= Solution
Use mean <incompressibility> to convert advection into momentum-flux divergence:
$$
UU_x+VU_y=\partial_x(U^2)+\partial_y(UV).
$$
Therefore the reduced <Reynolds-averaged momentum equation> gives
$$
\partial_x(U^2)+\partial_y\{UV+\overline{\hat u\hat v}\}=0.
$$
For a localized free jet, $UV$ and the <Reynolds stress> tend to zero in the quiescent ambient as $y\to\pm\infty$. Assume sufficient decay to differentiate its momentum integral. Integrating across the jet gives
$$
\boxed{\frac d{dx}\int_{-\infty}^{\infty}U^2\,dy
=-[UV+\overline{\hat u\hat v}]_{-\infty}^{\infty}=0.}
$$
Thus $M_0=\int U^2dy$ is constant. It is a specific momentum flux per unit span: multiplication by density gives the momentum flux. A clipped profile with nonzero stress just inside its artificial edge would fail this boundary-flux assumption unless an additional external momentum source were supplied.
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