Previous polynya flux models have specified a constant value for the collection thickness of frazil ice, H, at the polynya edge. In certain circumstances, this approach can cause the frazil ice depth, h, within the polynya, to exceed H, a result which violates assumptions made in the formulation of the ice flux balance equations at the polynya edge. To overcome this problem, a parameterization for H is derived in terms of the depth of frazil ice arriving at the polynya edge and the component, normal to the polynya edge, of the frazil ice velocity relative to the velocity of the consolidated ice pack. Thus, H is coupled to the unknown polynya edge. Using the new parameterization for H, an analysis of the unsteady one-dimensional opening of a coastal polynya is presented. Analytical solutions are also derived, using the new parameterization for H, for steady-state two-dimensional polynyas adjacent to a semi-infinite and finite-length coastal barrier, the latter case representing a prototype island. In all cases, the solutions show close qualitative and quantitative agreement with those derived using a constant value for H. However, the steady-state two-dimensional polynya edge can, in certain circumstances, exhibit a corner at the point where the offshore equilibrium width is reached. Precise conditions for the existence of a corner are derived in terms of the orientation of the frazil ice velocity (u) and the consolidated ice velocity (U). Upper and lower bounds are also obtained for the area of the steady-state island polynya, and it is shown that over a large range of orientations of u and U, the area exceeds that associated with the island polynya with constant H. Finally, two simulations of the St. Lawrence Island Polynya are presented using the new parameterization for H, and the results are compared with the H-constant theory.