Download PDF by G. Viennot: Algèbres de Lie libres et monoïdes libres : Bases des

By G. Viennot

ISBN-10: 0387090908

ISBN-13: 9780387090900

ISBN-10: 3540090908

ISBN-13: 9783540090908

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Example text

14) by writing [in polar coordinates (s, φ, z)]; B = Beφ + ∇ × (Aeφ ) = B eφ + BP , u = uP + U eφ . 90a) Δ− 1 s2 B. 90b) Further simplification ensues if we write A = χ/s, B = ψs, U = Ωs. 91b) with (Δ−(2/s)∂s )χ = ψ = 0 in c D and χ ∼ O(|x|−1 ) as |x| → ∞. It is notable that the toroidal field does not appear in the equation for χ. The analysis now proceeds © 2007 by Université Joseph Fourier 30 Michael P ROCTOR in a similar manner to that for the toroidal theorem. We form the poloidal “energy equation” 1d 2 dt χ2 dx = η D D χ Δ− 2 ∂ χdx = −η s ∂s R3 |∇χ|2 dx ≤ −η c23 χ2 dx .

0 0 1 2 3 4 5 k most easily seen when we set η = 0 (for small η the results are almost the same as long as τ is not too large). 98) which can be large for large τ . If we add (small) diffusion, we still get growth, provided τ is much less than the diffusion time. Then the second pulse can refold and stretch the field and give further enhancement. A more complicated version of this kind of flow is one that arises in thermal convection, where there is a homoclinic connection between two different planar flows.

It is worth stressing here that this mathematical description of the physical setup is oversimplified and further developments and refinements will be dicussed later in the book. Furthermore, we only consider here a linear problem. The currents here appear to grow indefinitely. This is because the Lorentz force acting on the disc to slow it down has been neglected. This force is the essence of a third setup that can also be constructed using such a disc configuration, which is the Barlow wheel.

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Algèbres de Lie libres et monoïdes libres : Bases des algèbres de Lie libres et factorisations des monoïdes libres by G. Viennot


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