Bottom-heavy spherical-cell orientation dynamics ($\dot{\mathbf{p}}=B^{-1}(\mathbf{k}-p_3\mathbf{p})+\mathbf{\omega }\times \mathbf{p}/2$)

For a spherical cell with its centre of mass displaced by $-{\ell }_g\mathbf{p}$ from its geometric centre, gravity produces torque $mg{\ell }_g\mathbf{p}\times \mathbf{k}$. Rotational resistance ${\zeta }_r$ gives angular velocity $\mathbf{\omega }/2+(mg{\ell }_g/{\zeta }_r)\mathbf{p}\times \mathbf{k}$. Taking its cross product with $\mathbf{p}$ gives the displayed orientation equation with $B={\zeta }_r/(mg{\ell }_g)$. Steady orientation therefore requires the two terms on the right to sum to zero; placing the vorticity term alone on the other side changes its sign.

This convention defines the gyrotactic reorientation time by an aligning rate $1/(2B)$. Multiplying the torque balance by $2B$ and perturbing about the upward direction gives ${\mathbf{p}}_{\perp }=B\mathbf{\omega }\times {\mathbf{e}}_z$. For an incompressible flow, its horizontal divergence is $-B{\nabla }^{2}w$, because $\nabla \times \mathbf{\omega }=-{\nabla }^2\mathbf{u}$. The parameter $B$ is half the reorientation parameter used in the convention with aligning rate $1/B$; writing the governing equation prevents a factor-of-two ambiguity.

Without rotational diffusion, $x=p_z$ follows $\dot{x}=(1-x^2)/B$, hence $x=(x_0+\tanh (t/B))/(1+x_0\tanh (t/B))$. Pushing forward the initial uniform orientation distribution gives $f=\alpha /[\pi (1+\alpha -(1-\alpha )x{)}^2]$, $\alpha =e^{-2t/B}$, per unit solid angle. Integrating $2\pi xf$ yields the displayed mean, interpreted continuously as zero at $t=0$. Its late deficit is $(4t/B-2)e^{-2t/B}+O((t/B)e^{-4t/B})$. At each finite time the density is smooth and normalized; the infinitely late distribution is a point mass at the upward pole.

In the convention with aligning rate $1/B$, the tilt angle in still fluid obeys $\dot{\theta }=-\sin \theta /B$, so $\tan (\theta /2)$ decays as $e^{-t/B}$. Some conventions instead put $1/(2B)$ in the orientation equation; their parameter has a different factor of two. Specify the dynamical equation when comparing reorientation times.

Close to the upward pole, the two Cartesian tilt components of bottom-heavy spherical-cell orientation dynamics obey the displayed Ornstein-Uhlenbeck process. Thus $d⟨|\mathbf{q}{|}^2⟩/dt=-2⟨|\mathbf{q}{|}^2⟩/B+4D_r$. The stationary spread is $2B{D}_r$, and late axisymmetric variance perturbations relax on $B/2$ in this weak-noise approximation. The zero-noise mean from deterministic alignment of an initially isotropic orientation distribution has the same exponential scale but a different transient prefactor. Reaching a tiny diffusion-supported mean-deficit floor $B{D}_r$ from an initially isotropic population takes order $B\log [(B{D}_r{)}^{-1}]$, with logarithmic corrections. Equality of limiting mean directions does not make the full transients identical, and the alignment time is not automatically the pure rotational-diffusion time $D_r^{-1}$.