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Scaling laws are an engineering tool to predict the performance of a system as a function of some significant parameters.
A scaling law is an engineering tool to predict the value of a system variable as a function of some other significant variables.
<ref>O.J.W.F. Kardaun, ''Classical methods of statistics: with applications in fusion-oriented plasma physics'', Springer Science & Business (2005) ISBN 3540211152</ref>
<ref>O.J.W.F. Kardaun, ''Classical methods of statistics: with applications in fusion-oriented plasma physics'', Springer Science & Business (2005) {{ISBN|3540211152}}</ref>
Its 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
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]]
* predict the performance of new (larger) devices, such as [[ITER]]
* summarize large amounts of experimental data
* summarize large amounts of experimental data
Line 7: Line 7:
* make educated guesses at local transport mechanisms
* make educated guesses at local transport mechanisms


The typical scaling law expression for a (dependent) variable ''y'' as a function of some (independent) system variables ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is:
 
== 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 ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is:
:<math>y = e^{\alpha_0} x_1^{\alpha_1} x_2^{\alpha_2} ...</math>
:<math>y = e^{\alpha_0} x_1^{\alpha_1} x_2^{\alpha_2} ...</math>
Here, the &alpha;<sub>i</sub> are the scaling parameters.
Here, the &alpha;<sub>i</sub> are the scaling parameters.
By taking the logarithm of this expression, it becomes linear and simple (multivariate) linear regression tools can be used to determine the parameters from a set of data.  
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:
However, a proper analysis requires:
* using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately)
* using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately)<ref>T.C. Luce, C.C. Petty, and J.G. Cordey,''Application of dimensionless parameter scaling techniques to the design and interpretation of magnetic fusion experiments'', [[doi:10.1088/0741-3335/50/4/043001|Plasma Phys. Control. Fusion, '''50''', 4 (2008) 043001]]</ref>
* guaranteeing the (linear) ''statistical independence'' of the independent variables (applying, e.g., [[:Wikipedia:Principal component analysis|Principal Component Analysis]])
* guaranteeing the (linear) ''statistical independence'' of the independent variables (applying, e.g., [[:Wikipedia:Principal component analysis|Principal Component Analysis]]), or at least taking mutual linear correlations into account <ref>S.M. Kaye et al, ''The role of aspect ratio and beta in H-mode confinement scalings'', [[doi:10.1088/0741-3335/48/5A/S44|Plasma Phys. Control. Fusion '''48''' (2006) A429]]</ref>
* 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|energy confinement time]]) can be cast into dimensionless forms that involve only three plasma parameters (apart from geometrical factors):
<ref name="ITER"/>
<ref>B.B. Kadomtsev,  Sov. J. Plasma Phys. '''1''' (1975) 295</ref>
 
:<math>\rho* = \frac{\rho_i}{a}</math>
 
:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>
 
:<math>\nu* \propto \nu_{ii}</math>
 
Here, &rho;<sub>i</sub> is the ion [[Larmor radius]] and &nu;<sub>ii</sub> the ion-ion collision frequency. Also see [[Beta|beta]] and [[Collisionality|collisionality]].
 
In dimensionless form, the diffusivities can be written as:
:<math>D = c_s \rho_s (\rho^*)^\alpha F(\nu^*,\beta,q, ...)\,</math>
When &alpha; = 0, the scaling is said to be of the Bohm type, and when &alpha; = 1, of the gyro-Bohm type.


== Confinement time scaling ==
== Confinement time scaling ==


The main performance parameter that is subjected to scaling law analysis is the [[Energy confinement time|energy confinement time]], &tau;<sub>E</sub>.
The main performance parameter that is subjected to scaling law analysis is the [[Energy confinement time|energy confinement time]], &tau;<sub>E</sub>.
The following are some of the most-used scalings for tokamaks:
The energy confinement time is expressed in engineering variables:
<ref>[http://dx.doi.org/10.1088/0029-5515/39/12/301 ITER Physics Expert Groups on Confinement Modelling and Transport, Confinement Modelling and Database, and ITER Physics Basis Editors, Nucl. Fusion '''39''' (1999) 2137]</ref>
* L-mode scaling (H<sub>89P</sub>)
* ELMy H-mode scaling (H<sub>H98</sub>(y,2))


For stellarators, a similar scaling has been derived.
:<math>\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}</math>
 
where
* I (MA) is the plasma current
* B (T) is the toroidal magnetic field
* <math>\overline{n}</math> (10<sup>19</sup> m<sup>-3</sup>) is the central line averaged density
* P (MW) is the absorbed power
* R (m) is the major radius
* &kappa; is the [[ellipticity|elongation]]
* &epsilon; is the inverse [[Toroidal coordinates|aspect ratio]]
* S<sub>cr</sub> 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.
<ref name="ITER">ITER Physics Expert Groups et al, ''ITER Physics Basis, Chapter 1'', [[doi:10.1088/0029-5515/39/12/301|Nucl. Fusion '''39''' (1999) 2137]] and
Ibid., ''Chapter 2'', [[doi:10.1088/0029-5515/39/12/302|Nucl. Fusion 48 (2008) 099801]]</ref>
<ref>J.G. Cordey, J.A. Snipes, M. Greenwald, et al., IAEA 20th Fusion Energy Conference, Vilamoura, Portugal, 2004, paper IAEA-CN-116/IT/P3-32, submitted to Nucl. Fusion.</ref>
<ref>P.N. Yushmanov, T. Takizuka, K.S. Riedel, et al., [[doi:10.1088/0029-5515/30/10/001|Nucl. Fusion 30 (1990) 1999]]</ref>
<ref>S.M. Kaye, et al., [[doi:10.1088/0029-5515/37/9/I10|Nucl. Fusion 37 (1997) 1303]]</ref>
 
{| class="wikitable sortable" border="1" cellpadding="4" cellspacing="0"
|+
|- style="background:#FFDEAD;"
!Scaling
!<math>C\times10^3</math>
!<math>\alpha_I</math>
!<math>\alpha_B</math>
!<math>\alpha_n</math>
!<math>\alpha_P</math>
!<math>\alpha_R</math>
!<math>\alpha_\kappa</math>
!<math>\alpha_\epsilon</math>
!<math>\alpha_S</math>
!<math>\alpha_M</math>
|-
|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 [[stellarator]]s, a similar scaling has been obtained (ISS).
<ref>[http://www.ipp.mpg.de/ISS ISS-IPP] and [http://iscdb.nifs.ac.jp/ ISS-NIFS] homepages</ref>
<ref>[http://www.ipp.mpg.de/ISS ISS-IPP] and [http://iscdb.nifs.ac.jp/ ISS-NIFS] homepages</ref>
<ref>[http://dx.doi.org/10.1088/0029-5515/47/9/025 A. Dinklage et al, ''Physical model assessment of the energy confinement time scaling in stellarators'', Nuclear Fusion '''47''', 9 (2007) 1265-1273]</ref>
<ref>A. Dinklage et al, ''Physical model assessment of the energy confinement time scaling in stellarators'', [[doi:10.1088/0029-5515/47/9/025|Nucl. Fusion '''47''', 9 (2007) 1265-1273]]</ref>
* ISS04v3
:<math>\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}</math>
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 [[:Wikipedia:Torus|torus]] (see [[Effective plasma radius]]).


== Power degradation ==
=== Power degradation ===


One of the remarkable and initially unexpected properties of magnetically confined plasmas is the reduction of the [[Energy confinement time|energy confinement time]] &tau;<sub>E</sub> as the heating power ''P'' is increased. Typically:
One of the remarkable and initially unexpected properties of magnetically confined plasmas is the reduction of the [[Energy confinement time|energy confinement time]] &tau;<sub>E</sub> as the heating power ''P'' is increased. Typically:
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:<math>\tau_E \propto P^{-\alpha}</math>
:<math>\tau_E \propto P^{-\alpha}</math>


where &alpha; has a value of 0.6 &plusmn; 0.1. The reason for this behaviour has not been fully clarified. However, 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. This then leads to an increase of transport, producing the observed confinement degradation.  
where &alpha; has a value of 0.6 &plusmn; 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|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 due to plasma [[Self-Organised Criticality|self-organisation]].
This phenomenon is therefore a form of plasma [[Self-Organised Criticality|self-organisation]].


== Dimensionless parameters ==
=== Size scaling ===


Assuming quasi-neutrality, the relevant scaling laws can be cast into dimensionless forms that involve only three plasma parameters (apart from geometrical factors):
The ELMy [[H-mode]] scaling is of the gyro-Bohm type (&alpha; = 1).
<ref>B.B. Kadomtsev,  Sov. J. Plasma Phys. '''1''' (1975)295</ref>
Gyro-Bohm scaling is what one would expect for diffusive transport based on a diffusive scale length proportional to &rho;<sub>i</sub> (the ion gyroradius).


:<math>\rho* = \frac{\rho_i}{a}</math>
By contrast, the L-mode scaling is of the Bohm type (&alpha; = 0), which suggests that transport may [[Non-diffusive transport|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).
<ref>A. Dinklage, ''Plasma physics: confinement, transport and collective effects'', Vol. 670 of Lecture notes in physics, Springer (2005) {{ISBN|3540252746}}</ref>
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|profile consistency]].


:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>
=== See also ===


:<math>\nu*</math>
* [[Fusion databases]]


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