On the association between northward turnings of the interplanetary magnetic field and substorm onsets

Authors


Abstract

[1] We re-examine whether substorms are triggered by solar wind fluctuations or an internal magnetospheric instability by comparing the statistical associations between substorm onsets and (1) an external trigger definition, (2) a simple internal trigger definition of only prior loading of solar wind energy that is a subset of the external trigger definition. Statistical associations are calculated both for observed substorm onsets and onsets generated by a Minimal Substorm Model in which substorms are purely internally triggered. Thence we argue that a minimum interval of prior loading is a necessary condition for substorm onset, a subsequent northward IMF turning is not necessary, and consequently that an internal trigger from a magnetospheric instability is a necessary and sufficient condition for substorm onset. We discuss how this result may explain a report that externally triggered substorms are systematically larger than non-externally triggered substorms.

1. Introduction

[2] The cause of substorm onset remains a central problem for magnetospheric physics. Substorm onset and the ensuing expansion phase is marked by the sudden release of energy extracted from the solar wind and stored in the magnetospheric tail lobes during a preceding growth phase [McPherron et al., 1973]. Explanations of when substorm onset occurs either invoke an external trigger such as a northward turning in the interplanetary magnetic field (IMF) [Lyons, 1995; Russell, 2000], or an internal trigger like the magnetic flux in the tail lobes reaching a critical level [Shukhtina et al., 2004]. Various physical models associated with these triggers have been proposed [e.g., Lyons, 1995; Baker et al., 1996; Russell, 2000; Lui, 2001].

[3] Clearly, the relative importance, or even veracity, of these models depends on ascertaining the likelihood that substorms are externally or internally triggered. For example, Lyons' [1995] model asserts that all substorms are caused by a reduction in the convection electric field, manifesting most commonly as a northward turning in the IMF, whereas the theory of Russell [2000] does not expect or require all substorms to be externally triggered. Though many authors have cited northward turnings as triggers for substorm onset, substorms have been observed without any external triggers being apparent [e.g., Henderson et al., 1996]. Determining whether a substorm onset is caused by an external or internal trigger requires the trigger to be identified objectively and reliably and to show that the simultaneous observation of onset and trigger is not merely coincidence.

[4] To this end, a set of numerical criteria has been defined to identify northward IMF turning triggers automatically [Lyons et al., 1997] that, when applied to high-quality, continuous IMF data, found over 80% of the trigger signatures identified by eye [Hsu and McPherron, 2003] (hereinafter referred to as HM2003). Using this automated technique (with some visual correction), it was found that there is a statistically significant association between the times of external northward IMF turning triggers and substorm onsets [Hsu and McPherron, 2002] (hereinafter referred to as HM2002), such that about half of all substorms are expected to be associated with a northward IMF turning trigger (HM2003).

[5] However, what of the alternative hypothesis that substorm onsets are caused by an internal trigger? A simple internal trigger definition would be that onsets are caused solely by a minimum interval of solar wind energy input into the magnetosphere, and that a northward IMF turning is not a necessary condition. But this is actually a subset of the external trigger definition, which defines its trigger as a minimum interval of southward IMF followed by a northward IMF turning [Lyons et al., 1997]. Thus, is the significant association between substorm onsets and external triggers found by HM2002 due to the inclusion of the prior loading in the external trigger definition, or is the additional requirement of a northward IMF turning actually necessary?

[6] In this paper, we answer this question by analyzing the HM2002 data set to compare the statistical associations between substorm onsets and (1) the external trigger definition of Lyons et al. [1997], (2) a simple internal trigger definition of prior loading only that is a subset of the external trigger definition. We repeat this analysis for another set of substorms generated by the Minimal Substorm Model (MSM) [Freeman and Morley, 2004] in which substorms are defined to be purely internally triggered. From these results, we show that an interval of prior energy input to the magnetosphere from the solar wind is a necessary condition for substorm onset, and that the additional requirement of a subsequent northward IMF turning is not necessary.

2. Method

2.1. Statistics of Point Processes

[7] To test the association of a substorm onset with a candidate trigger, we use a test based on the statistics of point processes [e.g., Cox, 1955; Brillinger, 1976]. This test has previously been applied to geophysical data [Mulargia, 1992]. A point process is a time series of discrete events, such as “spike trains” of neuronal firing [e.g., Brown et al., 2004] or, as here, substorm onsets and their candidate triggers (e.g., HM2002). The association between two point processes is assessed using the association number, calculated as follows. Suppose we have two point processes, A and B, with events at times ai (i ∈ {1, 2,… N}) and bj + u (j ∈ {1, 2,… M}), respectively, where u is a time lag. The individual association number ci is the number of events in series B that fall into a window of half-width h around ai and the association number n(u, h) is the summation of ci over all events ai

display math

where #{X} denotes the number of elements in the set X.

[8] We wish to test the null hypothesis that there is no association between point processes A and B for any lag u. The alternative hypothesis is therefore that there is an association. For independent processes, the association number is well estimated by the asymptotic association number n(u → ∞, h) because we expect that separating any two series by sufficient time will ensure independence between the points in the two series. Thus, in the case that the null hypothesis is correct, there will be no significant deviation of the association number from its asymptotic value at any lag. In the case that the alternative hypothesis is accepted, we expect to see a significant departure from the asymptotic level at some lag, relatively close to zero.

[9] Significance is evaluated using a non-parametric resampling technique called bootstrapping [e.g., Conover, 1999]. From equation 1 we see that n(u, h) is a summation of the N individual associations ci for given u, h. Using this set of individual associations, we can construct a new series, c*i, by drawing with replacement a random selection of N individual associations. Summing these N randomly-sampled associations gives a bootstrap estimate of the association number for given u, h. Repeating this for every lag u, we construct a bootstrap estimate of the association number with lag n*(u, h). Performing this bootstrapping procedure K times allows us to model the sampling variation in n(u, h). Here we set K = 4000. Taking the 2.5 and 97.5 percentiles of the bootstrapped estimates n*(u, h) at each lag u gives the 95% confidence limits for the sampling variation. In the event that our sampling variation is approximately normal, this corresponds to a 2σ deviation from the mean. Following Brillinger [1976], the observed association number n(u, h) is significant at the 5% level wherever the lower confidence limit of n*(u, h) lies above the asymptotic association number n(u → ∞, h).

[10] The bootstrap method of estimating statistical significance is favoured over that used by HM2002 that was based on the premise that at least one of the point processes under analysis is a Poisson process. As HM2002 pointed out, substorm onsets can occur both randomly and periodically, thus the assumption of a Poisson process is not applicable. Furthermore, contrary to their assertion, basic inspection of Lyons et al.'s [1997] trigger criteria show that this too cannot be a Poisson process. Firstly, a trigger cannot occur for at least 10 minutes after a previous trigger. This refractoriness, or memory effect, prohibits the process from being Poisson as it modifies the probability of encountering an event following a previous event. Secondly, a trigger can only occur after an extended interval of southward IMF which means that the probability of a trigger occurring in unit time is not constant as it must be for a Poisson process.

2.2. Identifying the Point Processes

2.2.1. Triggers

[11] A set of quantitative rules has been defined by Lyons et al. [1997] for identifying an external substorm trigger at time t in a set of measurements of the northward component Bz of the IMF sampled at discrete times ti. We adopt these rules and express them as follows assuming the time resolution Δt = ti+1ti = 60 s of the solar wind data used here.

[12] 1. The IMF must be southward for at least 22 of the preceding 30 minutes:

display math

where X(a: b) denotes all elements of X(xi) in the range a < xib.

[13] 2. The turning initiation must be rapid, so that

display math

[14] 3. The northward turning must be sustained, determined by the slope ∇Bz of the linear regression of Bz on ti between t and t +10 min:

display math

and Bz must be increasing so as to satisfy the requirements Bz (t: t +3) ≥ Bz (t) + 0.15 nT and Bz (t + 3: t +10) ≥ Bz (t) + 0.45 nT.

[15] 4. No other point in the previous 10 minutes should satisfy the listed criteria. That is, once a trigger is identified, any turnings satisfying the criteria in the following 10 minute interval are discounted.

[16] Criterion 1 specifies the pre-loading of the magnetosphere with energy from the solar wind. Criterion 4 represents the idea that after a trigger the magnetosphere will not respond to another within 10 minutes. Thus criteria 2 and 3 specify the actual IMF disturbance proposed to trigger a substorm, while criteria 1 and 4 define the necessary magnetospheric conditions for a substorm to occur. From these criteria we define two classes of trigger: 1. External trigger: Any point in the IMF satisfying criteria 1–4. This is the Lyons et al. [1997] trigger, which was also adopted by HM2002. 2. Internal trigger: Any point in the IMF satisfying criterion 1. This is a trigger based on a minimum interval of solar wind energy input.

[17] Using these definitions, trigger time series were identified in two IMF data sets. The first is the data set used by HM2002 comprising 65186 1 min-averaged IMF measurements made by the ISEE-2 spacecraft over October 1978 and September-October 1979. The second comprises 315453 1 min-averaged IMF measurements made by the MAG instrument on the NASA Wind spacecraft between 1 January 1995 and 1 July 1998. Wind was assumed to be in the solar wind whenever it was at a radial distance R > 30 RE and sunward of the earth (Xgse > 0). Data gaps of less than 5 min were filled by linear interpolation. To avoid larger data gaps, we only used intervals of data that were unbroken for 100 h or more.

2.2.2. Onsets

[18] Two substorm onset time series were used, corresponding to the same time intervals as the ISEE-2 and Wind trigger time series. For the ISEE-2 interval, we use a list of 258 substorm onset times provided by T. S. Hsu and previously described by HM2002. These were identified from a sharp decrease in the AL index accompanied within ±20 min by Pi2 pulsation bursts from at least three ground magnetometers within 3 h of midnight. For the Wind interval, we have used a substorm onset time series from the MSM using the Wind data as input [cf. Freeman and Morley, 2004]. This comprises 953 substorm onsets with the same probability distribution of waiting times as found for substorm onsets identified from energetic particle injections by Borovsky et al. [1993].

3. Analysis

3.1. Observed Onsets and External Triggers

[19] First we assess the association between the HM2002 substorm onset events and the ISEE-2 external triggers, which have previously been found to be significantly associated (HM2002). Figure 1a shows the association number, as a function of the lag u, summed over all onset times. If the association is no greater than that expected by chance, we expect that the association number will be similar at all lags. To aid comparison between each set of results, the left-hand axis is normalized to the asymptotic association number. The actual association numbers are marked on the right-hand axis. In all, we have N = 258 onsets and M = 598 triggers. At lags of ∣u∣ > 200 min the association number tends to a mean asymptotic association number of 31. The peak in association number is n(u, h = 10) = 57 at u = −8 and u = 6 min. The lower confidence interval is above the asymptotic association number for −20 < u < 50 min. Thus we can reject the null hypothesis at the 5% level and accept the alternative hypothesis that external triggers are associated with substorm onset.

Figure 1.

(a) Association number (thick red line) of candidate external triggers within a window of ±h of the onset time (summed over all onsets) as a function of the lag u. The association number n(u, h) is marked on the right-hand axis and normalized to the asymptotic association number of the left-hand axis. The thin red lines mark the 95% confidence limits and the dashed horizontal line indicates the expected association under independence. (b) Same as Figure 1a but for observed substorm onsets and internal triggers. Data from Figure 1a are overlaid in blue. (c) Same as Figure 1a but for MSM substorm onsets and external triggers, again with data from Figure 1a overlaid in blue. The MSM substorm curve (red) has been shifted by 17 min with respect to the observed substorm curve (blue).

3.2. Observed Onsets and Internal Triggers

[20] The necessity of a northward IMF turning for substorm onset is assessed in Figure 1b which shows the association number, as a function of lag, for the series of internal triggers. In this case we have N = 258 onsets and M = 25038 triggers. At lags of ∣u∣ > 200 min the association number tends to a mean asymptotic association number of 1001 and shows a peak n(u, h = 10) = 1671 at u = −6 and u = −2 min. The lower confidence interval is above the asymptotic association number for −60 < u < 110 min. Thus we can reject the null hypothesis at the 5% level and accept the alternative hypothesis that internal triggers are associated with substorm onset. The normalized association number is similar to that seen for the set of external triggers, which is shown on the same plot for comparison.

3.3. MSM Onsets and External Triggers

[21] The fact that both external and internal triggers are significantly associated with substorm onsets suggests that the growth phase requirement (criterion 1) common to both is a necessary condition for substorm onset and that the northward IMF turning requirement (criteria 2–4) need not be necessary. The necessity of a northward IMF turning can be tested using a series of substorm onset times generated by the MSM, in which (by design) there is no requirement for a northward IMF turning to trigger substorm onset. Instead, substorm onset occurs when solar wind energy input reaches a constant critical energy threshold following a prior substorm in which an amount of energy is lost that is proportional to the solar wind power input at the time of that substorm onset. The MSM displays the same distribution of inter-substorm timings as the observations of Borovsky et al. [1993], and is similar in mean occurrence rate (∼0.2 h−1) and inter-onset distribution to the HM2002 substorm onset data. The external triggers are identified in the real IMF data measured by the Wind spacecraft used to drive the MSM using the same Lyons et al. [1997] criteria as above, and as used in the HM2002 study.

[22] Figure 1c shows the number of external triggers (criteria 1–4) associated with the set of MSM substorm onsets. Here we have N = 953 onsets and M = 3363 triggers. At lags of ∣u∣ > 200 min the association number tends to a mean of 231. The peak in association number n(u, h = 10) = 398 is at u = 16 min. The lower confidence interval is above the asymptotic association number for −10 < u <100 min. Thus we can reject the null hypothesis at the 5% level and accept the alternative hypothesis that external triggers are associated with MSM substorm onsets. The normalized association number is identical, within uncertainties, to that seen for the association of external triggers with the observed onsets (Figure 1a), which is overplotted for comparison. The latter has been lagged by an additional 17 min as discussed below.

4. Discussion

[23] In the above analysis, we have tested the association of both observed and modelled substorm onsets with two candidate types of triggers in the observed IMF. The first type is the external trigger of Lyons et al. [1997] and the second type is an internal trigger that comprises a subset of the external trigger criteria.

[24] For both the external and internal triggers, the association with the observed substorm onsets was significant at the 5% level for a range of lags around zero. Furthermore, the variations of normalized association number with lag for the two cases are identical within uncertainties. This suggests that the association in both cases is due to an association with the common criteria of the two candidate trigger types, i.e., criterion 1 - the growth phase of the substorm. It may appear (Figure 1b) that the additional requirement of northward turning narrows the range of lags for which there is a statistically significant association and sharpens the start of significant association. However, the narrowing of the range of significant lags can be accounted for by the larger sampling error for the external trigger due to the lower number of such events. This reduces the lower 95% confidence curve and causes it to intersect the asymptotic association number line closer to zero lag. The apparent sharpening is because the internal trigger curve is a smoothed version of the external trigger curve. This smoothing is because the northward IMF turning criteria 2–4 cause the external triggers to be separated by at least 10 min, whereas the internal triggers have no such restriction.

[25] Consequently we conclude that a prolonged southward IMF interval (criterion 1) is necessary for substorm onset and that IMF northward turnings (criteria 2–4) are not necessary because inclusion of the IMF northward turning in the trigger criteria did not improve the statistical association between substorm onsets and triggers. Additional support for this is given by the result (not shown) that there is no significant association between observed substorm onsets and an external trigger based on a northward turning alone (criteria 2–4), i.e., without the requirement of a prior growth phase (criterion 1).

[26] The necessity of an IMF northward turning as a substorm trigger was also tested using a set of substorm onset times generated by the Minimal Substorm Model, in which there is no IMF northward turning criterion for triggering substorm onset. In this case (Figure 1c), the normalized association number curve of the external triggers and MSM onsets is identical (within uncertainties) to that of the external triggers and observed substorm onsets, provided that the latter curve is shifted by T ≈17 min. This delay may be broken down into three parts T = T1 + T2 + T3. T1 = (XsXo)/V takes into account the advection of the external trigger in the solar wind at speed V from a spacecraft at X = Xs to a position on the magnetopause at X = Xo, where X is the position coordinate on the Earth-Sun line. T2 = Ro/c is the time for the effect of the external trigger at the magnetopause at X = Xo to affect the central magnetotail and trigger substorm onset, where Ro is the effective radius of the magnetotail and c is the characteristic propagation speed, which we take to be the Alfvén speed. T3 is the time for the substorm onset to be detected on the ground. Assuming v = 400 km/s, 13 < Xs < 22 RE appropriate to the part of the ISEE-2 orbit in the solar wind, Ro = 25 RE and c = 500 km/s appropriate to the mid-tail (−10 > X > −20 RE) [Mazur and Leonovich, 2006], and T3 = 2 min [Samson, 1995] we find the T = 17 min delay would correspond to an effective substorm onset location at −16 > X > −25RE. This is comparable to the expected location of the near-earth neutral line at −20 > X > −30 RE [Baumjohann et al., 2000]. The similarity of the normalized association number curves for the modelled and observed substorm onsets supports our assertion that the association between the external triggers of Lyons et al. [1997] and the HM2002 substorm onsets arises from the growth phase requirement alone, and provides evidence that a minimum interval of prior loading may be a sufficient condition for onset. This is because the series of substorm onsets was generated using the Freeman and Morley [2004] minimal substorm model which has no requirement of a northward IMF turning for substorm onset. Thus the elevated and significant association around zero lag cannot be due to this requirement and must be attributable to a common factor of both the MSM substorm onset and the external trigger criteria. This common factor is the requirement that solar wind energy must accumulate in the magnetosphere before substorm onset.

[27] Thus again we conclude that an interval of solar wind energy input into the magnetosphere is a necessary condition for substorm onset and that a northward turning of the IMF is not necessary. Consequently, we argue that an internal magnetospheric instability is a necessary and sufficient condition for substorm onset. The MSM suggests the possible global properties of this instability-a constant energy threshold and an energy yield that is proportional to the solar wind power input at the point of instability.

[28] Interestingly, adopting the MSM may help to resolve a currently unsolved problem. [Hsu and McPherron, 2004] (hereinafter referred to as HM2004) found that substorms with an associated northward IMF turning trigger were systematically larger, in both Pi2 wave power and AL bay magnitude, than substorms not associated with such a trigger. This can be explained by: (1) a hypothesis H0 in which substorm size is proportional to the solar wind power input at substorm onset (as assumed in the MSM) or immediately prior to onset, H0: E = kP(t = image where k is a constant and (2) the solar wind power input being statistically higher at, or immediately prior to, substorm onsets associated with a northward IMF turning compared to onsets not so associated.

[29] Empirical evidence for (1) is provided by Kallio et al. [2000]. They showed that the substorm energy output dissipated in the ionosphere during the substorm expansion phase is better correlated with the solar wind energy input during the expansion phase, compared to integrated energy input over the growth phase or over the whole substorm. At first this seems to imply an alternative hypothesis H1 that substorm energy output is proportional to the average power input just after substorm onset, i.e., H1: E = kP(t = ti+). However, Kallio et al. [2000] also divide their results into “continued input” (CI) substorms, for which power input continues into the expansion phase P(t = image P(t = ti+), and “growth phase input” (GPI) substorms, for which power input substantially decreases near the end of the growth phase P(t = ti+) ∼ rP(t = image where r is a positive quasi-random variable much less than unity. Relative to the CI substorms, the GPI substorms were found to have (1) lower correlation between energy output and expansion phase input, and (2) larger energy output for given energy input [see Kallio et al., 2000, Figure 3 (middle)]. This can be explained under the hypothesis H0: E = kP(t = image in which EkP(t = ti+) for CI substorms but E = k/r P(t = ti+) for GPI substorms. Thus for GPI substorms the energy output E is larger and less well correlated with the power input after onset P(t = ti+) than for CI substorms, as Kallio et al. [2000] found. In contrast, hypothesis H1 gives the same relationship E = kP(t = ti+) for both CI and GPI substorms which is contrary to their results.

[30] We also find empirical evidence for (2) (not shown). Using the Wind database, we find that the solar wind power input prior to a northward IMF turning “trigger' (defined by rules 1–4) is statistically higher than that prior to an internal ”trigger' (defined by rule 1 only) because the IMF Bz is biased to be more negative immediately before a northward turning and rule 1 requires this bias to also have been sustained for some time prior to the turning. For example, the mean (median) of the distribution of IMF Bz values is −3.5 nT (−3.1 nT) immediately (within 10 min) prior to a northward IMF turning trigger compared with −2.5 nT (−2.2 nT) immediately prior to an internal trigger.

[31] Thus we explain the relationships reported by HM2004 using the Minimal Substorm Model. As they noted, this is not easily explicable using an alternative (external trigger) model. As we shall show in future work, the MSM also explains the statistical distribution of substorm magnitudes as measured by the amplitude of magnetic bays and the statistical variation of IMF Bz relative to substorm onset.

5. Conclusions

[32] 1. Observed substorm onsets are associated with external and internal triggers in a similar way and with high statistical significance. A similar association is obtained between external triggers identified in the IMF and substorm onsets generated by the Minimal Substorm Model, in which substorms are defined to be purely internally triggered.

[33] 2. An interval of prior energy input to the magnetosphere from the solar wind is shown to be a necessary condition for substorm onset and the necessity of any external trigger is cast into doubt.

[34] 3. Accordingly, the claim that any percentage of substorm onsets are externally triggered is not proven. Consequently, we argue that an internal trigger from a magnetospheric instability is a necessary and sufficient condition for substorm onset.

[35] 4. In fact, the internal trigger hypothesis can also account for a property of substorm size that is not easily explicable by the external trigger hypothesis.

Acknowledgments

[36] The authors are grateful to T.-S. Hsu and R.L. McPherron (UCLA) for providing the ISEE-2 data and the lists of onset and trigger times, R.P. Lepping and the MFI instrument team for IMF data from the NASA Wind spacecraft, G. Abel (BAS) for compiling these into a convenient database, and M. Sciffer (U. Newcastle, Australia) and L. Lyons (UCLA) for helpful discussions. SKM is the recipient of an Australian Research Council APD Fellowship and is supported by ARC Discovery DP0663643. MPF is supported by the BAS GSAC Natural Complexity programme.

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