We use the multi-spacecraft mission Cluster to make observational estimates of the local energy conversion across the dayside high-latitude magnetopause. The energy conversion is estimated during eleven complete magnetopause crossings under steady south-dawnward interplanetary magnetic field (IMF). We describe a new method to determine the reconnection rate from the magnitude of the local energy conversion. The reconnection rate as well as the energy conversion varies during the course of the eleven crossings and is typically much higher for the outbound crossings. This supports the previous interpretation that reconnection is continuous but its rate is modulated.
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 The efficiency of solar wind (SW)- magnetosphere (MSPH) coupling is thought to be controlled by the magnetic reconnection process at the magnetopause (MP) [Dungey, 1961]. One of the most important aspects of reconnection is the conversion between magnetic and particle energy. The conservation of electromagnetic energy is given by Poynting's theorem which can be written as,
using the electric field in the plasma rest frame (governed by the generalized Ohm's law), ′ = + × and neglecting the electric field energy density. Here = is the Poynting vector and · gives the local energy conversion rate. The first term on the right corresponds to energy conversion due to non-ideal magnetohydrodynamic (MHD) effects while the second term governs the ideal MHD conversion between magnetic and kinetic (thermal and bulk flow) energy. In the ideal MHD limit a positive value of ( × ) · corresponds to a load, conversion from magnetic to kinetic energy, and vice versa to a generator if it is negative. Figure 1 illustrates the predicted load and generator regions around the MP for southward IMF.
 Usually, the energy input to the MSPH is evaluated using empirical coupling functions [e.g., Akasofu, 1979] in the upstream SW. However, recently Rosenqvist et al.  estimated the global energy transfer across the MP based on direct measurements of the local energy conversion based on the physical relations above. This truly physical approach is optimized via the utilization of the Cluster multispacecraft mission. The local energy conversion Q is represented by
where vmp is the velocity of the magnetopause. The local energy conversion can be separated into three components · = n · mp + n · n + t · t ( = × ). The first term represents the energy involved in the motion of the MP while the second and third term involves the energy conversion due to the plasma flow and/or forces normal and tangential to the MP respectively. Current in-situ measurements are, due to their limited spatial extent, inadequate to evaluate the loss or gain of the first term. We thus neglect this term, assuming that the oscillatory motion of the MP is approximately an adiabatic process.
 In this paper we study the local energy conversion across the MP, Q, during multiple MP crossings on 26 January 2001. The Cluster spacecraft (SC) were located in the high-latitude northern duskside MP with an inter-SC separation of about 650 km with SC1-3 orientated tangential to a model MP and SC4 closer to Earth (see Figure 1). The IMF was remarkably steady for over two hours (south-dawnward). This event has previously been studied by Bosqued et al.  and Phan et al.  which both found convincing signatures of continuously active reconnection. Eleven complete MP crossings between 10:29 UT and 11:05 UT where identified by Phan et al.  during which high-speed plasma jets were detected by Cluster. This provides an excellent opportunity to study the energy conversion across the MP during an extended time interval when no influence from varying interplanetary conditions is expected. Our main goal is to study if, how and possibly why the energy conversion and reconnection rate varies in the course of the eleven crossings, as it should be important for our understanding of the solar wind driving of the SW-MSPH coupling.
2. Overview of Observations
 To justify the neglect of all non-ideal MHD contributions in Equation 1 we have compared observations from different instruments on Cluster [see Escoubet et al., 2001, and references therein]. We use data from the FGM, CIS, and EFW instruments. Plasma velocity data are obtained from CIS/CODIF on SC4 and CIS/HIA on SC1 with good agreement.
 An overview of the studied time interval is shown in Figure 2. The eleven complete MP crossings are coloured, blue marks the outbound MP crossings from MSPH to magnetosheath (MSH), and yellow marks the inbound crossings. The MP current layer is identified by a large change in Bz (Figure 2a). High-speed plasma jets are observed at all crossings except 10 (10 lacks ion measurements within the MP) and have been identified by Phan et al.  as alfvénic reconnection jets.
Figure 2c shows integrated values of the instantaneous power conversion ( × ) · (blue line) and · (green line) where and are obtained from FGM, from CIS and from EFW with 4 s resolution and assuming · = 0. A closer examination of the magnetic field data from all four SC reveals that sometimes the current sheet is thinner (<650 km) than the inter-SC separation, thus the curlometer method is not applicable for estimating for such crossings. Instead single-SC current estimations have been used based on a model boundary normal, n = [0.7, 0.43, 0.50] [Shue et al., 1998] and an average vmp = 40 km/s [Bosqued et al., 2001]. The assumption of vmp is not of critical importance as it affects the magnitude of both curves equally and we are interested in their difference. The iongyroradius is ∼50 km and thus we measure on a scale of a few iongyroradii which is smaller than the typical thickness of the current sheet, ∼1000 km [Bosqued et al., 2001]. Figure 2c shows the good agreement between the two different methods. Only at the end of the interval the differences approach 50%. Thus, it seems like the ( × ) · term is dominating the overall energy conversion. This is also supported by kinetic simulations of reconnection in the magnetotail [Birn and Hesse, 2005].
3.1. Local Energy Conversion
 The energy conversion is dominantly positive over the entire time period (Figure 2c), thus corresponding to a load in the GSE reference frame. To investigate the variation during the course of the eleven crossings we must evaluate the energy conversion at each crossing separately. As vn is small, associated with large uncertainties, and difficult to separate from vmp we neglect the contribution from n · n to Equation 2 in this paper. This is justified because for rotational discontinuities expected for MP crossings during ongoing reconnection Fn ≈ 0. However, in the few crossings where it is possible to estimate, Fn ≈ Ft (not shown). Although n · n can still be neglected since vt ≫ vn. Thus we need to determine the MP boundary normal for each crossing in order to estimate the velocity component tangential to the MP entering Equation 2.
Figure 3 shows observations from the first outbound crossing. Figure 3a shows the Bx for all four SC. It is seen that the current sheet thickness varies during the crossing, sometimes all SC can be inside the current sheet (thick current sheet, e.g., ∼10:31:50 UT) and sometimes the SC can be on opposite sides of the current sheet (thin current sheet, e.g., ∼10:31:35 UT). This indicates that the MP is rippled by bulges, as already suggested by Phan et al. . Thus, neither the 4-SC timing technique nor the curlometer method is applicable. However, single-SC current estimates obtained using MP normals from minimum variance analysis (MVA) can be used since the dependence on the MP velocity cancels as we integrate over the MP width in Equation 2. On the other hand, also the MVA proved to be difficult under such structured MP conditions. Therefore, we decided to perform the MVA on nested data segments centred at or near the actual current sheet crossing to investigate the time-stationarity of the normal vector. The SC (SC1 or SC4) that is least influenced by bulges during the crossing and consequently with the most reliable boundary normal estimations (highest ratio of the medium to minimum eigenvalues, L2/L3) is selected to evaluate the energy conversion. The normal direction is determined as the average over the time period when the nested analysis give similar results. Furthermore, we calculate the statistical uncertainty in the normal magnetic field component, Bn [Sonnerup and Scheible, 1998] and average the error over the nested segments. The relative error in Q is proportional to the relative error in Bn and considered to be the dominant uncertainty in the estimations. The first eight columns of Table 1 show for each of the eleven crossings, the SC used to evaluate Qi (where subscript i indicates the number of the crossing), the start time of the MP crossing and duration, the MP normal (GSE), L2/L3, the average Bn and also its relative error.
Table 1. Magnetopause Parameters for the Eleven Crossings
Figure 3b shows the magnetic field components for SC4 in the MP reference frame. It is evident that Bn is mostly less than zero consistent with reconnection taking place equatorward of Cluster. Bl changes from positive to negative, which indicates a crossing from the MSPH to the MSH. The current is mainly in the positive m-direction (Figure 3c). Note that we have assumed vmp = 40 km/s to estimate . The tangential velocity is mainly in the positive l-direction along the MP as shown in Figure 3d and consequently Q1 is positive (Figure 3e) as expected. The energy conversion (Q1) has been evaluated using both the model normal (green line) and the normal deduced from the nested variance technique (blue line) with good agreement. It is seen that the energy conversion mainly takes place at the centre of the current layer, thus the technique is not very sensitive to the definition of the start and end of the MP. Partial crossings (in/out) can also be seen in Q1 (e.g., between 10:31:10 and 10:31:35). However, the resulting effect does not significantly influence the final Q1 value.
 The result of the estimated MP energy conversion for each crossing is given in the 9th column of Table 1 and is summarized in Figure 4a together with the relative error. During crossing 4, 6 and 8 the MP is thick and it is possible to use the 4-SC timing method as well as the curlometer technique.
 It appears that the MVA is problematic for the first four crossings considering the very low L2/L3 value and rather large error bars. This sometimes results in an apparent flip of the m and n direction which is the case for crossing 4 and 8. Thus, we have used timing estimates for these crossings which seem to be more reliable than MVA. For the majority of the later crossings the MVA appears to provide more reliable results. Thus, we conclude that no technique works best for all crossings but instead the most reliable technique must be chosen on a case by case basis.
 By closer examination of these last crossings one finds that even when taking the error bars into account the energy conversion varies during the course of the crossings. This can be considered rather surprising as the interplanetary conditions are stable. Thus, some other mechanism must be responsible for the movement of the MP and the energy conversion variation.
3.2. Reconnection Rate
 The reconnection rate can be estimated from the observed Qi estimates obtained above. Considering the ideal MHD, steady state Poynting theorem in integral form:
assuming the same magnetic field magnitude on both sides of the current sheet, which is a good assumption in our case, and using = − × ⇒ E = −vinB0. Here, vin is the inflowing plasma velocity and B0 the background magnetic field. Furthermore ΔV = ΔS Δx, ΔS being the surface area element and Δx being the width of the magnetopause. Comparing the right-hand side of Equation 4 with Equation 2 we realize that it is equal to since Δx = vmpΔt. Now using the definition of the reconnection rate,
where vA,in is the Alfvèn velocity in the inflow region defined as, vA,in = , and vin = from Equation 4 we can express the reconnection rate as,
 Thus, we can calculate the reconnection rate based on the local energy conversion estimates for each crossing obtained in Section 4.1. The plasma density is obtained from the SC potential [Pedersen et al., 2001]. The results are shown in the second to last column in Table 1 and in Figure 4b. For comparison, the reconnection rate based on Bn observations, RBn = , is also shown in the last column in Table 1. RQ is found to vary with a mean for all crossings of 0.17. Considering the uncertainties in obtaining such values, they are in rather good agreement with previous values of ∼0.1 [e.g., Lindqvist and Mozer, 1990]. Furthermore, one can note that the reconnection rate is always higher for the outbound crossings. Active reconnection occurring equatorwards of the SC peals off magnetic flux from the dayside which should result in a contraction of the MSPH. The lower reconnection rate for the inbound crossings may consequently indicate a decrease in the reconnection rate resulting in a relaxation of the MSPH. However, other explanations are possible, for example, a compression of the MSH against the MP may cause a thinner MP current which could enhance the reconnection rate (and vice versa). However, it is beyond the scope of the paper to investigate this matter further.
 The advantage of using RQ lies in the possibility to directly obtain the tangential component of the Lorentz force by 4-sc techniques. If the tangential part is relatively large compared to the magnitude of the force it is not very sensitive to small variations in the normal direction. However, quantities that are small compared to the magnitude of the vector such as vin and Bn are sensitive to small variations in the normal direction. Therefore, the use of 4-sc techniques can significantly reduce the relative error of RQ in comparison to RBn. However, for the majority of the MP crossings we were forced to use single-SC current density estimates making RQ directly proportional to Bn. Thus, in this case the relative errors of RQ and RBn are the same. The estimations of RBn and RQ agree well in most crossings according to Table 1 except for crossing 8 and 10 where they differ by an order of magnitude. This is because the method to obtain RQ considers the real speed of the outflowing high-speed plasma instead of assuming the jets to be alfvénic as in RBn. At least, crossing 10 is characterised by the absence of high-speed flows. Thus, when there is a substantial deviation from alfvénic plasma flow the method based on the energy conversion estimates is more reliable despite the limitation of using single-SC current measurements. Note, however, that in the diffusion region MHD is not valid and thus both methods are inadequate there.
 The analysis of the multiple MP crossings on the 26 of January 2001 suggests the following:
 1. The energy conversion has been estimated for eleven complete MP crossings characterised by high-speed reconnection jets. In most cases Cluster observes a load region, conversion from magnetic to particle energy, as the reconnected field lines contract and accelerate the plasma.
 2. During this event the MP surface is rippled with the presence of bulges. We use different methods to estimate boundary properties and current densities. No method works best for all crossings but the most appropriate one should be chosen, depending on the nature of the crossing.
 3. The results indicate that the energy conversion rate change during the course of the eleven crossings. This supports the previous interpretation that reconnection is continuous, but its rate is modulated even when interplanetary conditions are steady.
 4. A new method to estimate the dimensionless reconnection rate based on the observed local energy conversion rate across the MP has been developed. The advantage of this method is that it does not rely on determining rather small quantities. It also takes into account that reconnection jets are not always alfvénic.
 5. The reconnection rate varies and is typically larger for the outbound crossings. One interpretation that is consistent with the observed magnetopause motion is that active reconnection peals off magnetic flux from the dayside resulting in a contraction of the MSPH. The subsequent MP relaxation could be a consequence of a decreased reconnection activity.
 LR and AR are supported by the Swedish National Space Board, and AV is supported by the Swedish Research Council.