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Neoclassical transport: Difference between revisions
→Brief summary of the theory
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:<math> | :<math> | ||
\frac{\partial f_\alpha}{\partial t} + v\cdot \nabla f_\alpha + F \frac{\partial f_\alpha}{\partial v} = C_\alpha(f) | \frac{\partial f_\alpha}{\partial t} + v\cdot \nabla f_\alpha + F \frac{\partial f_\alpha}{\partial v} = C_\alpha(f) + S_\alpha | ||
</math> | </math> | ||
where <math>\alpha</math> indicates the particle species, <math>v</math> is the velocity, | where <math>\alpha</math> indicates the particle species, <math>v</math> is the velocity, | ||
<math>F</math> is a force (the [http://en.wikipedia.org/wiki/Lorentz_force Lorentz force] acting on the particle) and <math>C_\alpha</math> the [[Collision operator|collision operator]]. | <math>F</math> is a force (the [http://en.wikipedia.org/wiki/Lorentz_force Lorentz force], <math>F = q(E + v \times B)</math> acting on the particle), <math>S_\alpha</math> a source and <math>C_\alpha</math> the [[Collision operator|collision operator]]. | ||
If the chosen collision operator is the Fokker-Planck operator, the equation is called the [http://en.wikipedia.org/wiki/Fokker-planck Fokker-Planck Equation]. | If the chosen collision operator is the Fokker-Planck operator, the equation is called the [http://en.wikipedia.org/wiki/Fokker-planck Fokker-Planck Equation]. | ||
The derivation of this collision operator is highly non-trivial and requires making specific assumptions; | The derivation of this collision operator is highly non-trivial and requires making specific assumptions; | ||
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The collision operator must also satisfy some obvious conservation laws (conservation of particles, momentum, and energy). | The collision operator must also satisfy some obvious conservation laws (conservation of particles, momentum, and energy). | ||
Once the collision operator is decided, the moments of the Kinetic Equation can be computed. | Once the collision operator is decided, the moments of the Kinetic Equation can be computed. | ||
These fluid moments are: | |||
:<math> | :<math> | ||
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(heat flux) | (heat flux) | ||
As an example, the evolution equation of the first moment becomes: | |||
:<math> | |||
\frac{\partial n}{\partial t} + \nabla \cdot (n u) = S_n | |||
</math> | |||
Similar conservation-type equations can be written down for the higher moments. | |||
The main goal of Neoclassical transport theory is to provide a closed set of equations for the time evolution of these moments, for each particle species. Since the determination of any moment requires knowledge of the next order moment, this requires truncating the set of moments (''closure'' of the set of equations). | The main goal of Neoclassical transport theory is to provide a closed set of equations for the time evolution of these moments, for each particle species. Since the determination of any moment requires knowledge of the next order moment, this requires truncating the set of moments (''closure'' of the set of equations). | ||
<ref>T.J.M. Boyd and J.J. Sanderson, ''The physics of plasmas'', Cambridge University Press (2003) ISBN 0521459125</ref> | <ref>T.J.M. Boyd and J.J. Sanderson, ''The physics of plasmas'', Cambridge University Press (2003) ISBN 0521459125</ref> | ||
It is customary to make a number of additional assumptions to facilitate the analysis: e.g., small gyroradius, nested magnetic surfaces, large parallel transport, Maxwellian distribution functions, etc. | |||
As a consequence of such assumptions, the equations can be restated to reflect the 'radial' transport (normal to the flux surfaces, and averaged over flux surfaces). | |||
Thus, the magnetic geometry is incorporated at an essential level in the theory. | |||
The theory takes account of all particle motion associated with toroidal geometry; specifically, ''∇ B'' and curvature drifts, and passing and trapped particles (banana orbits). | The theory takes account of all particle motion associated with toroidal geometry; specifically, ''∇ B'' and curvature drifts, and passing and trapped particles (banana orbits). | ||