= Solution
For $D_r>0$, the steady polar flux satisfies $\partial_\theta(\sin\theta j_\theta)=0$. Regularity at both poles makes its integration constant zero. Thus
$$
D_rf_{0,\theta}=-\frac{\sin\theta}{B}f_0,\qquad
f_0=C e^{\Lambda\cos\theta},\qquad \Lambda=(BD_r)^{-1}.
$$
Normalize with $x=\cos\theta$:
$$
1=2\pi C\int_{-1}^1e^{\Lambda x}dx=4\pi C\frac{\sinh\Lambda}{\Lambda}.
$$
The <zero-flow steady gyrotactic orientation distribution> is therefore
$$
\boxed{f_0(\theta)=\frac{\Lambda e^{\Lambda\cos\theta}}{4\pi\sinh\Lambda}.}
$$
Differentiating the logarithm of its normalization integral gives the <Langevin function> mean
$$
\langle\cos\theta\rangle_0=\frac{d}{d\Lambda}\log\frac{\sinh\Lambda}{\Lambda}
=\coth\Lambda-\frac1\Lambda,
\qquad
\boxed{\mathbf V_c=V_s\left(\coth\Lambda-\frac1\Lambda\right)\mathbf k.}
$$
As $\Lambda\to0$, $f_0\to1/(4\pi)$ and the mean tends to zero; as $\Lambda\to\infty$, the distribution concentrates at the upward pole and the mean tends to $V_s\mathbf k$. At zero <diffusivity> the pole-concentrated steady state is a probability measure, not a regular density of this formula.
Back to article page
