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Numerical modeling of sea ice in the seasonal sea ice zone
300 NUMERICAL MODELING OF SEA ICE IN THE SEASONALSEA ICE ZONE W.D. Hibler I l l USA Cold Regions Research and Engineering Laboratory, Hanover, N.H. 03755 INTRODUCTION In the polar regions the oceans are covered with a relatively thin, layer of sea ice.
variable thickness
Due to frequent deformation the thickness of this ice cover varies greatly,
and ice several years old, for example, may be located next to ice only a few days o l d .
The
behavior of this ice cover is coupled to both the thermodynamic and dynamic characteristics of the atmosphere and ocean.
In addition the ice cover can substantially modify the air-sea heat
and momentum fluxes, and thus change the behavior of the atmosphere and ocean. These feedbacks are particularly pronounced in the seasonal ice zone. Modeling this complex, coupled system has been the subject of considerable research over the last decade. Efforts to properly model the seasonal ice zone, however, are just beginning. In this paper progress in modeling sea ice and growth, d r i f t , and decay is b r i e f l y reviewed and particular problems associated with modeling the seasonal ice zone are discussed.
For this
purpose, the discussion is divided into three parts. I t should be noted that much of the material and Empirical
in the sections on Numerical Sea Ice Models
Studies of this paper has been abstracted from the modeling portions o f a more
complete review of sea ice growth d r i f t
and decay published elsewhere (Hibler,
Interested readers are referred to that review for more detail on ~ e l i n g
in press).
as well as a review
of other aspects of sea ice growth, d r i f t , and decay.
NUMERICAL SEA ICE MODELS A concept essential
to modeling sea ice is that
growth and decay are i n t r i n s i c a l l y
related.
the ice dynamics and the rates of ice
In p a r t i c u l a r ,
rates of growth and decay of the
ice cover depend on the d i s t r i b u t i o n of ice thicknesses which in turn depend on the ice deformation and transport patterns.
The ice d r i f t
is also modified by the ice thickness d i s t r i b u -
tion which affects the amount of stress the ice can transmit.
In a complete simulation model
these e f f e c t s are coupled by allowing the ice i n t e r a c t i o n to depend on the ice thickness chara c t e r i s t i c s , which, in turn, are functions of both the thermodynamics and dynamics. In more formal terms, construction of a coupled sea ice model ponents :
requires the following com-
301 I) lis
A momentum balance describing ice d r i f t which includes air and water stresses, Corio-
force, internal
tilt);
ice forces, inertial
forces, and steady current effects (including ocean
2) an ice rheology which relates (in some suitable space and time scale) the ice stress
to the ice deformation and strength; 3) an ice thickness distribution which accounts for the change of ice thicknesses due to thermodynamic and dynamic effects; 4) an ice strength determined primarily as a function of the ice thickness distribution; and 5) a thermodynamic code which specifies the growth and decay rates of various ice thicknesses in the environment of a variable thickness ice cover. A t a c i t assumption in l i s t i n g these components is that a description of sea ice as a twodimensional continuum is possible.
While this
is reasonable i t
results from studies of the kinematics of pack ice (e.g.,
is important
Hibler et al.,
to note that
1974a; Nye 1975;
Thorndike and Colony, in press) exhibit substantial deviations from a spatially smooth velocity field.
In practice, however, typical fluctuations are small enough so that on scales larger
than about 20 km they do not mask the continuum motion.
But their presence means that in
general, continuum motion has to be considered a s t a t i s t i c a l l y defined quantity for sea ice. Also, the physical structure, s a l i n i t y , and small scale strength of sea ice have purposely been excluded from the above outline. play a significant
role in local
Such physical
properties affect the thermodynamics and
force problems (e.g.,
Tryde, 1977; Croasdale, 1978).
addition, the salt expulsed from freezing sea ice plays a c r i t i c a l ling.
role in the ocean-ice coup-
However, for large scale ice modeling, the contribution of such small scale characteris-
tics to the ice interaction and d r i f t is f e l t to be of second order importance. of
In
For a review
small scale strength properties the interested reader is referred to Schwarz and Weeks
(1977). A variety of process studies and numerical simulations have been carried out to determine how to parameterize these five components. Numerical work to date can be broadly divided into two categories:
short term simulations spanning several months or less, and seasonal simula-
tions covering at least one annual cycle.
The short term simulations have been primarily
oriented toward determining proper treatments of the momentum balance, ice rheology and ice strength.
For this purpose these studies have concentrated on the simulation of ice d r i f t and
deformation characteristics.
The seasonal simulations on the other hand, have been more
oriented toward simulating ice thickness characteristics and more generally the air-sea heat and momentum exchanges. For these purposes all five components play a role, although particular emphasis has been placed on the thermodynamics and ice thickness distribution. In the remainder of this section a number of numerical studies which have yielded insight into modelling sea ice growth d r i f t and decay are b r i e f l y reviewed. into short term simulations and seasonal simulations.
These studies are divided
302 Short-Term Drift Simulations Free d r i f t studies The simplest type of short-term simulations make use of semi-empirical rules to estimate the ice d r i f t
from the local wind and current values.
The most widely used rule for this
purpose is that of Zubov (1943) which states that the ice drifts parallel to the isobars with about one-fiftieth of the geostrophic wind speed.
Ad hoc modifications to such rules include
seasonally modifying the proportionality constant and turning angle between ice d r i f t and wind, and spatially averaging the wind field in order to crudely take into account the affects of ice interaction.
Sobczak (1977) has compared several of these empirical
rules to observed ice
d r i f t in the Beaufort Sea. Most such empirical rules are essentially free d r i f t approximations in which the momentum balance without the internal ice stress and acceleration terms is used. Probably the most quantitative examination of free d r i f t is a study by McPhee (in press a) using summer ice d r i f t data taken during the 1975 AIDJEX field study.
In this analysis excel-
lent agreement between daily averaged predicted and observed d r i f t rates were obtained.
This
agreement, in turn, supports the u t i l i t y of existing wind and water boundary layer formulations for estimating ice d r i f t .
This type of free d r i f t approximation appears useful for determining
the average d r i f t of ice in the summer marginal ice zone (at least under low compactness conditions). Mechanistic model studies While s t i l l
considering only the momentum balance more complete mechanistic simulations
have been made by assuming certain spatially invariant mechanical properties of the ice cover. The primary goal in these studies was to determine the role of the ice interaction in modeling ice dynamics.
To aid in the discussion of these calculations a brief review of concepts of
rheology relevant to sea ice is included as an appendix. Exemples of mechanistic model studies using a given rheology with specified strengths are Newtonian viscous calculations by Laikhtman (1~54), Campbell (1965), and Egerov (1971); general linear viscous calculations (including both bulk and shear viscosities) by Hibler~(1974) and Hibler and Tucker (in press); calculations by Rothrock (Ig75a) assuming an incompressible ice cover with a spatially constant specified divergence rate; and near shore plastiC simulations by Pritchard (1978).
In all these mechanistic model studies spatially fixed ice strengths are
assumed, although in some cases temporal variations in the strengths are allowed (Hibler and Tucker, in press). These studies have yielded considerable insight into the im~rta~ce of ice rheology in modeling ice d r i f t .
The studies by Rothrock (1975a) and Hibler (1974) demonstrated the impor-
tance of including a compressive strength
in ice d r i f t
models as opposed to just a shear
303
strength as, for example, used in the Newtonian viscous models. Rothrock°s (1975a) study also indicated that i f shear strength effects were included in the ice rheology, some type of nonlinear behavior must be used near shore, strated that in the central
The study by Hibler and Tucker (in press) demon-
Arctic observed decreases in the turning angles and ice d r i f t
magnitude relative to the wind could be explained by a seasonally varying ice strength.
Prit-
chard's (1978) near shore study (using a plastic rheology) demonstrated the a b i l i t y of this rheology to create velocity discontinuities and shaded areas (due to irregular boundaries) of "dead ice" i f shear strengths were made large enough. Overall the results of these studies indicate that linear rheologies can produce reasonable results for ice d r i f t f a r
from shore, but that proper modeling of near-shore ice movement
requires the use of non-linear rheologies. Coupled short term simulations More complete model studies include coupling effects between the ice thickness distribution and the internal
ice stress.
One goal in these studies has been to determine how the ice
strength is related to the ice thickness distribution.
A useful concept in this regard has
been a suggestion by Rothrock (1975a) that the strength of the ice interaction might be related to the work done by ridging during deformation.
This concept has been incorporated into a
model for the dynamic and thermodynamic evolution of an ice thickness distribution by Thorndike et al. (1975). The most sophisticated coupled model studies have been made for the Beaufort Sea region in conjunction with the Arctic Ice Dynamics Joint Experiment (AIDJEX), for instance by Coon et al. (1976), Pritchard et al. (1977), and Coon and Pritchard (in press).
The model used in these
studies makes use of an elastic-plastic rheology coupled to a variable ice thickness distribution.
The numerical scheme is formulated in an irregular lagrangian grid which allows boundary
irregularities
to be taken into account.
advection effects are not included.
However, because of the lagrangian formulation,
In practice this model has been utilized together with
observed data buoy d r i f t data to supply a moving boundary condition. A fixed grid model similar to that used by Coon et al.
(1976) has been used by Hibler
(1979) for simulations (including near shore effects) over a seasonal cycle.
The model used by
Hibler (1979) employs a two level ice thickness distribution and makes use of a viscous-plastic numerical scheme for modeling plastic flow. An important feature of both these models is that the seasonal ice zone dynamics are, in principle,
naturally included by allowing
characteristics. zero naturally.
the ice strength
to depend on the ice thickness
Thus as the compactness becomes small near the ice edge the strength drops to
304
One difference between the Coon-Pritchard model and the Hibler model is the way in which the " r i g i d "
portion of the plastic rheology is treated.
Coon and Pritchard use an elastic-
plastic approach (Pritchard 1975) where the ice is assumed to behave elastically for certain low stress states.
Hibler (1979), on the other hand, uses a viscous-plastic approach.
In this
method the sea ice is considered to be a nonlinear viscous medium with the viscosities adjusted to yield plastic flow for normal strain rates. ice is assumed to behave as a very s t i f f
For very small deformation rates, however, the
linear viscous f l u i d .
Thus the rigid
low stress
behavior is approximated by a state of very slow flow. Other model studies employ a coupling only between the ice compactness and ice interaction. Examples of such studies are those of Udin and Ullerstig (1976) and Lepparante (1977) for the Bay of Bothnia in the Baltic; Kulakov et al. (in press) and Ling et al. (in press) for parts of the Antarctic; and Neralla and Liu (in press) and Doronin (1970) for parts of the Arctic.
Most
of these models make use of a Newtonian viscous theology dependent on the ice concentration together with
(Udin and Ullerstig,
prevent excessive convergence. field
1976; Doronin, 1970) a correction applied to reduce or
In particular, in Udin and Ullerstig (1976), the wind forcing
is reduced in regions of 100% compactness.
In Doronin (1970) the velocity field
is
i t e r a t i v e l y corrected to prevent convergence past 100% compactness. Udin and Ullerstig (1976) have also proposed a parameterization for tensile strength in shore fast ice.
In the Ling et
al. (in press) model a general linear viscous rheology is used but is considered to be spatially invariant so that the ice edge position has to be considered as part of the input data. Overall these various studies have demonstrated that, with adequate input wind data, short term d r i f t patterns and compactness evolution can be simulated well i f ice interaction effects are properly accounted for.
They have also confirmed the importance of a nonlinear rheelogy
for simulating ice d r i f t both near and far from shore.
However, precise details of the rela-
tionship between ice dynamics and ice thickness characteristics s t i l l the simulations by Pritchard et a l . ,
remain problematic.
In
(1977) and Coon et al. (1976) for example, strength esti-
mates based on theoretical energetic grounds (Rothrock, 1975b) were too small to obtain reasonable d r i f t
rates.
A specific problem here is how to treat the process of ridging in ice
models. Seasonal Simulations Simulations of spatially varying sea ice growth, d r i f t , and decay over an annual cycle have been performed both with observed forcing data (Washington et al., 1976; Parkinson and Washington, 1979; Hibler, 1979) and with data derived from global atmospheric-ice-ocean models (Bryan et al., 1975; Manabe et a l . , in press). in some fashion.
Thesemodels have included the seasonal sea ice zone
With the exception of the study by Hibler (1979), however, these studies have
been essentially thermodynamic in nature and have not made use of a constitutive law relating ice deformation and thickness to ice stresses.
For further
discussion
these studies are
divided into thermodynamic, dynamic-thermodynamic, and global ai~msphere-ice-ocean models.
305 I t should also be noted that process model studies over a seasonal cycle that do not include spatially varying effects or advection have been done by Maykut and Untersteiner (1971), Maykut (1979) and Thorndike et al. (1975).
In particular Maykut and Untersteiner (1971) pre-
sented a comprehensive thermodynamic model for level multi-year sea ice; Maykut (1978) numerically examined the thermodynamics of level thin ice; and Thorndike et al. (1975) presented a model for describing the dynamic thermodynamic evaluation of the ice thickness distribution. These studies have been reviewed by Hibler (in press).
The work by Maykut and Untersteiner
(1971) and Thorndike et al. (1975) have provided a substantial portion of the foundation used in the spatially varying studies described below. Thermodynamic simulations In order to examine spatially variable thermodynamic effects, Washington et al. carried out a global thermodynamic simulation
(1976)
using monthly averaged climatic forcing data
compiled by Crutcher and Meserve (1970) and by Taljaard, et al. (1969).
The idea in this work
was to generalize Maykut and Untersteiner's (1971) level ice thermodynamic model to include spatial variations.
To simplify these calculations they used a time independent thermodynamic
sea ice model developed by Semtner (1976) which was based on the work of Maykut and Untersteiner (1971).
As suggested by Semtner (1976) they included a 30-meter deep mixed layer in
the ocean to allow for zero ice thickness in the summer. With this approach, Washington et al. (1976) were able to simulate realistic seasonal advance and retreat of the ice margin as well as reasonable ice thicknesses in the central Arctic (see Fig. 1).
However, the regional varia-
tions of ice thickness and the total amount of ice production were less realistic.
Also, i t
should be remembered that by driving the simulation with observed air temperatures, which, in turn, are dependent on the actual ice conditions, proper results are partially forced. In the case of the Antarctic ice cover, the simulated ice extent (especially in summer) and thickness are much too large.
However by adding in a fraction of leads (allowed to increase
slowly under melt and decrease rapidly under growth to a fixed minimum of 2%), the simulation of extent is substantially improved, presumably due to increased melting due to shortwave radiation absorption (see Figure 2). Following
the Washington
et al. (1976) approach with climatologically
averaged data,
Parkinson and Washington (1979) have carried out seasonal simulations which to a limited extent include transport effects.
Normally Parkinson and Washington (1979) allow the ice to move in
the absence of internal ice stress effects.
However, when the fraction of open water reaches a
minimum value, the divergence rate is iteratively corrected to maintain a fixed fraction of leads.
This iteration, however, is performed without regard for conservation of momentum and
tends to excessively damp the velocity field.
A secondary problem is that the ice Is not
allowed to converge and increase in thickness by ridging as occurs in reality.
Because of
these ad hoc ice interaction corrections Parkinson and Washington's Arctic thickness contours (Fig. 3) differ little from those obtained by Washington et al. (1976) based on thermodynamic
306
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Figure 1. Themodynamically simulated Arctic ice thickness contours (from Washington et a l . , 1976). Dotted lines denote estimated observed extents. considerations alone; namely t h i c k e r ice near the pole with decreasing values as the coasts are approached.
Under summer melt conditions, however, the ad hoc v e l o c i t y correction employed by
Parkinson and Washington
(1979)
e f f e c t s are n~re reasonable.
is less often needed.
Consequently these s u ~ r
Because of t h i s , the simulated s u ~ r
transport
compactness e s p e c i a l l y near
shore tends to be considerably less than t h a t obtained from thernN)d3mmtic considerations alone, and in b e t t e r agree~nt vrith observed estimates.
Also, Parktnson and Washington (1979) are
able to c o r r e c t l y simulate a more compact summer ice cover in the Northern Hemisphere than in the Southern.
307
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Americo II .1
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"--'e.~..\~\
-
90 ° W
90OE
March 180 °
OO
90 ° W
90 ° E
180 °
Figure 2a. Thermodynamically simulated Antarctic ice thickness contours (from Washington et. al., 1976). Dotted lines denote estimated observed extents. Casewithout leads.
308
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South
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Figure 2b.
Sameas Figure 2a except with leads.
309
90 ° E
U
Asia
~
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/-
Europe
180°
0o
, 9(~W Figure 3.
L/
January
Simulated Arctic ice thickness contours (from Parkinson and Washington, 1979).
A dynamic-thermodynamic simulation A numerical investigation of the effects of ice dynamics on the seasonal growth and decay cycle of sea ice has been presented by Hibler (1979) in a dynamic thermodynamic simulation of the Arctic ice cover.
In this model Hibler (1979) assumed two categories of ice thickness
together with a plastic constitutive law relating ice strength to the ice thickness and fraction of thin ice.
A spatially invariant thermodynamic code, together with a time-dependent
wind stress field over a one year period were used to drive the model. With this model, reasonable fields of ice d r i f t velocity (including shear zone effects) and ice thickness values were obtained. Also, the volume of ice transport into the Greenland sea yields numbers compatible with existing estimates based on observations. Particularly notable are the thickness results (Figure 4) whic~ show a substantial build up of ice along the Canadian Archipelago in conjunction with a thinning of ice along the North Slope and Siberian coast.
The geographical shape of these dynamically forced contours agrees
better with the available observations than the Washington et al. Washington (1979) results.
(1976) and Parkinson and
The computed mean thickness averaged over the whole basin however,
appears to be low, probably as a result of the simplified two level assumption with regard to the ice thickness distribution.
310
12O
Caada q
. . . . . .
Gr~/a~td
Figure 4.
Average April Arctic Sea ice thickness contours from a dynamic.thennodynamic simulation {from Hibler, 1979).
This geographical thickness build up is primarily due to advection which tends to remove ice from the North Slope and Siberian coasts and either remove i t from the basin or deposit i t along the Canadian Archipelago,
As the simulation proceeds in time, the build up strengthens
the ice along the Archipelago while thinning weakens the ice off the other coasts.
The re-
sulting strength imbalance eventually counteracts the external forcing. The ice dynamics also causes a high rate of ice production in certain shelf regions induced by off-shore advection.
This effect is particularly pronounced in the Laptev Sea region.
A high rate of ice production implies a de-stabilization of the upper ocean, and a large heat flux into the atmosphere, which should have significant effects on oceanic and atmospheric circulation in these shelf regions. by McPhee (in press b).
Possible effects of this nature are discussed in a review
311 In the summer, off-shore ice advection creates a marginal ice zone in qualitative agreement with observations (Figure 5).
However, ice concentrations are too high near the coasts
and in the Chukchi Sea. These shortcomings of the model appear to be largely due to neglecting spatially varying thermodynamic effects.
The effects of ice rheology on this summer ice edge
are discussed more in the Modeling section of this review. This model also produced realistic shear zone velocity discontinuities off the North Slope of Alaska from time to time.
However, due to lack of resolution i t was not possible to examine
shading effects due to boundary irregularities. Global atmospheric-ice-ocean models. Using a full atmospheric circulation model coupled to an oceanic circulation model Bryan et al.
(1975) and Manabe et al. (in press) have carried out long-term simulations to equili-
12o
(
c5 .'l/a,ka
0.75
150
J
30°
0.9
~nada
.0o Greenland
Figure 5. Average August compactness from a dynamic-thermodynamic simul ation 1979).
( from Hibler,
312 brium conditions, with both constant and seasonally varying
insolation.
In these models a
single value is assigned to ice thickness and allowed to change in accordance with continuity and thermodynamic requirements.
Ice dynamics effects are approximated by allowing the ice to
d r i f t with the upper ocean until
i t reaches a fixed thickness of 4 m at which time the ice
motion is completely stopped. In the case of a constant solar heat input, the ice in the Arctic basin reaches unrealist i c a l l y large thicknesses (exceeding 20 meters in some locations) and ice velocities are far too slow.
In the case of a seasonally varying solar heat input, r e a l i s t i c ice thicknesses of
about 3 meters with an annual variation of 1 meter are obtained.
However, the computed ice
export into the Greenland Sea appears unrealistically small. EMPIRICAL STUDIES OF ICE EXTENTAND CONCENTRATION A class of studies particularly relevant to modeling the marginal ice zone on climatic time scales of several years, are empirical
statistical
or "stochastic" studies.
In these
studies correlations between ice parameters and atmospheric and oceanic variables are made, and conclusions drawn regarding the dominant factors.
In the most formal studies of this type,
f i r s t order Markov processes are used to describe the variables, with the various correlation values interpreted as feedback parameters (see e.g. Hasselmann, 1976). Studies of the spatial and temporal variations of the Arctic ice cover have been done by Sanderson (1975), Kukla (1978), Walsh and Johnson (in press) and Lemke (in prep).
Of these
studies, that of Walsh and Johnson (in press) covers the largest time period, namely about 25 years.
While there are a number of fluctuations, the ice extent record compiled by Walsh and
Johnson (in press), show the average extent of Arctic ice to have generally increased over the last 25 years (see Fig. 6).
Although analysis is not complete, this trend seems to agree with
decreases in Arctic surface temperatures between 1957 and 1975 (Walsh, i977).
Although based
on considerably less data similar conclusions about correlations between temperatures and ice extent have been reached by Budd (1975) for the Antarctic ice cover. Year to year variations in Arctic ice extent have generally been found to be quite asymmetric in space.
In particular, using data from 1969 to 1974 Sanderson finds heavy ice condi-
tions over east Greenland correlating with l i g h t ice conditions over the eastern Canadian area. Similar results were found by Lemke using data from 1966 to 1976.
Using a longer data set,
Walsh and Johnson (in press), on the other hand, find a smoother as~nmetric change, with ice in the North Atlantic having an anomalous change in extent opposite in sign to the changes over the remainder of the polar cap. Sanderson (1975) has suggested that these asymmetric effects may be due to shifts in the upper level westerlies. torical
This type of explanation (based on longer but more qualitative his-
data) has also been advanced by Kelly (1978) who suggests that Eastern Arctic
ice
313
3.0
N
E 0
20
m v
cO 0
E
1.0 0
L
", - I . 0 0 Q.
® -2.0 E3
-30 0
-4.0
'55
'60
~5
'70
'75
Yeor
Figure 6.
Time series of the departure from monthly means of the area covered by Arctic sea ice. The record has been smoothed with a 24 month running mean (from Walsh and Johnson, in press).
extent may correlate with the position of the Icelandic low.
In the Antarctic, similar asym-
metric changes have been reported in antarctic sea ice (although based on only a few years of data) by Ackley and Keliher (1976) and Zwally and Gloersen (1977). Another aspect of the ice extent is the degree of temporal persistence. Lemke (in press) has analyzed this persistence by assuming a f i r s t order Markov process.
His analysis yielded
typical persistence times of about 3 months for the whole Arctic ice cover. scales the persistence times were, however, significantly shorter.
On smaller space
More regional studies of ice extent have been made by Vinje (1976) and Skov (1970) for the North Atlantic and by Barnett (in press), and Rogers (1978) for the North Slope of Alaska and Canada.
Vinje's study emphasized the substantial variation of the North Atlantic ice edge
location from year to year.
Vinje's extreme ice extents for February and August for the time
period 1966 to 1974 are reproduced in Figure 7.
A prominent feature of these extents is the
ice free region west of Spitzbergen which is ostensibly caused by the warm west Spitzbergen current (see e.g. Aagaard and Greisman 1975). With respect to causes of the interannual ice edge variations here, correlations by Skov (1970) suggest that northward oceanic heat transport may be a primary factor.
314
oo
20° Figure
7a.
2oo
40°
Extreme sea ice conditions at the end of February from 1966-1974 (reproduced from V i n j e , 1976).
o
60° ~O
40¢ -4
o,
\ 40 °
/
August GfeeOlOOd 60 o
i
n,momEx,eo,
%
mRM 20° Figure 7b.
OQ
20o
40°
Extreme sea ice conditions at the end of August from 1966-1974 (reproduced from Vinje, 1976).
To examine north slope ice c o n d i t i o n s , Barnett constructed a 20 year h i s t o r i c a l severity index for the Barrow region.
A s a l i e n t c h a r a c t e r i s t i c of t h i s index is a q u a s i - p e r i o d i c f i v e
315
year oscillation.
Barnett found this
pressure in the central basin in April.
severity index correlated well
with the atmospheric
In a more detailed study over a shorter time period,
Rogers (1978), on the other hand, found that air temperature in the form of thawing degree days was the parameter that correlated best with summertime ice margin distance.
However, Rogers
also found that mild summers were often associated with pressure anomalies that resulted in more southerly surface winds advecting warmer air. MODELING THE SEASONALSEA ICE ZONE To properly model the seasonal ice zone requires that special attention be given to several components of sea ice models. Probably the most important component is the thermodynamic modification of the ice thickness distribution. A particular problem here is that pack ice at the ice edge is highly variable in thickness and may well contain a large fraction of open water.
Some of the thermal processes operable in such conditions have been reviewed by Roth-
rock (in press) and Wadhams (in press). A dominant problem at the summer ice edge, is the effect of open water on the adjacent pack ice.
During melting conditions the radiation can contribute to lateral melting at the
edge of ice floes. bottom ablation.
This heat also can be carried under the ice where i t will If
the ice cover is s u f f i c i e n c y disintegrated,
contribute to
some of the heat can be
stored in the mixed layer thus delaying autumn freezeup. I t is also l i k e l y that the open water and increased heat exchanges can modify one or both of the planetary boundary layers.
A number of possible effects of this nature in the oceanic
boundary layer are discussed in a companion review by McPhee (in press b).
The ice at the edge
is also subject to wave effects and ocean eddy effects which are reviewed by Wadhams (in press).
Overall i t may well be that the marginal ice zone is in effect an ocean front problem.
In the shear zone region, the continental shelf and coast create complex effects on the ocean current structure.
Due to the shallowness in these regions, the currents beneath the
boundary layer may be much more wind driven in nature than in the deeper ocean.
Because of
this the movement or lack of movement of the ice cover may affect the current structure here, which, in turn, may cause thermodynamic and dynamic changes in the ice cover.
Current effects
can also be induced at the summer ice edge due to horizontal density gradients there (see e.g. McPhee, in press).
I t is also possible for the boundary effects to couple with the ice rheology and modify the ice edge evolution in summer. To i l l u s t r a t e how drastic an effect t h i s can be, seasonal simulations were run using Htbler's (1979) dynamic themodynamic model with several d i f f e r e n t rheologies. Figure 8 shows the compactness time series at three 125 km grid c e l l s proceeding outward from Prudhoe Bay for three d i f f e r e n t ice rheologies: p l a s t i c , NevYconian viscous, and l i n e a r viscous.
In all of these rheologies the ice strength was taken to depend on the ice
316
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NViscous
!
I
0.4
Time(months)
Figure 8.
Time series of the simulated compactness at three grid cells (125 by 125 Km) progressing further from shore near Prudhoe Bay (Cell 4 closest, Cell 6 farthest away--see Hibler, 1979). The simulated results from these different rheologies are plotted. The simulations were carried out using the seasonal dynamic thermodynamic model of Hibler (1979).
317 thickness and compactness.
As can be seen, the Newtonian viscous rheology produces highly
unrealistic ice edge effects with the ice edge actually forming further off shore.
The linear
viscous and plastic both do well, with the plastic law creating much more of a near shore flow lead, whereas the viscous law tends to spatially smooth out the compactness variations. In practice, of course, model simulations may not be sensitive to all of the effects mentioned here.
However, their presence means that at least some quantitative studies need to be
done to assess their significance. CONCLUDING REMARKS In practice, building an ice model including the seasonal ice zone requires approaching this region with a hierarchy of models both deterministic and stochastic in nature, and progressing from crude and simple to more complete and sophisticated schemes. Simple models are often especially useful in helping to identify the relative importance of specific processes in the total system and can aid in the development of more complete simulation models. These more complete models can be used to test the adequacy of current understanding of the seasonal ice zone. Considerable progress has been made in developing decade.
numerical ice models over the last
The framework of these ice models in conjunction with existing ocean and atmospheric
models appears to be computationally
general enough to handle the seasonal sea ice zone.
However, how to properly parameterize the complex processes in the seasonal sea ice zone in a numerical model is an open question. ACKNOWLEDGEMENTS This work was supported in part by the Office of Naval Research and by the National Aeronautics and Space Administration. REFERENCES Aagaard, K. and P. Greisman, Toward new mass and heat budgets for the Arctic ocean, J. Geophys. Res., 80:3821-3827. Ackley, S.F. and T.E. Keliher, Antarctic sea ice dynamics and its possible climatic effects, AIDJEX Bulletin, 33:53-76. Barnett, D.S., A long-range ice forecasting method for the north coast of Alaska, ICSI/AIDJEX Symposium on Sea Ice Processes and Models, Univ. of Wash. Press, in press. Bryan, K., S. Manabe, and R.L. Pacanowski (1975), A global ocean-atmosphere climate model Part I I . The oceanic circulation. J. Phys. Oceanogr., 5:30-46. Budd, W.F. (1975), Antarctic sea-ice variations from s a t e l l i t e sensing in relation to climate, J. Glaciology, 15:417-428.
318 Campbell, W.J. (1965), The wind driven circulation of ice and water in a polar ocean, Geophysical. Res., 70:3279-3301.
J.
Coon, M.D., R. Colony, R.S. Pritchard, and D.A. Rothrock (1976), Calculations to test a pack ice inodel, Numerical Methods in Geomechanics, Vol. 2, (C.S. Desai, ed.), ASCE, New York, 1210-1224. Coon, M.D. and R.S. Pritchard, Glaciology, in press.
Mechanical energy considerations
in
sea ice
dynamics, J.
Croasdale, K.R. (1978), Ice engineering for offshore petroleum exploration in Canada, POAC77 Proceedings, Ocean Eng. Inf. Center, Mem. Univ. of Newfoundland, St. Johns, 1-32. Crutcher, H.L. and J.M. Meserve (1970), Selected level heights, temperatures and dew points for the northern hemisphere, NAVAIR Rep. 50-IC-52, revised, U.S. Nav. Weather Serv. Coamand, Washington, D.C. Ooronin, Y.P. (1970), On a method of calculating the compactness and d r i f t English translation in AIDJEX Bulletin, 3:22-39.
of ice floes,
Egorov, K.L. (1971), Theory of the d r i f t of ice fields in a horizontally inhomogeneous wind field, English translation in AIDJEX Bulletin, 6:37-45. Glen, J.W., Thoughts on a viscous model for sea ice, AIDJEX Bulletin, 2:18-27. Hasselmann, K. (1976), Stochastic climate models, Part I. Theory. Tellus 28:437-485. HiDler, W.O. I l l , (1974), Differential sea ice d r i f t II. Comparison of mesoscales strain measurements with linear d r i f t theory predictions, J. Glaciology, 13:457-471. Hibler, W.D. I l l , (in press).
(1979), A dynamic thermodynamic sea ice model, J. Physical Oceanography, 9,
Hibler, W.D. I I I , Sea ice growth d r i f t and decay, Chapter 3 in Dynamics of Snow and Ice Masses, (S. Colbeck, ed.), Academic Press, N.Y., in press. Hibler, W.D. I l l , W.F. Weeks, A. Kovacs, and S.F. Ackley (1974a), Differential sea-ice d r i f t I. Spatial and temporal variations in sea-ice deformation, J. Glaciology, 13:437-455. Hibler, W.D. I l l and W.B. Tucker, Some results from a linear viscous model of the Arctic ice cover, J. Glaciology, in press. Kelly, P.M. (1978), Forecasting the arctic years, Climate Monitor, 7:95-98.
sea ice on time scales of a few months to many
Kukla, G.J. (1978), Recent changes in snow and ice, in Climate Change (J. Gribbin, ed.) Cambridge Univ. Press, Cambridge, 114-129. Kulakov, I.Y., M.I. Maslovsky, and L.A. Timokhov, Seasonal variability of antarctic sea ice extent: Its numerical modeling, ICSI/AIDJEX Symposium on Sea Ice Processes and Models, Univ. of Wash. Press, in press. Laikhtman, D.L. (1964), Physics of the Boundary Layer of the Atmosphere, English translation published by U.S. Dept. of Commerce. Lemke, P. (in prep), A stochastic model of the coupled sea-ice atmosphere system. Lepparanta, M. (1977), On the dynamics of ice cover in the Bothnia Bay, Proceedings of 100 years of winter navigation in Finland, Oulu, Finland. Ling, C.H., L.A. Rasmussen and W.J. Campbell, A mesocale continuum sea ice model, ICSI/AIDJEX Symposium on Sea Ice Processes and Models, Univ. of Wash. Press, in press.
319 Manabe, S., K. Bryan, and M.J. Spelman, A global ocean-atmosphere climate model with seasonal variation for future studies of climate sensitivity, Dyn. Atmos. and Ocean, in press. McPhee, M., An analysis of pack ice d r i f t in summer, ICSI/AIDJEX Symposium on sea ice processes and models, Univ. of Wash. Press, (in press a). McPhee, M. (in press b) These proceedings. Maykut, S.A. (1978), Energy exchange over young sea ice in the central Res., 83:3646-3658.
arctic, J. Geophys.
Maykut, S.A. and Untersteiner, N. (1971), Some results from a time dependent thermodynamic model of sea ice, J. Geophys. Res., 76:1550-1575. Neralla, V.R. and W.S. Liu, A simple model to calculate the compactness of ice floes, J. Glaciology, in press. Nye, J.F. (1975), The use of ERTS photographs to measure the movement and deformation of sea ice, J. Glaciology, 15:429-436. Parkinson, C.L. and W.M. Washington, (1979), A large scale numerical model of sea ice, J. Geophys. Res., 84:311-337. Pritchard, R.S. (1975), An elastic-plastic constitutive law for sea ice, J. Appl. Mech., 42E, 379-384. Pritchard, R.S (1978), The effect of strength on simulation of sea ice dynamics, POAC 77 Proceedings, Ocean Eng. Inf. Center, Mem. Univ. of Newfoundland, St. Johns, 494-505. Pritchard, R.S., M.D. Coon, and M.S. McPhee (1977), Simulation of sea ice dynamics during AIDJEX, J. Pressure Vessel Technology, 99J, 49-497. Rogers, J.C~ (1978), The meteorological factors effecting intersummer variability of sea ice conditions in the Beaufort Sea, Mon. Weather Review, 106:890-897. Rothrock, D.A. (1975a), The steady d r i f t of an incompressible Arctic ice cover, J. Geophys. Res., 80:387-397. Rothrock, D.A. (1975b), The energetics of the plastic deformation of pack ice by ridging, J. Geophys. Res., 80:4514-4519. Rothrock, D.A., Modeling sea ice features and processes, J. of Glaciology, in press. Sanderson, R.M. (1975), Changes in the area of Arctic sea ice, 1966 to 1974, The Meteor. Magazine, 104:303-323. Schwarz, J. and W.F. Weeks (1977), 19:499-532.
Engineering properties
of
sea ice,
J.
Glaciology,
Semtner, A.J. Jr. (1976), A model for the thermodynamic growth of sea ice in numerical investigations of climate, J. Phys. Oceanogr., 6:379-389. Skov, N.A. (1970), The ice cover of the greenland Sea, Meddelesev om Gronland, Bd. 188, C.A. Reitzel s, Kopenhagen. Sobczak, L.W. (1977), Ice movements in the Beaufort Sea 1973-1975: Determination by ERTS imagery, J. Geophys. Res., 82:1413-1418. Taljaard, J.J., H. van Loon, H.L. Crutcher, and R.H. Jerun, (1969) Climate of the upper air, I, Southern hemisphere, vol. 1, Temperatures, Dew Points, and Heights at Selected Pressure Levels, ~VAIR Rep. 50-IC-55, 135 pp., U.S. Nav. Weather Serv. Command, Washington, D.C.
320
Thorndike, A.S., D.A. Rothrock, G.A. Maykut, and R. Colony (19755, The thickness distribution of sea ice, J. Geophys. Res., 80:4501-4513. Thorndike, A.S. and R. Colony, Large scale motion in the Beaufort Sea during AIDJEX April 1975-April 1976, ICSI/AIDJEX Symposium on Sea Ice Processes and Models, Univ. of Wash. Press, in press. Tryde, Per. (1977), Ice forces, J. Glaciology, 19:257-264. Udin, I. and A. Ullerstig (19765, A numerical model for forecasting the ice motion in the bay and sea of Bothnia. Research Report No. 18, Swedish Administration of shipping and Navigation, Norrkoping, Sweden. Vinje, T•E. (1976), Sea ice conditions in the European Sector of the marginal seas of the Arctic, 1966-1975. Norsk Polar Institute Arbok 1975, Oslo, 163-174. Wadhams, P. These proceedings, in press. Washington, W.M., A.J. Semtner, C. Parkinson and L. Morrison (1976), On the development of a seasonal change sea, ice model, J. Phys. Oceanogr., 6:679-685. Walsh, J.E. (1977), The incorporation of ice station data into a study of recent Arctic temperature N uctuations, Mon. Weather Rev., 105:1527-1535. Walsh, J.E. and C.M. Johnson, An analysis of Arctic sea ice fluctuations, 1953-77, J. Phys. Oceanography, in press. Zwally, H.J. and P. Gloersen (1977), Passive microwave images of the polar region and research applications, Polar Record, 18:431-450. Zubov, N.N. (1943), Arctic Ice, English translation U.S. Dept. of Commerce No. AD426972. APPENDIX: SEA ICE RHEOLOGY A general framework for examining sea ice rheologies has been outlined by Glen (1970). Based on earlier glacier flow concepts, Glen suggested a general consitutive law of the f o l lowing form might be applicable to sea ice:
° i j = 2n~ij + [({-n)(~xx + ~yy) - P]~ij
(15
where oij and ~ij are the two-dimensional stress and strain rate tensors, n, ~ and P are general functions of the two invariants of the strain rate tensor, and ~ij equals one for i equal to j or zero for i not equal to j .
These invariants can be taken as the principal values of
the strain rate tensor or alternatively as (~xx+Cyy5 and ~xx~xx+2~xy~xy -+~yy ~yyS. To obtain an internal ice force for the momentum balance, the stress tensor is differentiated, yielding @
Fx : ~
Fy = ~
@
(Oxx5 %-~ (axy) @
(Oxy5 ~
@
(Oyy5
where Fx and Fy are the components of the force transmitted through the ice.
(2)
(3)
321 Special cases of these rheologies often used in sea ice are a "linear viscous" rheology where the stress depends l i n e a r l y on the strain rates and an "ideal r i g i d plastic" rheology where the stress state is either indeterminate or independent of the magnitude of strain rates. In the linear viscous case, n and { are taken to be linear in cij and P is constant. results in a stress state l i n e a r l y dependent on the strain rates.
This
In the r i g i d plastic case
the stress state is taken to be fixed independent of the magnitude of the strain rate invariants provided the ratio of the invariants does not change. linear functions of the invariants. well defined for zero strain rates.
In this case n and ~ w i l l be non-
In the r i g i d plastic case, however, n and { may not be Under these conditions the ice is understood to move
r i g i d l y with the stress detemined by external balances. Some of the linear viscous and plastic rheologies proposed for sea ice are a Newtonian viscous f l u i d (n = constant; constant, P = constant).
{ = P = 0), and a general linear viscous f l u i d ({ = constant, n =
For an ideal plastic constitutive law the stress state l i e s on or
within some fixed y i e l d curve. state to the strain rates.
For plastic flow a rule is needed to uniquely relate the stress
The most common rule is to take the ratio of the strain rates for a
given stress state to be that of a vector normal to the surface. normal flow rule. Hibler (1979).
This is referred to as the
For e x p l i c i t stress strain-rate equations for an e l l i p t i c a l yield curve see
variable thickness
Due to frequent deformation the thickness of this ice cover varies greatly,
and ice several years old, for example, may be located next to ice only a few days o l d .
The
behavior of this ice cover is coupled to both the thermodynamic and dynamic characteristics of the atmosphere and ocean.
In addition the ice cover can substantially modify the air-sea heat
and momentum fluxes, and thus change the behavior of the atmosphere and ocean. These feedbacks are particularly pronounced in the seasonal ice zone. Modeling this complex, coupled system has been the subject of considerable research over the last decade. Efforts to properly model the seasonal ice zone, however, are just beginning. In this paper progress in modeling sea ice and growth, d r i f t , and decay is b r i e f l y reviewed and particular problems associated with modeling the seasonal ice zone are discussed.
For this
purpose, the discussion is divided into three parts. I t should be noted that much of the material and Empirical
in the sections on Numerical Sea Ice Models
Studies of this paper has been abstracted from the modeling portions o f a more
complete review of sea ice growth d r i f t
and decay published elsewhere (Hibler,
Interested readers are referred to that review for more detail on ~ e l i n g
in press).
as well as a review
of other aspects of sea ice growth, d r i f t , and decay.
NUMERICAL SEA ICE MODELS A concept essential
to modeling sea ice is that
growth and decay are i n t r i n s i c a l l y
related.
the ice dynamics and the rates of ice
In p a r t i c u l a r ,
rates of growth and decay of the
ice cover depend on the d i s t r i b u t i o n of ice thicknesses which in turn depend on the ice deformation and transport patterns.
The ice d r i f t
is also modified by the ice thickness d i s t r i b u -
tion which affects the amount of stress the ice can transmit.
In a complete simulation model
these e f f e c t s are coupled by allowing the ice i n t e r a c t i o n to depend on the ice thickness chara c t e r i s t i c s , which, in turn, are functions of both the thermodynamics and dynamics. In more formal terms, construction of a coupled sea ice model ponents :
requires the following com-
301 I) lis
A momentum balance describing ice d r i f t which includes air and water stresses, Corio-
force, internal
tilt);
ice forces, inertial
forces, and steady current effects (including ocean
2) an ice rheology which relates (in some suitable space and time scale) the ice stress
to the ice deformation and strength; 3) an ice thickness distribution which accounts for the change of ice thicknesses due to thermodynamic and dynamic effects; 4) an ice strength determined primarily as a function of the ice thickness distribution; and 5) a thermodynamic code which specifies the growth and decay rates of various ice thicknesses in the environment of a variable thickness ice cover. A t a c i t assumption in l i s t i n g these components is that a description of sea ice as a twodimensional continuum is possible.
While this
is reasonable i t
results from studies of the kinematics of pack ice (e.g.,
is important
Hibler et al.,
to note that
1974a; Nye 1975;
Thorndike and Colony, in press) exhibit substantial deviations from a spatially smooth velocity field.
In practice, however, typical fluctuations are small enough so that on scales larger
than about 20 km they do not mask the continuum motion.
But their presence means that in
general, continuum motion has to be considered a s t a t i s t i c a l l y defined quantity for sea ice. Also, the physical structure, s a l i n i t y , and small scale strength of sea ice have purposely been excluded from the above outline. play a significant
role in local
Such physical
properties affect the thermodynamics and
force problems (e.g.,
Tryde, 1977; Croasdale, 1978).
addition, the salt expulsed from freezing sea ice plays a c r i t i c a l ling.
role in the ocean-ice coup-
However, for large scale ice modeling, the contribution of such small scale characteris-
tics to the ice interaction and d r i f t is f e l t to be of second order importance. of
In
For a review
small scale strength properties the interested reader is referred to Schwarz and Weeks
(1977). A variety of process studies and numerical simulations have been carried out to determine how to parameterize these five components. Numerical work to date can be broadly divided into two categories:
short term simulations spanning several months or less, and seasonal simula-
tions covering at least one annual cycle.
The short term simulations have been primarily
oriented toward determining proper treatments of the momentum balance, ice rheology and ice strength.
For this purpose these studies have concentrated on the simulation of ice d r i f t and
deformation characteristics.
The seasonal simulations on the other hand, have been more
oriented toward simulating ice thickness characteristics and more generally the air-sea heat and momentum exchanges. For these purposes all five components play a role, although particular emphasis has been placed on the thermodynamics and ice thickness distribution. In the remainder of this section a number of numerical studies which have yielded insight into modelling sea ice growth d r i f t and decay are b r i e f l y reviewed. into short term simulations and seasonal simulations.
These studies are divided
302 Short-Term Drift Simulations Free d r i f t studies The simplest type of short-term simulations make use of semi-empirical rules to estimate the ice d r i f t
from the local wind and current values.
The most widely used rule for this
purpose is that of Zubov (1943) which states that the ice drifts parallel to the isobars with about one-fiftieth of the geostrophic wind speed.
Ad hoc modifications to such rules include
seasonally modifying the proportionality constant and turning angle between ice d r i f t and wind, and spatially averaging the wind field in order to crudely take into account the affects of ice interaction.
Sobczak (1977) has compared several of these empirical
rules to observed ice
d r i f t in the Beaufort Sea. Most such empirical rules are essentially free d r i f t approximations in which the momentum balance without the internal ice stress and acceleration terms is used. Probably the most quantitative examination of free d r i f t is a study by McPhee (in press a) using summer ice d r i f t data taken during the 1975 AIDJEX field study.
In this analysis excel-
lent agreement between daily averaged predicted and observed d r i f t rates were obtained.
This
agreement, in turn, supports the u t i l i t y of existing wind and water boundary layer formulations for estimating ice d r i f t .
This type of free d r i f t approximation appears useful for determining
the average d r i f t of ice in the summer marginal ice zone (at least under low compactness conditions). Mechanistic model studies While s t i l l
considering only the momentum balance more complete mechanistic simulations
have been made by assuming certain spatially invariant mechanical properties of the ice cover. The primary goal in these studies was to determine the role of the ice interaction in modeling ice dynamics.
To aid in the discussion of these calculations a brief review of concepts of
rheology relevant to sea ice is included as an appendix. Exemples of mechanistic model studies using a given rheology with specified strengths are Newtonian viscous calculations by Laikhtman (1~54), Campbell (1965), and Egerov (1971); general linear viscous calculations (including both bulk and shear viscosities) by Hibler~(1974) and Hibler and Tucker (in press); calculations by Rothrock (Ig75a) assuming an incompressible ice cover with a spatially constant specified divergence rate; and near shore plastiC simulations by Pritchard (1978).
In all these mechanistic model studies spatially fixed ice strengths are
assumed, although in some cases temporal variations in the strengths are allowed (Hibler and Tucker, in press). These studies have yielded considerable insight into the im~rta~ce of ice rheology in modeling ice d r i f t .
The studies by Rothrock (1975a) and Hibler (1974) demonstrated the impor-
tance of including a compressive strength
in ice d r i f t
models as opposed to just a shear
303
strength as, for example, used in the Newtonian viscous models. Rothrock°s (1975a) study also indicated that i f shear strength effects were included in the ice rheology, some type of nonlinear behavior must be used near shore, strated that in the central
The study by Hibler and Tucker (in press) demon-
Arctic observed decreases in the turning angles and ice d r i f t
magnitude relative to the wind could be explained by a seasonally varying ice strength.
Prit-
chard's (1978) near shore study (using a plastic rheology) demonstrated the a b i l i t y of this rheology to create velocity discontinuities and shaded areas (due to irregular boundaries) of "dead ice" i f shear strengths were made large enough. Overall the results of these studies indicate that linear rheologies can produce reasonable results for ice d r i f t f a r
from shore, but that proper modeling of near-shore ice movement
requires the use of non-linear rheologies. Coupled short term simulations More complete model studies include coupling effects between the ice thickness distribution and the internal
ice stress.
One goal in these studies has been to determine how the ice
strength is related to the ice thickness distribution.
A useful concept in this regard has
been a suggestion by Rothrock (1975a) that the strength of the ice interaction might be related to the work done by ridging during deformation.
This concept has been incorporated into a
model for the dynamic and thermodynamic evolution of an ice thickness distribution by Thorndike et al. (1975). The most sophisticated coupled model studies have been made for the Beaufort Sea region in conjunction with the Arctic Ice Dynamics Joint Experiment (AIDJEX), for instance by Coon et al. (1976), Pritchard et al. (1977), and Coon and Pritchard (in press).
The model used in these
studies makes use of an elastic-plastic rheology coupled to a variable ice thickness distribution.
The numerical scheme is formulated in an irregular lagrangian grid which allows boundary
irregularities
to be taken into account.
advection effects are not included.
However, because of the lagrangian formulation,
In practice this model has been utilized together with
observed data buoy d r i f t data to supply a moving boundary condition. A fixed grid model similar to that used by Coon et al.
(1976) has been used by Hibler
(1979) for simulations (including near shore effects) over a seasonal cycle.
The model used by
Hibler (1979) employs a two level ice thickness distribution and makes use of a viscous-plastic numerical scheme for modeling plastic flow. An important feature of both these models is that the seasonal ice zone dynamics are, in principle,
naturally included by allowing
characteristics. zero naturally.
the ice strength
to depend on the ice thickness
Thus as the compactness becomes small near the ice edge the strength drops to
304
One difference between the Coon-Pritchard model and the Hibler model is the way in which the " r i g i d "
portion of the plastic rheology is treated.
Coon and Pritchard use an elastic-
plastic approach (Pritchard 1975) where the ice is assumed to behave elastically for certain low stress states.
Hibler (1979), on the other hand, uses a viscous-plastic approach.
In this
method the sea ice is considered to be a nonlinear viscous medium with the viscosities adjusted to yield plastic flow for normal strain rates. ice is assumed to behave as a very s t i f f
For very small deformation rates, however, the
linear viscous f l u i d .
Thus the rigid
low stress
behavior is approximated by a state of very slow flow. Other model studies employ a coupling only between the ice compactness and ice interaction. Examples of such studies are those of Udin and Ullerstig (1976) and Lepparante (1977) for the Bay of Bothnia in the Baltic; Kulakov et al. (in press) and Ling et al. (in press) for parts of the Antarctic; and Neralla and Liu (in press) and Doronin (1970) for parts of the Arctic.
Most
of these models make use of a Newtonian viscous theology dependent on the ice concentration together with
(Udin and Ullerstig,
prevent excessive convergence. field
1976; Doronin, 1970) a correction applied to reduce or
In particular, in Udin and Ullerstig (1976), the wind forcing
is reduced in regions of 100% compactness.
In Doronin (1970) the velocity field
is
i t e r a t i v e l y corrected to prevent convergence past 100% compactness. Udin and Ullerstig (1976) have also proposed a parameterization for tensile strength in shore fast ice.
In the Ling et
al. (in press) model a general linear viscous rheology is used but is considered to be spatially invariant so that the ice edge position has to be considered as part of the input data. Overall these various studies have demonstrated that, with adequate input wind data, short term d r i f t patterns and compactness evolution can be simulated well i f ice interaction effects are properly accounted for.
They have also confirmed the importance of a nonlinear rheelogy
for simulating ice d r i f t both near and far from shore.
However, precise details of the rela-
tionship between ice dynamics and ice thickness characteristics s t i l l the simulations by Pritchard et a l . ,
remain problematic.
In
(1977) and Coon et al. (1976) for example, strength esti-
mates based on theoretical energetic grounds (Rothrock, 1975b) were too small to obtain reasonable d r i f t
rates.
A specific problem here is how to treat the process of ridging in ice
models. Seasonal Simulations Simulations of spatially varying sea ice growth, d r i f t , and decay over an annual cycle have been performed both with observed forcing data (Washington et al., 1976; Parkinson and Washington, 1979; Hibler, 1979) and with data derived from global atmospheric-ice-ocean models (Bryan et al., 1975; Manabe et a l . , in press). in some fashion.
Thesemodels have included the seasonal sea ice zone
With the exception of the study by Hibler (1979), however, these studies have
been essentially thermodynamic in nature and have not made use of a constitutive law relating ice deformation and thickness to ice stresses.
For further
discussion
these studies are
divided into thermodynamic, dynamic-thermodynamic, and global ai~msphere-ice-ocean models.
305 I t should also be noted that process model studies over a seasonal cycle that do not include spatially varying effects or advection have been done by Maykut and Untersteiner (1971), Maykut (1979) and Thorndike et al. (1975).
In particular Maykut and Untersteiner (1971) pre-
sented a comprehensive thermodynamic model for level multi-year sea ice; Maykut (1978) numerically examined the thermodynamics of level thin ice; and Thorndike et al. (1975) presented a model for describing the dynamic thermodynamic evaluation of the ice thickness distribution. These studies have been reviewed by Hibler (in press).
The work by Maykut and Untersteiner
(1971) and Thorndike et al. (1975) have provided a substantial portion of the foundation used in the spatially varying studies described below. Thermodynamic simulations In order to examine spatially variable thermodynamic effects, Washington et al. carried out a global thermodynamic simulation
(1976)
using monthly averaged climatic forcing data
compiled by Crutcher and Meserve (1970) and by Taljaard, et al. (1969).
The idea in this work
was to generalize Maykut and Untersteiner's (1971) level ice thermodynamic model to include spatial variations.
To simplify these calculations they used a time independent thermodynamic
sea ice model developed by Semtner (1976) which was based on the work of Maykut and Untersteiner (1971).
As suggested by Semtner (1976) they included a 30-meter deep mixed layer in
the ocean to allow for zero ice thickness in the summer. With this approach, Washington et al. (1976) were able to simulate realistic seasonal advance and retreat of the ice margin as well as reasonable ice thicknesses in the central Arctic (see Fig. 1).
However, the regional varia-
tions of ice thickness and the total amount of ice production were less realistic.
Also, i t
should be remembered that by driving the simulation with observed air temperatures, which, in turn, are dependent on the actual ice conditions, proper results are partially forced. In the case of the Antarctic ice cover, the simulated ice extent (especially in summer) and thickness are much too large.
However by adding in a fraction of leads (allowed to increase
slowly under melt and decrease rapidly under growth to a fixed minimum of 2%), the simulation of extent is substantially improved, presumably due to increased melting due to shortwave radiation absorption (see Figure 2). Following
the Washington
et al. (1976) approach with climatologically
averaged data,
Parkinson and Washington (1979) have carried out seasonal simulations which to a limited extent include transport effects.
Normally Parkinson and Washington (1979) allow the ice to move in
the absence of internal ice stress effects.
However, when the fraction of open water reaches a
minimum value, the divergence rate is iteratively corrected to maintain a fixed fraction of leads.
This iteration, however, is performed without regard for conservation of momentum and
tends to excessively damp the velocity field.
A secondary problem is that the ice Is not
allowed to converge and increase in thickness by ridging as occurs in reality.
Because of
these ad hoc ice interaction corrections Parkinson and Washington's Arctic thickness contours (Fig. 3) differ little from those obtained by Washington et al. (1976) based on thermodynamic
306
90°E
#"
'(
-.~..~' USSR ~
/
.
180"
~
."l
.,. "c-"
." 2
I
uss
R
".
-~,]/L.J/t .....o.,.
/~
<- 0 °
90" W
90° E
180 °
/ -'~'L\~L,,: J~.
j - - - ~ . ~ 4.-,o~- O
J
0°
" ' " ' °970 ° °E k - "~-'
Figure 1. Themodynamically simulated Arctic ice thickness contours (from Washington et a l . , 1976). Dotted lines denote estimated observed extents. considerations alone; namely t h i c k e r ice near the pole with decreasing values as the coasts are approached.
Under summer melt conditions, however, the ad hoc v e l o c i t y correction employed by
Parkinson and Washington
(1979)
e f f e c t s are n~re reasonable.
is less often needed.
Consequently these s u ~ r
Because of t h i s , the simulated s u ~ r
transport
compactness e s p e c i a l l y near
shore tends to be considerably less than t h a t obtained from thernN)d3mmtic considerations alone, and in b e t t e r agree~nt vrith observed estimates.
Also, Parktnson and Washington (1979) are
able to c o r r e c t l y simulate a more compact summer ice cover in the Northern Hemisphere than in the Southern.
307
O0 I
Americo II .1
-I /
"--'e.~..\~\
-
90 ° W
90OE
March 180 °
OO
90 ° W
90 ° E
180 °
Figure 2a. Thermodynamically simulated Antarctic ice thickness contours (from Washington et. al., 1976). Dotted lines denote estimated observed extents. Casewithout leads.
308
[
South
0o
..... 2.:i ~i!!!!
I America" r l
-
, ......
¢'
|
/
•
"-~-..'l\
March
18~1o
90 ° W
,
9 0 '=' E
18C)°
Figure 2b.
Sameas Figure 2a except with leads.
309
90 ° E
U
Asia
~
'l
/-
Europe
180°
0o
, 9(~W Figure 3.
L/
January
Simulated Arctic ice thickness contours (from Parkinson and Washington, 1979).
A dynamic-thermodynamic simulation A numerical investigation of the effects of ice dynamics on the seasonal growth and decay cycle of sea ice has been presented by Hibler (1979) in a dynamic thermodynamic simulation of the Arctic ice cover.
In this model Hibler (1979) assumed two categories of ice thickness
together with a plastic constitutive law relating ice strength to the ice thickness and fraction of thin ice.
A spatially invariant thermodynamic code, together with a time-dependent
wind stress field over a one year period were used to drive the model. With this model, reasonable fields of ice d r i f t velocity (including shear zone effects) and ice thickness values were obtained. Also, the volume of ice transport into the Greenland sea yields numbers compatible with existing estimates based on observations. Particularly notable are the thickness results (Figure 4) whic~ show a substantial build up of ice along the Canadian Archipelago in conjunction with a thinning of ice along the North Slope and Siberian coast.
The geographical shape of these dynamically forced contours agrees
better with the available observations than the Washington et al. Washington (1979) results.
(1976) and Parkinson and
The computed mean thickness averaged over the whole basin however,
appears to be low, probably as a result of the simplified two level assumption with regard to the ice thickness distribution.
310
12O
Caada q
. . . . . .
Gr~/a~td
Figure 4.
Average April Arctic Sea ice thickness contours from a dynamic.thennodynamic simulation {from Hibler, 1979).
This geographical thickness build up is primarily due to advection which tends to remove ice from the North Slope and Siberian coasts and either remove i t from the basin or deposit i t along the Canadian Archipelago,
As the simulation proceeds in time, the build up strengthens
the ice along the Archipelago while thinning weakens the ice off the other coasts.
The re-
sulting strength imbalance eventually counteracts the external forcing. The ice dynamics also causes a high rate of ice production in certain shelf regions induced by off-shore advection.
This effect is particularly pronounced in the Laptev Sea region.
A high rate of ice production implies a de-stabilization of the upper ocean, and a large heat flux into the atmosphere, which should have significant effects on oceanic and atmospheric circulation in these shelf regions. by McPhee (in press b).
Possible effects of this nature are discussed in a review
311 In the summer, off-shore ice advection creates a marginal ice zone in qualitative agreement with observations (Figure 5).
However, ice concentrations are too high near the coasts
and in the Chukchi Sea. These shortcomings of the model appear to be largely due to neglecting spatially varying thermodynamic effects.
The effects of ice rheology on this summer ice edge
are discussed more in the Modeling section of this review. This model also produced realistic shear zone velocity discontinuities off the North Slope of Alaska from time to time.
However, due to lack of resolution i t was not possible to examine
shading effects due to boundary irregularities. Global atmospheric-ice-ocean models. Using a full atmospheric circulation model coupled to an oceanic circulation model Bryan et al.
(1975) and Manabe et al. (in press) have carried out long-term simulations to equili-
12o
(
c5 .'l/a,ka
0.75
150
J
30°
0.9
~nada
.0o Greenland
Figure 5. Average August compactness from a dynamic-thermodynamic simul ation 1979).
( from Hibler,
312 brium conditions, with both constant and seasonally varying
insolation.
In these models a
single value is assigned to ice thickness and allowed to change in accordance with continuity and thermodynamic requirements.
Ice dynamics effects are approximated by allowing the ice to
d r i f t with the upper ocean until
i t reaches a fixed thickness of 4 m at which time the ice
motion is completely stopped. In the case of a constant solar heat input, the ice in the Arctic basin reaches unrealist i c a l l y large thicknesses (exceeding 20 meters in some locations) and ice velocities are far too slow.
In the case of a seasonally varying solar heat input, r e a l i s t i c ice thicknesses of
about 3 meters with an annual variation of 1 meter are obtained.
However, the computed ice
export into the Greenland Sea appears unrealistically small. EMPIRICAL STUDIES OF ICE EXTENTAND CONCENTRATION A class of studies particularly relevant to modeling the marginal ice zone on climatic time scales of several years, are empirical
statistical
or "stochastic" studies.
In these
studies correlations between ice parameters and atmospheric and oceanic variables are made, and conclusions drawn regarding the dominant factors.
In the most formal studies of this type,
f i r s t order Markov processes are used to describe the variables, with the various correlation values interpreted as feedback parameters (see e.g. Hasselmann, 1976). Studies of the spatial and temporal variations of the Arctic ice cover have been done by Sanderson (1975), Kukla (1978), Walsh and Johnson (in press) and Lemke (in prep).
Of these
studies, that of Walsh and Johnson (in press) covers the largest time period, namely about 25 years.
While there are a number of fluctuations, the ice extent record compiled by Walsh and
Johnson (in press), show the average extent of Arctic ice to have generally increased over the last 25 years (see Fig. 6).
Although analysis is not complete, this trend seems to agree with
decreases in Arctic surface temperatures between 1957 and 1975 (Walsh, i977).
Although based
on considerably less data similar conclusions about correlations between temperatures and ice extent have been reached by Budd (1975) for the Antarctic ice cover. Year to year variations in Arctic ice extent have generally been found to be quite asymmetric in space.
In particular, using data from 1969 to 1974 Sanderson finds heavy ice condi-
tions over east Greenland correlating with l i g h t ice conditions over the eastern Canadian area. Similar results were found by Lemke using data from 1966 to 1976.
Using a longer data set,
Walsh and Johnson (in press), on the other hand, find a smoother as~nmetric change, with ice in the North Atlantic having an anomalous change in extent opposite in sign to the changes over the remainder of the polar cap. Sanderson (1975) has suggested that these asymmetric effects may be due to shifts in the upper level westerlies. torical
This type of explanation (based on longer but more qualitative his-
data) has also been advanced by Kelly (1978) who suggests that Eastern Arctic
ice
313
3.0
N
E 0
20
m v
cO 0
E
1.0 0
L
", - I . 0 0 Q.
® -2.0 E3
-30 0
-4.0
'55
'60
~5
'70
'75
Yeor
Figure 6.
Time series of the departure from monthly means of the area covered by Arctic sea ice. The record has been smoothed with a 24 month running mean (from Walsh and Johnson, in press).
extent may correlate with the position of the Icelandic low.
In the Antarctic, similar asym-
metric changes have been reported in antarctic sea ice (although based on only a few years of data) by Ackley and Keliher (1976) and Zwally and Gloersen (1977). Another aspect of the ice extent is the degree of temporal persistence. Lemke (in press) has analyzed this persistence by assuming a f i r s t order Markov process.
His analysis yielded
typical persistence times of about 3 months for the whole Arctic ice cover. scales the persistence times were, however, significantly shorter.
On smaller space
More regional studies of ice extent have been made by Vinje (1976) and Skov (1970) for the North Atlantic and by Barnett (in press), and Rogers (1978) for the North Slope of Alaska and Canada.
Vinje's study emphasized the substantial variation of the North Atlantic ice edge
location from year to year.
Vinje's extreme ice extents for February and August for the time
period 1966 to 1974 are reproduced in Figure 7.
A prominent feature of these extents is the
ice free region west of Spitzbergen which is ostensibly caused by the warm west Spitzbergen current (see e.g. Aagaard and Greisman 1975). With respect to causes of the interannual ice edge variations here, correlations by Skov (1970) suggest that northward oceanic heat transport may be a primary factor.
314
oo
20° Figure
7a.
2oo
40°
Extreme sea ice conditions at the end of February from 1966-1974 (reproduced from V i n j e , 1976).
o
60° ~O
40¢ -4
o,
\ 40 °
/
August GfeeOlOOd 60 o
i
n,momEx,eo,
%
mRM 20° Figure 7b.
OQ
20o
40°
Extreme sea ice conditions at the end of August from 1966-1974 (reproduced from Vinje, 1976).
To examine north slope ice c o n d i t i o n s , Barnett constructed a 20 year h i s t o r i c a l severity index for the Barrow region.
A s a l i e n t c h a r a c t e r i s t i c of t h i s index is a q u a s i - p e r i o d i c f i v e
315
year oscillation.
Barnett found this
pressure in the central basin in April.
severity index correlated well
with the atmospheric
In a more detailed study over a shorter time period,
Rogers (1978), on the other hand, found that air temperature in the form of thawing degree days was the parameter that correlated best with summertime ice margin distance.
However, Rogers
also found that mild summers were often associated with pressure anomalies that resulted in more southerly surface winds advecting warmer air. MODELING THE SEASONALSEA ICE ZONE To properly model the seasonal ice zone requires that special attention be given to several components of sea ice models. Probably the most important component is the thermodynamic modification of the ice thickness distribution. A particular problem here is that pack ice at the ice edge is highly variable in thickness and may well contain a large fraction of open water.
Some of the thermal processes operable in such conditions have been reviewed by Roth-
rock (in press) and Wadhams (in press). A dominant problem at the summer ice edge, is the effect of open water on the adjacent pack ice.
During melting conditions the radiation can contribute to lateral melting at the
edge of ice floes. bottom ablation.
This heat also can be carried under the ice where i t will If
the ice cover is s u f f i c i e n c y disintegrated,
contribute to
some of the heat can be
stored in the mixed layer thus delaying autumn freezeup. I t is also l i k e l y that the open water and increased heat exchanges can modify one or both of the planetary boundary layers.
A number of possible effects of this nature in the oceanic
boundary layer are discussed in a companion review by McPhee (in press b).
The ice at the edge
is also subject to wave effects and ocean eddy effects which are reviewed by Wadhams (in press).
Overall i t may well be that the marginal ice zone is in effect an ocean front problem.
In the shear zone region, the continental shelf and coast create complex effects on the ocean current structure.
Due to the shallowness in these regions, the currents beneath the
boundary layer may be much more wind driven in nature than in the deeper ocean.
Because of
this the movement or lack of movement of the ice cover may affect the current structure here, which, in turn, may cause thermodynamic and dynamic changes in the ice cover.
Current effects
can also be induced at the summer ice edge due to horizontal density gradients there (see e.g. McPhee, in press).
I t is also possible for the boundary effects to couple with the ice rheology and modify the ice edge evolution in summer. To i l l u s t r a t e how drastic an effect t h i s can be, seasonal simulations were run using Htbler's (1979) dynamic themodynamic model with several d i f f e r e n t rheologies. Figure 8 shows the compactness time series at three 125 km grid c e l l s proceeding outward from Prudhoe Bay for three d i f f e r e n t ice rheologies: p l a s t i c , NevYconian viscous, and l i n e a r viscous.
In all of these rheologies the ice strength was taken to depend on the ice
316
1.0
-V
~7-----?d~-T~.=-.
~'4~-/-~-~
-~[~,,.
S/~ ~/ -
~' 0.8
7-.
E o 0.6
Plastic
4
0.4 Time (months)
IC
= o.~ 0.
E
8 0.6
4
Viscous
0.4~ Time (months)
1.0
~
0.E
I
R E
\X /
8o.6
~J6
NViscous
!
I
0.4
Time(months)
Figure 8.
Time series of the simulated compactness at three grid cells (125 by 125 Km) progressing further from shore near Prudhoe Bay (Cell 4 closest, Cell 6 farthest away--see Hibler, 1979). The simulated results from these different rheologies are plotted. The simulations were carried out using the seasonal dynamic thermodynamic model of Hibler (1979).
317 thickness and compactness.
As can be seen, the Newtonian viscous rheology produces highly
unrealistic ice edge effects with the ice edge actually forming further off shore.
The linear
viscous and plastic both do well, with the plastic law creating much more of a near shore flow lead, whereas the viscous law tends to spatially smooth out the compactness variations. In practice, of course, model simulations may not be sensitive to all of the effects mentioned here.
However, their presence means that at least some quantitative studies need to be
done to assess their significance. CONCLUDING REMARKS In practice, building an ice model including the seasonal ice zone requires approaching this region with a hierarchy of models both deterministic and stochastic in nature, and progressing from crude and simple to more complete and sophisticated schemes. Simple models are often especially useful in helping to identify the relative importance of specific processes in the total system and can aid in the development of more complete simulation models. These more complete models can be used to test the adequacy of current understanding of the seasonal ice zone. Considerable progress has been made in developing decade.
numerical ice models over the last
The framework of these ice models in conjunction with existing ocean and atmospheric
models appears to be computationally
general enough to handle the seasonal sea ice zone.
However, how to properly parameterize the complex processes in the seasonal sea ice zone in a numerical model is an open question. ACKNOWLEDGEMENTS This work was supported in part by the Office of Naval Research and by the National Aeronautics and Space Administration. REFERENCES Aagaard, K. and P. Greisman, Toward new mass and heat budgets for the Arctic ocean, J. Geophys. Res., 80:3821-3827. Ackley, S.F. and T.E. Keliher, Antarctic sea ice dynamics and its possible climatic effects, AIDJEX Bulletin, 33:53-76. Barnett, D.S., A long-range ice forecasting method for the north coast of Alaska, ICSI/AIDJEX Symposium on Sea Ice Processes and Models, Univ. of Wash. Press, in press. Bryan, K., S. Manabe, and R.L. Pacanowski (1975), A global ocean-atmosphere climate model Part I I . The oceanic circulation. J. Phys. Oceanogr., 5:30-46. Budd, W.F. (1975), Antarctic sea-ice variations from s a t e l l i t e sensing in relation to climate, J. Glaciology, 15:417-428.
318 Campbell, W.J. (1965), The wind driven circulation of ice and water in a polar ocean, Geophysical. Res., 70:3279-3301.
J.
Coon, M.D., R. Colony, R.S. Pritchard, and D.A. Rothrock (1976), Calculations to test a pack ice inodel, Numerical Methods in Geomechanics, Vol. 2, (C.S. Desai, ed.), ASCE, New York, 1210-1224. Coon, M.D. and R.S. Pritchard, Glaciology, in press.
Mechanical energy considerations
in
sea ice
dynamics, J.
Croasdale, K.R. (1978), Ice engineering for offshore petroleum exploration in Canada, POAC77 Proceedings, Ocean Eng. Inf. Center, Mem. Univ. of Newfoundland, St. Johns, 1-32. Crutcher, H.L. and J.M. Meserve (1970), Selected level heights, temperatures and dew points for the northern hemisphere, NAVAIR Rep. 50-IC-52, revised, U.S. Nav. Weather Serv. Coamand, Washington, D.C. Ooronin, Y.P. (1970), On a method of calculating the compactness and d r i f t English translation in AIDJEX Bulletin, 3:22-39.
of ice floes,
Egorov, K.L. (1971), Theory of the d r i f t of ice fields in a horizontally inhomogeneous wind field, English translation in AIDJEX Bulletin, 6:37-45. Glen, J.W., Thoughts on a viscous model for sea ice, AIDJEX Bulletin, 2:18-27. Hasselmann, K. (1976), Stochastic climate models, Part I. Theory. Tellus 28:437-485. HiDler, W.O. I l l , (1974), Differential sea ice d r i f t II. Comparison of mesoscales strain measurements with linear d r i f t theory predictions, J. Glaciology, 13:457-471. Hibler, W.D. I l l , (in press).
(1979), A dynamic thermodynamic sea ice model, J. Physical Oceanography, 9,
Hibler, W.D. I I I , Sea ice growth d r i f t and decay, Chapter 3 in Dynamics of Snow and Ice Masses, (S. Colbeck, ed.), Academic Press, N.Y., in press. Hibler, W.D. I l l , W.F. Weeks, A. Kovacs, and S.F. Ackley (1974a), Differential sea-ice d r i f t I. Spatial and temporal variations in sea-ice deformation, J. Glaciology, 13:437-455. Hibler, W.D. I l l and W.B. Tucker, Some results from a linear viscous model of the Arctic ice cover, J. Glaciology, in press. Kelly, P.M. (1978), Forecasting the arctic years, Climate Monitor, 7:95-98.
sea ice on time scales of a few months to many
Kukla, G.J. (1978), Recent changes in snow and ice, in Climate Change (J. Gribbin, ed.) Cambridge Univ. Press, Cambridge, 114-129. Kulakov, I.Y., M.I. Maslovsky, and L.A. Timokhov, Seasonal variability of antarctic sea ice extent: Its numerical modeling, ICSI/AIDJEX Symposium on Sea Ice Processes and Models, Univ. of Wash. Press, in press. Laikhtman, D.L. (1964), Physics of the Boundary Layer of the Atmosphere, English translation published by U.S. Dept. of Commerce. Lemke, P. (in prep), A stochastic model of the coupled sea-ice atmosphere system. Lepparanta, M. (1977), On the dynamics of ice cover in the Bothnia Bay, Proceedings of 100 years of winter navigation in Finland, Oulu, Finland. Ling, C.H., L.A. Rasmussen and W.J. Campbell, A mesocale continuum sea ice model, ICSI/AIDJEX Symposium on Sea Ice Processes and Models, Univ. of Wash. Press, in press.
319 Manabe, S., K. Bryan, and M.J. Spelman, A global ocean-atmosphere climate model with seasonal variation for future studies of climate sensitivity, Dyn. Atmos. and Ocean, in press. McPhee, M., An analysis of pack ice d r i f t in summer, ICSI/AIDJEX Symposium on sea ice processes and models, Univ. of Wash. Press, (in press a). McPhee, M. (in press b) These proceedings. Maykut, S.A. (1978), Energy exchange over young sea ice in the central Res., 83:3646-3658.
arctic, J. Geophys.
Maykut, S.A. and Untersteiner, N. (1971), Some results from a time dependent thermodynamic model of sea ice, J. Geophys. Res., 76:1550-1575. Neralla, V.R. and W.S. Liu, A simple model to calculate the compactness of ice floes, J. Glaciology, in press. Nye, J.F. (1975), The use of ERTS photographs to measure the movement and deformation of sea ice, J. Glaciology, 15:429-436. Parkinson, C.L. and W.M. Washington, (1979), A large scale numerical model of sea ice, J. Geophys. Res., 84:311-337. Pritchard, R.S. (1975), An elastic-plastic constitutive law for sea ice, J. Appl. Mech., 42E, 379-384. Pritchard, R.S (1978), The effect of strength on simulation of sea ice dynamics, POAC 77 Proceedings, Ocean Eng. Inf. Center, Mem. Univ. of Newfoundland, St. Johns, 494-505. Pritchard, R.S., M.D. Coon, and M.S. McPhee (1977), Simulation of sea ice dynamics during AIDJEX, J. Pressure Vessel Technology, 99J, 49-497. Rogers, J.C~ (1978), The meteorological factors effecting intersummer variability of sea ice conditions in the Beaufort Sea, Mon. Weather Review, 106:890-897. Rothrock, D.A. (1975a), The steady d r i f t of an incompressible Arctic ice cover, J. Geophys. Res., 80:387-397. Rothrock, D.A. (1975b), The energetics of the plastic deformation of pack ice by ridging, J. Geophys. Res., 80:4514-4519. Rothrock, D.A., Modeling sea ice features and processes, J. of Glaciology, in press. Sanderson, R.M. (1975), Changes in the area of Arctic sea ice, 1966 to 1974, The Meteor. Magazine, 104:303-323. Schwarz, J. and W.F. Weeks (1977), 19:499-532.
Engineering properties
of
sea ice,
J.
Glaciology,
Semtner, A.J. Jr. (1976), A model for the thermodynamic growth of sea ice in numerical investigations of climate, J. Phys. Oceanogr., 6:379-389. Skov, N.A. (1970), The ice cover of the greenland Sea, Meddelesev om Gronland, Bd. 188, C.A. Reitzel s, Kopenhagen. Sobczak, L.W. (1977), Ice movements in the Beaufort Sea 1973-1975: Determination by ERTS imagery, J. Geophys. Res., 82:1413-1418. Taljaard, J.J., H. van Loon, H.L. Crutcher, and R.H. Jerun, (1969) Climate of the upper air, I, Southern hemisphere, vol. 1, Temperatures, Dew Points, and Heights at Selected Pressure Levels, ~VAIR Rep. 50-IC-55, 135 pp., U.S. Nav. Weather Serv. Command, Washington, D.C.
320
Thorndike, A.S., D.A. Rothrock, G.A. Maykut, and R. Colony (19755, The thickness distribution of sea ice, J. Geophys. Res., 80:4501-4513. Thorndike, A.S. and R. Colony, Large scale motion in the Beaufort Sea during AIDJEX April 1975-April 1976, ICSI/AIDJEX Symposium on Sea Ice Processes and Models, Univ. of Wash. Press, in press. Tryde, Per. (1977), Ice forces, J. Glaciology, 19:257-264. Udin, I. and A. Ullerstig (19765, A numerical model for forecasting the ice motion in the bay and sea of Bothnia. Research Report No. 18, Swedish Administration of shipping and Navigation, Norrkoping, Sweden. Vinje, T•E. (1976), Sea ice conditions in the European Sector of the marginal seas of the Arctic, 1966-1975. Norsk Polar Institute Arbok 1975, Oslo, 163-174. Wadhams, P. These proceedings, in press. Washington, W.M., A.J. Semtner, C. Parkinson and L. Morrison (1976), On the development of a seasonal change sea, ice model, J. Phys. Oceanogr., 6:679-685. Walsh, J.E. (1977), The incorporation of ice station data into a study of recent Arctic temperature N uctuations, Mon. Weather Rev., 105:1527-1535. Walsh, J.E. and C.M. Johnson, An analysis of Arctic sea ice fluctuations, 1953-77, J. Phys. Oceanography, in press. Zwally, H.J. and P. Gloersen (1977), Passive microwave images of the polar region and research applications, Polar Record, 18:431-450. Zubov, N.N. (1943), Arctic Ice, English translation U.S. Dept. of Commerce No. AD426972. APPENDIX: SEA ICE RHEOLOGY A general framework for examining sea ice rheologies has been outlined by Glen (1970). Based on earlier glacier flow concepts, Glen suggested a general consitutive law of the f o l lowing form might be applicable to sea ice:
° i j = 2n~ij + [({-n)(~xx + ~yy) - P]~ij
(15
where oij and ~ij are the two-dimensional stress and strain rate tensors, n, ~ and P are general functions of the two invariants of the strain rate tensor, and ~ij equals one for i equal to j or zero for i not equal to j .
These invariants can be taken as the principal values of
the strain rate tensor or alternatively as (~xx+Cyy5 and ~xx~xx+2~xy~xy -+~yy ~yyS. To obtain an internal ice force for the momentum balance, the stress tensor is differentiated, yielding @
Fx : ~
Fy = ~
@
(Oxx5 %-~ (axy) @
(Oxy5 ~
@
(Oyy5
where Fx and Fy are the components of the force transmitted through the ice.
(2)
(3)
321 Special cases of these rheologies often used in sea ice are a "linear viscous" rheology where the stress depends l i n e a r l y on the strain rates and an "ideal r i g i d plastic" rheology where the stress state is either indeterminate or independent of the magnitude of strain rates. In the linear viscous case, n and { are taken to be linear in cij and P is constant. results in a stress state l i n e a r l y dependent on the strain rates.
This
In the r i g i d plastic case
the stress state is taken to be fixed independent of the magnitude of the strain rate invariants provided the ratio of the invariants does not change. linear functions of the invariants. well defined for zero strain rates.
In this case n and ~ w i l l be non-
In the r i g i d plastic case, however, n and { may not be Under these conditions the ice is understood to move
r i g i d l y with the stress detemined by external balances. Some of the linear viscous and plastic rheologies proposed for sea ice are a Newtonian viscous f l u i d (n = constant; constant, P = constant).
{ = P = 0), and a general linear viscous f l u i d ({ = constant, n =
For an ideal plastic constitutive law the stress state l i e s on or
within some fixed y i e l d curve. state to the strain rates.
For plastic flow a rule is needed to uniquely relate the stress
The most common rule is to take the ratio of the strain rates for a
given stress state to be that of a vector normal to the surface. normal flow rule. Hibler (1979).
This is referred to as the
For e x p l i c i t stress strain-rate equations for an e l l i p t i c a l yield curve see