46 
EQUATIONS FOR SEA ICE 
The complete sea ice model comprises both thermodynamics (growth, melting) and 
dynamics (deformation, drift). Both components are part of a two-way coupled ice-ocean 
model. The ice cover is described by the standard two variables of mean thickness and 
compactness. The amount of ice, which is governed by production and ablation, is 
modelled in terms of a heat budget, namely the budget of latent heat of fusion. As we use 
a common-type model, details of ice thermodynamics have been left out. 
Here we only expose some details of the ice mechanics model. As is expedient on a 
large scale, the fragmented ice is modelled as a continuum. The ice cover is considered 
as a thin skin in a state of plane stress. Our model is lent from Hibler (1979). The 
equations are listed in the following because, firstly, they are rarely found in terms of 
spherical co-ordinates and, secondly, the constitutive equation is a slight modification of 
standard. The momentum equations for (horizontal) ice drift read 
» .ÔW; 
R cos <p dA 
U : ÔV; 
ÔU, V; ÔU; 
R dtp 
tan cp 
R 
v i u i - 2co sin (p V( + g 
1 dÇ 
R cos (p dA 
)=TÏ+zî+F> 
, ,ÔV,. U; 
H O: ( + 
ôt R cos (p dA 
v,. dv tan <p 
+ ——— + — 
R dtp 
R 
- . 1 dÇ. 
+ 2cùsm cp u t +g——)-- 
R d(p 
ri + rf + FA 
The ice drift is forced by shear drag stresses at the surface (wind) and bottom (sea 
current) 
= c w p w ((u w -u i ) 2 +(v w -v i yy ,2 (u w -u i ) 
K =c w p w ((u w -u i y +(v w -v,.) : )’‘ 2 (v w -V,.) 
*a - C a p a {{u a -U t y +(v a -V ( .)T 2 K -W,) 
T a = C aPa(( U a ~ U if + ( V a ~ V i ) 2 )'' 2 ( V a ~ V i) 
and by forces due to internal stress 
FÂ = 
F f 
1 ,d. , d . .. tan^ 
(— (o-„) + — (cos (pv n )) —-cr n 
R cos cp dA dcp R 
1 , d , N d . .. tan <p 
— (—'( Cr i 2 ) + ^-( C0S <P^22))+——Cll 
Rcosç dA dcp R 
In this notation h is the mean ice thickness. Horizontal velocities (at the sea surface) are 
designated as u,v with subscripts a i w for air, ice and water, respectively. Accordingly, 
P a ,Pt>P v signifies the density. The quadratic parameterisation of drag stress stands for 
the turbulent shear layer at the air and water interfaces in the same way it does for wind 
drag in the open sea or bottom drag on the sea bed. The symbols c a ,c w denote the non- 
dimensional drag coefficients. 
What makes the ice special is its characterization as a continuously deforming substance 
with properties of mechanical response. On a large scale, sea ice is modelled as an 
equivalent continuum which exhibits a limit stress (yield stress) in compaction and shear 
but is unable to withstand tensile stress. In terms of continuum mechanics its main 
feature is plasticity.