We report the first spatially and temporally continuous observations of the upward electric field in the E-region using the Arecibo incoherent scatter radar. This is achieved by employing the dual beam incoherent scatter radar measurements of ion velocity and using a theoretical ion-neutral collision model. The derived daytime vertical electric field, from ∼105 km to 145 km, shows large height variation, as in previous nighttime rocket measurements. Assuming that the electric field along the field line is negligible, the height variation is the same as the horizontal variation at Arecibo. Although the height variation has been attributed to gravity waves in previous studies, this explanation is not consistent with the temporal characteristics in our observation. We further discuss the error sources that affect the measurement of the electric field.
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 Electric fields (or E-field, for short) play an important role in determining the dynamics of the ionosphere. Direct measurements of the E-field have been carried out through instrumented rocket flights [e.g.,Sangalli et al., 2009, and references therein]. This method, although limited to only snap shots of the electric fields, provides very good height resolution and is particularly useful to study the fine structures, such as those of the sporadic E. At the F-region heights, the ion drift velocity perpendicular to the geomagnetic field (B) is directly related to the E-field when the ion-neutral collision frequency is much smaller than the ion gyro-frequency. As a result, the F-region vector E-field is routinely obtained by incoherent scatter radars (ISR). In the E-region, Tsunoda et al.  reported E-field measurements from the rotation of the ion vector velocity. In addition to the ISR methods, spectral characteristics extracted from VHF coherent echoes have also been used to estimate vector E-field [e.g., Hysell et al., 2009]. The methods used by both Tsunoda et al.  and [Hysell et al. 2009] are limited to high latitudes because they require very large electric field either to make neutral wind negligible or to generate plasma instability. In all previous radar measurements, it has been assumed that the B-field line is equipotential, resulting in a height-invariant E-field. The height-invariant property has been often extended to E-region studies of the neutral wind [e.g., Zhou et al., 1997; Heinselman and Nicolls, 2008].
 In this paper, we outline a method using an ISR to derive the vertical E-field in the E-region without assuming that B is equipotential. The vertical E-field is particularly important in the E-region because it directly affects the vertical motion of the plasma. Our method is applicable to an altitude range of about 3-4 scale heights centered at the altitude where the ion gyro-frequency equals to the ion-neutral collision frequency. We present results from the Arecibo ISR, whose dual beam capability and high accuracy are most ideal for such observations. In discussing the derived results, we also discuss potential error sources of the method.
 The theoretical foundation of our method is the ion momentum equation. Under equilibrium in the E-region, the ion-momentum equation can be simplified as
where V, U, E, and B are the vector ion drift, neutral wind, E-field in the earth-fixed frame, and geomagnetic field, respectively [e.g., Heinselman and Nicolls, 2008]. Parameters e, mi and vin are the electron charge, ion mass, and ion-neutral collision frequency, respectively. Neglecting the vertical neutral wind, the vertical component equation is
where I is the dip-angle, Ve is the eastward ion drift velocity in the geomagnetic coordinate, and subscript z indicates vertical component with upward positive. Equation (2) can be further rewritten as
where ρ = is the ratio of ion-neutral collision frequency to ion gyro-frequency and Veb = VecosI. We may regard Vzρ as a scaled vertical velocity and Veb as the equivalent eastward velocity with the B field being horizontal. When ion-neutral coupling is weak, as in the F-region, we see that Ez/B = −Veb, which is a manifest of V = E × B/B2 in the zonal direction. By measuring the zonal and upward velocity and using a modeled ion-neutral collision frequency, we can thus derive the vertical E-field component.
 The Arecibo ISR is capable of measuring the radial ion velocity in two directions simultaneously, one by the line-feed and the other by the Gregorian feed. In our experiment, we pointed the line-feed in the vertical direction and the Gregorian feed 15° off zenith. The Gregorian feed was parked in the geomagnetic zonal plane most of the time. Assuming a horizontally uniform wind field within the beam separation distance of 30 km, the horizontal velocity, VH, is related to the upward (Vz) and oblique radial (Vor) velocities through the following equation:
where Za is the zenith angle of the oblique beam. When away is positive for the radial velocity, the sign of VH follows the projected direction of the oblique beam in the zonal plane.
 In order to obtain the vertical E-field, it is crucial to have accurate ion-neutral collision frequency. In the daytime E-region, the ion composition is largely NO+ and O2+. The ion-neutral collision frequency for NO+ is relatively simple because resonant collision can be largely neglected. The ion-neutral collision frequency for O2+ is complicated by the fact that O2-O2+ resonant collision is not well known for temperatures below 800 K, which applies to the altitude range of our interest. Since the non-resonant coefficient between O2+ and CO is as large as 5.63 × 10−10, we therefore adopt an ad hoc equivalent coefficient of 6.0 × 10−10 for O2+-O2 “resonant” collision. Assuming the ionosphere has equal amount of NO+ and O2+, and using the non-resonant coefficients given by Schunk and Nagy , the average ion neutral collision frequency is
where  is number density in cm−3. The averaged ion-neutral collision frequency is a few percent off of a pure NO+ or O2+ ionosphere. The ratio of ion-neutral collision frequency to ion gyro-frequency at altitude z (in km) can be expressed as
where z0 is 123.7 km. At the altitude of zo, ρ = 1 and the scale height is 9.4 km.
3. Results and Discussion
 The E-region data at Arecibo was taken using the coded-long pulse technique as described by Sulzer . Further E-region data processing to obtain geophysical parameters is described by Zhou et al. . Here we are only interested in the ion drift. In Figure 1 (top), we present the measured vertical ion drift. Eastward drift, as calculated from equation (4), is shown in Figure 1 (middle). The calculated vertical E-field scaled by B is shown in Figure 1 (bottom). The magnitude of B is about 3.4 × 10−5 tesla at 120 km and varies very little throughout 105 to 150 km. For a velocity of 40 m/s, which is approximately the largest Ez/B, the corresponding vertical E-field is 1.36 mv/m.
 The E-field is seen to be well correlated with the eastward velocity Ve. The general trend is that Ez/B has the opposite sign of Ve × B, which points upward for a positive Ve. The magnitude of Ez/B is mostly smaller than ∣Veb∣ (=0.7∣Ve∣). This is expected of the dynamo effect generated by the neutral wind. As the Lorenz force due to the neutral wind drives the ions and electrons in opposite directions, a polarized E-field is generated. This E-field, in the direction largely perpendicular to B, will impede the motion of the ions in the vertical direction. Since the horizontal velocity is the driving mechanism, we thus expect ∣Veb∣ to be larger than either ∣Vz∣ or ∣Ez/B∣. The sign of Ve is expected to be largely the same as that of Vz and opposite to that of Ez/B.
 To examine the relationships among Veb, ρVz, and Ez/B in a more quantitative manner, we plot them at 3 heights in Figure 2a. For easier comparison, both in terms of trend and magnitude, we plot −Ez/B in Figure 2a so that it is largely in phase with the eastward velocity. The middle plot is for 124 km where the normalized collision frequency is unity while the other two plots are one scale height above and below 124 km, respectively. At 115 km, where the collision frequency is larger than the gyro-frequency, ρVz follows Veb very closely most of the time. At 133 km, where the collision frequency is smaller than the gyro-frequency, −Ez/B and Veb are almost identical. The high altitude behavior is the same as in the F-region where ion motion across the field line can only be caused by E × B drift. Our measurement here gives the applicable lower altitude boundary at about 133 km for this statement to be valid. This boundary is about the same as obtained from rocket measurements at high latitudes [Sangalli et al., 2009].
Figure 2b shows the altitudinal variation of the normalized upward E-field along with Veb, Vz, and the scaled vertical ion drift for three time intervals. As discussed before, if the Lorentz force is the driving mechanism and E-field is a response to it, we expect the magnitude of Ez/B to be smaller than Veb and their signs to be opposite (note that −Ez/B is plotted). Furthermore, we expect that an eastward drift corresponds to an upward vertical velocity. We see that this is largely the case between 115 and 125 km. At other altitudes, we see that ∣Ez/B∣ can be larger than ∣Veb∣, or Ez/B and Veb have the same sign. This is most obvious below 115 km in Figure 2. This may indicate that the total E-field dominates that of the polarization field. However, it is also possible that the ion-neutral collision frequency may be erroneous. If the ion-neutral collision frequency is reduced by a factor of two, effectively making ρ = 1 at 118.3 km without changing the scale height, −Ez/B will be very much in phase with Veb.
 To further explore the correlations between Ve and −Ez/B, and Ve and Vz, we plot in Figure 3 their correlation coefficients (without subtracting the mean). The thick solid curve is the correlation coefficient between Ve and the vertical ion drift. The cross-correlation coefficient is larger than 0.8 in the region between 112 and 125 km, indicating that Vz and Ve are proportional to each other. Further examination shows that ρVz has a larger amplitude than that of Ez/B, as indicated in Figure 2, at 115 and 124 km. This implies that the total E-field plays a relatively less important role between these two altitudes. It should be noted, however, that the negligibility of the E-field depends on the choice of ρ although the correlation coefficient between Ve and ρVz does not. If the E-field is proven to play a minor role, it has important implications on our understanding of the E-region electrodynamics.
 The thick dotted line in Figure 3 is the correlation between Ve and −Ez/B with the ion-neutral collision frequency expressed as in equation (5) and the number densities determined from the MSIS-E-90 model. Above 133 km, we see that Ve and −Ez/B are the same, resulting in a perfect correlation. Below 120 km, the correlation drops very fast. At 112 km, the correlation coefficient becomes negative. The correlation of Ve and −Ez/B, however, depends strongly on the ion neutral collision frequency. If we reduce the ion neutral collision frequency by a factor of 2, the new correlation between Ve and −Ez/B is shown as a thin dash-dot curve. This artificial reduction of the ion-neutral collision frequency moves the negative correlation to a much lower altitude where the measurement uncertainty becomes large. This exercise highlights the importance of ion-neutral collision frequency for the measurement of the E-field.
 Since this appears to be the first time that the E-region E-field has been derived from an ISR without assuming equipotential B lines, the validity of the underlying assumptions and potential error sources need to be carefully examined. The first error source we consider is the measurement error of the ion drift. During the experiment, the Arecibo line-feed was pointed in the vertical direction and the Gregorian feed was pointed 15 degree off of the zenith. The line-of-sight velocity error is largely determined by the spectral width in the altitudes of interest. At 600 m and a few minutes of time resolution, the line-of-sight velocity error is about 2 m/s at 110 km and increases to 7 m/s at 140 km. Since the horizontal drift depends on the amplified velocity of the obliquely pointing Gregorian feed as well as that of the vertically pointing line-feed, its error is about 1.2/sin(Za)∼ 4.7 times larger than that of the vertical velocity. The error in the E-field at the upper altitude range (above ∼120 km) shown in the previous figures is dictated by the error in the horizontal ion velocity. At an altitude of ∼115 km, it depends both on the horizontal and vertical velocity error. At lower altitudes, the E-field error is mostly due to vertical velocity error as it is amplified by ρ.
 The ion momentum equation (1) neglects gravity and ambipolar diffusion as in previous studies [e.g., Davies et al., 1997; Heinselman and Nicolls, 2008]. A non-negligible error source is the vertical neutral winds associated with gravity waves. Hooke  showed that the perturbed electric field due to gravity waves is negligible in the F-region. Unfortunately, his analysis, requiring ρ to be smaller than ∼0.1, is not applicable in the altitude range discussed here. From equation (1), the error in Ez/B caused by neutral vertical wind is Uzρ, which is expected to be generally larger at lower altitudes because of the exponential variation of ρ as a function of altitude. The fluctuations with a period of ∼1 hr shown Figure 1 (bottom) may not be accurate because of negligence of the neutral wind. Nevertheless, because the gravity wave periods are typically less than 90 minutes at Arecibo [Zhou, 2000; Djuth et al., 2010], its magnitude is relatively small with a moderate average (say, 60 min). The profiles averaged over about 160 min shown in Figure 2b are expected to be largely free from the effect of gravity waves. A further error is caused by the horizontal inhomogeneity induced by gravity waves, which affects the calculation of Ve. This error is, however, not expected to be large because the horizontal wavelength is typically much larger than the dual beam separation distance of 30 km [Djuth et al., 2010].
 Ion neutral collision frequency is the most uncertain factor in the inaccuracy of the E-field derived. The variation of the ion-neutral collision frequency is largely dictated by that of the neutral atmosphere and is expected to be smooth. Assuming vin has a relative error of α, then the error in Ez/B, Δ(Ez/B), is α∣Vz∣ρ. At the same altitude, the accuracy of Ez is proportional to the amplitude of Vz. The deduced E-field is independent of the ion-neutral collision frequency in the region where Vz is 0 (or more accurately Vz − Uz = 0). This largely applies to the 8:00-12:00 LT period above 112 km. This characteristic suggests that one can accurately determine the electric field within sporadic E layers, where Vz is expected to be zero. The Ez/B error due to a 50% inaccuracy in νin is shown in Figure 2 as error bars.
 It is of interest to put our results into the context of comparable measurements made previously. Despite that there were a number of rocket launches previously at Arecibo and other mid-latitude stations, practically all those launches were at nighttime to study sporadic E or field aligned irregularities [e.g., Pfaff et al., 1998]. Typical rocket flights measure the E-field in the horizontal plane. If we assume that the E-field is perpendicular to the field line, the vertical E-field, at a dip angle of 45°, is the same as the horizontal northward E-field. While the largest E-field in our observations has a magnitude well less than 2 mv/m, rocket results show values more than 10 mv/m [Pfaff et al., 1998]. One potential reason for the difference is that the high conductance during the daytime makes the occurrence of strong plasma instability unlikely while rockets were typically launched during the period of very intensive plasma instability at night. At high latitudes, several studies reported measurement of ion-neutral collision frequency by assuming that the E-field can be mapped along the field line from the F-region down to the lower part of the E-region [e.g., Davies et al., 1997]. In order to make neutral winds negligible, these measurements were typically made when E-field was very strong.
 One notable aspect of our observations is the large vertical variation of the E-field. This characteristic does not dependent on the choice of ρ. As ρ is reduced, the Ez/B curve in Figure 2b moves closer to the Veb curve, which also has strong vertical variation. Because of the high conductance along the field lines, it is generally assumed that the E-field is perpendicular to B and height invariant along the same field line. Since the vertical E-field measured here is not along the same field line, the height variation may be a reflection of the horizontal variation caused by gravity waves. As the dip angle is about 45° at Arecibo, the scale size of the variation in the upward direction is the same as that in the horizontal direction, implying a horizontal scale size of ∼20 km. Such an explanation may be plausible for the nighttime ionosphere, as simulated by Yokoyama et al. , when sporadic E's and quasi-periodic coherent echo regions are observed to have small horizontal scales [e.g., Hysell et al., 2004]. It is, however, inconsistent with the temporal characteristics in our data. Horizontal variation would be observed as temporal variation in our data. We see that the horizontal velocity is of the order of 1.4*30 m/s∼40 m/s in the afternoon at 124 km, a 20 km horizontal variation corresponds to a time scale about 10 minutes. The time-integrated E-field shown in Figure 2b would be essentially zero if horizontal variation is responsible. As an alternative to the horizontal variation explanation, we speculate that the E-field along the field line, although small, may no longer be negligible in the E-region. This would lead to the conclusion that the electric current is substantial along B since specific conductivity is still many times larger than either the Hall or Pedersen conductivities at 100 km. Conceivably, an instrumented rocket launched along B would determine whether field-line current or gravity wave is more viable to explain the vertical variation in the E-region electric field.
4. Summary and Conclusions
 We have reported a method of using the Arecibo dual beam capability to measure the vertical E-field in the E-region. The vertical E-field is measured via the ion drift in the vertical and zonal directions and by assuming a known ion-neutral collision frequency. The method provides continuous upward E-field observations for the altitude range from about 105 km to 145 km during the daytime. The limit for the upper altitude is dictated by the radar line of sight measurements while the limit for the lower altitude is determined by the accuracy of the ion-neutral collision frequency.
 Independent of ρ, the ratio of ion-neutral collision frequency to ion gyro-frequency, we find: 1) There is nearly a perfect correlation between the Lorentz force and the vertical ion drift around 118 km. This suggests that the electric fields play a smaller role than the Lorentz force in determining the motion of the plasma. 2) There is a significant gradient in Ez/B in the altitude of 105 to 140 km, especially in the afternoon sector. Vertical variations in horizontal electric fields were previously observed by rocket measurements and were thought to be caused by gravity wave perturbation. Such an explanation, however, is not consistent with the temporal characteristic of our observation. With the ρ given in Section 2, we find: 1) Ez/B during our observation is significantly smaller than ρVz at around 118 km. This implies that Ez can be neglected at this altitude. 2) V = E × B/B2 is valid at altitudes above 133 km. This result is consistent with the rocket results presented by Sangalli et al. .
 This work is partially supported by NSF grant ATM-0633418 to Miami University. Arecibo Observatory is managed by Cornell University under a cooperative agreement with the National Science Foundation.