The Butterfly Effect

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Let Ek ( ) denote the kinetic energy per unit wave number of the system at wave number k −3/2 −1/2 Eddy "turn over" time τ ( k) : k E ( k) <strong>The</strong> time taken for some initial small-scale error to grow and nonlinearly infect some large-scale component k , N powers of 2 larger in scale, N n ( N) τ ( 2 k ) Ω =∑ n= 0 L L : :
−5/3 2/3 If Ek ( ) k then τ − : ( k) : k an N ∑ n Ω ( N) = τ (2 k ) n= 0 L N 2 /3 ( ) ∑ − n L 2 : τ L) n= 0 = τ k ( k as N →∞ d
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The Butterfly Effect

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