Stratified mixing by microorganisms

Vertical mixing is of fundamental significance to the general circulation, climate, and life in the ocean. In this work we consider whether organisms swimming at low Reynolds numbers might collectively contribute substantially to vertical mixing. Scaling analysis indicates that the mixing efficiency $\eta$, or the ratio between the rate of potential energy conversion and total work done on the fluid, should scale with $\eta \sim (a / \ell)^3$ as $a / \ell \to 0$, where $a$ is the size of the organism and $\ell = \left ( \nu \kappa / N^2 \right )^{1/4}$ is an intrinsic length scale of a stratified fluid with kinematic viscosity $\nu$, tracer diffusivity $\kappa$, and buoyancy frequency $N^2$. A regularized singularity model demonstrates this scaling, indicating that in this same limit $\eta \approx 1.2 \left ( a / \ell \right )^3$ for vertical swimming and $\eta \approx 0.14 \left ( a / \ell \right )^3$ for horizontal swimming. The model further predicts the absolute maximum mixing efficiency of an ensemble of randomly oriented organisms is around 6\% and that the greatest mixing efficiencies in the ocean (in regions of strong salt-stratification) are closer to 0.1\%, implying that the total contribution of microorganisms to vertical ocean mixing is negligible.