So Anderson et al proposed a model to explain the production of Sub Auroral Ion Drifts. In the paper they talk about how the decrease in ionospheric conductance results in an increase in the electric field, but why is this so?
I understand that decrease in conductance is an increase in resistivity. By Ohm's law $V = IR$ and thus $V$ increases, but since $E = -\displaystyle\frac{dV}{dx}$ shouldn't $E$ decrease if conductance decreases ?
In the paper they talk about how the decrease in ionospheric conductance results in an increase in the electric field, but why is this so?
This just results from Ohm's law and some assumptions about the system. The generalized Ohm's law can be written as: $$ \mathbf{E} + \mathbf{v} \times \mathbf{B} \approx \frac{ \mathbf{j} \times \mathbf{B} }{ n \ e } - \frac{ \nabla}{ n \ e } \cdot \left( \mathcal{P}_{e} + \frac{ m_{e} }{ m_{i} } \mathcal{P}_{i} \right) + \eta \ \mathbf{j} + \frac{ m_{e} }{ n \ e^{2} } \frac{ d \mathbf{j} }{ d t } \tag{0} $$ where $\mathbf{j}$ is the total current density, $n$ is the total number density (assuming quasi-neutrality, i.e., $n_{e} = n_{i}$), $e$ is the fundamental charge, $\mathcal{P}_{s}$ is the pressure tensor of species $s$, $m_{s}$ is the mass of species $s$ ($s$ can be $e$ for electron or $i$ for ion), and $\eta$ is the scalar electrical resistivity (see also https://physics.stackexchange.com/a/438272/59023 or https://physics.stackexchange.com/a/363523/59023 or https://physics.stackexchange.com/a/261223/59023 for more on Ohm's law and conductivities). Typically, many of these terms are small enough to be negligible and one can approximate the electric field as: $$ \mathbf{E} \approx \eta \ \mathbf{j} \tag{1} $$ Note that $\eta^{-1} = \sigma$ which is the electrical conductivity. So for situations where the current is held constant but $\sigma$($\eta$) decreases(increases), the magnitude of $\mathbf{E}$ must increase accordingly. Situations like this can arise when there's a constant input source for the current from an external driver.
As an aside, one generally does not drop several of the terms I ignored here in generalized Ohm's law for the ionosphere because they are not always negligible in this region. I merely did so to help simplify the point.
It's probably this:
To sustain a field requires a potential difference between two points. if the path between those two points is conductive, a small current will flow between them, the potential difference gets bled away, and the strength of the field diminishes. If the conductance between them is low, no current will flow, the charge imbalance will be sustained, and the strength of the field is not diminished.
