Geoscience Reference
In-Depth Information
Fig. 7.1
Staggered grid scheme. Mean properties are evaluated on the
zz
grid, flux and gradient
relatedquantities onthe
z
grid
wheresubscriptsdenotepartial differentiation,
θ
is agenericconservativeproperty
ofthefluid,
F
θ
istheverticalfluxof
,and
Q
θ
isasourceterm.Withtheleap-frog-
in-timescheme,thecorrespondingdifferenceequationis
θ
F
i
+
1
θ
−
F
i
θ
i
,
j
+
1
−
θ
i
,
j
−
1
2
=
−
(
)
Q
θ
+
(7.2)
∆
t
∆
z
i
wherefluxesareevaluatedonastaggeredgrid(Fig.7.1).Thegridischosensothata
conservedquantity(where
mightrepresenttemperature,salinity,ormomentum)is
evaluatedatpointsonthe
zz
grid,where
zz
i
liesmidwaybetweenpoints
z
i
and
z
i
+
1
in
the
z
grid, on which variables dependenton gradients(fluxes, buoyancyfrequency,
eddyviscosity,etc.)areevaluated.
First-order closure relates flux to the mean quantity vertical gradient by means
ofaneddydiffusivityappropriateto
θ
θ
F
θ
=
−
K
θ
θ
(7.3)
z
withcorrespondingdifferenceequation
K
i
F
i
=
−
zz
i
−
1
(
θ
i
−
θ
i
−
1
)
(7.4)
∆
