Observations and Modelling of Fronts and Frontogenesis

We assume that the depths of layers 1 and 3 are
continuous across the juncture. This assumption should be
appropriate initially, when the uppermost interior layer
first vanishes, and will remain so if the derived conditions
maintain continuity of the solutions.
We require that ageostrophic (that is, cross-interface)
momentum in each layer be continuous across the juncture.
Since layer depths are continuous, this implies continuity of
velocity:
[v] = [v3] 0, y y. (20)
where for any function ji,
[]
(2la)
= urn [(y y, y < y2) 1, (2lb)
lim [(y y > y2) ]. (21c)
Because (10) holds at y y, (20) represents only one
independent condition.
We have assumed that the layer depths are continuous at
y(t). In order that they remain so, we must have,
d[h1]/dt d[h3]t/dt 0. (22)
Using (8) and (21), (22) becomes,
[(huvl)yit = [we] + [hlylidy2/dt. (23a)
[(h3v3)y] (We'1 + [h3y]1dy2/dt. (23b)
Here We' We in the two-layer region, but We' = 0 in the
three-layer region, where layer 3 is not in contact with the
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