Download - CMSO (PPPL) Solitary Dynamo Waves Joanne Mason (HAO, NCAR) E. Knobloch (U.California, Berkeley)
CMSO (PPPL)
Solitary Dynamo Waves
Joanne Mason (HAO, NCAR)
E. Knobloch (U.California, Berkeley)
• Large-scale solar dynamo theory
• -dynamo
• Mean-field electrodynamics
• Long wave dynamo instability
• Nonlinear evolution mKdV equation solitary wave solutions
The dynamo
BBBuB 2
t
time
lati
tud
e
(Courtesy HAO)
CMSO (PPPL)
effect effect
PT BBB
• Spatially localised and (Moffatt 1978; Kleeorin & Ruzmaikin 1981; Steenbeck & Krause 1966)
•
CMSO (PPPL)
The Model
)0(ˆ)(,ˆˆWrite yTP zuBA yuyyBBB
Bx
A
dz
duD
t
B 2
ABt
A 2
20
3000
zG
Dnumberdynamo
-effect -effect
1 z z
11 Lz
02 Lz
CMSO (PPPL)
Linear Theory
• Seek travelling wave solutions
• Apply continuity in A and B, matching conditions and boundary conditions
dispersion relation
ipikxptzbzaBA exp],[],[
0,0 2,12,1
Lzz
ALzB
22
212122
where
02sinh12sinhsinh4
kpq
qLLqikDLLqq
Mason, Hughes & Tobias (2002)
CMSO (PPPL)
Most unstable mode
• Marginal stability (=0)
• Set
• Dynamo waves set in for with O() wavenumber and O() frequency
1
112
22110
LL
LLDk c
kk c 101
2211
10122
3
LLL
1k
cDD
CMSO (PPPL)
Nonlinear theory – mKdV equation
functions of only
• Consider
• Solve dynamo equations at each order in
• Inhomogeneous problems require solvability condition
• Modified Korteweg-de Vries equation for
22
11),(1
1Bz
zxB
z
)1(~,0 0
1
10 OctxAB
AA
B
A
431
43, TTTtXx
0ˆ
ˆˆˆˆ
0203
03
0
3
0
A
AbA
aA
aT
A
:)( 3O
)( ctx
Jepps (1975) Cattaneo & Hughes (1996)
/ˆ00 AA
ba, 1021 ,, LL
22
20 , dDDD
CMSO (PPPL)
Solutions to mKdV• Solutions depend upon signs of a and b
• kinks:
• solitary waves:
• Snoidal and cnoidal waves also exist
b
vaN
a
bNC
)(6,
6
N-sech 2
2/12
a
v
b
vaNNC 1
2
1,
)(3,tanh 22 v
0,0 vaa
0,0 vaa
1
2
L
Ll
a b
21 L
CMSO (PPPL)
The perturbed mKdV equation•On longer times forcing enters the description
•The perturbation selects the amplitude :
•Amplitude stability:
• solitary waves are unstable
)( 223
3
OfC
bCC
aC
aC
)( ctx dD 2
)(, 0
OAC
)(ˆ Oft
N
22,1212
2 ,,,/130 DLahhbhdDLaLN
f N
)0,0,0( 2 dDha
212102
42
135
2ˆ
LLa
hNb
N
f
N v
dd
21 lL
CMSO (PPPL)
Physical manifestation of solution
• Reconstruct the fields from
Solitary Waves:
Kinks:
,110 ,, BAA C
/||,/ PBB
/||,/ PBB
2,2,1,0 1 lLdz
1,2,1,0 1 lLdz
CMSO (PPPL)
Conclusions
• Mean-field dynamo equations with -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)