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Neoclassical transport: Difference between revisions
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The '''Neoclassical Transport Model''' is one of the pillars of the physics of magnetically confined plasmas. | The '''Neoclassical Transport Model''' is one of the pillars of the physics of magnetically confined plasmas. | ||
<ref>F.L. Hinton and R.D. Hazeltine, [[doi:10.1103/RevModPhys.48.239|Rev. Mod. Phys. '''48''', 239 (1976)]]</ref> | <ref>F.L. Hinton and R.D. Hazeltine, [[doi:10.1103/RevModPhys.48.239|Rev. Mod. Phys. '''48''', 239 (1976)]]</ref> | ||
<ref>P. Helander and D.J. Sigmar, ''Collisional Transport in Magnetized Plasmas'', Cambridge University Press (2001) ISBN 0521807980</ref> | <ref>P. Helander and D.J. Sigmar, ''Collisional Transport in Magnetized Plasmas'', Cambridge University Press (2001) {{ISBN|0521807980}}</ref> | ||
It provides a model for the transport of particles, momentum, and heat due to Coulomb collisions in confined plasmas in complex magnetic geometries, assuming that the plasma is in a quiescent state. | It provides a model for the transport of particles, momentum, and heat due to Coulomb collisions in confined plasmas in complex magnetic geometries, assuming that the plasma is in a quiescent state. | ||
Thus, transport due to fluctuations lies outside of the scope of the theory. | Thus, transport due to fluctuations lies outside of the scope of the theory. | ||
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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. | 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. | ||
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* Maxwellianity. This assumption implies that a certain minimum level of [[Collisionality|collisionality]] is needed in order to ensure that Maxwellianisation is effective. The strong drives and resulting gradients that characterise fusion-grade plasmas often lead to a violation of this assumption. | * Maxwellianity. This assumption implies that a certain minimum level of [[Collisionality|collisionality]] is needed in order to ensure that Maxwellianisation is effective. The strong drives and resulting gradients that characterise fusion-grade plasmas often lead to a violation of this assumption. | ||
* A fixed geometry. Neoclassical transport is calculated in a static magnetic geometry. In actual experiments (especially Tokamaks), the magnetic field evolves along with the plasma itself, leading to a modification of net transport. While a slow evolution (with respect to typical transport time scales) should not be problematic, rapid changes (such as magnetic reconnections) are outside of the scope of the theory. | * A fixed geometry. Neoclassical transport is calculated in a static magnetic geometry. In actual experiments (especially Tokamaks), the magnetic field evolves along with the plasma itself, leading to a modification of net transport. While a slow evolution (with respect to typical transport time scales) should not be problematic, rapid changes (such as magnetic reconnections) are outside of the scope of the theory. | ||
* The linearity of the model. Neoclassical theory is a linear theory in which profiles are computed from sources, boundary conditions, and transport coefficients (that depend linearly on the profiles). No non-linear feedback of the profiles on the transport coefficients is contemplated. However, there are many experimental studies that show that the profiles feed back non-linearly on transport (via [[TJ-II:Turbulence|turbulence]]), leading to some degree of [[Self-Organised Criticality|self-organisation]]. | * The linearity of the model. Neoclassical theory is a linear theory in which profiles are computed from sources, boundary conditions, and transport coefficients (that depend linearly on the profiles). No non-linear feedback of the profiles on the transport coefficients is not usually contemplated. However, there are many experimental studies that show that the profiles feed back non-linearly on transport (via [[TJ-II:Turbulence|turbulence]]), leading to some degree of [[Self-Organised Criticality|self-organisation]]. | ||
* Locality. Neoclassical theory is a theory of diffusion, and therefore it assumes that particle motion between collisions is small with respect to any other relevant spatial scales. This assumption then allows writing down differential equations, expressing the fluxes in terms of ''local'' gradients. This basic assumption is violated under specific conditions, which may include: (a) the low-collisionality limit, (b) any situation in which the gradient scale length is very small (e.g., [[Internal Transport Barrier]]s), (c) locations close to the plasma edge<ref>T. Fülöp, P. Helander, [[doi:10.1063/1.1372179|Phys. Plasmas 8, 3305 (2001)]]</ref><ref>V. Tribaldos and J. Guasp, ''Neoclassical global flux simulations in stellarators'', [[doi:10.1088/0741-3335/47/3/010|Plasma Phys. Control. Fusion '''47''' (2005) 545]]</ref>, and (d) particles transported in ''streamers''. Such phenomena could give rise to [[Non-diffusive transport|super-diffusion]]. Points (a) through (c) can be handled by using a Monte Carlo or Master Equation approach instead of deriving differential equations. | * Locality. Neoclassical theory is a theory of diffusion, and therefore it assumes that radial particle motion between collisions is small with respect to any other relevant spatial scales. This assumption then allows writing down differential equations, expressing the fluxes in terms of ''local'' gradients. This basic assumption is violated under specific conditions, which may include: (a) the low-collisionality limit, (b) any situation in which the gradient scale length is very small (e.g., [[Internal Transport Barrier]]s), (c) locations close to the plasma edge<ref>T. Fülöp, P. Helander, [[doi:10.1063/1.1372179|Phys. Plasmas 8, 3305 (2001)]]</ref><ref>V. Tribaldos and J. Guasp, ''Neoclassical global flux simulations in stellarators'', [[doi:10.1088/0741-3335/47/3/010|Plasma Phys. Control. Fusion '''47''' (2005) 545]]</ref>, and (d) particles transported in ''streamers''. Such phenomena could give rise to [[Non-diffusive transport|super-diffusion]]. Points (a) through (c) can be handled by using a Monte Carlo or Master Equation approach instead of deriving differential equations. | ||
* Markovianity. A second assumption underlying diffusive models (including Neoclassics) is Markovianity, implying that the motion of any particle is only determined by its current velocity and position. However, there are situations, such as stochastic magnetic field regions, persistent turbulent eddies, or transport barriers, where this assumption may be violated (due to trapping effects, so that the preceding history of the particle trajectory becomes important). Typically, this would then give rise to [[Non-diffusive transport|sub-diffusion]]. | * Markovianity. A second assumption underlying diffusive models (including Neoclassics) is Markovianity, implying that the motion of any particle is only determined by its current velocity and position. However, there are situations, such as stochastic magnetic field regions, persistent turbulent eddies, or transport barriers, where this assumption may be violated (due to trapping effects, so that the preceding history of the particle trajectory becomes important). Typically, this would then give rise to [[Non-diffusive transport|sub-diffusion]]. | ||
== References == | == References == | ||
<references /> | <references /> | ||