Tamtam Proceedings - lamsin

64 Benzekri et al.1. Global controlIn Refs.[1, 2], it was proposed a model describing this phenomena based on the followingstreamline :H(x, y, t) = ɛ sin(x + B sin t) sin y, (1)where :ɛ is the maximal vertical velocity in the flow, B is the amplitude of the lateral oscillationsof the velocity field. Let us recall the global control theory as explained in Refs.[3, 4].For a fixed Hamiltonian H 0 , we define the linear operator {H 0 } by :{H 0 }H = {H 0 , H}, (2)where {., .} is the Poisson bracket. We consider a pseudo-inverse of {H 0 }, i.e. a linearoperator Γ such that :We define the resonant operator R as :{H 0 } 2 Γ = {H 0 }. (3)R = 1 − {H 0 }Γ. (4)We note that equation (3) becomes {H 0 }R = 0. A consequence is that any element RVis constant under the flow of {H 0 }.We assume that H 0 is integrable with action-angle variables (A, θ) ∈ R L ×T L , where T Lis an L-dimensional torus i.e. H 0 = H 0 (A). The Poisson bracket between two functionsH and V is given in the usual form :The operator {H 0 } acts on V given by :V = ∑aswhere{H, V } = ∂H∂A · ∂V∂θ − ∂H∂θ · ∂V∂A . (5)H 0 V (A, θ) = ∑k∈Z L V k (A)e iθ.k , (6)k∈Z L iω(A).kV k (A)e iθ.k , (7)ω(A) = ∂H 0∂A . (8)TAMTAM –Tunis– 2005