Chemical Abundances in a Turbulent Medium -- H$_2$, OH$^+$, H$_2$O$^+$,
  ArH$^+$
  Supersonic turbulence results in strong density fluctuations in the
interstellar medium (ISM), which have a profound effect on the chemical
structure. Particularly useful probes of the diffuse ISM are the ArH$^+$,
OH$^+$, H$_2$O$^+$ molecular ions, which are highly sensitive to fluctuations
in the density and the H$_2$ abundance. We use isothermal magnetohydrodynamic
(MHD) simulations of various sonic Mach numbers, $\mathcal{M}_s$, and density
decorrelation scales, $y_{\rm dec}$, to model the turbulent density field. We
post-process the simulations with chemical models and obtain the probability
density functions (PDFs) for the H$_2$, ArH$^+$, OH$^+$ and H$_2$O$^+$
abundances. We find that the PDF dispersions increases with increasing
$\mathcal{M}_s$ and $y_{\rm dec}$, as the magnitude of the density fluctuations
increases, and as they become more coherent. Turbulence also affects the median
abundances: when $\mathcal{M}_s$ and $y_{\rm dec}$ are high, low-density
regions with low H$_2$ abundance become prevalent, resulting in an enhancement
of ArH$^+$ compared to OH$^+$ and H$_2$O$^+$. We compare our models with
Herschel observations. The large scatter in the observed abundances, as well as
the high observed ArH$^+$/OH$^+$ and ArH$^+$/H$_2$O$^+$ ratios are naturally
reproduced by our supersonic $(\mathcal{M}_s=4.5)$, large decorrelation scale
$(y_{\rm dec}=0.8)$ model, supporting a scenario of a large-scale turbulence
driving. The abundances also depend on the UV intensity, CR ionization rate,
and the cloud column density, and the observed scatter may be influenced by
fluctuations in these parameters.
