Fluid flows in which the particles passively advect and diffuse are common throughout nature.

There are large volumes of work in the area of advection-diffusion however, despite the many

applications to fluids flowing with a reaction component, no clear model of capturing front dynamics

exists. Fronts propagating in two-dimensional advection-reaction-diffusion systems exhibit rich

topological structure just like their advection-diffusion counterparts. It has been shown that

invariant manifolds form barriers to passive transport in time-dependent or time-periodic fluid flows.

Recently, analogous manifolds termed burning-invariant-manifolds (BIMs), have been shown to

form one-sided barriers to reaction fronts in ARD systems. Here we apply the BIMs model to that

of time-periodic flows. When the underlying fluid flow is time-periodic, the reaction front will often

mode-lock to the driving frequency. This mode-locking phenomenon can be described by BIMs.

In fact, the mode-locking front is exactly the BIM attached to a relative periodic orbit (RPO) of

the front element dynamics. Changes in the type of mode-locking (and the loss of mode-locking)

can be understood in terms of local and global bifurcations of these RPOs. In the first part of this

thesis we illustrate the above concepts numerically using a chain of alternating vortices in a channel

geometry.

In nature most flows are not time-independent or time-periodic but may have an unknown or

aperiodic time dependence. Therefor we can no longer talk about infinite time structures like

invariant manifolds and BIMs. In the field of advection-diffusion there exists a model for extracting

coherent structures in time-aperiodic flows called Lagrangian Coherent Structures (LCSs). Recent

theoretical work in ARD systems has suggested that similar one-sided barriers, termed burning

Lagrangian coherent structures (bLCSs), exist for fluid velocity data prescribed over a finite time

interval. In the second part of this thesis, we apply the bLCS model to a numerically generated flow

with an aperiodic wind. The wind is used to generate time dependence in a double-vortex channel

flow where we can then demonstrate that the (locally) most attracting or repelling curves are the

bLCSs.

Finally we model an experimental flow performed by Solomon et. al [23]. The experiment

utilizes a single vortex in a windy channel. In the reference frame of the vortex, a constant wind is

blowing from left to right, while a swirling wind blows in the lateral direction. First we extract the

BIMs for the steady, constant, left to right wind, then we show the time evolved BIMs for the fully

unsteady case and comment on the validity of evolving the BIMs.