Download Solitary Dynamo Waves

Solitary Dynamo Waves
Joanne Mason (HAO, NCAR)
E. Knobloch (U.California, Berkeley)
CMSO (PPPL)
• Large-scale solar dynamo theory
• aW-dynamo
a-effect
a
B  BT  B P
W
latitude
The aW dynamo
time
(Courtesy HAO)
W-effect
• Mean-field electrodynamics
B
   u  B     aB    2 B
t
• Long wave dynamo instability
• Nonlinear evolution  mKdV equation  solitary wave
solutions
CMSO (PPPL)
The Model
z  L1  1
a   z -1
W   z 
z  L2  0
• Spatially localised a and W
(Moffatt 1978; Kleeorin & Ruzmaikin 1981; Steenbeck & Krause 1966)
• Write B  B P  BT    Ayˆ  Byˆ , u  u ( z )yˆ ( y  0)
W-effect
a-effect
A
 aB   2 A
t
a 0G0 z0 3
dynamo number D 
0 2
B
du A
D
 2 B
t
dz x
CMSO (PPPL)
Linear Theory
• Seek travelling wave solutions
[ A, B]  [az , bz ] exp  pt  ikx
p    i
• Apply continuity in A and B, matching conditions and
boundary conditions
Bz  L1, 2   0,
A
z  L1,2   0
z
 dispersion relation
4q 2 sinh 2 qL1 - L2   ikD sinh 2qL1 - 1sinh 2qL2   0
where
q2  p  k 2
Mason, Hughes & Tobias (2002)
CMSO (PPPL)
Most unstable mode
• Marginal stability (=0)
k  1 Dc  -
10 L1 - L2 2
L2 L1 - 1
• Set k  e  1
k  1  c  10k
10 
3
2
22 L1 - 1 - L1  L2 
• Dynamo waves set in forD  Dc with O(e) wavenumber
and O(e) frequency
CMSO (PPPL)
Nonlinear theory – mKdV equation
a
 z - 1

  z - 1 1 - B 2
1  B 2 ( x, z )
• Consider D  D0  e 2 D2     ,

Jepps (1975)
Cattaneo & Hughes (1996)
  de 2
 A   A0   A1 
      e    , A0 e x  ct  ~ O(1)
 B   0   B1 
 x  e X ,  t  eT1  e 3T3  e 4T4
• Solve dynamo equations at each order in e
• Inhomogeneous problems require solvability condition
3
• O(e ) : Modified Korteweg-de Vries equation forAˆ0  A0 / 
ˆ
Aˆ 0
Aˆ 0
 3 Aˆ 0

A
2
0
ˆ
-a
-a

b
A
0
0
3
T3



CMSO (PPPL)
a, b functions of
  e ( x  ct )
L1 , L2 , 10 only
Solutions to mKdV
• Solutions depend upon signs of a and b
b
a
L1  2
l-
• a  0, a  v  0 kinks:
C  N tanh  ,
N2 
L2
L1
3(a  v)
1 v
,  2  1  
b
2 a
• a  0, a  v  0 solitary waves:
 - N 2b 1/ 2 
6(a  v)
 , N 2 
C  N sech 
b
 6a 

• Snoidal and cnoidal waves also exist
CMSO (PPPL)
   - v
The perturbed mKdV equation
•On longer times forcing enters the description
C
C
 3C
2 C
-a
- a 3  bC
 ef  O(e 2 )




C  ,   A0  O(e )
  e ( x  ct )
D2  d
•The perturbation f selects the amplitude N :
N ˆ
2
 f  O(e )  N  30aL2 L1 -1D2  d  / bh,
t
h  ha, L1, 2 , D2 
v
N
L1  l  2
d
•Amplitude stability:
d
fˆ
2b 2 N 4 h
2
N
135a 210 L2 - L1 
• solitary waves are unstable (a  0, h  0, D2  d  0)
CMSO (PPPL)
Physical manifestation of solution
• Reconstruct the fieldsA0 , A1 , B1,  fromC
Solitary Waves:
B / e , | BP | / e
Kinks:
B / e ,  | BP | / e


z  0, d  1, L1  2, l  2
z  0, d  1, L1  2, l  1
CMSO (PPPL)
Conclusions
• Mean-field dynamo equations with a-quenching
possess solitary wave solutions
• Leading order description is mKdV equation.
Correction that includes effect of forcing and
dissipation leads to pmKdV. Allows identification
of N(d), v(d).
• Solutions will interact like solitons do  modify
butterfly diagram
References: Mason & Knobloch (2005), Physica D, 205, 100
Mason & Knobloch (2005), Physics Letters A (submitted)
CMSO (PPPL)