36
3 Basics of Nonhydrostatic Modelling
Note that, for convenience, the author has also included the nonlinear terms in
the momentum equations. By splitting pressure into parts according to Eq. (3.13),
these equations take the form:
∂u
∂t
+ u
∂u
∂ x
+ w
∂u
∂z
= −
1
ρ o
∂( p + q)
∂ x
∂w
∂t
+ u
∂w
∂ x
+ w
∂w
∂z
= −
1
ρ o
∂q
∂z
∂ρ
∂t
+ u
∂ρ
∂ x
+ w
∂ρ
∂z
=
∂
∂ x
K h
∂ρ
∂ x
+
∂
∂z
K z
∂ρ
∂z
(3.38)
∂u
∂ x
+
∂w
∂z
= 0
∂q s
∂t
= −ρ o g
∂(h u)
∂ x
The pressure term p refers to the hydrostatic dynamic pressure with reference to
an undisturbed sea level and is given by:
∂ p
∂z
= −
(ρ − ρ o )
ρ o
g
(3.39)
with p = 0 at the sea surface. This ignores effects due to atmospheric pressure variations, which usually can be neglected. Notice that the reduced-gravity force (often
symbolised in short as g
) has disappeared from the vertical momentum equation,
but, in fact, this force has been shifted into the hydrostatic pressure part and density
effects appear now in the horizontal momentum equations.
3.6.2 Discretisation of the Advection Terms
The advection equation for a variable B subject to flow with components u and w
in the vertical ocean slice is given by:
∂ B
∂t
= −u
∂ B
∂ x
− w
∂ B
∂z
(3.40)
Using the product rule of differentiation, this equation can be reformulated as:
∂ B
∂t
= −
∂(u B)
∂ x
−
∂(w B)
∂z
+ B
∂u
∂ x
+
∂w
∂z
(3.41)
where the last term vanishes with insertion of the continuity equation (Eq. 3.17).