electric fields - What is the physical basis for Pedersen and Hall currents in an atmosphere?

There are three components of a conductivity tensor in atmospheric electrodynamics: the parallel conductivity along the direction of the magnetic field, the Pedersen conductivity (in the direction of the polarization electric field but perpendicular to the magnetic field), and the Hall conductivity (perpendicular to both the polarization electric field and the magnetic field).

It makes sense that if there’s an electric field along the magnetic field, it will induce a current in that direction. But what about the Pedersen and Hall currents? What is going on? What causes them?

Answer

It makes sense that if there’s an electric field along a magnetic field line, it will induce a current in that direction. But what about the Pedersen and Hall currents? What is going on? What causes them?

The three ionospheric conductivities derive from electric fields at an oblique but not exactly perpendicular angle to the background magnetic field (i.e., Earth's magnetic field).

The Hall term derives from the Hall effect, namely the drift induced by a finite value for $\mathbf{E} \times \mathbf{B}$, called the ExB-drift. That it is a finite and not infinite conductivity results from charge-neutral, charge-charge collisions (i.e., Coulomb collisions between charged and neutral and/or charged species).

The Pederson conductivity does correspond to the component parallel to $\mathbf{E}$, but it results from a combination of Coulomb collisions (similar to the Hall term) and drifts induced by $\mathbf{E}$. To understand this more clearly, we need to examine their mathematical form.

Suppose we define the total collision frequency of species $s$ as:

$$\nu_{s} = \nu_{s n} + \nu_{s s'} \tag{1}$$ where $\nu_{s n}$ is the collision frequency between species $s$ and neutral particles and $\nu_{s s'}$ is the collision frequency between species $s$ and $s'$, where $s'$ can be the same as $s$ or different (generally collisions between particles of the same species have lower collision rates than those for different species).

From a dimensional analysis point of view (and one of the simplest form of Ohm's law), a conductivity can be expressed as:

$$\sigma_{s} = \frac{ n_{s} \ q_{s}^{2} }{ m_{s} \ \nu_{s} } \tag{2}$$ where $n_{s}$ is the number density of species $s$, $q_{s}$ is the charge of species $s$ (including sign and charge state), $m_{s}$ is the mass of species $s$, and $\nu_{s}$ is the total collision frequency of species $s$.

The Hall and Pederson conductivities (assuming $n_{e} = n_{i}$) can be expressed as (after a bunch of algebra):

$$\begin{align} \sigma_{P} & = n \ e^{2} \left( \frac{ \nu_{e} }{ m_{e} } \frac{ 1 }{ \nu_{e}^{2} + \Omega_{ce}^2 } + \frac{ \nu_{i} }{ m_{i} } \frac{ 1 }{ \nu_{i}^{2} + \Omega_{ci}^2 } \right) \tag{3a} \\ \sigma_{H} & = n \ e^{2} \left( \frac{ \Omega_{ce} }{ m_{e} } \frac{ 1 }{ \nu_{e}^{2} + \Omega_{ce}^2 } + \frac{ \Omega_{ce} }{ m_{e} } \frac{ 1 }{ \nu_{e}^{2} + \Omega_{ce}^2 } \right) \tag{3b} \end{align}$$ where $\Omega_{cs} = \tfrac{ q_{s} B }{ m_{s} }$ is the cyclotron frequency of species $s$. It is generally the case that $\Omega_{ce} \gg \nu_{e}$, thus we can simplify Equations 3a and 3b to: $$\begin{align} \sigma_{P} & \approx \frac{ n \ e^{2} }{ m_{i} } \left( \frac{ \nu_{i} }{ \nu_{i}^{2} + \Omega_{ci}^2 } \right) \tag{4a} \\ \sigma_{H} & \approx \frac{ n \ e }{ B } \left( \frac{ \nu_{i}^{2} }{ \nu_{i}^{2} + \Omega_{ci}^2 } \right) \tag{4b} \end{align}$$

We can now see some of the physical implications of these conductivities by looking for the conditions that maximize them. One can see that $\sigma_{P}$ maximizes when $\nu_{i} \sim \Omega_{ci}$, or the ions experience roughly one collision per gyration which randomize their trajectories and lead to a net bulk flow along $\mathbf{E}$. Conversely, $\sigma_{P}$ increases with increasing $\nu_{i}$, i.e., for large $\nu_{i}$ the ions collide so frequently that they cannot undergo an ExB-drift causing them to stay nearly stationary while the electrons freely undergo an ExB-drift. Thus, the electrons carry the current in the opposite direction to that of $\mathbf{E} \times \mathbf{B}$.

Finally, the parallel conductivity is largely a result of the higher mobility of particles along vs. across $\mathbf{B}$. $\sigma_{\parallel}$ goes to infinity in the limit that $\nu_{i} \rightarrow 0$, though on average $\sigma_{\parallel}$ is already much much larger than both $\sigma_{P}$ and $\sigma_{H}$. Thus, it is generally approximated to infinite for simplicity.