Energy exchange rate for the equatorial electrojet: Test of the model of two-stream processes that includes thermal corrections

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1 GEOPHYSICAL RESEARCH LETTERS, VOL. 34, L20806, doi: /2007gl030903, 2007 Energy exchange rate for the equatorial electrojet: Test of the model of two-stream processes that includes thermal corrections L. M. Kagan 1 and R. S. Kissack 1 Received 5 June 2007; revised 30 August 2007; accepted 14 September 2007; published 25 October [1] The experiment of Balsley and Farley (1971) in Jicamarca, Peru has been the only multi(3)-frequency investigation of the equatorial electrojet. The radar was operated sequentially at frequencies of 16.25, and MHz and probed the same ionospheric volume. The most prominent feature of these observations was the increase in the phase velocity V ph of Farley-Buneman (FB) waves with increasing radar frequency while the classical fluid theory gives V ph independent on their wavelength and the simple kinetic theory accounts for only a small increase in V ph. Our recent theory predicts a wavenumber dependence of the FB wave phase velocity. The rate of increase depends on the altitude of the electrojet and seems to match the observed wave behavior. The most important byproduct of this case study is our conclusion that the energy exchange rate in inelastic electron-neutral interactions is about two-tothree times higher that it has been assumed in estimates. Citation: Kagan, L. M., and R. S. Kissack (2007), Energy exchange rate for the equatorial electrojet: Test of the model of two-stream processes that includes thermal corrections, Geophys. Res. Lett., 34, L20806, doi: /2007gl Introduction [2] Farley-Buneman waves are observed to move with a phase speed of the order of the ion-acoustic speed c s,i which, in standard classical theory, is the speed of waves moving at their linear instability threshold. Both linear and nonlinear approaches of ever-increasing sophistication have been developed to explain this. Non-linear theories have focused on the mechanisms which produce FB waves, while linear theories have aimed at providing more refined expressions for the instability threshold speed as well as understanding the attendant physics. There have been two distinct approaches to tackle the non-isothermal electron corrections in Farley-Buneman linear instability calculations. In the series of papers written by Dimant and Sudan, the focus has been on an improved kinetic starting point for the instability calculations [Dimant and Sudan, 1995, 1997]. The other approach based on Grad s set of fluid equations closed at the heat flow level has been used in work of Kissack et al. [1995, 1997], St.-Maurice and Kissack [2000], and Kagan and St.-Maurice [2004]. Our advanced linear theory self-consistently describes the effects of collisions, using Burgers [Burgers, 1969] expressions for collision integrals. A detailed discussion comparing the two approaches are given by Kagan and St.-Maurice [2004]. 1 Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada. Copyright 2007 by the American Geophysical Union /07/2007GL [3] Here we will concentrate on theoretical predictions for the phase velocity of two-step type-i waves as labeled by St.-Maurice et al. [2003]. Based on the theory for a linear two-stream instability involving thermal corrections [St.-Maurice and Kissack, 2000], St.-Maurice et al. [2003] were the first to explain the puzzling behavior of two-step type-i waves in the lower electrojet moving at speeds up to 50% higher than the isothermal ion acoustic speed. Encouraged by this success, we looked into a non-zero aspect angle situation, taking neutral winds into consideration [Kagan and St.-Maurice, 2004]. The predictions for aspect sensitivity of Farley-Buneman waves to be less than matched well to that observed by Kudeki and Farley [1989]. [4] Our further recent development of the theory included allowance for arbitrary flow angle, which is essentially the angle between the radar wave vector and the E B drift, and therefore should be capable of explaining observations from off vertical transmissions (R. S. Kissack et al., Thermal effects on Farley-Buneman waves for nonzero aspect and flow angles, 1, Dispersion relation; 2, Threshold analysis, submitted to Physics of Plasmas, 2007, hereinafter referred to as Kissack et al., submitted manuscript, 2007a and 2007b, respectively). A significant step forward by Kissack et al. (submitted manuscript, 2007a, 2007b) compared to our previous work was the presentation of the thermal corrections themselves which now were written in the way that allows to see contributions from each physical process. In particular we showed that any changes in the phase velocity for off-vertical transmissions (compared to the vertical ones) are largely due to the Dimant-Sudan thermal instability [Dimant and Sudan, 1997]. Therefore for the daytime equatorial electrojet the Farley-Buneman waves, of the same wavelength, would have a smaller threshold velocity for negative flow angles (a radar looking east) than for positive flow angles (a radar looking west). Because the Dimant-Sudan instability has a cut off at longer wavenumbers (higher radar frequencies) and higher altitudes (due to increased ion magnetization resulting in reduced polarization electric field) the east-west asymmetry should be more pronounced at 16 MHz and be almost unnoticeable at 150 MHz and at higher altitudes (see Kissack et al., submitted manuscript, 2007b, for more details). Two case studies for west-off-zenith multi-frequency transmissions at Jicamarca [Balsley and Farley, 1971; R. A. Cuevas et al., manuscript in preparation, 2007] are discussed in detail by L. M. Kagan et al. (manuscript in preparation, 2007). The next logical step was to test our theoretical predictions for frequency/radar wavevector dependences of phase velocities, so we needed to find multi-frequency observations. The most perfect was the 38-year old set of L of7

2 Figure 1. Experimental results for three-frequency radar observations at Jicamarca on May 15 and 16, 1969, from Balsley and Farley [1971]. Radar zenith angle was 60 for magnetic east transmissions. The radar transmitted at 16.25, and MHz in sequence. data by Balsley and Farley [1971] for three-frequency observations eastward off zenith. 2. Case Study [5] The data we are to analyze were observed by Balsley and Farley [1971] on May 1969 and up to date have been the only three-frequency probing of equatorial electrojet. The radar backscatter was received from 60 east off vertical at 16, 50 and 146 MHz in sequence and presented the signal averaged over entire electrojet. In Figure 1 we reproduce data of Balsley and Farley [1971] for May 15 and 16, The upper panels show four successive spectra of a phase velocity for radar frequencies of 146, 16, 50 and 146 MHz on May 15 and of 16, 50, 146 and 16 MHz on May 16. Notice that in both cases the two spectra at the same frequency (146 MHz on May 15 and 16 MHz on May 16) are almost identical. This allowed Balsley and Farley to conclude that the ionospheric conditions were about the same during each set of multi-frequency observations. In the two lower panels of Figure 1 we reproduce the local time dependence of the phase velocities for these three radar frequencies and indicate the time interval corresponding to the spectra on the respective upper panel. [6] The most prominent feature of the Balsley and Farley experiment was a higher phase velocity for the Farley- Buneman waves with a higher wave number, while the classical fluid theory predicts the phase velocity to be the isothermal ion-acoustic speed, which is independent of wavelength and the simple kinetic theory accounts for only a small increase in V ph [Farley, 1963]. One can also see that an increase in V ph with LT in the afternoon was not constant but fluctuating near a constant d V ph /dlt. This latter effect is possibly attributable to neutral motions, gravity waves in particular [see Hysell et al., 2007]. Since radar backscattered signal was averaged over entire electrojet in our analysis below we concentrate on explanation of the observed alignment V 146 ph > V 50 ph > V 16 ph using our recent theories [Kagan and St.-Maurice, 2004; Kissack et al., submitted manuscript, 2007a, 2007b]. [7] Based on the fact that the thermal correction to the frequency of FB waves are less than 3% in most cases [Kagan and St.-Maurice, 2004; see also Kissack et al., submitted manuscript, 2007b] we can significantly simplify equations describing the phase velocity of Farley-Buneman waves by Kissack et al. (submitted manuscript, 2007b) and write V ph for non-zero flow angle as: V ph ¼ u i0 c s;i k? k vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u T e0 c 1 þ DS ð2=3þh ½ CD e? k? 2 þ hšþ ð1 þ gþ2 A 2 ½y T = ð1 þ y T ÞŠ 2 u t n o 2 0 k2 : ð1þ ðt e0 þ T i0 Þ ð3=2þ½y T = ð1 þ y T ÞŠ 2 u 2 0 k2 þ ð2=3þ ½ CD e? k? 2 þ hš2 2of7

3 [8] Such a presentation allows easy tracking the dependence of the wave phase velocity on its wave number. In line with our previous work we also write equation (1) in a way that makes it easy to see both the thermal corrections to the classical expression, contributions from non-zero flow angles and the dominating physical processes, such as thermal conduction and inelastic electron cooling. [9] Here n e, n in, w e, and w i are the electron and ion collision and cyclotron frequencies, respectively; u 0 = V i0 V e0 is a current velocity, i.e. a relative velocity between ions and electrons; k T is the Boltzmann constant; d e n e is the inelastic volume electron-neutral energy exchange rate, where the dimensionless energy exchange factor d e is essentially pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi constant over the altitudes of interest; c s,i = ðt e0 þ T i0 Þ=M is the isothermal ion-acoustic speed; m and M are the electron and ion mass respectively, k is the magnitude of the wave vector and k? is the magnitude of the wave vector component perpendicular to the geomagnetic field B; T e0 and T i0 are background electron and ion temperatures respectively; u i0 is essentially the ion drag by a zonal wind; and the coefficients are C k2 k k 2? g ¼ A k2 k k 2?! ¼ 1 þ ða gþ, ð1 þ gþ b ½1 þ abš; ð2þ! ¼ 5=2 þ 2g þ g 2 ðg 5=2Þb gð1 þ gþ T e0 n e ð1 bþa k2 k k? e T e0 ; 1 b ¼ ð1 þ 2g=5Þ k 2 k w2 e k 2? n2 en! a ¼ 1 þ 2gð1 þ gþ=5; y T ¼ y 0 ð1 þ abþ; y 0 ¼ n en in ; w e w i h ¼ d e n e ; D e? ¼ k T T e0 n e =mw 2 e : ð3þ ð4þ ð5þ c DS ¼ ½ð2 þ gþð1 þ gþa 3gs Šs ð6þ is the term describing Dimant-Sudan instability where the parameter s ¼ k2? u2 0 sin q f cos q f D e? k 2 h n e y 0 w e ð1 þ y T Þ is a function of a flow angle q f resulting in zeroing the Dimant-Sudan instability effects at 0 and 90 flow angles. [10] Wavenumber dependence of the phase velocity is in the second term under the square root of equation (1) which describes the thermal corrections. All effects of the thermal ð7þ processes are in the second term under the square root, where the first term in the numerator corresponds to the Dimant-Sudan instability [Dimant and Sudan, 1997] that manifests only at non-zero flow angles. In an isothermal treatment equation (1) reduces to the classical expression in which the phase velocity does not depend on the transmitted frequency. [11] From equation (1) it is clear that the only way to match the observations of Balsley and Farley [1971] that V ph increased with an increasing radar frequency, so that V 146 ph > V 50 ph > V 16 2 ph, is that h should exceed the term CD e? k? describing thermal diffusion and thermal conduction processes. When the electron temperature is close to the neutral temperature (as usually has been considered for the lower E region for simplicity) h may be written as introduced in (5), h = d e n e where d e is constant. [12] Based on Kagan and St.-Maurice [2004] who showed that thermal corrections to V ph are noticeable for the aspect angles less than 0.25 and are maximum for a 0 aspect angle, we ran our full computer code from Kissack et al. (submitted manuscript, 2007b) to find V 146 ph, V ph, V ph for a zero aspect angle (vertical transmissions) for three values of d e : 0.003, and 0.01 (see Figure 2). Note that of the three abovementioned d e = has been routinely used in such estimates in order to avoid complex calculations. Similar to Kagan and St.-Maurice [2004] in our evaluation we (1) have used g = 5/6; (2) have found the ionosphere parameters from the MSIS and IGRF models for the time and location of the Balsley and Farley experiment; and (3) calculated collisional frequencies using formulas from Schunk and Nagy [2000]. [13] In Figure 2 the phase velocities at (red), (green) and (blue) MHz are plotted for the electron energy exchange rate of inelastic interactions d e = (the left upper plot), d e = (the left column, middle) and d e = 0.01 (the lower plot on the left) and the aspect angles of 0 (left column), 0.1 (right column, middle plot for d e = and the lower plot for d e = 0.01) and 0.2 (the upper plot in the right column). Phase velocities in the right column are for non-zero aspect angles. It is clearly seen that 146 > in case of zero aspect angles only d e = 0.01 meets V ph V 50 ph > V 16 ph observed by Balsley and Farley [1971]. Note also that the effect disappears for the larger aspect angles in accordance with the Kagan and St.-Maurice [2004] prediction that V ph approaches c s,i at large aspect angles. [14] In case of off-vertical transmissions (a non-zero flow angle) the Dimant-Sudan thermal instability [Dimant and Sudan, 1997] contributes to the threshold phase velocity of Farley-Buneman waves by introducing asymmetry for east and west transmissions. This instability occurs for negative flow angles corresponding to the radar looking east for a daytime equatorial electrojet. The instability is easier to generate for the longer waves, so we would expect lower magnitudes of V ph at a lower frequency and therefore further splitting and perhaps making easier to satisfy the condition V 146 ph > V 50 ph > V 16 ph. [15] For positive flow angles corresponding to west transmissions, the Dimant and Sudan [1997] thermal instability might still be induced but requires significantly higher threshold velocities than the ion-acoustic speed [Dimant and Sudan, 1997]. Thus if there are any radar returns observed the phase velocity of type-1 irregularities would 3of7

4 Figure 2. Theoretical predictions for vertical transmissions (zero flow angle) at (red), (green), and (blue) MHz based on the theory of Kagan and St.-Maurice [2004] for 12:45 LT on May 15, be higher than for the east and vertical (if the last are not smeared by gradient-drift processes) transmissions. [16] There is only a slight difference in our predictions for V ph calculated in the absence of a neutral wind for May 16, 1969 compared to May 15, 1969, mostly caused by the different local time of observations: 12:45 LT on May 15 and 11:00 LT on May 16 (a bit higher velocities in afternoon). Note that of two data sets in Figure 1 the data 4of7

5 Figure 3. Theoretical predictions for the phase velocity of Farley-Buneman waves for 60 -east ( 60 flow angle) transmissions at MHz (blue), MHz (green), and MHz (red) for (top) d e = 0.01, (middle) d e = 0.007, and (bottom) d e = on 16 May 1969, 11:00 LT. Solid and dashed lines correspond to 0 and 0.2 aspect angles, respectively. Shaded areas between solid and dashed lines of the same color denote the predicted magnitudes of phase velocity at a given frequency as function of altitude. from May 15 show more altered behavior than on May 16, most probably caused by neutral motions (winds and/or gravity waves; see Hysell et al. [2007]). To avoid distortions introduced by neutral motions, we next compare V ph predicted by our theory with the ones observed on May 16, [17] In Figure 3 we present V 146 ph, V 50 ph and V 16 ph calculated based on the Kissack et al. (submitted manuscript, 2007b) code for the case where the radar beam is tilted 60 east ( 60 flow angle) corresponding to the Balsley and Farley data for 11:00 LT on May 16, 1969 from Figure 1. We plot the predicted phase velocities as functions of altitudes in blue, green and red for transmissions at MHz, MHz and MHz correspondingly for d e = 0.01 (upper panel), d e = (middle panel) and d e = We consider three possible values of d e : (1) (lower panel) that have been commonly used in estimates; (2) (middle panel) which is an average of d e = that are expected at these altitudes and location (see Discussions for more details); and (3) 0.01 that follows from our simulations for a zero flow angle. [18] From the experiment of Kudeki and Farley [1989] and theory of Kagan and St.-Maurice [2004] one would expect that backscattered signals are observed within aspect angles of 0 0.2, which we mark with solid (0 aspect) and dashed (0.2 aspect) lines for each frequency. With shaded areas between solid and dashed lines of the same color we show the magnitudes of phase velocity of Farley- Buneman waves that one would expect to observe at each frequency near a given altitude based on our theoretical predictions. 5of7

6 Figure 4. Altitude dependences of dimensionless parameters x AT (dashed lines) and x T (solid lines) that define dominating physical process for d e = (black lines) and d e = (gray lines) for each of three radar frequencies. Calculations are done with the code from Kissack et al. (submitted manuscript, 2007b) revised for the time and location of the Balsley and Farley [1971]. [19] Note that the echoes in Figure 1 are averaged over entire electrojet and we in fact, don t know whether the observed backscatter at all three frequencies came from the same altitude. The observed velocities on May 16, 1969 were V 16 ph ffi 320 m/s, V 50 ph ffi 360 m/s, V 146 ph ffi 395 m/s. From middle panel of Figure 3 (d e = 0.007) such velocities correspond to echoes at 16 MHz coming from km, at 50 MHz coming from km and at 146 MHz coming from km, which seem to match well the observations and correspond well to the observed differences in phase velocities at different frequencies. 3. Discussions and Conclusions [20] We have applied our recently developed theory [Kagan and St.-Maurice, 2004; Kissack et al., submitted manuscript, 2007a, 2007b] to multi-frequency observations of Farley-Buneman waves over Jicamarca [Balsley and Farley, 1971] and have found that the rate of inelastic electron cooling should be more than twice as high than that most scientists, us included, used for estimates [e.g., see St.-Maurice et al., 2003; Kagan and St.-Maurice, 2004]. One exception we know was Gurevich [1978] who used d e [21] To verify our observation-based conclusion about more than the double electron energy exchange rate that it has been routinely used in the estimates of electron cooling, we have done simplified but yet complex calculations of electron energy exchange rate h using the procedure suggested by Schunk and Nagy [2000], in which we took care of the known typos. In our calculations we account for rotational and vibrational (negligible in our case) excitation of N 2 and O 2 and for fine structure of atomic oxygen. To be consistent with our assumption T e T n and to allow for an energy exchange process in our calculations, we manually set T e = T n K. In our assumption of such particular difference between T e and T n we use Gurevich [1978] as a reference. In our evaluation of h we used the same ionosphere parameters from MSIS and IGRF models for 12:45 LT on May 15, 1969 and 11:00 LT on May 16, Assuming h = d e n e and dividing h by n e (which as mentioned above was calculated based on Schunk and Nagy [2000]) and ionosphere parameters from the MSIS model for the time and location of the experiment) we found d e As we have shown above the phase velocities in Figure 3 calculated for d e = (unlike those for d e = 0.003) fit the observations of Balsley and Farley [1971]. [22] In Figure 4 we plot altitude dependences of dimensionless parameters x AT =3(w r k u 0? )/2d e n e (dashed curves) and x T = CD e? k 2 /d e n e (solid curves) from Kissack et al. (submitted manuscript, 2007b) that define the dominating physical process for d e = (black lines) and d e = (gray lines) for each of three radar frequencies. It is clear that more than doubled d e could lead to a different dominating process at the lower frequencies of 16 and 50 MHz and significantly extend the altitude range where inelastic electron cooling plays crucial role defining transitional process between super-adiabatic and isothermal regimes. At 16 MHz, for example all altitude range is transitional from super-adiabatic to isothermal for d e = 0.007, while for d e = transitional process dominates only above 104 km and is super-adiabatic below. At 50 MHz the altitude range dominated by the transitional process becomes significantly shorter and starts at higher altitudes. At 146 MHz thermal conduction and Doppler terms significantly exceed inelastic electron cooling, therefore actual value of d e does not play any role, resulting in very rapid decrease in V ph with altitude and switching to the isothermal regime directly from the super-adiabatic. Examples of such behavior can be found in the V ph observations at 430 MHz (L. M. Kagan et al., manuscript in preparation, 2007). [23] Finally, we would like to mention that the rate with which V ph changes with frequency largely depends on the altitudes of backscatter as could be easily seen from our Figure 2 for vertical transmissions and Figure 3 for radar pointing 60 east. In Figure 3, middle panel (d e = 0.007) for example, the phase velocities between 102 and 103 km increase significantly at 16 MHz, much less so for 50 MHz and only slightly at 146 MHz. Altitude of backscatter would be also crucial in the sign of the change (increase/decrease). While V 16 ph always increases with altitude, V 50 ph and V 146 ph start to decrease starting from altitudes of 104 and 103 km respectively. 6of7

7 [24] Acknowledgment. This research has been supported by the Canadian Natural Sciences and Engineering Research Council. References Balsley, B. B., and D. T. Farley (1971), Radar studies of the equatorial electrojet at three frequencies, J. Geophys. Res., 76, Burgers, J. M. (1969), Flow Equations for Composite Cases, Academic, New York. Dimant, Y. S., and R. N. Sudan (1995), Kinetic theory of the Farley- Buneman instability in the E region of the ionosphere, J. Geophys. Res., 100, 14,605 14,626. Dimant, Y. S., and R. N. Sudan (1997), Physical nature of new cross-field instability in the lower ionosphere, J. Geophys. Res., 102, Farley, D. T. (1963), A plasma instability resulting in field-aligned irregularities in the ionosphere, J. Geophys. Res., 68, Gurevich, A. V. (1978), Nonlinear Phenomena in the Ionosphere, Springer, New York. Hysell, D. L., J. Drexler, E. B. Shume, J. L. Chau, D. E. Scipion, M. Vlasov, R. Cuevas, and C. Heinselman (2007), Combined radar observations of equatorial electrojet irregularities at Jicamarca, Ann. Geophys., 25, Kagan, L. M., and J.-P. St.-Maurice (2004), Impact of electron thermal effects on Farley-Buneman waves at arbitrary aspect angles, J. Geophys. Res., 109, A12302, doi: /2004ja Kissack, R. S., J.-P. St.-Maurice, and D. R. Moorcroft (1995), Electron thermal effects on the Farley-Buneman fluid dispersion relation, Phys. Plasmas, 2, Kissack, R. S., J.-P. St.-Maurice, and D. R. Moorcroft (1997), The effect of electron-neutral energy exchange on the fluid Farley-Buneman instability threshold, J. Geophys. Res., 102, 24,091 24,116. Kudeki, E., and D. T. Farley (1989), Aspect sensitivity of equatorial electrojet irregularities and theoretical implications, J. Geophys. Res., 94, Schunk, R. W., and A. F. Nagy (2000), Ionospheres: Physics, Plasma Physics and Chemistry, Cambridge Univ. Press, New York. St.-Maurice, J.-P., and R. S. Kissack (2000), The role played by thermal feedbacks in heated Farley-Buneman waves at high latitudes, Ann. Geophys., 18, St.-Maurice, J.-P., R. K. Choudhary, W. L. Ecklund, and R. T. Tsunoda (2003), Fast type-i waves in the equatorial electrojet: Evidence for nonisothermal ion-acoustic speeds in the lower E region, J. Geophys. Res., 108(5), 1170, doi: /2002ja L. M. Kagan and R. S. Kissack, Department of Physics and Astronomy, University of Western Ontario, Richmond Street, London, ON, Canada N6A 3K7. (lkagan@uwo.ca) 7of7