Bohm diffusion
Encyclopedia
Bohm diffusion is the diffusion
of plasma
across a magnetic field
.
,
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
. 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.
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
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 inverse of the collision frequency 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.
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
of plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
across a magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
.
Description
Bohm diffusion is charactrerized with a diffusion coefficient equal to,where B is the magnetic field strength, T is the temperature, and e is the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
.
It was first observed in 1949 by David Bohm
David Bohm
David Joseph Bohm FRS was an American-born British quantum physicist who contributed to theoretical physics, philosophy, neuropsychology, and the Manhattan Project.-Youth and college:...
, E. H. S. Burhop, and Harrie Massey
Harrie Massey
Sir Harrie Stewart Wilson Massey FRS was an influential Australian mathematical physicist. He worked primarily in the fields of atomic and atmospheric physics.- Life and career :...
while studying magnetic arcs for use in isotope separation
Isotope separation
Isotope separation is the process of concentrating specific isotopes of a chemical element by removing other isotopes, for example separating natural uranium into enriched uranium and depleted uranium. This is a crucial process in the manufacture of uranium fuel for nuclear power stations, and is...
. 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
Lyman Spitzer
Lyman Strong Spitzer, Jr. was an American theoretical physicist and astronomer best known for his research in star formation, plasma physics, and in 1946, for conceiving the idea of telescopes operating in outer space...
considered this fraction as a factor related to plasma instability.
Generally diffusion can be modelled as a random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
of steps of length δ and time τ. If the diffusion is collisional, then δ is the mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
and τ is the inverse of the collision frequency. The diffusion coefficient D can be expressed variously as
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 inverse of the collision frequency 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
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
. Regions of higher or lower electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
result in eddies
Eddy (fluid dynamics)
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid flows past an obstacle. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object...
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.