  (Abridged.) The mean-square current quadrupole moment associated with
vorticity fluctuations in high-Reynolds-number turbulence in a differentially
rotating neutron star is calculated analytically, as are the amplitude and
decoherence time of the resulting, stochastic gravitational wave signal. The
calculation resolves the subtle question of whether the signal is dominated by
the smallest or largest turbulent eddies: for the Kolmogorov-like power
spectrum observed in superfluid spherical Couette simulations, the wave strain
is controlled by the largest eddies, and the decoherence time approximately
equals the maximum eddy turnover time. For a neutron star with spin frequency
$\nu_s$ and Rossby number $Ro$, at a distance $d$ from Earth, the
root-mean-square wave strain reaches $h_{RMS} \approx 3\times 10^{-24} Ro^3
(\nu_s / 30 Hz)^3 (d/1 kpc)^{-1}$. A cross-correlation search can detect such a
source in principle, because the signal decoheres over the time-scale $\tau_c
\approx 10^{-3} Ro^{-1} (\nu_s / 30 Hz)^{-1} s$, which is adequately sampled by
existing long-baseline interferometers. Hence hydrodynamic turbulence imposes a
fundamental noise floor on gravitational wave observations of neutron stars,
although its polluting effect may be muted by partial decoherence in the
hectohertz band, where current continuous-wave searches are concentrated, for
the highest frequency (and hence most powerful) sources.
