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Stability lecture 17 - Hydrodynamic Stability Lecture 17...

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Hydrodynamic Stability Lecture 17 Dispersion relation for double diffusion stability problem. ( ) ( ) ( ) ( ) ( ) 2 2 3 2 2 4 6 2 2 2 2 2 2 Pr Pr 1 Pr Pr Pr Pr 0 where a p k p k p Ra Rs p k Rs Ra k k a n π τ τ τ τ τ π + + + + + + + + = = + 2 a π See figure 8.2, pg. 256 Turner. ( ) 4 4 line: Pr 27 1 1 Pr 1 Pr 4 line: 27 4 x w Ra Rs x z Rs Ra τ τ π τ π τ + = + + + + = + 1
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For Static Stability or or T T S S T S T S Rs Ra α β α α α < − < > β Salt fingers set in for Rs slightly above critical value. Motion is direct, exponential growing mode. imag P 0 = For large , Rs Ra Ra τ < Most unstable mode predicts cells that are very tall and thin ( ) ( ) 1 4 large or small wavelength. ah Ra ah π Diffusive regive when salt heat 1 K K τ τ = ± Instability occurs in oscillatory mode. ( ) 4 1 2 2 0 0 Pr 27 1 Pr 4 1 frequency 3 Pr 1 where low freqency oscillation. i P Ra
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