Transcript Уравнения Паркера в двухслойной среде

Modeling of the solar interface dynamo
Vienna, EGU-2012
Artyushkova M. E.
Schmidt Institute of Physics of the Earth
of the Russian Academy of Sciences
Based on “Popova H., Artyushkova M. and Sokoloff D. The WKB approximation
for the interface dynamo // Geophysical & Astrophysical Fluid Dynamics.
2010. V.104, № 5. P. 631-641. “
Single-layer dynamo model proposed by Parker in 1955.
Parker, E.N., Hydromagnetic dynamo models// Astrophysical
Journal. 1955, 122, 293–314.
A
2 A
  ( ) B  2 ,
t

B
A  2 B
 D cos  
 2
t
t 
B(r , , t )
A (r , , t )
- toroidal magnetic field
- toroidal component of vectorial potential,
poloidal magnetic field

- latitude (so    / 2 corresponds to the poles).
 ( )   sin ( )
D - dynamo number combined with α-effect amplitude, differential rotation Ω
and coefficient of diffusion. So the equation system is dimensionless.
WKB-approach for solving migratory dynamo model
Kuzanyan, K.M. and Sokoloff, D.D., A dynamo wave in an inhomogeneous medium.
Geophys. Astrophys.Fluid Dyn. 1995, 81, 113–129.
Bassom, A.P., Kuzanyan, K.M., Sokoloff, D. and Soward, A.M.,
Non-axisymmetric 2-dynamo waves in thin stellar shells//
Geophys. Astrophys. Fluid Dyn. 2005, 2, 309–336.
The WKB solution is looked for in the form of waves traveling
in the -direction ~
e
1
2
iD 3 S D 3 t

where S  kd  action
k corresponds to the wave vector, or impulse

k, 
1
- complex growth rate, Im{ }gives the length of the activity cycle
derived from the Hamilton-Jacobi equation (  ik  k 2 ) 2  ik cos   0
According this method the solution must decay towards
the boundaries of the domain - the poles and equator
Two-layers dynamo model
Parker’s system of equations:
Boundary conditions for
r  0:
b  B,
B
 B,
t
A
B  A,
t
b
a
 D cos    b,
t
t
a
 a.
dt
a  A,
b

r
a

r
B
,
r
A
.
r
α-effect localized first layer and the differential rotation Ω localized in the second layer
B (r ,  , t ), A (r , , t )
b (r,  , t ), a (r ,  , t )
- magnetic field, potential in the first layer
- magnetic field, potential in the second layer
 ( )   sin ( )
D
- dynamo number combined with
α-effect amplitude, Ω
  ratio of the turbulent diffusivity
coefficients
in the first and second layers
Applying the WKB-approach for two-layers model
B
1
2
1
3
3
3
 eiD S D t iD m1r ,
1
2
1
iD 3 S D 3 t iD 3 m1r
A (
,
1r )e
1
2
1
2

iD 3 S D 3 t iD 3 m2 r
b (  D 3 1r )e
,
1
2
1
iD 3 S D 3 t iD 3 m2 r
a e
.


2
D 3



, ,  ,  ,  , 1 , 1 , m1 , m2 
slowly varying functions
The Hamilton–Jacobi equation
( 


 k 2     k 2 )(   k 2     k 2 ) 


(  ik  k 2 ) 2  ik cos   0
i  k   cos 
4    k 2  

 k2

0
- Hamilton–Jacobi equation for one-layer dynamo
for comparison
~
D

16
D
, ( D  0)
Squared the Hamilton–Jacobi equation with
a6  k 6  a5  k 5  a4  k 4  a3  k 3  a2  k 2  a1  k  a0  0
a6     4   3   2
a5  8i 2ˆ  8iˆ
a4  4 2  2   2   2  4   3 2   2
a3  8iˆ  8i 2ˆ  16i
a2  16 2  2 3 3  2 3   2 3  2 2 3
a1  8i 2ˆ  8i 2ˆ
a0   2 4  2 4   4
where ˆ   cos
Solving the Hamilton–Jacobi equation
H ( , k ,  )  0,
H k' ( , k ,  )  0,
condition that k has to be continuous from pole to equator,
particulary k has to be continuous at a turning point  ,*
where two roots of the Hamilton–Jacobi equation coincide
Roots of the Hamilton–Jacobi equation
for various values of  as points in the complex k plane
Single-layer model
Two-layers model,
  1,5
(Kuzanyan, K.M. and Sokoloff, D.D, 1995)
Im {k}
Re {k}
Numbers 1,2,3,4 enumerate the branches of the various roots.