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 /( 4t )1 / 2
u 2
0
2B

e
du )
0
Solution for current density:
2 B0
 x 2 /( 4t )
j( x, t )  (0,
e
,0)
1/ 2
  (4t )
• 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