By Ph. Cardin and L.F. Cugliandolo (Eds.)
This ebook is a set of lectures given in July 2007 on the Les Houches summer time college on "Dynamos". . presents a pedagogical advent to issues in Dynamos . Addresses every one subject from the foundation to the newest advancements . Covers the lectures by way of internationally-renowned and prime specialists
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Extra resources for Dynamos
2. One dimensionless parameter In addition to ρ, ν, σ and μ, we add just one of U , H or B in turn. In each case, there is one pertinent dimensionless parameter according to Vashy-Buckingham: not surprisingly, this is the magnetic Prandtl number identified previously. tex; 22/05/2008; 17:25 18 1 2 3 4 5 6 7 8 9 10 11 12 p. 18 T. Alboussière The question whether these five parameters can describe an interesting problem can be asked. At least, when U is the additional parameter, the answer is definitely yes.
Those dimensional parameters involve four fundamental units (m, kg, s, A). They are not dimensionally independent though, as 1/(μσ ) has the same dimension of ν. It is called the magnetic diffusivity (m2 s−1 ). 2. One dimensionless parameter In addition to ρ, ν, σ and μ, we add just one of U , H or B in turn. In each case, there is one pertinent dimensionless parameter according to Vashy-Buckingham: not surprisingly, this is the magnetic Prandtl number identified previously. tex; 22/05/2008; 17:25 18 1 2 3 4 5 6 7 8 9 10 11 12 p.
With six parameters and four units, two independent dimensionless parameters must be identified. parameters U and H : a common choice is Pm and the magnetic Reynolds number: Rm = μσ U H. 4) 33 However, one can also take (Re = U H /ν, Rm ) as governing parameters, or (Re, Pm ), or (Re2 , Rm + Re), . . With the following scales: 34 x ←→ H H t ←→ U √ B ←→ U ρμ u ←→ U 36 37 38 p ←→ ρU 2 U2 j ←→ ν 35 39 ρ . tex; 22/05/2008; 17:25 Fundamentals of MHD 1 p. 5) −1 2 = ∇ × (u × B) + Rm ∇ B. 7) 27 28 29 30 31 32 33 34 35 36 37 38 39 40 8 9 10 11 12 14 15 16 17 18 20 21 22 who is a Reynolds √ number based on the thickness of the Hartmann layer (see later) δ = 1/B ρν/σ .