  The kagome lattice can host exotic magnetic phases arising from frustrated
and competing magnetic interactions. However, relatively few insulating kagome
materials exhibit incommensurate magnetic ordering. Here, we present a study of
the magnetic structures and interactions of antiferromagnetic
Na$_2$Mn$_3$Cl$_8$ with an undistorted Mn$^{2+}$ kagome network. Using
neutron-diffraction and bulk magnetic measurements, we show that
Na$_2$Mn$_3$Cl$_8$ hosts two different incommensurate magnetic states, which
develop at $T_{N1} = 1.6$ K and $T_{N2} = 0.6$ K. Magnetic Rietveld refinements
indicate magnetic propagation vectors of the form $\mathbf{q} =
(q_{x},q_{y},\frac{3}{2})$, and our neutron-diffraction data can be well
described by cycloidal magnetic structures. By optimizing exchange parameters
against magnetic diffuse-scattering data, we show that the spin Hamiltonian
contains ferromagnetic nearest-neighbor and antiferromagnetic third-neighbor
Heisenberg interactions, with a significant contribution from long-ranged
dipolar coupling. This experimentally-determined interaction model is compared
with density-functional-theory simulations. Using classical Monte Carlo
simulations, we show that these competing interactions explain the experimental
observation of multiple incommensurate magnetic phases and may stabilize
multi-$\mathbf{q}$ states. Our results expand the known range of magnetic
behavior on the kagome lattice.
