Plasma turbulence spectrum in the lower ionosphere at the heights 80-100 km

A. V. Gurevich ^*, K. Rinnert and K. Schlegel

Max-Planck-Institut fur Aeronomie, Katlenburg-Lindau, Germany

Abstract

Introduction

Density fluctuations were measured using a retarding potential analyzer (RPA) operated in the saturation mode [Schlegel, 1992]. The relative density changes Dn/n were considered to be proportional to the saturation current Dj/j. The absolute values of the electron density are difficult to estimate with this technique, but relative density changes are easy to measure. The measured current in the saturation mode (10^-10 to 10^-6 A) was amplified by a fast electrometer and then sampled every 0.5 ms in order to retrieve the density fluctuation spectrum up to 1 kHz. For more details, see Schlegel [1992].

The electric field measuring instrument was a floating Langmuir double-probe system primarily designed to measure the dc electric field. Two pairs of booms were utilized which had gold-plated spherical probes (50 mm diameter) mounted on the ends. Each deployed boom pair placed the probes in opposite directions at a distance of 1.8 m from the payload axis. Hence the probe separation of each boom pair was 3.6 m. One boom pair was mounted at the lower level of the payload, and the other pair was mounted at an upper level 1.8 m apart from the lower level and rotated by 90^o. Thus the four probes on the deployed booms formed the corners of a tetrahedron. All six differential voltages between any two probes were samples simultaneously at a rate of 4 kHz. The corresponding six electric field components are transformed into a three-dimensional electric field vector in the payload frame of reference and finally into an Earth-fixed frame of reference. For more details, see Rinnert [1992].

As an example, Figure 1 exhibits both the dc and the ac electric field data obtained during Flight 4. The lower frame of Figure 1 shows the dc electric field component perpendicular to the Earth's magnetic field versus flight time. It shows a fairly stable electric field primarily southward directed and slightly increasing during the flight from about 35 mV m^-1 to about 50 mV m^-1. The dc electric field in magnitude and direction is confirmed by STARE measurements and EISCAT measurements of the plasma drift at 150-km altitude just above the apogee of the payload trajectory [Kohl et al., 1992]. The dots indicate the payload altitude versus flight time. The apogee was reached at 123 km about 180 s after launch. In the upper frame of Figure 1 an ac signal is presented. Plotted are the contours of constant amplitude (voltage between the upper two probes) versus frequency and flight time. The spectra are 1-s averages. The fluctuations occur in a well-defined altitude region, and the payload passed this instability region twice, during upleg and during downleg. The ac electric field activity started at about 80-km altitude with low frequencies and at about 95 km with higher frequencies and stopped just above 115 km. The width of the instability region as well as the intensity is somewhat larger during downleg than during upleg because of the larger dc electric field.

An important feature is the occurrence of intense fluctuations in the frequency band below 50 Hz in the lower part of the instability region between about 80- and 100-km altitude. The high-frequency oscillations are practically absent in this region, but they dominate at higher heights. This is an indication that two different mechanisms are responsible for these oscillations in the lower and upper part. The fluctuations in the higher region are certainly related to the Farley-Buneman  (FB) or "two-stream instability" [Farley, 1963]. In the lower region the "gradient drift instability" is usually considered as a source of plasma oscillations [Fejer and Kelley, 1980], but it strongly depends on the local gradients of plasma density. On the other hand, the fact that the lower-altitude range is dominated by low-frequency fluctuations was found for all four flights under quite different conditions. This finding is therefore interesting to evaluate with regard to the turbulence of the neutral atmosphere, which is another possible source of low-frequency plasma fluctuations in the ionosphere.

Measurements

Figure 4 exhibits some relevant height dependences during Flight 4: v_r is the rocket velocity and v_{r B} is the rocket velocity perpendicular to B, v_Drift is the electron drift velocity deduced from the measured dc electric field according to v_Drift = ([E, B]/B²) [1/(1 + y)], C_s is the speed of sound and is assumed to be constant (C_s= 300 m s^-1 for the altitudes below ~110 km), f_c is the critical frequency according to (1), and l₀ is the characteristic scale of neutral turbulence:

(1)

Here n_i is the collision frequency of ions, t is the lifetime of electrons, y is a nondimensional factor, determining the collisional damping of the drift velocity [Fejer and Kelley, 1980]. If the drift velocity exceeds the sound velocity (v_Drift > C_s), which happens quite suddenly at about 95 km, then the FB instability is excited for the frequencies f > f_c. Thus below 95 km, FB waves are not excited and the observed fluctuations must have another source.

On the downleg the payload flew almost parallel to B, and therefore v_{r B} is then much smaller than during the upleg. Assuming that the atmospheric conditions in both altitude regions were the same during the upleg and downleg (during the downleg the probed region was about 40 km further north and it happened about 3 min later), this allows some information on the directional dependence of these structures with respect to B.

The three ac components dE_east, dE_north, and dE_parallel with respect to the Earth's magnetic field have been Fourier analyzed with a frequency resolution of 3.906 Hz using overlapping time windows of 256 ms. The spectra were combined to provide an estimate of the total spectral energy density dE².

Figure 5 shows some measured power spectra from Flight 4. The energy density is plotted versus a normalized wave number x = kl₀ = 2p fl₀/v; v is rocket velocity, and l₀ is the characteristic scale of neutral turbulence and is a function of altitude. According to Tatarski [1961] it can be expressed as

(2)

(in meters). Here N_m is the number density of the neutral atmosphere (in cm^-3), T is the neutral gas temperature (in kelvins), and e the turbulent energy dissipation parameter. We used the value e = 0.1 W kg^-1 in our estimates [e.g., Rüster}, 1984]. In the range of interest, l₀ is of the order of 1-10 m (see  Figure 4). (The heavy dotted curves at Figure 5 represent the theoretical power spectrum after formula (3) below. The constant C in (3) is chosen to fit the low-frequency part of the observed spectra.)

The transformation from the frequency scale to the kl₀ scale leads to an altitude dependent position of the measured spectra with respect to the kl₀ scale.

Plotted in Figure 5 are spectra from various height regions. The top frames are from the upleg, and the bottom frames are from the downleg. At the lower altitudes the measured spectra are steeper than the theoretical spectrum, and at higher altitudes they are flatter. The sudden change in the measured spectra at the heights higher then 95 km (upleg) is clearly seen. The additional contribution at high frequencies (or large k) is attributable to the FB instability. During the downleg this change happens at a somewhat lower altitude. As the dc electric field is larger during the downleg, the height range of the FB instabilities is extended (see Figures 1 and 4).

Figure 6 represents the same measured spectra that are now plotted versus kl₀ = 2pl₀ f/v_{r B}. Here one can see better agreement with the form of the theoretical spectrum at lower altitudes.

No data exist from below about 77 km because the booms were not released during the downleg and the payload became unstable. Unfortunately, there are no neutral wind data. Neutral winds, of course, have an impact on the transformation from the frequency into the k domain as the wind velocity adds to the rocket velocity v_r, but as v_r is large enough (see Figure 4), the role of neutral winds is clearly unimportant.

Qualitatively, the same behavior of turbulent spectra is found for the data from Flight 1. The dc electric field or the drift velocity was lower, and consequently the destabilization of FB waves was less. The trajectory of F1 was such that there was not a big difference in v_{r B} for the upleg and downleg, and therefore the differences in k, k _B up and k _B down is less pronounced. Figures 7 and 8 present the F1 data corresponding to Figures 5 and 6, respectively. Data analogous to those presented in Figures 5, 6, 7, and 8 were obtained during other flights, which demonstrates the general stability of the shape of the electric field fluctuations spectra.

Comparison With Theory

On the other hand, fluctuations observed at lower heights (z 95 km) cannot be excited by the FB instability, and so another process must be responsible for these oscillations. A theory describing such a process was developed recently which considered the electric field and plasma density fluctuations induced at low altitudes (z ~ 80 to 100 km) due to the turbulence of the neutral atmosphere [Gurevich et al., 1997]. We will compare the observational data with the predictions of this theory.

It should be emphasized that the theory of induced turbulence gives a quantitative description of plasma turbulence. It means that one is able to compare with the theory not only the wave vector k (or frequency f) dependence of the observed spectrum, but also its anisotropy (which is determined mostly by the Earth's magnetic field B) and, what is especially remarkable, its absolute values.

The averaged frequency spectrum of electric field fluctuations according to Gurevich et al. [1997] has the form

(3)

Here v is rocket velocity, l₀ is the characteristic length of neutral turbulence (see (2)), and C is a normalizing constant. The spectrum (3) is compared with the experimental data at different heights in Figures 5 to  8. There is general agreement between the theory and observational data at low heights. At higher altitudes (z > 95-100 km) there is no agreement, as could be expected owing to excitation of the FB instability. Note that the 9 observed spectrum has a tendency to become steeper than (3) at low heights (z ~ 80 km).

It should be emphasized that the theory predicts the existence of an asymmetry of the spectrum with respect to the magnetic field direction: part of the inhomogeneities are strongly elongated along the Earth's magnetic field B. In the observational data there seems to be an indication of such an anisotropy. If the inhomogeneities are strongly elongated along B, only the component of the rocket velocity perpendicular to B} (v_r B) should be used in (3) instead of the full velocity v. Especially for lower altitudes there is better agreement with observations with respect to k_} = 2pf/v_{r B} ( Figures 6 and  8). In the same time that the anisotropy is not too strong, what can be seen from comparison of downleg spectra at Figure 6 and Figure 5 for the heights 90.66-94.48 km: note that velocity v_{r B} was very small in this case, and so the role of the k_parallel component became important.

It is significant that the theory predicts a universal dependence of directional distribution of electric field fluctuations with respect to B: the dependence on the angle between k and B is the same for all values of k and varies only slightly with altitude. It allows us to average the observed angular dependence over k values and over the heights in the lower ionosphere (80  z  100 km). The result of this averaging is compared with the predictions of the theory in Figure 9. One can see quite good agreement between the theory and the observations.

Analysis of the density observations leads to analogous results. To compare the experimental data with the theory, the following formula for d | Dn/n | /df was obtained using the density spectrum d á | Dn/n |² ñ/dk, presented by Gurevich et al. [1997]:

(4)

where v again is rocket velocity, l₀ is the characteristic length of neutral turbulence, and C₁ is a normalization constant. The results of (4) for appropriate heights have been included in Figure 2 (top panel) and Figure 3 as a dashed line.

So we deduce that at low heights, shape and asymmetry of the spectra of the observed electric field and plasma density fluctuations are in agreement with the theory of neutral atmosphere induced plasma turbulence. A discrepancy with the theory is seen, however, when we compare the normalization constant C for electric field fluctuations. The observed values of this constant are approximately 10³ times larger than those obtained in the theory.

It is important to note here that the theory by Gurevich et al. [1997] in fact establishes only direct connection between the spectrum of induced plasma turbulence and the spectrum of neutral turbulence. The latter (according to Carter and Balsley [1982], Rüster [1984], and Manson et al. [1987]) can be regarded as Kolmogoroff's, which means that it is fully determined by the only parameter e that characterizes the energy dissipation in neutral turbulence and is generally proportional to v_0³, where v₀ is the characteristic value of neutral wind velocity fluctuations. The intensity of fluctuations v₀ has significant seasonal, daily, and temporal variations (depending, for example, on gravity waves). Because of this even the time-averaged values of e can differ by an order of magnitude. As follows from observations, the averaged value of parameter e grows slightly with height in the upper atmosphere [Manson et al., 1981]; it can also be essentially amplified (up to 1 order of magnitude and even more) in the region of gradient temperature inversion in the height profile (z ~ 80-85 km). In our analysis here we do not take into account all these variations of neutral turbulence, using for parameter e its characteristic value e = 0.1 W kg^-1.

This normalization is denoted at C = 1. It follows from the previous discussion that when the rocket measurements of the turbulent spectra in given concrete conditions are compared with the theory, the changes of normalization constant C at least within 1 order of magnitude should be considered as quite normal fluctuations. For example, the change of fluctuations in one and the same realization of turbulent spectra are usually more than 1 order of magnitude, which is directly seen from Figures 5, 6, 7, 8, 10, and  11. Regardless of these fluctuations, the established strong amplification of the observed turbulent spectra C ~ 10³ is regular and definitely has real significance.

Discussion

We further emphasize that as the velocity of turbulent fluctuation v₀ in atmospheric conditions is much smaller than the sound velocity C_s, the compression and heating effects, which determine the temperature and density fluctuations in the neutral atmosphere, are small [Tatarski, 1961].

It is important that not only the neutral gas motion, but also the Earth's magnetic field B and electric field E, in the ionospheric plasma, have a strong influence on the motion of charged particles. Therefore the charged components at the ionospheric heights 70-100 km are not only transported by the turbulent motion of neutrals, but accomplish their own motion. This motion becomes compressible owing to the action of the magnetic field B and is thus essentially different from the motion of the neutral component. Compressibility leads to the creation of plasma density fluctuations and to the generation of random electric fields and currents. In this sense the plasma turbulence in the lower ionosphere induced by the noncompressible turbulence of the neutral gas is considered in the theory of Gurevich et al. [1997].

The interaction with neutral particles has a crucial influence on the motion of the charged components. The interaction between charged particles and excited plasma waves is much smaller. Because of these facts, the plasma turbulence in the lower ionosphere can be described as the superposition of the linear low-frequency plasma waves, excited by turbulent motions of the neutral gas. Neutral turbulence acts as a driving force for the natural ionospheric plasma modes. The key role in the induced turbulence is played by the drift mode. Note that exactly the same mode is excited in the Farley-Buneman instability, in the gradient-drift instability [Fejer and Kelley, 1980], and in the thermodiffusive instability [Gurevich and Karashtin, 1984].

The resulting k spectrum of induced plasma turbulence obtained by Gurevich et al. [1997] has some interesting peculiarities, depending on the angles between k and the directions of the electron drift velocity v_e and the magnetic field B. The dependence on the angle between k and B is shown in Figure 9. One can see that the spectrum of induced plasma turbulence is essentially anisotropic.

The integration over the angles gives the dependence of the spectrum on the absolute value of k. The spectrum for the density fluctuations Dn at the low height of z  95 km is

(5)

Here the characteristic scale l₀ and the function E(x) are determined according to equations (2) and (3). The parameter Q₀ depends substantially on height z and is given in Table 2 of Gurevich et al. [1997]. We emphasize that in the inertial interval kl₀ <<1, the spectrum (5) is proportional to k^-8/3. A comparison of the spectrum with density fluctuations at mesospheric heights obtained by Royrvik and Smith [1984] during quiet ionospheric conditions is presented in Figure 10. One can see good agreement between the theory and the observations (for 86.5 km, Q₀ ~ 3 x 10^-7, l₀ 3.3 m).

This agreement is present not only in the shape of the spectrum, but also in the absolute values of the fluctuations. The discrepancy is small enough and could be corrected with the chosen atmospheric model and the actual local value of the turbulence dissipative factor as indicated with the dashed line in Figure 10a. One can notice also that the amplification of Figure 10a, can be attributed to the region of the temperature gradient inversion in the height profile. As was already mentioned before, the amplification of fluctuations is often observed in the mesopause near this region (see, for example, Karashtin et al., [1997a]), and it is also clearly seen in Figure 2 of Royrvik and Smith [1984]. Note that gradual growth of the averaged value of | Dn/n | with height up to 95-98 km,which is seen in this figure is also in agreement with the theory of Gurevich et al. [1997].

Let us return to the comparison of the theory with our experimental data. We have seen that in the lower ionosphere the shape of the spectra of the density and of the electric field fluctuations presented in Figures 2, 3, 5, 6, 7, and 8 is in agreement with the theory. We especially emphasize the good agreement in the predicted and observed asymmetry of the spectra ( Figure 9): the spectrum is not isotropic, as it should be for neutral turbulence; nor is it strongly anisotropic, as in the case of a plasma-excited instability.

The absolute values of á |Dn/n | ñ were not measured in our experiment. The absolute values of the electric field fluctuations d á |DE |² ñ df, however, were measured, and we have seen that the observed values are amplified by the factor C ~ 10³ in comparison with the predictions in the theory.

One can suggest the following main reasons for the observed amplification of the low-frequency oscillations:

1.  The theory is developed for a homogeneous medium. The real ionospheric plasma, however, is height inhomogeneous, and this inhomogeneity leads to WKB amplification of the oscillations in the reflection region at the lower boundary of the ionosphere (see Gurevich et al., [1995]; Karashtin et al., [1997b]).

2.  Our measurements were made during strongly disturbed conditions, when the FB instability was excited. This instability, being excited at higher altitudes, possibly could amplify the oscillations at lower heights also. For example, in the equatorial electrojet under conditions when the FB instability was excited, the electric field fluctuations at low heights compared with the theory of induced turbulence have the same spectral form and the same values (C ~ 10³) as in our experiments [Pfaff, 1991; Pfaff et al., 1987]. Quite analogous comparison with measurements in the absence of or with very weak FB instability gives a C value of 10-30 times less (see Figure 11, where a direct comparison of formula (3) with observations is presented; the exact value C = 1 from Gurevich et al. [1997] is used here to calculate the theoretical curve).

3.  The vertical electric currents, which usually exist in disturbed conditions, could also lead to the amplification of electric field oscillations (see Karashtin et al., [1997b]).

We have to add also that on the basis of the observations [Carter and Balsley, 1982; Manson et al., 1981; Rüster, 1984], it was supposed in the theory that the neutral turbulence is isotropic and has Kolmogoroff's spectrum. This statement for the real upper atmosphere does not seem to be proven well enough and should be studied more.

One can conclude that there is general agreement of the theory of induced turbulence with observational data in the lower ionosphere. It is significant that the observed shape of low-frequency spectra in the equatorial and auroral region are alike. On the one hand, this is indicative of a common mechanism for these phenomena and could easily be understood if the excitation of plasma oscillations at low heights is mainly determined by neutral plasma turbulence, which has a unique spectrum. On the other hand, it must be emphasized that in disturbed conditions the amplitudes of the observed electric field fluctuations are strongly amplified in comparison with the one predicted by the theory of induced turbulence.