Potential coauthors: Maria-Theresia Walach, Kyle Murphy...
The solar wind interplanetary electric field $(E_M)$ (Stauning, 2013) and auroral electrojet index $(AE)$ are found to have linear dependence that saturates at high electric field values (Weimer et al., 1990). The electric field drives day-side reconnection, which determines the amount of energy stored in the magnetotail. While the auroral electrojet can be thought of as currents driven by this magnetic energy as a response of the Magnetosphere-ionosphere system to the above driving.
Weimer et al., 1990 identified the above result by calculating $<AE|E_M>$ vs. $E_M$using data from 1978 to 1985, corrected for seasonal variations. Error bars on $<AE|E_M>$ is derived from the standard error of the quantity $AE|E_M$, which is a measure of its variance. The expectation of this quantity is assumed to provide variation of $AE$ with $E_M$, as it smooths out the random noise in $AE$ caused by the inaccuracies in the measurement method (e.g., the auroral electrojet position might vary relative to the magnetometer arrays measuring the current).
The posterior distribution $f_{AE|E_M}(AE|E_M)$ contains all knowledge we have about $AE$, and hence the best estimate of $AE$, $\hat{AE}$ is the $<AE|E_M=e_m>$ in terms of minimizing the mean squared error. And for this reason, the above method is good in estimating the response of $AE$ to $E_M$, with a caveat that there is no uncertainty in $E_M$ (any ref?).
However, since $E_M$ is measured at the L1 point, about 200 $R_E$ away from the nose of the magnetopause, there is some uncertainty in $E_M$. If this uncertainty is large, then $<AE|E_M=e_m>$ is no longer a good estimate of $AE$. We demonstrate how we can go about getting a better estimate, by measuring the variance in $E_M$, variance in $AE$, $<AE|E_M>$ and $<E_M|AE>$. [See Methods section].
In fact, if the signal-to-noise ratio (SNR) or mean to variance ratio of $E_M$ is large and constant, we can show that the response of $AE$ to $E_M$ is near-linear, and likely does not saturate (See Figure 2).
Note: The main source of error in the measurement of $E_M$is spatio-temporal variations in the solar wind that leads to the satellite at L1 measuring different values than the actual electric field at the nose of the magnetopause that is driving the magnetosphere. And it seems that the error in $E_M$ reduces at high values of $E_M$. This could be because at high driving, the solar wind structures become larger, and spatio-temporal variance decreases. As a result, the above effect of more linear-response is probably visible only at low or moderate $E_M$, where the true response of $AE$ is different from what $<AE|E_M>$ suggests due to the larger variance in $E_M$. And conversely, at high $E_M$, $AE$ saturates as described by Weimer et al., 1990. This needs to be verified since the error estimates we have are not constant for a particular $E_M$ magnitude. They are probabilistic.
Using OMNI and superMag database, we calculate $E_M$ and $SML$ (a proxy for the auroral electrojet index) respectively for each minute during the years 1995 to 2019. Using this data we can calculate $<E_M|SML>$ and $<SML|E_M>$.
Instead of plotting the expectations, if we plot the raw data, you see what appears to be a saturation of $SML$ (~2000 $mA/m^2$) towards high $E_M$ values (25-40 $mV/m$). And it is clear to see that the conditional expectations $<E_M|SML>$ and $<SML|E_M>$ will be different from each other, particularly due to the variance in either quantity given one of them as a conditional variable (i.e. variance in $E_M|SML$ and $SML|E_M$.
