Long-wave bioconvection threshold in weak stratification ($R_c\sim 720/h$)

For upward swimming with $h=V_{s}H/D\to 0$, no-slip impermeable plates, and stationary long-wave modes with horizontal wavenumber $\kappa =h{\kappa }^{\prime }\ne 0$, the leading concentration mode is constant and the vertical velocity mode is $W_0=-{\kappa }^{\prime 2}{R}_0{z}^2(z+1{)}^2/24$. The next concentration solvability condition is ${\kappa }^{\prime 2}+{\int }_{-1}^0{W}_{0}dz=0$. Since $\int z^2(z+1{)}^{2}dz=1/30$, this gives $R_0=720$, independently of the leading gyrotactic parameter. The exactly uniform horizontal mode ${\kappa }^{\prime }=0$ instead changes conserved total cell number and does not select the instability threshold.

For the constant leading cell-concentration mode normalized to one, the leading vertical velocity mode in shallow long-wave bioconvection solves $W_1^{(4)}=-R_0{\alpha }^2$ and $W_1=W_1^{\prime }=0$ at both $z=0,1$. The displayed quartic satisfies all four conditions and is unique because a homogeneous cubic with those conditions is zero. Its integral is $-R_0{\alpha }^2/720$. The next cell-conservation solvability condition is ${\alpha }^2+{\sigma }_2=-\int W_{1}dz$, hence ${\sigma }_2={\alpha }^2(R_0/720-1)$. The leading threshold is independent of the distinguished leading gyrotactic coefficient; a horizontally uniform conservation mode does not select it.