5. MOVEMENT OF PACK ICE
6.2. ENERGY DISSIPATION WITHIN PACK ICE
According to Rothrock (1975a), kinetic energy is dissipated within pack ice through (i) generation of gravitational potential energy in ridging,
(ii) frictional losses in the rubble piles of ridges during ridging, (iii) fracture of ice sheets,
(iv) frictional losses in shearing between floes.
We can still add to these
(v) frictional losses in overriding of ice sheets.
For the Arctic seas, Rothrock concluded that the sinks (i) and (ii) due to ridging are nearly equal and argued that (iii) is small compared to (i) and (ii), while (iv) is not greater than (i)
and (ii). The point (v) is probably meaningful only in rafting of very thin ice.
The expression for the potential energy per unit area P of the ice-water system relative to the sea surface was derived by Rothrock (1975a). It consists of the potential energy of the ice and the energy required to displace the water
P = 1/2 Qig h,2 + ½ (Qw — Qi) g h"2
where h' and h" are the freeboard and draft of the ice, respectively. This eq. can be written as
P = %a
ei
(ew —Q) g h 2 + '/2Qwg (h" — Qi h)2 Qwwhere the latter term gives the energy due to the departure from isostasy and will be neglected below. It can be meaningful (Rothrock 1975a), but its estimation requires information not currently available to us.
The potential energy of a ridge is defined here as the sum of the energies of the rubble piles above and below the level ice sheet. The quantities in the following formulas are explained in Figs. 37 and 38. The potential energy per unit width of the ridge sail is
ds y tan~Ps
Ps = (1 — v) 2 J J Qsg (z + h') dxdz
(6.5)
= (1 — v) esg (% hs + h' hs2 ).
tan cp
71
– ---.- sea
surface
Figure 37. The quantities for determing the potential energy of an ice ridge.
Figure 38. Integration of the potential y
energy of a ridge sail.
The ridge keel is treated similarly. Using the isostatic conditions
h' = (1 — ~t ) h, h" = ~1 h, (6.6.a)
ew Qw
y Def hk = IS tan (Pk) ,2 (6.6. b)
hs lew — Qk tan (Ps
the potential energy of the ridge per unit width becomes
Pr=PS+P' =(1—v)• Qsg •( 1—Yh3 +hh2 ). (6.7)
tan cp s \ 3
The term hhs arises from our definition in which the rubble piles above and below the level ice sheet are considered. It would vanish if the potential energy of the ridge had been
72
defined relative to the sea surface; consequently, the sail height would then be the elevation above the sea surface. Eq. (6.7) includes four ridge characteristics, which are chosen as follows: v = 0.4 and QS = 870 kg m-3 (Keinonen 1977); Y = 6.9 (from Palosuo 1975, see section 3.1);
cps =arrctan (bohs ),
bo
= 0.442 m —' (6.8)(see section 3.1).
The potential energy of ridges per unit ice-covered area in a pack ice particle is Pr = Lr P'Ai 1 ,
where L, is the total length of ridges and A, the area of the particle. Using the formulas (3.3.b), (3.4), (6.7) and (6.8), and neglecting the spatial variation of ice thickness in integration, we have
Pr = n N(1—v) g
{ 3
[h2 + (hs —hso 2+ hhs }.
2 b0
Insertion of the values for the parameters gives
Pr = 18.2 kJ m-3 x { 2.63
p hs
+ (hs — hso )2] + hhs } µ . (6.9) Rothrock (1975a) studied the frictional losses in the rubble pile during ridge formation on the basis of a simple Coulomb friction model. He obtained the following expression for the frictional force per unit width of a ridge:_ h2 k
F
fr kfr g 2 tancpkHere
k f
, is the dynamic coefficient of friction; Rothrock usedk f ,
N 0.1-0.4. Ryvlin (1973) concluded that ki, for ice on ice and steel on ice are about equal and recommended the valuekf
, = 0.1, which we shall take here. Using Eq. (6.6. b) the force Ffr can be expressed in terms of the sail characteristics through__ h2
F
fr kfr Qs g2 tancpsThe work done per unit width against friction in building up a ridge is
Wf r =F fr (hs(Y)) dy , 1r (6.10)
where 1, is the length of the ice sheet needed for the rubble pile and h,(Y) is the instantaneous sail height when the ice sheet has advanced the length y in the ridge, i.e. h5(° = 0 and h,(',) _ h, (the final sail height ). It is assumed that the geometric structure of the ridge is similar during the whole ridging process, i.e. cp,(Y) = artan (hse') bo ). The conservation of mass implies that
hy = A,6') = x h,(y) / b,, ,
where Ar(Y) is the instantaneous cross-sectional area of the ridge. Hence h50'> = hybo/ x and It = x hs/boh, and we can integrate Eq. (6. 10) directly:
w
= kfr Qs g : hs 3 (6.11)2 tancps 3 boh
The ratio of the work done against friction to the potential energy is, from Eqs. (6.7) and (6.11)
kfr
6 (1—v) boh \ 3 hs/
E.g., with h, h, "V2 m the ratio is 0.5 and increases to 0.7 as hs - —. Exactly by the same way as in the case of the potential energy we can integrate Eq. (6.11) to have the work done against friction per unit area in a pack ice particle. The result is written
Wfr fr = nµ 2 kfr Osg . x [ h2 ~s s + () —hsoJ 2 tan cps 3bo h
)21
The rest of the energy dissipation is a big question mark. Parmerter & Coon (1972) showed with their pressure ridging model that the energy loss due to fracture of ice is small compared to the potential energy. Frictional losses in shearing between floes are considered meaningful (e.g. Rothrock 1975a) but there is no estimate, empirical or theoretical, of their magnitude.
It was shown in the previous section that the rate of dissipation of kinetic energy per unit area in internal deformation processes is given by tr(E • E). Assuming the viscous constitutive law (Eq. 4.4), the dissipation becomes
74
tr(r • I) = tr [(4 tri) E - I + 2
= 4EI + '1£II ,
which is a quadratic expression of the strain-rate invariants. In contrast, the plastic constitutive law of Coon et al. (1974) gives a linear form.
Kri
g 30
EIcefree time
February— March
5 10 15 20
wind speed (ms' )
Figure 39. The northeast sea surface tilt in the Bothnian Bay (after Lisitzin 1957).