A Numerical Model for the Ice/Ocean Boundary Layer - Air-Ice-Ocean Interaction - page 134

whichincorporatesboththedynamicaspectof ice inertiaand thewind stress mod-
ified by the ice Coriolis force.Substituting into the momentumdifferenceequation
forthefirstgridpoint:

u 1 , j − 1 = − τ 2 2

∆

z 1 + τ 1 2

t ∆ t

t Q u 1

u 1 , j + 1 −

z 1 +

2 ∆ (7.20)

∆ ∆

where τ 1 = − τ w ,providingamodifiedversionof(7.12)

A 1

=

E 1 (7.21)

h ice
∆

A 1 − 1 +

z 1

u 1 , j − 1 1

∆ z 1

t − τ m Q 1 u

h ice

− + 2 ∆

∆ z 1 +

− θ , j −

1 1

D 1 =

h ice
∆

A i − 1 −

z 1

7.2.4 Flux of θ Specified at the Bottom of Model Domain

Forafluxconditionspecifiedatthelowerboundary(includingzeroflux)

F b −

F θ kb − 1

2 ∆ t

tQ θ kb − 1

θ kb − 1 − θ kb − 1 , j − 1 = + 2 ∆ (7.22)

∆ z kb − 1

substitutingfor F kb − 1 andrearrangingterms

t Q θ kb − 1 +

F b θ

C kb − 1 θ kb − 2 =( C kb − 1 − 1 ) θ kb − 1 + 2 ∆ + θ kb − 1 , j − 1 (7.23)

∆ z kb − 1

but since = +

θ kb − 2 D kb − 2 E kb − 2 θ kb − 1 wehave

t Q θ kb − 1 +

F b θ
∆

2 ∆ + θ kb − 1 , j − 1 − C kb − 1 D kb − 2

z kb − 1

θ kb − 1 = (7.24)

+ ( − )

1 C kb − 1 E kb − 2 1

whichthenbeginstheiterationbackupthroughtheIOBLasindicatedby(7.6).

7.2.5 θ Specified at the Bottom of the Model Domain

Figure 7.1 illustrates that zz kb is beyond the model depth domain. Eddy viscos-
ity/diffusivityisevaluatedonthe z (flux)gridanditslastvalueis K kb ,sofrom(7.5),
A and the fields that depend on it are evaluated at indices from 1 to kb -1. Thus to
specifythe valueatthebaseofthedomain,set

θ kb = θ b ,fromwhich

θ kb − 1 = D kb − 1 + E kb − 1 θ b (7.25)

whichagainstartstheiterativeevaluationof(7.6).