By Plummer H. C. 1875
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Additional info for An Introductory Treatise On Dynamical Astronomy
89) 2μ0 V The MHD dynamos that will be the focus of much of what follows are fluid systems that convert mechanical energy into magnetic energy, through the agency of the u · F term on the RHS of Eq. 87). 10 Magnetic Flux Freezing and Alfvén’s Theorem Let us return to the differential form of Faraday’s law: ∇×E=− ∂B . 90) Project now each side of this expression onto a unit vector normal to some surface S fixed in space and bounded by a closed countour γ, integrate over S, and apply Stokes’ theorem to the LHS: (∇ × E) · nˆ dS = γ S E·d =− S ∂B ∂t · nˆ dS .
Fields showing symmetry with respect to an axis, usually rotational. Working in spherical polar coordinates (r, θ, φ) with the polar axis coinciding with the field’s symmetry axis, the most general axisymmetric (now meaning ∂/∂φ = 0) magnetic field can be written as: B(r, θ, t) = ∇ × (A(r, θ, t)ˆeφ ) + B(r, θ, t)ˆeφ . , the component contained in meridional (r, θ) planes. The azimuthal component B is often called the toroidal field. 2) satisfies the constraint ∇ · B = 0 by construction, and another great advantage of this mixed representation is that the MHD induction equation for the vector B can be separated in two equations for the scalar components A and B.
2 The Challenges Copper disks and sliding contacts being a rather sparse commodity in the universe, we must now figure out to apply the general idea of a dynamo to astrophysical fluids. 59). In its simplest form, the dynamo problem consists in finding a flow field u that can sustain a magnetic field against Ohmic dissipation. We will encounter in the following chapter flows that can amplify a magnetic field during a transient time interval, after which B decays again. So we tighten our definition of the dynamo problem by demanding that a flow be a dynamo if it can lead to E B > 0 for times much larger than all relevant advective and diffusive timescales of the problem.
An Introductory Treatise On Dynamical Astronomy by Plummer H. C. 1875