  Disclination configurations of a nematic liquid crystal are studied within a
self-consistent molecular field theory. The theory is based on a tensor order
parameter, and can accommodate anisotropic elastic energies without the known
divergences in the Landau-de Gennes formulation. Our results agree with the
asymptotic results of Dzyaloshinsky for the Frank-Oseen energy and far from the
defect core, but reveal biaxial order at intermediate distances from the core,
crossing over to uniaxial, but isotropic configurations as the core is
approached. The elastic terms considered in our energy allow for separate
control of surface tension, anchoring, and elasticity contrast, and are used to
analyze recent results for lyotropic chromonic liquid crystals. The latter
display unusually large defect cores (on the order of tens of microns) which
can be used for a quantitative comparison with the theory. Both $\pm 1/2$
disclination configurations are well reproduced by our calculations. Elastic
anisotropy is also shown to lead to qualitative changes in the disclination
polarization, a quantity that is proportional to the active stress in models of
active matter.
