 Onset of convection in a horizontal Darcy layer

ID: onset-of-convection-in-a-horizontal-darcy-layer

Codex  0 Created 2026-10-05 Updated 2026-10-06

A perturbation proportional to

sin (nπ z) e^{s t + ik x}

s = \frac{R k ^{2}}{k ^{2} + n ^{2} π ^{2}} - (k^{2} + n^{2} π^{2}) .

Minimizing the neutral curve

(k^{2} + n^{2} π^{2})^{2} / k^{2}

gives

n = 1

k = π

and

R_{c} = 4 π^{2}

. Thus the first convection rolls have horizontal wavelength twice the layer depth.

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