Magnetized Limit

Let us now examine the magnetized limit,

$${\mit\Omega}_i\,\tau_i, \,\,\,{\mit\Omega}_e\, \tau_e \gg 1,$$ (4.95)

in which the electron and ion gyroradii are much smaller than the corresponding mean-free-paths. In this limit, the two-Laguerre-polynomial Chapman-Enskog closure scheme yields

$${\bf F} = n\,e\left(\frac{{\bf j}_\parallel}{\sigma_\parallel} +\frac{{\b... ...frac{3\,n}{2\,\vert{\mit\Omega}_e\vert\,\tau_e}\,{\bf b}\times\nabla_\perp T_e,$$ (4.96)

$$w_i = \frac{3\,m_e}{m_i} \frac{n\,(T_e-T_i)}{\tau_e},$$ (4.97)

$$w_e = -w_i + \frac{ {\bf j}\cdot {\bf F} }{n \,e}.$$ (4.98)

Here, the parallel electrical conductivity, $\sigma_\parallel$, is given by Equation (4.84), whereas the perpendicular electrical conductivity, $\sigma_\perp$, takes the form

$$\sigma_\perp = 0.51\,\sigma_\parallel = \frac{n\,e^2\,\tau_e}{m_e}.$$ (4.99)

Note that $\nabla_\parallel(\cdots) \equiv [{\bf b}\cdot\nabla (\cdots)]\,{\bf b}$ denotes a gradient parallel to the magnetic field, whereas $\nabla_\perp \equiv \nabla-\nabla_\parallel$ denotes a gradient perpendicular to the magnetic field. Likewise, ${\bf j}_\parallel \equiv ({\bf b}\cdot{\bf j})\,{\bf b}$ represents the component of the plasma current density flowing parallel to the magnetic field, whereas ${\bf j}_\perp \equiv {\bf j} - {\bf j}_\parallel$ represents the perpendicular component of the plasma current density.

We expect the presence of a strong magnetic field to give rise to a marked anisotropy in plasma properties between directions parallel and perpendicular to ${\bf B}$, because of the completely different motions of the constituent ions and electrons parallel and perpendicular to the field. Thus, not surprisingly, we find that the electrical conductivity perpendicular to the field is approximately half that parallel to the field [see Equations (4.96) and (4.99)]. The thermal force is unchanged (relative to the unmagnetized case) in the parallel direction, but is radically modified in the perpendicular direction. In order to understand the origin of the last term in Equation (4.96), let us consider a situation in which there is a strong magnetic field along the $z$-axis, and an electron temperature gradient along the $x$-axis. (See Figure 4.1.) The electrons gyrate in the $x$-$y$ plane in circles of radius $\rho_e\sim v_e/\vert{\mit\Omega}_e\vert$. At a given point, coordinate $x_0$, say, on the $x$-axis, the electrons that come from the right and the left have traversed distances of approximate magnitude $\rho_e$. Thus, the electrons from the right originate from regions where the electron temperature is approximately $\rho_e\,\partial T_e/\partial x$ greater than the regions from which the electrons from the left originate. Because the friction force is proportional to $T_e^{-1}$, an unbalanced friction force arises, directed along the $-y$-axis. (See Figure 4.1.) This direction corresponds to the direction of $-{\bf b}\times\nabla T_e$. There is no friction force along the $x$-axis, because the $x$-directed fluxes are associated with electrons that originate from regions where $x=x_0$. By analogy with Equation (4.85), the magnitude of the perpendicular thermal force is

$${\bf F}_{T\perp} \sim \frac{{\rho}_e}{T_e}\frac{\partial T_e}{\pa... ...sim \frac{n}{\vert{\mit\Omega}_e\vert\,\tau_e} \frac{\partial T_e}{\partial x}.$$ (4.100)

The effect of a strong magnetic field on the perpendicular component of the thermal force is directly analogous to a well-known phenomenon in metals called the Nernst effect (Rowe 2006).

Figure 4.1: Origin of the perpendicular thermal force in a magnetized plasma.

In the magnetized limit, the electron and ion heat flux densities become

$${\bf q}_e = -\kappa_\parallel^e\,\nabla_\parallel T_e -\kappa_\perp^e\, \na... ...ac{3\,T_e}{2\,\vert{\mit\Omega}_e\vert\,\tau_e\,e}\,{\bf b}\times{\bf j}_\perp,$$ (4.101)

$${\bf q}_i = -\kappa_\parallel^i\,\nabla_\parallel T_i -\kappa_\perp^i\, \nabla_\perp T_i +\kappa_\times^i\,{\bf b}\times\nabla_\perp T_i,$$ (4.102)

respectively. Here, the parallel thermal conductivities are given by Equations (4.89)–(4.90), and the perpendicular thermal conductivities take the form

$$\kappa_\perp^e = 4.7\,\frac{n\,T_e}{m_e\,{\mit\Omega}_e^{2}\,\tau_e},$$ (4.103)

$$\kappa_\perp^i = 2\, \frac{n\,T_i}{m_i\,{\mit\Omega}_i^{2}\,\tau_i}.$$ (4.104)

Finally, the cross thermal conductivities are written

$$\kappa_\times^e = \frac{5\,n\,T_e}{2\,m_e\,\vert{\mit\Omega}_e\vert},$$ (4.105)

$$\kappa_\times^i =\frac{5\,n\,T_i}{2\,m_i\,{\mit\Omega}_i}.$$ (4.106)

The first two terms on the right-hand sides of Equations (4.101) and (4.102) correspond to diffusive heat transport by the electron and ion fluids, respectively. According to the first terms, the diffusive transport in the direction parallel to the magnetic field is exactly the same as that in the unmagnetized case: that is, it corresponds to collision-induced random-walk diffusion of the ions and electrons, with frequency $\nu$, and step-length $l$. According to the second terms, the diffusive transport in the direction perpendicular to the magnetic field is far smaller than that in the parallel direction. To be more exact, it is smaller by a factor $(\rho/l)^2$, where $\rho$ is the gyroradius, and $l$ the mean-free-path. In fact, the perpendicular heat transport also corresponds to collision-induced random-walk diffusion of charged particles, but with frequency $\nu$, and step-length $\rho$. Thus, it is the greatly reduced step-length in the perpendicular direction, relative to the parallel direction, that ultimately gives rise to the strong reduction in the perpendicular heat transport. If $T_e\sim T_i$ then the ion perpendicular heat diffusivity actually exceeds that of the electrons by the square root of a mass ratio: that is, $\kappa_\perp^i/\kappa_\perp^e\sim \sqrt{m_i/m_e}$.

The third terms on the right-hand sides of Equations (4.101) and (4.102) correspond to heat fluxes that are perpendicular to both the magnetic field and the direction of the temperature gradient. In order to understand the origin of these terms, let us consider the ion flux. Suppose that there is a strong magnetic field along the $z$-axis, and an ion temperature gradient along the $x$-axis. (See Figure 4.2.) The ions gyrate in the $x$-$y$ plane in circles of radius $\rho_i\sim v_i/{\mit\Omega}_i$, where $v_i$ is the ion thermal velocity. At a given point, coordinate $x_0$, say, on the $x$-axis, the ions that come from the right and the left have traversed distances of approximate magnitude $\rho_i$. The ions from the right are clearly somewhat hotter than those from the left. If the unidirectional particle fluxes, of approximate magnitude $n\,v_i$, are balanced, then the unidirectional heat fluxes, of approximate magnitude $n\,T_i\,v_i$, will have an unbalanced component of relative magnitude $(\rho_i/T_i)\,\partial T_i/\partial x$. As a result, there is a net heat flux in the $+y$-direction (i.e., the direction of ${\bf b}\times\nabla T_i$). The magnitude of this flux is

$$q_\times^i \sim n\,v_i\, \rho_i\,\frac{\partial T_i}{\partial x} ... ... \frac{n\,T_i}{m_i\,\vert{\mit\Omega}_i\vert}\,\frac{\partial T_i}{\partial x}.$$ (4.107)

There is an analogous expression for the electron flux, except that the electron flux is in the opposite direction to the ion flux (because the electrons gyrate in the opposite direction to the ions). Both the ion and electron fluxes transport heat along isotherms, and do not, therefore, give rise to any change in plasma temperature.

Figure 4.2: Origin of the convective perpendicular heat flux in a magnetized plasma.

The fourth and fifth terms on the right-hand side of Equation (4.101) correspond to the convective component of the electron heat flux density, driven by motion of the electrons relative to the ions. It is clear from the fourth term that the convective flux parallel to the magnetic field is exactly the same as in the unmagnetized case [see Equation (4.87)]. However, according to the fifth term, the convective flux is radically modified in the perpendicular direction. Probably the easiest method of explaining the fifth term is via an examination of Equations (4.81), (4.87), (4.96), and (4.101). There is clearly a very close connection between the electron thermal force and the convective heat flux. In fact, starting from general principles of the thermodynamics of irreversible processes—the so-called Onsager principles (Reif 1965)—it is possible to demonstrate that an electron frictional force of the form $\alpha\,(\nabla\,T_e)_\beta\,{\bf i}$ necessarily gives rise to an electron heat flux of the form $\alpha\,(T_e\,j_\beta/n\,e)\,{\bf i}$, where the subscript $\beta$ corresponds to a general Cartesian component, and ${\bf i}$ is a unit vector. Thus, the fifth term on the right-hand side of Equation (4.101) follows by Onsager symmetry from the third term on the right-hand side of Equation (4.96). This is one of many Onsager symmetries that occur in plasma transport theory.

In order to describe the viscosity tensor in a magnetized plasma, it is helpful to define the rate-of-strain tensor

$$W_{\alpha\beta} = \frac{\partial V_\alpha}{\partial r_\beta} + \f... ...}{\partial r_\alpha} - \frac{2}{3} \,\nabla\cdot{\bf V}\, \delta_{\alpha\beta}.$$ (4.108)

Obviously, there is a separate rate-of-strain tensor for the electron and ion fluids. It is easily demonstrated that this tensor is zero if the plasma translates, or rotates as a rigid body, or if it undergoes isotropic compression. Thus, the rate-of-strain tensor measures the deformation of plasma volume elements.

In a magnetized plasma, the viscosity tensor is best described as the sum of five component tensors,

$$\mbox{\boldmath$\pi$} = \sum_{n=0,4} \mbox{\boldmath$\pi$} _n,$$ (4.109)

where

$$\mbox{\boldmath$\pi$} _0 = - 3\,\eta_0\,\left({\bf b}{\bf b} - \frac{1}{3}\,{\bf I}\right) \left({\bf b}{\bf b} - \frac{1}{3}\,{\bf I}\right): \nabla {\bf V},$$ (4.110)

with

$$\mbox{\boldmath$\pi$} _1 =- \eta_1\left[{\bf I}_\perp \cdot{\bf W}\cdot{\bf I}_\perp + \frac{1}{2}\,{\bf I}_\perp\,({\bf b}\cdot{\bf W}\cdot{\bf b})\right],$$ (4.111)

and

$$\mbox{\boldmath$\pi$} _2 = -4\,\eta_1\,\left({\bf I}_\perp\cdot {\bf W}\cdot{\bf b}{\bf b} + {\bf b}{\bf b}\cdot{\bf W} \cdot{\bf I}_\perp\right).$$ (4.112)

plus

$$\mbox{\boldmath$\pi$} _3 = \frac{\eta_3}{2}\,\left( {\bf b}\times {\bf W}\cdot{\bf I}_\perp - {\bf I}_\perp\cdot{\bf W}\times{\bf b} \right),$$ (4.113)

and

$$\mbox{\boldmath$\pi$} _4 = 2\,\eta_3\,\left({\bf b} \times{\bf W} \cdot {\bf b}{\bf b} - {\bf b}{\bf b} \cdot{\bf W} \times{\bf b}\right).$$ (4.114)

Here, ${\bf I}$ is the identity tensor, and ${\bf I}_\perp = {\bf I} - {\bf b}{\bf b}$. The previous expressions are valid for both electrons and ions.

The tensor $\pi$$_0$ describes what is known as parallel viscosity. This is a viscosity that controls the variation along magnetic field-lines of the velocity component parallel to field-lines. The parallel viscosity coefficients, $\eta_0^e$ and $\eta_0^i$, are specified in Equations (4.93)–(4.94). The parallel viscosity is unchanged from the unmagnetized case, and is caused by the collision-induced random-walk diffusion of particles, with frequency $\nu$, and step-length $l$.

The tensors $\pi$$_1$ and $\pi$$_2$ describe what is known as perpendicular viscosity. This is a viscosity that controls the variation perpendicular to magnetic field-lines of the velocity components perpendicular to field-lines. The perpendicular viscosity coefficients are given by

$$\eta_1^e = 0.51\, \frac{n\,T_e}{{\mit\Omega}_e^{2}\,\tau_e},$$ (4.115)

$$\eta_1^i = \frac{3\, n\,T_i}{10\,{\mit\Omega}_i^{2}\,\tau_i}.$$ (4.116)

The perpendicular viscosity is far smaller than the parallel viscosity. In fact, it is smaller by a factor $(\rho/l)^2$. The perpendicular viscosity corresponds to collision-induced random-walk diffusion of particles, with frequency $\nu$, and step-length $\rho$. Thus, it is the greatly reduced step-length in the perpendicular direction, relative to the parallel direction, that accounts for the smallness of the perpendicular viscosity compared to the parallel viscosity.

Finally, the tensors $\pi$$_3$ and $\pi$$_4$ describe what is known as gyroviscosity. This is not really viscosity at all, because the associated viscous stresses are always perpendicular to the velocity, implying that there is no dissipation (i.e., viscous heating) associated with this effect. The gyroviscosity coefficients are given by

$$\eta_3^e = -\frac{n\,T_e}{2\,\vert{\mit\Omega}_e\vert} ,$$ (4.117)

$$\eta_3^i = \frac{n\,T_i}{2\,{\mit\Omega}_i}.$$ (4.118)

The origin of gyroviscosity is very similar to the origin of the cross thermal conductivity terms in Equations (4.101)–(4.102). Both cross thermal conductivity and gyroviscosity are independent of the collision frequency.