  Prior mathematical work of Constantin and Iyer (2008, 2011) has shown that
incompressible Navier-Stokes solutions possess infinitely-many stochastic
Lagrangian conservation laws for vorticity, backward in time, which generalize
the invariants of Cauchy (1815) for smooth Euler solutions. We simplify this
theory for the case of wall-bounded flows by appealing to the Kuz'min
(1983)-Oseledets (1989) representation of Navier-Stokes dynamics, in terms of
the vortex-momentum density associated to a continuous distribution of
infinitesimal vortex rings. The Constantin-Iyer theory provides an exact
representation for vorticity at any interior point as an average over
stochastic vorticity contributions transported from the wall. We discuss
relations of this Lagrangian formulation with the Eulerian theory of Lighthill
(1963)-Morton (1984) for vorticity generation at solid walls, and also with a
statistical result of Taylor (1932)-Huggins (1994), which connects dissipative
drag with organized cross-stream motion of vorticity and which is closely
analogous to the "Josephson-Anderson relation" for quantum superfluids. We
elaborate a Monte Carlo numerical Lagrangian scheme to calculate the stochastic
Cauchy invariants and their statistics, given the Eulerian space-time velocity
field. The method is validated using an online database of a turbulent
channel-flow simulation (Graham et al. 2016), where conservation of the mean
Cauchy invariant is verified for two selected buffer-layer events corresponding
to an "ejection" and a "sweep". The variances of the stochastic Cauchy
invariants grow exponentially backward in time, however, confirming earlier
observations of Lagrangian chaos in channel-flow turbulence.
