Fluctuations in the direction of propagation of intermittent low-frequency ionospheric waves

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011ja017242, 2012 Fluctuations in the direction of propagation of intermittent low-frequency ionospheric waves H. Sato, 1 H. L. Pécseli, 1 and J. Trulsen 2 Received 10 October 2011; revised 8 January 2012; accepted 23 January 2012; published 27 March [1] Low-frequency (8 28 Hz), long-wavelength electrostatic waves in the ionospheric E region over northern Scandinavia are studied by using data obtained from an instrumented rocket having four probes mounted on two perpendicular booms. Two data sets are available, one for upleg and one for downleg conditions with somewhat different ionospheric parameters. The ionospheric plasma is unstable with respect to the electrostatic Farley-Buneman instability in both cases, but the DC electric field is somewhat enhanced during the downleg part of the flight. We find that the direction of wave propagation as given by the local normalized fluctuating electrostatic field vector varies randomly within an interval of aspect angles. The distribution of the directional change per time unit is determined. The waves propagate predominantly in the electrojet direction, but significant variations in directions can be found, both with respect to the magnetic field (the aspect angle) and with respect to the electrojet direction. Some of our results are in variance with related radar observations in the electrojet near the equator. Indications of significant spatial intermittency of the signal is demonstrated. Large-amplitude electrostatic fluctuations are confined to spatially localized regions and have a narrower aspect angle distribution with reduced directional fluctuations. We introduce an intermittency measure based on average excess time statistics for the record for the absolute value of the detected time-varying electric fields. We thus determine the average of time intervals spent above a prescribed amplitude threshold level. The results are compared with an analytical expression obtained for a reference nonintermittent Gaussian signal. The general analysis requires the joint probability density of signal amplitude and its time derivative to be known. The analytical models for quantifying the intermittency effects were tested by synthetic time series allowing study of the transition from non-gaussian to Gaussian random signals. Citation: Sato, H., H. L. Pécseli, and J. Trulsen (2012), Fluctuations in the direction of propagation of intermittent low-frequency ionospheric waves, J. Geophys. Res., 117,, doi: /2011ja Introduction [2] Low-frequency longitudinal waves can be excited by plasma instabilities in the ionospheric E region, in the equatorial as well as the polar regions. The waves were detected first by radar scattering [Olesen and Rybner, 1958; Bowles et al., 1963; Balsley, 1969] and later studied by instrumented rockets [Prakash et al., 1971; Olesen et al., 1976; Bahnsen et al., 1978; Pfaff et al., 1985] with a summary and bibliography given by Kelley [1989]. Studies of the ionosphere by radar and in situ rocket measurements are to some extent complementary. While radar scattering relies on a wave vector matching condition being fulfilled [St.-Maurice and Schlegel, 1983] without particular conditions on the amplitudes of the 1 Department of Physics, University of Oslo, Oslo, Norway. 2 Institute of Theoretical Astrophysics, University of Oslo, Oslo, Norway. Copyright 2012 by the American Geophysical Union /12/2011JA plasma oscillations, the analysis of rocket data will give results dominated by the largest-amplitude components of the signal with less restrictions on their frequency/wave number distributions. Also, a rocket is ideally making a point measurement (at least for waves with length scales much larger than the rocket), while a radar averages over a scattering volume. In the present analysis we report results based on studies of data from the ROSE rocket campaign carried out in northern Scandinavia [Rose et al., 1992; Rinnert, 1992]. [3] The aim of the present study is to analyze the direction of electrostatic wave propagation in high-latitude E region turbulence [Pfaff et al., 1997]. Bulk variations in the direction of propagation with altitude were illustrated by Krane et al. [2000] with some earlier qualitative studies presented by Rinnert [1992]. The results by, for instance, Krane et al. [2000] were obtained by a local cross-correlation technique, which involves averaging over an altitude interval. Here we want to obtain results with the sampling resolution of the rocket instruments and also to analyze the time variability of the directions of the electric field vectors. The probe 1of20

2 configuration on the ROSE rockets is standard, and the results concerning field resolution obtained here can have general interest. [4] Two frames of reference are distinguished, the moving rocket frame and a fixed Earth or ground frame. For the former, it is most natural to take the z axis along the rocket axis; for the latter we choose the z axis along the Earth s magnetic field B 0 and the x direction being west east. [5] Ionospheric waves are in many cases propagating in the direction approximately perpendicular to the local magnetic field, i.e., k? B 0. For electrostatic waves in which the fluctuating electric field ee is parallel to k (apart from a sign), this implies that also ee? B 0. The associated density fluctuations will give rise to coherent radar backscatter from these waves only when the radar beam is also directed perpendicular to B 0. For cases with many wave components with distributed wavelengths propagating in a narrow cone of directions close to being perpendicular to B 0, the radar is in effect performing a Fourier transform of the medium [Bekefi, 1966; St.-Maurice and Schlegel, 1983] and selecting only those components satisfying the wave vector matching condition mentioned before. For these cases a narrow aspect sensitivity of the radar [Kudeki and Farley, 1989] indicates electrostatic wave propagation almost perpendicular to B 0. For a wavefield composed of many plane electrostatic waves we have ZZZ Er; ð tþ ¼ Ek ð Þe iðwðkþt krþ d 3 k ZZZ ¼ EðkÞ k k e i ðwðkþt krþ d 3 k; ð1þ where Ek ð Þ k and the dispersion relation w ¼ wðkþ is assumed to be known. The integral gives the weighted average of the electric field vector at a space-time position ðr; tþ and at the same time also a correspondingly averaged direction of propagation, where EðkÞ enters as a weight function for the direction of the unit vector k=k. Any wavefield can be decomposed into Fourier components, but the model (1) in addition assumes that a dispersion relation wðkþ can be defined locally. Using this assumption, we imply that the scale lengths for variations in plasma parameters, such as density, temperature, and collision frequencies, are large compared to the wavelengths included in the integration. [6] We use (1) to obtain the expression for the time derivative of the electric field in the form ZZZ deðr; tþ ¼ i wðkþeðkþ k dt k e i ðwðkþt krþ d 3 k: ð2þ For stationary random processes we have a simple relation between the power spectrum of the electric field and the corresponding spectrum of its time derivative. As an order of D magnitude Eestimate, expression (2) will indicate ðdeðr; tþ=dtþ 2 w 2 E 2 ðr; tþ. [7] While the probe measurements on the rocket give an estimate for the local electric fields, the radar scattering relies on density fluctuations. For short-wavelength fluctuations propagating in the direction perpendicular to B 0 we have good indications that the electrons are not in a local Boltzmann equilibrium [Mikkelsen and Pécseli, 1980; Krane et al., 2000]. For long wavelengths, low frequencies, this approximation can, however, be justified [Krane et al., 2000]. We can thus assume n 0 þ en n 0 exp ef=t e e or en=n 0 ef=t e e, implying that ee ken for this limit. For small k even a large density variation will then only give a modest electric field, and vice versa for large k. Also in this respect we find the radar and rocket diagnostics to be supplementary. [8] Data from the ionospheric plasma would be best studied by probe configurations having scale sizes small compared to the characteristic scales of the plasma disturbances, so that we can effectively treat the results as originating from a point measurement. Unfortunately, this condition is only rarely fulfilled, and a part of our study addresses the problems related to finite probe separations on the rocket. The electric field magnitude and direction estimated from the probe measurements will generally be different from the true values, and we discuss these errors. These discussions will have a general nature, and the conclusions will be relevant for other similar probe configurations. [9] Intermittency effects are often found to be an important characteristic of turbulent fluctuations in plasmas as well as in fluids. For short-wavelength E region irregularities we have investigated such phenomena using standard structure function analysis [Dyrud et al., 2008]. Our present study contains a study of intermittency for the long-wavelength part of the spectrum. For this limit we introduce a different, alternative, intermittency definition which we believe to have general interest and applicability. [10] The paper is organized as follows: In section 2 we describe the conditions for the rocket experiment as well as the basic observations. Intermittent features in the data are tested in section 3. In section 4 we suggest a plasma instability that can account for the basic features of the observations. Section 6 contains our conclusions. The main text of the paper addresses observations, while Appendix A contains analytical models for the intermittency effects that we observe together with a test of intermittency using synthetic signals. 2. Electrostatic Waves Observed in the Ionospheric E Region [11] During the ROSE rocket campaign [Rose et al., 1992; Rinnert, 1992], an instrumented payload F4 was launched 9 February 1989, 23:42:00 UT from Kiruna, Sweden. The apogee of this flight was km. The declination of the momentum vector (within 2 this is parallel to the rocket axis) was 66.6 at launch, to vary slowly with time. Good quality data were obtained on the upleg as well as on the downleg parts of the flight with 20 km horizontal separation in the E region. The DC electric field strength was changing during the flight (typically E 0 40 mv m 1 upleg and E 0 60 mv m 1 downleg), so in reality we have data from two independent experiments. (Related numerical studies of Dyrud et al. [2008] use the maximum peak value of E 0 70 mv m 1 to emphasize the nonlinear features.) The changes in E 0 are mostly in magnitude: the direction is almost constant. The rocket uses 10 s to traverse the active part of the E region. Summary figures for the changes in E 0 are presented by Rinnert [1992] and Dyrud et al. [2006]. A simultaneous measurement of the E 0 B 0 =B 2 0 drift by 2of20

3 geometry also after launch, ignoring also possible vibrations in the booms. [13] We analyzed the fluctuating signals U 6 (t) =f 1 (t) f 2 (t); U 5 (t) =f 4 (t) f 3 (t); U 4 (t) =f 1 (t) f 4 (t); U 3 (t) = f 2 (t) f 3 (t); U 2 (t) =f 1 (t) f 3 (t); and U 1 (t) =f 2 (t) f 4 (t), where f j (t) forj = 1, 2, 3, 4 denotes the potential on the jth probe with respect to a suitably defined common ground. There is a redundancy in the available signals, which can be used to check the performance of individual probes. For wavelengths much larger than the probe separations, the potential difference signals can be used to estimate the fluctuating electric fields, ee. The signals were digitized with 12 bit resolution. The space-time varying electric field fluctuations were sampled at time intervals of 0.5 ms, giving a Nyquist frequency of 1000 Hz. The electric circuits give an effective frequency limitation closer to 600 Hz. Samples of raw data are shown by Krane et al. [2000]. The DC electric field E 0 was measured by the same set of probes. A summary of the electron drift velocity deduced from DC electric field variations during the flight is given in Figure 2. The altitude variation of the DC plasma density is shown in Figure 3. [14] Many basic tests can be carried out to determine the reliability of the data. The easiest analysis consists of basic check sums: inspection of Figure 1 shows that sums of selected signals should ideally vanish such as, for instance, U 6 (t) +U 3 (t) U 2 (t) = 0. We have made these checks and Figure 1. Schematic diagram for positioning of the probes on the ROSE F4 rocket. EISCAT is consistent with the results from the ROSE F4 rocket [Dyrud et al., 2006]. A summary of our previous results from studies of these data are given by Krane et al. [2010]. [12] The ionospheric conditions and details of the instrumentation relevant for the present data set were discussed in a special issue of Journal of Atmospheric and Terrestrial Physics (54, , 1992) and also in a detailed report [Rose et al., 1990]. For completeness, we here summarize some of the basic parameters of the flight. The ROSE F4 rocket was launched in a direction perpendicular to the Hall current of the auroral electrojet. The spin period of the rocket was 0.55 s. The corresponding time for the coning motion was 5.8 s with a cone angle of 2. The ELF signals analyzed were obtained by gold-plated spherical probes of 5 cm diameter [Rinnert, 1992], mounted on two pairs of booms, one near the top of the payload (labeled 1 and 2) and the other 185 cm lower (labeled 3 and 4), oriented at an angle of 90 with respect to the first pair, as illustrated schematically in Figure 1. The length of each boom was 180 cm, giving a probe separation of 360 cm on each boom pair. We denote the boom length a and the separation along the rocket axis as b in the following. The following analysis assumes that the booms and probes are positioned in this ideal Figure 2. Diagrammatic representation of the ROSE4 rocket trajectory, shown by the dots [Rose et al., 1990; Dyrud et al., 2006]. The dashed lines give the direction of the magnetic field. The apogee of this rocket was 123 km. Arrows indicate the direction and magnitude of the E 0 B 0 drifts deduced from the DC electric fields detected by the rocket. See also a related figure presented by Rose et al. [1992, Figure 3] and Rinnert [1992, Figure 1]. The circle with an arrow indicates an EISCAT observation. 3of20

4 Figure 3. Altitude variation of the DC plasma density during the ROSE F4 flight. The drawing is made on the basis of data from Rose et al. [1990]. Note linear scale on both axes. In the active part of the E region, km altitude, we estimate the vertical scale length for the density variation to be L 10 km. find them to be satisfied within 3% accuracy. The deviations have no correlations with the amplitudes of the probe signals. [15] The basic ionospheric parameters, such as collision frequencies and temperatures vary with altitude as illustrated by Krane et al. [2000] or Dyrud et al. [2006] for a standard ionosphere. In the active part of the E region the sound speed increases with 10%, while the variations in the difference between the ion and electron drift velocities is insignificant. Electron drifts obtained from the present experiments are shown in Figure 2. [16] The electron and ion temperature measurements on the rocket gave only qualitative results, but measurements of the neutral gas density and temperature [Friker and Lübken, 1992] indicated a neutral temperature of K, which can be used as an approximation also to the ion temperature. [17] Studies of the ROSE F4 data demonstrated that a broad band spectrum of low-frequency electrostatic waves propagate in the E 0 B 0 direction with phase velocities in the range of m s 1 as shown by Iranpour et al. [1997], Krane et al. [2000] and Dyrud et al. [2006]. Similar rocket experiments were performed in Greenland [Bahnsen et al., 1978]. Local cross correlations of data from one of these rockets with RMS fluctuation levels of 4 8 mvm 1 were presented by Pécseli et al. [1993]. The velocities deduced from these data also show wave propagation in the direction perpendicular to the rocket. This velocity has a magnitude and also altitude variation very similar to those found in our present work, although those features were not discussed by Pécseli et al. [1993]. [18] The observed wave frequency range on the ROSE F4 rocket is Hz. Use of the experimentally obtained dispersion relation [Iranpour et al., 1997; Krane et al., 2000; Dyrud et al., 2006] implies that this frequency range corresponds to wavelengths in the range of m. The shortest wavelengths are strongly filtered by the two-point probe sampling of the electrostatic field [Kelley and Mozer, 1973; Pfaff et al., 1984; Krane et al., 2000]. The data can thus approximate the fluctuating electric fields only for wavelengths significantly exceeding the probe separation (see Figure 1), corresponding to frequencies below 28 Hz, approximately, when we use the observed propagation velocity for an estimate. We filter the data correspondingly and consider only a limited spectral range. (In previous related studies by Rinnert [1992], the speed of propagation was not known, and too short wavelengths were included in the data analysis.) We filter the data with a band-pass filter 8:28 Hz, whereby we at the same time remove the rocket spin frequency and its first harmonics, and also ensure that the filter bandwidth is larger than the average filter frequency. (For narrow filter bandwidths, it will be this bandwidth that determines the time variability of the signal output.) The filtered data do not contain any contribution from the induced U R B 0 electric field, with U R being the rocket velocity vector Electric Field Estimates [19] Our electric field estimates are based on the following probe combinations: E x = U 6 /2a, E y = U 5 /2a. Ideally, we could use E z = (U 3 + U 4 )/2b just as well as E z = (U 1 + U 2 )/2b, the two signals being identical. However, due to imperfections in the setup there are small differences up to at most 5%, so we here use the average value E z = (U 3 + U 4 + U 1 + U 2 )/4b, with the z axis being along the rocket axis. These combinations would give the exact result for a constant electric field (like the ambient electric field E 0 ) in an arbitrary direction. We will now analyze the geometrical errors made by using these estimates for a spatially varying potential field. [20] We can take a plane wave propagating in an arbitrary direction as a model for the electrostatic potential variation, i.e., f(x, y, z, t) =A cos(k x x + k y y + k z z wt + y), where y is a phase. The total phase addition to k r is wt + y, so without loss of generality we can take t = 0 and let y represent all of the phase. We obtain E x ¼ A a sin k zb 2 þ y sinðk x aþ; E y ¼ A a sin k zb 2 y sin k y a ; E z ¼ A b cos k zb 2 y k z b cos k y a cos 2 þ y cosðk x aþ : ð3þ The combination (U 1 + U 2 )/2b gives the same result for E z as in (3). The detected electric field components vanish for any phase value y if some simple geometric conditions are fulfilled, for instance, when k x = k y = 0 and k z b =2pn 4of20

5 Figure 4. Illustration of directional error (with color scale in degrees) when using the approximation (3) to represent the electrostatic electric field. Each sphere corresponds to one wave number, i.e., one wavelength. The thin lines indicate the relative positions of the booms and probes and also define the coordinate system. The relative lengths and separations are in correct scale. The vertical line presents the rocket axis. We average over phases y in the plane wave model and give results for four wavelengths, l = 35, 23.3, 17.5, and 14 m from the top left sphere to the bottom right sphere. with n = 1, 2,, with obvious generalizations [Kelley and Mozer, 1973; Pfaff et al., 1984]. The result has a simple geometrical interpretation: the condition implies that the probe separations 2a and b are multipliers of half wavelengths. [21] Note that by the definitions in (3) we effectively consider the rocket as a point probe, and information concerning phase differences from the probe sets giving U 1 and U 2 is lost. Similarly, they are lost for the probe sets giving U 3 and U 4. When available, this phase information can be utilized to estimate the components of the propagation velocity that is perpendicular to the rocket axis. Thus, Bahnsen et al. [1978] used this information to obtain the velocity components perpendicular to the rocket body. Later studies [Iranpour et al., 1997; Pécseli et al., 1989, 1993] used a cross-correlation technique to calculate the time delay of propagation in the direction perpendicular to the rocket and thereby found a characteristic average phase velocity component. It can also be noted that the signal sets {U 1, U 2 } and {U 3, U 4 } are in quadrature and give orthogonal velocity components, so that the speed and direction of propagation can be obtained at the same time [Krane et al., 2000, 2010]. With launches perpendicular to the electrojet as here (see Figure 2) and expected wave propagation along the E 0 B 0 direction, this analysis will basically give the entire phase velocity, not only a component. When the phase velocity is perpendicular to the rocket momentum vector (as observed here), the contribution from the Doppler shift is negligible. The methods outlined here work best when the phase changes from, e.g., U 1 and U 2 are large, implying wavelengths being short or comparable to the probe separations. [22] The true electric n field o in the plane wave model has components Ex t ; Et y ; Et z ¼ A k x ; k y ; k z sinðk r þ yþ, where the propagation direction is given by the k vector. At the rocket reference position (the geometrical center of the probes) we have {E t x, E t y, E t z }=A{k x, k y, k z }sin(y), to be compared with (3). Using sin(k x a)/a k x for a 0, etc., it is easily seen that the differences between the two fields E and E t vanish in the limit where a 0 and b 0, i.e., when the probe separations are much shorter than the wavelength. Since an arbitrary electric field variation can be described by a superposition of plane waves, we can use this single wave as an adequate model. In particular, we can give results for the error that we make concerning the electric field direction and magnitude by using the estimates (3) instead of the true electric field. We here define the error in direction by the angle arccosðe E t = jejje t jþ. [23] It is important to emphasize that, for instance, signal combinations such as { U p 6 /2a, U 5 /2a, U 1 /b} and U 6 =2a; U 5 =2a; U 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a 2 þ b 2 will not reproduce the correct field components in the long-wavelength limit. The purpose of the study summarized in their work was, however, different: the important information was contained in the phase difference for the signal as it propagated from, e.g., probe set (1 3) to (2 4). [24] The basic parameters for the present problem are {k x, k y, k z } and y. For each wave vector k, the phase y can take any value in the range {0, 2p}. Figure 4 shows the average error in the direction of propagation when using (3) to represent the true electrostatic electric field E t in the ionosphere. Each sphere corresponds to one wave number, i.e., one wavelength with an averaging over all y. A point on the spheres correspond to a direction for the wave propagation given by k and the color coding gives an indication of the error, with scales given by the color bar. We show results for four frequencies f = 10, 15, 20, 25 Hz which translate to four wavelengths l = 14, 17.5, 23.3, and 35 m by use of a characteristic phase velocity of u f = 350 m s 1. The frequencies are within the band-pass filter, and the phase velocity can be taken as representative for the observations for these conditions as reported by Iranpour et al. [1997], Krane et al. [2000] and Dyrud et al. [2006]; see also summary by Krane et al. [2010]. [25] The results in Figures 4 give the average over all phases y for the wavelength determined by each sphere. For individual y values, the error can be larger, in particular when y 0orpwhere {E t x, E t y, E t z } {0, 0, 0}. For vanishing electric fields (at phases y =0orp) the field direction is thus undefined and the error becomes large. In particular, we might experience that a local small electric field can be detected as having the opposite direction. [26] While Figure 4 gives the average angular error for a given direction and wavelength, Figure 5 gives the average and the variance in the angular error distribution for a given phase and wavelength but now as an average over all directions of propagation, i.e., averaging over the corresponding spheres in Figure 4. For phases y different from 0 and p we find that for large wavelengths in the interval considered, the error in estimating direction and magnitude by (3) is negligible and errors are moderate even for the shortest wavelengths in the given range (see Figure 5). Considering, 5of20

6 Figure 5. (top) Average and (bottom) variance in the distribution of directional errors for given wavelength and for varying phase y obtained for each y by averaging over all directions of propagation for a given wavelength, i.e., average over the surface of the spheres in Figure 4. The curves are for the four wavelengths used in Figure 4, with the top curve corresponding to the shortest l value and subsequently longer wavelengths corresponding to the lower curves. for instance, the longest wavelength, we can obtain an average error below 10 for a phase angle interval of 140. Even for the shortest wavelength considered, the average error is below 10 for a phase interval of 90.We extended the analysis and found that larger errors occur when we had half wavelengths l/2 comparable to or shorter than the double probe separation, where we note that the resolution along the rocket axis is almost twice as good as in the perpendicular direction with the given geometry of the probe positions. [27] As expected, we find the largest errors at phase values y 0 and p, where the electric field amplitude at the rocket position vanishes. This result indicates that it is an advantage to omit the smallest field magnitudes from the analysis when considering estimates of local electric field directions and intensities. By doing this we reduce the probability of including phase values y close to 0 or p, giving a better estimate of the distributions in the local electric field directions in the parts of the signal that remain. [28] For presenting the uncertainty on the amplitude estimates, we take the difference between the estimate and the true value and then divide by correct wave amplitude and give the result for the error in percent. The true value is Ak obtained directly from je t j for the phase y = 0, while the q amplitude ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi error estimate is found from (3) by maximizing Ex 2 þ E2 y þ E2 z by varying the phase y. In Figure 6 we summarize the error in the field amplitude estimate relative to the actual electric field amplitude. The explanation for the presentation of Figure 6 is the same as for Figure 4. For the wavelength interval determined by the filtering of data, we found that errors in the amplitude estimates are generally below 10%, so only the directional errors are significant. The smallest relative errors in both Figures 4 and 6 are found for propagation along the rocket axes, i.e., the direction where the probes are closest (1.85 m versus 3.6 m). [29] The approximate rocket velocity at launch is 1kms 1. At the ionospheric E region it is 500 m s 1. The observed frequencies are due to a combination of actual wave frequency and a Doppler shift, k U R. The waves analyzed by Iranpour et al. [1997], Krane et al. [2000] and Dyrud et al. [2006] propagated perpendicular to the rocket axis (which, within 2, is the angular momentum vector), so Doppler shifts are immaterial for those results. The present analysis also finds wave vectors with a distribution of directions, where Doppler shifts can be relevant. To have a Doppler shift correction of 10 Hz, i.e., near the middle of our frequency band, we need a wavelength of 50 m or shorter for waves propagating along the U R direction. For the downleg part of the flight, this will correspond almost to magnetic fieldaligned wave propagation (see Figure 2). If these fieldaligned wave components propagate up and down along the field lines with equal probability, the up shift and down shift in frequencies will be asymmetric in the rocket frame. Figure 6. Illustration of the error (with color scale in percent) in the relative electric field amplitude when using (3) to represent the electrostatic electric field. For each wavelength the amplitude is found by maximizing over all phases y. See Figure 4. The error has a negative sign since the estimate always gives an E field magnitude smaller than the actual one. 6of20

7 Figure 7. Sample of data taken in the rotating rocket frame of reference, showing electric field vectors represented by lines with correct relative magnitudes in arbitrary units placed along the time axis. Times are given in seconds after launch. The three x, y and z vector components are all in the same arbitrary units. Only every fifth time step is shown. The data are for downleg conditions. The time interval shown corresponds approximately to two spin periods of the rocket. The consequence of such Doppler shifts will affect only the frequency distribution of the waves and not the electric field amplitudes and directions assigned these fields. In particular, the averaging implied in expression (1) will remain accurate, but we cannot determine the range of wave vectors k entering the integral. This estimate can be made only for waves propagating approximately perpendicular to U R (and thus for the B 0 direction on the downleg part of the flight), since we know a characteristic velocity of propagation for this direction from a local cross-correlation analysis [Iranpour et al., 1997; Krane et al., 2000] Rocket Frame Analysis [30] The simplest data analysis refers to the rotating rocket frame, where the electrostatic potentials are detected. This frame was used in earlier studies [Bahnsen et al., 1978; Pécseli et al., 1989; Iranpour et al., 1997; Pécseli et al., 1993], where the average phase velocity of the waves was estimated. By analysis of the frequency variation of the cross-phase signal, wave dispersion could be estimated [Krane et al., 2000]. Within the experimental uncertainty, no evidence for significant dispersion was found. [31] Figure 7 shows variations of the estimated electric field vector with time, as detected in the spinning rocket frame. The lines represent the magnitude and direction of the local electric field vectors. The selected time window is in units of time after launch and covers approximately two spin periods. The electric field vectors in this rotating frame are transformed to a fixed ground frame of reference for obtaining absolute vector directions. [32] Some quantities, such as the altitude variation of the RMS field amplitudes, are independent of the frame of reference. Others, such as the directional variations of the electric fields, are relevant only in the fixed ground frame Fixed Ground Frame Analysis [33] Any point in the active ionosphere can be assigned an electric field vector at any time, as illustrated in Figure 7. The electric field can be fluctuating in amplitude as well as direction. Part of the observed variation in the rocket frame is due to the rocket spin, which can be removed by proper transformations, while parts have origins in the plasma dynamics. Two extremes can be considered: the variations can be (1) purely temporal or (2) purely spatial and observed as temporal variations only due to the movement of the rocket, in which case any observed frequency is due to a Doppler shift. This latter limit is often called the assumption of frozen turbulence, or Taylor s hypothesis in studies of neutral turbulence [Shkarofsky, 1969]. It is not logically possible to distinguish these two extreme states by a single point measurement, with a moving detector. Since, however, the phase change analysis of the high-frequency component by cross correlation showed a well-defined propagation velocity in the direction perpendicular to the rocket axis, it is reasonable to assume that the plasma is in a dynamic state. [34] Part of the time variations is due to the rocket spin, and therefore it is physically irrelevant. As long as we can consider the rocket as a point-like detector, there is no ambiguity in the magnitude and direction of the local electric field, so we can transform the field vector to a fixed frame. The data are thus transformed to a fixed ground frame, with the z axis parallel to the ambient magnetic field. The transformation is performed by using the proper rotation matrices [Temple, 1960]. The transformation matrices use analytical approximations for the rotation and coning phases of the rocket motion in the form of polynomial approximations with time after launch as variable Directional Distributions [35] The electric field amplitude variations with altitude have been illustrated previously [Iranpour et al., 1997; Krane et al., 2000]. Here we study primarily the variations in field directions. In principle, two angles suffice to specify the direction of a vector. We use (1) the angle between the electric field and B 0 and (2) the angle between west east direction and the projection of the electric field vector on the plane perpendicular to B 0 (see Figure 8 for definitions). These choices are justified by the expected properties for plasma instabilities driven by the electrojet current, giving E 0 B 0 as the preferred direction, which is here close to west-east. [36] To quantify the distribution in directions, Figure 9 shows the probability density (PDF) for the angle q B between the local average electric field vector as defined by (1) and estimated by (3), for the spectral range defined before, and the magnetic field B 0, for both upleg and downleg conditions. 7of20

8 Figure 8. Definition of the two angles q B and q WE analyzed in the present study, using a representation with B 0 along the z axis, while the x axis is in the west east direction. In this presentation we have shown the south north vector pointing into the plane. Figure 10. Probability density for the angle q WE between west east direction and the projection of the electric field vector on the plane perpendicular to B 0. (left) Upleg and (right) downleg conditions with threshold values as in Figure 9. The angle 0 corresponds to due east. Figure 9. Probability densities of the angle q B between the fluctuating electric field vector and B 0, for (left) upleg and (right) downleg conditions for several electric field amplitude threshold levels, i.e., no threshold, 15%, 30% and 45%. The vertical axis is probability density. The threshold level is inserted in terms of percent of the maximum value E m of the record and also in related Figures [37] Figure 10 shows the probability density for the angle q WE between west east direction and the projection of the electric field vector on the plane perpendicular to B 0. [38] The basic analysis includes all electric field vectors, irrespective of magnitude. More information can be gained, however, by a selective study, where only a subset of vectors are included in the analysis, i.e., cases where jej exceeds some reference level. Results for several amplitude thresholds are shown, i.e., cases without threshold, one where only signals above 15% of the maximum electric fields are included in the analysis, similarly for 30% and 45%. If we increase the threshold level to be above 60%, the number of data becomes too small to allow a reliable estimate for the distribution. For the downleg conditions, we note the presence of a distinct narrow peak corresponding to B 0 - perpendicular propagation. For the upleg conditions (with the reduced value for E 0 ) the result is not nearly as obvious. Here several sporadic peaks in the probability density can be seen, although we also here find components with B 0 -perpendicular propagation, and observe in particular an overall narrowing of the distribution with increasing threshold values. Figures 9 13 show normalized probability densities. [39] When dealing with Gaussian random processes, the RMS electric field would be the most natural reference 8of20

9 Figure 11. Probability densities of the changes in direction of propagation Dq B of low-frequency electrostatic waves in the ionospheric E region in the fixed frame of reference. The change in direction Dq B is shown with respect to the magnetic field lines within one sampling time Dt for (left) upleg and (right) downleg conditions. Threshold values are as in Figure 9. To obtain the approximation dq B /dt Dq B /Dt, we divide the abscissa values by Dt = 0.5 ms. amplitude. Since, however, our results demonstrate that we have a non-gaussian process, we prefer to use the maximum electric field found in the relevant record. [40] The distribution of angles contains only a part of the information of the electric field properties. The fact that for instance the angle q B is randomly distributed does not contain information on the time variability of the angle: the extremes could be either a slow, large-scale variation, or alternatively a rapid irregular variation in direction. The variation can be different for the two angles q B and q WE specified before. [41] To quantify the variability of the electric field direction with time, we obtain results for the change in the local average wave propagation direction with respect to a fixed direction in space, which for these ionospheric conditions is taken to be along B 0. A related problem was analyzed in numerical simulations of fluid turbulence by Pécseli and Trulsen [2007], who found some unexpected probability distributions. The probability density (PDF) for the variation in electric field direction Dq B with respect to B 0 from one sampling time step to the next is shown in Figure 11. Also this variation can be determined for various electric field amplitude thresholds. As the amplitude threshold value is increased we find a narrowing of the distribution of Dq B both for upleg and downleg conditions, although the effect is again strongest for downleg. This result demonstrates that the largest amplitude electric fields are also those with the most steady direction. [42] To investigate more whether the observed variations were related to the electric field q amplitude ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we obtain the correlation coefficient hjejdq B i= hjejidq 2 2 B between the fluctuating electric field amplitude and Dq B, obtaining the values 0.32 and 0.38 for upleg and downleg conditions, respectively. These are nontrivial correlation values, which seem to be largest when the ambient electric field E 0 is largest. To test that the effects of finite record length are unimportant for these results, we made a test by random number generators, producing synthetic data samples of same length. Ideally, these two signals should be independent with a vanishing correlation. Due to the finite record lengths we find also in this case a finite correlation coefficient. It is, however, of the order of 10 3, which we consider to be insignificant, thus demonstrating that our results are robust, and the observed correlations are not due to finite record lengths. [43] We analyzed also the changes in direction between EðÞand t Eðt þ DtÞ. The temporal relative variation of the direction of propagation is obtained from the data by the simple formula EðÞE t ðt þ DtÞ Dq E ¼ arccos ; ð4þ jet ðþjjetþ ð DtÞj giving the change in q E (t) within one sampling time interval, Dt. In this case it is not meaningful to distinguish the sign of the angular change, so here we have all q E (t) 0. This result is shown in Figure 12. With Dt being small compared to characteristic times for changes in q E, we can approximate Eðt þ DtÞ EðÞþDt t ðdeðþ=dt t Þ and simplify (4) by a series expansion, which can be used for analytical estimates. [44] We note that the distribution of directional changes gets narrower as we increase the acceptance threshold, both for the changes in direction with respect to B 0 (see Figure 11) and for the relative directional changes between EðÞ t and Eðt þ DtÞ. It is, however, apparent that the distribution in Figure 12 is wider for the largest threshold than the similar distribution in Figure 11. This indicates that the largestamplitude electric field vectors are confined closely to the plane perpendicular to B 0, while their directions can vary somewhat more freely in direction with respect to a vector along the electrojet. Here we note that as the magnetic field lines are not strictly vertical, the direction determined by the E 0 B 0 vector is not strictly horizontal. [45] In Table 1 we summarize the results of upleg and downleg conditions of the angular variation considered here by presenting (Dq B ) 2 Dq B 2, (Dq E ) 2 Dq E 2 and Dq E. For symmetry reasons we should expect that Dq B = 0, and indeed we find values of the order of 0.05, which are within the statistical uncertainty. The data in Table 1 give a clear illustration of the narrowing of the distributions of the temporal angular variations as the threshold value of the amplitudes is increased. The change is of the order of 50%. The observed distributions represent an 9of20

10 Table 1. Standard Deviations of the Angular Variations of Electric Field Vector Direction for Upleg and Downleg Conditions for Different Amplitude Threshold Values a E 0 (mv m 1 ) Threshold (%) s B s E Dq E a We have s 2 B (Dq B ) 2 Dq B 2 and s 2 E (Dq E ) 2 Dq E 2. Ideally, we have Dq B =0. Figure 12. Probability densities of the changes in relative direction of propagation of low-frequency electrostatic waves in the ionospheric E region. Dq E is obtained by (4). Also here the results are conditional, the imposed conditions being that the electric field magnitude is now above 15%, 30% and 45% of the peak value. To obtain the approximation dq E /dt Dq E /Dt, we also divide the abscissa values by Dt = 0.5. Results are again shown for both (left) upleg and (right) downleg conditions. [47] To illustrate the intermittency effects observed in this study, Figure 13 shows the probability density for the time intervals DT where the signal is above the mentioned 15%, 30% and 45% levels. For amplitudes below the mentioned thresholds we find a few long time intervals, in one case with duration up to 0.5 s. The results demonstrate that the largest amplitudes are confined to narrow time intervals, typically of the order of ms, corresponding to m spatial intervals, where we use a typical rocket velocity of 500 m s 1 to relate temporal and spatial separations. The large-amplitude regions are clustered in the sense that the probability density of their separation has a long tail. We can find large time intervals (typically 250 ms, corresponding to 250 m spatial intervals) where the signal remains below a moderate level. [48] Figure 13 shows the time interval probability densities for the selected threshold values. To obtain a reduced average over the active part of the ionospheric E region. It is plausible that the distribution of directional variability obtained for a local altitude interval can be narrower. If we restrict our analysis to a shorter time sequence, i.e., smaller altitude interval, the statistical uncertainty will increase due to the reduction in the amount of data. Consequently, we are not able to estimate a reliable altitude variation of the width of the directional distribution. 3. Intermittency Effects [46] Analyzing the structure function of the potential fluctuations, Dyrud et al. [2008] found that the signal has significant intermittent features, with localized bursts of intense wave activity intermixed with somewhat more quiescent regions. This feature is most pronounced for the downleg part of the flight where E 0 is largest. Also, Fejer et al. [1975] report spatially intermittent wave activity. From radar scattering using the Jicamarca radar, they found that particularly at night the echoes were concentrated in many thin layers up to altitudes of 130 km. Their spatial resolution was 1 km. Figure 13. Probability densities of the time intervals DT where the fluctuating electric fields exceed the 15%, 30% and 45% of the peak signal levels. The data are for both (left) upleg and (right) downleg conditions. 10 of 20

11 Gaussian signals, as discussed in more detail also in Appendix A. [50] The present model [Kristensen et al., 1991] for describing intermittent features in a random process F(t) is tested in Appendix A. The model contains the basic probability density functions P(F), P(F ), and P(F, F ), where F df(t)/dt. The rocket data were analyzed for obtaining these quantities. Our general analysis is applicable for any process F(t); in the following, we set F(t) = E(t). Figure 14 thus shows P( E ) as well as P( E ). Figure 15 shows the estimate for the joint probability density P( E, E ) for downleg conditions. It is readily seen that P( E, E ) P( E ) P( E ), demonstrating that E and E are not statistically Figure 14. Estimates for the probability densities of (a) E and (b) d E /dt for downleg conditions. and more transparent information we take averages of the excess time intervals for varying threshold levels. The corresponding analysis follows [Kristensen et al., 1991] (see also Appendix A). This interpretation of intermittency differs from what is traditionally used in the fluid literature [Davidson, 2004], where these effects are discussed in terms of the structure functions as also used by Dyrud et al. [2008] in their analysis of these rocket data. [49] The term intermittency is often used rather vaguely, to state that some random or turbulent phenomena are of a bursty nature, exhibiting irregular spikes, localized in space or in time. In a general sense, however, any random signal will have intermittent effects! A somewhat pragmatic definition of intermittency was given by Rollefson [1978, p. 1426], stating that a variable with zero mean will be called intermittent if it has a probability distribution such that extremely small and extremely large excursions are more likely than in a normally distributed variable. In this sense, a random Gaussian process serves as the reference nonintermittent case. It is possible to find many types of signals with distributed separations between large- and smallamplitude regions. The important feature in our case is the difference in physical characteristics of the data in the largeand small-amplitude regions, respectively. When amplitudes are large, exceeding 20% of the maximum amplitude in the corresponding record, there is a tendency for these amplitudes to be confined to narrow spatial regions as compared to Figure 15. Estimate for the joint probability density P( E, d E /dt). For clarity, we show both (a) a threedimensional version and (b) a contour version: fine details at low levels are seen better in the latter. Data are for downleg conditions. For upleg conditions the details (i.e., the spikes seen on the top figure) disappear. 11 of 20

12 Figure 16. Estimate for the joint probability density P( E, d E /dt) for upleg conditions to be compared with Figure 15 obtained for downleg conditions. independent variables. Already this information suffices to prove that E(t) does not constitute a Gaussian random process. Also, for comparison, Figure 16 shows the similar results for upleg conditions. We note a significant difference in the fine scale structure of the two probability densities. The only known significant difference between the upleg and downleg plasma conditions is the magnitude E 0 of the DC electric field. It therefore seems that the intermittency features as described by P( E, E ) are sensitive indicators for the large-scale DC electric fields. [51] Figure 17 shows one of the major results for describing intermittent features of the observed turbulence. The solid line gives the variation of DT for varying threshold levels, measured in percent of the maximum value E m in the record. A dotted lines give the results obtained for a Gaussian random process with the average and standard deviations for E (i.e., E and s E 2 E 2, which are expressed in terms of E m ) and for d E /dt (i.e., E =0 and s ) obtained from the data. We find significant deviations from the Gaussian case, both for large and for small E field threshold levels. In particular, we have in our data an excess Figure 17. Solid line shows the variation of the average time interval DT spent above threshold for varying threshold level, measured here in percent of the maximum value E m in the record. Dotted line shows the analytical result (A4) for Gaussian random processes. The inset shows the similar results for upleg conditions. Figure 18. Solid line shows the variation of the average time interval DT b spent below threshold for varying threshold level. Data are for downleg conditions with the inset showing results for upleg conditions. of time intervals spent over large (> 30% of E m ) threshold values as compared to the Gaussian limit. We also have an excess for threshold values below 5% of E m. For an intermediate threshold level of 10 30% of the maximum E m we thus find noticeably shorter excess time intervals as compared to the Gaussian reference. The small threshold level result is, however, of minor significance since the Gaussian model will allow for unphysical negative values of E, even when large values of E are used. The Gaussian model is inaccurate for small E. While Figure 17 gives the average value DT for a range of excess levels, we find samples (for given threshold) of the corresponding probability densities P(DT) in Figure 13. The Figure 17 inset shows conditions for upleg conditions. Also here we recover intermittency features. [52] As well as showing results for average time intervals DT = DT a spent above a certain threshold, we study also time intervals DT = DT b spent below. The two results are related, since we have N DT a þndt b ¼T, where we took the number N of time intervals above and below to be the same. For long time series T, where we can ignore end effects this is a reasonable assumption. For long time series, we also expect N to be independent of DT a as well as DT b,soupon averaging we have hdt a iþ hdt b i ¼T= hn i,wheret = hn i can be directly obtained from (A2). Since hn idepends on the threshold level as in (A2), this relation is not trivial. [53] Also, for completeness, Figure 18 shows results for the average times DT b spent below varying reference levels, with inset showing results for upleg conditions. These time intervals represent the separations between the largeamplitude bursty electric field regions. [54] We are not aware of any theoretical analysis directly addressing the intermittency definition used in our study. Our results demonstrate that such a theory should address not only the E field probability density but rather the full joint probability density for E and its time derivative. 4. Ionospheric Plasma Instability [55] So far we have not addressed the origin of the observed waves. Here we argue that the most likely origin of 12 of 20