Scaling law - Revision history

Admin at 10:43, 26 January 2023

2023-01-26T10:43:16Z

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	A scaling law is an engineering tool to predict the value of a system variable as a function of some other significant variables.		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>
	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		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]]
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	By contrast, the L-mode scaling is of the Bohm type (α = 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).		By contrast, the L-mode scaling is of the Bohm type (α = 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>		<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]].		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]].

Admin: /* Confinement time scaling */

2017-08-23T06:59:01Z

Confinement time scaling

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	The following tables shows some of the most generally used sets of scaling parameters for the ELMy [[H-mode]] and L-mode.		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		<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
	[[doi:10.1088/0029-5515/39/12/302\|~~Ibid~~.~~, ''Chapter 2''~~]]</ref>		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>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>P.N. Yushmanov, T. Takizuka, K.S. Riedel, et al., [[doi:10.1088/0029-5515/30/10/001\|Nucl. Fusion 30 (1990) 1999]]</ref>

Admin: /* Size scaling */

2017-08-22T11:46:23Z

Size scaling

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	<ref>A. Dinklage, ''Plasma physics: confinement, transport and collective effects'', Vol. 670 of Lecture notes in physics, Springer (2005) ISBN 3540252746</ref>		<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]].		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]].

			=== See also ===

			* [[Fusion databases]]

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

Schmittj: /* Confinement time scaling */

2015-03-12T15:53:56Z

Confinement time scaling

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	\|-		\|-
	\|ITER-89P		\|ITER-89P
	\|48		\|38
	\|0.85		\|0.85
	\|0.20		\|0.20

Admin: /* Confinement time scaling */

2015-01-16T10:52:14Z

Confinement time scaling

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	* R (m) is the major radius		* R (m) is the major radius
	* κ is the [[ellipticity\|elongation]]		* κ is the [[ellipticity\|elongation]]
	* ε is the inverse aspect ratio		* ε is the inverse [[Toroidal coordinates\|aspect ratio]]
	* S<sub>cr</sub> is the cross sectional area		* S<sub>cr</sub> is the cross sectional area
	* M is the hydrogen isotope mass		* M is the hydrogen isotope mass

Admin: /* General method */

2015-01-15T10:48:42Z

General method

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	However, a proper analysis requires:		However, a proper analysis requires:
	* 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>		* 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]]), 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>		* 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		* considering errors in the predictor variables and the propagation of these errors in order to define confidence intervals for the resulting predictions

Admin: /* General method */

2015-01-15T10:48:08Z

General method

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	== General method ==		== General method ==

	The typical scaling law expression for a (dependent) variable ''y'' as a function of some (positive and independent) system variables ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is:		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 α<sub>i</sub> are the scaling parameters.		Here, the α<sub>i</sub> are the scaling parameters.
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	However, a proper analysis requires:		However, a proper analysis requires:
	* 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>		* 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 ==		== Dimensionless parameters ==

Admin: /* Confinement time scaling */

2014-06-11T08:32:47Z

Confinement time scaling

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	* P (MW) is the absorbed power		* P (MW) is the absorbed power
	* R (m) is the major radius		* R (m) is the major radius
	* κ is the elongation		* κ is the [[ellipticity\|elongation]]
	* ε is the inverse aspect ratio		* ε is the inverse aspect ratio
	* S<sub>cr</sub> is the cross sectional area		* S<sub>cr</sub> is the cross sectional area

Admin: /* General method */

2014-06-11T08:03:32Z

General method

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	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.		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)<ref>~~[http://dx.doi.org/10.1088/0741-3335/50/4/043001~~ 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'', Plasma Phys. Control. Fusion, '''50''', 4 (2008) 043001]</ref>		* 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]])

Admin: /* Confinement time scaling */

2014-06-11T08:02:44Z

Confinement time scaling

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	For [[stellarator]]s, a similar scaling has been obtained (ISS).		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		* 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>		:<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>