SOME OCEAN MODEL FUNDAMENTALS
53
Forces due to the contact stresses at the west and east sides are given
by
CONTACT FORCE ON WEST SIDE = -
dy ds z , ~
(k T - kp)
(96)
SS
CONTACT FORCE ON EAST SIDE =
dy ds zjs (k
- k p )
with similar results at the north and south sides. At the top of the
cell, dA(fi) n = Vs dx dy whereas dA(q n = -Vs dx dy at the bottom.
Hence,
CONTACT FORCE ON CELL BOTTOM = -
dy d~ Zls (VS ' 7 - p VS). (99)
SS
Bringing these results together, and taking limit as the time independent
horizontal area dxdy t 0, leads to the thickness weighted budget for
the momentum per horizontal area of an interior grid cell
at (dzpv) = - V,. [ d z u ( p v ) ]
-
Note that both the time and horizontal partial derivatives are for p~
sitions fixed on a constant generalized vertical coordinate surface. Additionally, we have yet to take the hydrostatic approximation, so these
equations are written for the three components of the vertical velocity.
The first term on the right hand side of the thickness weighted m e
mentum budget (100) is the convergence of advective momentum fluxes
occurring within the layer. We discussed the analogous flux convergence for the tracer and mass budgets in Section 3.3. The second and
third terms arise from the transport of momentum across the upper and
lower constant s interfaces. The fourth and fifth terms arise from the
53
Forces due to the contact stresses at the west and east sides are given
by
CONTACT FORCE ON WEST SIDE = -
dy ds z , ~
(k T - kp)
(96)
SS
CONTACT FORCE ON EAST SIDE =
dy ds zjs (k
- k p )
with similar results at the north and south sides. At the top of the
cell, dA(fi) n = Vs dx dy whereas dA(q n = -Vs dx dy at the bottom.
Hence,
CONTACT FORCE ON CELL BOTTOM = -
dy d~ Zls (VS ' 7 - p VS). (99)
SS
Bringing these results together, and taking limit as the time independent
horizontal area dxdy t 0, leads to the thickness weighted budget for
the momentum per horizontal area of an interior grid cell
at (dzpv) = - V,. [ d z u ( p v ) ]
-
Note that both the time and horizontal partial derivatives are for p~
sitions fixed on a constant generalized vertical coordinate surface. Additionally, we have yet to take the hydrostatic approximation, so these
equations are written for the three components of the vertical velocity.
The first term on the right hand side of the thickness weighted m e
mentum budget (100) is the convergence of advective momentum fluxes
occurring within the layer. We discussed the analogous flux convergence for the tracer and mass budgets in Section 3.3. The second and
third terms arise from the transport of momentum across the upper and
lower constant s interfaces. The fourth and fifth terms arise from the