Profile consistency

Profile consistency (or profile resilience or stiffness) is the observation that profiles (of temperature, density, and pressure) often tend to adopt roughly the same shape (in tokamaks), regardless of the applied heating and fueling profiles. The resulting (stiff) profiles are known as canonical profiles. This phenomenology is due to plasma self-organisation, i.e., the feedback mechanism regulating the profiles (by turbulence) is often dominant over the various source terms.

Observations

Tokamaks

• TFTR
• ASDEX
• Various devices

Stellarators

• W7-AS
• Various devices

Proposed explanations

The phenomenon is not fully understood, but attempts at explanation have been made. These include:

• Marginal stability
• Critical gradient models
• Energy pinch
• Turbulent equipartition

Quantification methods

Ad-hoc transport models

It is customary to introduce an ad-hoc transport model with a critical gradient (sharply enhanced transport above a critical value of the local gradient) to attempt to quantify the 'criticality' of transport:

$\chi =\chi _{0}+\chi _{1}\xi \left({\frac {R}{L_{T}}}-{\frac {R}{L_{T,crit}}}\right)^{\alpha }H\left({\frac {R}{L_{T}}}-{\frac {R}{L_{T,crit}}}\right)$

Here, H is a step function (to activate supercritical transport), L_T = T/∇ T is the temperature gradient scale length, and χ is the heat transport coefficient (χ₀ and χ₁ being the sub- and super-critical transport coefficients, and ξ the 'stiffness factor'). This sharply non-linear dependence of the transport coefficient on the relevant profile parameter (L_T) makes the profiles 'stiff' in the sense that the gradients (L_T) will change little in response to a large change in drive (the heat source) in the appropriate parameter range.

The degree of stiffness can then be gauged by fitting the predictions of the ad-hoc model to experimental results, involving different heating schemes and/or heating modulation.

Directly measuring stiffness

However, it is possible to devise methods for the objective quantification of profile stiffness that do not depend so much on the introduction of any ad-hoc model, simply by making this idea of stiffness explicit (i.e., by measuring the response of the gradient to a change in drive or heat source).

The general definition of stiffness of a system is

$\kappa ={\frac {\Delta F}{\delta }}$

i.e., the stiffness κ is the applied force change Δ F divided by the system response (displacement) δ. In the case at hand, the (thermodynamic) force or drive is the heat flux Q, whereas the system response is the (thermodynamic) gradient ∇ T (but see below).

A useful measure of stiffness should depend on the quantities (Δ F and δ) in such a way that the extreme case of a totally stiff system would correspond to κ = ∞ (δ = 0). Thus, assuming that profile stiffness is best evidenced in the normalized gradient (or inverse gradient length) ∇ T / T (based on both experimental observation and, e.g., ETG instability theory), an appropriate stiffness definition for the temperature profile could be:

$\kappa ={\frac {\Delta (Q/nT)}{\Delta (\nabla T/T)}}$

where the heat flux Q has been normalized by the pressure nT so that κ has the dimension of a heat diffusivity. A dimensionless stiffness measure is obtained by normalizing κ to the background heat diffusivity χ: C = κ/χ. A non-stiff situation would correspond to C = 1, whereas a stiff situation would yield C >> 1. One concludes that the stiffness can be measured directly by simply observing the behaviour of the gradients as the drive (Q) is changed.