 Travelling and standing convection rolls

ID: travelling-and-standing-convection-rolls

At fixed orientation, opposite travelling amplitudes can obey

\dot{Z}_{\pm} = (r + i ω_{0}) Z_{\pm} - (g ∣ Z_{\pm} ∣^{2} + h ∣ Z_{\mp} ∣^{2}) Z_{\pm}

. For

r > 0

and

g_{r} = Re g > 0

, a travelling branch has one nonzero intensity

r / g_{r}

and is stable to its counter-propagating amplitude when

h_{r} > g_{r}

. A standing branch has both intensities

r / (g_{r} + h_{r})

and is stable within the pair when

g_{r} > h_{r}

and

g_{r} + h_{r} > 0

. These follow by linearizing the real intensity equations. Spatial modulation and other orientations remain separate stability tests.

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