A multi-shock model for the density variance of anisotropic,
  highly-magnetised, supersonic turbulence
  Shocks form the basis of our understanding for the density and velocity
statistics of supersonic turbulent flows, such as those found in the cool
interstellar medium (ISM). The variance of the density field,
$\sigma^2_{\rho/\rho_0}$, is of particular interest for molecular clouds (MCs),
the birthplaces of stars in the Universe. The density variance may be used to
infer underlying physical processes in an MC, and parameterises the star
formation (SF) rate of a cloud. However, models for $\sigma^2_{\rho/\rho_0}$
all share a common feature -- the variance is assumed to be isotropic. This
assumption does not hold when a trans/sub-Alfv\'enic mean magnetic field,
$\vec{B}_0$, is present in the cloud, which observations suggest is relevant
for some MCs. We develop an anisotropic model for $\sigma_{\rho/\rho_0}^2$,
using contributions from hydrodynamical and fast magnetosonic shocks that
propagate orthogonal to each other. Our model predicts an upper bound for
$\sigma_{\rho/\rho_0}^2$ in the high Mach number $(\mathcal{M})$ limit as
small-scale density fluctuations become suppressed by the strong $\vec{B}_0$.
The model reduces to the isotropic $\sigma_{\rho/\rho_0}^2-\mathcal{M}$
relation in the hydrodynamical limit. To validate our model, we calculate
$\sigma_{\rho/\rho_0}^2$ from 12~high-resolution, three-dimensional,
supersonic, sub-Alfv\'enic magnetohydrodynamical (MHD) turbulence simulations
and find good agreement with our theory. We discuss how the two MHD shocks may
be the bimodally oriented over-densities observed in some MCs and the
implications for SF theory in the presence of a sub-Alfv\'enic $\vec{B}_0$. By
creating an anisotropic, supersonic density fluctuation model, this study paves
the way for SF theory in the highly anisotropic regime of interstellar
turbulence.
