Classical Closure Scheme

The so-called classical asymptotic closure scheme for the electron and ion fluid equations, (2.26)–(2.31), due to Braginskii [6], is premised on the assumption that both fluids are highly magnetized, which means that the electron and ion gyro-radii are much smaller than the corresponding mean-free-paths between collisions. In other words,

$$\frac{\rho_e}{l_e} \ll 1,$$ (2.32)

$$\frac{\rho_i}{l_i} \ll 1.$$ (2.33)

As is clear from Table 2.2, the electron and the ion fluids in tokamak fusion reactors are indeed highly magnetized, which implies that particle, momentum, and energy flows perpendicular to magnetic field-lines are quite different to those parallel to magnetic field-lines [18].

For particle, momentum, and energy flows perpendicular to magnetic field-lines, the small (compared to unity) dimensionless expansion parameters upon which the classical asymptotic closure scheme is based are $\rho_e/a$ and $\rho_i/a$. Here, we are assuming that the variation length-scales of quantities perpendicular to magnetic field-lines are of order the plasma minor radius, $a$. As is apparent from Table 2.2, the expansion parameters $\rho_e/a$ and $\rho_i/a$ are indeed small compared to unity in tokamak fusion reactors.

For particle, momentum, and energy flows parallel to magnetic field-lines, the small (compared to unity) expansion parameters upon which the classical asymptotic closure scheme is based are $l_e/L_c$ and $l_i/L_c$. Here, we are assuming that the variation length-scales of quantities parallel to magnetic field-lines are of order the connection length, $L_c = 2\pi\,q\,R_0$, which is the typical distance a magnetic field-line has to travel in order to fully traverse a magnetic-flux surface. As is clear from Table 2.2, the expansion parameters $l_e/L_c$ and $l_i/L_c$ are actually both large compared to unity in tokamak fusion reactors. It follows that the classical asymptotic closure scheme fails in such reactors (because the confined plasmas are not sufficiently collisional in nature). Nevertheless, in the following, we shall describe the classical asymptotic closure scheme, before attempting to repair it.

Table: 2.2 Classical fluid closure parameters in a low-field and a high-field tokamak reactor. Here, $B$ is the toroidal magnetic field-strength, $\rho _e$ the electron gyro-radius, $\rho _i$ the ion gyro-radius, $l_e$ the electron mean-free-path, $l_i$ the ion mean-free-path, and $L_c$ the connection length (calculated with $q=3$).

	Low-Field	High-Field
$B({\rm T})$	5.0	12.0
$\rho_e/l_e$	$8.0\times 10^{-9}$	$1.9\times 10^{-8}$
$\rho_i/l_i$	$3.8\times 10^{-7}$	$9.2\times 10^{-7}$
$\rho_e/a$	$2.2\times 10^{-5}$	$2.2\times 10^{-5}$
$\rho_i/a$	$1.5\times 10^{-3}$	$1.5\times 10^{-3}$
$l_e/L_c$	$49$	$21$
$l_i/L_c$	$70$	$29$

According to the classical asymptotic closure scheme [6],

$${\bf F}_e = {\bf F}_u + {\bf F}_T,$$ (2.34)

$${\bf F}_u = n_e\,e\left(\frac{{\bf j}_\parallel}{\sigma_\parallel} +\frac{{\bf j}_\perp}{\sigma_\perp}\right),$$ (2.35)

$${\bf F}_T =-0.71\,n_e\,\nabla_\parallel T_e +\frac{3\,n_e}{2\,{\mit\Omega}_e\,\tau_e}\,{\bf b}\times\nabla_\perp T_e,$$ (2.36)

$$W_i = \frac{3\,m_e}{m_i} \frac{n_e\,(T_e-T_i)}{\tau_e},$$ (2.37)

$$W_e = -W_i + \frac{ {\bf j}\cdot {\bf F}_e }{n_e \,e}.$$ (2.38)

Here, ${\bf b}={\bf B}/B$ is a unit vector parallel to the magnetic field, and

$${\bf j}= n_e\,e\,({\bf V}_i-{\bf V}_e)$$ (2.39)

is the plasma current density. Moreover, the parallel electrical conductivity is given by [6,47]

$$\sigma_\parallel = 1.96\,\frac{n_e \,e^2\,\tau_e}{m_e}.$$ (2.40)

whereas the perpendicular electrical conductivity takes the form [6]

$$\sigma_\perp = \frac{n_e\,e^2\,\tau_e}{m_e}.$$ (2.41)

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.

It can be seen that parallel component of the friction force density, ${\bf F}_u$, is smaller than the perpendicular component by a factor $1.96$; this is a consequence of the fact that the collision frequency decreases with increasing velocity ( $\nu_e\propto v^{\,-3}$), causing the distribution of electrons with large parallel velocities to be more distorted from a Maxwellian distribution than that of slower electrons [30]. The thermal force density, ${\bf F}_T$, is also a consequence of the velocity dependence of the collision frequency [18]. (See Section 2.23.)

The electron and ion heat fluxes are written [6]

$${\bf q}_e ={\bf q}_{\parallel\,e}+ {\bf q}_{\times\,e} + {\bf q}_{\perp\,e} +{\bf q}_T ,$$ (2.42)

$${\bf q}_i ={\bf q}_{\parallel\,i}+ {\bf q}_{\times\,i} + {\bf q}_{\perp\,i},$$ (2.43)

respectively, where

$${\bf q}_{\parallel\,s} = -\kappa_{\parallel\,s}\,\nabla_\parallel T_s,$$ (2.44)

$${\bf q}_{\times\,s} = \kappa_{\times\,s}\,{\bf b}\times\nabla_\perp T_s,$$ (2.45)

$${\bf q}_{\perp\,s} = -\kappa_{\perp\,s}\,\nabla_\perp T_s,$$ (2.46)

$${\bf q}_T = - 0.71\,\frac{T_e}{e}\,{\bf j}_\parallel+ \frac{3\,T_e}{2\,{\mit\Omega}_e\,\tau_e\,e}\,{\bf b}\times{\bf j}_\perp$$ (2.47)

are known as the parallel, cross, perpendicular, and thermal heat fluxes, respectively. Here, the parallel thermal conductivities, which control the diffusion of heat parallel to magnetic field-lines, are given by [6]

$$\kappa_{\parallel\,e} = 3.16\,\,\frac{n_e\,\tau_e\,T_e}{m_e},$$ (2.48)

$$\kappa_{\parallel\,i} = 3.9\,\,\frac{n_e\,\tau_i\,T_i}{m_i},$$ (2.49)

Moreover, the cross thermal conductivities, which control the (non-diffusive) flow of heat within magnetic flux-surfaces, perpendicular to magnetic field-lines, are written [6]

$$\kappa_{\times\,e} = \frac{5\,n_e\,T_e}{2\,m_e\,{\mit\Omega}_e},$$ (2.50)

$$\kappa_{\times\,i} =\frac{5\,n_e\,T_i}{2\,m_i\,{\mit\Omega}_i}.$$ (2.51)

(See Section 2.11.) Finally, the perpendicular thermal conductivities, which control the diffusion of heat perpendicular to magnetic flux-surfaces, take the forms [6]

$$\kappa_{\perp\,e} = 4.66\,\frac{n_e\,T_e}{m_e\,{\mit\Omega}_e^{\,2}\,\tau_e},$$ (2.52)

$$\kappa_{\perp\,i} = 2\, \frac{n_e\,T_i}{m_i\,{\mit\Omega}_i^{\,2}\,\tau_i}.$$ (2.53)

Note that $\kappa_{\perp\,s} \sim (\rho_s/l_s)\,\kappa_{\times\,s}$ and $\kappa_{\times\,s}\sim (\rho_s/l_s)\,\kappa_{\parallel\,s}$. In other words, in a highly magnetized plasma (i.e., $\rho_s/l_s\ll 1$), the species-$s$ perpendicular thermal conductivity is much less than the cross conductivity, which, in turn, is much less than the parallel conductivity.

According to the previous expressions, the diffusion of heat parallel to magnetic field-lines is characterized by the diffusivities

$$\chi_{\parallel\,e} \equiv \frac{\kappa_{\parallel\,e}}{n_e} = 1.58\,\nu_e\,l_e^{\,2},$$ (2.54)

$$\chi_{\parallel\,i} \equiv \frac{\kappa_{\parallel\,i}}{n_e} = 1.95 \,\nu_i\,l_i^{\,2}.$$ (2.55)

These diffusivities clearly correspond to collision-induced random-walk motions of electrons and ions, parallel to magnetic field-lines, with a step-frequency of order $\nu_{e,i}$ and a step-length of order $l_{e,i}$ [18,41]. On the other hand, the diffusion of heat perpendicular to magnetic field-lines is characterized by the diffusivities

$$\chi_{\perp\,e} \equiv \frac{\kappa_{\perp\,e}}{n_e} = 2.33\,\nu_e\,\rho_e^{\,2},$$ (2.56)

$$\chi_{\perp\,i} \equiv \frac{\kappa_{\perp\,i}}{n_e} = \nu_i\,\rho_i^{\,2}.$$ (2.57)

These diffusivities clearly correspond to the collision-induced random-walk motions of electrons and ions, perpendicular to magnetic field-lines, with a step-frequency of order $\nu_{e,i}$ and a step-length of order $\rho_{e,i}$ [18,41].

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

$$W_{s\,jk} = \frac{\partial V_{s\,j}}{\partial r_k} + \frac{\partial V_{s\,k}}{\partial r_j} - \frac{2}{3} \,\nabla\cdot{\bf V}_s\, \delta_{jk}.$$ (2.58)

It is easily demonstrated that this tensor is zero if the species-$s$ fluid translates, or rotates as a rigid body, or if it undergoes isotropic compression. Thus, the rate-of-strain tensor measures the deformation of species-$s$ fluid volume elements.

In a highly magnetized plasma, the viscosity tensor is conveniently described as the sum of three component tensors [6]:

$$\mbox{\boldmath$\pi$} _s = \mbox{\boldmath$\pi$} _{\parallel\,s} + \mbox{\boldmath$\pi$} _{\times\,s}+ \mbox{\boldmath$\pi$} _{\perp\,s},$$ (2.59)

where

$$\mbox{\boldmath$\pi$} _{\parallel\,s} = - 3\,\eta_{\parallel\,s}\,\left({\bf b}{\bf b} - \frac{1}{3}\,{... ...I}\right) \left({\bf b}{\bf b} - \frac{1}{3}\,{\bf I}\right): \nabla {\bf V}_s,$$ (2.60)

$$\mbox{\boldmath$\pi$} _{\times\,s} = \frac{\eta_{\times\,s}}{2}\,\left( {\bf b}\times {\bf W}_s\cdot... ...W}_s \cdot {\bf b}{\bf b} - {\bf b}{\bf b} \cdot{\bf W}_s \times{\bf b}\right),$$ (2.61)

$$\mbox{\boldmath$\pi$} _{\perp\,s} =- \eta_{\perp\,s}\left[{\bf I}_\perp \cdot{\bf W}_s\cdot{\bf I}_... ..._s\cdot{\bf b}{\bf b} + {\bf b}{\bf b}\cdot{\bf W}_s \cdot{\bf I}_\perp\right).$$ (2.62)

Here, ${\bf I}$ is the identity tensor, and ${\bf I}_\perp = {\bf I} - {\bf b}{\bf b}$.

The tensor $\pi$$_{\parallel\,s}$ describes what is known as parallel viscosity; this is a viscosity that controls the diffusion of parallel (to the magnetic field) momentum along magnetic field-lines. The parallel viscosity coefficients are given by [6]

$$\eta_{\parallel\,e} = 0.73\,n_e\,\tau_e\,T_e,$$ (2.63)

$$\eta_{\parallel\,i} = 0.96\,n_e\,\tau_i\,T_i.$$ (2.64)

Moreover, the tensor $\pi$$_{\times\,s}$ describes what is known as gyro-viscosity; 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 gyro-viscosity coefficients are given by [6]

$$\eta_{\times\,e} = \frac{n_e\,T_e}{2\,{\mit\Omega}_e} ,$$ (2.65)

$$\eta_{\times\,i} = \frac{n_e\,T_i}{2\,{\mit\Omega}_i}.$$ (2.66)

Finally, the tensor $\pi$$_{\perp\,s}$ describes what is known as perpendicular viscosity; this is a viscosity that controls the diffusion of perpendicular momentum perpendicular to magnetic field-lines. The perpendicular viscosity coefficients take the forms [6]

$$\eta_{\perp\,e} = 0.51\, \frac{n_e\,T_e}{{\mit\Omega}_e^{\,2}\,\tau_e},$$ (2.67)

$$\eta_{\perp\,i} = \frac{3\, n_e\,T_i}{10\,{\mit\Omega}_i^{\,2}\,\tau_i}.$$ (2.68)

Note that $\eta_{\perp\,s} \sim (\rho_s/l_s)\,\eta_{\times\,s}$ and $\eta_{\times\,s}\sim (\rho_s/l_s)\,\eta_{\parallel\,s}$. In other words, in a highly magnetized plasma (i.e., $\rho_s/l_s\ll 1$), the species-$s$ perpendicular viscosity is much less than the gyroviscosity, which, in turn, is much less than the parallel viscosity.

Table: 2.3 Classical diffusivities in a low-field and a high-field tokamak reactor. Here, $B$ is the toroidal magnetic field-strength, $\chi_{\parallel\,e}$ the parallel electron energy diffusivity, $\chi_{\parallel\,i}$ the parallel ion energy diffusivity, $\chi_{\perp\,e}$ the perpendicular electron energy diffusivity, $\chi_{\perp\,i}$ the perpendicular ion energy diffusivity, ${\mit\Xi}_{\parallel\,e}$ the parallel electron momentum diffusivity, ${\mit\Xi}_{\parallel\,i}$ the parallel ion momentum diffusivity, ${\mit\Xi}_{\perp\,e}$ the perpendicular electron momentum diffusivity, and ${\mit\Xi}_{\perp\,i}$ the perpendicular ion momentum diffusivity.

	Low-Field	High-Field
$B({\rm T})$	5.0	12.0
$\chi_{\parallel\,e}({\rm m^2/s})$	$5.5\times 10^{11}$	$9.6\times 10^{10}$
$\chi_{\parallel\,i}({\rm m^2/s})$	$1.4\times 10^{10}$	$2.5\times 10^{9}$
$\chi_{\perp\,e}({\rm m^2/s})$	$5.2\times 10^{-5}$	$5.2\times 10^{-5}$
$\chi_{\perp\,i}({\rm m^2/s})$	$1.1\times 10^{-3}$	$1.1\times 10^{-3}$
${\mit\Xi}_{\parallel\,e}({\rm m^2/s})$	$1.3\times 10^{11}$	$2.2\times 10^{10}$
${\mit\Xi}_{\parallel\,i}({\rm m^2/s})$	$3.5\times 10^{9}$	$6.1\times 10^{8}$
${\mit\Xi}_{\perp\,e}({\rm m^2/s})$	$5.8\times 10^{-6}$	$5.8\times 10^{-6}$
${\mit\Xi}_{\perp\,i}({\rm m^2/s})$	$1.6\times 10^{-4}$	$1.6\times 10^{-4}$

According to the previous expressions, the diffusion of parallel momentum parallel to magnetic field-lines is characterized by the diffusivities

$${\mit \Xi}_{\parallel\,e} \equiv \frac{\eta_{0\,e}}{n_e\,m_e} = 0.37\,\nu_e\,l_e^{\,2},$$ (2.69)

$${\mit \Xi}_{\parallel\,i} \equiv \frac{\eta_{0\,i}}{n_e\,m_i} = 0.48 \,\nu_i\,l_i^{\,2}.$$ (2.70)

As before, these diffusivities correspond to collision-induced random-walk motions of electrons and ions, parallel to magnetic field-lines, with a step-frequency of order $\nu_{e,i}$ and a step-length of order $l_{e,i}$ [18,41]. On the other hand, the diffusion of perpendicular momentum perpendicular to magnetic field-lines is characterized by the diffusivities

$${\mit\Xi}_{\perp\,e} \equiv \frac{\eta_{1\,e}}{n_e\,m_e} = 0.26\,\nu_e\,\rho_e^{\,2},$$ (2.71)

$${\mit\Xi}_{\perp\,i} \equiv \frac{\eta_{1\,i}}{n_e\,m_i} = 0.15\,\nu_i\,\rho_i^{\,2}.$$ (2.72)

Again, these diffusivities correspond to the collision-induced random-walk motions of electrons and ions, perpendicular to magnetic field-lines, with step-frequency of order $\nu_{e,i}$ and a step-length of order $\rho_{e,i}$ [18,41].

Table 2.3 shows estimates for the classical heat and momentum diffusivities in a tokamak fusion reactor. It can be seen that the electron parallel diffusivities are much larger than the ion parallel diffusivities, but that both are extremely large. On the other hand, the ion perpendicular diffusivities are much larger than the electron perpendicular diffusivities, but both are much smaller than the experimentally observed perpendicular diffusivities, which are all approximately $1\,{\rm m^2/s}$ [52].