Yearly Comparison of Magnetic Cloud Parameters, Sunspot Number, and Interplanetary Quantities for the First 18 Years of the Wind Mission

Abstract

In the scalar part of this study, we determine various statistical relationships between estimated magnetic cloud (MC) model fit-parameters and sunspot number (SSN) for the interval defined by the Wind mission, i.e., early 1995 until the end of 2012, all in terms of yearly averages. The MC-fitting model used is that of Lepping, Jones, and Burlaga (J. Geophys. Res. 95, 11957 – 11965, 1990). We also statistically compare the MC fit-parameters and other derived MC quantities [e.g., axial magnetic flux (Φ_O) and total axial current density (J _O)] with some associated ambient interplanetary quantities (including the interplanetary magnetic field (B _IMF), proton number density (N _P), and others). Some of the main findings are that the minimum SSN is nearly simultaneous with the minimum in the number of MCs per year (N _MC), which occurs in 2008. There are various fluctuations in N _MC and the MC model-fit quality (Q′) throughout the mission, but the last four years (2009 – 2012) are markedly different from the others; Q′ is low and N _MC is large over these four years. N _MC is especially large for 2012. The linear correlation coefficient (c.c.≈0.75) between the SSN and each of the three quantities J _O, MC diameter (2R _O), and B _IMF, is moderately high, but none of the MC parameters track the SSN well in the sense defined in this article. However, there is good statistical tracking among the following: MC axial field, B _IMF, 2R _O, average MC speed (V _MC), and yearly average solar wind speed (V _SW) with relatively high c.c.s among most of these. From the start of the mission until late 2005, J _O gradually increases, with a slight violation in 2003, but then a dramatic decrease (by more than a factor of five) occurs to an almost steady and low value of ≈ 3 μA km⁻² until the end of the interval of interest, i.e., lasting for at least seven years. This tends to split the overall 18-year interval into two phases with a separator at the end of 2005. There is good tracking between 2R _O and the total axial current density, as expected. The MC duration is also correlated well with these two quantities. Φ_O shows marked variations throughout the mission, but has no obvious trend. N _P, B _IMF, V _MC, Q′, and V _SW are all quite steady over the full 18 years and have markedly low relative variation. Concerning vector quantities, we examine the distribution of MC type for the 18 years, where type refers to the field directional change through a given MC starting at first encounter (i.e., North-to-South, or South-to-North, All South, All North, etc.). Combining all 18 years of MC types shows that the occurrence rate varies strongly across the various MC types, with N-to-S being most prevalent, with a 27 % occurrence rate (of all MCs), and S-to-N being second, with a 23 % occurrence. Then All N and All S come next at 16 % and 10 % occurrence rate, respectively. All others are at 7 % or lower. For the variation of MC types with time, the southern types (i.e., those that start with a southern magnetic field, a negative B _Z in geocentric-solar-ecliptic coordinates) decrease, as the northern types (i.e., those that start with a northern field) increase, apparently consistent with the specific timing of the polarity change of the solar magnetic field, as predicted by Bothmer and Rust (in Crooker, N., Joselyn, J., Feynman J. (eds), Geophys. Monogr., 139 –  146, 1997).

Appendices

Appendix A: Formulation Used in Magnetic-Cloud Fitting

A useful and generally adequate analytical approximation of the magnetic field of the MCs considered here is the static, constant-α, force-free, cylindrically symmetric configuration (Burlaga 1988; Goldstein 1983), given by the Lundquist solution

$$ \nabla^{2} \mathbf{B} = - \alpha^{2} \mathbf{B}, $$

(10)

which results from assuming J=α B and the use of Maxwell’s equations (Lundquist 1950), which we express in the International System (SI) of units. Other MC models have considered the possibility that MCs expand as they move away from the Sun (Burlaga et al. 1981; Farrugia et al. 1992; Osherovich, Farrugia, and Burlaga 1993; Marubashi 1997; Berdichevsky, Lepping, and Farrugia 2003) and/or the possibility of a violation of cylindrical symmetry (e.g., Vandas, Romashets, and Watari 2005; Marubashi and Lepping 2007). We fit the Lundquist (1950) solution of Equation (10) to averages of B (in GSE coordinates). This is a Bessel-function solution for the separate cylindrical field components, axial, tangential, and radial, as

$$ B_{\mathrm{A}} = B_{\mathrm{O}} J_{0} (\alpha r),\qquad B_{\mathrm{T}} = B_{\mathrm{O}} H J_{1}(\alpha r),\quad \mbox{and}\quad B_{\mathrm{R}} = 0, $$

(11)

where for a given MC the values for B _O, α, and H are to be determined, along with four other parameters described below. The seven MC fit-parameters are

• B _O, H, R _O, all defined in Table 1, and

• φ _A,θ _A, the longitude and latitude, respectively, of the MC axis (GSE coordinates)

• t _O, the center time (i.e. the closest approach time); and

• Y _O, the closest approach (CA) distance (often called the impact parameter) and generally given below as a percentage of R _O.

A reduced χ ² deviation to the fit, χ ²/(3N−n) (called $\chi_{\mathrm{R}}^{2}$), where N is the number of time-averaged points and n=5 is the number of parameters (R _O, φ _A, θ _A, t _O, and Y _O) in this part of the fitting, is calculated to help us measure the quality (Q _O) of the fit. The χ ² parameter is dimensionless since the magnetic field is normalized to unity at this point. Note that ideally we choose the boundary of the MC such that the magnetic field becomes purely azimuthal there, i.e. where r=R _O. Hence, the argument of the Bessel functions at that point is αr=2.40 (i.e. R _O=2.40/α at the MC boundary), to three-place accuracy. See Appendix A of Lepping, Berdichevsky, and Ferguson (2003, 2004) for details on how the MC parameter fitting technique is carried out. The analysis interval chosen for the fitting procedure is the MC duration (see Table 1). By allowing t _O to be a free parameter, the model will attempt to find a best estimate of the center time, even in cases where ΔT/2 may not have been well chosen. In many cases any natural distortion of the MC will cause t _O and ΔT/2 to differ. We measure the difference by defining a percentage asymmetry factor (asf) by

$$ \operatorname{asf} = \bigl|(1 - 2t_{\mathrm{O}} /\Delta T)\bigr| \times100. $$

(12)

Note that a full measure of the quality of the MC’s parameter fitting depends on many factors as well as on |χ _R|, as described in Appendix A of Lepping et al. (2006), which defines quality as Q _O. The size of the asf is one of the most important considerations in defining Q _O, which can be 1 (excellent), 2 (fair), or 3 (poor).

Since J=α B, the magnitude the axial current density, J _O=αB _O, and therefore,

$$ J_{\mathrm{O}} = 2.40 B_{\mathrm{O}} /R_{\mathrm{O}}, $$

(13)

according to our choice of boundary, α=2.40/R _O, given above. Since B _O and R _O are quantities that are estimated directly by the data-fitting process, J _O is a direct by-product of the fitting process. The axial magnetic flux, Φ_O, is defined as

$$ \Phi_{\mathrm{O}} = \int\mathbf{B} \cdot\mathrm{d}\mathbf{S} = B_{\mathrm{O}} \int J_{0} (\alpha r)2\pi r\,\mathrm{d}r = 1.36 B_{\mathrm{O}} R_{\mathrm {O}}^{2}. $$

(14)

Furthermore, with this model it is easy to show that the formula for total axial current is

$$ I_{\mathrm{T}} = 3.88 R_{\mathrm{O}} B_{\mathrm{O}} $$

(15)

(e.g. Lepping et al. 1996). We also use the MC duration (ΔT) directly to make a second estimate of the MC radius (called R _ΔT ), as well as the model estimate (R _O), by the following

$$ R_{\Delta T} = \bigl[Y_{\mathrm{O}}^{2} + \bigl[(V \Delta T/2)(\sin \beta_{\mathrm{CA}})\bigr]^{2}\bigr]^{1/2}, $$

(16)

where V is the average speed of the MC, and β _CA is the cone angle given by Lepping et al. (2006) as

$$ \beta_{\mathrm{CA}} = \cos^{-1}\bigl|(\cos \varphi _{\mathrm{A}} \cos \theta_{\mathrm{A}})\bigr|. $$

(17)

We can then make a relative check of the reliability of our estimate of the MC radius by comparing the results of Equation (16) with the model radius (R _O). We define this comparison by

$$ \operatorname{Check} = (R_{\Delta T} - R_{\mathrm{O}} )/R_{\mathrm{O}}, $$

(18)

which is usually given as a percentage. Obviously, smaller Checks are desired.

Appendix B: Data Simulations Showing the Differences Between c.c. and S

Here we show eight simulated data sets, with 18 points each (i.e., using the same number as the actual data sets in Section 3), of generally very different characteristics, to gain a better understanding of the differences between c.c. and S, especially in terms of the magnitude of Variations in the quantities. All quantities are defined in Section 2; note that all quantities are positive at all points, consistent with our actual physical data. The simulated data are defined analytically by the following:

$$ \begin{aligned}[c] &\mathrm{F}1_{i} = 10 + 200 \sin\bigl[80^{\circ} f(i) + 80^{\circ}\bigr]&&\mbox{(part of a sinusoid)},\\ &\mathrm{F}2_{i} = 50 + 700 \sin\bigl[90^{\circ} f(i)\bigr]&& \mbox {(part of a sinusoid)}, \\ &\mathrm{F}3_{i} = 400 - 8 i && \mbox{(straight line of negative slope, with large}\\ &&&\mbox{intercept)}, \\ &\mathrm{F}4_{i} = 4 + 9 i&& \mbox{(straight line of positive slope, with small}\\ &&&\mbox{intercept)}, \\ &\mathrm{F}5_{i} = 200 + 0.000004 i&& \mbox{(almost constant)}, \\ &\mathrm{F}6_{i} = 300 + 0.000004 i&& \mbox{(almost constant, but different from F5),} \\ &\mathrm{F}7_{i} = 550 - 100 \sin\bigl[75^{\circ}(17-i) +40^{\circ}\bigr],&& \mbox{(3.5 cycles of a sinusoid)},\\ &\mathrm{F}8_{i} = 100 + 25 i&& \mbox{(straight line of different positive slope and}\\ &&&\mbox{intercept from F4)}, \end{aligned} $$

(19)

for i: 0,…,17, and where f(i)≡(17−i)/17. The data are plotted in Figure 7. The resulting values of c.c., S and Variation are given in Table 6. First, we stress that when Variation is small for both quantities in a pair, as it is for pair F1:F7 in Table 6, for example, its respective S value (a relatively high value 0.706 in this case) must be high, even if the c.c. value is low (near zero in this case), and therefore S is less useful for measuring tracking in such cases. There are about ten cases in this table where both Variations are small (i.e., 0.261 or less, and sometimes much less) and in every such case S is large, or relatively large, i.e., all are above 0.700 except one for pair F1:F3 with S=0.626. Especially interesting are pairs with one large Variation and one small/medium one. Consider, for example, case F1:F4 with large c.c.=0.948, medium S=0.651 and with a large Variation on F4 of 0.597 and small/medium Variation on F1 of 0.261. Intuitively, it appears that the c.c. tracks better than S, in this case. By comparison, case F1:F5 shows a reversal in c.c. (= 0.480) and S (= 0.747), where the Variation dropped from 0.597 to 0.000 from F4 to F5, again both Variations are small/moderate. An interesting case is F4:F8 where the c.c. is unreasonably high at 1.000 and S has a good (and gives a useful) value of 0.835; note that both Variations are quite high at about 0.60 and 0.43, respectively.

Figure 7

Plots of simulated data using 18 points with the purpose of examining how c.c., S, and Variation relate to each other (see Equations (8), (4), and (7), respectively).

We should point out that all straight-line comparisons, or nearly straight lines, give c.c.s of near 1.000 or −1.000, and therefore give limited information, which is essentially only the difference in sign of the slopes of the straight-lines of this case. But the S value may give some indication of the relative differences of the lines. For example, consider F3:F4 and F3:F8, as expected, the c.c.s are both −1.000, but the S value is significantly higher for F3:F8, i.e. the Ss are 0.295 vs. 0.460, respectively. Because even though neither S is especially very good, at least the S for F3:F8 shows that the elements of this pair are nearly at the same level. There are many cases in Table 6 where S and c.c. give relatively similar results, at least semi-quantitatively. Some examples of this are F4:F6, F5:F8, F6:F8, and F2:F3, where we have ignored all negative c.c. cases. In all of these cases there is a big difference in the Variation of the two elements of the pair. However, F4:F8 is a good example of similar c.c. and S, with c.c.=1.000, S=0.835 (good), but with similar Variations, i.e., $\operatorname{Variation}X = 0.597$ and $\operatorname{Variation}Y= 0.427$.

It is important to realize that for S to have a chance of being high or moderately high, and have a high information content (meaning that it is not simply due to low Variations for both quantities in the pair), both Variations must be at least similar in value and (in an absolute sense) these values should be between moderate and high.

All aspects considered, the c.c., although limited in its value for tracking, is usually better than S in determining how similar two quantities are. But it can be misleading when it is near 1.000 or −1.000 and has no ability to compare the average of the two elements of the pair.