34
STEPHEN GRIFFIES
Most notably, these coordinates, when used with the familiar horizontal
coordinates (x, y), form a non-orthogonal triad, and thus lead to some
unfamiliar relationships. To proceed in this section, we present some
salient results of the mathematics of generalized vertical coordinates,
and reserve many of the derivations for Griffies, 2004.
When considering generalized vertical coordinates in oceanography,
we always assume that the surfaces cannot overturn on themselves. This
constraint means that the Jacobian of transformation between the generalized vertical coordinate
s = s(x, y, z, t)
(11)
and the geopotential coordinate z, must be one signed. That is, the
specific thickness
∂z
∂s
= z ,s
(12)
is of the same sign throughout the ocean fluid. The name specific thickness arises from the property that
,s ds
(13)
is an expression for the thickness of an infinitesimal layer of fluid bounded
by two constant s surfaces.
Deriving the bottom kinematic boundary condition in s-coordinates
requires a relation between the vertical velocity component used in geopotential coordinates, w = dz/dt, and the pseudo-velocity component
ds/dt. For this purpose, we refer to some results from Section 6.5.5
of Griffies, 2004. As in that discussion, we note isomorphic relations
dz/dt = z ,t + u · ∇ s z + z ,s ds/dt
(14)
ds/dt = s ,t + u · ∇ z s + s ,z dz/dt,
(15)
with rearrangement leading to
dz/dt = z ,s (d/dt − ∂ t − u · ∇ z ) s.
(16)
This expression is relevant when measurements are taken on surfaces
of constant geopotential, or depth. To apply this relation to the ocean
bottom, which is generally not a surface of constant depth, it is necessary
to transform the constant depth gradient ∇ z to a horizontal gradient
taken along the bottom. We thus proceed as in Section 6.5.3 of Griffies,
2004 and consider the time-independent coordinate transformation
(x, y, z, t) = (x, y, −H(x, y), t).
(17)
dz = z
STEPHEN GRIFFIES
Most notably, these coordinates, when used with the familiar horizontal
coordinates (x, y), form a non-orthogonal triad, and thus lead to some
unfamiliar relationships. To proceed in this section, we present some
salient results of the mathematics of generalized vertical coordinates,
and reserve many of the derivations for Griffies, 2004.
When considering generalized vertical coordinates in oceanography,
we always assume that the surfaces cannot overturn on themselves. This
constraint means that the Jacobian of transformation between the generalized vertical coordinate
s = s(x, y, z, t)
(11)
and the geopotential coordinate z, must be one signed. That is, the
specific thickness
∂z
∂s
= z ,s
(12)
is of the same sign throughout the ocean fluid. The name specific thickness arises from the property that
,s ds
(13)
is an expression for the thickness of an infinitesimal layer of fluid bounded
by two constant s surfaces.
Deriving the bottom kinematic boundary condition in s-coordinates
requires a relation between the vertical velocity component used in geopotential coordinates, w = dz/dt, and the pseudo-velocity component
ds/dt. For this purpose, we refer to some results from Section 6.5.5
of Griffies, 2004. As in that discussion, we note isomorphic relations
dz/dt = z ,t + u · ∇ s z + z ,s ds/dt
(14)
ds/dt = s ,t + u · ∇ z s + s ,z dz/dt,
(15)
with rearrangement leading to
dz/dt = z ,s (d/dt − ∂ t − u · ∇ z ) s.
(16)
This expression is relevant when measurements are taken on surfaces
of constant geopotential, or depth. To apply this relation to the ocean
bottom, which is generally not a surface of constant depth, it is necessary
to transform the constant depth gradient ∇ z to a horizontal gradient
taken along the bottom. We thus proceed as in Section 6.5.3 of Griffies,
2004 and consider the time-independent coordinate transformation
(x, y, z, t) = (x, y, −H(x, y), t).
(17)
dz = z