Wind Magnetic Clouds for 2010 – 2012: Model Parameter Fittings, Associated Shock Waves, and Comparisons to Earlier Periods

Abstract

We fitted the parameters of magnetic clouds (MCs) as identified in the Wind spacecraft data from early 2010 to the end of 2012 using the model of Lepping, Jones, and Burlaga (J. Geophys. Res. 95, 1195, 1990). The interval contains 48 MCs and 39 magnetic cloud-like (MCL) events. This work is a continuation of MC model fittings of the earlier Wind sets, including those in a recent publication, which covers 2007 to 2009. This period (2010 – 2012) mainly covers the maximum portion of Solar Cycle 24. Between the previous and current interval, we document 5.7 years of MCs observations. For this interval, the occurrence frequency of MCs markedly increased in the last third of the time. In addition, over approximately the last six years, the MC type (i.e. the profile of the magnetic-field direction within an MC, such as North-to-South, South-to-North, all South) dramatically evolved to mainly North-to-South types when compared to earlier years. Furthermore, this evolution of MC type is consistent with global solar magnetic-field changes predicted by Bothmer and Rust (Coronal Mass Ejections, 139, 1997). Model fit parameters for the MCs are listed for 2010 – 2012. For the 5.7 year interval, the observed MCs are found to be slower, weaker in estimated axial magnetic-field intensity, and shorter in duration than those of the earlier 12.3 years, yielding much lower axial magnetic-field fluxes. For about the first half of this 5.7 year period, i.e. up to the end of 2009, there were very few associated MC-driven shock waves (distinctly fewer than the long-term average of about 50 % of MCs). But since 2010, such driven shocks have increased markedly, reflecting similar statistics as the long-term averages. We estimate that 56 % of the total observed MCs have upstream shocks when the full interval of 1995 – 2012 is considered. However, only 28 % of the total number of MCLs have driven shocks over the same period. Some interplanetary shocks during the 2010 – 2012 interval are seen to apparently occur without an obvious MC-driver, probably indicating an encounter with a distant flank of a MC-driven shock. Some of these may be driven by a different kind of structure, however.

Appendix: Formulation Used in Fitting a Magnetic Cloud

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

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

(3)

which results from assuming $\boldsymbol{J} = \alpha \boldsymbol{B}$ and the use of Maxwell’s equations (Lundquist 1950), and 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 (3) to averages of $\boldsymbol{B}$ (in GSE coordinates). This is a Bessel function solution for the separate cylindrical field components (i.e. in terms of axial (A), tangential (T), and radial (R) components) by the following:

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

(4)

where for a given MC the values for $B_{0}$, $\alpha$, and $H$ are to be determined, along with four other parameters described below. The seven MC fit parameters then are

• $B_{0}$, $H$, $R_{0}$, all defined in Table 1, and

• $\varphi _{\mathrm{A}}$, $\theta_{\mathrm{A}}$, the longitude and latitude, respectively, of the MC axis (GSE coordinates)

• $t_{0}$, the center time (i.e. the closest approach time); and

• $Y_{0}$, the closest approach [CA] distance (often called the impact parameter) and generally given below as a percentage of $R_{0}$.

A “reduced” $\chi^{2}$-deviation to the fit, $\chi^{2} /(3N - n)~(\equiv \chi_{\mathrm{R}}^{2})$, where $N$ is the number of time-averaged points and $n = 5$ is the number of parameters (being $R_{0}$, $\varphi _{\mathrm{A}}$, $\theta_{\mathrm{A}}$, $t_{0}$, and $Y_{0}$) in this part of the fitting, is calculated to help us measure the quality [$Q_{0}$] of the fit. The $\chi^{2}$-parameter is dimensionless since the magnetic field was unit normalized up to this point; $| \chi_{\mathrm{R}} | \equiv [ \chi^{2} /(3N - n)]^{1/2 }$ is displayed. We note that ideally we choose the boundary of the MC such that the magnetic field becomes purely azimuthal there, i.e. where $r = R_{0}$. Hence, the argument of the Bessel functions at that point is $\alpha\mathrm{r} = 2.40$ (i.e. $R_{0} = 2.40/\alpha$ at the MC boundary), to three-place accuracy. The analysis interval chosen for the fitting procedure is the “duration” (see Table 1). By allowing $t_{0}$ to be a free parameter, the model will attempt to find a “best estimate” of the center time, even in cases where $\Delta T/2$ may not have been well chosen. In many cases, any natural distortion of the MC will cause $t_{0}$ and $\Delta T/2$ to differ. We measure the difference by defining an asymmetry factor [asf] by

$$ \mathrm{asf} = \bigl|(1 - 2t_{0} /\Delta T)\bigr| \times 100~\%. $$

(5)

We note that a full measure of the “quality” of the MC parameter fitting depends on many factors as well as on $|\chi_{\mathrm{R}}|$, as described in Appendix A of Lepping et al. (2006), which defines quality, called $Q_{0}$. The size of the asf is one of the most important considerations of these other parameters in defining $Q_{0}$, which can be 1 (excellent), 2 (fair), or 3 (poor).

Since $\boldsymbol{J} = \alpha \boldsymbol{B}$, then the axial current density ($J_{0} = \alpha B_{0}$) and therefore,

$$ J_{0} = 2.40 B_{0} /R_{0}, $$

(6)

according to our choice of boundary, $\alpha = 2.40/R_{0}$, given above. Since $B_{0}$ and $R_{0}$ are quantities that are directly estimated by the data-fitting process, $J_{0}$ is a direct by-product of the fitting process. The axial magnetic flux [$\Phi_{0}$] is defined as

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

(7)

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

$$ I_{\mathrm{T}} = 3.88 R_{0} B_{0} $$

(8)

(e.g. Lepping et al. 1996). We also directly use the duration [$\Delta T$] of the spacecraft passage to make a second estimate of C’s radius [called $R_{\Delta T}$], as well as the model estimate [$R_{0}$], by the following

$$ R_{\Delta T} = \sqrt{\bigl[Y_{0}^{2} + \bigl[(V_{\mathrm{MC}} \Delta T/2) (\sin \beta_{\mathrm{CA}}) \bigr]^{2}\bigr]}, $$

(9)

where $V_{\mathrm{MC}}$ is the average speed of the MC and $\beta$ _CA, the cone angle, is given by

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

(10)

(Lepping et al. 2006). We can then make a relative check of the reliability of our estimate of the MC radius by comparing the results of Equation (9) with the model radius [$R_{0}$]. We define this comparison by

$$ \mbox{``Check''} = (R_{\Delta T} - R_{0})/R_{0}, $$

(11)

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

Finally, we find it useful to define a magnetic cloud (Cl) coordinated system In the Cl system the $\boldsymbol{X}_{\mathrm{Cl}}$-axis is along the MC axis, positive in the direction of the positive polarity of the axial magnetic field, the $\boldsymbol{Z}_{\mathrm{Cl}}$-axis is the projection of the trajectory of the spacecraft (relative to the MC velocity, which is approximately along the $\boldsymbol{X}_{\mathrm{GSE}}$-axis) onto the cross section of the MC, and $\boldsymbol{Y}_{\mathrm{Cl}} = \boldsymbol{Z}_{\mathrm{Cl}} \times \boldsymbol{X}_{\mathrm{Cl}}$. Ideally then, $\langle B_{\mathrm{X}}\rangle_{\mathrm{Cl}}$ (i.e. the average of $B_{\mathrm{X}, \mathrm{Cl}}$) should always be positive and $\langle B_{\mathrm{Y}} \rangle_{\mathrm{Cl}}$ should be zero (or near zero in practice) because of the fundamental field structure of the force-free model (solutions given by Equation (4)) and this definition of the Cl coordinate system.