  In rough-wall boundary layers, wall-parallel non-homogeneous mean-flow
solutions exist that lead to so-called dispersive velocity components and
dispersive stresses. They play a significant role in the mean-flow momentum
balance near the wall, but typically disappear in the outer layer. A
theoretical framework is presented to study the decay of dispersive motions in
the outer layer. To this end, the problem is formulated in Fourier space, and a
set of governing ordinary differential equations per mode in wavenumber space
is derived by linearizing the Reynolds-averaged Navier-Stokes equations around
a constant background velocity. With further simplifications, analytically
tractable solutions are found consisting of linear combinations of $\exp(-kz)$
and $\exp(-Kz)$, with $z$ the wall distance, $k$ the magnitude of the
horizontal wavevector $\mathbf{k}$, and where $K(\mathbf{k},\textit{Re})$ is a
function of $\mathbf{k}$ and the Reynolds number $\textit{Re}$. Moreover, for
$k\rightarrow \infty$ or $k_1\rightarrow 0$, $K\rightarrow k$ is found, in
which case solutions consist of a linear combination of $\exp(-kz)$ and
$z\exp(-kz)$, and are Reynolds number independent. These analytical relations
are verified in the limit of $k_1=0$ using the rough boundary layer experiments
by Vanderwel and Ganapathisubramani (J. Fluid Mech. 774, R2, 2015) and are in
good agreement for $\ell_k/\delta \leq 0.5$, with $\delta$ the boundary-layer
thickness and $\ell_k = 2\pi/k$.
