By John Earman
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Extra info for A primer on determinism
30) for a single ¯uid plasma model with the speci®cation of the physical meaning of each term: ¶ 3 nT r Á 32 nTu ÿnTr Á u ÿ r Á Q ZJ 2 Paux ¶t 2 Pconvection Pcompress Pconduction POhm Paux Te ÿ Ti Te ÿ Tn ÿ ÿ teqYi teqYn Pexch e±i Pexch e±n 3 ÿ wi ne niÿ1 hsviI ÿ 2 Tne ni hsviR Pioniz ÿ P Trad X Copyright © 2001 IOP Publishing Ltd Precomb 24 Description of Continuous Media To establish the energy loss per second of con®ned plasma, it is convenient to introduce the times of energy con®nement and recycling tE E Y Q dS tErecycle 3 S 2 nTv ZJ 2 E Paux dV 2X35 with the de®nitions of total thermal energy E v 3 2 nT dV 2X36 and bolometric power Pbolo v Prad Pion Pe±i Pcompr dVX 2X37 One obtains the global power balance & ' 1 dE 1 1 Pbolo ÿ X E dt tErecycle tE E 2X38 What is wanted is the con®nement time and what is measurable is the recycling time tErecycle % h32 nTiV X Vloop I hPaux Vi 2X39 Just in steady state, one has 1 1 Pbolo X ÿ tE tErecycle E 2X40 The search of empirical scaling laws of this tE may make sense.
1) is transformed into ! 1) and matching powers of t0 ÿ t m ÿ1a 1 ÿ pY n 12 1 md 3X3X5 and the following non-linear ODE is obtained for f x ! g. (a) p 2, d 1 or (b) p 3Y d 2. s. 6) vanishes and x x. In case (a), the equation is ! ¶ ¶f f 3X3X7 f ÿ f2 ¶x ¶x and in case (b), ! 8) can be integrated by multiplying both sides by f ¶fa¶x and f2 ¶fa¶xY respectively. 7) is expressed implicitly as: x ÿ x0 Æ p f 6 f df q 2 f0 f3 ÿ f3 ÿ 3 f4 ÿ f4 c 4 3X3X9 c where fc is a constant of integration.
A) p 2, d 1 or (b) p 3Y d 2. s. 6) vanishes and x x. In case (a), the equation is ! ¶ ¶f f 3X3X7 f ÿ f2 ¶x ¶x and in case (b), ! 8) can be integrated by multiplying both sides by f ¶fa¶x and f2 ¶fa¶xY respectively. 7) is expressed implicitly as: x ÿ x0 Æ p f 6 f df q 2 f0 f3 ÿ f3 ÿ 3 f4 ÿ f4 c 4 3X3X9 c where fc is a constant of integration. Choosing fc 0 or fc 4a3Y the result is identical. 1) n xY t Copyright © 2001 IOP Publishing Ltd r!
A primer on determinism by John Earman