An assessment of Joule heating associated with electric field variability at high latitudes previously made by Codrescu et al. (1995) is refined with accounting for realistic probability density functions (PDFs) and root-mean-square (RMS) values of variable electric fields. It is shown that the observed PDFs constructed as functions of x ≡ Er/rms, Er being the amplitude and rms the root-mean-square amplitude of the residual field, which is left after subtracting the mean field Em from the data, can be fitted with a Castaing distribution. The parameter λ in this distribution that characterizes the degree of non-Gaussianity of data appears ∼0.7. Joule heating patterns are demonstrated for the electric fields observed in two passes of the low-altitude polar-orbiting Dynamic Explorer 2 (DE2) satellite over the high-latitude region in the Northern Hemisphere, one under positive IMF Bz in the presence of IMF By, and the other under negative IMF Bz. The relative contributions to Joule heating of the mean field, here considered as 1024 km running average of samples, and of the residual fields are estimated.
 Joule heating is one of the major energy sources in the upper atmosphere. Accurate knowledge of the magnitude of this source is necessary to set correctly the magnetospheric driver in the ionosphere-thermosphere global circulation models. In a number of studies it has been shown that simulated ionospheric and thermospheric responses when driven by average electric field patterns are systematically biased as compared to observations, and this can be attributed to insufficient Joule heating [Matsuo and Richmond, 2008, and references therein].
 The importance of accounting for electric field variability in the calculations of Joule heating in the high-latitude ionosphere was first pointed out in a theoretical study by Codrescu et al. . It was shown that the Joule heating calculated as proportional to the square of the average electric field Em is lower than the observed values, and that the inclusion of E-field variability which is commonly observed at high latitudes can significantly increase the amount of Joule heating. As discussed by Codrescu et al. , the understanding of this fact is quite important for bringing into consistence the energy budget in the upper atmosphere.
 Since at that time little was known about the actual distributions of variable electric fields at high latitudes, Codrescu et al.  in their study considered an idealized probability density function (PDF) of E-field fluctuations. Specifically, it was assumed that the field can take with equal probability any value within Em ± rms, the rms being the root-mean-square amplitude of the variable field superimposed on the mean field Em, and no values beyond this range.
 Recently, Matsuo and Richmond  have conducted a comprehensive study of the effects of high-latitude electric field variability on Joule heating, assuming Gaussian distribution of the variable fields. They generated normally distributed random electric fluctuations, specifying their variance from DE2 electric field observations. These fluctuations were included to the electric field driver for the Thermosphere Ionosphere General Circulation Model (TIGCM) and associated thermospheric and ionospheric response was simulated. A noticeable increase in Joule heating due to electric field variability was revealed.
 As has been demonstrated, e.g., by Tam et al. , and Kozelov and Golovchanskaya , the actual PDFs of electric field fluctuations are non-Gaussian distributions with enhanced tails. When considered on different scales s, the fluctuating fields exhibit a power-law (or broken power-law) scaling, i.e., their rms value varies as ∼sγ, γ being the scaling exponent. These findings motivate us to study the effect of E-field variability on Joule heating for actually observed characteristics of variable electric fields in the high-latitude ionosphere.
 Following Codrescu et al. , we consider that the Joule heating rate Q is determined by the average of the square of the E field, which is the sum of the mean field Em and residual field Er. Then, denoting x ≡ Er/rms, Er being the amplitude and rms the root-mean-square amplitude of the residual field, one can write
where essentially meridional closure of field-aligned currents associated with variable electric fields is suggested [Sugiura et al., 1982], ΣP is the ionospheric Pedersen conductance, and PDF the probability density function of the residual field. The first term in the RHS of (1) yields the contribution to Q from the mean field Em, the second term is zero for the PDF(x) symmetric with respect to x = 0, and the third term is due to the electric field variability.
 From (1) it is seen that to find Q, one should know, in addition to ΣP and Em, the RMS value and probability density function (PDF) of the residual field. Further these quantities will be estimated from DE2 observations in two passes over the high-latitude ionosphere. In section 2, a pass corresponding to northward IMF Bz conditions is considered, when the RMS value of the residual field is expected to exceed the mean field Em, thus providing a dominant contribution to the RHS of (1). A pass that is considered in section 3 was during southward IMF Bz conditions, for which it is hard to predict in advance the relation between the mean field Em and RMS value of the residual field. The results are summarized in section 4.
2. High-Latitude Joule Heating Pattern Under Positive IMF Bz
 In a high-latitude pass of the Dynamic Explorer 2 (DE2) spacecraft (Figure 1a) during northward IMF Bz and strong IMF By, (the hourly values taken from the OMNI database [http://omniweb.gsfc.nasa.gov/html/omni2_doc.html] were, respectively, +13.4 nT and −17.9 nT), highly variable along-track electric fields were observed (Figure 1b). The fields were measured by orthogonal double probes of the Vector Electric Field Instrument (VEFI), described in detail by Maynard et al. . Here, the data with resolution 1/16 s (∼500 m) are used, and the field variations are considered as purely spatial throughout the paper (for discussion see, e.g., Kozelov et al. ). Simultaneously, the magnetometer on DE2 registered across-track variations in the magnetic field (Figure 1c), which correlated well with those in the electric field.
 To estimate Joule heating associated with the field shown in Figure 1b, one should define what is the mean field and what are fluctuations in the observed field pattern. From the theory of large-scale electric fields at high latitudes, which are generated as a result of solar wind-magnetosphere interaction, it is known that such fields are non-uniform. This manifests in the existence of the large-scale field-aligned currents/convection velocity shears, whose characteristic scale size is typically considered ∼500 – 1000 km at ionospheric level. Moreover, as shown on large statistical sample by Golovchanskaya et al. , the non-uniformity of the large-scale convection is crucial for the occurrence of electric field variability at smaller scales. Taking this into account, the mean electric field Em is here considered as 1024 km running average of the data (shown by the dash-dot line in Figure 1b). The RMS value of the residual fields, i.e., those left after subtracting the mean field from the observations, appears 31 mV/m and is considered the same for the whole sample.
Figure 1d shows the probability density function P of the residual fields Er normalized by the RMS value. The PDF was calculated for absolute values ∣Er∣ and then mirrored into the range Er < 0.
where parameter λ characterizes the degree of non-Gaussianity of data (for Gaussian fluctuations λ = 0). The variance of unity for the distribution given by (2) is provided by choosing ln α0 equal to −λ2. Substituting the Castaing approximation for the PDF to (1), we find that the integral x2PDF(x)dx tends to unity with decreasing λ and can be taken ∼1 for λ < 0.8 (in our case λ = 0.7). For the PDF considered by Codrescu et al. , this integral is equal to 0.333. For Gaussian fields it is exactly unity.
 Now, to compute the Joule heating rates from (1), the values of ΣP are needed. As has recently been shown by Kozelov et al. , under a number of simplifying assumptions, these can be obtained by applying the wavelet transform to simultaneous electric and magnetic observations, and taking the ratio of magnetic to electric wavelet magnitudes on different scales. It was demonstrated by Kozelov et al.  that the Pedersen conductance calculated in such a manner is scale-dependent, i.e. decreases with decreasing scale, which is typically interpreted as due to non-perfect mapping of the magnetospheric electric fields down to the ionosphere [e.g., Forget et al., 1991]. This effect is only essential at scales <30 km and has been ignored in the present study. As seen from the plots shown in Figure 1e, the Pedersen conductance computed from the wavelet spectra of the electric and magnetic fields here considered, behaves in a similar way at scales 32, 64, and 128 km. The along-track distribution of ΣP obtained by averaging over those presented in Figure 1e was applied in (1).
Figure 1f shows the Joule heating rates Q calculated according to (1) in the considered pass of DE2 over the high-latitude ionosphere during northward IMF Bz. The dash-dot curve indicates the Joule heating associated with the mean electric field, and the solid curve shows the contribution of the variable fields. One can see that the increase in Q due to the inclusion of electric field variability is significant and can reach an order of magnitude.
3. Case of Negative IMF Bz
Figures 2b and 2c are electric and magnetic observations of DE2 in a pass over the polar region under southward IMF Bz conditions. The orbit of the spacecraft is shown in Figure 2a.
 Under southward IMF conditions an extended polar cap forms, so that the regions of open and closed magnetic field lines can be easily discriminated. For a given pass, this has previously been done by Kozelov et al.  by using particle precipitation data from the Low-Altitude Plasma Instrument on DE2. The polar cap was identified with the region that was traversed by DE2 between UT = 0050 and UT = 0102. The morning (evening) auroral zone was crossed between UT = 0046 and UT = 0050 (UT = 0102 and UT = 0105). For the purposes of the present study such discrimination is important for two reasons. First, the RMS values of the field fluctuations in the polar cap and in the auroral zone are expected to be different, and this should be included in the estimation of Joule heating. Second, there is a possibility that a Castaing distribution, which is typically interpreted as a manifestation of the intermittent turbulence [e.g., Chang et al., 2004], might come about due to a summation of different local Gaussian distributions.
 The analysis performed separately for the auroral zone (the observations in the morning and evening auroral zone were combined into one data series) and the polar cap shows that a Castaing distribution, though with different λ, is observed in either of the two regions (Figures 2d and 2e). This is consistent with the finding of Abel et al. , who demonstrated a scale-free spatial structure of ionospheric velocity fluctuations in regions of both open and closed magnetic field lines by radar observations.
 By applying the same technique as in section 2, the Joule heating rates Q along the spacecraft track were computed. The RMS values of the residual fields were found to be 7 mV/m in the polar cap and 40 mV/m in the auroral zone. From Figure 2g it is seen that the contributions of the mean field and of the residual field in the auroral zone are close to each other. In the polar cap, the effect of electric field variability on Joule heating is small and can be neglected.
 As already mentioned, the fields under study exhibit scaling properties. For a given pass, this is illustrated in Figure 3, where the RMS values of residual field fluctuations on different scales are shown. The fluctuations were calculated as dEr = Er(p + s) − Er(p), p being the position in data series, and s the scale on which the fluctuation is considered. A clear power-law scaling is observed for the fields in the auroral zone (Figure 3a), while in the polar cap it is less evident on the largest scales (Figure 3b).
 The effect of variable electric fields on high-latitude Joule heating has been examined with including realistic PDFs of the fields.
 Two case studies here presented suggest differences in the Joule heating pattern due to IMF conditions. In the event during northward IMF Bz (in the presence of IMF By), the contribution of the E-field variability to Joule heating was found to be dominant. In the case of southward IMF Bz conditions, an increase in Joule heating associated with the variable electric fields was by a factor of ∼2 in the auroral zone. In the polar cap, the rms value of the field fluctuations was small in comparison to the mean E-field, resulting in a negligible contribution to Joule heating. Further events should be investigated to verify these findings.
 In our analysis we used the hourly values for the IMF, ignoring the IMF variability. Earlier, Golovchanskaya et al.  searched for the effect of the IMF variability on the amplitudes of small-scale electric fields at high latitudes and did not find it to be strong. However, this point should be addressed more carefully in future studies.
 The variable electric fields at high latitudes, which have recently been a subject of intense study, are typically derived from original measurements by high-pass filtering. In doing that, the scaling properties of the fields demonstrated in section 3 above should be taken into account. In the present consideration a spatial cutoff was chosen at 1024 km. Different choice of this cutoff can change the relative contributions of the mean E-field and of the electric field variability to Joule heating.
 Throughout the present paper it has been implied that the variable electric fields are produced by a magnetospheric source and not by variations in local ionospheric conductance. In the latter case, a reliable assessment of Joule heating can hardly be made because the true variability of ionospheric conductance is not known well. Here we refer to the study of Codrescu et al. , where a theoretical consideration of both possible sources of the E-field variability has been performed.
 Author acknowledges the NASA National Space Science Data Center and Nelson C. Maynard for DE2 VEFI data used. This work was supported by the Presidium of the Russian Academy of Sciences (RAS) through the basic research program “Solar activity and physical processes in the Sun-Earth system” and by the Division of Physical Sciences of RAS through the program “Plasma processes in the solar system.”