig. 8.Height dependent apparent full wave vector of the Ne-TID witha wave period of 44min.Upper left panel: magnitude of the horizontalwave vector; upper right panel: azimuth of the full wave vector; lowerleft panel: magnitude of the full wave vector; lower right panel:elevation angle of the full wave vector

166 S. Y. Ma et al.: Case studies of the propagation characteristics of auroral TIDS with EISCAT CP2 data

TID. The apparent horizontal wave number for thesecond TID is about 0:015 kmÿ1 between 200 and300 km altitude, corresponding to a horizontal wavelength of about 420 km and a phase speed of 160m/s.The vertical wave-number at 250 km altitude is about0:005 kmÿ1, corresponding to a vertical wavelength of125 km and a phase speed of about 50m/s. This wavepropagates mainly southwestwards and can be charac-terized as typical medium-scale TID.

Acknowledgements. The authors thank the director and sta� ofEISCAT for operating the facilities and providing the data. TheEISCAT Scienti®c Association is funded by scienti®c agencies ofFinland (SA), France (CNRS), Germany (MPG), Japan (NIPR)Norway (NF), Sweden (NFR) and the United Kingdom (PPARC).This work is supported by China National Foundations of NaturalScience. One of the authors S. Y. Ma appreciated the hospitality ofthe Max-Planck Institut fuÈ r Aeronomie during a stay in Germanysupported by the DFG.

Topical Editor D. Alcayde thanks G. Kirchengast and B. S.Lanchester for their help in evaluating this paper.

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