  We examine the vertical mixing induced by the swimming of microorganisms at
low Reynolds and P\'eclet numbers in a stably stratified ocean, and show that
the global contribution of oceanic microswimmers to vertical mixing is
negligible. We propose two approaches to estimating the mixing efficiency,
$\eta$, or the ratio of the rate of potential energy creation to the total
rate-of-working on the ocean by microswimmers. The first is based on scaling
arguments and estimates $\eta$ in terms of the ratio between the typical
organism size, $a$, and an intrinsic length scale for the stratified flow,
$\ell = \left ( \nu \kappa / N^2 \right )^{1/4}$, where $\nu$ is the kinematic
viscosity, $\kappa$ the diffusivity, and $N$ the buoyancy frequency. In
particular, for small organisms in the relevant oceanic limit, $a / \ell \ll
1$, we predict the scaling $\eta \sim (a / \ell)^3$. The second estimate of
$\eta$ is formed by solving the full coupled flow-stratification problem by
modeling the swimmer as a regularized force dipole, and computing the
efficiency numerically. Our computational results, which are examined for all
ratios $a/\ell$, validate the scaling arguments in the limit $a / \ell \ll 1$
and further predict $\eta \approx 1.2 \left ( a / \ell \right )^3$ for vertical
swimming and $\eta \approx 0.15 \left ( a / \ell \right )^3$ for horizontal
swimming. These results, relevant for any stratified fluid rich in biological
activity, imply that the mixing efficiency of swimming microorganisms in the
ocean is at very most 8\% and is likely smaller by at least two orders of
magnitude.
