Scaling law

A scaling law is an engineering tool to predict the value of a system variable as a function of some other significant variables. Their extended use in magnetic confinement physics reflects the fact that detailed transport calculations or predictions on first principles are difficult in this field. In the latter context, they are mainly used to

• predict the performance of new (larger) devices, such as ITER
• summarize large amounts of experimental data
• make performance comparisons between devices
• make educated guesses at local transport mechanisms

General method

The typical scaling law expression for a (dependent) variable y as a function of some (independent) system variables x₁, x₂,... is:

$y=e^{\alpha _{0}}x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}...$

Here, the α_i are the scaling parameters. By taking the logarithm of this expression, it becomes linear in the parameters and simple (multivariate) linear regression tools can be used to determine the parameters from a set of data. However, a proper analysis requires:

• using dimensionless variables (easily achieved by normalizing all quantities appropriately)
• guaranteeing the (linear) statistical independence of the independent variables (applying, e.g., Principal Component Analysis)

Dimensionless parameters

Assuming quasi-neutrality, the relevant scaling laws can be cast into dimensionless forms that involve only three plasma parameters (apart from geometrical factors):

$\rho *={\frac {\rho _{i}}{a}}$

$\beta ={\frac {\left\langle p\right\rangle }{B^{2}/2\mu _{0}}}$

$\nu *\propto \nu _{ii}$

Here, ρ_i is the ion Larmor radius and ν_ii the ion-ion collision frequency. Also see beta.

Confinement time scaling

The main performance parameter that is subjected to scaling law analysis is the energy confinement time, τ_E. The following are some of the most-used scalings for tokamaks:

• L-mode scaling
• ELMy H-mode scaling (IPB98(y,2))

For stellarators, a similar scaling has been derived (ISS).

Power degradation

One of the remarkable and initially unexpected properties of magnetically confined plasmas is the reduction of the energy confinement time τ_E as the heating power P is increased. Typically:

$\tau _{E}\propto P^{-\alpha }$

where α has a value of 0.6 ± 0.1. The reason for this behaviour has not been fully clarified. Qualitatively, it seems obvious that an increase of P will lead to an increase of (temperature and density) gradients, and thus an increase of "free energy" available to instabilities and turbulence. These instabilities may grow by feeding on the "free energy", which may lead to an increase of transport, producing the observed confinement degradation. This phenomenon is therefore a form of plasma self-organisation.

Size scaling

It is convenient to express the diffusivities in dimensionless form:

$D=c_{s}\rho _{s}(\rho ^{*})^{\alpha }F(\nu ^{*},\beta ,q,...)\,$

When α = 0, the scaling is said to be of the Bohm type, and when α = 1, of the gyro-Bohm type.

The L-mode scaling is of the Bohm type, while the ELMy H-mode scaling is of the gyro-Bohm type. Gyro-Bohm scaling is what one would expect for diffusive transport based on a diffusive scale length proportional to ρ_i (the ion gyroradius). Bohm scaling, however, suggests that transport may not be diffusive and is not characterized by a typical scale length, i.e., is dominated by the scale length corresponding to the machine size (non-local).