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Profile consistency (view source)

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<ref>[http://link.aip.org/link/?PHPAEN/8/4096/1 F. Jenko et al, ''Critical gradient formula for toroidal electron temperature gradient modes'', Phys. Plasmas '''8''' (2001) 4096]</ref>
<ref>[http://link.aip.org/link/?PHPAEN/8/4096/1 F. Jenko et al, ''Critical gradient formula for toroidal electron temperature gradient modes'', Phys. Plasmas '''8''' (2001) 4096]</ref>


== Observations ([[Tokamak|tokamaks]]) ==
== Observations ==
 
=== [[Tokamak]]s ===


* TFTR <ref>[http://dx.doi.org/10.1088/0029-5515/26/7/002 E.D. Fredrickson, J.D. Callen, et al., ''Heat pulse propagation studies in TFTR'',  Nucl. Fusion '''26''' (1986) 849]</ref>
* TFTR <ref>[http://dx.doi.org/10.1088/0029-5515/26/7/002 E.D. Fredrickson, J.D. Callen, et al., ''Heat pulse propagation studies in TFTR'',  Nucl. Fusion '''26''' (1986) 849]</ref>
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* Various devices <ref>[http://dx.doi.org/10.1088/0741-3335/35/10/002 F. Wagner and U. Stroth, ''Transport in toroidal devices-the experimentalist's view'', Plasma Phys. Control. Fusion '''35''' (1993) 1321]</ref><ref>[http://dx.doi.org/10.1088/0741-3335/43/12A/325 F. Ryter, C. Angioni, et al., ''Experimental studies of electron transport'', Plasma Phys. Control. Fusion '''43''' (2001) A323]</ref>
* Various devices <ref>[http://dx.doi.org/10.1088/0741-3335/35/10/002 F. Wagner and U. Stroth, ''Transport in toroidal devices-the experimentalist's view'', Plasma Phys. Control. Fusion '''35''' (1993) 1321]</ref><ref>[http://dx.doi.org/10.1088/0741-3335/43/12A/325 F. Ryter, C. Angioni, et al., ''Experimental studies of electron transport'', Plasma Phys. Control. Fusion '''43''' (2001) A323]</ref>


== Observations ([[Stellarator|stellarators]]) ==
=== [[Stellarator]]s ===


* W7-AS <ref>[http://dx.doi.org/10.1088/0741-3335/40/1/002 U. Stroth, ''A comparative study of transport in stellarators and tokamaks'', Plasma Phys. Control. Fusion '''40''' (1998) 9]</ref>
* W7-AS <ref>[http://dx.doi.org/10.1088/0741-3335/40/1/002 U. Stroth, ''A comparative study of transport in stellarators and tokamaks'', Plasma Phys. Control. Fusion '''40''' (1998) 9]</ref>
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== Quantification methods ==
== 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:
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:
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<ref>[http://dx.doi.org/10.1088/0741-3335/46/9/002 X. Garbet, P. Mantica, F. Ryter, et al., ''Profile stiffness and global confinement'', Plasma Phys. Control. Fusion '''46''' (2004) 1351]</ref>
<ref>[http://dx.doi.org/10.1088/0741-3335/46/9/002 X. Garbet, P. Mantica, F. Ryter, et al., ''Profile stiffness and global confinement'', Plasma Phys. Control. Fusion '''46''' (2004) 1351]</ref>


:<math>\chi = \chi_0 + \chi_1 \xi \left ( \frac{R}{L_T}-\frac{R}{L_{T,crit}}\right )^\alpha H_c</math>
:<math>\chi = \chi_0 + \chi_1 \xi \left ( \frac{R}{L_T}-\frac{R}{L_{T,crit}}\right )^\alpha H_{crit}</math>


Here, ''H<sub>c</sub>'' is a step function (to activate supercritical transport), ''L<sub>T</sub> = T/&nabla; T'' is the temperature gradient scale length, and &chi; is the heat transport coefficient (&chi;<sub>0</sub> and &chi;<sub>1</sub> being the sub- and super-critical transport coefficients, and &xi; the 'stiffness factor').
Here, ''H<sub>crit</sub>'' is a step function (to activate supercritical transport), ''L<sub>T</sub> = T/&nabla; T'' is the temperature gradient scale length, and &chi; is the heat transport coefficient (&chi;<sub>0</sub> and &chi;<sub>1</sub> being the sub- and super-critical transport coefficients, and &xi; the 'stiffness factor').
This sharply non-linear dependence of the transport coefficient on the relevant profile parameter (''L<sub>T</sub>'') makes the profiles 'stiff' in the sense that the gradients (''L<sub>T</sub>'') will change little in response to a large change in drive (the heat source) in the appropriate parameter range.
This sharply non-linear dependence of the transport coefficient on the relevant profile parameter (''L<sub>T</sub>'') makes the profiles 'stiff' in the sense that the gradients (''L<sub>T</sub>'') 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).
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).
<ref>[http://www.jspf.or.jp/PFR/PFR_articles/pfr2008S1/pfr2008_03-S1070.html B.Ph. van Milligen et al, ''Quantifying profile stiffness'', Plasma and Fusion Research, '''3''' (2008) S1070]</ref>
<ref>[http://www.jspf.or.jp/PFR/PFR_articles/pfr2008S1/pfr2008_03-S1070.html B.Ph. van Milligen et al, ''Quantifying profile stiffness'', Plasma and Fusion Research, '''3''' (2008) S1070]</ref>
The [[:Wikipedia:Stiffness|general definition of stiffness of a system]] is
:<math> \kappa = \frac{\Delta F}{\delta}</math>
i.e., the stiffness ''&kappa;'' is the applied force change ''&Delta; F'' divided by the system response (displacement) ''&delta;''.
In the case at hand, the (thermodynamic) force or drive is the heat flux ''Q'', whereas the system response is the (thermodynamic) gradient ''&nabla; T'' (but see below).
Another issue to take into account is that a useful measure of stiffness should depend on the quantities (''&Delta; F'' and ''&delta;'') in such a way that the extreme case of a totally stiff system would correspond to ''&kappa; = &#8734;'' (''&delta; = 0'').
Thus, assuming that profile stiffness is best evidenced in the normalized gradient (or inverse gradient length) ''&nabla; T / T'' (based on both experimental observation and, e.g., ETG instability theory), an appropriate stiffness definition for the temperature profile could be:
:<math>\kappa = \frac{\Delta (Q/nT)}{\Delta (\nabla T / T)}</math>
where the heat flux ''Q'' has been normalized by the pressure ''nT'' so that ''&kappa;'' has the dimension of a heat diffusivity.
The stiffness can thus be measured directly by observing the behaviour of the gradients as the drive (''Q'') is changed.


== References ==
== References ==
<references />
<references />
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