Analysis of the step response function relating the interplanetary electric field to the dayside magnetospheric reconnection potential

Transcription

1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009ja014681, 2010 Analysis of the step response function relating the interplanetary electric field to the dayside magnetospheric reconnection potential G. T. Blanchard 1 and K. B. Baker 2 Received 24 July 2009; revised 18 December 2009; accepted 30 December 2009; published 18 May [1] We present a statistical analysis of the response of the magnetic reconnection rate between the interplanetary magnetic field and the magnetosphere to southward turnings of the interplanetary magnetic field. The magnetic reconnection rate is calculated from Super Dual Auroral Radar Network (SuperDARN) measurements. The polar cap boundary is identified as the offset circle (3 toward midnight from the magnetic pole) that best separates high spectral width backscatter, indicative of open magnetic field lines, from low spectral width backscatter. The electric potential on the boundary is determined from the best fit of the line of sight F region plasma velocity to an eighth order spherical harmonic function of ionospheric electrical potential. The reconnection rate is determined from the electric potential on the boundary and the expansion rate of the polar cap. Solar wind data (velocity and magnetic field) are obtained from Wind spacecraft measurements and propagated to the magnetopause. The relationship between interplanetary electric field and the magnetic reconnection rate is analyzed by calculating linear response functions. We determine that the convection term in the reconnection rate measurement exhibits a unimodal response to the interplanetary electric field, whereas the boundary motion term in the reconnection rate measurement exhibits a bimodal response, with a positive mode (increase in polar cap expansion rate) followed by a negative mode. The response of the dayside reconnection rate to a step change in the interplanetary electric field exhibits a transient response of one hour duration followed by a steady state response at a level that is half of the peak response. Citation: Blanchard, G. T., and K. B. Baker (2010), Analysis of the step response function relating the interplanetary electric field to the dayside magnetospheric reconnection potential, J. Geophys. Res., 115,, doi: /2009ja Introduction 1 Department of Chemistry and Physics, Southeastern Louisiana University, Hammond, Louisiana, USA. 2 Division of Atmospheric and Geospace Sciences, National Science Foundation, Arlington, Virginia, USA. Copyright 2010 by the American Geophysical Union /10/2009JA [2] Reconnection between the interplanetary magnetic field (IMF) and the magnetosphere applies the interplanetary electric field to the magnetosphere, resulting in magnetospheric convection [Dungey, 1961], increased magnetic field and particle energy derived from the solar wind kinetic energy [Cowley, 1980], and increased energy transfer to the ionosphere and atmosphere. Processes internal to the magnetosphere, and thus the magnetospheric response to reconnection, can be accurately modeled, but without accurate measurements of the reconnection rate along the entire separatrix between open and closed magnetic field lines, the modeled magnetospheric response will be inaccurate. Vasyliunas [1984] laid out the theory that allows remote measurements of the reconnection rate by ionospheric radar. Integration of Faraday s law of induction over the separatrix between open and closed field lines yields dl ð rþ ¼ ðdl BÞðU VÞ; ð1þ x line ionosphere where U represents the velocity of the separatrix and V represents the velocity of the plasma. We refer to the lefthand side of equation (1) as the magnetic reconnection potential F rec, which is the rate at which magnetic flux is opened on the dayside of Earth, and which is the quantity of interest in this study. As depicted in Figure 1, the integration in the ionosphere is carried out along an arc of the polar cap boundary. Note that on the portions of the boundary referred to by Siscoe and Huang [1985] as adiaroic, the boundary convects with the plasma (i.e., U = V); therefore, the integrand of the right hand side of equation (1) is exactly zero there. Therefore, the position of the endpoints of the path of integration on the boundary may vary. We choose to integrate from 0600 MLT to 1800 MLT, since this magnetic local time range should accommodate the motion of the flux gap caused by changes in IMF B y [Idenden et al., 1996]. 1of10

2 Figure 1. Diagram of the expanding polar cap adapted from the work of Siscoe and Huang [1985]. Local noon is at the top of the diagram. The polar cap boundary comprises the flux gap, which is the projection of the magnetospheric X line onto the ionosphere, and the adiaroic line, which convects with the ionospheric plasma. The shaded arrow from dawn to dusk indicates the path of integration for the right hand side of equation (1). [3] Distributing the terms in the right hand side of equation (1) yields rec ¼ ð V BÞdl B ðu dlþ; ð2þ ionosphere which can be rewritten as ionosphere rec ¼ C þ B I ds dt ; where F C represents the convection electric field potential difference between the endpoints of the path of integration and ds/dt is the area swept out by the motion of the path of integration per unit time, which is equal to the time rate of change of the area S of the sector of the polar cap defined by the arc of the integration path. We refer to F C as the convective potential, and for the sake of uniformity of notation, we refer to the last term in equation (3) as the inductive electric potential F I, in which case, equation (3) may be written as rec ¼ C þ I : Therefore, the measurable quantities required in order to infer the reconnection potential are the electric potential due to ionospheric convection between two points on the polar cap boundary and the time rate of change of the area ð3þ ð4þ of the corresponding sector of the polar cap. Both of these quantities can be calculated on the basis of data obtained by the Super Dual Auroral Radar Network (SuperDARN) [Greenwald et al., 1995; Chisham et al., 2007] as described in section 2. We use the method described in section 2 to calculate an ensemble of 13,599 individual magnetospheric reconnection rate measurements during an experiment lasting from 1 September 2006 to 30 September 2006, inclusive. [4] Our purpose in making these measurements is to investigate the response of the dayside reconnection rate to a southward turning of the IMF. There have been many studies of the response of the ionospheric convection potential to southward turnings of the IMF [e.g., Ridley et al., 1998; Ruohoniemi and Greenwald, 1998; Jayachandran and MacDougall, 2000; Lu et al., 2002]. The consensus result from these studies is that the ionospheric convection potential begins to change promptly upon the arrival of a southward turning at the magnetopause, but the potential requires tens of minutes to reach its new final state. However, these studies were only concerned with the ionospheric convection potential F C, and not the potential attributable to the separatrix motion, F I, and therefore do not provide complete information on the dayside reconnection potential F rec, which is the subject of this study. [5] Another aspect of this study is that whereas most previous studies of the magnetospheric response to southward turnings of the IMF have been based on case studies, 2of10

3 we apply linear prediction analysis [e.g., Makhoul, 1975] to calculate the response of the reconnection potential to a southward turning of the IMF. The advantage of this method over the method of case study is that the response to a southward turning is reconstructed from all available data, not just the data in the vicinity of the southward turning, which yields a much more general result. Using linear prediction analysis, we accomplish the purpose of investigating the response of the dayside reconnection rate to a southward turning of the IMF by calculating the unit step response function of the measured reconnection potential to the eastward rectified interplanetary electric field vb S incident upon the magnetopause, where v is the solar wind bulk flow speed and B S = B z for B z < 0 and B S = 0 for B z 0in the geocentric solar magnetic (GSM) coordinate system. We choose the solar wind coupling function vb S instead of other coupling functions that include B y for purposes of comparing our results to other studies that use this function explicitly and to studies of the magnetospheric response to southward turnings of the IMF. The unit step response function is so called because it describes the response of the output function, in this case F rec, to a hypothetical unit step in the input, in this case a 1 mv/m change in vb S. The response is linear such that the response of F rec to an arbitrary step change in vb S may be determined by scaling the response to the unit step. [6] The unit step response function is calculated through the intermediary of the unit impulse response function [e.g., Bendat and Piersol, 1986]. The unit impulse response function is the kernel g(t) of a convolution integral relating the input I(t) of a system to its output O(t): ½Ot ðþ O Š ¼ gðþ½it ð Þ I Šd; ð5þ where I and O represent the mean values of the input and output, respectively, over the time period of interest. The impulse response function is so called because it describes the response of the output to a hypothetical unit impulse input. This can be demonstrated by equating the input to a Dirac delta function d(t), ½It ðþ IŠ ¼ ðþ; t which when substituted into equation (5) yields ½Ot ðþ OŠ ¼ gt ðþ Similar to the unit impulse response function, the unit step response function h(t) represents the response of the system to a hypothetical unit step input; that is, ht ðþ¼ ð6þ ð7þ gðþht ð Þd; ð8þ where H(t) is the Heaviside step function, which yields the following relationship between the unit impulse and unit step response functions: t ht ðþ¼ 0 gðþd: ð9þ The unit step response function completely characterizes the linear relationship between F rec and vb S. Therefore, knowledge of the unit step response allows prediction of the reconnection rate based on measurements of the interplanetary electric field as well as presenting new observations pertinent to theories of steady state magnetospheric dynamics. 2. Method [7] To obtain the reconnection rate measurements necessary for this study, we use data from the SuperDARN radars while operating in common mode, a sequential 16 beam scan synchronized to start every 2 min with an integration time of 7 s per beam. From the transmission of a multiple pulse scheme in the high frequency (HF) band, the SuperDARN radars measure the autocorrelation function (ACF) of the backscattered signal at several distances in range gates of 45 km. The backscattered signals are produced by coherent scatter by field aligned electron density irregularities. In each range gate, this ACF is routinely analyzed by a basic method [Villain et al., 1987; Baker et al., 1995, Appendix A] which extracts the power, the lineof sight mean Doppler velocity of the irregularities, and the spectral width of the Doppler power spectrum directly from the ACF. The ionospheric electric potential F(, ), where is colatitude and =(MLT 12 h )p/12 h is determined from the best fit of the F region line of sight mean Doppler velocity to an eighth order spherical harmonic function of ionospheric electrical potential [Ruohoniemi and Baker, 1998]. The location of the open closed field line separatrix is determined from the Doppler spectral width as described in the following paragraph. [8] Chisham et al. [2005a, 2005b, 2005c] have shown that there is a good correlation between the spectral width boundary and the open closed field line separatrix when the spectral width boundary is determined by the following procedure [Chisham and Freeman, 2003]. The backscatter spectral width is first median smoothed across three adjacent beams and subsequently smoothed across five consecutive times. Finally, data are only used from those range gates where beam direction is within 30 of the magnetic meridian. When processed in this way, the 150 m/s spectral width contour is correlated with the open closed field line separatrix at all magnetic local times. [9] For the purposes of this study, we model the colatitude b ( ) of the polar cap boundary as a circle of radius Q offset by 0 = 3 degrees in magnetic latitude toward 2400 MLT ( 0 = p), where Q is a free parameter. Therefore, ð b ð Þ ¼ þ 3 Þð 3 Þ þ 3 cos 180 ; ð10þ and the area S of the dayside polar cap is S ¼ =2 =2 1 cosð b ð ÞÞ R 2 E d ð cos Þd : ð11þ Holzworth and Meng [1975, 1984] demonstrated that equation (10) is an unbiased model of the polar cap boundary on time scales long enough that B y 0. This is the 3of10

4 Figure 2. Example of measurements of the polar cap radius Q (thin line) and estimated true polar cap radius ~ (thick line) on 1 September simplest model of the polar cap for which F I is not identically equal to 0. During initial investigations, we considered a refinement of this model in which 0 and 0 were allowed to vary as free parameters as well as Q. This refined model did not significantly affect the covariance between F I and vb S while exponentially increasing the computation time required to fit the free parameters. Therefore, we use the parsimonious model of the polar cap boundary expressed by equation (10) for this study. [10] We find the best fit of the polar cap boundary to the 150 m/s spectral width contour by maximizing the number of echoes on the correct side of the boundary, correct meaning that high spectral width (>150 m/s) backscatter, indicative of open magnetic field lines, should be poleward of the boundary, and low spectral width should be equatorward of the boundary. Figure 2 shows an example of the polar cap radius Q as determined by this method over a 24 h period. We note that in order to ensure that there is sufficient data to determine the ionospheric electric potential and the location of the polar cap boundary, we restrict our analysis to those 2 min intervals in which the total number of dayside echoes is greater than or equal to 50, which accounts for the data gaps seen in Figure 2. It can also be seen in Figure 2 that the determination of the polar cap radius is noisy. The noise is due to the sporadic nature of the coherent backscatter process. The appearance and disappearance of regions of HF backscatter in the vicinity of the polar cap boundary may cause the identified boundary to move by a few degrees in a short period of time. To estimate the noise in the measurement of the polar cap radius, we first use a Savitzky Golay smoothing filter [Savitzky and Golay, 1964] with width = 30 and order = 3 to calculate an estimate of the true polar cap radius ~. The estimated true polar cap radius is also plotted in Figure 2. We take the uncertainty (noise) in the polar cap radius dq to be equal to the residual standard deviation ~ = 2.1. [11] From the measured values of Q, dq, and F, we calculate F C, df C, F I, and df I. The electric potential difference between 0600 MLT and 1800 MLT on the polar cap boundary determines the convective potential F C, C ¼ ððþ 3 Þð 3 Þ=; =2Þ ððþ 3 Þð 3 Þ=;=2Þ: ð12þ The uncertainty in F C, df C, is therefore sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ 2 : ð13þ Since F/ Q = E ( /m ) = v B I ( m/ ), we use the characteristic values v = 500 m/s and B I = T to calculate df C = 8.2 kv. Figure 3 shows an example of measurements of F C over a 24 h period. The magnitude of the measured convective potential is positive and of the same order as typical polar cap potentials (tens of kilovolts), which is as expected. The inductive potential F I is determined by the product of the ionospheric magnetic field strength and the time rate of change of the dayside polar cap area S using a three point centered difference, Stþ ð 2 minþ St ð 2 minþ I ¼ B I : ð14þ 4 min The uncertainty in F I is therefore sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I ¼ B 2 ð1 R Þ ; ð15þ where R Q = 0.45 is the autocorrelation of Q(t) at a lag of 4 min. For the average polar cap radius = 17.6, equation (15) yields df I = 318 kv. Figure 3 also shows an example of measurements of F I over a 24 h period. It is apparent that the expected signal, which is of the order of tens of kilovolts is buried in noise propagating from the noise in Q(t). As described in section 3 and the appendix, we 4of10

5 Figure 3. Example of measurements of (top) convective reconnection potential, F C, and (bottom) inductive reconnection potential, F I, on 1 September will use correlation analysis applied to a large data set to extract the buried signal from the noise. The sum of F C (t) and F I (t) constitute the dayside reconnection rate F rec (t) (equation (4)), whose response to the solar wind electric field is the subject of this study. [12] To obtain the solar wind electric field data necessary for this study, we use data from the Wind spacecraft solar wind experiment [Ogilvie et al., 1995] and magnetic field investigation [Lepping et al., 1995]. Solar wind data (bulk velocity and magnetic field components) are obtained from the Wind spacecraft measurements and propagated ballistically to the magnetopause as provided by the OMNI database [King and Papitashvili, 2005]. Figure 4 shows an example of the solar wind data and the solar wind input function vb S (t) over a 24 h period. We present our method analyzing the response of the reconnection potential to the solar wind input function in section Analysis [13] Having thus obtained input, vb S (t), and output, F C (t) and F I (t), data, we proceed to calculate the unit step response functions relating F rec to vb S. To calculate the step response function, we begin with the impulse response function (equation (5)). Since the input and output time series are both discretely sampled with a cadence Dt, we substitute idt for t and jdt for t, which yields Oit ð Þ O ¼ Xn j gjt ð Þ Iðði jþtþ I t: ð16þ Equation (16) demonstrates the basic operation of the unit impulse response function. The output time series is represented as the sum of n copies of the input time series, each lagged in time by jdt and scaled by the unit impulse response function at that particular lag, g(jdt). To calculate the unit impulse response function from time series that contain data gaps, we proceed as follows. We multiply both sides of equation (16) by the input time series at time (i + k)dt, sum over index i, and divide by N: 1 X N N i ¼ XN j Oit ð Þ O Iððiþ kþtþ I ( ) 1 X N gjt ð Þ Iðði jþtþ I Iððiþ kþtþ I t: N i ð17þ The left hand side of equation (17) is the definition of the input output cross covariance, C, and the term in curly brackets on the right hand side of equation (17) is the definition of the input autocovariance, A. Therefore, equation (17) may be written as Ckt ð Þ ¼ X j gjt ð ÞAððk jþtþ: ð18þ As in equation (16), equation (18) demonstrates that the unit impulse response function is used to represent the input output cross covariance function as the summation of lagged and scaled copies of the input autocovariance function. The benefit of using the covariance functions is that we are able to calculate regularly sampled covariance functions from time series that are irregularly sampled due to data gaps. Using matrix notation, equation (18) can be written as C k ¼ A kj g j t; ð19þ where the summation over j is implied. Solving equation (19) for g j yields kj C k g j ¼ A 1 t : ð20þ 5of10

6 Figure 4. Example of measurements of (top) solar wind bulk flow speed, v, (middle) z component of the interplanetary magnetic field B z, and (bottom) the solar wind input function, vb S, on 1 September We use the method of singular value decomposition to invert the autocovariance matrix [e.g., Press et al., 1992]. When using singular value decomposition, a decision must be made as to the number of singular values to retain, which corresponds to the number of free parameters in g. Retention of more singular values increases the standard error in g, while retention of fewer singular values increases the discrepancy between C k and A kj g j Dt. We apply the criterion of Morozov [1966], which states that the discrepancy between A kj g j Dt and C k should be less than the uncertainty dc k ; that is, A kj g j t C k Ck : ð21þ Therefore, we retain the smallest number of singular values such that g satisfies equation (21). This procedure is applied to both F C (t) and F I (t) to determine their respective impulse response functions. [14] The final step in the analysis is to calculate the unit step response function that relates the interplanetary electric field vb S to the reconnection rate F rec. We sum the component impulse response functions g C (t) and g I (t) to obtain the total impulse response function g rec (t). We integrate g rec (t) according to equation (9) to obtain the unit step response function h(t). We apply these methods to the data set presented above. We present the results of our analysis in section Results [15] We calculate the auto and cross covariance functions necessary for calculation of the unit impulse response functions from the ensemble of available measurements. The input autocovariance A is calculated from 17,669 measurements of the interplanetary electric field. The inputoutput cross covariance C C of the interplanetary electric field and the convective reconnection potential is calculated from 11,339 simultaneous measurements of these quantities, while the input output cross covariance C I of the interplanetary electric field and the inductive reconnection potential is calculated from 11,839 simultaneous measurements of these quantities. These three covariance functions are plotted in Figure 5. Of note is the fact that the decorrelation time of the inductive reconnection potential and the interplanetary electric field is distinctly shorter than the decorrelation time of the interplanetary electric field alone, whereas the decorrelation time of the convective reconnection potential and the interplanetary electric field is comparable to the decorrelation time of the interplanetary electric field alone. A and C C both decrease to one half of their respective maximum values by t 40 min, while C I decreases to one half of its maximum value by t 20 min. The similarity of C C and the function A, which will be used to construct it (equation (18)), implies qualitatively that the response of F C to vb S comprises a single positive response mode, while the initially positive covariance C I followed by rapid decorrelation implies qualitatively that the response of F C to vb S comprises at least two response modes: an initially positive response mode followed by a negative response mode which causes the rapid decorrelation of F I from vb S. [16] We now proceed to calculate the response functions quantitatively. We invert the autocovariance matrix A kj = A((k j)dt) using singular value decomposition and apply the Morozov [1966] criterion as described above to determine the number of singular values that should be retained in the calculation of g C and g I, respectively. The results indicate that five singular values should be retained to calculate g C, and four singular values should be retained to calculate g I. The unit impulse response functions that are thereby obtained are plotted in Figure 6. Figure 6 shows that the response of the convective potential to an impulse is unimodal (i.e., one significant peak occurs at t = 8 min). This peak is fairly broad, and all values between t = 0 min 6of10

7 Figure 5. Autocovariance of (top) the interplanetary electric field, A, (middle) cross covariance of the interplanetary electric field with the convective reconnection potential, C C, and (bottom) cross covariance of the interplanetary electric field with the inductive reconnection potential, C I versus lag, t. Error bars represent the standard error of the covariance functions. and t = 14 min are consistent with the peak value. After t = 14 min, the convective potential tapers off until it reaches a value that is consistent with zero after t = 40 min. This result is consistent with the results of Ridley et al. [1997, 1998], Ruohoniemi and Greenwald [1998], and Ruohoniemi et al. [2001], who studied the response of the ionospheric convection potential to southward turnings of the interplanetary magnetic field. The conclusion of those studies is that the global convection potential responds in less than 2 min to the arrival of an interplanetary magnetic field southward turning at the magnetosphere. The convection continues to respond for 10 to 13 min and then the change in the convection potential decreases. It should be noted that Ridley et al. [1997, 1998], Ruohoniemi and Greenwald [1998] Ruohoniemi et al. [2001], and this study each use a different methodology to arrive at this conclusion. [17] Figure 6 also shows that the response of the inductive reconnection potential to an impulse is bimodal (i.e., one Figure 6. Unit impulse response function of (top) convective reconnection potential, g C, and (bottom) inductive reconnection potential, g I (bottom) versus lag, t. Error bars represent the standard error of the impulse response functions. 7of10

8 Figure 7. Unit step response function, h, relating the total magnetospheric dayside reconnection potential to the interplanetary electric field versus lag, t. significant positive peak occurs at t = 8 min, and another significant negative peak occurs at t = 52 min). After t = 70 min, the inductive potential is consistent with zero. The negative peak represents the effect of nightside reconnection closing the magnetic flux opened by the impulse on the dayside. This result is consistent with the results of Bargatze et al. [1985] and Blanchard and McPherron [1995], who observed that the response of the AL index to vb S is bimodal, with the first peak at 20 min and the second peak at 60 min. Both Bargatze et al. [1985] and Blanchard and McPherron [1995] speculated that the mechanism leading to the second peak in the response function is due to a linear relationship between nightside reconnection and the interplanetary electric field. The additional 10 min delay observed in the response of AL index to vb S is consistent with the time scale of magnetosphere ionosphere coupling. The presence of the second, negative peak is also qualitatively consistent with the relatively short decorrelation time of F I from vb S as discussed above. The combination of a positive and negative response mode causes F I to decorrelate from vb S more rapidly than does F C, which has only a single, positive response mode. [18] Finally, we calculate the unit step response function, h(t). By calculating the total unit impulse response function, g C (t) +g I (t), and integrating as in equation (9), we obtain the unit step response function shown in Figure 7. The result indicated by Figure 7 is that following a step change in vb S, the magnetospheric reconnection rate F rec exhibits a transient response with a peak magnitude of m (i.e., 29 kv mv 1 m 1 ). This transient response peaks at t = 32 min. The reconnection potential then declines to a steady state level of approximately matt = 60 min. Although we do not speculate on the mechanism responsible for this behavior, we note that in steady state, the dayside reconnection rate, the rate of flux transport to the magnetotail, the nightside reconnection rate, and the rate of magnetic flux return to the dayside must all be in equilibrium. The low steady state reconnection rate relative to the transient peak indicates that one of the other processes must be the limiting factor in steady state magnetospheric convection. 5. Conclusions [19] In this article, we present a method for calculating the rate of magnetic reconnection between the interplanetary magnetic field and the magnetosphere and analyze the response of the reconnection rate to the eastward rectified interplanetary electric field. To calculate the reconnection rate, we use data from the Super Dual Auroral Radar Network alone. With regard to the reconnection rate measurements generated by this method, we conclude that the noise level overwhelms any individual measurement. Nevertheless, by constructing a sufficiently large ensemble of measurements and integrating over this large ensemble to calculate the covariance between the reconnection rate and the interplanetary electric field, we are able to obtain the covariance functions necessary to analyze the linear response of the reconnection rate to the interplanetary electric field. [20] We observe that the unit impulse response of the convective reconnection potential consists of a single positive mode that peaks at a lag of 8 min relative to the interplanetary electric field incident upon the magnetopause. We also observe that the unit impulse response of the inductive reconnection potential consists of one positive mode that peaks at a lag of 8 min relative to the interplanetary electric field and one negative mode that peaks at a lag of 52 min. The negative mode corresponds to a closing of magnetic flux by reconnection on the nightside of the Earth and indicates a linear relationship between that quantity and the interplanetary electric field. Such a linear relationship was proposed by Bargatze et al. [1985] and Blanchard and McPherron [1995]. Our results constitute direct evidence confirming their hypothesis. Finally, we observe that the unit step response of the dayside reconnection potential to the interplanetary electric field comprises a transient response of 60 min duration and a peak value of m 8of10

9 followed by a steady state response of m. From this we conclude that in the case of steady driving of the magnetosphere by the interplanetary electric field, the magnetosphere requires 60 min to reach steady state. Once steady state is reached, either magnetic flux transport to the magnetotail, nightside reconnection or magnetic flux return to the dayside constitutes the process that limits steady magnetospheric convection. Appendix A: Effect of Noise, Bias, and Sample Size on Covariance [21] If we represent a measured output O(t) time series as the sum of the true output time series ~O(t), a noise time series N(t) which has a mean value of 0 and is independent of I(t), and a constant bias b, that is, Ot ðþ¼~o ðþþnt t ðþþb; then the input output cross covariance is C ¼ C ~ O þ C N þ C b : ða1þ ða2þ Since I(t) and N(t) are independent time series, the expected value of C N is 0. The covariance C b of any time series I(t) with any constant b is identically equal to 0. Therefore, neither noise nor bias contribute to the input output cross covariance, that is, C ¼ C ~ O : ða3þ The noise does contribute, however, to the uncertainty dc in the input output cross covariance, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ C 2 ~O þ C2 N þ C2 b: ða4þ Since C b is identically equal to 0, the uncertainty in that term dc b is also identically equal to 0. For the remaining two terms in equation (A4), we use the standard error. Therefore, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ 2 I ~O n þ 2 IN n ; ða5þ where n is the sample size. Since I(t) and N(t) are independent time series and the mean value of N(t) is 0, we can expand the second term inside the radical as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 I þ 2 t I ~O C ¼ n þ I 2 N : ða6þ n As a concrete example, we take the case of C I (0), the cross correlation between F I and vb S, the value of which is equal to (4.8 ± 2.3) V 2 /m. Assuming that all of the uncertainty in C I (0) is due to noise, the first term under the radical in equation (A6) can be neglected relative to the second term. Using the values I = vb S = mv/m, s I = s vbs = mv/m, s N = df I = 318 kv, and n = 11,839, equation (A6) predicts dc = 2.0 V 2 /m, which is only slightly smaller than the experimental value of 2.3 V 2 /m. [22] Acknowledgments. This study was supported by National Science Foundation grants ATM and ATM Super- DARN data were provided by R. Greenwald. Assistance with SuperDARN data processing was provided by R. Barnes. OMNI data were obtained from the GSFC/SPDF OMNIWeb interface at [23] uyin Pu thanks Gareth Chisham and another reviewer for their assistance in evaluating this paper. References Baker, K. B., J. R. Dudney, R. A. Greenwald, M. Pinnock, P. T. Newell, A. S. Rodger, N. Mattin, and C. I. Meng (1995), HF radar signatures of the cusp and low latitude boundary layer, J. Geophys. Res., 100, 7671, doi: /94ja Bargatze, L. F., D. N. Baker, E. W. Hones Jr., and R. L. McPherron (1985), Magnetospheric impulse response for many levels of geomagnetic activity, J. Geophys. Res., 90, 6387, doi: /ja090ia07p Bendat, J. S., and A. G. Piersol (1986), Random Data: Analysis and Measurement Procedures, 2nd ed., John Wiley, New York. Blanchard, G. T., and R. L. McPherron (1995), Analysis of the linear response function relating AL to VB S for individual substorms, J. Geophys. Res., 100, 19,155, doi: /95ja Chisham, G., and M. P. Freeman (2003), A technique for accurately determining the cusp region polar cap boundary using SuperDARN HF radar measurements, Ann. Geophys., 21, 983. Chisham, G., M. P. Freeman, T. Sotirelis, R. A. Greenwald, M. Lester, and J. P. Villain (2005a), A statistical comparison of SuperDARN spectral width boundaries and DMSP particle precipitation boundaries in the morning sector ionosphere, Ann. Geophys., 23, 733. Chisham, G., M. P. Freeman, T. Sotirelis, and R. A. Greenwald (2005b), The accuracy of using the spectral width boundary measured in offmeridional SuperDARN HF radar beams as a proxy for the open closed field line boundary, Ann. Geophys., 23, Chisham, G., M. P. Freeman, M. M. Lam, G. A. Abel, T. Sotirelis, R. A. Greenwald, and M. Lester (2005c), A statistical comparison of Super- DARN spectral width boundaries and DMSP particle precipitation boundaries in the afternoon sector ionosphere, Ann. Geophys., 23, Chisham, G., et al. (2007), A decade of the Super Dual Auroral Radar Network (SuperDARN): Scientific achievements, new techniques and future directions, Surv. Geophys., 28, 33, doi: /s Cowley, S. W. H. (1980), Plasma populations in a simple open model magnetosphere, Space Sci. Rev., 26, 217, doi: /bf Dungey, J. W. (1961), Interplanetary magnetic field and the auroral zones, Phys. Rev. Lett., 6, 47, doi: /physrevlett Greenwald, R. A., et al. (1995), DARN/SuperDARN: A global view of the dynamics of high latitude convection, in The Global Geospace Mission, Space Sci. Rev, vol. 71, edited by C. T. Russell, pp , Kluwer Acad., Norwell, Mass. Holzworth, R. H., and C. I. Meng (1975), Mathematical representation of the auroral oval, Geophys. Res. Lett., 2, 377, doi: / GL002i009p Holzworth, R. H., and C. I. Meng (1984), Auroral boundary variations and the interplanetary magnetic field, Planet. Space Sci., 32, 25, doi: / (84) Idenden, D. W., R. J. Moffett, S. Quegan, and T. J. Fuller Rowell (1996), Time dependent convection at high latitudes, Ann. Geophys., 14, Jayachandran, P. T., and J. W. MacDougall (2000), Central polar cap convection response to short duration southward interplanetary magnetic field, Ann. Geophys., 18, 887, doi: /s z. King, J. H., and N. E. Papitashvili (2005), Solar wind spatial scales in and comparisons of hourly Wind and ACE plasma and magnetic field data, J. Geophys. Res., 110, E12S10, doi: /2005je Lepping, R. P., et al. (1995), The WIND magnetic field investigation, Space Sci. Rev., 71, 207, doi: /bf Lu, G., T. E. Holzer, D. Lummerzheim, J. M. Ruohoniemi, P. Stauning, O. Troshichev, P. T. Newell, M. Brittnacher, and G. Parks (2002), Ionospheric response to the interplanetary magnetic field southward turning: Fast onset and slow reconfiguration, J. Geophys. Res., 107(A8), 1153, doi: /2001ja Makhoul, J. (1975), Linear prediction: A tutorial review, Proc. IEEE, 63, 561, doi: /proc Morozov, V. A. (1966), On the solution of functional equations by the method of regularization, Sov. Math. Dokl., 7, 414. Ogilvie, K. W., et al. (1995), SWE, A comprehensive plasma instrument for the WIND spacecraft, Space Sci. Rev., 71, 55, doi: /bf Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling (1992), Numerical Recipes in C: The Art of Scientific Computing, Cambridge Univ. Press, New York. 9of10

10 Ridley, A. J., G. Lu, C. R. Clauer, and V. O. Papitashvili (1997), Ionospheric convection during nonsteady interplanetary magnetic field conditions, J. Geophys. Res., 102, 14,563, doi: /97ja Ridley, A. J., G. Lu, C. R. Clauer, and V. O. Papitashvili (1998), A statistical study of the ionospheric convection response to changing interplanetary magnetic field conditions using the assimilative mapping of ionospheric electrodynamics technique, J. Geophys. Res., 103, 4023, doi: /97ja Ruohoniemi, J. M., and K. B. Baker (1998), Large scale imaging of highlatitude convection with Super Dual Auroral Radar Network HF radar observations, J. Geophys. Res., 103, 20,797, doi: /98ja Ruohoniemi, J. M., and R. A. Greenwald (1998), The response of highlatitude convection to a sudden southward IMF turning, Geophys. Res. Lett., 25, 2913, doi: /98gl Ruohoniemi, J. M., R. J. Barnes, R. A. Greenwald, S. G. Shepherd, and W. A. Bristow (2001), The response of the high latitude ionosphere to the CME event of April 6, 2000: A practical demonstration of space weather nowcasting with the SuperDARN HF radars, J. Geophys. Res., 106, 30,085, doi: /2000ja Savitzky, A., and M. J. E. Golay (1964), Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem., 36, 1627, doi: /ac60214a047. Siscoe, G. L., and T. S. Huang (1985), Polar cap inflation and deflation, J. Geophys. Res., 90, 543, doi: /ja090ia01p Vasyliunas, V. M. (1984), Steady state aspects of magnetic field line merging, in Magnetic Reconnection in Space and Laboratory Plasmas, Geophys. Monogr. Ser, vol. 30, edited by E. W. Hones Jr., 25 pp., AGU, Washington, D. C. Villain, J. P., R. A. Greenwald, K. B. Baker, and J. M. Ruohoniemi (1987), HF radar observations of E region plasma irregularities produced by oblique electron streaming, J. Geophys. Res., 92, 12,327, doi: / JA092iA11p K. B. Baker, Division of Atmospheric and Geospace Sciences, National Science Foundation, 4201 Wilson Blvd., Arlington, VA USA. (kbaker@nsf.gov) G. T. Blanchard, Department of Chemistry and Physics, Southeastern Louisiana University, SLU 10878, Hammond, LA , USA. (gblanchard@selu.edu) 10 of 10