An empirical phase space analysis of ring current dynamics' Solar wind control of injection and decay

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 15, NO. A4, PAGES , APRIL 1, 2 An empirical phase space analysis of ring current dynamics' Solar wind control of injection and decay T. Paul O'Brien and Robert L. McPherron Institute of Geophysics and Planetary Physics, University of California, Los Angeles Abstract. This empirical analysis of the terrestrial ring current, as measured by Dst, uses conditional probability density in Dst phase space to determine the evolution of the ring current. This analysis method does not assume a dynamic equation, but merely requires that the evolution of Dst depends on Dst and the solar wind. Our simple model, with seven nontrivial parameters, describes the dynamics of 3 years of hourly Dst with solar wind data provided by the OMNI database. The solar wind coupling is assumed to be determined by VBs. We arrive at a dynamic equation nearly identical to the Burton equation (Burton et al., 1975) with a slight correction. The method is restricted to Dst > -15 nt owing to the rarity of larger excursions. We show that the ring current decay lifetime varies with VBs but not with Dst, and we relate this variation to the position of convection boundaries in the magnetosphere. Convection boundaries closer to the Earth result in shorter charge exchange decay times owing to the higher neutral density near the Earth. The decay time in hours varies as T exp [9.74/(4.69+ VBs)] with VBs in millivolts per meter. We also show that the energy injection function as derived by Burton et al. is essentially correct. The injection Q is zero for I/Bs < E.:.49 mv/m, and it is Q = -4.4(VBs- E.) for VBs > E.. We derive the correction for magnetopause contamination: Dst* = Dst-7.26P m + 11 nt, where P is solar wind dynamic pressure in nanopascals. Finally, we apply the model to a moderate storm and to an intense storm. We demonstrate that, in spite of the fact that spacecraft observe compositional changes in the ring current at intense Dst, the dynamics of the two storms are not obviously different in the context of our model. We demonstrate that the generally observed dependence of the decay parameter on Dst is actually an alias of the coincidence of intense Dst and intense VBs. 1. Introduction Dst*(t) 2E(t) The ring current is a toroidal current that flows in the Earth's magnetosphere between 2 and 1 Earth radii (R/,:)[Gonzalez e! Here B and E, representhe magnetic field at the surface of the al., 1994]. The balance of the current is carried by energetic ions, Earth and the magnetic energy in the Earth's field above the mostly protons with energies between 3 and 2 kev. During surface, respectively. From there, assuming the change in total magnetic storms, the ring current is greatly enhanced, and its particle kinetic energy is a combination of a source U and energy budget contains a significant contribution from oxygen proportional loss, ions [Daglis e! al., 1999]. The exact mechanism that adds particles to the ring current is not completely understood [Daglis de(t) E(t) et al., 1999], but the loss mechanisms are fairly well documented. = W(t)-, (2) dt r The primary loss mechanism is commonly held to be charge exchange, wherein an energetic ion exchanges an electron with a we arrive at the Burton equation [Burton e! al., 1975], which we cold neutral and, having lost its charge, escapes the magnetic write in a slightly different form: field of the Earth, carrying most of its kinetic energy with it [Daglis e! al., 1999]. Although we understand the mechanism ddst* Dst* (t) -O(t)- (3) well, there is considerable uncertainty in the specific features of dt r the decay during different phases of a magnetic storm. In this study, we will attempt to develop an empirical model of the This particular equation performs quite well for various dynamics of the storm time ring current, including both injection definitions of Dst*, Q, and ' [e.g., Burton et al., 1975; Feldstein et and decay. al., 1984; Vasyliunas, 1987]. Dst* is one of various corrections to Our study of ring current dynamics will focus on an analysis Ds! which will be discussed below. Q is an injection term, which of the Dst index. While directly measuring the magnetic field of we will treat as being uniquely determined by VBs, the the toroidal current flowing in the magnetosphere, Ds! is also a interplanetary electric field in geocentric solar magnetospheric measure of the kinetic energy E of the particles that make up the (GSM) coordinates. We define VBs to be positive in terms of the ring current. This is stated formally in the Dessler-Parker- solar wind parameters in GSM: Sckopke relation [Dessler and Parker, 1959; Sckopke, 1966]: (4) Copyright 2 by the American Geophysical Union. o B._>O Paper number 1998JA //1998JA The decay time T is presumed to be a result of charge exchange loss through collisions with the neutral geocorona 777

2 778 O'BRIEN AND MCPHERRON: PHASE SPACE ANALYSIS OF Dst [Daglis et al., 1999]. On the basis of the observation that this recovery time is much shorter during very intense storms, as identified by large, negative Dst, it has been postulated that 'r depends on Dst. Essentially three mechanisms have been suggested to explain how 'r could vary during a storm. Assuming 'r is related to charge exchange lifetimes, we would write an effective decay time: '--(ncrv)-' COS 35+2 &, (5) They were basically attempts to provide a continuous functional form that would have shorterecovery time at larger, negative Dst. One recent approach that deserves detailed comment is that of Vassiliadis et al. [1999a, b] (also D. Vassiliadis et al., A nonlinear dynamical model of Ds! geomagnetic disturbances from high-time-resolution data, submitted to Journal of Geophysical Research, 1998). They have suggested that the Burton equation be replaced with a second-order equation: where n, is the density of hydrogen in the geocorona, c is an d2dst ddst --+ v + f22dst(t)= O(t). effective charg exchange cros section, v is an effective particle dt 2 dt (9) velocity, and k,, is an effective mirror latitude [Smith and Bewtra, 1978]. The geocorona density falls off with distance from the In their analysis, they used state-local linear filters, a method by Earth L, in RE, giving the following approximate dependence to which a typical linear filter is built out of state space neighbors of n, [Smith and Bewtra, 1978]' the current state of the magnetosphere. For example, the parameters of (9) would be determined from a historical sample n, oc e -' /' ' (6) of times when Dst, ddst/dt, and VBs resembled the current values of Dst, ddst/dt, and VBs. Using this approach, Vassiliadis et al. Here L is a scale height determined by the mass and temperature [1999a] concluded that "decay time depends on the presence of of the atmosphere, by the gravitational pull of the Earth, and by the approximate location of the quie time convection boundary. The first mechanism for varying 'r assumes that there are two the solar wind input, presumably FBs can change the magnetospheric electric field and modify the ring current decay." Below we will reinforce and quantify this result. ring currents, an inner one and an outer one. The inner one would In a similar vein to the work of Vassiliadis et al. is that of have a larger effective n and would therefore decay faster. As the inner current decayed, the apparent decay rate would seem to slow as the outer current became the dominant contributor to the Dst index [,4kasofu et al., 1963]. Satellite experiments have Klimas et al. [1997, 1998]. Instead of using state-localinear filters, however, they use time-localinear filters. The process is similar, and the fundamental equation is essentially given by (9); however, instead of using the state of the magnetosphere find shown that there are not two spatially distinct ring currents neighbors, a time window is centered around each observation, [Hamilton et al., 1988], but the idea of radial dependence on decay time will be used below. The second mechanism for varying 'r assumes that wave activity is very intense at the storm peak. During enhanced wave activity, particles are constantly being scattered into states that allow them to travel very close to the Earth during their bounce motion [Daglis eta[., 1999]. By traveling closer to the Earth, the particles experience higher effective mirror latitude ),, and therefore shorter z [Smith and Bewtra, 1978]. The final mechanism, and by far the most popular, assumes a significant change in the composition of the ring current during the most intense part of the storm. Specifically, satellite measurements show that during intense storms, a significant fraction, sometimes over 5%, of the ring current energy is and the parameters of (9) are calculated from only those neighbors. This approach has yielded some insight into the dynamics of Dst but will not be directly used here. We will, however, make use of the idea, referred to by Klimas et al. as "closure relations," that analytical functions ought to be used to organize the variations in model parameters. However, rather than using arbitrary mathematical expressions, we will attempto derive our analytical functions from physical properties of the system. Since it is such a crucial aspect of ring current dynamics, there have been many attempts to explain and model the recovery rate. There are so many models that any new model is obliged not merely to fit some data, but also to explain how other models would also be able to fit the data. After we have done some data carried by O + ions [Hamilton et al., 1988]. Having a relatively analysis, we will return briefly to the history of the variable 'r larger cross section c around 1 kev [Smith and Bewtra, 1978], these ions have a much shorter lifetime to charge exchange than H + ions, which are usually the dominant contribution to the ring current [Hamilton e! al., 1988]. By shifting the balance of energy models to put our results in their most immediate context. We will derive our model from the data, then we will analyze the deviations from the model, and finally we will use our model to reproduce the observed dependence of : on Dst. among various species, it is possible to change the effective decay time of the ring current. There have been numerous other corrections to the Burton 2. Method of Analysis equation. Some authors [Gonzalez et al., 1989; Prigancova and Feldstein, 1992] have suggested that z drops to an hour or less during very intense activity. Other authors have been concerned with a contamination of Dst by the tail current [Alexeev et al., 1996]. Recently, Kamide et al. [1998] have suggested that a storm is actually composed of a two-step development where the sudden injection of oxygen provides a second intensification. In some cases it has been suggested that the decay time of the ring current can be parameterized in terms of Dst. Mac-Mahon and Gonzalez [1997] suggested the function: roldxtl -"'2 (7) Valdivia et al. [ 1996] suggested another function: r /(1 -.12Dst). (8) Typically, the analysis of any dynamic system begins with a look at the system's phase space. For Dst this is a nontrivial task because hourly changes in Ds! are often lost in the noise background. Therefore we have chosen to analyze the probability density in phase space. Specifically, we are interested in the probability of a given hourly change in Dst, signified by ADst, for a given Dst and Q; we write this probability P(ADstIDst, Q). Recall that Q represents the injection of new energy into the ring current and that Q, being unknown, is assumed to be uniquely determined by?bs. The discrete form of the Burton equation suggests that the probability density in phase space will be focused on a line: ADst* = (t + l)- Dst* [Q(t)- Ds ]N. ( )

3 O'BRIEN AND MCPHERRON: PHASE SPACE ANALYSIS OF Dst 779 This forward difference is permitted owing to a time delay in the magnetospheric response. The slope of such a line would be related to x and the offset would be related to Q according to offset = QAt (11) Various values of b and ½ have been calculated [e.g., Burton et al., 1975' Feldstein et al., 1984; Pudovkin et al., 1985], but disagreement exists over the precise values. In the statistical averaging that we will perform, the correction to Dst becomes a constant: 3. Pressure Correction At slope = --- (12) It is common when studying Dst to remove the contribution of the magnetopause current system. When the solar wind dynamic pressure is enhanced, the magnetopause moves closer to the Earth, and the currents associated with it contaminate Dst. The formal equation for this correction is given by Burton et al. [1975]' Dst* = Dst-b,w +c. (13) = (Ost [ - c' \ I I (), Da I In phase space, we plot P(ADst Dst, Q) versus ADst and Dst for a specified VBs. (Throughout this paper, we will use a foreward first difference to approximate time derivatives.) In order to get quality statistics, we restrict our analysis to Dst > -15 nt. This This constant c' will be defined as the value of Dst for which ADst is zero when Q is zero. As there is no reason to believe that the hourly changes in P.,.w approach is completely general, where the only assumptions should be correlated with the magnitude of Dst nor with VBs, we about the dynamics are as follows: (1) Q is uniquely determined by VBs, with Q being zero when VBs is zero; (2) the spread of the can reasonably assume that the effects of hourly changes in should average to zero in our statistical analysis: probability density is caused by processes which are completely random with respecto Dst and VBs; and (3) the evolution of the system is otherwise entirely determined by Dst and Q. We do not assume a priori that the system tbllows the Burton equation. We can therefore use uncorrected ADst for statistically averaged Assumptions 1 and 2 will be discussed in more detail below. dynamics: Assumption 3 allows us to generalize statistics on 3 years of hourly Dst transitions to the dynamics of individual storms. For this analysis, we will use the OMNI database [King, 1977] O,Dal \ IIQ.Dal of hourly solar wind and Dst measurements. These solar wind measurements are calculated as hourly averages from spacecraft For these reasons, it is not necessary to know either b or c a near the Earth but outside the magnetosphere. For some priori. It should be noted that these are largely technical details, conditions, propagation is performed from the spacecraft location provided here to satisfy the skeptical reader, but that the analysis to the magnetopause. We use the years ; during this is not appreciably affected by this correction. interval, there are many times when no solar wind data are available. We must, therefore, omit these times from our statistical analysis. In total, the database contains nearly 3, 4. Analysis in Phase Space hours of Dst data. After removing times when no solar wind data Having laid the statistical groundwork, we will now turn to the are available, we are left with 13, hours when VBs is actual analysis of data. Figure 1 shows the phase space available and 125, hours when VBs and dynamic pressure probability density for three values of VBs. It is of supreme are available. At final accounting, our data set includes hundreds importance that in each case the probability density is clustered of storms of various sizes and many hours of quiescent behavior, around a linear trajectory. This tells us explicitly that the Burton all taken from various stages of the solar cycle. equation is the simplest description of the dynamics. It also tells Since we will be computing probability densities from discrete us that for a given VBs, x, as given by (12), does not change with samples, we will perform some binning of the data. In phase Dst because the slope of the trajectory does not change with Dst. space, this binning amounts to calculating average quantities in a We do see that the slope and offset change for different values of neighborhood of Q and Dst values. We will denote this type of lbs. The most general conclusion of this observation is that x and average of a variable X as <X>IQ,D.,.,. In cases where we have a Q depend on VBs but not on Dst. Before we begin to analyze the large enough sample size, we will use medians rather than means forms of these dependencies on VBs, we should take some time to to eliminate the influence of large outliers. This averaging will be address some sources of error and to test their randomness. useful in generalizing with regard to the standard technique for removing the magnetopause contribution to Dst. 5. Errors (A.,,, >[e.,,, =. (16) There are numerous sources of error, most of which are believed to be small and random with respecto the influence of injection and decay. One source of error is the calculation of Dst itself. Since Dst is calculated as the arithmetic average of AH values measured at four stations around the Earth, weighted by the reciprocal of the average of the stations' magnetic latitudes [Sugiurand Kamei, 1991], there is some inherent uncertainty Dst. Assuming that this uncertainty is spread in a Gaussian around the true value of Dst, we can characterize the uncertainty in Dst as CrD.,,(t) z ASY(t)/ x. VBs(t), (18) Here c contains not only an adjustment for the quiet time magnetopause, but it may also contain an offset in the calculation where N is the sample size (4), ASY is the asymmetry index, of Dst itself. This definition of Dst* gives rise to a modification of which measures the range of M-/values contributing to Dst, and the ADst equation: asymmetry is related to VBs according to Clauer et al. [1983]. This suggests that more intense FBs corresponds to more ADst* = ADst - bax/p,w (14) uncertainty in Dst.

4 771 O'BRIEN AND MCPHERRON: PHASE SPACE ANALYSIS OF Dst I P(ADstIDst,Q ) for Dst transitions (VBs = ) i / i I! 886O/o '\'",,\...,, I I l,/ "/'/' i! \ \ "' X3 i 3 r...,-" [ ", " ' -oo "",, \ x \ i -/',... -%..... ',,, j,,'...i....< -.., a) Dst(t+ 1 h rs)-dst(t) 15 P(ADstlDst,Q) for Dst transitions (VBs- 2 +/- 1. mv/m) \ '", 35.8%,\!. o ;,,! /..:, i,.,...!... ½ { ',,, ',,,,... / / \... / t... /...;... i b) Dst(t+ lhrs)-dst(t) c) -5 P(ADstlDst,Q) for Dst transitions (VBs- 4 +/- 1. mv/m) Dst(t+ 1 h rs)-dst(t) Figure 1. Probability density in phase space for three values of VBs: (a), (b) , and (c) mv/m. Here the FBs bin boundaries are indicated by + 1 mv/m The function is normalized so that horizontal bands integrate to 1%. It is clear that the trajectories are lines, but the slope and offsets of these lines change with VBs. The rectangle in the upper right-hand corner indicates the bin size. The stability line ADst- is also provided. A linear fit to the median values (x) is provided. Another source of error is the hourly averaging performed in the construction of the OMNI database. First, this contributes to an error in the computation of gbs from hourly averaged g and hourly averaged Bz: (V)(Bz)., (FBs). (19) These quantities are not exactly equal and are particularly unequal when Bz is nearly zero. It is simple to understand that at least 3 rain of southward Bz may be lost in this approximation. Next, we must consider rapid dynamic pressure changes that occur on a scale of less than 1 hour. These can give rise to unexpected behavior in the Dst index due to a similar inequality between hourly averaged quantities. A rapid change in dynamic pressure may not be evident in the hourly averages of solar wind density and velocity, which contribute to dynamic pressure, but may be evident in the hourly averages of the AH values which are used in computing Dst. Finally, we must consider hourly boundaries in the data points. Since the OMNI database does not completely account for propagation of the solar wind, it is easy to see how a change in the solar wind may occur earlier or later in the database than its effects appear in the Dst index. Taken together, these sources of error should be random and small in relation to the influence of injection and decay. However, at small values of Dst or for weak injection, these errors may likely dominate hourly changes in Dst. If the errors described above represent the sum of many random deviations from the actual values of Dst, they should result in a Gaussian distribution of ADst values for a given Dst and Q [Hogg and Craig, 1995]. There will be some difficulty in measuring the distribution of ADst values because standard Dst is reported on an integer scale, in nanoteslas, and this rounding will mask the precise distribution of the errors. For a given set of preconditions, Dst, and Q, we observe a nearly Gaussian distribution of values of ADst. This allows us to assume an approximately Gaussian distribution for error analysis; however, to gain physical insight, we would like to constrain the likelihood that this distribution of ADst values does indeed come from a Gaussian source distribution. That is, we would like to know if the deviations from the mean ADst can be explained as the superposition of many small deviations which are not uniquely determined by Dst and Q. Only in the case of many, rather than few, superposed sources of error will a Gaussian distribution arise, according to the central limit theorem [Hogg and Craig, 1995]. We wish to test the hypothesis that our measured distribution of ADst comes ti'om a Gaussian whose mean and standard deviation are the same as the distribution we measured. Unfortunately, statistical tests can only tell us the certainty with which a sample does not come from a hypothetical distribution. The statistical measure is called the significance, and it represents the probability that the ditl 2rences between the ADst distribution and the Gaussian distribution are the result of an underlying difference between the two distributions rather than simply the result of finite sample size. A significance of 1 tells us that the ADst values did not come t?om a Gaussian distribution, and a significance of tells us that we cannot prove that the ADst values did not come from a Gaussian distribution. To test the Gaussian hypothesis, we will use a direct comparison of the measured ADst distributions to those of integer-rounded Gaussian distributions. We will employ the Kolmogorov-Smirnov (KS) test [Press et al., 1986], which measures the similarity of two distributions based on the maximum separation of their cumulative distribution functions. The cumulative distribution function F(x) is the probability that a sample measured from the distribution will have a value less than or equal to x. The KS test is accurate to significances of.1 or

5 O'BRIEN AND MCPHERRON' PHASE SPACE ANALYSIS OF Dst 7711 smaller for sample sizes of 2 or larger [Press et al., 1986]. Figure 2 shows that for VBs = 2 mv/m the difference between the measured ADst distribution and a Gaussian is significant for small values of Dst and not clearly significant tbr large values of Dst. This result is typical for all values of VBs. With such high significances for rejection of the Gaussian hypothesis at small Dst, we conclude that the distribution of ADst for a given Dst and Q does not stem from the superposition of many sources of error, but rather, it stems from at most a few sources of error, at the l- hour resolution. We will revisit the importance of this result when we perform the pressure correction. 6. How Injection and Decay Depend on VBs From the phase space plots it is clear that the errors are not systematic with Dst or VBs; therefore we can safely say that the systematic features of the ring current are described by Q and T. We have seen that these vary with VBs; so we must characterize this variation. It is odd to think of features of internal magnetospheric dynamics varying with VBs, which is an external parameter. However, we will describe below that VBs is merely a proxy for internal parameters of the magnetosphere. To calculate precise values of Q, we must accommodate the fact that a given calculation of the phase space trajectory is biased because of binning in VBs. In Figure 1, 1 mv/m bins were used. This is a fine approximation for values of VBs below 5 mv/m, but above that threshold the sparseness of VBs values requires us to enlarge our VBs bins. Large VBs bins will be heavily biased toward the smaller VBs values because of the rapid occurrence dropoff with larger VBs. Therefore we must accommodate this dropoff by correcting the bin bias. We will use the slope and offset of the phase space fits to estimate Q and z according to (11) and (12). We representhe measured value of Q as VBs o +b' / 2 VBs o - rs' / 2 VBs o +b' / 2 VBx o - b' / 2 (2) where values with [VBs-VBso[ < 15/2 were used, and N(x) is the fraction of data points at a given VBs such that the integral of N over the entire bin is 1. If we expand Q(VBs) to first order about VBso, we have We can now remove Q(VBso) and Q'(VBso) from the integrand: VB. b + b' / 2 VBso +b' / 2.[ VB.% - ' / 2 +Q'WBs,,) (x- rbs,,)n(x) VBso -b' / 2 Since N integrates to 1, this becomes ( O(r s,, )) = O( s,, ) (22) +Q'(VBs,, )((VBs),. - VBs,, )' (23) This equation has two unknowns, Q(VBso) and Q'(VBso). If we repeat this process for several values of 15, we will have an overdetermined linear system and we can perform a least squares regression to find Q and Q'. Recall that each <Q>a is measured as the offset of a linear fit to the medians of the ADst distributions for a range of Dst with Q constrained by [VBs- VBsol < 15/2. By 1 K-S Test that ADst is Not Gaussian (VBs - 2 +/- 1. mv/m) o9.6 o.5 o.4 o :i:;i:ii::i::ii i i:.:: _.:ii:i:?iii;!i i." ::::::::::::::::::::::... :::::::.::::.:::. ::::::::::::::::::::::::::::::::... ::;.:::... ::......, :gff_.? ½ i!iii½ :-:,!½... :::::::::::::::::::::::::: ½.- i ½2 5:,... :!:iiiiii:iiii!ii ii '... :::::::::::::::::::::::::: * ;... - ::::-...,: : Dst (nt) Figure 2. The KS test used to determine if the distribution of ADst is Gaussian. The significance level is the probability that the observed difference between the ADst distribution and a Gaussian is real rather than an artifact of finite sample size. The asterisks indicate occasions when the sample size for the KS test was smaller than 2, where very low significances are poorly determined.

6 7712 O'BRIEN AND MCPHERRON: PHASE SPACE ANALYSIS OF Dst using this technique, we can extend our estimation of Q to values of l/bs up to about 9 mv/m. It is difficult to extend beyond that threshold because of a scarcity of data. However, because Q turns out to be very linear in l/bs, we are able to trust values slightly beyond that limit. As well as extending the region of l/bs over which we can measure the phase space fit parameters, this have described are not simply the superposition of many random deviations. This new information does not tell us what kind of deviations we are seeing, but it does tell us that there are few rather than many sources of error contributing to individual 1- hour measurements. technique has also given us much more stable measurements compared to using the uniform bin size of 1 mv/m. 8. Hypothesis on VBs Control of Decay Figure 3 shows the estimates of Q we obtain from this process. It is clear that Q is a linear function of VBs. Since a positive Q is The interesting shape of the decay parameter x versus VBs nonphysical, we define Q in terms of l/bs as suggests that an analytical function could fit much of the variation. A mathematical fit, however, would have two shortcomings. First, it would tell us little about the underlying Q(nT/h)={a(VB;-E ) l/bs VBs>E < ' (24) physical mechanism which governed the control of decay by?bs. Second, we could not reliably extrapolate a simple mathematical where a is -4.4 nt/h(mv/m) ' and EL is.49 mv/m; these values fit beyond VBs = 9 mv/m. If we can use some physical insight to are similar to those provided by Burton et al. [1975], provide the functional form of the 'r-vbs relationship, we could -5.4 nt/h(mv/m) - and.5 mv/m, respectively. both gain some insight into the mechanism for ß variation and We can also obtain from the value of Q() the correction have some confidence when extrapolating beyond the region of parameter c introduced in (13). Since Dst should be defined such fit. that no input results in Dst-, the corrected Dst should be Akasofu [1981] suggested a tabular variation of T with the e adjusted according to (13) and (15) by a constant offset c' of parameter but offered no physical justification, aside from the.1 nt. This offset is so small, however, that it can be taken to be observation that was about 2 hours for e < 5x1 8 ergs and that essentially zero, since Dst is only measured on a 1-nT resolution. ß was about 1 hour for e > 5x118 ergs. Pudovkin et al. [1988] Because c' is essentially zero, there is no appreciable difference derived a relation which roughly matches the 'r-vbs curve in in Q or T whether calculated from Dst or Dst* by our method. Figure 4, but they were not able to adequately fit the transition We can apply the same technique to the slope of the phase from long to short recovery time. Their functional dependence space fits to obtain T as a function of l/bs. Figure 4 shows that was given by this relationship is by no means linear. The value of 7.7 hours calculated by Burton et al. [1975] corresponds roughly to a VBs of about 4 mv/m, a typical storm time value. In a later section we r(hours) = e-"/4. (26) will discuss one way of fitting this relationship to an analytic F = x 1 -' V (.5or - B, ) function. For now it is sufficient to note that Q is linear in VBs and that z is nonlinear in VBs. Here V is in kilometers per second, B: and cy are in nanoteslas, and F is in nt/h(mv/m) -l. The standard deviation cy of & in the 1-7. Calculation of Pressure Correction hour sample interval is included to account for periods of southward B.. which are lost in the 1-hour average. We now return to the issue of contamination of Dst by the To explain the shape of the 'r-vbs curve, we suggest that VBs magnetopause currents and the associated pressure correction. controls the position of the ring current by controlling the We argued above that ignoring pressure fluctuations was positions of convection paths and boundaries. In the legitimate because the influence of those fluctuations averages to approximation that there is a global electric field from dawn to zero. However, if we choose to fix pressure fluctuations to a dusk throughout the magnetosphere, the hot ions that make up the particular range and then recalculate the linear fits in phase space, ring current will set up a convection pattern [Southwood and we can estimate the pressure correction parameter b introduced in Kaye, 1979]. At a given energy and equatorial pitch angle, the (13). We extend the definition of the offset to be convection pattern will have both closed and open trajectories. Nominally, the ring current is made up only of the particles on (25) closed trajectories. The position of the boundary between open and close drift orbits defines the outer edge of the ring current. A linear fit of the residual phase space offset after the removal of This position depends on the strength of the dawn-dusk electric Q to the change of the square root of the dynamic pressure yields field and on particle energy and equatorial pitch angle a pressure correction term b of 7.26 nt(npa) - /2. Burton et al. [Southwood and Kaye, 1979]. This is consistent with in situ [1975] provided 16 nt(npa) - /2. Figure 5 shows how the dynamic observations, which showed that the energy density peaks closer pressure affects the phase space offset by fitting the residual to to the Earth during the most intense part of the February 1986 the hourly change in the square root of the dynamic pressure. great storm [Hamilton et al., 1988]. If the ring current is closer to Now that we have obtained b and c ', we can recover c, as defined the Earth, the particles that make up the ring current will be in (13) and (15). We obtain a value tbr c of 11 nt, whereas exposed to a higher neutral density. This higher neutral density Burton et al. [1975] obtained 2 nt. will increase the effective charg exchange rate, or decrease the Now that we know how to correctly remove the contamination effective charge exchange lifetime in (5), of the ring current. The by the magnetopause current, we can reevaluate the randomness final piece of the puzzle is the fact that the dawn-dusk electric of our model errors. The error distributions are still roughly field is related to VBs [Reiffet al., 1981]. We can follow this line Gaussian; so we can still use the Gaussian assumption in our error of reasoning to determine the functional form of the 'r-vbs analysis. However, to evaluate the physical interpretation of the relationship, and then we can fit that form to the curve we have errors, we have performed the KS analysis after the pressure derived from our data analysis. correction. The resulting distribution of ADst appears to be The following derivation roughly follows that of Southwood statistically less Gaussian than the uncorrectedistribution. This and Kaye [ 1979]. We begin the derivation by assuming that a suggests that after removal of the magnetopause contamination, given particle convects on contours of constantotal energy, a the remaining deviations from the deterministic equations we combination of electrostatic potential energy and kinetic energy.

7 _ O'BRIEN AND MCPHERRON' PHASE SPACE ANALYSIS OF Dst Injection (Q) vs VBs i i i i i i I... O O Offsets in Phase Space - _ + + Points Used in Fit ' -2-3o ß , -5-6 Ec = I I I I I VBs (mv/m) Figure 3. Injection Q versus?bs. Q is calculated from a set of linear fits to the phase space trajectories indicated by the probability density P(ADst Dst, Q). Q is clearly linear in?bs beyond the injection cutoff E,.. Some data were not used in determining the best fit: low values of?bs were removed owing to the injection cutoff, and high values of VBs were removed owing to the scarcity of data beyond?bs = 9 mv/m. We will perform this derivation for an equatorially mirroring particle with kinetic energy given by (27) Here p is the dipole moment of the gyrating particle and B is the local magnetic field strength. We use the dipole magnetic field approximation' = (28) Z Bo is the magnetic field at the surface of the Earth, and L is radial distance measured in Earth radii. We employ a simple electric field E according to Folland [ 1979] and Stern [ 1977]' E = -EoR L p sin ½. (29) Eo is the field strength, q) is the angular local time, and p is a shielding parameter. When there is no electric field shielding by the plasmasphere, p is 1. A value for p of 2 may also be 2... Decay Time (- )vs VBs ' '1 o "-" 1 - I I I I VBs (m V/m) Figure 4. Decay time z versus?bs. Here z is calculated from a set of linear fits to the phase space trajectories indicated by the probability density P(ADst Dst, Q)' ß is clearly nonlinear?bs.

8 7714 O'BRIEN AND MCPHERRON: PHASE SPACE ANALYSIS OF Dst 6 (Phase-Space Offset) -Q vs A[P 1/2] i i I i i I o 6 o o -64:) ½) - - (PS Offset) - Q Best Fit - (7.26)A[P 1/2] -1() I I I I I A[p 1/2] (npal/2/h) Figure 5. Residual phase space offset plotted against chm ges in the square root of the solar wind dynamic pressure P. Here we have removed the part of the phase space offsets due to Q. We perform a linear fit of the residual to AP ]/2. This reveals the coupling parameter b. appropriate, if the cold plasma diminishes penetration of the electric field into the inner magnetosphere. We will continue the derivation without specifying p in order to maintain generality. We now calculate an effective electrostatic potential (I), which is the total energy per charge q: el) = -EoR/: L/' sin cp + (3) ql. ' Equatorially mirroring particles drift on contours of constant. In order to calculate the drift boundary, we calculate the critical point L.,. of ß with the magnetic moment held constant: (c )=peor/,,l/'-'-3/tb - -L =O ',, ql 4 ß (31) Solving this equation for L,, after substituting for p, we have L,=I 3Wz 1 l//' (32) L qpe,,r/,. Physically, L, is a point on the dawnside of the Earth at which equatorially mirroring ions do not drift. This is known as the stagnation point, and it is part of the convection or trapping boundary. We will use this point as a representative of the entire trapping boundary. Next we must replace Eo with some relation to VBs. We can assume that the convection electric field is proportional to the polar cap potential drop, and then we can use the relation from Reiff et al. [1981] that relates the polar cap potential drop to VBs: Now we can combine this result with (5) and (6) to achieve an equation for z as a function of VBs: r oc e (35) Here ot is a constant resulting from the combination of the preceding equations. We have performed a least squared error fit of the measured z for VBs values between and 9 mv/m. The database was too sparse to construct phase space density plots with median?bs values above 9 mv/m; therefore the points above 9 mv/m are suspect owing to extreme extrapolation in the estimation technique described in (2)-(23) above. We performed the fit for various values of the shielding parameter p and found that p = 1, no shielding, provides a marginally better fit to the data. This fit is provided in Figure 6: r(hours) - 2.4e ' 74/(469+VB,) (36) We are quite satisfied with the quality of this fit. Statistically we can say that the likelihood that this fit occurred by chance is less than.1%. However, the statistical argument does not remove the possibility that a different physical mechanism creates the observed relationship between z and l/bs. In this fit, VBs is given in millivolts per meter. The value a'-4.69 mv/m we have achieved through our fit disagrees with the value of 1.1 provided by Reiff et al. [ 1981 ]. One physical implication of our fit relative to that of Reiff et al. [1981] is a weaker penetration of the interplanetary electric field into the magnetosphere than into the polar cap region. E o oc q3/, c a o + a VBs. (33) Normalized by a/, this gives rise to an overall dependence of L,. oc (a'+ VBs) -'/p (34) 9. Comments on VBs Control Hypothesis We have suggested a physical mechanism that fits the observed data. However, we have not provided any collateral

9 - O'BRIEN AND MCPHERRON: PHASE SPACE ANALYSIS OF Dst Decay Time ( ) vs VBs 16 o + ' from Phase-Space Slope -I- Points Used in Fit ' : 1.97e 11.18/(4.98+VBs) 14 - _ 2 i i i, i VBs (my/m) Figure 6. Least squares fit of decay time (l/bs) according to the trapping boundary hypothesis. physical evidence that the mechanism is correct. We would like to suggest ways to test this hypothesis in more detail, and we would also like to put our model in the context of other recent models. First, we address the issue of testability. The mechanism we have described depends specifically on the variation of nh with altitude. We mentioned that the scale height L varies with the temperature of the exosphere. The temperature of the exosphere varies with solar cycle. We can therefore construct a test of how the fit parameters change with solar cycle. Another method of testing our hypothesis is energetic neutral atom (ENA) imaging. ENAs are the byproducts of charge exchange and can be used to create images of charge exchange loss throughout the magnetosphere [Roelof, 1987; Jorgensen et al., 1997]. If the charge exchange lifetime of ring current particles is indeed shorter for increased VBs because the ring current is closer to the Earth, we should see the effects in the ENA image. Specifically, more ENAs should originate closer to the Earth at times of enhanced VBs compared to times of weak VBs. This correlation would, however, be influenced by Dst because there would be more charged source particles at larger Dst. These tests may be within reach of our current data resources. Second, we would like to address some of the immediate consequences of the fit we have performed. Spacecraft have observed that the ring current composition changes during storms [Hamilton et al., 1988]. Specifically, oxygen makes up a larger fraction of the total energy budget in the ring current for the most intense values of Dst. This leads us to conclude that, since we do not have to account for this compositional change in our fit of z to VBs, the effective charge exchange cross-section (x does not change significantly with composition. This conclusion is certainly counterintuitive and leaves the door open for a compositional, rather than a convective, mechanism to explain the dependence of z on VBs. In this study we have neglected a lot of potentially relevant physical processes by assuming that all ring current decay can be explained by charge exchange. Simulations are able to include more physical processes, and they can give us insight as to the credibility of our assumption. Kozyra et al. [1998b] studied the great magnetic storm of February 1986 using a bounce-averaged drift loss model. They concluded that charge exchange alone was not sufficient to explain the decay of the ring current but that O + was a significant contributor to rapid charge exchange loss immediately after the main phase. In a separate study, Kozyra et al. [1998a] studied the November 1993 storm using a similar model. Their simulation suggested that changes in the plasma density of the solar wind could influence ring current dynamics. In another study [Kozyra et al., 1997] the influence of electromagnetic ion cyclotron waves was analyzed by using a similar model. They concluded that enhanced wave activity can increase the rate of pitch angle diffusion and thereby increase the effective mirror latitude *,, in (5), causing a shorter decay time. In none of these simulation studies is an explicit 'r-l/bs parameterization provided. However, all of them implicitly relate solar wind conditions to a variable decay lifetime. In each case, the route from increased FBs to shorter z is more arduous than what we have outlined above. Until other evidence is brough to bear, we suggesthat our explanation, being the simplest, ought to be used. While many have observed that the decay time z is shorter at larger Dst, we do not find any z dependence on Dst. We will show in the next section that the apparent dependence of z on Dst is actually an alias of the typical concurrence of large FBs and large Dst. This result will affect all alternate models for the u-fbs relationship. Finally, we note that a dependence of the decay time on the cross-tail electric field suggests that our calculation of the offset might be influenced by the quiet time ring current. That is, the quiet time ring current is created by the quiet time cross-tail electric field: its intensity is determined by the balance between quiet time input and charge exchange decay. The quiet time ring current is removed in the calculation of Dst, which is why,

10 7716 O'BRIEN AND MCPHERRON: PHASE SPACE ANALYSIS OF Dst generally, Q does not include the quiet time term a' above. However, if the decay lifetime suddenly drops because of a nonzero VBs, the quiet time ring current will decay away from its value for VBs =, while simultaneously new energy is being injected by the nonzero VBs. The decay of the quiet time ring current and the new injection both contribute to the phase space offset, and therefore our measured Q-VBs relation. As it tums out, the reciprocal of T is nearly linear for VBs < 5 mv/m, and so it just adds another small linear term to the measured Q. This term, in turn, contributes to the observed offset E,. in the Q-VBs relation. We do not know if this term entirely explains E,. or merely contributes to it. 1. Recreation of Decay Dependence on Dst We now turn to directly assessing why we observe an apparent T-Dst relationship when our analysi suggests that there is indeed no such relationship. Having demonstrated that the Burton equation is appropriate if the T-l/Bs relationship is accommodated, we will use it to demonstrate how a T-Dst relationship might arise. If we solve the discrete Burton equation for T, we have r= Dst* Q- ADst* At (37) are infinite and negative. It is impossible to accommodate the divergence of this equation through direct averaging. A more stable technique for calculating T is linear regression fitting of ADst to Dst as in (1). The slope of this fit is given by (12). However, if this method is not performed for restricted values of VBs, then the calculated value of T will depend on the range of VBs values that were used in the fit. This gives rise to the typically observed T dependence on Dst because of the coincidence of large VBs and large Dst. In other words, T appears to depend on Dst because large Dst is correlated with large VBs, and T does depend on VBs. Our phase space analysis performed above is highly analogous to this least squares fitting technique, excepthat by fixing VBs, we were able to remove the aliasing created by the coincidence of large FBs and large Dst. Figure 7 demonstrates the effects of using various ranges of Dst in calculating T. This analysis requires the use of very large bins in Dst because we are fitting to Dst in each bin, and the trend would be lost in the noise if small bins were used. Figure 7a shows the usual result: if larger values of Dst are used, a shorter decay time is calculated. However in Figure 7b, it is clear that when VBs is restricted, the same T is calculated regardless of what range of Dst values are used. This proves that the dependence of T on Dst is an alias effect. Formally, the conceptual error is seen in the partial derivatives that make up the T-Dst relationship: This equation is disastrously unstable, considering that typical ADst values are 3 nt in an hour but that the uncertainty Dst is typically at least 8 nt. This error gives rise to values of T which dr c?r c?dst ) ' (38) for various ranges of Dst (without specification of VBs) VBs = l,;,'o ^" vso 18 VBs = 2 for various ranges of Dst (with specification of VBs) , 1- lo ' -2 I I i I 4 I I I I I Dst Range (nt) Dst Range (nt) a) b) Figure 7. Decay time T calculated for various ranges of Dst using least squared error fits of ADst to Dst: (a) T calculated without specification of FBs; (b) the same calculation with l/bs fixed at three different values. In Figure 7a, the often observed T-Dst dependence is reproduced because calculations which include larger values of Dst result in shorter decay times. However, in Figure 7b it is clear that fixing?bs obliterates that dependence because at any fixed value of?bs the same T is calculated for any range of Dst. The endpoints of the lines indicate the Dst range used, and the point in the middle of each line indicates the median Dst value in that range.

11 O'BRIEN AND MCPHERRON' PHASE SPACE ANALYSIS OF Dst 7717 The total derivative is what we typically measure, and it is typically assumed that the right-hand side of the equation is dominated by the first term. However, we have shown that this term is actually zero and that the measured variation is due to the second term. The partial with fixed Dst is clearly not zero in Figure 7: when the same range of Dst is used to calculate z but?bs is fixed to a different value, we calculate a different z. The partial with fixed z is an odd quantity, since the relationship between Dst and?bs is given in terms of the time change of Dst rather than Dst itself. However, because Dst and VBs generally change smoothly over several hours, there is a residual correlation between Dst and?bs. In our data set, the statistical correlation coefficient of Dst and VBs is, in fact, This storm and 97.7% for the larger storm; the root-mean-squarerrors (RMSE) are 4.2 nt and 8.8 nt, respectively. It should be noted, however, that the measures of prediction efficiency are misleading because "persistence," a model that assumes Dst does not change at all in an hour, provides upward of 95% prediction efficiencies. The "skill score," which measures how much more of the variation in Dst is provided by our model relative to persistence, gives a better idea of the performance of a 1-hour step model. The skill scores are 36.7% and 31.6% for the smaller and larger storms, respectively. Effectively, our model accounts for about a third of the hourly change in Dst. The remaining portion is assumed to be related to the error sources described above. analysis clarifies the nature of the z-dst relationship: z appears to At the peak of the large storm, there is presumably a larger depend on Dst because Dst is weakly related to?bs, but z component of O + in the ring current energy budget. If the actually depends only on?bs. compositional change caused a change in decay rate, we would see our model consistently overestimating the decay time, which would result in 1-hour step model values of Dst too far from zero. 11. Application to Two Sample Storms However, there is no such consistent error, indicating that intense Dst is not the source of rapid decay time. At this point, we have completed a thorough analysis of the qualitative and quantitative aspects of our model in a statistical sense. However, it is necessary at this time to turn to modeling storms as they occur: individually. In the interest of brevity, we will model only two sample storms, one very large storm and one small storm. It is important to note that our model is only a single-step model of Dst and that it makes no corrections for missing solar wind data. We have chosen two storms for which there is good solar wind coverage. We have chosen the moderate storm that occurred on October 11, day 285, of 198 and the large storm that occurred on March 2, day 61, of Figure 8 We provide additional details of our model performance in Figure 9, which shows how the model errors relate to Dst. It is clear that the model commits a few large errors for each storm, committing more large errors for the larger storm. However, the large errors do not occur consistently at large Dst, indicating that the model is performing well in that region. Again, this shows that the presumed compositional change at large Dst does not significantly alter the evolution of the Dst index. In an operational setting, ground-based estimates of Dst are typically not available until hours, if not days, behind real time. Solar wind data, however, are now available in real time through compares our model (solid line) to the standard Dst index (dots) the Advanced Composition Explorer Real-Time Solar for these two storms. It is obvious that our model does quite well Wind service from the Space Environment Center for 1-hour steps: the prediction efficiency is 97.2% for the smaller ( html). Therefore an Dst Comparmon for storm O -loo -15 Dst Comparison for storm _ I' I- M delo (um It,-step) 'o 5 1 ' ; 4, 6 ; 8 ; ,, 4 E lo a)..._._. E = 49 mv/m Epoch Hours b) o o Epoch Hours E = 49 mv/m L 1 '" Figure 8. Dst track and YBs for two storms of different sizes: (a) storm a moderate storm; (b) storm , a large storm. The dots indicate actual Dst measurements. The solid line indicates the 1-hour step model. The dashed line is obtained by using the model recursively starting with Dst at epoch time zero: standard Dst at epoch time zero is used, but thereafter Dst is calculated only from model results and the solar wind data. Gaps in the solar wind data are indicated by the absence of Dst dots. It is clear that the model does quite a good job of matching Dst for both magnitudes of storms. The model performsimilarly on other storms for which solar wind data are generally available.

12 7718 O'BRIEN AND MCPHERRON: PHASE SPACE ANALYSIS OF Dst 2O Dst Transitions for Dst Transitions for E rro r x x VBs > E c VBs > 5-15 a) -12 'o -25 ; ;, i Error. ModeI-Dst (nt) b) Error: Uodel-Dst (nt) Figure 9. Errors of the 1-hour step model for (a) storm and (b) storm The solid line connects points that are adjacent in time. The fact that the errors are clustered around zero with a few outliers indicates that the model is performing well. The facthat subsequent errors alternate in sign indicates that the hourly boundary problem is important. There is no evident systematic error for the intense portion of storm ; this contradicts the notion that large storms have special dynamics at large Dst. operational application of our model would require the use of multistep evolution from the last available Dst measurement forward in time to the latest available solar wind measurements. That is, from an initial Dst value provided from ground measurements taken several hours in the past, we could use more recent hourly spacecraft measurements of solar wind to advance Dst to the present time. In this case, after the last hour when ground-based Dst data are available, we would have to use the model output from the previous hour as the starting Dst value. This recursive use of our model can be a significant source of error. We have simulated this process in Figure 8 as the multistep model (dashed line). It is apparenthat the trace of the model in multistep mode only separates fi'om Dst when either Dst experiences new injection without an accompanying increase in VBs or there is a data gap. The resulting prediction efficiencies are 8.7% (RMSE 11. nt) for the smaller storm and 87.6% (RMSE 2.4 nt) for the larger storm. We interpret this as excellent performance, considering that the 1-hour step predictions are only about 33% better than persistence. The viability analysis of our model in an operational setting has not been performed, but the improve definition of T should yield improved performance over existing models that do not include such a 'r-vbs dependence. 12. Conclusions We have shown that the Burton equation, with only slight modification, does accurately describe the dynamics of the ring current index Dst. We did not assume this a priori, but rather demonstrated it through probabilities of 1-hour transitions in Dst phase space. We have calculated the injection function Q in terms of the interplanetary electric field VBs. Q is linear in VBs with a cutoff for small VBs. We have also calculated a pressure correction coefficient and a small olivet correction for Dst. Our model provides values similar to those of Burton et al. [1975]. These results apply for Dst > -15 nt. We have shown that the deviations from the deterministic behavior are probably not Gaussian. Non-Gaussian error distributionsuggesthat the standard techniques for handling noise, specifically, least squared error fitting, may not be appropriate. We have used medians whenever possible as a more stable alternative to means. We have demonstrated that the decay time of Dst does not explicitly depend on Dst itself. Instead, it depends on VBs, a proxy for either the injection rate or the convection electric field. We have been able to reproduce the observedependence of on Dst as a consequence of the coincidence of large VBs and Dst. We have suggested one mechanism that can produce the observed functional dependence of on VBs. This mechanism stipulates that the recovery rate is increased for larger VBs because the ring current is confined to lower altitudes by the convection electric field. The increased neutral density at these lower altitudes gives rise to a shorter effective charg exchange lifetime. Although compositional changes are observed by spacecraft [Hamilton et al., 1988], it has not been necessary to explicitly account for such changes in our model. This suggests that the effective charge exchange cross section of the ring current does not depend significantly on composition in the context of our mechanism. We have also suggested some ways in which our mechanism can be independently tested. Over the years, models of Dst have become increasingly complex. Some of this may be attributable to the problem of determining the general dynamics of a noisy system from a small set of events. This study, unlike many of its predecessors, uses a large database of ring current and solar wind parameters, covering hundreds of storms. Any study of individual storms is highly susceptible to the uncertainty inherent in the Dst index. We have shown, however, that all available Dst data can be organized according to our slight modification of the Burton equation. That is, allowing the decay time to vary with VBs is the only modificationecessary. This study marks a rare step backward in the complexity of ring current modeling, and we have done so without employing any highly sophisticated mathematics. In the interest of brevity, we leave it to another investigation to demonstrate how second-order models, like the model of Vassiliadis et al. [ 1999a, b], simulation models, like the model of Kozyra et al. [1998b], and the host of other more complicated models can explain the systematic relationship between and VBs which we observe in the OMNI data. In the interest of full disclosure, we must admit that the simplicity of our results and of our analysis technique is somewhat unintended. When we began our study we intended to apply some sophisticated probabilistic modeling techniques to the time evolution of Dst. However, even from the beginning of our

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