I'm having trouble understanding the way the plasma fluid equations are normalized in Ch. 4 of Fitzpatrick's "Plasma Physics: An Introduction." Here's a a link to the page in question. Equations 4.147-49 are the electron fluid equations in a magnetized plasma. The Braginskii/Chapman-Enskog closure is used to get the higher-order velocity moment quantities i.e. viscocity tensor $\pi_{\alpha \beta}$, heat flux vector $q$, and the collisional force and work, F and W. The subscript U and T denote thermal and convective quantities. The formulas for all the closure terms are here. Three dimensionless quantities are assumed to be small: the ratio of gyroradius to variational length ($\delta_e$), the ratio of gyroradius to mean-free path ($\zeta_e$), and the ratio of square-root of the electron to ion mass ratio $\mu$.

The main idea is to normalize the fluid equations so that we can identify terms that can be neglected in particular regimes. However, I'm having trouble understanding how the normalization constants are chosen i.e. where those 18 formulas come from at the bottom of the page on the first link. From what I understand, we first assume that we know characteristic values for n, $v_e$ (thermal velocity), $l_e$ (mean-free path), B, and $\rho_e$ (gyroradius). Also assume that the typical flow rate is $\lambda_e v_e$. Fitzpatrick claims that the chosen normalization constants result in all normalized quantities being of O(1). It's pretty obvious how this works for the first 7 quantities, but I can't understand where the factors of $(1 + \lambda_e^2)$ come from. I also can't figure out why a factor of $\lambda_e$ is needed in the $\pi_e$ term in order to make it order 1.

In particular, if anybody could help me understand the $\lambda_e$ factors in the expressions for U (which is the $e^-$ velocity relative to ion velocity $U = V_e - V_i$) and/or $\pi_{\alpha \beta}$, I think I could figure out the rest.

I also have a more general question about this type of normalization. Is it the case that all of these quantities are not necessarily order 1, but when we restrict to certain parameter ranges they end up being order 1? Or are these terms necessarily order 1 as a result of our assumptions about known characteristics values? I think it's the latter, but I'm not exactly sure.

Also, most of this material can be found on this page of Fitzpatrick's online notes.

For the purposes of this question, it should be fine to use only the $\pi_0$ contribution to the viscosity tensor (since this is the highest order term). This $\pi_0$ is the parallel viscosity that is referenced in the last paragraph on that first page.

Thanks very much for your help, let me know if I can provide any other helpful info.

  • $\begingroup$ What is $\lambda_{e}$? The Debye or inertial length? $\endgroup$ – honeste_vivere Dec 15 '18 at 16:05
  • $\begingroup$ It's neither, it's defined by saying that the characteristic flow rate is $\lambda_e v_e$ where $v_e$ is the characteristic thermal velocity. In other words,we're assuming that $v_e$ and the characteristic flow rate are known, and then defining $\lambda_e$ as their ratio. From what I gather from the text that I'm using, we're not assuming anything about the size of $\lambda_e$ at this point, so that later we can look at different limits such as $\lambda_e$ small, large, or ~1. $\endgroup$ – Ricky P Dec 18 '18 at 19:48

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