  A nonlocal Landau-type free energy functional of the spin density is
developed to model the large variety of magnetic states which occur in the
magnetic field-temperature phase diagram of magnetoelectric CuFeO$_2$.
Competition among long-range quadratic exchange, biquadratic anti-symmetric
exchange, and trigonal anisotropy terms, consistent with the high-temperature
rhombohedral R$\bar{3}$m crystal symmetry, are shown to all play important
roles in stabilizing the unusual combination of commensurate and incommensurate
spin structures in this highly frustrated triangular antiferromagnet. It is
argued that strong magnetoelastic coupling is largely responsible for the
nonlocal nature of the free energy. A key feature of the analysis is that an
electric polarization is induced by a canting of the non-collinear
incommensurate spin structure. Application of the model to ordered spin states
in the triangular antiferromagnets MnBr$_2$ and NaFeO$_2$ is also discussed.
