Gyrotactic reorientation time ($B={\zeta }_r/(mg{\ell }_g)$)

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 $D_r^{-1}$.