2. Secondary Gradient Drift Waves in the Ionospheric shape E Region

[10]  Traditionally, electric fields have been assumed as a source for both the Farley-Buneman [Buneman, 1963; Farley, 1963; Sudan, 1983] and gradient drift [e.g., Fejer and Kelley, 1980; Rogister and D'Angelo, 1970; Sudan et al., 1973] instabilities. For a long time neutral winds had been assumed to have magnitudes of about 50 m s^-1 or less. This might have been in part because there has been no theory either predicting or explaining winds as strong as 100 m s^-1 or higher, although sounding rockets have measured neutral winds using a chemical release as a tracer since 1958. The majority of these observations were not widely published, but instead presented either in internal institution reports or conference proceedings. The measurements of wind velocities of about 150 m s^-1 near 106 km altitude during the SEEK campaign [Larsen et al., 1998], and the publication of the theory of the neutral wind-driven gradient drift instability [Kagan and Kelley, 1998] immediately following these measurements, turned minds toward the role of neutral motions in E -region small-scale structuring. Recently, Larsen [2002] reviewed all sources of wind measurements over 40 years and showed that the neutral wind velocity exceeded 100 m s^-1 in more than 60% of observations and that the wind maximum was located between 100 and 110 km altitude. Such winds may be a source of free energy themselves for gradient drift [Kagan, 2002; Kagan and Kelley, 1998] and for thermal processes [Kagan and Kelley, 2000; Kagan et al., 2000a].

[11]  The true driving force for these instabilities is a polarization electric field appearing as a result of local charge separation (electron and ion) [Kagan, 2002]. This polarization field may be much stronger than the background electric field and at midlatitudes may map along the geomagnetic field line for large distances [Cosgrove and Tsunoda, 2001; Yokoyama et al., 2003, 2004].

[12]  Sudan et al. [1973] showed that 3-m gradient drift irregularities observed with 50-MHz radars cannot be induced directly and were the result of secondary gradient drift processes. In the Sudan et al. [1973] scenario the gradients, necessary for the secondary gradient drift instability (GDI) to develop, were provided by primary gradient drift waves. These primary waves, larger-scale inhomogeneities with the wavelength l_pr, produced the polarization electric field, which played the role of an external electric field for secondary plasma perturbations with wave length l _sec l_pr

[13]  McDonald et al. [1975] studied numerically the generation of secondary small-scale gradient drift irregularities and found that irregularities with l _sec <28 m are excited only after the amplitude of the larger-scale primary waves reached 4% of the background plasma density. This result is consistent with the theory of small-scale irregularity generation due to secondary gradient drift processes.

[14]  Although there are fewer rocket measurements of midlatitude small-scale irregularities than in the equatorial E region, this two-step process seems to be the case for midlatitudes as well. Thus, for example, Bowhill [1966] and Itoh et al. [1975] detected small-scale plasma irregularities associated with midlatitude sporadic E with rocket-borne Langmuir probes. Bowhill's [1966] estimates gave a vertical scale of these irregularities of about 25 m and a broadband amplitude of the irregularities of about 5-10%; Itoh et al. [1975] reported the broadband amplitude of 4-100 m waves to be about 1-5% of the background density; Kelley et al. [1995] observed a burst of waves with an estimated wavelength of 1-100 m near 110 km altitude and they reported the presence of few kilometer-scale waves throughout the entire flight. The amplitude of the 60-600 m primary waves was about 7% of the background density.

[15]  In their theory, Sudan et al. [1973] assumed that the background electric fields were the only instability source. Cooperation of neutral winds and electric fields for primary wave generation is discussed by Kagan [2002].

[16]  The development of the theory of secondary gradient drift waves presented herein follows the lines of the linear theory developed for the primary waves by Kagan and Kelley [1998] and similar to that suggested by Sudan et al. [1973] for electric field-driven processes.

[17]  Under routine operation most coherent scatter radars observe backscatter in the meridional plane perpendicular to the geomagnetic field lines and at frequencies close to 50 MHz. Thus the detected signal is backscattered by irregularities with a typical scale close to 3 m. The meter-scale irregularities cannot be generated directly and are the result of secondary gradient drift processes in which the primary waves serve a double purpose for these secondary waves: they act as the regular gradient and they create a polarization electric field playing the role of the external electric field. For this routine geometry of the experiment the radar measures backscatter from meter-scale (secondary) irregularities with the wave vector along the radar line of sight in the meridional plane. Thus the primary waves for these meter-scale irregularities should have the wave vector perpendicular to both the wave vector of the secondary waves ( k_y ^sec below) and the geomagnetic field, i.e., in the zonal direction. These primary waves can be considered as a regular plasma gradient for secondary gradient drift waves if

Figure 1

(1)

where k_y ^sec is the radar line-of-sight wave vector of the secondary wave and l_x^pr is the zonal wavelength of the primary wave [Sudan et al., 1973]. Here we have chosen a Cartesian coordinate system relative to the routine operation of most radars (the MU radar in particular) (Figure 1), i.e., the z axis is along the geomagnetic field B₀, the x axis is eastward, and the y axis is aligned with the radar line of sight (southward and downward).

[18]  The dimensionless plasma density n=N/N₀ ( N is the plasma density and N₀ is the background plasma density) for the secondary processes may be written as

(2)

where n_pr is the dimensionless plasma density fluctuation in the primary wave and n₁ is the dimensionless plasma density.

[19]  The primary irregularities embedded in the background plasma create a background for the secondary processes. The meridional Doppler velocity observed by the radar is the phase velocity of the secondary gradient drift waves and is due to electron drift in the crossed electric and geomagnetic fields and ion motion driven by neutral wind. The expression for the line-of-sight phase velocity of the secondary wave (in the following, if not stated otherwise, by "line-of-sight velocity" we mean the line-of-sight velocity routinely observed by CSRs, i.e., in the meridional plane) is similar to that for the primary wave,

(3)

[20]  Here V_{e, sec, y} is the electron drift velocity in the crossed total imposed electric (  E_0x+ E _pol,x below) and geomagnetic B₀ fields; V_{i, sec} represents ion drag by moving neutrals; and

(4)

where n_en, n_in and w_e, w_i are the collisional and gyrofrequencies of the electrons and the ions. In equation (4) and below we omit the subscript "sec" for the wave vector of the secondary waves.

[21]  The polarization electric field E _pol,x, originated in the primary processes, may be quite strong and together with the background electric field E_0x acts as an external electric field for the secondary gradient drift waves. The line-of-sight electron and ion velocities of the secondary waves in equation (3) are

(5)

(6)

respectively, where u_n is the neutral wind velocity. The polarization electric field may be written as [Kagan and Kelley, 2000]:

(7)

where D= (D_iz + D_ez) cos² a + (D_i + D_e) sin² a is the diffusion coefficient; D_iz, D_i and D_ez, D_e are the field-aligned and field-perpendicular diffusion coefficients of ions and electrons, respectively; a is the angle between k and B₀; k_T is the Boltzmann constant; T is the temperature, which at the altitudes of interest may be considered the same for electrons, ions and neutrals; e is the magnitude of the elementary charge; u₀ = V_i0- V_e0 is the background current velocity, i.e., the relative velocity between electrons and ions; and V_i0 and V_e0 are the zero-order ion and electron velocities, respectively.

[22]  The aspect sensitivity angle, which is the complementary angle to a, is measured to be less than 1^o. For E -region altitudes this gives D (k_T T / M n_in)(1+ y). The zonal component of the polarization electric field (7) contributing to the meridional electron velocity (5) is

(8)

where the zonal zero-order current velocity is

(9)

(10)

[23]  Note that for the lower E region (isotropic ions, q_i 1) the zonal current velocity is defined by the electron drift in the crossed meridional electric and geomagnetic fields and by the ion drag by a zonal neutral wind.

[24]  Equations (8)-(10) lead to

(11)

[25]  Equation (3), taking into account (5), (6) and (11), gives the following phase velocity along the radar line of sight for the secondary gradient waves

(12)

[26]  In equation (12) we have neglected the term y (q_i E_0y -q_i² E_0x)/ (1+q_i²) compared to E_0x.

[27]  Note that for the lower E -region case ( q_i 1 ), and for zero neutral winds, (12) gives the expression for the phase velocity of secondary waves derived by Sudan et al. [1973].

[28]  There is a very important notion related to the observational use of the above formulas, equations (3) and (12). Coherent radars observe the line-of-sight Doppler velocity, which is the phase velocity of irregularities backscattering the radar signal. Independently of the nonlinearity of the processes producing the small-scale irregularities, this line-of-sight phase velocity, to leading order, is the same as for linear processes. Moreover, it is the same for the different instabilities: gradient drift, thermal and Farley-Buneman. However, for the latter process one should keep in mind that, although the phase velocity is described by the same formula, observations have shown that Farley-Buneman waves always move at their threshold speed, which is close to the ion acoustic speed c_x (due to the factor adjustments).

[29]  In more sophisticated experimental procedures when CSR observes backscatter signal from small-scale irregularities perpendicularly to the geomagnetic field line but the radar beam does not lie in the meridional plane, the line-of-sight phase velocity is

(13)

where b is the angle between the radar line-of-sight and zonal directions; V_{ph, sec, y} is defined by (12) and V_{ph, sec, x} is:

(14)

and

(15)

[30]  The growth rate of the secondary GDI is

(16)

where k _los is the radar line-of-sight wave vector of the secondary waves. For the routine observational geometry k _los = k_y.

[31]  The first term in brackets describes the two-stream processes. For the gradient drift instability the velocities are much less than the ion acoustic speed and the two-stream term may be neglected. For the routine radar geometry at the condition of marginal stability

(17)

The right-hand part of (17) is defined by the ionospheric parameters y, n_in, w_i, and c_s, which are functions of altitude, location and time. These parameters may be found using the models. Neither of the values on the left-hand side of (17) is known. However, to satisfy our initial assumption k_y ^sec l^pr 1 and based on observations [Kelley et al., 1995], one may suppose l^pr to be of the order of 100 m for 3-m irregularities observed by the 46.5-MHz MU radar ( l ^sec 3.2 m).

[32]  It should be mentioned that the primary waves, which provide a regular plasma density gradient, are not necessarily caused by gradient drift processes. The only requirement is that these primary waves should have a wave vector perpendicular to the geomagnetic direction and appear due to the same source as the secondary waves. Thus the role of primary waves may be played by the irregularities excited by thermal processes [Dimant and Sudan, 1997; Kagan and Kelley, 2000], for example. The wave vector of these thermal irregularities is perpendicular to B₀, thus providing the equatorial-like geometry for secondary GDI generation. The wave vector of the primary waves is perpendicular to both the wave vector of the secondary waves and to the geomagnetic field. Thus, for the secondary waves observed in the meridional plane, the primary waves should have a wave vector in the zonal direction.

[33]  Note that the scenario considered does not discuss the energy transfer from long wavelengths to short ones. For stationary turbulence the energy input at long wavelengths is balanced by energy dissipation at short wavelengths. To find the power spectrum of stationary turbulence, one may follow the scheme proposed by Sudan [1983] for the equatorial electrojet, with appropriate corrections from neutral wind inputs into the linear GDI growth rate [see Kagan and Kelley, 1998]. The effects of turbulence on the width of signal spectra received by radars operated in the frequency range between 2 MHz and several hundred megahertz are reviewed by Hocking [1985].