# 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, regardless of the applied heating and fueling profiles.
^{[1]}
^{[2]}
The resulting (stiff) profiles are known as *canonical* profiles.
^{[3]}
This phenomenology is due to plasma self-organisation,
^{[4]}
i.e., the feedback mechanism regulating the profiles (by turbulence) is often dominant over the various source terms.
^{[5]}

## Contents

## Observations

### Tokamaks

- TFTR
^{[6]} - ASDEX
^{[7]} - Various devices
^{[8]}^{[9]}

### Stellarators

- W7-AS
^{[10]} - Various devices
^{[11]}

## Proposed explanations

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

- Marginal stability
^{[12]} - Critical gradient models
^{[13]} - Energy pinch
^{[14]}^{[15]}^{[16]} - Self-Organised Criticality
- Turbulent equipartition
^{[17]}^{[18]}

## 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:
^{[19]}
^{[20]}

- $ \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 (χ

_{0}and χ

_{1}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*) makes the profiles 'stiff' in the sense that the gradients (

_{T}*L*) will change little in response to a large change in drive (the heat source) in the appropriate parameter range.

_{T}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).
^{[21]}

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.

## References

- ↑ B. Coppi,
*Nonclassical Transport and the "Principle of Profile Consistency"*, Comments Plasma Phys. Cont. Fusion**5**, 6 (1980) 261-270 - ↑ Yu.N. Dnestrovsky et al, Sov. J. Plasma Phys.
**16**(1990) 120 - ↑ Yu.N. Dnestrovsky et al,
*Canonical profiles in tokamak plasmas with an arbitrary cross section*, Plasma Physics Reports**28**, 11 (2002) 887-899 - ↑ Yu.N. Dnestrovsky et al,
*Self-organization of plasma in tokamaks*, Plasma Physics Reports**31**, 7 (2005) 529-553 - ↑ F. Jenko et al,
*Critical gradient formula for toroidal electron temperature gradient modes*, Phys. Plasmas**8**(2001) 4096 - ↑ E.D. Fredrickson, J.D. Callen, et al.,
*Heat pulse propagation studies in TFTR*, Nucl. Fusion**26**(1986) 849 - ↑ G. Becker,
*Electron temperature profile invariance in OH, L- and H-mode plasmas and consequences for the anomalous transport*, Nucl. Fusion**32**(1992) 81 - ↑ F. Wagner and U. Stroth,
*Transport in toroidal devices-the experimentalist's view*, Plasma Phys. Control. Fusion**35**(1993) 1321 - ↑ F. Ryter, C. Angioni, et al.,
*Experimental studies of electron transport*, Plasma Phys. Control. Fusion**43**(2001) A323 - ↑ U. Stroth,
*A comparative study of transport in stellarators and tokamaks*, Plasma Phys. Control. Fusion**40**(1998) 9 - ↑ Yu.N. Dnestrovsky et al, IAEA Fusion Energy Conference, Geneva (2008) TH/P8-24
- ↑ P.H. Diamond and T.S. Hahm,
*On the dynamics of turbulent transport near marginal stability*, Phys. Plasmas**2**(1995) 3640 - ↑ F. Imbeaux and X. Garbet,
*Analytical solutions for the propagation of heat pulses with temperature gradient length-dependent diffusion coefficient*, Plasma Phys. Control. Fusion**44**(2002) 1425 - ↑ P. Mantica, G. Gorini, G.M.D. Hogeweij, N.J. Lopes Cardozo, and A.M.R. Schilham,
*Heat Convection and Transport Barriers in Low-Magnetic-Shear Rijnhuizen Tokamak Project Plasmas*, Phys. Rev. Lett.**85**(2000) 4534–4537 - ↑ X. Garbet, N. Dubuit, E. Asp, Y. Sarazin, C. Bourdelle, P. Ghendrih, and G.T. Hoang,
*Turbulent fluxes and entropy production rate*, Phys. Plasmas**12**(2005) 082511 - ↑ P. Mantica, A. Thyagaraja, J. Weiland, G.M.D. Hogeweij, and P.J. Knight,
*Heat Pinches in Electron-Heated Tokamak Plasmas: Theoretical Turbulence Models versus Experiments*, Phys. Rev. Lett.**95**(2005) 185002 - ↑ V. Naulin, A.H. Nielsen, and J. Juul Rasmussen,
*Turbulence spreading, anomalous transport, and pinch effect*, Phys. Plasmas**12**(2005) 122306 - ↑ Lu Wang and P.H. Diamond,
*Kinetic theory of the turbulent energy pinch in tokamak plasmas*, Nucl. Fusion**51**(2011) 083006 - ↑ F. Imbeaux, F. Ryter, and X. Garbet,
*Modelling of ECH modulation experiments in ASDEX Upgrade with an empirical critical temperature gradient length transport model*, Plasma Phys. Control. Fusion**43**(2001) 1503 - ↑ X. Garbet, P. Mantica, F. Ryter, et al.,
*Profile stiffness and global confinement*, Plasma Phys. Control. Fusion**46**(2004) 1351 - ↑ B.Ph. van Milligen et al,
*Quantifying profile stiffness*, Plasma and Fusion Research,**3**(2008) S1070