## Wave-mean flow interactions: from nanometre to megametre scales

##### Abstract

Waves, which arise when restoring forces act on small perturbations, are ubiquitous in fluids. Their counterpart, mean flows, capture the remainder of the motion and are often
characterised by a slower evolution and larger scale patterns. Waves and mean flows,
which are typically separated by time- or space-averaging, interact, and this interaction
is central to many fluid-dynamical phenomena. Wave-mean flow interactions can be
classified into dissipative interactions and non-dissipative interactions. The former is
important for small-scale flows, the latter for large-scale flows. In this thesis these
two kinds of interactions are studied in the context of microfluidics and geophysical
applications.
Viscous wave-mean flow interactions are studied in two microfluidic problems. Both
are motivated by the rapidly increasing number of microfluidic devices that rely on the
mean-flow generated by dissipating acoustic waves - acoustic streaming - to drive small-scale
flows. The first problem concerns the effect of boundary slip on steady acoustic
streaming, which we argue is important because of the high frequencies employed. By
applying matched asympototics, we obtain the form of the mean flow as a function of
a new non-dimensional parameter measuring the importance of the boundary slip. The
second problem examined is the development of a theory applicable to experiments and
devices in which rigid particles are manipulated or used as passive tracers in an acoustic
wave field. Previous work obtained dynamical equations governing the mean motion of
such particles in a largely heuristic way. To obtain a reliable mean dynamical equation
for particles, we apply a systematic multiscale approach that captures a broad range of
parameter space. Our results clarify the limits of validity of previous work and identify
a new parameter regime where the motion of particles and of the surrounding fluid are
coupled nonlinearly.
Non-dissipative wave-mean flow interactions are studied in two geophysical fluid
problems. (i) Motivated by the open question of mesoscale energy transfer in the ocean,
we study the interaction between a mesoscale mean flow and near-inertial waves. By
applying generalized Lagrangian mean theory, Whitham averaging and variational calculus,
we obtain a Hamiltonian wave-mean flow model which combines the familiar
quasi-geostrophic model with the Young & Ben Jelloul model of near-inertial waves.
This research unveils a new mechanism of mesoscale energy dissipation: near-inertial
waves extract energy from the mesoscale
ow as their horizontal scale is reduced by differential
advection and refraction so that their potential energy increases. (ii) We study
the interaction between topographic waves and an unidirectional mean flow at an inertial
level, that is, at the altitude where the Doppler-shifted frequency of the waves match
the Coriolis parameter. This interaction can be described using linear theory, using a
combination of WKB and saddle-point methods, leading to explicit expressions for the
mean-flow response. These demonstrate, in particular, that this response is switched
on asymptotically far downstream from the topography, in contrast to what is often
assumed in parameterisation.