38
STEPHEN GRIFFIES
kinematic boundary condition
ρ (∂ t + u · ∇) η = ρ w q w + ρ w
at z = η
(33)
which can be written in the material form
ρ
d(z − η)
dt
= −ρ w q w
at z = η.
(34)
Contrary to the solid earth condition (10), where z + H is materially
constant, permeability of the ocean surface leads to a nontrivial material
evolution of z − η.
To derive the analogous s-coordinate boundary condition, we proceed
as for the bottom. Here, the coordinate transformation is time dependent
(x, y, z, t) = (x, y, η(x, y, t), t).
(35)
The horizontal gradient and time derivative operators are therefore related by
∇ z = ∇ z + (∇ η) ∂ z
(36)
∂ t = ∂ t + η ,t ∂ z .
(37)
Hence, the relation (16) between vertical velocity components takes the
following form at the ocean surface
w = z ,s (d/dt − ∂ t − u · ∇ z ) s + (∂ t + u · ∇)η
at z = η.
(38)
Substitution of the z-coordinate kinematic boundary condition (33) leads
to
ρ z ,s (d/dt − ∂ t − u · ∇ z ) s = −ρ w q w
at s = s top
(39)
where s top = s(x, y, z = η, t) is the value of the generalized vertical
coordinate at the ocean surface. Reorganizing the result (39) leads to
the material time derivative form
ρ z ,s
d(s − s top )
dt
= −ρ w q w
at s = s top
(40)
which is analogous to the z-coordinate result (34). Indeed, it can be
derived trivially by noting that dz/dt = z ,s ds/dt. Even so, it is useful
to have gone through the previous manipulations in order to garner
experience and confidence with the formalism. Such confidence becomes
of particular use in the next section focusing on the dia-surface flux.