# 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 the basis of 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 or response) variable y as a function of some (positive and independent) system (or predictor) variables x1, x2,... is:

$y = e^{\alpha_0} x_1^{\alpha_1} x_2^{\alpha_2} ...$

Here, the αi 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), or at least taking mutual linear correlations into account 
• considering errors in the predictor variables and the propagation of these errors in order to define confidence intervals for the resulting predictions

## Dimensionless parameters

In the magnetic confinement context, and assuming quasi-neutrality, the relevant scaling laws (mainly, for the energy confinement time) 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, ρi is the ion Larmor radius and νii the ion-ion collision frequency. Also see beta and collisionality.

In dimensionless form, the diffusivities can be written as:

$D = c_s \rho_s (\rho^*)^\alpha F(\nu^*,\beta,q, ...)\,$

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

## Confinement time scaling

The main performance parameter that is subjected to scaling law analysis is the energy confinement time, τE. The energy confinement time is expressed in engineering variables:

$\tau_E = C I^{\alpha_I} B^{\alpha_B} \overline{n}^{\alpha_n} P^{\alpha_P} R^{\alpha_R} \kappa^{\alpha_\kappa} \epsilon^{\alpha_\epsilon} S_{cr}^{\alpha_S} M^{\alpha_M}$

where

• I (MA) is the plasma current
• B (T) is the toroidal magnetic field
• $\overline{n}$ (1019 m-3) is the central line averaged density
• P (MW) is the absorbed power
• R (m) is the major radius
• κ is the elongation
• ε is the inverse aspect ratio
• Scr is the cross sectional area
• M is the hydrogen isotope mass

The following tables shows some of the most generally used sets of scaling parameters for the ELMy H-mode and L-mode.    

Scaling $C\times10^3$ $\alpha_I$ $\alpha_B$ $\alpha_n$ $\alpha_P$ $\alpha_R$ $\alpha_\kappa$ $\alpha_\epsilon$ $\alpha_S$ $\alpha_M$
ITERH-98P(y,2) 56.2 0.93 0.15 0.41 -0.69 1.97 0.78 0.58 - 0.19
ITPAH-04P(y,1) 22.8 0.86 0.21 0.40 -0.65 0.32 - -0.99 0.84 0.08
ITPAH-04P(y,2) 19.8 0.85 0.17 0.26 -0.45 -0.04 - -1.25 0.82 0.11
ITPAH-04P(y,3) 88.0 0.90 - 0.30 -0.47 1.73 - 0.43 - -
ITER-89P 38 0.85 0.20 0.10 -0.50 1.50 0.50 0.30 - 0.50
ITERL-96P(th) 23 0.96 0.03 0.40 -0.73 1.83 0.64 -0.06 - 0.20

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

• ISS04v3
$\tau_E = 0.148 R^{0.64} a^{2.33} \bar n_{20}^{0.55} B^{0.85} \iota_{2/3}^{0.41} P^{-0.61}$

There is an ongoing discussion on whether to replace the plasma size quantifiers (a,R) by (S,V) (surface and volume), which might be more appropriate for stellarator flux surfaces, deviating strongly from a torus (see Effective plasma radius).

One of the remarkable and initially unexpected properties of magnetically confined plasmas is the reduction of the energy confinement time τE 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 (anomalous) transport (i.e., more than the expected -diffusive- increase due to the increased gradient alone), producing the observed confinement degradation. This phenomenon is therefore a form of plasma self-organisation.

### Size scaling

The ELMy H-mode scaling is of the gyro-Bohm type (α = 1). Gyro-Bohm scaling is what one would expect for diffusive transport based on a diffusive scale length proportional to ρi (the ion gyroradius).

By contrast, the L-mode scaling is of the Bohm type (α = 0), which suggests that transport may not be diffusive and not characterized by a typical scale length, i.e., it is dominated by the scale length corresponding to the machine size (non-locality).  One possible explanation of this behaviour is Self-Organised Criticality, i.e., the self-regulation of transport by turbulence, triggered when a critical value of the gradient is exceeded. As a corollary, this mechanism might also explain the phenomenon of profile consistency.