[6] We propose a theoretical boil vertical propagation model based on the assumption that the boils observed in COHSTREX were surface eruptions of hairpin vortices. Without evidence to the contrary, we assume that the boil vorticity loops originate at the abrupt bathymetry change at the crest of the submerged rocky sill. Boils generated there are carried horizontally with the river flow and upwards by self-advection due to the vortex loop self-interaction.

[7] The vertical velocity of a boil, *w*_{b}, due to a three-dimensional vorticity loop is simplified as the self-advection of an irrotational vortex dipole in the vertical plane [*Batchelor*, 1967],

where Γ_{b} is the boil circulation and *d* is the distance between vortex pair centers. As shown later, the lateral scale of the boils observed on the surface is similar to the depth of the shear layer, so we approximate *d* and the vortex diameter as the observable surface boil diameter. Circulation for each vortex in the pair is the product of the vorticity, *ω*, and the vortex cross-section, *dA*,

As is discussed in section 3, the sill extends over 2/3 of the water column during the observation period. Thus, the vorticity generated by shear at the sill crest is expected to be significantly greater than vorticity carried into this region from the bottom boundary layer upstream of the sill. We assume, therefore, that *ω* is generated at the sill crest, i.e., *ω* ≈ , where *u* is the streamwise velocity, *z* is the vertical coordinate, and vertical velocity is assumed negligible. We further simplify and express the shear in terms of bulk parameters, *ω* ≈ , where *U* is the surface velocity over the sill and *h* is the depth of water over the sill crest. Hence, the predicted distance downstream from the sill to the eruption point at the surface, *L*_{1}, is to first order

assuming *w*_{b} is constant and the boil travels downstream at constant velocity *U*. Combining equation (1) through equation (3) results in a simple expression for the downstream boil eruption point,

which depends only on boil diameter and the sill depth, as *U* has been eliminated due to kinematics defined in terms of streamwise velocity. Following a similar derivation, but substituting the measured *U* into equation (3) and the measured vertical gradient in streamwise velocity *U*_{z} for *ω* in equation (2), *L* can also be estimated as