5 Solar Wind Coupling Functions, the Importance of Averaging and Allowance for Data Gaps

As discussed in Section 2.3, geomagnetic activity is enhanced when the northward component of the interplanetary magnetic field (IMF), in the GSM frame of reference, BZ, is increasingly negative. As a result, a half-wave rectified form of BZ is often used to predict geomagnetic activity, such as Bs, where Bs = − BZ when BZ ≤ 0 and Bs = 0 when BZ > 0. Because Bs is discontinuous in slope around Bs = 0, a form such as B sin4(𝜃∕2) is often preferred, where the IMF “clock angle” 𝜃 = tan −1(B ∕B ) Y Z, B Y being the dawn-to-dusk component of the IMF in GSM. This has a very similar form to Bs at large |Bs | but is continuous in slope around zero. The power density in the solar wind at Earth is dominated by the kinetic energy of the bulk flow of the particles and so the square of the solar wind velocity, VSW, is another important factor. Hence a simple “coupling function”, designed to quantify the effect of the solar wind on geomagnetic activity, is 2 Bs × V SW. A great many such coupling functions have been proposed and tested. One widely-used example is the “epsilon parameter”, 2 4 2 𝜖 = (4π ∕μo)VSWB sin (𝜃 ∕2)lo where μo is the magnetic permeability of free space and lo is a scaling factor that allows for the cross-sectional area of the geomagnetic field presented to the solar wind. In practice this area reduces with increased solar wind dynamic pressure (P = m N V 2 SW SW SW SW, where m SW is the mean solar wind ion mass and NSW is the number density of solar wind ions) but a constant value of lo = 7 RE, where RE is a mean Earth radius, is often used. However 𝜖 is based on the energy density in the solar wind hitting the Earth’s space environment being in the form of Poynting flux, which is not correct because by far the largest energy density in the undisturbed solar wind is in the form of the ions’ bulk-flow kinetic energy (which is converted into Poynting flux by the currents that flow in Earth’s bow shock and magnetopause, see Cowley, 1991; Lockwood, 2004Jump To The Next Citation Point). A correct version was provided by Vasyluinas et al. (1982Jump To The Next Citation Point) who applied dimensional analysis as well as the energy flow equations and used pressure balance on a hemispherical dayside magnetopause to compute lo. From this they derived the coupling function P α, which is the power coupled into the magnetosphere. It is the product of the power density in the solar wind, times the cross sectional area of the magnetosphere presented to the solar wind, times the fraction of the incident power that crosses the magnetopause, tr
3 2 P α = (mSWNSWV SW )∕2 × (πlo) × tr . (1 )

From pressure balance at the nose of the magnetosphere, and assuming the dayside magnetosphere is hemispherical in shape we have

[ 2 ]1∕6 lo = k1 M E ∕(PSW μo) , (2 )
where ME is the Earth’s magnetic moment and k1 is the blunt-nose shape factor for flow around the magnetosphere. Vasyluinas et al. (1982) noted that the transfer function tr must be dimensionless and proposed a form
−2α 4 tr = k2M A sin (𝜃∕2), (3 )
where α is a free fit parameter which arises from the unknown dependence of the coupling on the solar wind Alfvén Mach number, MA. Combining Equations (1View Equation), (2View Equation), and (3View Equation) yields
[ (1∕3−α) ] 2∕3 (2∕3−α) 2α (2∕3−α) (7∕3−α) 4 P α = k1k2π∕(2μ o ) M E m SW B NSW VSW sin (𝜃∕2). (4 )
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Figure 8: Correlations between the Am geomagnetic activity index and a number of solar wind coupling functions, as a function of averaging timescale, T. Upper and lower graphs are identical, other than that the upper graph displays timescale linearly, whereas the lower graph displays it logarithmically. The coloured lines give the results for: P α (dark blue), V2SWB (light blue), V 2SWBS (green), 𝜖 (red), VS2W (olive), PSW (magenta), and B (black). Image reproduced by permission from Finch and Lockwood (2007Jump To The Next Citation Point), copyright by EGU.

Finch and Lockwood (2007Jump To The Next Citation Point) studied the interplanetary medium coupling functions P α, 2 V SWB, V 2SWBS, 𝜖, VS2W, PSW, and B by evaluating their correlations with various geomagnetic activity indices on a range on averaging timescales, T between 1 day and 1 year. The results for the Am global index are shown in Figure 8View Image.

A factor that should not be neglected is that although the geomagnetic data are essentially continuous, the same is far from true of the interplanetary data. Since 1995 the WIND and the ACE spacecraft have provided almost 100% coverage, but before then coverage had sometimes been lower than 50% in any one year. Finch and Lockwood (2007Jump To The Next Citation Point) showed that ignoring these data gaps has a considerable effect at given averaging timescale T and can even change which coupling function performs best. Hence before making the correlation, Finch and Lockwood (2007) piece-wise removed both interplanetary and geomagnetic data for which there was a gap in the interplanetary data of duration one hour or greater during a 3-hour geomagnetic data interval (allowing for the predicted propagation lag between the interplanetary monitoring spacecraft and the dayside magnetopause). Figure 8View Image shows that for the full range of T, P α (as given by Equation (4View Equation) and shown in dark blue) performs best for the range index Am, although for T > 27 days the much simpler function 2 VSWB (light blue) gives correlations as high (or even slightly higher). The 𝜖 parameter (red) performs as well as VS2WB (but less well than Pα and VS2WBS) at low T and is considerably poorer at high T, reflecting its nonphysical basis. For the case of Am shown, functions that do not combine both IMF and solar wind speed (V 2 SW in olive, P SW in magenta, and B in black) do not perform as well as those that do. There has been much discussion about the precise form of coupling function that performs best, but these discussions almost invariably neglect the facts that this conclusion depends on T and on which activity index is considered. The importance of this is discussed in Section 6. Note that in Figure 8View Image the high-performing indices P α and V 2 B SW reach correlation coefficients near 0.97 at T = 1 yr and that these do not fluctuate with the precise T value used to anything like the same extent as do the others.

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Figure 9: The northward component of the near-Earth interplanetary magnetic field (IMF) in the GSM frame, BZ, from the OMNI2 composite dataset. The left-hand plot shows the temporal variations of ⟨BZ ⟩T for 1966 – 2013 at different averaging timescales, T. The right-hand plot shows the corresponding probability density functions over the same interval. Note the difference in the vertical scales. In both panels light blue is for T = 1 h, dark blue for T = 1 day, red for T = 27 days and black for T = 1 yr.

One noticeable feature of Figure 8View Image is that for large T, 2 VSWB performs as well as P α which is interesting because it does not contain the sin4(𝜃∕2 ) IMF orientation factor that equation 4View Equation shows is part of Pα. Figures 9View Image and 10View Image explain why this is the case. The left-hand panel of Figure 9View Image shows the northward component of the IMF in the GSM reference frame, BZ between 1966 and 2012 (inclusive). The different colours identify the timescale T on which the data are averaged before they are plotted. For T = 1 h (light blue), values vary between –30 nT and +30 nT (off scale), and periods of smaller and larger excursions, both positive and negative, are seen (corresponding to the peaks and minima of the solar cycle). The same is true for T = 1 day (dark blue) but the range of variation is reduced as intervals of opposite B Z cancel to a great extent for the larger T. For T = 27  days (in red) the fluctuation level is very small and it has almost disappeared for T = 1 yr (black). The distributions of BZ values over the interval are shown for each case in the right-hand plot. It can be seen that the averaging almost completely removes the orientation factor such that BZ tends to zero for large T.

The effect of averaging is demonstrated by Figure 10View Image which shows scatter plots of 4 B sin (𝜃∕2 ) as a function of the IMF magnitude B for the 1966 – 2012 data. Part (a) is for hourly observations. It can be seen that there is large scatter between B sin4(𝜃∕2) = B (when the field points directly southward so sin4(𝜃∕2) = 1) and B sin4(𝜃∕2) = 0 (when the field points directly northward so sin4(𝜃∕2) = 0). Part (b) is the same for 1-year averages. In this case there is a good linear relationship, with some scatter. Hence, on timescales of T = 1 yr, the IMF orientation factor is averaged out and the average southward IMF component and, hence, the level of geomagnetic activity, is proportional to B, as first noted by Stamper et al. (1999). This is the basic reason that we are able to make deductions about B from geomagnetic activity when averaging is done on annual timescales. There is some information that could be extracted at higher time resolution, but Figure 9View Image shows that even for T = 27 days scatter will be introduced because this T is not sufficient to average out as much of the IMF orientation factor and at yet smaller T this scatter would render the results completely meaningless. In this review we restrict our attention to using T = 1 yr, for which the orientation factor is almost completely averaged out and for which correlations between the better coupling functions and geomagnetic activity of 0.97 can be obtained (as shown by Figure 8View Image). Part (c) of Figure 10View Image shows the distribution of annual values of the ratio B ∕[B sin4(𝜃∕2)]. The mean value of this distribution is 3.251 and the standard deviation is 0.369. This distribution will be used in Section 9.4 in a quantitative analysis of the uncertainties in reconstructions.

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Figure 10: Scatter plots of the half-wave rectified southward component of the IMF in the GSM frame, B sin4(𝜃∕2 ) as a function of the IMF magnitude, B, for (a) hourly observations and (b) 1-year averages. Panel (c) shows a histogram of the distribution of annual values of the ratio 4 B ∕[B sin (𝜃∕2)], which has a mean value of 3.251 and a standard deviation of 0.369. The thin line is the normal distribution with the same mean and standard deviation.

In addition, in order to derive the open solar flux, the modulus of the radial component of the IMF, away from the Sun (Br, which is the same as − BX in the GSM frame) is usually used (see Section 7.3). |Br| can be obtained from B, again because of the effect of averaging. Because in one year roughly as much “Toward” IMF (Br < 0) flux will be seen as “Away” (Br > 0) flux, ⟨Br⟩ T will tend to zero when averaged over T = 1 yr and so |Br |, or some equivalent which does not cancel Toward and Away flux (see Section 7.5), is needed. The orientation angles of the IMF, both the clock angle 𝜃 and the garden-hose angle ζ = tan− 1(BX ∕BY ), vary considerably on short time scales. Figure 9View Image shows that the long-term average of 𝜃 is 90° and that of ζ is given by Parker spiral theory (Parker, 1958, 1963) which predicts for near-Earth interplanetary space

−1 −1 ζ = tan (VSW ∕r1ω) = sin (|Br |∕B ), (5 )
where r1 is the mean Earth-Sun distance (r1 = 1 Astronomical Unit, AU), and ω is the angular rotation velocity of the solar atmosphere with respect to the fixed stars. Equation (5View Equation) shows that the ratio |Br |∕B can be predicted for a given VSW. The left hand panel of Figure 11View Image shows a scatter plot of the values of |Br |∕B predicted by Equation (5View Equation) against the observed values for hourly means (T = 1 h). It can be seen that the large variations of 𝜃 and ζ on hourly timescales mean that there is no relationship between the observed and predicted values. On the other hand, the right hand plot shows the scatter plot for annual means. (Note that the modulus of Br has here been taken of means over 1 day, i.e., ⟨|Br|1d⟩1yr is used). For this timescale there is a linear relationship between the observed and predicted values. The observed values are lower than the predicted ones because of the use of T = 1 day in taking the modulus which means there is some cancellation of Toward and Away field: this issue is discussed further in Section 7.5. Figure 11View Image demonstrates that Parker spiral theory can be used to predict |Br| from B if the solar wind speed VSW is known and an annual averaging timescale is used.
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Figure 11: Scatter plots of the predicted ratio |Br|∕B from Parker spiral theory, where B is the IMF magnitude and |Br| is the modulus of its radial component, as a function of the observed value of that ratio. The left-hand plot is for hourly observations, the right-hand plot for 1-year averages. In the case of the annual means, the modulus of daily means of Br observations, i.e., ⟨|Br |1d⟩1yr, is used.

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