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Planning Project 2: D-region Conductance & Multiple substorms

The formulas for calculating parallel, Hall, and Pedersen conductivity are described in section \ref{section: conductance}. Developed in the 1920s-1950s, they are derived from mean-free path theory and assuming the plasma is in thermodynamic equilibrium. These conductivity's are therefore only accurate to first order. To calculate the ion-neutral collision frequencies ($nu_{in}$) we used the formula $\nu_{in} = K_{in} n_n$, for nonresonant ion-neutral interactions. With $K_in$ values from the table 4.4. in \cite{Schunk2009} multiplied by $10^{-16}$ to convert the coefficients into SI units. And for resonant ion-neutral interactions, we used formulas from table 4.5 in \cite{Schunk2009}. The electron collision frequency $\nu_e$ can be thought of as a contribution from electron-neutral interactions ($\nu_{en}$) and electron-ion interactions ($\nu_{ei} (See equation \ref{eq: electron collision}). The electron-neutral collision frequencies $\nu_{en}$ were derived from table 4.6 of \cite{Schunk2009} and $\nu_{ei}$ from equation 2.29 b of \cite{Kelley2009}, rewritten here as equation

\ref{eq: electron-ion collision}.

$$ \nu_e = \nu_{ei} + \nu_{en} $$

\label{eq: electron collision}

$$ \nu_{ei} = [34 + 4.18 \ln (T_e^3/n_e)] n_e T_e^{-3/2} $$

\label{eq: electron-ion collision}

Notice that these equations depend on neutral density ($n_n$), ion temperature ($T_i$), electron temperature ($T_e$), electron density ($n_e$), and ion concentration ($C_i$). In the D-region, it is possible to measure $n_e$, $T_e$, $T_i$, using ISR. However, we do not fit the D-region spectra here and instead rely on ISR only for estimating the electron density ($n_e$). Algorithms that can reliably fit the ISR power spectra in the D-region ionosphere are being developed at SRI International. Until then, we fall back on using the International Reference Ionosphere Model (IRI-2016) for the ion composition and temperatures, and the Mass Spectrometer Incoherent Scatter model (MSIS-00E) for neutral atmosphere composition. Based on the uncertainties in the neutral atmosphere and collision frequency estimates, conductance estimated by different authors has a variance of about 100\% \citep{Brekke1993}.

All inputs into the formula are based on physical or empirical models, except for the electron density, which is estimated by ISR. As described in \ref{chapter5}, by using the electronically-steerable phased-array radar - PFISR, we can develop a 3-D volumetric profile of conductivity in the lower-ionosphere, which varies with time at a resolution of 15 seconds. Altitude slices of the time-varying 3-D data set produce a 2-D map of conductivity per measurement instance. The process for developing such maps is summarized in figure \ref{fig: ch6 methodology}.

\label{fig: ch6 methodology}

Process of developing small-scale ionosphere conductivity maps from PFISR.

Process of developing small-scale ionosphere conductivity maps from PFISR.

Figure \ref{fig: ch6 methodology}a, shows the beam pattern in the sky of the Sporadic04 mode that uses the barker code to make measurements across the different beam positions several times in 15 seconds. An average of these measurements provides us with estimates of electron density along the beam (26 profiles, one for each beam). These beams are not field-aligned. Since precipitating electrons that cause the ionization are field-aligned, we interpolate the 3-D data set on to a set of points along the field line. This provides us with 26 altitude profiles of electron density, each parallel to the magnetic field line estimated from the IGRF model. After this, the electron density profiles are plugged into the conductivity formulas shown in figure \ref{fig: ch6 methodology} and equations \ref{eq:conductivity_pedersen}-\ref{eq:conductivity_hall}. No direct validation of conductivity estimates is currently possible. However, we verified our estimates for the period with estimates derived by \cite{Yu2018a} independently, and it is within 100\%. This error bar is considered acceptable, given our lack of precise measurements of $T_i$,$T_e$,$N_n$ and $C_i$.

There is an additional issue while estimating the D-region conductivity. The chemical composition of IRI is insufficient, as it does not include D-region chemistry. The D-region has complex chemistry of ions, negative ions, cluster ions with recombination reactions, electron attachment, reciprocal neutralization, and electron detachment (See section \ref{section: ionosphere chemistry}). Since we do not have reliable estimates of the collision frequencies of the negative and cluster ions, incorporating this complex chemistry is challenging even with the availability of D-region chemistry models such as the Sodankylä Ion Chemistry model. From rocket measurements, it is clear that the maximum $N^-/N^+$ ratio is about 1 \citep{Arnold1971, Amemiya1996}. This implies to maintain quasi-neutrality in the D-region $N^+ = N_e + N^-$. However, ISR directly measures electron density, as electrons cause the bulk of the back-scattered radiation. And electron mobility is the dominant factor influencing electric conductivity in the D-region below $\sim$ 97 km. Therefore, negative and cluster ion mobility, which is likely lower than positive ion mobility due to its higher collisional cross-section and mass, is unlikely to alter our estimate of conductivity in the D-region. In other words, in the D-region, the neutral density is high enough that ions, both negative and positive, are bound to the neutrals, and electrons are the only charge carriers. However, if we calculate conductivity from satellite measurements of energetic precipitation, we will need to use an ion chemistry model to estimate the resulting electron density caused by the precipitating particles. In this case, not incorporating the complex D-region chemistry can cause an error in conductivity estimates of about 20\% \citep{Yu2018a}.