  The family of metallic kagome compounds $A$V$_3$Sb$_5$ ($A$=K, Rb, Cs) was
recently discovered to exhibit both superconductivity and charge order. The
nature of the charge-density wave (CDW) phase is presently unsettled, which
complicates the interpretation of the superconducting ground state. In this
paper, we use group-theory and density-functional theory (DFT) to derive and
solve a phenomenological Landau model for this CDW state. The DFT results
reveal three unstable phonon modes with the same in-plane momentum but
different out-of-plane momenta, whose frequencies depend strongly on the
electronic temperature. This is indicative of an electronically-driven CDW,
stabilized by features of the in-plane electronic dispersion. Motivated by the
DFT analysis, we construct a Landau free-energy expansion for coupled CDW order
parameters with wave-vectors at the $M$ and $L$ points of the hexagonal
Brillouin zone. We find an unusual trilinear term coupling these different
order parameters, which can promote the simultaneous condensation of both CDWs
even if the two modes are not nearly-degenerate. We classify the different
types of coupled multi-$\bf{Q}$ CDW orders, focusing on those that break the
sixfold rotational symmetry and lead to a unit-cell doubling along all three
crystallographic directions, as suggested by experiments. We determine a region
in parameter space, characterized by large nonlinear Landau coefficients, where
these phases - dubbed staggered tri-hexagonal and staggered Star-of-David - are
the leading instabilities of the system. Finally, we discuss the implications
of our results for the kagome metals.
