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Let $f$ be probability density per unit solid angle, with ${\int }_{S^2}fd\Omega =1$ and isotropic initial condition $f(\mathbf{p},0)=1/(4\pi )$. In still fluid, bottom-heavy spherical-cell orientation dynamics gives $\dot{\mathbf{p}}=B^{-1}[\mathbf{k}-(\mathbf{k}\cdot \mathbf{p})\mathbf{p}]$. Adding isotropic rotational diffusion gives the gyrotactic orientation Fokker-Planck equationThe derivatives are tangential to the unit sphere. Rotational symmetry about $\mathbf{k}$ is preserved by both the equation and the initial condition, so $f$ depends only on $\theta ,t$. Since $\dot{\theta }=-\sin \theta /B$, the polar probability flux is $j_{\theta }=-\sin \theta f/B-D_r{f}_{\theta }$. Expanding ${\partial }_{t}f+{\sin }^{-1}\theta {\partial }_{\theta }(\sin \theta j_{\theta })=0$ yieldsRegular solutions have no probability flux through the poles. The mean swimming velocity is ${\mathbf{V}}_c=V_s⟨\mathbf{p}⟩$, with only a vertical component by symmetry.

Use the rotation convention of part (a), so the orientation drift is $B^{-1}({\mathbf{e}}_3-p_3\mathbf{p})-\Omega {\mathbf{e}}_2\times \mathbf{p}$. The steady gyrotactic orientation Fokker-Planck equation on the unit sphere isLet $\eta =(B\Omega {)}^{-1}$, $\lambda =(B{D}_R{)}^{-1}$, and introduce the rotation generator $\mathcal{A}=({\mathbf{e}}_2\times \mathbf{p})\cdot {\nabla }_p$. Dividing by $\Omega$ givesThe leading isotropic density is $f_0=1/(4\pi )$. At first order, writing $f=f_0+\eta f_1+\cdots$, rotational diffusion makes no contribution because ${\Delta }_p{f}_0=0$. The tangential drift is the spherical gradient of $p_3$, whose divergence is ${\Delta }_p{p}_3=-2p_3$. ThereforeIn the polar convention of the paper, $p_1=\sin \theta \cos \varphi$, and $\mathcal{A}{p}_1=p_3$. Thus, in the specified first-harmonic form,The correction integrates to zero, preserving normalization. The first correction has no dependence on $\lambda$ because the isotropic leading density has zero spherical Laplacian; $\lambda$ enters the next-order equations.

Spherical symmetry gives $\int p_i{p}_{j}d\Omega =4\pi {\delta }_{ij}/3$. Consequently the rapid-rotation mean gyrotactic orientation isIt is this small horizontal swimming bias, rather than a mean vertical orientation at leading order, that enters the concentration equation of part (c).