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)