SOME OCEAN MODEL FUNDAMENTALS
37
We now introduce a more convenient expression for the mass transport
across the surface by exploiting our assumption that the ocean surface
has no overturns. For this purpose, define
q w dA = ˆ
n η · ˆ
n w (P − E + R) dA η ,
(27)
where
dA = dx dy
(28)
is the horizontal projection of the surface area element dA η . The volume per time per horizontal area of fluid crossing the ocean surface is
therefore defined by q w
q w =
ˆ
n η · ˆ
n w (P − E + R) dA η
dA
=
(volume/time) through free surface
horizontal area under free surface
.
(29)
This is the surface water flux that appears in ocean model budgets for
mass, tracer, and momentum.
As discussed in Section 3.4.3 of Griffies, 2004, the mass budget per
horizontal area of a column of fluid extending from the ocean surface to
its bottom is given by
∂ t
⎛
⎝
η
−H
dz ρ
⎞
⎠ = −∇ ·
⎛
⎝
η
−H
dz ρ u
⎞
⎠ + q w ρ w .
(30)
This budget says that the time tendency of the total fluid mass per
unit horizontal area within a column (left hand side) is balanced by the
convergence of mass into the column (first term on the right hand side)
and transport across the upper ocean surface (second term on the right
hand side). To develop the upper ocean kinematic boundary condition,
perform the derivatives in equation (30), keeping in mind Leibnitz’s Rule
when differentiating an integral. This step then leads to
[ρ (∂ t +u·∇) η] z=η +[ρ ∇H ·u] z=−H +
η
−H
dz [ρ ,t +∇·(ρ u)] = ρ w q w . (31)
Use of the mass conservation equation (4) yields
[ρ (η ,t + u · ∇η − w)] z=η + [ρ (w + ∇H · u)] z=−H = ρ w q w .
(32)
The solid earth kinematic boundary condition (9) allows us to cancel
the second term on the left hand side, thus leading to the surface ocean
37
We now introduce a more convenient expression for the mass transport
across the surface by exploiting our assumption that the ocean surface
has no overturns. For this purpose, define
q w dA = ˆ
n η · ˆ
n w (P − E + R) dA η ,
(27)
where
dA = dx dy
(28)
is the horizontal projection of the surface area element dA η . The volume per time per horizontal area of fluid crossing the ocean surface is
therefore defined by q w
q w =
ˆ
n η · ˆ
n w (P − E + R) dA η
dA
=
(volume/time) through free surface
horizontal area under free surface
.
(29)
This is the surface water flux that appears in ocean model budgets for
mass, tracer, and momentum.
As discussed in Section 3.4.3 of Griffies, 2004, the mass budget per
horizontal area of a column of fluid extending from the ocean surface to
its bottom is given by
∂ t
⎛
⎝
η
−H
dz ρ
⎞
⎠ = −∇ ·
⎛
⎝
η
−H
dz ρ u
⎞
⎠ + q w ρ w .
(30)
This budget says that the time tendency of the total fluid mass per
unit horizontal area within a column (left hand side) is balanced by the
convergence of mass into the column (first term on the right hand side)
and transport across the upper ocean surface (second term on the right
hand side). To develop the upper ocean kinematic boundary condition,
perform the derivatives in equation (30), keeping in mind Leibnitz’s Rule
when differentiating an integral. This step then leads to
[ρ (∂ t +u·∇) η] z=η +[ρ ∇H ·u] z=−H +
η
−H
dz [ρ ,t +∇·(ρ u)] = ρ w q w . (31)
Use of the mass conservation equation (4) yields
[ρ (η ,t + u · ∇η − w)] z=η + [ρ (w + ∇H · u)] z=−H = ρ w q w .
(32)
The solid earth kinematic boundary condition (9) allows us to cancel
the second term on the left hand side, thus leading to the surface ocean