Cold plasma approximation leads to infinite coupling

Question

In plasma physics, there is an approximation called cold plasma approximation according to which, the temperature of a plasma species in an electron-ion plasma, say ions ( $T_i$ ) is assumed to be zero. As far as I have understood this approximation is valid when the average temperature of electrons ($T_e$) is much larger than that of ions ( $T_e>>T_i$).

For any plasma species, one can define Coulomb coupling parameter($\Gamma$), $$\Gamma=\frac{q^2}{4\pi\epsilon_0ak_BT_i},$$ which is basically the ratio of average interparticle potential energy to average thermal energy, where $q,\;a\;\text{and} \;T_i$ denotes charge, average interparticle distance and average temperature of ions respectively.

So, if one assumes ions to be cold, he/she gets infinite coupling for ions ( as $T_i\to0$, $\Gamma\to\infty).$

I don't understand the physical meaning of infinte coupling. Is this result ( i.e, $\Gamma\to\infty$ as $T_i\to 0)$ correct physically ?

Answer 1

As far as I have understood this approximation is valid when the average temperature of electrons ($T_{e}$) is much larger than that of ions ($T_{e} \gg T_{i}$).

No, this is not really necessary. The cold plasma approximation is more of a statement that the finite temperature effects are not important for the phenomena in question. Sometimes this is thought of in terms of the plasma beta, defined as: $$ \beta_{s} = \frac{ 2 \ \mu_{o} \ n_{s} \ k_{B} \ T_{s} }{ B_{o}^{2} } \tag{0} $$ where $\mu_{o}$ is the permeability of free space [$T \ m \ A^{-1}$], $n_{s}$ is the number density [$\# \ m^{-3}$] of species $s$, $k_{B}$ is the Boltzmann constant [$J \ K^{-1}$], $T_{s}$ is the average temperature [$K$] of species $s$, and $B_{o}$ is the quasi-static magnetic field magnitude [$T$].

In a plasma where $\beta_{s} \ll 1$, one can usually assume that thermal contributions are not as important as magnetic ones (e.g., this works well for whistler waves).

I don't understand the physical meaning of infinte coupling. Is this result (i.e., $\Gamma \rightarrow \infty$ as $T_{i} \rightarrow 0$) correct physically?

Physically correct? No, not really. Though when $\Gamma \ll 1$ (e.g., most space plasmas), the approximation of an ideal gas is much more accurate for a plasma than when $\Gamma \gg 1$ (e.g., many types of high energy lab plasmas like inertial fusion experiments). The reason being is that the weakly coupled plasma cares more about the electrostatic and electromagnetic interactions than strict binary particle-particle collisions. Or another way to think about it is that when $\Gamma \gg 1$, you are packing the particle constituents so tightly together that it disrupts the usual collective plasma behavior because the Coulomb fields are so strong, they dominate the particle dynamics.