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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] [az , bz ] exp pt ikx p i • Apply continuity in A and B, matching conditions and boundary conditions Bz L1, 2 0, A z L1,2 0 z dispersion relation 4q 2 sinh 2 qL1 - L2 ikD sinh 2qL1 - 1sinh 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 22 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 eT1 e 3T3 e 4T4 • 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 -1D2 d / bh, t h ha, L1, 2 , D2 v N L1 l 2 d •Amplitude stability: d fˆ 2b 2 N 4 h 2 N 135a 210 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)