  Motivated by the complex behavior of the frustrated magnet $\beta$-TeVO$_4$,
we study an anisotropic Heisenberg model for coupled spin-$1/2$ zigzag chains.
Using cluster mean field approach to capture quantum correlations we find, upon
reducing temperature in the absence of applied field, (i) a partially ordered
state, (ii) a collinear antiferromagnetic phase, and (iii) an elliptical spiral
state characterized by finite vector chirality. For finite fields, we find
metamagnetic response close to saturation magnetization. We show via explicit
calculations that the quadrupolar order parameter is finite in the metamagnetic
regime. The exchange parameters reported in the ab-initio study of
$\beta$-TeVO$_4$ are used in our study. We compare our results with those
reported in recent experiments on $\beta$-TeVO$_4$ and highlight similarities
as well as differences between experimental results and our cluster mean field
calculations.
