Observations and Modelling of Fronts and Frontogenesis

mixed layer. As is the case with (20), only one of (23) is
an independent condition, since (10) and (19) hold.
The condition (23) may appear artificial at first sight,
but in a simplified form it can be motivated by consideration
of the dynamics. If h3y = h3y, (23b) becomes,
(h3V3)y+ (h3v3)y we (23c)
This form of the condition may be derived by formulating an
equation similar to (16) that is valid for a composite upper
layer that includes both the mixed layer and layer 2, and
then letting the width of the transition in the composite
layer between the mixed layer and layer 2 approach zero. We
give the details of this derivation in Appendix A. It may
also be argued that since a discontinuity in layer depths at
i
would necessitate an unphysical infinite geostrophic shear
by the thermal wind relation, the only dynamically consistent
condition must be exactly that which maintains continuity of
the layer depths.
The third condition is a regularity condition. Using
(10) and expanding terms, (l6b) may be written as,
g(p0f)1(p3 p2)[h2v2 + 2h2yv2y + h2v2 + (h1v1)]
(f u2y)v2 + (f - u3y)v3 0. (24)
A point where h2 - 0 is a singular point of (24), and thus of
the coupled set (16), since the coefficient of the highest
derivative of v2 vanishes there. Solution of (16) by power
series shows that three regular solutions and one singular
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