# Significance of the ion Hall parameter

For an ionized Helium gas, for example, MHD Hall generators rely on the mobility ratios between electrons and ions to function and produce a Hall current. As far as I can tell, intuitively, increasing the electron Hall parameter arbitrarily high should result in increasing efficiency for a Hall generator. But is this necessarily true?

In this paper, an effective Hall parameter, $$\beta_e'$$ is defined in (14.45) where $$\beta_e' = \frac {\beta_e}{1 + \beta_e \beta_i}$$, where $$\beta_e$$ and $$\beta_i$$ are electron and ion Hall parameters, respectively. The significance of $$\beta_e'$$ is displayed on Figure 14.8, where the Hall generator's efficiency goes to 0 as $$\beta_e'$$ goes to 0. But in the above equation, $$\beta_e'$$ goes to 0 as $$\beta_i$$ becomes increasingly large, irrespective of $$\beta_e$$. Even if $$\beta_e > \beta_i$$, in the limit where both of them are large (e.g. very high magnetic field), $$\beta_e'$$ goes to 0 and so does efficiency. But why is this? What is physically happening such that the ion beta increasing (even with electron beta higher!) can reduce the Hall current and thus efficiency?

• Isn't the $\mathbf{j} \cdot \mathbf{E}$ term dependent upon the $\beta_{e}'$ (I think squared from Equation 14.42) parameter? If so, then the plot in Figure 14.8 looks like Y(x) ~ $x^{1/2}$, does it not? Or are you asking why the $\mathbf{j} \cdot \mathbf{E}$ term depends upon $\beta_{e}'$ at all? – honeste_vivere Oct 30 '18 at 13:30
• I am asking why ${\bf j \cdot E}$ depends upon upon $\beta_e'$ at all. I understand why the local efficiency increases as the electron Hall parameter increases. But don't quite understand why efficiency would decrease if the ion Hall parameter was to become arbitrarily large (even with the condition that $\beta_e > \beta_i$ always). Essentially, Hall current is transported by electrons, and if electron mobility is always greater than ion mobility, shouldn't you still have good efficiency? – Mathews24 Oct 30 '18 at 13:48
• Not if the ions get really hot, then their thermal speeds can be comparable to the electrons (e.g., similar order of magnitude, not necessarily the same). In real plasmas (i.e., not MHD), high beta often coincides with very turbulent flows that would affect transport terms like diffusion, advection, etc. This would affect currents, which in turn, would affect the Hall term. I did not read through the whole document, but I am fairly confident if you are clever with Google you can find a newer and better document on the subject. – honeste_vivere Oct 30 '18 at 14:32
• I agree in general that if the electrons and ions are moving comparably, then it is as you say. But if you look at the formula, even if the electrons are always moving significantly faster than ions, if $\beta_i$ is simply an arbitrarily large value, then efficiency goes to 0. But that is impossible, no? If you could possibly comment upon the formula's dependence on $\beta_i$ and/or recent derivations that provide and explain similar results, that would be greatly appreciated as I have not come across literature that accounts for the ion Hall parameter like Sutton. – Mathews24 Oct 31 '18 at 4:03

# Definitions

First, let's start by defining some parameters:

• $$\mu_{o}$$ is the permeability of free space
• $$\varepsilon_{o}$$ is the permittivity of free space
• $$k_{B}$$ is the Boltzmann constant
• $$\mathbf{E}$$ is the 3-vector electric field
• $$\mathbf{B}$$ is the 3-vector magnetic field with a quasi-static magnitude of $$B_{o}$$
• $$\mathbf{j}$$ is the 3-vector electric current density
• $$m_{s}$$ is the mass of particle species $$s$$
• $$q_{s}$$ is the charge of particle species $$s$$
• $$n_{s}$$ is the number density of particle species $$s$$ (i.e., number per unit volume)
• $$T_{s}$$ is the scalar temperature of particle species $$s$$
• $$\mathbf{v}_{s}$$ is the bulk flow velocity of particle species $$s$$
• $$V_{Ts} = \sqrt{\tfrac{k_{B} \ T_{s}}{m_{s}}}$$ is the rms thermal speed of a 1D Gaussian of particle species $$s$$
• $$\Omega_{cs} = \tfrac{q_{s} \ B_{o}}{m_{s}}$$ is the nonrelativistic, angular gyrofrequency of particle species $$s$$
• $$\omega_{ps} = \sqrt{\tfrac{n_{s} \ q_{s}^{2}}{m_{s} \ \varepsilon_{o}}}$$ is the angular plasma frequency of particle species $$s$$
• $$\rho_{cs} = \tfrac{V_{Ts}}{\Omega_{cs}}$$ is the nonrelativistic thermal gyroradius of particle species $$s$$
• $$\mathcal{P}_{s}$$ is the pressure tensor of species $$s$$ with a scalar magnitude of $$P_{s}$$
• $$L$$ is the relevant scale size of a system
• $$\delta = \tfrac{\rho_{ci}}{L}$$ is the magnetization parameter of the plasma

The generalized Ohm's law, from which the issue at hand arises, is given by: $$\mathbf{E} + \mathbf{v} \times \mathbf{B} \approx \frac{ \mathbf{j} \times \mathbf{B} }{ n \ e } - \frac{ \nabla}{ n \ e } \cdot \left( \mathcal{P}_{e} + \frac{ m_{e} }{ m_{i} } \mathcal{P}_{i} \right) + \eta \ \mathbf{j} + \frac{ m_{e} }{ n \ e^{2} } \frac{ d \mathbf{j} }{ d t } \tag{1}$$ 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, $$m_{s}$$ is the mass of species $$s$$ ($$s$$ can be $$e$$ for electron or $$i$$ for ion), and $$\eta$$ is the scalar electrical resistivity.

The resistivity due to collisions between species $$s$$ and $$s'$$ can be expressed by: $$\eta_{ss'} = \frac{ m_{s} \ \nu_{ss'} }{ n_{s} \ q_{s}^{2} } \tag{2}$$ where $$\nu_{ss'}$$ is the collision frequency between species $$s$$ and $$s'$$ and is given in geneneral form at https://physics.stackexchange.com/a/268594/59023.

As far as I can tell, intuitively, increasing the electron Hall parameter arbitrarily high should result in increasing efficiency for a Hall generator. But is this necessarily true?

The efficiency defined in the linked article is the ratio of the power done on the system by electromagnetic fields (i.e., $$\eta \ \mathbf{j}$$ in Equation 1 above) to the bulk flow work required to overcome the Lorentz force (i.e., inner product between $$\mathbf{v}$$ and Hall term). Thus, increasing the Hall term without increasing the work done by electromagnetic fields to confine the plasma will reduce the efficiency.

The Hall parameter of species $$s$$, $$\beta_{s}$$, is related to the ratio of $$\Omega_{cs}$$ to $$\nu_{ss'}$$ for $$s' \rightarrow s$$, i.e., the ratio of the gyro-to-collision frequency. Thus, the following is generically true for ideal, isotropic plasmas $$\beta_{s} \propto T_{s}^{3/2}$$. Another way of expressing this is to say that the resistivity is inversely proportional to Hall parameter (being very loose and casual here to illustrate a point), i.e., increasing the temperature of a confined plasma tends to increase the conductivity.

Even if $$\beta_{e} > \beta_{i}$$, in the limit where both of them are large (e.g. very high magnetic field), $$\beta_{e}'$$ goes to 0 and so does efficiency. But why is this? What is physically happening such that the ion beta increasing (even with electron beta higher!) can reduce the Hall current and thus efficiency?

Note the parameter $$\delta$$ above in the Definitions section. If you have a plasma with $$T_{e} = T_{i}$$, then the ion gyroradius will be ~43 times larger than the electron gyroradius (assuming protons where $$\sqrt{\tfrac{m_{i}}{m_{e}}}$$ ~ 43). Therefore, we need only be concerned with ensuring that the ions are magnetized. This is why $$\delta$$ does not depend upon $$\rho_{ce}$$, i.e., because it is almost always going to be so much smaller than the system that we can neglect it.

The issue at hand here is a loss of confinement due to something called ion slip currents and it arises when the following is not satisified $$\delta \ll 1$$. We know that $$\rho_{ci} \propto T_{i}^{1/2}$$ for a constant $$B_{o}$$, thus the gyroradii increases with increasing ion temperature. Eventually, the gyroradii could become so large that $$\delta$$ ~ 1 at which point the efficiency goes to zero.

We could just as easily define $$\delta = \tfrac{V_{Ti}}{\Omega_{ci} \ L}$$ and see that increasing the ion thermal speed eventually will lead to a scenario where the ions move so fast that they can traverse any relevant gradient (e.g., magnetic field gradients) in the system faster than they gyrate, thus demagnetizing them and possibly losing them (e.g., to the wall in a plasma experiment).

A final note is that if one makes the ions too hot, then it is possible that their gyroradii could become comparable to the physical apparatus used in the experiment. Generally this is nearly impossible to do as the fields are quite large in lab plasmas and the heating times to such high energies are much longer than loss time scales.

The overall point is that if the ions are demagnetized, then the assumptions and approximations used to derive the efficiency in the linked article break down. It's another way of saying that you cannot impart momentum and energy to the plasma using electromagnetic fields if the ions are not confined to the quasi-static magnetic field, e.g., if you try to "push" them along $$\mathbf{B}$$ but they drift across it faster, no net transport along $$\mathbf{B}$$ will occur.

# Update 1

The ion slip parameter adds an extra term to the generalized Ohm's law in Equation 1 above proportional to $$\beta_{i} \left( \mathbf{j} \times \mathbf{B} \right) \times \mathbf{B}$$, where $$\beta_{i}$$ is the ion slip parameter. One can see that the current due to this effect is orthogonal to $$\mathbf{B}$$, i.e., it is a cross-field current. There are also terms in the ion slip current that depend upon the electron collision rate and the electron gyrofrequency [Ghara et al., 2012] (Note I chose this article because there was no paywall).

# Update 2

The derivation of the extra term in the generalized Ohm's law including ion slip -- relative motion of ions with respect to fluid center of mass -- can be found in the work by Kunkel . They do not derive the current term, but rather the extra electric field term. Regardless, the approach is the same and it has some good explanations based upon physically significant arguments.

# References

• Ghara, N., et al., "Effects of Hall Current and Ion-Slip on Unsteady MHD Couette Flow," Open Journal of Fluid Dynamics 2, pp. 1--13, doi:10.4236/ojfd.2012.21001, 2012.
• Kunkel, W.B., "Generalized Ohm's law for plasma including neutral particles," Phys. Fluids 27(9), pp. 2369--2371, doi:10.1063/1.864903, 1984.
• Thank you for the answer. Could you possibly define/explain ion slip current and why it leads to loss of confinement? Hall generators extract energy via the electrons which carry high current due to their increased mobility; I'm missing the significance of ion currents here since they appear negligible compared to electron currents (even at high temperatures). – Mathews24 Nov 2 '18 at 15:24
• The entire apparatus only works if you can control the currents and the flow of plasma, right? If your ions run away to or hit the walls of the apparatus, you will develop a charge imbalance that will affect the electron current flow, right? So it will start to suppress the Hall electron current at some point. – honeste_vivere Nov 2 '18 at 16:02
• Yes, but if the ions are simply hitting the surfaces randomly (as it appears to be) since the ion Larmor radius is on the order of the system size, the charge accumulation will be fairly uniform/isotropic, no? To understand the phenomenon better, is there a defined functional form for calculating ion slip currents? – Mathews24 Nov 2 '18 at 20:45
• @Mathews24 - Yes, there are gyroradii- and Coulombic-resistivities in devices such as these and the $\delta$ parameter to which I eluded above is generally used as a guide to determine what will dominate the dynamics. The electron gyroradii are so small in these systems they can be effectively ignored but the ions can be problematic if one is not careful, which is why that effective Hall parameter's influence on the efficiency only depends upon the ions. – honeste_vivere Nov 2 '18 at 21:25
• To finalize the answer, could you just define in words the ion slip current and provide a mathematical definition of $\bf J_{ion \ slip}$ or provide references that do? – Mathews24 Nov 2 '18 at 22:08