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Normalisation and Resolution

Source: PHY653B Ch. 1

Intuition

Before a plasma simulation can be written it must be decided what it will resolve. Plasma physics spans an obscene range of scales — electron gyroperiod to confinement time is twelve orders of magnitude — and no code resolves all of them. Choosing which scales to keep is the single most consequential design decision, and normalised units are how that decision gets written down.

The scales

Quantity Symbol Tokamak core value
Debye length \(\lambda_D\) ~70 µm
Electron gyroradius \(r_{Le}\) ~50 µm
Ion gyroradius \(r_{Li}\) ~3 mm
Machine size \(a\) ~1 m
Electron plasma period \(\omega_{pe}^{-1}\) ~2 ps
Ion gyroperiod \(\Omega_{ci}^{-1}\) ~20 ns
Confinement time \(\tau_E\) ~1 s

Resolving \(\lambda_D\) and \(\tau_E\) simultaneously in one dimension would need \(\sim10^4\) cells per metre and \(\sim10^{12}\) steps. This is not an engineering problem to be solved with a bigger machine; it is why the method hierarchy exists at all.

Resolution constraints

For an explicit electrostatic PIC code, two hard rules:

\[\Delta x \lesssim \lambda_D, \qquad \omega_{pe}\Delta t \lesssim 0.2\]

Violate the first and you get numerical grid heating: the finite-difference field solver misrepresents the short-wavelength response, energy is spuriously injected, and the plasma warms until \(\lambda_D\) grows to match \(\Delta x\) — the simulation "fixes" the violation by destroying your temperature. Violate the second and the plasma oscillation is unstable.

The consequences are worth internalising because they are silent. A grid-heated run does not crash; it produces a smooth, plausible, entirely wrong temperature history.

Normalised units

Pick a reference length, time and velocity and divide everything by them. For electrostatic work, the canonical choice is \(\omega_{pe} = 1\), \(\lambda_D = 1\), so \(v_{th} = 1\) and the equations become

\[\frac{\partial f}{\partial t} + v\frac{\partial f}{\partial x} - E\frac{\partial f}{\partial v} = 0, \qquad \frac{\partial E}{\partial x} = 1 - \int f\,dv\]

with every physical constant gone. Three benefits, in increasing order of importance:

  1. No overflow. Nothing is \(10^{-31}\) or \(10^{20}\) any more.
  2. Bugs become visible. In normalised units a correct answer is usually order 1. A result of \(10^{7}\) is obviously wrong, whereas in SI you would have to work it out.
  3. The parameters that matter become explicit. After normalisation the surviving free parameters are the dimensionless groups the physics depends on — \(k\lambda_D\), \(m_i/m_e\), \(\beta\), \(S\), \(Re\). Two runs with the same dimensionless parameters are the same simulation, whatever they claim to model.

That third point is the real one. Normalisation is dimensional analysis applied to your code, and it tells you how many runs you actually need.

Reduced mass ratio: the standard lie

Full \(m_i/m_e = 1836\) forces you to resolve electron timescales while waiting for ion ones. Many codes use 100, or 25, and it is a defensible cheat — provided you say so and check the result is not sensitive to it. Run the same case at two mass ratios; if the answer moves, the physics depends on the separation of scales and your reduced ratio has changed it.

Common mistakes

  • Reporting results in normalised units without stating the normalisation. A growth rate of 0.05 is meaningless without knowing 0.05 of what.
  • Assuming a stable run is a resolved run. Grid heating is stable. So is an under-resolved boundary layer. Stability is not accuracy.
  • Scanning dimensional parameters. Varying \(n\) and \(T\) separately when only \(k\lambda_D\) matters wastes a factor of ten in compute and hides the trend.

Knowledge graph position

Prerequisites: plasma parameters, Debye shielding. Leads to: PIC, Vlasov solvers, gyrokinetic ordering.

Quiz

Q1 (conceptual). What is numerical grid heating and why is it dangerous?

Answer

When \(\Delta x > \lambda_D\) the field solve misrepresents short-wavelength shielding and spuriously pumps energy into the particles. The plasma heats until \(\lambda_D\) grows to match \(\Delta x\) — self-consistently satisfying the constraint by ruining the temperature. It is dangerous because it is stable and smooth: nothing crashes, and the output looks fine.

Q2 (computational). A PIC run has \(\lambda_D = 70\) µm and must simulate a 1 cm region for 1 µs. Roughly how many cells and steps in 1-D?

Answer

Cells: \(10^{-2}/7\times10^{-5} \approx 140\). With \(n = 10^{20}\), \(\omega_{pe} \approx 5.6\times10^{11}\) s⁻¹, so \(\Delta t \lesssim 0.2/\omega_{pe} \approx 3.6\times10^{-13}\) s, needing \(\approx 2.8\times10^{6}\) steps. Cheap in 1-D — and the same calculation in 3-D is what makes full-device kinetic simulation impossible.

Q3 (MCQ). The main scientific benefit of normalised units is that:

  • (a) numbers are smaller
  • (b) the free parameters that survive are the dimensionless groups the physics actually depends on
  • (c) the code runs faster
  • (d) results are automatically validated
Answer

(b). It converts a large dimensional parameter space into the much smaller dimensionless one that governs the answer — the same logic as the Buckingham Pi theorem, applied to a code.