Scaling law: Difference between revisions

<ref>O.J.W.F. Kardaun, ''Classical methods of statistics: with applications in fusion-oriented plasma physics'', Springer Science & Business (2005) ISBN 3540211152</ref>

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:

:<math>y = e^{\alpha_0} x_1^{\alpha_1} x_2^{\alpha_2} ...</math>

Here, the α<sub>i</sub> are the scaling parameters.

* 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>

<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, ρ<sub>i</sub> is the ion [[Larmor radius]] and ν<sub>ii</sub> the ion-ion collision frequency. Also see [[Beta|beta]] and [[Collisionality|collisionality]].

:<math>D = c_s \rho_s (\rho^*)^\alpha F(\nu^*,\beta,q, ...)\,</math>

When α = 0, the scaling is said to be of the Bohm type, and when α = 1, of the gyro-Bohm type.

The main performance parameter that is subjected to scaling law analysis is the [[Energy confinement time|energy confinement time]], τ<sub>E</sub>.

<ref name="ITER">[http://dx.doi.org/10.1088/0029-5515/39/12/301 ITER Physics Expert Groups et al, ''ITER Physics Basis, Chapter 1'', Nucl. Fusion '''39''' (1999) 2137] and [http://dx.doi.org/10.1088/0029-5515/39/12/302 Ibid., ''Chapter 2'']</ref>

:<math>\tau_E^L = 0.048 I_M^{0.85} R_0^{1.2} a^{0.3} \kappa^{0.5} \bar n_{20}^{0.1} B_0^{0.2} A^{0.5} P_M^{-0.5}</math>

:<math>\tau_E^H = 0.145 I_M^{0.93} R_0^{1.39} a^{0.58} \kappa^{0.78} \bar n_{20}^{0.41} B_0^{0.15} A^{0.19} P_M^{-0.69}</math>

where ''I<sub>M</sub>'' is given in MA, ''P<sub>M</sub>'' in MW, ''n<sub>20</sub>'' in 10<sup>20</sup> m<sup>-3</sup>, and ''B<sub>0</sub>'' in T.

<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>

:<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>

One of the remarkable and initially unexpected properties of magnetically confined plasmas is the reduction of the [[Energy confinement time|energy confinement time]] τ<sub>E</sub> as the heating power ''P'' is increased. Typically:

:<math>\tau_E \propto P^{-\alpha}</math>

where α has a value of 0.6 ± 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.  

The ELMy [[H-mode]] scaling is of the gyro-Bohm type (α = 1).

Gyro-Bohm scaling is what one would expect for diffusive transport based on a diffusive scale length proportional to ρ<sub>i</sub> (the ion gyroradius).

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>

<references />