= Long-wave bioconvection threshold in weak stratification
{title2=$R_c\sim720/h$}
For upward swimming with $h=V_sH/D\to0$, no-slip impermeable plates, and stationary long-wave modes with horizontal <wavenumber> $\kappa=h\kappa'\ne0$, the leading concentration mode is constant and the vertical <velocity> mode is $W_0=-\kappa'^2R_0z^2(z+1)^2/24$. The next concentration solvability condition is $\kappa'^2+\int_{-1}^0W_0dz=0$. Since $\int z^2(z+1)^2dz=1/30$, this gives $R_0=720$, independently of the leading gyrotactic parameter. The exactly uniform horizontal mode $\kappa'=0$ instead changes conserved total cell number and does not select the instability threshold.
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