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Prior mathematical work of Constantin and Iyer (2008, 2011) has shown thatincompressible Navier-Stokes solutions possess infinitely-many stochasticLagrangian conservation laws for vorticity, backward in time, which generalizethe invariants of Cauchy (1815) for smooth Euler solutions. We simplify thistheory for the case of wall-bounded flows by appealing to the Kuz'min(1983)-Oseledets (1989) representation of Navier-Stokes dynamics, in terms ofthe vortex-momentum density associated to a continuous distribution ofinfinitesimal vortex rings. The Constantin-Iyer theory provides an exactrepresentation for vorticity at any interior point as an average overstochastic vorticity contributions transported from the wall. We discussrelations of this Lagrangian formulation with the Eulerian theory of Lighthill(1963)-Morton (1984) for vorticity generation at solid walls, and also with astatistical result of Taylor (1932)-Huggins (1994), which connects dissipativedrag with organized cross-stream motion of vorticity and which is closelyanalogous to the "Josephson-Anderson relation" for quantum superfluids. Weelaborate a Monte Carlo numerical Lagrangian scheme to calculate the stochasticCauchy invariants and their statistics, given the Eulerian space-time velocityfield. The method is validated using an online database of a turbulentchannel-flow simulation (Graham et al. 2016), where conservation of the meanCauchy invariant is verified for two selected buffer-layer events correspondingto an "ejection" and a "sweep". The variances of the stochastic Cauchyinvariants grow exponentially backward in time, however, confirming earlierobservations of Lagrangian chaos in channel-flow turbulence.

Stochastic Lagrangian Dynamics of Vorticity. I. General Theory