GoKawiilThe most common symbols used are explained in the Extended Data Table 2.
Solar wind measurement uncertainty
There are several solar wind coupling functions that attempt to quantify the driving, one important one is the solar wind merging electric field or merging geoeffective field \({E}_{{\rm{m}}}^{* }\) (refs. 4,24).
$${E}_{{\rm{m}}}^{* }={V}_{\mathrm{sw}}{B}_{{\rm{T}},\mathrm{sw}}{\sin }^{2}({\theta }_{\mathrm{sw}}/2)$$ (1.1)
This field is also known as the Kan–Lee electric field and is calculated from solar wind parameters at Lagrange point L1 alone24. In equation 1.1, V sw is the solar wind speed in km s−1, \({B}_{{\rm{T}},\mathrm{sw}}={({B}_{y}^{2}+{B}_{z}^{2})}^{1/2}\) is the transverse magnitude of the interplanetary magnetic field (IMF) in nT, and θ sw = tan−1(B y /B z ) is the transverse IMF clock angle in radians. E m is a positive-valued quantity with the units of the electric field (mV m−1), reflecting the assumption that energy flows only from the solar wind to the magnetosphere. The above quantities are measured in geocentric solar magnetospheric (GSM) coordinates using spacecraft orbiting the L1 point approximately 230 R E upstream from Earth. The GSM coordinates are convenient for studying the effects of the IMF components on magnetospheric and ionospheric phenomena. Its x axis points towards the Sun, and the z axis is the projection of Earth’s magnetic dipole axis onto the plane perpendicular to the x axis. For a purely southwards IMF, the merging electric field \({E}_{{\rm{m}}}^{* }\) is equal to the solar wind electric field E sw = V x,sw B z , where V x,sw is a component of the solar wind velocity along the x direction.
The dawn–dusk portion of the shocked solar wind convection electric field maps along the equipotential magnetic-field lines and drives the plasma convection down in the polar cap ionosphere. The convection corresponds to an electric field across the polar cap in the rest-frame of the Earth (E PC ). The PCI is a measure of this field and part of the energy input into the Earth’s magnetosphere. The index has gone through many iterations over the past 50 years, but it is essentially the maximum amplitude of variations observed from magnetometers near the South and North Poles3,25,26. The current version of the index is scaled on a statistical basis of magnetic variations to the merging electric field, such that the index is highly correlated to the merging electric field. This makes the PCI independent of daily and seasonal variations and the local ionospheric properties. PCN is the index derived from the magnetometer in the northern polar cap and PCS is derived from the southern polar cap. We use the PCI as a proxy for E PC (that is, PCI ≈ E PC ), as it combines both indices to provide a positive-valued index, which is more accurate than the individual indices26. PCI = 0.5(PCS + PCN), with a condition that negative values of either PCS or PCN are set to 0. PCI or E PC also has the units of the electric field, and several statistical analyses in the literature reveal that it is linearly proportional on average to the electric potential across the polar cap, called the cross-polar cap potential11,27. Fluctuations in PCI can occur owing to nightside magnetosphere processes28, but the above relation is still maintained on average.
The Kan–Lee electric field \({E}_{{\rm{m}}}^{* }\) is only an approximation of the true driver of the magnetosphere, that is, the shocked solar wind plasma \({E}_{{\rm{m}}}^{\mathrm{sh}}={V}_{\mathrm{sh}}{B}_{\mathrm{sh}}{\sin }^{2}({\theta }^{{\prime} }/2)\), the value of which is not easily available to us (Supplementary Methods 2a). This leads to the uncertainty and nonlinear regression bias discussed in this work. Instead, we have only an erroneous estimate of the true coupling function approximately propagated to the polar cap, \({E}_{{\rm{m}}}^{* }\). Hence, we need to reinterpret the literature as saying that low values of the erroneous estimate of the solar wind strengths \({E}_{{\rm{m}}}^{* }\) (not \({E}_{{\rm{m}}}^{\mathrm{sh}}\)) correlate linearly with the polar cap potential. And at high values of \({E}_{{\rm{m}}}^{* }\), the polar cap potential saturates.
We calculate \({E}_{{\rm{m}}}^{* }\) by using WIND satellite measurements published in the OMNIWeb database29. The database provides the values corrected for the propagation delay of the solar wind from L1 to the bow-shock nose30,31,32. Then as frequently done, we apply a further correction of a constant delay of about 17 min to account for the propagation delay from the nose to the polar cap ionosphere16,33. Previous literature that discusses the polar cap potential saturation problem estimates \({E}_{{\rm{m}}}^{* }\) similarly. We use the WIND data from 1995 to 2019 and the polar cap indices from the same time range. Both data are 1-min averages; larger time averages will lead to additional uncertainties34. We only use data samples when both WIND measurements and polar cap indices are available.
\({E}_{{\rm{m}}}^{* }\) can differ from the shocked solar wind driver in the magnetosheath \({E}_{{\rm{m}}}^{\mathrm{sh}}\) in two ways. One is through a consistent deterministic bias that is determined by the bow-shock that slows down the plasma. This deterministic bias is nearly zero as the tangential (and the largest) component of the electric field across the bow-shock remains unchanged. We do not concern ourselves with this deterministic bias, as it will manifest in the data as a physical effect anyway. However, \({E}_{{\rm{m}}}^{* }\) can also differ randomly from the true solar wind driver \({E}_{{\rm{m}}}^{\mathrm{sh}}\), owing to fluctuations in the plasma properties caused by physical processes between L1 and the reconnection site, acting in random directions. These random fluctuations contribute to the uncertainty in \({E}_{{\rm{m}}}^{* }\) when we use it as a proxy for the true driver \({E}_{{\rm{m}}}^{\mathrm{sh}}\). This random error is concerning, as it does not average away, and manifests as a regression bias in data analysis and is easily confused as a physical effect, hence we calculate and correct for it.
We categorize the random uncertainty in \({E}_{{\rm{m}}}^{* }\) relative to the true value \({E}_{{\rm{m}}}^{\mathrm{sh}}\) into three primary sources.
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