1. CMSO (PPPL) Solitary Dynamo Waves Joanne Mason (HAO, NCAR) E. Knobloch (U.California, Berkeley)

2. Large-scale solar dynamo theory -dynamo Mean-field electrodynamics Long wave dynamo instability Nonlinear evolution mKdV equation solitary wave solutions The dynamo time latitude (Courtesy HAO) CMSO (PPPL) effect

3. Spatially localised and (Moffatt 1978; Kleeorin & Ruzmaikin 1981; Steenbeck & Krause 1966) CMSO (PPPL) The Model -effect

4. CMSO (PPPL) Linear Theory Seek travelling wave solutions Apply continuity in A and B, matching conditions and boundary conditions dispersion relation Mason, Hughes & Tobias (2002)

5. CMSO (PPPL) Most unstable mode Marginal stability ( =0) Set Dynamo waves set in for with O( ) wavenumber and O( ) frequency

6. 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 Jepps (1975) Cattaneo & Hughes (1996)

7. CMSO (PPPL) Solutions to mKdV Solutions depend upon signs of a and b kinks: solitary waves: Snoidal and cnoidal waves also exist

8. CMSO (PPPL) The perturbed mKdV equation On longer times forcing enters the description The perturbation selects the amplitude : Amplitude stability: solitary waves are unstable

9. CMSO (PPPL) Physical manifestation of solution Reconstruct the fields from Solitary Waves: Kinks:

10. 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)