R. André and M. Pinnock
British Antarctic Survey, Natural Environment Research Council, Cambridge, UK
Laboratoire de Physique et Chimie de l'Environnement, Centre National de la Recherche Scientifique, Orléans Cedex, France
Laboratoire de Sondage de l'Environnement Electromagnétique Terrestre, Centre National de la Recherche Scientifique, Université de Toulon et du Var, La Garde Cedex, France
SuperDARN is a network of HF radars which determines amongst other things, the large-scale convection pattern in a plane perpendicular to the magnetic field [Greenwald et al., 1985] over the ionospheric auroral zone. From the transmission of a multiple pulse scheme, the radars measure the autocorrelation (ACF) of the signal backscattered at several distances (range gates) from the radar by field-aligned concentration irregularities. For each range gate, this ACF is routinely analyzed by a basic method (FITACF) [Baker et al., 1995; Villain et al., 1987] which extracts the power, the line-of-sight Doppler velocity of the irregularities and the spectral width. This Doppler spectral width is a measure of ionospheric plasma turbulence caused by the structured energetic precipitation or by electric field variations within each range-beam cell of the radar.
Baker et al. [1990, 1995], and Rodger et al.  have shown that the radar spectral width, and more generally the ACF characteristics have some particular properties inside the cusp. They have used low-altitude satellite data and optical imagers to show that HF data recorded in the cusp are characterized by a high-spectral width. Several studies have used this parameter to identify the cusp, and to derive more information about solar wind-magnetosphere-ionospheric coupling, such as the temporal evolution of the reconnection rate at the magnetopause [Baker et al., 1997; Pinnock et al., 1999]. Recently, André et al.  have shown that wave activity in the Pc1 frequency band can explain all the ACF characteristics observed in the cusp.
On the nightside, some studies have related the increase of spectral width with latitude with the central plasma sheet (CPS)/boundary plasma sheet (BPS) boundary [Dudeney et al., 1998]. This property has been used to observe the boundary motion during substorm events [Lewis et al., 1997].
Few studies have explored the physical processes which give rise to the spectral width. Grésillon et al. , Hanuise et al.  and later Villain et al.  have applied collective scattering theory to the HF radar data. They have shown that the shape of the ACF power variation and its spectral width are related to characteristics of the turbulent motion of density irregularities.
In this paper, we review the physical origin of the radar spectral width, and discuss factors that can affect it. The next section presents a general view of the spectral width behavior from a statistical point of view. The third part discusses the link between the spectral width and the micro-scale ionospheric turbulence and makes some comments about the interaction between radar waves and field-aligned irregularities. The effect of wave activity routinely seen by low-altitude satellites will be examined in the fourth section. The next two sections evaluate the impact of the large-scale convection pattern, and the effect of meso-scale inhomogeneities, like vortices, which can be present in the convection. The last part summarizes these effects and their importance in our understanding of the spectral width measured by HF radars.
To have a global view of the spectral width over the whole polar ionosphere, we have conducted a statistical study. We have built a large database, compiled from northern hemisphere radar data recorded between October 1996 and March 1997. This database contains only ionospheric scatter from the F region selected by its range (900 km < range < 3300 km) and characterized by a large signal-to-noise ratio ( > 3 dB). These data are localized in a grid defined in MLT (Magnetic Local Time) and MLAT (Magnetic Latitudes) which has a resolution of 30 minutes and 1o, respectively.
Figure 1 shows the average spectral width observed in the low-altitude ionosphere by HF radars in the northern hemisphere. Its main characteristic is that the spectral width is low at low latitudes and inside the polar cap ( 150 m s -1 ), but increases up to 200 m s -1 in the auroral oval, and is at a maximum inside the cusp ( > 250 m s -1 ). These results have been determined before (J. P. Villain and R. A. Greenwald, unpublished data, 1994) using a similar statistical database. A more detailed presentation of these two statistical studies is the subject of a forthcoming paper [Villain et al., manuscript in preparation, 1999]. Here, the physical mechanisms that lead to these high-spectral widths are reviewed and discussed.
The spectral width is usually described in terms of turbulent processes and instability mechanisms. It arises from interactions between irregularities, or from micro-scale electric fields.
In this description, we assume that all irregularities are moving with a large-scale velocity V0, and have a random displacement d due to this micro-scale electric field. The signal backscattered by the electron density irregularities can be written as equation (1), where G( k) is the form factor which mainly depends on the number of irregularities and their amplitude [Grésillon et al., 1992].
The ACF phase is controlled by the second term, and gives access to the line-of-sight Doppler velocity. The last term, which determines the spectral width, is the Fourier transform of the probability distribution of irregularity displacement during the time t ( P(d, t) ), and is also called the statistical characteristic. It depends on statistical properties of the micro-scale displacements kd(t) . The average expressed by the brackets is taken over the whole range gate and integration time (typically 7 s). In equation (1), the second term refers to large-scale processes (greater than a range-beam cell) and the third one to micro-scale processes (size of the irregularities).
Grésillon et al.  and Hanuise et al.  consider the limiting cases where the correlation length of the irregularity motion is in turn much shorter and much longer than the observation wavelength, to derive P(d, t) and thus derive the ACF power. When this correlation length is much shorter than the radar wavelength, P(d, t) can be represented by a random walk and hence a Gaussian function. The resulting ACF power is an exponential function characterized by a decay which is mainly controlled by a diffusion coefficient. Grésillon et al.  have estimated this diffusion coefficient in a tokamak and found a very good agreement with those estimated by the classical method. In the other limit, the displacement depends linearly on the velocity. Under the assumption that the velocity distribution function is Gaussian, they obtained the usual Gaussian Doppler spectrum which depends on the microscopic velocity fluctuations observed during the integration process. Hanuise et al.  have already shown that the ACF power variation observed in HF radar data has these shapes, but more often has an intermediate state, with an ACF starting with a Gaussian shape and ending with an exponential one.
Extending this study to a more general case, Cabrit , followed by Villain et al. , have derived the general analytical function under the assumption of a Gaussian distribution function P(d, t). This general function, displayed in equation (2), depends on both the diffusion coefficient ( D ) and the correlation time of irregularity motions ( T ).
These last two parameters can be estimated by fitting equation (2) to ACFs observed by HF radars. Figure 2 shows a typical example of the temporal evolution of the ACF power recorded at the Halley station in the southern hemisphere. The best fit is shown by the line, and uses a diffusion coefficient, D, and a correlation time, T, of 90 m 2 s -1 and 5.5 ms, respectively. The spectral width determined by FITACF is 125 m s -1. This function represents well the recorded data, with a Gaussian decay on the first lags and an exponential decay on the last ones.
By making a large statistical study of the diffusion coefficient values in the auroral F region, André et al.  showed that its distribution function is frequency dependent. Its maximum is found around 100 m 2 s -1 and 400 m 2 s -1 for radar frequencies of 12 and 9 MHz, respectively. More precisely, they have shown that the distribution functions obtained with a radar frequency above 11 MHz are very similar, whereas the increase in diffusion coefficient occurs for frequencies below 10 MHz.
A diffusion coefficient of 100 m 2 s -1 is comparable to the Bohm diffusion coefficient in the F region (125 m 2 s -1 ), and to the ion cross B field diffusion coefficient in the collisional E region, but is much higher than the typical ambipolar diffusion coefficient (1 m 2 s -1 ). This suggests an overestimation of D in the F region.
At low radar frequencies, André et al.  showed that this distribution depends also on the length of the radar wave propagation path. This suggests an interaction between the radar wave and the field-aligned irregularities in the ionosphere. This interaction decreases the wavefront coherence during its propagation, and so artificially increases the spectral width.
These studies have shown a clear correspondence between the spectral width and micro-scale turbulent processes. They have estimated some typical values of the turbulent transport experienced by ionospheric irregularities which have a wavelength around 10 m. Although the ACF shapes agree well with their theoretical counterparts, the estimated diffusion coefficient is higher than the expected values. This suggests that some other effects contribute to spectrum broadening.
Baker et al.  have shown a clear correspondence between high spectral width and the particle signature of the low-altitude cusp. Figure 1 shows that, on average, spectral width values in the cusp, centered at about 73o and 11 MLT, are higher than 250 m s -1. More precisely, spectra recorded inside the cusp contain several components, and both the velocity and spectral width determined by FITACF are highly variable [Baker et al., 1995; Pinnock et al., 1995].
As observed by low-altitude satellite, the cusp region is characterized by a sharp increase in low frequency wave activity (0.1-10 Hz) [Erlandson and Anderson, 1996; Matsuoka et al., 1993]. A part of these electric and magnetic field variations arises from field-aligned currents. However, a significant component arises from a mixture of downgoing and upgoing Alfvén waves. These waves are generated at the dayside magnetopause, during reconnection processes or by the upgoing accelerated ions at the poleward edge of the cusp [e.g. Dyrud et al., 1997]. Their amplitudes are sufficiently high (a few mV m -1 ) to modulate the large-scale velocity field and hence generate temporal variations of the macroscopic contribution to the ACF in equation (1).
Recently, André et al.  have evaluated the effect of such electric field variations on the ACF recorded by HF radars. They have simulated this effect on the radar data, under the assumption that there is no turbulence in the ionosphere: The expected ACF power variation should have no decay because all the irregularities are always totally correlated. Figure 3 shows an example of their simulation. The upper panels show the electric field spectrum (left) and the corresponding temporal variation of the velocity field (right). The lower panels show the temporal evolution of the ACF phase (left) and ACF power (right). The dashed line shows the expected ACF without the wave. Here, the plasma is moving with a constant velocity of 100 m s -1, modulated by a wave characterized by an amplitude of 40 ms -1 and a frequency of 0.5 Hz.
This example shows that the temporal evolution of both the phase and the power are modulated. This implies that the associated spectrum contains more than one component, as recorded in the cusp. The power decays despite the fact that there are no turbulent processes included in this simulation. If the velocity deduced by FITACF is representative of the input field, then the spectral width obtained is very high, greater than 300 m s -1. They have also shown that under these conditions, and even with a low-amplitude wave, the velocity and spectral width determined from these ACFs are highly variable, and that the spectral width is very high.
To produce a high-spectral width, the wave amplitude has to be greater than a threshold which is frequency dependent (R. André et al., Identification of the low altitude cusp by SuperDARN radars: A physical explanation for the empirically derived signature, Submitted to J. Geophys. Res., Nov. 1999, hereinafter referred to as André, 1999). For example, the minimum value is 20 ms -1 (which corresponds to an electric field amplitude of 1 mV m -1 ) at 0.5 Hz, and only 2 m s -1 (0.1 mV m -1 ) at 5 Hz. These amplitudes are low compared to those usually recorded by satellites [Maynard et al., 1991].
All the waves which have a frequency lower than the Nyquist frequency (4 mHz in the common radar's running mode, and 50 mHz in the high-resolution mode) are correctly resolved by the radar, but the highest frequency waves are undersampled (André, 1999). In this case, the Doppler spectrum contains several components. These supplementary components are caused by the radar technique and are artefacts. At last, These results are independent of the background line-of-sight velocity (André, 1999).
Low frequency wave activity is also observed along magnetic field lines in the auroral oval [Gurnett, 1991]. Hence, one can probably use the same mechanism to explain the high spectral widths found over the whole auroral oval (see Figure 1), and especially in the nightside where a smooth increase of the spectral width has been found together with both the central plasma sheet/boundary plasma sheet boundary layer and with an increase in the wave activity [Dudeney et al., 1998].
Following Gary et al. , this wave activity over the auroral zone could be the signature of boundaries in the large-scale field-aligned current system. Therefore, because low-frequency waves (0.1-10 Hz) strongly perturb the ACF, leading to multi-component spectra and very high spectral width, one can probably use these characteristics to map boundaries of the large-scale current system, however rigorous testing of this idea has yet to be carried out.
The large-scale plasma motion at high latitudes is driven by the solar wind-magnetosphere-ionosphere system, and is usually composed of two, three or four cells depending on the interplanetary magnetic field orientation [Cowley and Lockwood, 1992]. This ionospheric convection results in several gradients in the velocity field over the whole auroral oval, which are sufficiently sharp to affect the velocity distribution experienced by the irregularities.
For example, one can expect that inside a convection reversal, one part of a radar range gate is sensitive to an electric field directed in one direction whereas the other part is mainly dominated by an electric field directed in the opposite direction. Thus, the measured ACF should reflect these two different velocities, and the spectrum should contain two components. Because the FITACF method is not well adapted to these kind of ACFs, the computed spectral width is expected to be higher.
Such a situation regularly appears in the radar data. For example, Barthes et al.  have applied the high-resolution spectral analysis method MUSIC (MUltiple SIgnal Classification) to radar data. They have found that the probability of finding multi-component spectra is strongly enhanced in convection reversals.
By considering a realistic convection model [e.g. Rich and Maynard, 1989] derived from Heppner and Maynard , one can assign an electric field vector to each point in the high latitude ionosphere. One can also define a grid in a given radar range gate which has a spatial resolution (1 km) much smaller than the gate dimensions (45 km long, 100 km wide on average). Thus, at any particular time, one can compute the line-of-sight velocity at each grid point, and then compute the velocity distribution found over the whole range gate. Assuming there is no turbulent effect, the ACF spectrum should reflect this distribution, and thus its width should be the spectral width recorded by the radar. Thus, at one particular time and for a given range gate, one can estimate the spectral width induced by the large-scale ionospheric convection pattern.
Figure 4 (left panel) shows the estimated spectral width in the Stokkseyri radar field of view at 1900 UT. At this time, the radar is looking mainly at the convection reversal. Black lines represent isocontours of the convection model defined by Rich and Maynard . Two examples of the computed velocity distribution are also shown. When the radar is looking in a region where there is no large velocity gradient (upper right panel), the velocity distribution is narrow, and the associated spectral width is negligible. By contrast, when this range gate is inside a large velocity gradient (lower right panel), the spectral width can be as large as 250 m s -1.
One can also see that the distribution shown in the lower right hand panel is not totally flat but has a minimum value near 0. If the velocity gradient increases in this convection reversal, one can expect a better separation between the positive and negative values of the velocity. Then, one can obtain a clear multi-component spectrum. Despite the fact that this ionospheric convection model is based on a representative description of the real convection pattern [Heppner and Maynard, 1987], it smoothes the velocity gradients. Thus one can conclude that convection reversals can naturally introduce some multi-component spectra, and that they should induce a high spectral width value.
Because the radar records only the line of sight velocity ( k v ), the results presented in Figure 2 can be different when looking in the same geophysical region with another radar. For example, when considering a radar which is mainly looking poleward, the gradients will be smaller, even in the convection reversal. In this case, the deduced spectral width is also much smaller.
We have estimated the contribution of the convection pattern to the spectral width by using the statistical ensemble (radar/beam/gate) found in Figure 1. Because radars in the northern hemisphere have different orientations with respect to a magnetic coordinate system, and most of them are oriented northward, the maximum spectral width found in the convection reversals has a value of only 150 m s -1. In other geophysical regions, the spectral width is lower than 50 m s -1. This clearly shows that this effect cannot be used to explain the high spectral values recorded in the cusp and along the auroral oval, and even at low latitudes where spectral width is of the order of 150 m s-1 (see Figure 1).
One can conclude that velocity gradients induced by the large-scale convection pattern can affect the radar spectral width. This effect becomes important when the gradient is orthogonal to the radar beam direction, but it cannot be used to explain the spectral width usually recorded by the HF radars.
If the ionospheric convection leads to large-scale velocity gradients ( 100 km), specific events in the solar wind-magnetosphere-ionosphere system can induce smaller-scale perturbations in the ionospheric convection, such as flux transfer events or travelling convection vortices. To clearly evaluate their effects on the ACF, we have conducted a realistic simulation of the radar processing technique. We first apply the simulation to a velocity shear that could be induced by either a large-scale convection reversal, as shown previously, or by an auroral arc. Secondly, the simulation is applied to meso-scale vortices ( 10 km).
To evaluate more precisely the ACF resulting from meso-scale inhomogeneities in one range gate, we have to write equation (1) as (3), where Pj is the backscattered power from a single irregularity, Vj is its velocity, r0j+ d rj is its initial position, and where the average is made over the integration time, which represents about 65 individual ACFs.
To simplify this description, we remove turbulent effects (first term of (3)), and define a velocity profile onto a grid r0j which has a spatial resolution of 2 km. The defined range gate is 45 km long and 90 km wide, and corresponds to a gate located at about 1500 km from the radar.
To take into account the various positions of the scatterers, we have applied a small perturbation d r0j to their initial position on the grid. This perturbation is linked to the fact that during two successive individual ACFs, the irregularities have moved, producing a phase shift in (3) which can be simulated by a small random number which has a maximum value of 10 meters, much smaller than the grid resolution (2 km). One has to note that this initial perturbation does not vary during the construction of each individual ACF and thus cannot be compared to the displacement due to turbulent processes d(t).
Finally, the power Pj is linked to the irregularity backscattering cross section, their number and amplitude. The backscattering cross section depends on the geophysical conditions which give rise to the irregularities and should increase in their source region.
The grid coordinates are defined such that the directions parallel and perpendicular to the wavevector are in the Y and X directions, respectively. The center of this coordinate system corresponds to the gate center.
The first application is a velocity shear, which may result from a more realistic large-scale convection pattern, i.e., with a sharper gradient. In this example, the velocity shear is defined by (4) and is superimposed on a background velocity of 50 m s -1 in both the X and Y direction. The backscattered power associated with the plasma outside the structure is 3 dB
Figure 5a shows the resulting velocity vectors of the plasma flow and the backscattered power coded in gray scale. Most power is defined to come from the velocity shear itself; the velocity shear instability [Kintner, 1976] can increase both the irregularity number and their amplitude. In this simulation, the maximum power due to this gradient is defined to be 20 dB. One has to note that ACF characteristics do not strongly depend upon this arbitrary value.
Figures 5b and 5c show the temporal evolution of the ACF phase and power as deduced by this simulation. The FITACF method gave a line-of-sight velocity of 120 m s -1 and a spectral width of 170 m s -1. This value is high despite no turbulent effects being included in this simulation.
The dashed line shows the temporal evolution of the ACF phase which corresponds to this velocity. Although a general agreement is found, the simulated phase shows a non linear behavior which cannot be fully reproduced by the FITACF method. Because this modulation is also seen in the power, the spectrum associated with this ACF contains more than one component. This result is similar to that found by Barthes et al. , who showed an increase in the probability of finding several components (in the velocity spectrum) inside a convection reversal.
In this part, we simulate the effect of a filamentary field aligned current (of the order of the range/beam cell) [Borovsky, 1993]. The current closure in the conducting layer will generate a divergent electric field structure, and hence a small-scale vortex.
A vortex in a plane perpendicular to the magnetic field can be described by equation (5) [e.g. Pudovkin et al., 1997], where r is the distance from the vortex center, defined at (X,Y)=(-5,-5), normalized over the radius of a charged cylinder (10 km). The background velocity is directed in the Y direction, with an amplitude of 50 m s -1.
Depending of its direction, the current that sustains the vortex can generate irregularities by the current convective instability [Ossakow and Chaturvedi, 1979] and thus could increase the backscattered power in the vortex center.
The resulting plasma flow and the phase and power of the recorded ACF are shown in Figures 6a, 6b and 6c, respectively. Again, the phase does not present a linear behavior, the power is characterized by a very high spectral width, and both suggest a multi-component spectrum.
Applying a high-resolution spectral analysis method (maximum entropy) to the radar data, Schiffler et al.  have found a large number of double-peaked spectra in the low latitude boundary layer (LLBL). They suggested that these double-peaked spectra could arise from filamentary currents generated by structured soft electron precipitation ( E 100 eV) observed by satellites at the same time. These currents could generate small-scale vortices (scale size of around 10 km), such as the one shown in Figure 6a.
The non-linearity seen in the phase is reduced when decreasing either the maximum velocity in the vortex center or the vortex size. For example, a vortex that has a typical size less than 5 km does not generate a clear multi-component spectrum in the radar data. On the other hand, structured precipitation might generate a series of vortices in the radar gate, which would strongly increase the non-linearity, but would not necessarily generate a double-peaked spectrum; they would more likely generate a spectrum that has more than 2 components.
We have described, through simulations, a number of factors which condition the line-of-sight Doppler spectral width as determined by SuperDARN HF radars.
The spectral width is related to microscopic properties of the displacement of irregularities resulting from a turbulent electric field. More precisely, the ACF power is linked to the correlation time of the irregularity motion and typical diffusion coefficient. This is supported by the observed shape of the ACF power variation. The estimated diffusion coefficient in the ionospheric F region is found to be of the order of the maximum diffusion allowed in a magnetized plasma (Bohm diffusion). This very high value suggests it is an overestimation. Therefore, although the microscopic physics can provide the main contribution to the spectral width, there are other factors which can be equally important.
When the radar frequency approaches the typical plasma frequency in the ionosphere, the radar wavefront can interact with field-aligned irregularities. This results in a decrease of the wavefront coherence and hence an apparent increase of the measured spectral width. This interaction is supported by the increase in the diffusion coefficient when both the radar frequency is low and the distance from the radar increases.
Considering meso-scale contributions to the spectral width, simulations indicate that any inhomogeneities in the velocity flow in the probed area would increase the spectral width. Moreover, this can easily introduce several components in the recorded spectrum. Because the analysis method that is used to derive the spectral width (FITACF) is not well adapted to these spectra, the resulting spectral width will be overestimated. Such multi-component spectra have already been found in large-scale velocity shears and in the dayside ionosphere. Such an effect can also be achieved by taking into account temporal variations of the large-scale electric field. In this situation, the velocity spectrum is expected to have several components, and the recorded spectral width should be very high and variable. These characteristics are routinely observed when the radar is probing the ionospheric cusp.
Thus, we have demonstrated that the Doppler spectral width determined by the SuperDARN HF radars is a complex convolution of (i) Pc1/Pc2 wave activity, (ii) geometry of the radar with respect to the large scale convection pattern, (iii) the presence of velocity shears of the scale size of a range/beam cell and (iv) the microscale turbulence ( 10 m). We have shown that wave activity is the dominant parameter in areas such as the cusp. At other longitude and magnetic local times the other three effects make a significant contribution.
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