36
STEPHEN GRIFFIES
facilitating a mathematical description analogous to the ocean bottom
just considered. The vertical coordinate takes on the value
z = η(x, y, t)
(23)
at this idealized ocean surface.
We furthermore assume that density of the water crossing the ocean
surface is ρ w , which is a function of the temperature, salinity, and pressure. Different water densities can be considered for precipitation, evaporation, runoff, and ice melt, but this level of detail is not warranted for
present purposes. The mass transport crossing the ocean surface can be
written
(mass/time) through surface = ˆ
n η · ˆ
n w (P − E + R) ρ w dA η . (24)
In this expression, P > 0 is the volume per time per area of precipitation
entering the ocean, E > 0 is the evaporation leaving the ocean, and
R > 0 is the river runoff and ice melt entering the ocean. The unit
normal
ˆ
n η =
∇ (z − η)
|∇ (z − η)|
(25)
points from the ocean surface at z = η into the overlying atmosphere,
whereas the unit normal ˆ
n w orients the flow of the water mass transported across the ocean surface (see Figure 4). Finally, the area element
dA η measures the infinitesimal area on the ocean surface z = η, and it
is given by (see Section 20.13.2 of Griffies, 2004)
dA η = |∇(z − η)| dx dy.
(26)
Figure 4. Schematic of the ocean’s upper surface with a smoothed undulating surface
height at z = η(x, y, t), outward normal direction ˆ
nη, and freshwater normal direction
ˆ
nw. Undulations of the surface height are on the order of a few meters due to tidal
fluctuations in the open ocean, and order 10m-20m in certain embayments (e.g., Bay
of Fundy in Nova Scotia). When imposing the weight of sea ice onto the ocean surface,
the surface height can depress even further, on the order of 5m-10m, with larger values
possible in some cases. It is important for simulations to employ numerical schemes
facilitating such wide surface height undulations.
STEPHEN GRIFFIES
facilitating a mathematical description analogous to the ocean bottom
just considered. The vertical coordinate takes on the value
z = η(x, y, t)
(23)
at this idealized ocean surface.
We furthermore assume that density of the water crossing the ocean
surface is ρ w , which is a function of the temperature, salinity, and pressure. Different water densities can be considered for precipitation, evaporation, runoff, and ice melt, but this level of detail is not warranted for
present purposes. The mass transport crossing the ocean surface can be
written
(mass/time) through surface = ˆ
n η · ˆ
n w (P − E + R) ρ w dA η . (24)
In this expression, P > 0 is the volume per time per area of precipitation
entering the ocean, E > 0 is the evaporation leaving the ocean, and
R > 0 is the river runoff and ice melt entering the ocean. The unit
normal
ˆ
n η =
∇ (z − η)
|∇ (z − η)|
(25)
points from the ocean surface at z = η into the overlying atmosphere,
whereas the unit normal ˆ
n w orients the flow of the water mass transported across the ocean surface (see Figure 4). Finally, the area element
dA η measures the infinitesimal area on the ocean surface z = η, and it
is given by (see Section 20.13.2 of Griffies, 2004)
dA η = |∇(z − η)| dx dy.
(26)
Figure 4. Schematic of the ocean’s upper surface with a smoothed undulating surface
height at z = η(x, y, t), outward normal direction ˆ
nη, and freshwater normal direction
ˆ
nw. Undulations of the surface height are on the order of a few meters due to tidal
fluctuations in the open ocean, and order 10m-20m in certain embayments (e.g., Bay
of Fundy in Nova Scotia). When imposing the weight of sea ice onto the ocean surface,
the surface height can depress even further, on the order of 5m-10m, with larger values
possible in some cases. It is important for simulations to employ numerical schemes
facilitating such wide surface height undulations.