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CMSO (PPPL)

CMSO (PPPL). Solitary Dynamo Waves Joanne Mason (HAO, NCAR) E. Knobloch (U.California, Berkeley). a. W. CMSO (PPPL). The aW dynamo. Large-scale solar dynamo theory aW -dynamo Mean-field electrodynamics Long wave dynamo instability

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CMSO (PPPL)

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

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

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

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

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

  6. CMSO (PPPL) Nonlinear theory – mKdV equation Jepps (1975) Cattaneo & Hughes (1996) • Consider • Solve dynamo equations at each order in e • Inhomogeneous problems require solvability condition • Modified Korteweg-de Vries equation for functions of only

  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. 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) Conclusions

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