  The efficiency of radial transport of icy solid material from outer disk to
the inner disk is currently unconstrained. Efficient radial transport of icy
dust grains could significantly alter the composition of the gas in the inner
disk. Our aim is to model the gaseous CO$_2$ abundance in the inner disk and
use this to probe the efficiency of icy dust transport in a viscous disk.
Features in the simulated CO$_2$ spectra are investigated for their dust flux
tracing potential. We have developed a 1D viscous disk model that includes gas
and grain motions as well as dust growth, sublimation and freeze-out and a
parametrisation of the CO$_2$ chemistry. The thermo-chemical code DALI was used
to model the mid-infrared spectrum of CO$_2$, as can be observed with
JWST-MIRI. CO$_2$ ice sublimating at the iceline increases the gaseous CO$_2$
abundance to levels equal to the CO$_2$ ice abundance of $\sim 10^{-5}$, which
is three orders of magnitude more than the gaseous CO$_2$ abundances of $\sim
10^{-8}$ observed by Spitzer. Grain growth and radial drift further increase
the gaseous CO$_2$ abundance. A CO$_2$ destruction rate of at least $10^{-11}$
s$^{-1}$ is needed to reconcile model prediction with observations. This rate
is at least two orders of magnitude higher than the fastest known chemical
destruction rate. A range of potential physical mechanisms to explain the low
observed CO$_2$ abundances are discussed. Transport processes in disks can have
profound effects on the abundances of species in the inner disk. The
discrepancy between our model and observations either suggests frequent shocks
in the inner 10 AU that destroy CO$_2$, or that the abundant midplane CO$_2$ is
hidden from our view by an optically thick column of low abundance CO$_2$ in to
the disk surface XDR/PDR. Other molecules, such as CH$_4$ or NH$_3$, can give
further handles on the rate of mass transport.
