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The most commonly used coordinate system on the sphere is the geographic system ( λ , φ ) . The curvilinear nature of these coordinates on the sphere lead to some “metric” terms in the component momentum equations. Under the thin-atmosphere and hydrostatic approximations these terms are discretized:

(2.101)
¶
A
w
Δ
r
f
h
w
G
m
e
t
r
i
c
u
=
¯
¯
u
i
a
tan
φ
A
c
Δ
r
f
h
c
¯
v
j
i

(2.102)
¶
A
s
Δ
r
f
h
s
G
m
e
t
r
i
c
v
=
−
¯
¯
u
i
a
tan
φ
A
c
Δ
r
f
h
c
¯
u
i
j

(2.103)
¶
G
m
e
t
r
i
c
w
=
0

where a is the radius of the planet (sphericity is assumed) or the radial distance of the particle (i.e. a function of height). It is easy to see that this discretization satisfies all the properties of the discrete Coriolis terms since the metric factor u a tan φ can be viewed as a modification of the vertical Coriolis parameter: f → f + u a tan φ .

δ
i
:
δ
i
Φ
=
Φ
i
+
1
/
2
−
Φ
i
−
1
/
2

−
i
:
¯
Φ
i
=
(
Φ
i
+
1
/
2
+
Φ
i
−
1
/
2
)
/
2

δ
x
:
δ
x
Φ
=
1
Δ
x
δ
i
Φ

¯
∇
= horizontal gradient operator :
¯
∇
Φ
=
{
δ
x
Φ
,
δ
y
Φ
}

¯
∇
⋅
= horizontal divergence operator :
¯
∇
⋅
→
f
=
1
A
{
δ
i
Δ
y
f
x
+
δ
j
Δ
x
f
y
}

¯
∇
2
= horizontal Laplacian operator :
¯
∇
2
Φ
=
¯
∇
⋅
¯
∇
Φ

The equations of motion integrated by the model involve four prognostic equations for flow, u and v , temperature, θ , and salt/moisture, S , and three diagnostic equations for vertical flow, w , density/buoyancy, ρ / b , and pressure/geo-potential, ϕ h y d . In addition, the surface pressure or height may by described by either a prognostic or diagnostic equation and if non-hydrostatics terms are included then a diagnostic equation for non-hydrostatic pressure is also solved. The combination of prognostic and diagnostic equations requires a model algorithm that can march forward prognostic variables while satisfying constraints imposed by diagnostic equations.

Since the model comes in several flavors and formulation, it would be confusing to present the model algorithm exactly as written into code along with all the switches and optional terms. Instead, we present the algorithm for each of the basic formulations which are:

In all the above configurations it is also possible to substitute the Adams-Bashforth with an alternative time-stepping scheme for terms evaluated explicitly in time. Since the over-arching algorithm is independent of the particular time-stepping scheme chosen we will describe first the over-arching algorithm, known as the pressure method, with a rigid-lid model in
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. This algorithm is essentially unchanged, apart for some coefficients, when the rigid lid assumption is replaced with a linearized implicit free-surface, described in
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. These two flavors of the pressure-method encompass all formulations of the model as it exists today. The integration of explicit in time terms is out-lined in
Section 2.5
and put into the context of the overall algorithm in
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and
Section 2.8
. Inclusion of non-hydrostatic terms requires applying the pressure method in three dimensions instead of two and this algorithm modification is described in
Section 2.9
. Finally, the free-surface equation may be treated more exactly, including non-linear terms, and this is described in
Section 2.10.2
.

The horizontal momentum and continuity equations for the ocean (
(1.98)
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), or for the atmosphere (
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), can be summarized by: