## 1. Introduction

[2] 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].

[3] 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.* [1995]. It was shown that the Joule heating calculated as proportional to the square of the average electric field E_{m} 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.* [1995], the understanding of this fact is quite important for bringing into consistence the energy budget in the upper atmosphere.

[4] Since at that time little was known about the actual distributions of variable electric fields at high latitudes, *Codrescu et al.* [1995] 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 *E*_{m} ± *rms*, the *rms* being the root-mean-square amplitude of the variable field superimposed on the mean field *E*_{m}, and no values beyond this range.

[5] Recently, *Matsuo and Richmond* [2008] 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.

[6] As has been demonstrated, e.g., by *Tam et al.* [2005], and *Kozelov and Golovchanskaya* [2006], 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 ∼*s*^{γ}, *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.

[7] Following *Codrescu et al.* [1995], 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 E_{m} and residual field E_{r}. Then, denoting *x* ≡ *E*_{r}/*rms*, *E*_{r} 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], Σ_{P} 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 E_{m}, 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.

[8] From (1) it is seen that to find *Q*, one should know, in addition to Σ_{P} and *E*_{m}, 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 E_{m}, 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 E_{m} and RMS value of the residual field. The results are summarized in section 4.