Gyrotaxis

Gyrotaxis is the orientation response of swimming cells to the competition between gravity-induced torque and fluid rotation. A bottom-heavy spherical cell tends to point upward in still fluid, while ambient vorticity rotates it. The response changes the mean swimming direction and can focus cells or contribute to bioconvection.

For a bottom-heavy spherical swimmer in axial flow $u_z=W_0(-1+r^2+w)$, the bottom-heavy spherical-cell orientation dynamics with aligning rate $1/B$ gives $\sin \theta =-\lambda (2r+w^{\prime })$, $\lambda =B{W}_0/(2R)$. On the upward-stable branch $\cos \theta >0$, the cells tilt inward in a downward Poiseuille flow. Zero radial cell flux with scalar diffusivity gives $n^{\prime }=\chi n\sin \theta$, $\chi =V_{c}R/D$. For $w=0$, integration and the average normalization $2{\int }_0^{1}nrdr=1$ give the displayed concentration. A steady deterministic orientation requires $|\lambda (2r+w^{\prime })|<1$; beyond this shear threshold the stationary orientation closure fails.

Let weak excess-cell buoyancy be $\gamma \ll 1$ and the swimming stresslet coefficient be $\sigma =\gamma {\sigma }_1$. Set $w=\gamma w_1$ and use the leading gyrotactic focusing in a downward pipe flow density $n_0=A{e}^{-a{r}^2}$. With active stress entering as $+Sn\mathbf{pp}$, the axial momentum equation integrates from the regular axis to the displayed derivative. The wall condition gives $w_1(r)=-{\int }_r^1{w}_1^{\prime }(s)ds$. Both terms of the integrand are positive for ${\sigma }_1\ge 0$, so the correction increases downward flow. A negative stresslet strength can oppose this effect; reversal at a given radius requires ${\sigma }_1$ smaller than minus the ratio of the integrated buoyancy and stresslet coefficients. This convention must be specified before attaching extensile or contractile sign labels.

For fluid velocity $\Omega (-x_3,0,x_1)$ and $B\Omega <1$, the stable deterministic orientation is $\mathbf{p}=(-B\Omega ,0,\sqrt{1-B^2{\Omega }^2})$. Once this orientation has relaxed, swimming at speed $V_s$ shifts the centre of the circular material trajectories to $(-V_s\sqrt{1-B^2{\Omega }^2}/\Omega ,-V_{s}B)$ in the $(x_1,x_3)$ plane. Its distance from the cylinder axis is $V_s/\Omega$. General initial orientations produce an orientation transient before these circular paths.

For orientation probability density $f$ on the unit sphere, deterministic drift $\dot{\mathbf{p}}$ and isotropic rotational diffusion coefficient $D_R$ give ${\partial }_{t}f+{\nabla }_p\cdot (f\dot{\mathbf{p}}-D_R{\nabla }_{p}f)=0$. The derivatives are tangential to the sphere, and normalization is $\int fd\Omega =1$. The spherical divergence of $\mathbf{k}-p_3\mathbf{p}$ is $-2p_3$.

For aligning rate $1/B$, positive rotational diffusivity $D_r$ and $\Lambda =(B{D}_r{)}^{-1}$, the zero-current steady Fokker-Planck equation gives $f_{0,\theta }=-\Lambda \sin \theta f_0$. Normalization over the unit sphere gives the displayed density per unit solid angle. Its mean direction is $⟨\mathbf{p}⟩=(\coth \Lambda -1/\Lambda )\mathbf{k}$, the Langevin function. As $B{D}_r\to 0$, it concentrates weakly into a point mass at the upward pole; the limit is not an ordinary bounded probability density.

For rigid fluid rotation with vorticity $-2\Omega {\mathbf{e}}_2$, $B\Omega \gg 1$, and $B{D}_R$ of order one, the normalized steady density is $f=(4\pi {)}^{-1}[1-2p_1/(B\Omega )]+O((B\Omega {)}^{-2})$. The result follows by acting with the rotation generator on $p_1$ and using $\int p_i{p}_{j}d\Omega =4\pi {\delta }_{ij}/3$. Reversing the imposed rotation reverses the mean horizontal swimming direction.

With ${\beta }^2=\Omega R^2/D$ and $\epsilon =2V_{s}R/(3B\Omega D)$, the first angular harmonic of the steady cell conservation equation satisfies $a^{\prime \prime }+a^{\prime }/r-a/r^2-i{\beta }^{2}a=0$ and $a^{\prime }(1)=-1$. The regular solution is $a=-I_1(qr)/[q{I}_1^{\prime }(q)]$, $q=e^{i\pi /4}\beta$. At large $\beta$ it gives the displayed layer with width $1/\beta$. The exact Modified Bessel function of the first kind expression remains regular at the axis, unlike extrapolation of the boundary-layer formula there.

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 $D_r^{-1}$.