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Gyrokinetic Ordering

Source: PHY653B Ch. 6

Intuition

Tokamak turbulence happens on timescales far longer than the gyroperiod, at frequencies far below the cyclotron frequency. Resolving every gyro-orbit to study it is like integrating molecular vibrations to model weather. Gyrokinetics removes the gyration analytically, reducing six dimensions to five and lifting the time-step constraint by the mass ratio — the single biggest win in computational plasma physics.

The ordering

Every gyrokinetic derivation assumes one small parameter, \(\varepsilon = \rho_i/L \sim 10^{-3}\) in a tokamak, and orders everything against it:

\[\frac{\omega}{\Omega_{ci}} \sim \varepsilon,\qquad \frac{k_\parallel}{k_\perp} \sim \varepsilon,\qquad \frac{\delta f}{f_0} \sim \varepsilon,\qquad \frac{e\phi}{T} \sim \varepsilon,\qquad k_\perp\rho_i \sim 1\]

Read the last one carefully — it is the clever part. Perpendicular wavelengths are comparable to the gyroradius, not much larger. Gyrokinetics keeps finite-Larmor-radius physics in full while eliminating the fast gyration. It is not a long-wavelength approximation; it is a low-frequency one.

What survives the gyro-average

Averaging over gyrophase replaces each particle with a charged ring of radius \(\rho\). The ring, not the point, is what responds to the field, so a particle samples the potential around its whole orbit. In Fourier space the gyro-average is multiplication by \(J_0(k_\perp\rho)\) — a Bessel function that suppresses the response to structures finer than a gyroradius.

This has a real physical consequence, not merely a computational one: the plasma responds to the field averaged over the gyro-orbit, which is why turbulent eddies at \(k_\perp\rho_i \sim 1\) are the ones that matter for transport.

The field equation changes too. Quasi-neutrality is imposed directly (no Poisson equation), and the difference between the gyro-averaged and true densities produces the polarisation density — the gyrokinetic analogue of a dielectric response.

What you gain and what you lose

Gain: six phase-space dimensions become five (\(\mathbf{x}\), \(v_\parallel\), \(\mu\), with \(\mu\) a conserved parameter rather than a coordinate). The time step is set by \(\omega \ll \Omega_{ci}\) rather than by the gyroperiod. Together this is several orders of magnitude — the difference between impossible and routine.

Lose: all cyclotron-frequency physics. Ion and electron cyclotron heating, cyclotron resonances, anything with \(\omega \sim \Omega_c\) — invisible by construction. Gyrokinetics cannot be used to study RF heating, and no amount of resolution changes that. It is a model choice, not a numerical approximation.

Common mistakes

  • Applying it where \(\omega \sim \Omega_{ci}\). The ordering is violated and the results are meaningless — not inaccurate, meaningless.
  • Assuming it is a long-wavelength theory. \(k_\perp\rho_i \sim 1\) is central; FLR physics is retained in full.
  • Forgetting \(\mu\) is a coordinate, not a constant of the simulation. It is conserved per particle, which is exactly what lets it be a parameter rather than a dimension.

Knowledge graph position

Prerequisites: guiding-centre drifts, adiabatic invariants, Vlasov–Poisson. Leads to: ITG modes and zonal flows, GENE/GS2/CGYRO-class production codes.

Quiz

Q1 (conceptual). Why is \(k_\perp\rho_i \sim 1\) (rather than \(\ll 1\)) essential to gyrokinetics?

Answer

Because the turbulence that drives transport lives at exactly those perpendicular scales. Ordering \(k_\perp\rho_i \ll 1\) would throw away finite-Larmor-radius effects and the physics of interest with them. Gyrokinetics is a low-frequency expansion that deliberately retains full FLR structure through the \(J_0(k_\perp\rho)\) gyro-average.

Q2 (computational). Estimate the speed-up from removing the gyromotion constraint for ions in a tokamak with \(\Omega_{ci}/\omega_{\rm turb} \sim 10^3\).

Answer

The time step can grow by roughly that factor, ~\(10^3\), and one velocity dimension is eliminated. Combined, the saving is several orders of magnitude — which is why gyrokinetic turbulence simulation is routine while full-orbit kinetic simulation of the same problem is not.

Q3 (MCQ). Gyrokinetics cannot be used to study:

  • (a) ion-temperature-gradient turbulence
  • (b) zonal flows
  • (c) ion cyclotron resonance heating
  • (d) transport driven by drift waves
Answer

(c). Cyclotron-frequency physics is averaged away by construction. This is a model limitation, not a resolution limitation — refining the grid cannot recover it.