Induction Equation Clare E Parnell School of Mathematics and Statistics Induction Equation B 2 ( v B ) B t Advection term Diffusion term If Rm >> 1 then magnetic field advected: B ( v B) t If Rm << 1 then magnetic field diffuses: B 2 B t Diffusion Equation B 2 B t Consider what happens to the following magnetic field after diffusion B( x) (0,0, B0 ( x)) B0 , x 0 B0 ( x) B0 , x 0 Diffusion Equation Solution for magnetic field: B( x, t ) (0,0, x /( 4t )1 / 2 u 2 0 2B e du ) 0 Solution for current density: 2 B0 x 2 /( 4t ) j( x, t ) (0, e ,0) 1/ 2 (4t ) • Diffusion - Evolution of the magnetic field lines • Diffusion - Evolution of Bz versus x • Diffusion - Evolution of jy versus x Advection Equation B ( v B) t Consider what happens to the following magnetic field B moved by the flow v B0 B( x, t ) (0,0, ) 2 1 x v ( x,0, z ) Advection Equation Solution for magnetic field: t B0e B( x, t ) (0,0, ) t 2 1 ( xe ) Solution for current density: 3t 2 B0e x j( x, t ) (0, ,0) t 2 2 (1 ( xe ) ) • Advection - Evolution of the magnetic field lines • Advection - Evolution of Bz versus x • Advection - Evolution of jy versus x Importance • On Sun vast majority of plasma is advected • except where small length scales are created, e.g., in current sheets. Combination of advection and diffusion known as magnetic reconnection Importance • Magnetic reconnection important in: – – – – Powering solar flares Heating solar corona Driving solar wind Triggering CMEs