Summary

Context :

Goal :

Utility :

Method :

Features of the Earth's magnetosphere such as 1) the position of the magnetopause, 2) the size of the polar cap, and 3) the hemispherical auroral precipitation flux are primarily driven by the interaction of solar wind with the Earth's magnetic field, and the flow of plasma and energy within the magnetosphere-ionosphere system. However, decades of analyses suggest that solar wind parameters$^1$ have a causal connection to this magnetospheric properties$^2$. For example, the strength and orientation of the solar wind magnetic field $B_{IMF}$ seem to control the day-side reconnection rate, and the driver function $v_{sw} B_z$ seems to control the amount of energy stored in the magnetotail. Many researchers have analyzed the individual effect of these parameters on different aspects of the magnetosphere and mapped out how for example auroral precipitation changes with the driver function [Citation - Newell].

• We'd like to derive driving parameters that affect these features of the Earth's magnetosphere
• We will come up with state parameters that describe features of the magnetosphere (e.g., magnetopause distance, or magnetopause direction of motion)
• So that we can predict these features by only knowing the solar-wind parameters
• We don't know yet if we can use just Canonical Correlation Analysis to determine the magnetosphere state parameters, or do we need to use other techniques.
• We want to know whether these state parameters include parameters with time history, or are they independent of time history?

The issue of correlated driving parameter & multiple driving parameters leading to the same effect:

However, these parameters are correlated with each other, and it isn't easy to directly study the effect of each parameter on the magnetosphere. Coming to conclusions based on such studies that analyze the effect of one driving parameter on the magnetosphere, keeping all else constant or varying - ignores the contributions from the other parameters. Unfortunately, this will add uncertainty to our conclusions. One way to address this problem is to use canonical correlation analysis (CCA) to derive composite indices from the driving parameters uncorrelated with each other but correlate with the properties of the magnetosphere. CCA is a technique that allows us to evaluate the degree of correlation between two multivariate data sets by rotating the principle-coordinate representation of the parameters. This change in coordinates in the driving parameter-space will provide a way to naturally evaluate the effect of the solar wind parameters on the magnetosphere without making unrealistic assumptions on how the system evolves. An additional reason why studying the effect of single driving parameters and their effect on the magnetosphere is challenging is that a combination of different driving parameters can have the same effect. CCA condenses all that set of parameters with the same effect on the magnetosphere into one set of composite indices.

Utility of composite indices?

The construction of these composite indices will depend on the aspect of the magnetosphere we decide to focus on. For example, it is possible that we can derive composite indices from solar wind parameters that correlate the most with the magnetopause location and the polar-cap size. The output will be a set of composite solar indices $S_1$ and $S_2$ that correlate with $E_1$ and $E_2$ respectively, such that $S_1 \text{ and } S_2$ are a function of solar wind parameters and uncorrelated with each other, while $E_1 \text{ and } E_2$ are a function of magnetopause location and polar-cap size and uncorrelated with each other.