Bohm diffusion

The diffusion of plasma across a magnetic field was conjectured to follow the Bohm diffusion scaling as indicated from the early plasma experiments of very lossy machines.
Description

Bohm diffusion is characterized with a diffusion coefficient equal to

\( D_{Bohm} = \frac{1}{16}\,\frac{k_BT}{eB}, \)

where B is the magnetic field strength, T is the temperature, and e is the elementary charge.

It was first observed in 1949 by David Bohm, E. H. S. Burhop, and Harrie Massey while studying magnetic arcs for use in isotope separation.[1] It has since been observed that many other plasmas follow this law. Fortunately there are exceptions where the diffusion rate is lower, otherwise there would be no hope of achieving practical fusion energy. In Bohm's original work he notes that the fraction 1/16 is not exact; in particular "the exact value of [the diffusion coefficient] is uncertain within a factor of 2 or 3." Lyman Spitzer considered this fraction as a factor related to plasma instability.[2]

Generally diffusion can be modelled as a random walk of steps of length δ and time τ. If the diffusion is collisional, then δ is the mean free path and τ is the inverse of the collision frequency. The diffusion coefficient D can be expressed variously as

\( D = \frac{\delta^{2}}{\tau}=v^{2}\tau=\delta\,v,\)

where v = δ/τ is the velocity between collisions.

In a magnetized plasma, the collision frequency is usually small compared to the gyrofrequency, so that the step size is the gyroradius ρ and the step time is the collision time, τ, which is related to the collision frequency through τ = 1/ν, leading to D = ρ²ν. If the collision frequency is larger than the gyrofrequency, then the particles can be considered to move freely with the thermal velocity vth between collisions, and the diffusion coefficient takes the form D = vth²/ν. Evidently the classical (collisional) diffusion is maximum when the collision frequency is equal to the gyrofrequency, in which case D = ρ²ωc = vth²/ωc. Substituting ρ = vth/ωc, vth = (kBT/m)1/2, and ωc = eB/m, we arrive at D = kBT/eB, which is the Bohm scaling. Considering the approximate nature of this derivation, the missing 1/16 in front is no cause for concern. Therefore, at least within a factor of order unity, Bohm diffusion is always greater than classical diffusion.

In the common low collisionality regime, classical diffusion scales with 1/B², compared with the 1/B dependence of Bohm diffusion. This distinction is often used to distinguish between the two.

In light of the calculation above, it is tempting to think of Bohm diffusion as classical diffusion with an anomalous collision rate that maximizes the transport, but the physical picture is different. Anomalous diffusion is the result of turbulence. Regions of higher or lower electric potential result in eddies because the plasma moves around them with the E-cross-B drift velocity equal to E/B. These eddies play a similar role to the gyro-orbits in classical diffusion, except that the physics of the turbulence can be such that the decorrelation time is approximately equal to the turn-over time, resulting in Bohm scaling. Another way of looking at it is that the turbulent electric field is approximately equal to the potential perturbation divided by the scale length δ, and the potential perturbation can be expected to be a sizeable fraction of the kBT/e. The turbulent diffusion constant D = δv is then independent of the scale length and is approximately equal to the Bohm value.

The theoretical understanding of plasma diffusion especially the Bohm diffusion remained elusive until the 1970s when Taylor and McNamara[3] put forward a 2d guiding center plasma model. The concepts of negative temperature state,[4] and of the convective cells[5] contributed much to the understanding of the diffusion. The underlying physics may be explained as follows. The process can be a transport driven by the thermal fluctuations, corresponding to the lowest possible random electric fields. The low-frequency spectrum will cause the ExB drift. Due to the long range nature of Coulomb interaction, the wave coherence time is long enough to allow virtually free streaming of particles across the field lines. Thus, the transport would be the only mechanism to limit the run of its own course and to result in a self-correction by quenching the coherent transport through the diffusive damping. To quantify these statements, we may write down the diffusive damping time as τD=1/k⊥2D, where k⊥ is the wave number perpendicular to the magnetic field. Therefore, the step size is cδE τD/B, and the diffusion coefficient is

\( D=\left\langle \frac {\Delta x^2}{\tau_D}\right\rangle \sim \frac {c^2 \delta E^2}{B^2k_{\perp}^2\,D} \sim \frac {c \delta E}{Bk_{\perp}} .\)

It clearly yields for the diffusion a scaling law of B⁻¹ for the 2d plasma. The thermal fluctuation is typically a small portion of the particle thermal energy. It is reduced by the plasma parameter ε_p≡(n₀λ_D³)⁻¹<<1, and is given by |δE|²~ε_pn₀k_BT/π^1/2~4π^1/2n₀q²λ_D⁻¹, where n₀ is the plasma density, λ_D is the Debye length, and T is the plasma temperature. Taking 1/k_⊥~λ_D and substituting the electric field by the thermal energy, we would have D~(2cqπ^1/4/B)(λ_Dn₀)^1/2~(ε_p)^1/2ck_BT/qB/2π^3/4.

The 2D plasma model becomes invalid when the parallel decoherence is significant. A mechanism of Hsu diffusion proposed in 2013 by Hsu, Wu, Agarwal and Ryu.[6] predicts a scaling law of B−3/2.
See also

Classical diffusion
Hsu diffusion
Plasma diffusion

References

Bohm, D. (1949) The characteristics of electrical discharges in magnetic fields, A. Guthrie and R. K. Wakerling (eds.), New York: McGraw-Hill.
Spitzer, L. (1960). "Particle Diffusion across a Magnetic Field". Physics of Fluids 3 (4): 659–651. Bibcode:1960PhFl....3..659S. doi:10.1063/1.1706104.
Taylor, J. B. (1971). "Plasma Diffusion in Two Dimensions". Physics of Fluids 14 (7): 1492–1491. Bibcode:1971PhFl...14.1492T. doi:10.1063/1.1693635.
Montgomery, D. (1974). "Statistical mechanics of "negative temperature" states". Physics of Fluids 17 (6): 1139–1131. Bibcode:1974PhFl...17.1139M. doi:10.1063/1.1694856.
Dawson, J.; Okuda, H.; Carlile, R. (1971). "Numerical Simulation of Plasma Diffusion Across a Magnetic Field in Two Dimensions". Physical Review Letters 27 (8): 491. Bibcode:1971PhRvL..27..491D. doi:10.1103/PhysRevLett.27.491.
Hsu, Jang-Yu; Wu, Kaibang; Agarwal, Sujeet Kumar; Ryu, Chang-Mo (2013). "The B−3/2 diffusion in magnetized plasma". Physics of Plasmas 20 (6): 062302. Bibcode:2013PhPl...20f2302H. doi:10.1063/1.4811472.

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