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Naval Postgraduate School

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A Thermodynamic Model of a

Central Arctic Open Lead

by

Richard Harris Schaus

Thesis Advisor:

J.

A.

Gait

September 1971

Apptiov&d ^oh. pub tic nzlzabz; dtbtlibution im&lrrUXzd.

A Thermodynamic Model of a Central Arctic Open Lead

by

Richard Harris Schaus
it Commander, United S1
B.S., University of Michigan, 1962

Lieutenant Commander, United States Navy

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

from the

NAVAL POSTGRADUATE SCHOOL

September 1971

1. 1

ABSTRACT

A time-dependent, two-dimensional thermodynamic model of an open lead
in central arctic sea ice is presented. The model is generated by opening
a lead of finite width and infinite extent in the equilibrium sea-ice cover.
From this initial condition, the model is integrated numerically as a sea-
ice cover is reestablished over the lead. The effects of various representa-
tive advective parameterizations, and temperature and salinity profiles in
the ocean's surface layer, on the heat flux through the lead and on the
nature of ice formation are investigated as the lead closes by thermal
processes alone. Continuity equations involving a horizontal advection
term and a vertical diffusion term govern heat and salt transport in the
water. Cooling-induced convective overturn as a mechanism for vertical
heat and salt transport in the water column is treated through an artifice
of the vertical diffusion term.

TABLE OF CONTENTS

I . INTRODUCTION 8

A. POLAR AIR-SEA INTERACTIONS 8

B. SURVEY OF PREVIOUS RESEARCH 9

C. PRESENT RESEARCH DIRECTION 12

D. CONCEPTUALLY-RELATED STUDIES - 13

E. THESIS OBJECTIVE 14

II. NATURE OF THE PROBLEM 15

A. DYNAMICS OF THE SEA-WATER FREEZING PROCESS - 15

B. DISTRIBUTION OF VARIABLES 18

C. HYDRODYNAMIC PROCESSES — 19

D. MASS AND ENERGY BALANCE 19

III. EXPERIMENTAL PROCEDURE 22

A. FORMULATION OF THE MODEL 22

1. Establishment of a Representative Cross Section 22

2. Adaptation of the Continuity Equation to the Model 22

3. Initial Conditions 24

4. Boundary Conditions 25

5. The Finite-Difference Scheme 27

6. Representation of Physical Processes 31

7. Physical Quantities and Relations 32

B. ENVIRONMENTAL DATA INPUTS 34

C. MODEL INTEGRATION PLAN 35

D. MODEL LIMITATIONS 37

IV. PRESENTATION OF DATA 40

A. TEST CASES 40

B. REAL CASES 41

V. CONCLUSIONS 62

VI. RECOMMENDATIONS FOR FURTHER RESEARCH - 69

APPENDIX A COMPUTER PROGRAM DESCRIPTION 70

COMPUTER PROGRAM 77

BIBLIOGRAPHY - 82

INITIAL DISTRIBUTION LIST 84

FORM DD 1473 85

LIST OF TABLES

I. Environmental Variable Profiles 35

II. Schedule of Runs and Variable Combinations 38

III. Ice Growth Rates, Test Cases 42

IV. Ice Thickness Profiles, Test Cases 45

V. Surface Heat-Loss Densities, Test Cases 48

VI. Mean Surface Heat-Loss Densities, Test Cases 51

VII. Maximum Depth of Penetration of Convective

Overturn, Test Cases 51

VIII. Ice Growth Rates, Real Cases 52

IX. Ice Thickness Profiles, Real Cases 54

X. Surface Heat-Loss Densities, Real Cases 56

XI. Mean Surface Heat-Loss Densities, Real Cases 58

XII. Maximum Depth of Penetration of Convective

Overturn, Real Cases 58

XIII. Ice Growth Rate, Run 13 59

XIV. Mean Surface Heat-Loss Densities, Run 13 59

LIST OF FIGURES

1. Temperatures of Sea-Water Maximum Density and

Freezing Point vs Chlorinity 16

2. Pictorial Depiction of Open Lead 23

3. Description of Processes Considered in Model 28

4. Model Cross Section with Governing Continuity Equations and
Boundary Conditions 29

5. Ice Growth Rates, Test-Case Runs 1-4 43

6. Ice Growth Rates, Test-Case Runs 5-8 44

7. Ice Thickness Profiles, Test-Case Runs 1-4 46

8. Ice Thickness Profiles, Test-Case Runs 5-8 47

9. Surface Heat-Loss Densities, Test-Case Runs 1-4 49

10. Surface Heat-Loss Densities, Test-Case Runs 5-8 - 50

11. Ice Growth Rates, Real Cases JJ

12. Ice Thickness Profiles, Real Cases 55

13. Surface Heat-Loss Densities, Real Cases 57

14. Ice Growth Rate, Run 13 - — 60

15. Mean Surface Heat-Loss Densities, Run 13 - 61

16. Descriptive Flow Diagram of Program 74

ACKNOWLEDGEMENTS

The topic for this thesis was suggested by Assistant Professor Jerry A.
Gait of the Naval Postgraduate School, Monterey, California. His invaluable
experience in the fields of arctic dynamics and the physical oceanography
of the arctic, and his guidance in developing the conceptual model contri-
buted greatly to the successful completion of the research. Associate
Professor Warren W. Denner and Assistant Professor Ken L. Davidson, also
of the Naval Postgraduate School, provided helpful ideas during the research,
Associate Professor J. J. Von Schwind of the Naval Postgraduate School
contributed valuable time and thought to reading the manuscript. To these
people the author expresses his sincere thanks.

I. INTRODUCTION

A. POLAR AIR-SEA INTERACTIONS

Heat lost from polar regions together with the heat source of the
tropics form the basis of the global general circulation. The flux of heat
into polar regions from lower latitudes results from the general meridional
temperature gradient from polar regions toward the equator due to unequal
latitudinal heating of the earth's surface. Most of the heat advected into
polar regions by the atmosphere is lost to space by long wave radiation,
thus forming a basic mechanism for maintaining planetary radiative
equilibrium.

Heat is also advected into polar regions by ocean currents, but due
to the relative inefficiency of air-sea heat transfer and the relative
slowness of ocean currents, this transfer is roughly one-tenth that
attributed to the atmosphere.

Sea ice is a dominant feature of both arctic and antarctic seas; however,
due to major differences in the physical configuration of sea and land
masses of these regions, it plays decidedly different roles. The arctic,
being essentially an ice-covered ocean, is extremely sensitive to climatic
perturbations since the areal extent and depth of sea ice is critical to the
regional energy balance. A surplus of about one-third of the annual energy
influx would remove the arctic ice pack [Maykut and Untersteiner 1969, 1971 J.
On the other hand, the antarctic continental mass, covered by 2000 to 4000
meters of ice, is relatively insensitive to small energy flux perturbations,
although its sea-ice cover would be affected similarly to that of the arctic.

Of major import to polar air-sea interactions is the degree to which
the sea surface is covered by ice. Sea ice acts analogously to a 'lid'
on the sea, modifying the dynamics and continuity of the sea and the air-
sea system. Sea-ice cover dramatically reduces the heat transfer between
ocean and atmosphere by effectively limiting that transfer to a process of
molecular conduction. Further, by intercepting the vertical flux of
momentum transferred from the mean wind field to the sea surface by the
Reynold's stress, it suppresses wind mixing. Finally, it reflects a
considerable portion of the incoming short-wave radiation, imposes an
upper limit on the sea's surface-boundary temperature, reduces latent heat
transfer by impeding evaporation, and, through its changes of phase, serves
as a buffer, or damping mechanism. Alternatively, heat transfer processes
are significantly enhanced over open-water areas in proximity to large ice
masses, particularly during the winter, as surface air temperatures drop
far below surface water temperatures. Wind and water stresses, in addition
to directly affecting ocean-atmosphere heat transfer processes, keep the
ice in nearly continuous motion, thus locally affecting heat transfer as
well as producing mechanical stresses resulting in the formation of pressure
ridges, polynya, and open leads.

Polar air-sea interactions are therefore characterized by complex feed
back wherein the ocean and atmosphere individually and collectively influence
the thickness and areal extent of the ice, which in turn has a pronounced
effect on the exchange in question.

B. SURVEY OF PREVIOUS RESEARCH

Range and depth of interest in arctic research has expanded considerably
since Nansen's epic drift in the FRAM (1893-1896). From oioneering studies

until only recently, research emphasis has been primarily of a descriptive
nature, and it was not until the Soviet establishment of the drifting
station "North Pole 1 (NP-1)" in 1937 that a systematic study of the
arctic interior was initiated. Launching of a comparable long-range
arctic research program by the United States would await the establishment
of "Fletcher's Ice Island (T-3)" in 1952.

Attempts to understand the response of sea ice to environmental
changes has long stimulated the interest of arctic researchers. Quantita-
tive prediction of these responses has classically involved two primary
approaches, the empirical and the theoretical. Although providing minimal
insight to the dynamics of physical processes occurring between air, ice,
and sea, empirical formulae relating sea-ice growth processes to observed
surface temperatures have accomplished basic predictive goals. The most
sophisticated example of this approach is perhaps one advanced by Bilello
[1961 j. Theoretical studies involved analytical solutions and were there-
fore severely limited in scope and complexity. One of the foremost of
these analytical treatments is attributed to Kolesnikov [U.S. Naval Civil
Engineering Laboratory 1966].

In the early 1950' s the tenor of arctic research began to reflect
increased emphasis on the theoretical approach to arctic dynamics as
comprehensive data from systematic studies became available. It was at
this point that studies relating the arctic energy balance to the mass
budget and subsequent research directed toward heat transfer processes
began to emerge. Development of physical relationships was confined to
empirical and statistical approaches, or theoretical problems to which
simple, analytical, approximate solutions could be developed.

10

Classic arctic heat budget studies were presented by several authors.
Badgley [1961, 1966] summarized and collated available data to deduce a
typical heat budget at the surface of the central Arctic Ocean. Fletcher
[1966] discussed annual patterns of atmospheric heat loss for each component
of the heat budget in relation to the general atmospheric circulation, for
both ice-covered oceans and ice-free oceans. Vowinckel [Arctic Meteorology
Research Group 1964] independently investigated the amount of heat conducted
through the Arctic Ocean ice to check previously calculated values for the
heat released from the Arctic Ocean. Doronin [1966] presented an analysis
of heat-balance components of the surface layer in the Arctic Ocean.
Coachman [1966] proposed a model to explain the observed features of the
oceanic regime of the Arctic surface layer during the winter. Muench
[Baffin Bay - North Water Project 1971] discussed the physical oceanography
of the northern Baffin Bay region in considerable detail.

It was not until the 1960's and the attendant widened use of the high-
speed digital computer as a research tool that physical processes could be
modeled free of constraints imposed by analytical solutions. Using finite
difference schemes, a system of integral-differential equations could now
be solved rapidly within acceptable error limitations. Typical of first-
generation numerical models in the field is that developed by Untersteiner
[1966], which predicted thickness and temperature of sea ice, thus allowing
further insight into the relationship of climatic change to ice production,
ice decay, and equilibrium ice thickness. The advanced one-dimensional
thermodynamic model of central Arctic sea ice developed by Maykut and
Untersteiner [1969, 1971] is a prime example of the present theoretical
approach.

11

C. PRESENT RESEARCH DIRECTION

A great deal of research effort is presently being directed toward
developing numerical models of the atmosphere and of the ocean as well
as atmosphere-ocean models, with the objective of ultimately producing
an efficient, reliable global system model. Problems inherent in the
realization of this objective are numerous and include not only a lack of
refinement in respective models of the atmosphere and ocean, but also a
lack of refinement in the technique of coupling the respective models.
Further, observational input data is in general deficient for these models,
particularly in the Arctic. Present state-of-the-art examples of models
of this type are well represented by the sophisticated models of Manabe
[1969] and Bryan [1969].

One of the more ambitious research projects in recent years is the
Arctic Ice Dynamics Joint Experiment. AIDJEX is essentially a United
States - Canadian cooperative effort to gain quantitative understanding
of the interaction between atmospheric sea ice and fluid ocean fields of
motion. In progress at the time of this writing, AIDJEX is unique in its
observational approach since, in order to gain a full understanding of
interacting fields, it involves the collaboration of many research groups
in making coordinated measurements of interacting fields of motion, stress,
and strain - observing appropriate time and space scales - over a minimum
period of one year. The fruits of this ambitious and imaginative study,
in terms of fresh insights, controlled data, and international cooperation
appear to be vast already.

12

D. CONCEPTUALLY-RELATED STUDIES

Studies of Arctic dynamics, energy balances and mass budgets range
over a wide spectrum. Scientific literature reflects numerous experimental
and theoretical models of microstructure and micro-scale energy transfer
processes between ocean and atmosphere, ice and atmosphere, and ice and
ocean, essentially occurring in boundary-layer regions. Pertinent to the
work presented herein is research performed by Foster [1968, 1969] in which
haline convection induced by the freezing of sea water was studied, and by
Lake and Lewis [1970] who studied salt rejection by sea ice during growth.
Work performed by Lewis and Walker [1970] illustrates seasonal changes in
the temperature and salinity profiles beneath annual sea ice cover, and
Stern and Turner [1969] reported on the formation of salt fingers and
convecting layers.

Directly related to the model proposed herein is the macroscale advanced
model derived by Maykut and Untersteiner [1969, 1971] to forecast ice
temperature and thickness, which defines the role of each component of the
energy budget in relation to its effects upon the ice. Although comprehen-
sive in its treatment of physical processes related to components of the
energy budget, the model admittedly has several shortcomings. Physical
limitations are due to uncertainties in environmental data and include
the necessary assumption of a constant oceanic heat flux and generally
course albedo and energy-flux data. Limitations, arising from an inability
to model certain processes include: 1) the neglect of mechanical stresses
produced by winds and surface currents, 2) treatment of turbulent fluxes
at boundaries as being independent of growth rates or the physical states
of ice and snow, 3) imprecise treatment of downward heat transport from

13

melting snow, 4) undefined shapes of salinity profiles for various ice
conditions, and 5) neglect of heat storage by melt ponds. Leads were
considered only so far as they indirectly affected the energy fluxes over
the bare ice. In spite of its limitations, this will certainly be a
cornerstone for future thermodynamic ice models and provides a number of
conceptual bridges used in the present work.

E. THESIS OBJECTIVE

It is known that significant heat transfer occurs between the atmosphere
and the ocean through open leads in the Arctic sea ice cover. As found
experimentally by Badgley [1961], sensible heat loss is at least two orders
of magnitude greater from open leads than from perennial sea ice.

With the need for further refinement in models of the ocean and the
atmosphere, as well as in model coupling schemes, the requirement for
mesoscale examination of ocean-ice-atmosphere heat- transfer processes is
immediate and apparent. With appropriate observational input data becoming
increasingly more abundant, informative numerical models of mesoscale
processes appear promising for furthering the understanding of physical
processes and for realizing control of mass-energy fluxes in the arctic.

The objective of this study was to develop a thermodynamic model of an
open lead in central arctic sea-ice in which hydrodynamic processes of
convection, advection, and diffusion of heat and salt were represented.
By integrating the model numerically, the effects of various representative
advective parametizations, and temperature and salinity profiles in the
ocean's surface layer, on the heat flux through the lead and on the nature
of ice formation, could be investigated as the lead reclosed by thermal
processes alone.

14

II. NATURE OF THE PROBLEM

Since sea ice essentially insulates the ocean from the atmosphere and
suppresses important thermodynamic processes, it, along with heat-advecting
ocean currents, becomes a primary factor in determining the vertical
structure in Arctic seas. However, the dynamic transition from ice-free to
ice-covered seas is not fully understood. In particular, the nature of
the fluxes of salt and heat, as well as the structure and depth of unstable
convective overturn during initial phases of the freezing process, require
study.

A. DYNAMICS OF THE SEA WATER FREEZING PROCESS

The structure of the active surface layer of the Arctic Ocean is
generally horizontally uniform with a very stable density gradient. How-
ever, during the freezing process the density gradient may be significantly
altered by three primary mechanism: 1) progressive cooling of surface
waters from above, 2) brine expulsion by the ice during the freezing pro-
cess, and 3) advection of heat and/or salt by surface currents.

The fundamental physical relation which makes the sea water freezing
process unique may be understood by examining the curves of maximum density
and freezing point for sea water, plotted with respect to temperature versus
chlorinity or salinity (Figure 1). Specifically, for waters of salinity
greater than 24.70 parts per thousand, the temperature of maximum density of
the liquid state will always be lower than the liquid's freezing-point
temperature under pressures existing in nature. Any process that tends to
produce a vertical density instability will induce convective overturn in

15

TEMP

1 "

-1

-2

-3

-4

Figure 1.

Temperatures of Sea Water

Maximum Density and

Freezing-Point

vs
Chlorinity

after Sverdrup, Johnson and Fleming [1942]

Maximum Density

Freezing-Point

16

the water column, resulting in mixing of the unstable segment such as
would be produced in the freezing process by surface cooling and freezing-
associated brine expulsion. Convective overturn will therefore be continuous
in the unstable segment of the liquid water column from the onset of surface
cooling to the point of total change-of -phase of the water column. A some-
what less obvious but straightforward point should be mentioned in qualifi-
cation of the foregoing. From an arbitrary water temperature above the
freezing point, down to, but not including the freezing point, convection
is temperature controlled during cooling since there is no ice formed,
hence, no brine expulsion. During cooling of surface waters at the freezing
point for existing surface salinities, convection is primarily salinity
controlled since brine is being continuously expelled from newly forming
ice. Within this latter water temperature regime, temperature changes play
a minor role in effecting changes in surface water densities.

As convective overturn is a mechanism for inhibiting freezing by
continuously transferring heat from depth to the active boundary at which
freezing is taking place, so it is also a mechanism for effectively removing
heat from depths significantly greater than could be effected by molecular
diffusion. An equilibrium ice thickness is reached when the resultant heat
flux upward through convection, advection and diffusion in the water column
balances the upward resultant heat flux through the ice, at the ice-water
boundary. Prior to this point, ice is accreted in depth since the upward
heat flux through the ice exceeds the upward heat flux in the water column
arriving at the ice-water boundary.

17

B. DISTRIBUTION OF VARIABLES

The distribution of temperature, salinity and velocity are considered
within a volume of water under a refreezing lead. The water is assumed to
be inviscid, horizontally homogeneous, horizontally isotropic, irrotational ,
and incompressible. Accelerations resulting from the coriolis effect and
horizontal pressure gradients are assumed to be negligible. Momentum, heat
energy, and salt mass are conserved within the volume such that no velocity,
temperature, or salinity sources or sinks exist except at the volume's
boundaries. Furthermore, currents are assumed to be steady-state and
uniform both horizontally and vertically.

Using the classic box-model approach, these conservative fields are
represented continuously within the volume by individual continuity
equations. For the vector field of velocity, where K is the diffusivity
and Sw is a source term.

DV/_ = V-KW+ S (1)

Dt v

which reduces to

v(t,x,y,z) = v = constant (2)

i.e., the velocity is specified as uniform and steady.

Similarly, the scalar fields of heat and salt, here collectively
depicted by the scalar quantity 0, are represented by

DJ3/Dt = v. K v 0 + S (3)

whe

re K is the diffusivity and S. is a source term.

18

This reduces to

*%t = " V'V(3 + Khv20 +Vv ' K(z) vv0 (4)

where K is the horizontal diffusivity and vv is the vertical del operator
since,

K(x) = K(y) = KH = constant

The local, or time-rate-of-change, is therefore equal to the sum of the field-
rate-of-change and the diffusive-rate-of-change.

C. HYDRODYNAMIC PROCESSES

Mechanisms for the hydrodynamic transport of heat and salt within the
lead are advection and diffusion. However, from the dynamics of the sea-
water freezing process, it is known that convection plays a major role in
transporting both heat and salt in the water column. Although it is riot
represented by a discrete term in the foregoing continuity equation, the
convective transport term may be thought of as simply a temporary (when
vertical density instabilities arise) modification of the vertical diffusion
term taking the form

'im *A «■)""'*, (5)

K(*)-«

which in essence results in free vertical (convective) transport.

D. MASS AND ENERGY BALANCE

With the foregoing analyses of continuity relations and mechanisms of
hydrodynamic transport existing within the box model of the open lead, the
mass and energy balances existing at the volume's boundaries may be
examined.

19

Over an arbitrary surface area of a reclosing lead, a given change of
heat, AQ, at the water surface must equal the heat added by advection and
diffusion at upstream boundaries, Q , plus the heat added from depth in the
lead by convection, advection, and diffusion, Qz, plus the heat added by
surface ice accretion, Qf, plus the open-water heat loss at the surface by
conduction, evaporation and radiation, Q +Q +Q , plus the heat loss through
the ice cover by conduction, Q. , minus the heat diffused across vertical
boundaries normal to advective transport, Qn, minus the heat transported
across the downstream boundaries by advection and diffusion, Q^, such that
AQ = Qu + Qz + Qf + W(QS+Qe+Qr) + Qi (1-W) - Qn - Qd (6)

Where W is the fraction of open water comprising the surface of the lead at
any given time. The heat added from depth in the lead is heat ultimately
derived from the Bering Sea Water (in the Canadian Basin) and from the
Atlantic layer.

By similar analysis, a given change of salinity, aS, at the water's
surface must equal the salt added by advection and diffusion at upstream
boundaries, Su, plus the salt added from depth in the lead by convection,
advection, and diffusion, Sz, plus the salt added by surface ice accretion
(salt rejection), Sf, plus the salt lost through surface ice melting (salt
dilution), Sm, plus the salt added by surface evaporation, Se, minus the
salt diffused across vertical boundaries normal to advective transport,
Sn, minus the salt transported across downstream boundaries by advection and
diffusion, S^, such that

AS = su + sz + (i-w)(sf+sm) + w(se) - sn - sd (7)

where again W is the fraction of open water comprising the surface of the
lead at any given time. The salt added from depth in the lead is derived
from the generally strong positive halocline. existing at depths below 25 to
50 meters.

20

Qualitatively then, before the equilibrium sea ice cover has been
reestablished, there will exist differentials in the temperature and
salinity profiles of water columns. A primary process of interest in the
refreezing lead is that of surface ice formation, which lends a unique
character to lead hydrodynamics. The heat added/lost by surface ice
accretion/melting is a function of the incremental change in the mass of
ice,AI, and the heat of formation of sea ice, L . , such that

Qf = -Qm = L-AI (8)

Coupled directly to this process is the process of brine expulsion (salt
rejection) by the sea ice during growth. The salt added/lost by surface ice
accretion/melting is a function of the incremental change in the mass of ice,
AI, the salinity of the sea ice, S^ , (which in turn is dependent upon rate-
of-freezing, ice temperature, and age of the ice), and the salinity of the
parent sea water, Sw, such that

Sf = -Sm = f(Al,Si,Sw) (9)

Qualitative analysis of the salt rejection process by constituent salts is
not appropos "to either the scale or the nature of the study at hand; there-
fore salt rejection was treated simply as a process of water-salinity
modification.

21

III. EXPERIMENTAL PROCEDURE

A. FORMULATION OF THE MODEL

1 . Establishment of a Representative Cross Section

In deciding upon what physical dimensions would be appropriate for
the model, it appeared most suitable to represent a small open lead in
central arctic sea ice such as one would expect to observe between early
to late winter. Therefore, the width of the lead was chosen to be 150
meters, with an additional 50 meters under equilibrium sea-ice cover on
the downstream side of the model lead to facilitate studying resultant
thermodynamics and hydrodynamics in that region.

The model's depth extent was set at 50 meters based upon lower
boundary condition considerations. Over much of the Arctic Basin the
vertical temperature gradient at that depth is zero or yery slightly posi-
tive. Since a no-flux lower boundary condition was desirable in order to
better study the effects of advection of conservative constituents in the
lead, that depth was chosen.

It was felt that the model could be simplified without the loss of
any essential physics if derivatives along the lead (Y-axis) were set to
zero. This reduced the problem to one of looking at a cross-sectional
distribution in the lead. Figure 2 is a pictorial depiction of the open
lead.

2. Adaptation of the Continuity Equations to the Model

Using the simplification of the previous paragraph which allows the
use of a cross-sectional area rather than a true volume to represent the

22

Figure 2.

Pictorial Depiction of Open Lead

WATER

LEAD CROSS-SECTION

S

^x

rrc

/"

>

150 m.

50 m,

23

lead, the horizontal advection and horizontal diffusion terms describing
changes along the Y-axis were eliminated from equation (4).

The problem was further simplified by assuming that all horizontal
diffusion was negligible relative to horizontal advection, and that verti-
cal advection could be disregarded.

Applying the foregoing simplifications to the expanded form of
equation (4), and choosing the state variables of temperature and salinity
as the relevant dependent variables, the final form of the equations used
to describe the conservative scalar fields of temperature and salinity
within the lead was

d0/dt = - udp/ax + a/az K(*)d0/Sz do)

where,

0 = 0 (t,x,z)

3. Initial Conditions

Sea-water temperatures and salinities within the lead at the
instant that'the lead is generated are represented by local stable tem-
perature and salinity profiles previously existing beneatii the equilibrium
sea-ice cover. The temperature and salinity fields are both horizontally
uniform and vary only with depth in the lead such that

T(0,X,Z) = T(Z) (11)

S(0,X,Z) = S(Z) (12)

The surface of the newly-generated lead is initially taken to be
ice-free over the left (upstream) 150 meters, and ice-covered with equili-
grium thickness ice over the right (down-stream) 50 meters. Surface
temperatures are initially at the freezing point, in temperature equilibrium
with the preexisting sea-ice cover.

24

4. Boundary Conditions

The model's left, or upstream boundary, may be regarded as a
source boundary across which sea water having a given temperature and
salinity profile flows into the lead. Sea-water temperature and salinity
profiles are taken as steady-state functions of depth alone, identical to
the lead's initial temperature and salinity profiles such that

T(t,0,Z) = T(Z) (13)

S(t,0,Z) = S(Z) (14)

The model's surface boundary is the primary region of mass and
energy fluxes and is therefore the most dynamically active boundary. The
boundary condition on temperature across the entire surface is that of an
upward (or downward) diffusive flux of heat across the water surface such
that

%x T(t,X,0) = f(AT,Zi,Si,Kw,Ki) (15)

where aT is the air-water temperature difference (an externally specified
parameter), Z-j is the surface ice thickness, S- is the surface ice salinity
(an externally specified constant parameter), and Kw and K-j are the co-
efficients of thermal eddy diffusivity of sea water and thermal conducti-
vity of sea ice respectively. When no ice is present at the surface, the
surface heat flux is taken as the cumulative open-water sensible, latent,
and radiative heat fluxes, which are specified in the model as external
parameters according to the air-water temperature difference. Since the
model's surface at all times represents the water surface, the fluxes at
that boundary are matched as ice is formed, such that the flux through the
water surface is always equal to the flux through the ice to the atmosphere,
but at no time does it exceed the maximum open-water flux to the atmosphere.

25

The boundary condition on sea water salinity at the surface
specifies a no-diffusive flux condition between air and water as well as
between ice and water and ice and air such that

%t S(t.x.o) = 0 (16)

However, salt rejection from ice to water during the freezing process (as
well as salt dilution during melting) is represented as a discrete process
of physically adding the change in salinity produced by sea-ice accretion
or melting to the surface waters such that

S(t,X,o)=» S(t,X,o) + f(Al,AS) (17)

where AI is the incremental change in the mass of sea ice, and AS is
the salinity differential existing between the sea ice and its parent sea
water.

The bottom boundary, as previously discussed, is characterized
as a no-flux boundary, thus

*/^m T(t,X,h) = 0 (18)

%t S(t,X,h) = 0 (19)

where h is the model's depth extent.

The right, or downstream boundary, is also characterized as a
no-diffusive flux boundary where whatever temperature-salinity field is
generated by the thermodynamic processes of the lead is simply advected
out. However, as mentioned previously, the right 50 meters of the lead
model's surface is overlain by an equilibrium sea-ice cover. This boundary
condition is therefore representative of a return towards the equilibrium
vertical profiles. At the model's right boundary then

26

a/dX T(t,W,Z) = 0 (20)

^d X S(t,W,Z) = 0 (21)

where W is the model's width.

Figure 3 describes processes considered in the model cross section.
Figure 4 schematically represents the model cross section with its governing
continuity equations and boundary conditions. In this figure, the surface
boundary condition is simply a statement of the balance of fluxes at the
ice-water boundary. With no ice cover (5=0), the upward heat flux through
the water surface, Fz is equal to the sum of open-water sensible (F ), latent
(F ), and radiative (F ) heat fluxes. When an ice cover is present (5>0), the
upward heat flux through the water surface (* =0+) is equal to the upward heat
flux through the ice ( * =0") minus the heat added by surface ice accretion.
The above relations are presented in terms of temperature gradients in Figure
4 as they are applied to the governing continuity equation for temperature.
Symbols not previously defined are p., volume density of sea ice; q . , heat of
formation of sea ice; 5 ice thickness (volume of ice per unit surface area).
5. The Finite-Difference Scheme

A purely explicit finite difference scheme was selected to numeri-
cally represent the linear parabolic partial differential equation (10).
The time rate-of -change term was represented by forward differencing, the
field rate-of-change term by backward (upstream) differencing, and the
diffusive rate-of change term by central differencing. Thus over the interior
(non-boundary) region of the model

Ck - "j.r . . »j.k -»j-i,k + K g",k+r2gj/gj,k-i m

At AX (AZ)2

27

Figure 3.

Description of Processes Considered in Model

x = 0
z = 0

(1)

z = h

(2)

(3)

x = w

V^XX.

(5)

-* X

(A)

(1) Advection of steady-state temperature-salinity field into the
lead from under equilibrium ice pack.

(2) Heat loss to the atmosphere as a function of ice thickness,
ice accretion/melting, and salt rejection/dilution.

(3) Heat and salt transport by convective overturn, advection and
eddy diffusion.

(4) No transport across boundary.

(5) Advection of temperature-salinity field generated by lead
thermodynamics out of the lead model.

28

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29

where,

0n = 0nAt , (23)

■j,k JAX, kAZ ^6>

At the left boundary, equation (22) was inapplicable since tempera-
ture and salinity were both steady-state functions of depth alone. There-
fore, at this boundary at all times

C - 0S,k (24)

At the surface boundary, the diffusive term for temperature was
modified in equation (22) since

Tn -Tn

'j,k+l 'j,k-l

- = ? T 25

2AZ

or,

since

•j,k-i - 'j,k+i 2AI^v n U&J

The diffusive salt term at the surface was modified in equation (22)

Sn - sn

J>k+1 J'""1 = 0 (27)

2 A I

or,

Sj ,k-l = sj,k+l (28)

At the bottom boundary, the diffusive rate-of-change terms of both
temperature and salinity were modified in equation (22) since

n an

Pj,k+1 " "j,k-l _ (29)

2AZ

= 0

30

or

0j,k+l - »j.k-l

(30)

Finally, at the right boundary, equation (22) remains unmodified
since advection is the only transport mechanism operating across it.

Stability analysis of the finite difference equation using the
Fourier series method [Smith 1965] shows that

n+1

s.k[»B-2*y

(31)

where e represents an arbitrary error introduced into the problem. There-
fore, the finite-difference equation (22) is stable when the absolute value
of the amplification factor remains less than or equal to unity

Ati

U

K

1

< 1

(32)

6. Representation of Physical Processes

Convective overturn is treated nominally through an artifice of the
vertical diffusion term as in equation (5). Numerically however, convective
overturn and its associated mixing is accomplished through arithmetic
averaging of temperatures and salinities within regions of vertical density
instability as determined by sigma-t values, followed by recomputation of
the resultant sigma-t value for the newly mixed (and neutrally stable)
region. Subroutine OVRTRN accomplishes convective overturn in the model.

Advection of heat and salt (temperature and salinity) in the lead is
driven by a current, u, in the positive-x direction. Values and depth pro-
files of the current as used in the model are given in subparagraph B of this
section.

31

Diffusion of heat and salt in the vertical are determined by the

coefficients of vertical temperature and salinity diffusivities, both set

2 -1

at constant values of 10 cm sec in the model.

Formation of sea ice is accomplished as in equation (8). More
specifically however, when a given mass of water, Aw, is cooled AT degrees
centigrade below its salinity dependent freezing-point temperature, the
incremental mass of ice formed, A I, is

A I = Cp(AW.AT)/, (33)

where Cp is the specific heat of sea water and L- is the heat of formation
of sea ice.

Salt rejection (or dilution) resulting from surface sea ice accretion

(or melting) is described in equation (17). Specifically, if A I is the

incremental change in the mass of sea ice and AS is the salinity differential

existing between a unit mass of parent sea water and a unit mass of its

generated sea ice, then the incremental salinity change, ASr, in the surface
waters resulting from that freezing or melting process is

ASR = A I. AS (34)

7. Physical Quantities and Relations

Sea water chlorinity is related to salinity [Knudsen 1902] by the

relation

Cl=(s - 0.030)/l .8050 (35)

Freezing-point depression is empirically related to sea water

chlorinity by Thompson [Sverdrup, Johnson and Fleming 1942] by the relation

Tr = -0.0966'Cl - 0.0000052-C13 (36)

fP

32

The density of sea water as a function of salinity and temperature
(sigma-t) [Knudsen 1902] is empirically computed through

*0 = ^(Cl) (37)

AT = '2(T) (38)

BT = '3(T) (39)

2T = f4(T) (40)

't = f5(2T» VAT'BT> <41>

The specific heat at constant pressure of sea water is empirically
related to its salinity at normal temperature and pressure by Kuwahara
[Sverdrup, Johnson and Fleming 1942] by the relation

C = 1.005 - 0.004136-S + 0.0001098-S2

- 0.000001324-S3 (42)

The thermal conductivity of sea ice was determined using the relation-
ship established by Untersteiner [Maykut and Untersteiner !969, 1971].

where,

ki = kf h- ^f^ , T<0 (43)

the subscript f refers to pure ice,

kf = 0.00486 cal/cm-sec-°C,

p
b = 0.28 cal-cm /g-sec,

T = temperature of ice in degrees Celsius,

S(z) = ice salinity at depth z in g/cm .

The heat of formation of sea ice was represented by the simplified

relationship [Pounder 1965]

where

L^Lf.O-f-) (44)

Lf = 79.77 cal/g,

<* = salinity of sea ice, g/kg,

S = salinity of parent sea water, g/kg.

33

B. ENVIRONMENTAL DATA INPUTS

Environmental data used in the model fell into two basic categories:
that which remained constant throughout all runs of the model, and that
which was variable from run-to-run. Representative of the former were
initial equilibrium-ice thickness, set at 200 cm, an ice salinity of 8.0
g/kg, and an ice density of 0.91 g/cm3. Further, representative open water
sensible, latent, and radiative heat flux densities as presented by Muench
[Baffin Bay-North Water Project 1971] were applied in the model. However,
as ice formed at the surface, the latent heat flux density was modified so
as to decrease exponentially with increasing ice thickness. The radiative
heat flux density was modified so as to decrease from open water values as
ice formed, approaching a constant value representative of the radiative
flux density over pack ice as also presented by Muench. The sensible heat
flux density was modified internally by the relation shown in Figure 4 as
surface ice thicknesses increased. All surface heat flux densities were
representative of an air-water temperature difference of 25°C with no wind.
Finally, the coefficients of vertical temperature and salinity eddy diffusi-

9

vity were set at 10.0 cm / sec.

Primary variable parameters in the model were horizontal current magni-
tudes, vertical temperature profiles, and vertical salinity profiles.
Currents applied in the model were within the range of zero to 10.0 cm/sec.
Extreme test case values were 0.0 and 10.0 cm/sec. Additional test case
values of 2.0 and 4.0 cm/sec were used to study one particular temperature-
salinity field combination with its associated hydrodynamics in further
detail. A value of 7.0 cm/sec was chosen to represent a typical horizontal
current occurring in the lead, and was subsequently used as the real case
value.

34

Test case values of vertical temperature and salinity profiles
representative of extremes in vertical gradient were applied. Profiles
A represented strong gradients, while profiles B represented weak
gradients (Table I). Real case temperature and salinity profiles,
qualitatively representative of early winter (profile C) and late
winter (profile D) conditions in the central arctic [Coachman and
Barnes 1961, 1962], were chosen to represent the typical lead.

Table I summarizes the profile designations of the three variable
parameters and their various values.

C. MODEL INTEGRATION PLAN

In order to gain maximum insight to physical processes occurring in
the lead, as well as to study the effects of current magnitude, vertical
temperature gradient, and vertical salinity gradient on lead thermodynamics,
trie i>erieb 01 cesx Vdi lauki was run in nic mouci • rvcprcacHteu uj iuh;>
one through eight, these test cases involved running all possible combina-
tions of current magnitude (u) profiles A and B, temperature (T) profiles
A and B, and salinity (S) profiles A and B. These test cases were
integrated over a lead reclosure period of two days.

As a result of the interesting dynamics occurring in run 6, additional
runs 9, 10, and 14 were scheduled using temperature and salinity profiles
identical to run 6 (and 2), but with current magnitude profiles C, D, and
F respectively, in order to further investigate these phenomena. Integra-
tion was again for a period of two days.

Finally, the real-case variables, representative of naturally
occurring parameters, were put into the model and integrated over two-day
periods in runs 11 and 12, and over a 30 day period in run 13.

35

Table I. Environmental Variable Profiles

Horizontal

Current

Magnitud

es (cm/sec)

Depth
(m)

Profile

A

B

C

D

E

F

0 - 50

10.0

1.0

2.0

4.0

7.0

0.0

Water Temperatures (QC)

Depth
(m)

Profile

A

0

-1.50

5

-1.40

10

-1.30

15

-1.20

20

-1.10

25

-1.00

30

-0.90

35

-0.80

40

-0.70

45

-0.60

50

-0.50

■1.50
■1.49
•1.43
■1.47
■1.46
■1.45
•1.44
•1.43
■ 1 42

■1.41
■1.40

-1.68
-1.68
-1.68
-1.67
-1.66
-1.65
-1.64
-1.63
-1 .62
-1.61
-1.60

D

■1.82

■1.82
•1.82
•1.82

■1.81
■1.80
■1.79
•1.78
■1 .77
■1.76
■1.75

Water Salinities (g/kg)

Depth
(m)

Profile

A

B

C

D

0

28.00

28.00

31.00

33.51

5

28.50

28.01

31.01

33.52

10

29.00

28.02

31.02

33.53

15

29.50

28.03

31.06

33.55

20

30.00

28.04

31.15

33.57

25

30.50

28.05

31.25

33.60

30

31.00

28.06

31.35

33.63

35

31.50

28.07

31.45

33.66

40

32.00

23.08

31.55

33.69

45

32.50

28.09

31.65

33.72

50

33.00

28.10

31.75

33.75

36

Table II presents a summary of runs conducted with their respective
variable combinations.

D. MODEL LIMITATIONS

In addition to the assumptions made earlier in arriving at the
simplified model of an open lead, basic limitations in the model's
capabilities are: 1) the atmosphere is treated as an external parameter
such that it affects lead dynamics but cannot be affected in turn by the
lead; 2) sea ice is treated as a quasiexternal parameter since it is
depicted as a simple, uniform mass floating on the water's surface.
Density and salinity are constant throughout this mass in both space and time,
and the vertical temperature gradient is taken to be linear throughout. Its
thermal conductivity is determined only at the ice-water boundary, which is
warmest and therefore limiting through the relation given in equation (43),
and an upper limit is applied to the upward heat- flux density through the
ice such that at no time is the flux through the ice allowed to exceed that
over open water; 3) mechanical lead closure (such as by wind-drifted ice)
is disallowed; 4) snow cover over the lead's surface is not modelled; 5)
super-cooling of surface waters [Coachman 1966] is not portrayed; 6) as
convection-induced mixing occurs instantaneously in the model, so also
does surface ice melting whenever heat is brought to the surface by
convection. This results in an artificiality in that, although heat and
salt are conserved in the model, heat brought to the surface under an ice
cover immediately melts ice until no further surface heat is available for
melting. Therefore, under this limitation, it is not possible for surface
heat to be advected downstream under an ice cover to promote downstream
melting. Only in cases when no ice is present to be melted may this heat

37

Table II. Schedule of Runs and Variable Combinations

Run

Variable Profile

Period of Integration
(days)

U

T

S

1

A

A

A

2

2

A

A

B

2

3

A

B

A

2

4

A

B

B

2

5

B

A

A

2

6

B

A

B

2

7

B

B

A

2

8

B

B

B

2

9

C

A

B

2

10

D

A

B

2

11

E

C

C

2

12

E

D

D

2

13

£

n

n

30

14

F

A

B

2

38

be advected downstream; 7) in situations wherein vertical salinity gradients
existing in the water column are weaker than a certain minimum gradient,
the model will over-mix the water column in depth. Specifically, the model
forms ice in response to surface cooling, thereby rejecting salt into the
surface water layer of depth Z. The model then mixes the induced un-
stable region as determined by relative sigma-t values of density. If
however, there is heat available at some shallower depth than that to
which the water column was mixed which could have decreased the initial
amount of ice formed, then although salt, ice, and heat are ultimately
conserved, potential energy is not conserved. Initial freezing and resul-
tant salt rejection have induced convection to a greater depth than would
have actually been the case in a continuous process. Therefore, for
vertical salinity gradients weaker than a certain minimum gradient, the
model's numerical convective process acts as a potential energy source.
Quantitatively, the limiting minimum vertical salinity gradient which can
be accurately treated by this model, in g/kg per vertical grid space, is

Pi . (Sw - S^ • AI/A-Z (45)

where p. is the ice density, Sw is the salinity of sea water, S.. is the
salinity of sea ice, A I is the incremental change in the volume of ice,
and AZ is the vertical grid spacing. In this model, the limiting gradient
is in the neighborhood of 1 X 10"3 g/kg per vertical grid space, which is
lower by an order of magnitude than the weakest salinity gradient used.

39

IV. PRESENTATION OF DATA

Some results of integrating the model using the combinations of
environmental variables summarized in sections III . B. and III .C. are
presented in tabular and graphic form. Three primary output variables are
treated: surface ice thickness, surface heat loss to the atmosphere, and
depth of penetration of surface-freezing induced convective overturn.
Throughout the section, these three output variables are related by run
number (section III.C.) to the input variables of current magnitude,
vertical temperature profile and vertical salinity profile (section III.B.).

A. TEST CASES

Ice thicknesses are treated in two ways: by time growth rate and by
spatial profile across the lead. Values for growth rates were arrived at
by tabulating the maximum ice thickness occurring anywhere in the lead as
a function of. time. Values for spatial profiles were taken as the ice
thicknesses across the model's surface (200 meters) after 48 hours of
freezing. Values over the right 50 meters of the model's surface, initially
covered by 200 cm of ice, are tabulated to show the downstream surface
effects of lead thermodynamics on the equilibrium ice cover.

Surface heat losses to the atmosphere are presented as cumulative heat-
loss densities (heat loss per unit surface area) occurring at each grid
point over the surface of the initially open lead after 48 hours of lead
reclosure. Additionally, these surface heat-loss densities are averaged and
tabulated separately as mean surface heat-loss densities.

40

Finally, maximum depths of penetration of surface-freezing induced
convective overturn are tabulated. Due to the numerical nature of the
model however, these depths should be treated as bounding depths rather
than precise depths as would be indicated for continuous schemes.

B. REAL CASES

Ice thicknesses, surface heat losses to the atmosphere, and depths of
penetration of surface-freezing induced convective overturn are treated in
an analogous manner. Additionally however, ice thickness and mean surface
heat-loss densities are presented as functions of time for run number 13,
in which the model was integrated to represent 30 days of lead reclosure.
Values for surface ice thicknesses were determined analogously to those of
48-hour ice-growth rates.

41

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48

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fO o

CO *

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o

CO

o

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o

( uiD/LBD 20L *) S9!4LSUaQ SS01-^P3H

a

CD

KCi

Table VI. Mean Surface Heat-Loss Densities, Test Cases,

48-Hour Mean-Surface Heat Losses
(x 102 cal/cm2)

Run

Number

A0

1

14.9

2

17.0

3

13.7

4

14.9

5

14.9

6

23.2

7

13.7

8

14.5

Table VII. Maximum Depth of Penetration of Convective Overturn,
Test Cases.

Run

Depth

Number

(m)

1

5 •

2

10

3

5

4

10

5

5

6

20

7

5

8

20

51

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UD

- CO

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ID

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c;-5

Table IX. Ice Thickness Profiles, Real Cases

Ice Thicknesses (cm) Across Lead Surface
48 Hours After Lead Generation.

X

(m)

Run Number

11

12

10

24.7

24.6

20

24.7

24.6

30

24.7

24.6

40

24.7

24.6

50

24.6

24.3

60

24.3

24.0

70

24.4

24.2

80

24.4

24.2

90

24.4

24.1

100

24.4

23.9

110

24.3

23.9

120

24.2

23.6

130

24.0

23.5

140

24.0

23.4

150

203.

203.

160

203.

203.

170

203/

203.

180

203.

203.

190

203.

203.

200

203.

203.

54

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to

<v

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o

, —

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(UIO) SS9U>|01L|1 931

55

Table X. Surface Heat-Loss Densities, Real Cases

48-Hour Surface Heat Losses
(x 102 cal/cm2)

X
(m)

Run Number

11

12

10

13.6

13.9

20

13.6

13.9

30

13.6

13.9

40

13.6

13.9

50

13.7

14.3

60

14.2

14.7

70

14.0

14.4

80

13.9

14.3

90

13.8

14.3

100

13.8

14.3

110

13.8

14.3

120

13.8

14.3

130

13.8

14.3

140

13.8

14.2

150

1.99

2.01

160

1.99

2.01

170

1.99

2.01

180

1.99

2.01

190

1.99

2.01

200

1.99

2.01

56

C\J i —

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GO

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ro

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cu

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(2UD/|.e3 Z0L><) S9L^LSU9Q SS01-^8H

57

Table XI. Mean Surface Heat-Loss Densities, Real Cases.

48-Hour Mean Surface Heat Losses (x 102 cal/cm2)

Run
Number

aQ

11
12

13.3
14.2

Table XII. Maximum Depth of Penetration of Convective Overturn,
Real Cases.

Run

Number

Depth
(m.)

11
12

15
15

58

Table XIII. Ice Growth Rate, Run 13.

Maximum Ice Thickness (cm) Occurring in the
Lead's Surface at Five-Day Intervals.

Ice

Day

' Thickness

5

40.6

10

59.6

15

74.9

20

88.3

25

101.

30

112.

Table XIV. Mean Surface Heat-Loss Densities, Run 13.

9 9

Mean Surface Heat-Loss Densities (xlO^ cal/cnf)
at Five-Day Intervals.

Day

aQ

5

23.5

10

34.7

15

43.7

20

51.6

25

58.8

30

65.5

59

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61

V. CONCLUSIONS

Through the preceding data and the time-dependent temperature-salinity-
density fields generated in the lead model, numerous insights to the process
of lead reclosure were afforded, as well as to the hydrodynamic validity of
the model under certain conditions.

Perhaps most pronounced was the very strong role played by the near-
surface vertical salinity gradient in maintaining water-column stability.
A strong halocline near the surface tended to insulate deeper waters in the
lead from the thermodynamic activity occurring at the surface. Surface
waters whose densities had been increased by salt rejected by newly forming
ice, was not able to mix vertically until density instabilities occurred in
the surface region, thus inducing convective overturn. Since the presence
of a strong surface halocline would require higher surface-salinity accumula-
tions and hence permit more ice formation and associated salt rejection
before this surface layer would become unstable, a strong halocline would
be expected to inhibit convective overturn, both in frequency of occurrence
and depth of penetration in the lead. Added to this insulating effect is
the fact that, as surface salinities increase, surface freezing temperatures
correspondingly decrease such that freezing rates are proportionately lowered.

The role played by the vertical temperature gradient in lead thermodynamics
was strongly dependent upon the existing vertical salinity gradient. Since
density changes in sea water at temperatures near the freezing point are
almost totally a function of salinity changes, the insulating qualities of
the thermocline are undefinable when treated independently of the halocline.

However, in situations wherein weak vertical salinity gradients (poor in-
sulation) and strong vertical temperature gradients coexisted, convective
overturn resulted in relatively large quantities of heat being transported
to the surface. Freezing and associated salt rejection were thus inhibited
and surface melting was induced. In this manner, the temperature stratifi-
cation was superimposed on the primary control exerted by the vertical
salinity gradient.

The effect of horizontal current magnitude on lead stability appeared,
as in the case of the thermocline, to be secondary to the effect of the
halocline. Strong currents will advect surface salt accumulations out of
the lead relatively quickly, and will in general allow less time for lead
thermodynamics to affect a given parcel of water advectively transiting
the lead's width. This allows less change in the parcel's physical charac-
teristics. Through these effects, the current also acts as an insulating
agent.

Ice growth rates were clearly dependent upon combinations of the above
three variable parameters. Neglecting spacial ice profiles across the lead
temporarily and examining only the maximum ice thicknesses occurring on the
lead's surface as a function of time, the highest growth rates were exhibited
for cases where a strong current, a weak thermocline, and a weak halocline
occurred together. Conversely, lowest growth rates resulted from the co-
existence of a strong current, a strong thermocline, and a strong halocline.
Very little difference however, was noted in the high growth-rate case
when currents were weak instead of strong, nor in the low growth-rate case
when currents were weak or when currents were weak and the halocline was
weak. Analysis of these results showed that high growth rates were in all
cases associated with v/eak vertical temperature gradient: and low growth
rates with strong vertical temperature gradients. Further, that current

• 63

strength was relatively unimportant in changing growth rates when either
strong haloclines were present, or when weak temperature gradients together
with weak salinity gradients were present.

The effect of the near-surface vertical temperature gradient appears
then to dominate in determining surface freezing rates. Strong gradients
resulted in lower rates of ice formation than in the case of weak gradients.
Vertical transport of heat from depth in the lead by both eddy diffusion and
convective overturn is then modified by both current strength and the near-
surface halocline acting as insulating agents.

Ice thickness profiles across the lead's width varied considerably with
different combinations of the three basic variable parameters. Uniform or
nearly-uniform ice thicknesses across the lead were in all cases associated
with strong near-surface haloclines. Variations occurred where weak halo-
clines existed, growing larger when weak haloclines and strong thermoclines
occurred together. The greatest surface irregularity, both by amplitude of
variation as well as by spacial frequency of variation, was found when weak
currents, weak haloclines, and strong thermoclines occurred together.

Analysis of these cases indicated that the halocline was the dominant
variable in determining uniformity of ice thickness across the lead through
insulation of the surface from the heat at depth in the lead. When convective
overturn occurred as a result of increases in surface salinity, surface freezing
rates were altered at the point of convective overturn and produced surface
irregularities. Since rejected salt is being continuously advected down-
stream, there will exist horizontal salinity gradients which, when exceeding
the magnitudes of vertical salinity gradients, will induce convective overturn
and resultant surface melting at that point. Downstream of this point,

64

surface waters are diluted due to the upstream melting, thus experiencing
increased freezing rates since their freezing point has effectively been

raised.

The near surface thermocline appeared to affect surface-ice uniformity
through determining the amount of heat transported to the surface during
convective overturn, and thus the amount of melting and dilution occurring.
Finally, current strength appeared to act analogously to the halocline as an
insulating agent, though less effectively. By flushing surface-salt accumu-
lations out of the lead more rapidly, a strong current appears to inhibit
convective overturn and hence variations in ice uniformity. By decreasing
the amount of change a given parcel of water experiences over a given hori-
zontal distance in the lead, a strong current appears to produce less
intensive but more extensive lee effects from surface dynamics.

Surface heat-loss densities varied across the lead's surface in response
to ice thickness profiles. Since open-water sensible, latent, and radiative
heat-flux densities used in this work totalled 2.00 x 10" cal/cm2-sec as
compared to a corresponding value over the 200 cm of ice of 1 .15 x 10"
cal/cm2-sec, resultant heat-loss densities were considerably larger over
regions of the lead surface covered with relatively thinner ice. Uniformity
of surface heat-loss densities over the lead's surface may then be directly
related to the uniformity of the ice profile over the surface, while relative
magnitudes of heat-loss densities may be inversely related to relative ice

thicknesses.

The extreme case of lead instability and non-uniformity (weak current,
strong thermocline, and weak halocline) requires special comment. A high

mean surface heat-loss density characterized this case, which was expected as

a result of its highly non-uniform surface-ice profile. However, the lack

fiB

of evidence of these highly non-uniform ice covers occurring in nature in the
absence of mechanical deformation forces prompted further analysis. By
studying model output fields of runs 2, 6, 9, 10, and 14 (particularly the
horizontal density gradients generated in the lead), an upstream pressure
force was found to exist in the vicinity of the source boundary which was
capable of performing on the order of ten ergs of work against a unit mass
of water entering the lead at the source boundary. Since a thermodynamically
generated force of this type would completely alter the hydrodynamics
occurring in the lead for low current magnitudes, the model appears to be non-
representative of actual lead thermodynamics when using very low current
magnitudes with strong thermoclines and weak haloclines. Stronger currents,
on the other hand, had generally weaker horizontal density gradients associated
with them and were also proportionately less affected by these pressure forces
since their capacity to do work against the horizontal pressure force in-
creases as the square of their magnitude.

Results of run 14 proved to be of particular interest through the in-
sights they afforded to the process of convective overturn. Since no current
was present, there was no horizontal coupling between model grid points. As
a result, although freezing progressed uniformly over the surface of the lead,
strong differential horizontal density gradients developed along the 'source'
boundary as cooling progressed. Of prime interest however, was the graphic
illustration of the role played by the current as a convective-overturn
scaling factor. With zero current, a strong temperature gradient and a
weak salinity gradient, freezing-induced convective overturn penetrated to
a depth of 35 meters roughly 18 hours after lead generation. Without the
flushing action of a horizontal current or the insulating effect of a strong

66

halocline, the effects of surface thermodynamics were felt much deeper in
the lead. It appears probable then, that in cases wherein weak currents and
weak haloclines exist in open leads (marginal vertical stability), surface
freezing will induce, through convective overturn and vertical mixing, sub-
stantial vertical and horizontal gradients in both temperature and salinity.
As a result, significant fluxes of heat, salt, and momentum would be genera-
ted in the waters of the lead, and sophistication beyond that possessed by
the work at hand would be required to accurately represent the resultant

lead hydrodynamics.

Summarizing the analysis of results from integrating the open-lead model
over the experimental ranges of variables, the near-surface vertical salinity
gradient appears to exert the dominant force in maintaining vertical density
stability in the water column through its insulating effect. By inhibiting
convective overturn, a strong halocline was associated with horizontally-
uniform ice thickness profiles and consequently uniform surface heat-flux
densities. Current strength and the vertical temperature gradient appeared
to be secondary as insulating agents.

The near-surface temperature gradient appeared to be the dominant
variable in controlling the rate of ice formation since its strength
directly influences the amount of heat transported to the ice-water boun-
dary from depth by both eddy diffusion and convective overturn. In this
case, the halocline and the current strength appeared secondary in their
effects on the freezing rate.

Mean surface heat-loss densities were higher for more non-uniform
surface ice thicknesses, which in turn may be inversely related to the
strength of the near-surface halocline. This conclusion could prove to be

67

of significance to critical examinations of the heat budget over large
areas of the arctic in which significant regional variations in the near-
surface halocline might reasonably be expected.

Representative freezing rates, ice-thickness profiles, and surface
heat-loss densities found in integrating the model using real-case variables
agreed well with values in the literature. Due to the similarity of current
strengths, near-surface thermoclines, and near-surface haloclines, the fore-
going relations were generally not discernable, although the effect of down-
stream salt accumulation was noticed in slightly decreasing downstream ice
thicknesses.

Finally, it is concluded that, with the exception of cases in which
weak horizontal currents occur together with strong thermoclines and weak
haloclines, the model presented herein provides an accurate physical
representation of the thermodynamics occurring in an open lead in central
arctic sea ice over the experimental range of variables.

68

VI. RECOMMENDATIONS FOR FURTHER RESEARCH

Although the surface boundary of the model could be considerably more
sophisticated in its treatment of energy fluxes and transport processes,
models already exist which have this sophistication, notably that of Maykut
and Untersteiner [1969, 1971]. Coupling the model at hand to their model
would therefore seem a more reasonable approach to refinement of the surface
boundary in order to model air-ice-sea interactions and mass-energy balances.

The hydrodynamics occurring in a refreezing lead however, are still not
accurately modelled, nor are they fully understood despite the insights
afforded by the research performed. Extended development of this work
would promote further conceptual understanding of these processes as well
as provide a more physically-accurate model. Incorporation of accelerations
resulting from thermodynamically-generated horizontal pressure gradients,
as well as the coriolis acceleration, would considerably refine the model.
Additional refinement would be gained by removing previously discussed
limitations on: 1) the minimum vertical salinity gradient able to be
accurately treated, and 2) the preclusion of downstream advection of warmer-
then-freezing-point waters under an ice cover.

In summary then, further research could profitably be applied to the
work at hand to further refine the hydrodynamic accuracy of the model, as
well as to couple it to larger-scale models of air-ice-sea interactions and
mass-energy balance predictive schemes.

£Q

APPENDIX A
COMPUTER PROGRAM DESCRIPTION

The computer program is written in FORTRAN IV and was used with the
IBM 360/67 computer system at the W.R. Church Computer Center, Naval
Postgraduate School. The program has been generalized such that sea
water temperature and salinity profiles used as initial and left boundary
fields are treated as input data. A design feature of the program allows
the use of an internally variable time step to facilitate adjusting the
model's temporal resolution. Additionally, the model allows variable grid
spacing as well as variable lead dimensions, requiring changes only in the
DIMENSION, COMMON and FORMAT statements. Although all currents used in
the model were constant in time and space, the model allows use of both a
depth-dependent and time-dependent horizontal current, again internally
variable. With only minor modifications, the program could be easily ex-
tended to include a bottom-boundary heat and salt flux, more sophisticated
treatment of air-sea interactions at the surface boundary, and feedback of
accelerations, resulting from horizontal density gradients generated by lead
thermodynamics, to the motion of water through the lead.

The program is divided into two basic sections: the main program and
subroutine OVRTRN. The main program is further subdivided into seven parts:
problem initialization, data input, computation of time-dependent temperature
and salinity fields, ice-accretion/melting and salt rejection/dilution, with
vertical stability check and computation of resultant heat flux densities,
filling temperature and salinity arrays with resultant values, computation
of applied surface heat flux densities and conversion to an integrated surface

70

temperature gradient, and writing output. Subroutine OVRTRN accomplishes
vertical convective overturn and mixing of unstable regions in the water

column.

FORTRAN IV symbols for primary program parameters are defined below:
AH Downstream-advected heat-flux density (cal/cm -sec)

AHF Applied surface heat-flux density (cal/cm -sec)

BTC Sea ice thermal conductivity constant (cal-cm /g-sec)

2
CAH Cumulative downstream-advected heat density (cal/cm )

CAHF Cumulative applied surface heat density (cal/cm )

CHC Cumulative subsurface cooling heat density (cal/cm )

CHF Cumulative applied surface heat density less cumulative

downstream-advected heat density (cal/cm2)

CHIF Cumulative heat density of surface-ice formation (cal/cm )

CI Thermal conductivity of sea ice (cal/cm-sec-°C)

CL Sea-water chlorinity (g/kg)

CO Thermal conductivity of pure ice (cal/cm-sec-°C)

CP Specific heat of sea water (cal/g-°C)

DI Rate of surface ice formation (cm3/cm2-sec)

DSALT Salt rejected per cm3 of ice formed (g/kg)

DTAU Time step (sec)

DTEMP Air-water temperature difference (°C)
DX Horizontal grid spacing (cm)

DZ Vertical grid spacing (cm)

EPS Allowable rate-of-freezing error (cnrVcm -sec)
ERR Allowable ice thickness error (cm /enr)
GRADI Vertical ice temperature gradient (°C/cm)

71

H Model depth extent (cm)

2

HC

Subsurface cooling heat-flux density (cal/cm -sec)
HF Applied surface heat-flux density less downstream-advected

heat flux density (cal/cm -sec)
HIF Heat-flux density of surface-ice formation (cal/cm -sec)
HO Heat of formation of pure ice (cal/g)

HS Heat of formation of sea ice (cal/g)

MM Number of grid columns
MV1 Number of grid rows
MVH1 Total number of grid points
QI Heat-flux density through sea ice to the atmosphere

(cal/cnr-sec)
QL Open-water latent heat-flux density to the atmosphere

(cal/cnr-sec)

QR Open-water radiative heat-flux density to the atmosphere

(cal/cnr-sec)
QS Open-water sensible heat-flux density to the atmosphere

(cal/cm -sec)

RHO Density of sea water (g/cm )

RHOI Density of sea ice (g/cm3)

S Sea-water salinity (g/kg)

SALI Sea-ice salinity (g/kg)

SI Sea-ice salinity (g/cm3)

SIGT Sigma-t

SSCHG Salinity change in surface layer of depth DZ (g/kg)

72

SSFLUX Salinity change in one cm3 of sea water at the ice-water

boundary (g/kg)
STGD Total applied surface temperature gradient at the water's

surface (°C/cm)
T Sea-water temperature (°C)

Tau Time (sec)

TF Sea-water freezing temperature (°C)

TGRADL Surface temperature gradient due to latent heat-flux

density (°C/cm)
TGRADR Surface temperature gradient due to radiative heat- flux

density (°C/cm)
TGRADS Surface temperature gradient due to sensible heat-flux

density (oc/cm)
TI Temperature of lower ice surface (°C)

U Horizontal current magnitude (cm/sec)

VEC Vertical eddy conductivity (cal/cm-sec-°C)

VSK Vertical salt eddy diffusivity (cnf/sec)

VTK Vertical temperature eddy diffusivity (cm /sec)

W Lead width (cm)

Z Depth (cm)

11 Surface ice thickness (cm3/cm2)

A descriptive flow diagram of the program is shown in Figure 16.

73

Figure 16. Descriptive Flow Diagram of Program.

Problem Initialization:
DIMENSION/COMMON
Function statements,
Constants

Input of Initial Conditions:
Temperature TNEW,
Salinity SNEW,
Sigma-t SIGT,
Equilibrium Ice Thickness ZI

Computation of Time-Dependent
Temperature and Salinity Fields
TNEW, SNEW

Computation of Surface Temperature Chances
THF

(202)— — — —

74

Figure 16 (continued)

202

"1

Computation of:
Surface freezing temperatures TF,
Surface temperature changes due to ice
formation TIF,

Iterative rate of ice formation HICE,
Iterative volume of ice formed ZICE,
Iterative salt rejected SSCHG,
Resultant surface temperature and salinity
fields (if melting occurred as a result of
convective overturn, the resultant dilution
is distributed over the mixed region of the
water column) TNEW, SNEW

Computation of Surface Ice Thicknesses ZI
(iteration to account for freezing-point
changes as a result of salinity changes)

1

Vertical Stability Check

OVRTRN

Check for Surface Heat Brought
From Depth Which Would Promote
Melting

J

Computation of:

Rate of ice formation,
Water-column heat-flux densities,
Cumulative water column heat densities

7k

1

Fill Temperature and Salinity Arrays
T
S

Confutation of Surface Heat-Flux Densities and
Conversion to an Integrated Surface Temperature
Gradient

'-i+inn H,

©^

76

CENTRAL ARCTIC OPEN-LEAD MODEL* ,„ ,

DIMENSION T(232) ,S(232) ,PUF<22) fAH(22) ,CAH(22) ,
1H F(22,CH H22 DK22I Z I < 22 ) , TF < 22 ) , HF < 22 ) , CHF( 22 ) ,
2STGD(22),HC(22),CHC(22>,AHF(22),CAHF(22),HUF(22)
CCMMON TMEW(232) ,SNEW(232) , SIGT<232)

AT1(bI=1.-DTAU*(U(B)/DX+2.*VTK/DZ**2)
AS1(B)=1.-DTAU*(U(B)/DX+2.*VSK/DZ**2)

AT2(B)=OTAU*U( B)/DX
AS2(5)=OTAU*U( e )/DX
AT3(D)=D*VTK/DZ**2

AS3(D)=D*VSK/DZ^"2

SOu7 = -0^5&9 + lo4708*A-O.OCa57G*A**2+OcG000398*A**3
AT (A =A*(4o78O7-G.G98135*A+Jc0C10843*A**2)*9o001
BT(A)=A*( 18.U3O0. 8164*A+0.01667*A**2)*J. OOOOOl
ST(A)=-( A-3o^8 )**2/5C3.570*( A-»-283.)/(A + 67e26)
SGTU)=A+(SIG0+O.1324)*(1.0-ATMP+(i}TMP*<SIG0-0.1324) ) )
SH( A )= 1.005-0. 0041 3 &*A+C. 000 l098*A**2-0. 000001324* A**3

DE(A)=1.0+< A/1000. )

F< A )=0.G-O.G966*A+0. 0000052* A**3)

R(C)=DSALT*C

ERR(D)=D*EPS

BTC=0.28

C0-0.0C486

DTAU=9C0

DTEMP=25.

DX=1000.0

DZ=5C0.0

EPS=1.0E-8

H=5C"0.0

H0=79.77

QL=3oOE-3

QR=2.0E-3

QS = 1.5E-2

RH0I=0.91

A L 1 — o . v
SI=RHOI*SALI/1000.0
VSK=10oO
VTK=10.0
W= 20000.0
MH=IFIX(W/OX)
MV=IFI X(H/DZ)
MV1=MV+1
KF1=MH+1
MVH1=MV1*MH1

1 READ( 5t 100JIRUN

IF( IRUN.EQ.99)G0 TO 1000

READ (5, 15';) (TNEWd ) ,I = 1,MV1)

R E AD ( 5 , 1 50 ) ( SN E vi ( I ) » I = 1 » M V 1 )

CL=C(SNEW(1 ) )

TF( 1)=F(CL)

TNEW(1)=TF(1)

M=MV1

DO 3 J=2,MH1

DO 2 K=ltMVl

I=M+K

TNEW( I )=TNEW(K)

T( I )=TMEW(K)

SNEWd )=SNEW(K)

S( I )=SNEW(K)

2 CONTINUE
M=M+MV1

3 CONTINUE

DO 4 1=1, MV1
CL=C(SNEW( I) )
SIGO = Sr (CD
ATMP=AT(TNEW( I ) )
BTMP=BT (TNEW( I ) )
SMT=ST(TNEW( I) )
SIGT( I l=SGT(SMTJ
T( I)=TNEW( I)

77

S( I)=SNEW( I)

4 CONTINUE

DO 5 I=12MH1
AH< I )=O.C
AHF(I)=0«0
CAH( I)=0.0
CAHF( I )=C„0
CHC(I)=0.0
CHF( I)=0.0
CHIF( I )=2.0
DI (I )=uoO
HC( I)=0.0
HF( I)=OoO
HIFU )=0o0
ZI( I )=0.0

5 CONTINUE
J=MHl-(MH/4)
DO 6 I=J,MH1
ZI(I)=20CoO

6 CONTINUE
TAU=O*0
GO TO 201

101 TAU=TAU+DTAU
DO 10 2 I=2»MH1
AHF( I)=HUF( I )

CAHF( I)=CAHF( I ) + PUF( I)

102 CONTINUE
J=MV1+1

DO 103 1=2, MH

Z=O.C

Jl= J— MV1

TNEW( J)=AT1(Z)*TU ) +AT2C Z ) *T( Jl )
1+AT3(DTAU)*2.*(T( J+1)-DZ*STGD( I ) J

SNEW(J)=AS1(Z)*S(J)+AS2(Z)*S(J1)
l+AS3(DTAU)*2o*S(J+l)

Z = H

O — U T|-| V

J1=J-MV1
TNEW(J)=AT1(Z)*T(J)+AT2(Z)*TU1)

1+AT3(DTAU)*2.*T( J- 1 )

SNEW( J)=AS1(Z)*S(J )+AS2(Z)*S(Jl)
l+AS3(DTAU)*2o*S( J-l >

J = J + 1

103 CONTINUE
Z=0.0
K=MVH1-MV

TNEw'(K) = ATl<Z)*T(K)-i-AT2(Z)*T(J} + AT3(DTAU}*2.*

1SNEW(K)=ASl(Z?*s'(K)+AS2(Z)^S(J)+AS3(DTAU)*2,*S(K+l)

Z=H
K=MVH1

J— |/_ V)W 1

TNEW'(K)=AT1(Z)*T(K)+AT2(Z)*T(J)+AT3(DTAU)*2«*T(K-1 j
SNEW(K)=ASl(Z)*S(K)+AS2(Z)*S(J)+AS3(DTAU)*2e*S(K-l)

DO 105 J=2,MH
Z = DZ

Kl=( ( J-l )*MVl)+2
K2=( J*MV1)-1

DO 104 K=K1,K2

TNEw7k)=AT1(Z)*T(K)+AT2< Z)*T (Jl )+AT3(DTAU)*

1SNFwTk)=ASi!z)-S(K)+AS2(Z)*S(J1)+AS3(DTAU)*
1(S(K-1)+S(K+1) )
Z=Z+DZ

104 CONTINUE

105 CONTINUE
K1=1+(MVH1-MV)
K2=MVH1-1

Z = DZ

DO 106 K=K1,K2

7R

JrrK-MV1

TNEW(Ki = ATKZ)*T(K) + AT2(Z)*T(J) + AT3(DTAU)*

1SNEw7k)=ASi!z)*S(K)+AS2(Z)*S(J)+AS3(DTAU)^
1(S(K-1 )+S(K+l> )

Z=Z+DZ
1C6 CONTINUE
2C1 J=l

DO 213 I=2,MH1

J=J+MV1

J3=J+MV

MELT=G

HIF(I)=OoC , i%1

THF=DZ/2o*(T(J )-TNEW(J) )
2C2 CL=C(SNEW( J) )

TF( I)=F(CL)

TTEKP=TNEW(J)

IF(TTEMPoLEoTF(I ))TTEMP=TF(I )

I^TTD^l:Si:^i!Tl^:fM«I..Lt.ERR!DTAU.».T!F=C,0

CP=SH(SNEM( J) )
SIGOSC-(CL)
ATMP=AT(TTEMP)
BTMP=BT(TTEMP)

SMT=ST(TTEMP)
SIGT( J)=SGT(SMT)
RHO=DE( SIGT( J) )
ZKW=RHO*CP/DTAU
SRATIO=SALI/SNEW(J)
HS=HC*( l.O-SRATIO)
HICE=(TIF*ZKW)/(RHOI*HS)
GO TO 204

203 HICE=-(ZI( I )/DTAU)
TIF*=HICE*RhOI*HS/ZKW

204 Gl=-(DTAUvHlCEJ
IF(Gl.GT.ZKI) >G0 TO 203
DSALT-RHOI*(SNEW(J 5-SALI )
ZICE=DTAU*HICE
SSFLUX=R(ZICE)
SSCHG=SSFLUX/DZ
IF(MELT.E0«C)G0 TO 2 06
SDIST=SSCHG/JSA

Kl=J+( JSA-1)

DO 205 K=JtKl

SNEW(K)=SNEW(K)+SDIST

205 CONTINUE
GO TO 207

206 SNEW( J)=SNEW( J J+SSCHG

2C7 TNE W ( J ) = TNE W( J ) + ( T I F*2 . /DZ )

ZI( I)=ZI (I )+DTAU*HICE , ,.,,,,,. , r

IF(ZI( I ) el_ToERR( DTAO.AND. ( HICE.LE.O.o ) ) Z H I )=>./* 0

HIF( IJ=HIF( I )+TIF*ZKW

IF(ABS(HICE)oGTcEPS)GO TO 2-J2

CL=C(SNEW( J) )

TF( 1)=F(CL)

DO 208 K=J,J3

CL=C(SNEW(K) )

SIGC=SO(CL)

ATMP=AT(TNEW(K) )

BTMP=BT(TNEW(K) )

SMT=ST(TNEW(K) )

SIGT(K)=SGT(SMT)

208 CONTINUE
IF(MELT,EQol)GO TO 212
J2=J3+1

209 DO 210 K=1,MV

IF(SIGT( J2-K).GE.SIGT( J3-K) ) GO TO 210

L=J2-K

CALL OVRTRN (J,L)

GO TO 209

210 CONTINUE
CL=C(SNEW( J) )

79

TF( I)=F(CL)

IF(TNEW(J),LE.TF( I) ) GO TO 212

IF(ZI ( I ).L6oERR(DTAU) ) GO TO 212

MELT=1

DO 211 K=1,MV

IF(SNEW( J+K).EQoSNEW(J) ) GO TO 211

JSA=K

GO TO 202

211 CONTINUE
JSA=MV1
GO TO 202

212 OK I)=HIF( I )/( RHOI*HS)
CHIFU ) = CH1F( I)+HIF( I)*DTAU
HF( I)=THF*ZKW

CHF( I)=CHF( I) + HF(I )*DTAU

AH( I)=AHF( l)-HF{ I)

CAH( I)=CAH( I )+DTAU*AH( I )

HC( I)=HF< I)-HIF( I )

CHC( I)=CHF( I)-CHIF( I)

213 CONTINUE
I1=MV1+1

DO 301 I=I1,MVH1
T(I )=TNEW( I )
S( I) = SNEW( I )
301 CONTINUE
J=MV1+1

DO 403 I=2,MH1
RHO=OE(SIGT( J) )
CP = SH(S< J) )
VEC=VTK*RHO*CP

IF(ZI( I ).LE0ERR(DTAU) ) GO TO 401
TI=TF(I )
CI=CO+BTC*SI/TI
GRADI=DTEMP/ZI ( I )
QI=CI*GRADI
TGRADS=QI/VEC

TGRADR=0.35*QR/VEC
TGOW=QS/VEC

IF(TGRADSoGT.TGOW)GO TO 401
GO TO 402

401 TGRADS=QS/VEC
TGRADL=QL*EXP(-0.5*ZI( I) )/VEC
TGRA0R=(0. 35*QR+0a65*QR*EXP(-C

402 STGD( I )=TGRADS+TGRADL+TGRADR
HUF( I )=STGD( I )*VEC
PUF( I )=HUF( I )*DTAU
J=J+MV1

403 CONTINUE
IF(AM0D(TAU,36'JJo )
GO TO 101

501 Jl=l
TIME=TAU/3600a
J2=MVH1-MV

WRITE(6,200)TIME,IRUN
WRITE ( 6, 25C ) (TF(K) ,K= It MH1 )
DO 502 1=1, ^Vl
WRITF.< 6,300)(T( J) ,
WRITE(6,35, ) (S( J),
WRITE (6, 400) (SIGT(
J1=J1+1
J2=J2+1

502 CONTINUE
WRITE (6,450)
DO 50 3 I=2,«H1
WRITE(6,5CMHIF(I)

1HF(I) ,CHF( I),AH( I)

503 CONTINUE
IF(TIME.GEo
GO TO 101

100 F0RMATU2)
150 FORMAT ( 11F7

5*ZI(I) ) )/VEC

LT. l.E-4) GO TO 501

J=J1 ,J2,MV1)
J=J1, J2,MV1)
J ), J=J1, J2,MV1)

,CHIF(I) ,DI (I ),ZI(I) ,HC( I),CHC(I),
,CAH(I) ,AHF( I) ,CAHF( I)

48, )G0 TO 1

2)

80

200 FORMAT ( US 'PROFILE ( X-Z PLANE) OF LEAD AT TIME=f,F5.1

1 , 10X, 'RUN NUMBER' ,13)
2 50 FORMAT ( ■ 0« , ' TF ' , 3X , 21 ( 1 X ,F5. 2 ) )
300 FORMAT ( «C» ,'T' ,4X,21( 1X.F5.2) )
350 FORMAT (• • , • S ■ ,4X, 2 1 ( i X , F5. 2 ) Y
400 FORMAT (• ' , • ST ' , 3X , 21 ( 1 X, F 5. 2 ) )
450 FORMAT (//, ' G ' , 3X , » H I F • , 7 X, » CHI F ■ , 6X, • DI « , SX , « Z I ■ , 8X ,

l'HC ,8X, 'CMC , 7X,«HF' , 8X,*CHF« ,7X,»AH' ,8X, 'CAH' ,7X,

2«AHF« ,7X, 'CAHF', /)
500 FORMAT (• « , 1 PE9. 2, 1 1 ( 1 X , E9.2 ) )
10C0 STOP
END

SUBROUTINE CVRTRN (I,K)

CCMMON TNEW (232 ) , SNElH 232) ,SIGT( 232)

C(A) = ( A-0.030)/1.8( 5(x

S0( A )=-C .069+1. 47Q8*A-0«0C'l 570* A**2+0o0GC 0398* A**3

AT ( A ) = A* ( 4 o 786 l-r ■ . 098 1 8 5*A + 0e 00 1 Q843*A** 2 ) *C . 001

BT<A)=A*(18o03r-.?o8164*A+0.01667*A*#2)*Cc0CC0Gl

ST(A)=-( A-3«98)**2/503.570*( A+283. ) /(A+67.26)

SGT(A)=A+(SIGO+G.1324)*(1.0-ATMP+(BTMP*( STGO-G. 1324) ) )

K1=K-1

J$=l

L=0

DO 1 M=I ,K1

L = L + 1

IF(SIGT(K-L)oGE.SIGT(K))GO TO 1

L1=1+(K-L)

J$=2

GO TO 2

1

CONTINUE

2

CTEMP=C.O

CSALT=O.C

IFUS.EQ.l )L1=I

DO 3 N= 1. 1 , K

CTEMP=CTEMP+TNEW(N5

CSALT=CSALT + SNEVv'(N)

CONTINUE

NN=1+(K-L1 )

ATEMP=CTcMP/FLOAT( NN)

ASALT=CSALT/FLOAT(NN)

CL=G( ASALT)

SIGO=SO(CL )

ATMP=AT(ATFMP)

BTMP=BT(ATEMP)

SMT=ST(ATEMP)

ASIGT=SGT(SMT)

DO 4 M=L1,K

TNEW(M)=ATEMP

SNEW(M)=ASALT

SIGT(M)=ASIGT

CONTINUE

RETURN

END

ftl

BIBLIOGRAPHY

Arctic Meteorology Research Group, Department of Meteorology, McGill
University, Montreal, Publication in Meteorology No. 70, Scientific
Report No. 12, Heat Flux Through the Polar Ocean Ice, by E. Vowinckel ,
15 pp., September 1964.

Badgley, F.I., "Heat Balance at the Surface of the Arctic Ocean,"
Proc. Western Snow Conference, Spokane, Washington, 101-104, 1961.

Badgley, F.I. 9 "Heat Budget at the Surface of the Arctic Ocean,"
Proc S.ymp. Arctic Heat Budget and Atmospheric Circulation,
The Rand Corporation, RM-5233-NSF, 257-277, 1966.

Baffin Bay-North Water Project Scientific Report No. 1, The Physical
Oceanography of the Northern Baffin Bay Region, by R.D. Muench,
150 pp., The Arctic Institute of North America, January 1971.

Bilello, M. , "Formation, Growth and Decay of Sea Ice in the Canadian
Arctic Archipelago," Arctic, v. 14, 3-24, March 1961.

Bryan, K. , "Climate and the Ocean Circulation, Part III: The Ocean Model,"
Monthly Weather Review, v. 97, 806-827, November 1969.

Coachman, L.K., "Production of Supercooled Water During Sea Ice Formation,"
Proc Symp. Arctic Heat Budget and Atmospheric Circulation, The Rand
CorpoFation, RM-5233-NSF, 497-529, 1966.

Coachman, L.K. and Barnes, C.A. , "The Contribution of Bering Sea Water
to the Arctic Ocean," Arctic, v. 14, 147-161, September 1961.

Coachman, L.K. and Barnes, C.A., "Surface Water in the Eurasian Basin
of the Arctic Ocean," Arctic, v. 15, 251-277, December 1962.

Doronin Y.P., "Characteristics of the Heat Exchange," Proc. Symp.
Arctic Heat Budget and Atmospheric Circulation, The Rand Corporation,
RM-5233-NSF, 247-266, 1966.

Fletcher, J.O., "The Arctic Heat Budget and Atmospheric Circulation,"
Proc. Symp. Arctic Heat Budget and Atmospheric Circulation, The Rand
Corporation, RM-5233-NSF, 23-43, 1966.

Foster, T.D., "Haline Convection Induced by the Freezing of Sea Water,"
Jour. Geophys. Res., v. 73, 1933-1938, March 1968.

Foster, T.D., "Experiments on Haline Convection Induced by the Freezing
of Sea Water," Jour. Geophys. Res., v. 74, 6967-6974, December 1969.

Knudsen, M. , Hydrographical Tables , p. I-V, reprinted by Tutein and
Koch, Copenhagen 1959, lyuz.

Lake, R.A. and Lewis, E.L., "Salt Rejection by Sea Ice During Growth,"
Jour. Geophys. Res., v. 75, 583-597, January 1970.

Lewis E L and Walker, E.R., "The Water Structure Under a Growing Sea
Ice'sheet," Jour. Geophys. Res., v. 75, 6836-6845, November 1970.

Manabe, S. , "Climate and the Ocean Circulation, Part I: The Atmospheric
Circulation and the Hydrology of the Earth's Surface; Part II: The
Atmospheric Circulation and the Effect of Heat Transfer by Ocean
Currents," Monthly Weather Review, v. 97, 739-805, November 1969.

Maykut, G.A. and Untersteiner, N. , Numerical Prediction of the Thermo-
dynamic Response of Arctic Sea Ice to hrivnronm2ntal Changes, 173 pp.,
The Rand Corporation, RM-6093-PR, November 1969.

Maykut, G.A. and Untersteiner, N. , "Some Results from a Time-Dependent
Thermodynamic Model of Sea Ice," Jour. Geophys. Res., v. 76, 1550-1 57j,
February 1971 .

Pounder, E.R., The Physics of Ice, pp. 125-127, Pergamon, 1965.

Smith, G.D., Numerical Solution of Partial Differential Equations, pp.
70-71, Oxford University, 1965.

c+orn M r ?^<-\ Tnvpor J.S.. "Salt Finoers and Convecting Layers, weep
Sea Research, v. 15, 497-511, 1969.

Sverdrup, H.U. , Johnson, M.W. and Fleming, R.H., The Oceans, p. 61, 66,
Prentice-Hall, 1942.

Untersteiner, N. , "Calculating Thermal Regime and Mass Budget of Sea Ice,"
Proc. Svmp_^_Arctic Heat Budget and Atmospheric Circulation, The Rand
Corporation, RM-5233-NSF, 203-213, 1966.

U S. Naval Civil Engineering Laboratory Technical Report R-497, Ice
Engineering - Analysis of the Growth of Sea Ice, by A.J. Mettler and
N.S. Stehle, 46 pp., November 1966.

83

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 2
Cameron Station

Alexandria, Virginia 22314

2. Dr. Ned A. Ostenso 1
Office of Naval Research, Code 480D

Arlington, Virginia 22217

3. U.S. Naval Arctic Research Laboratory 1
Attention: Librarian

Barrow, Alaska 99723

4. The Arctic Institute of North America 1
1619 New Hampshire Avenue, NW

Washington, D.C. 20009

5. University of Michigan 1
Attention: Library

Ann Arbor, Michigan 48108

6. Library, Code 0212 2
Naval Postoraduate School

nun lci cy , ua i i i ui n i d jj^u

7. Department of Oceanography 3
Naval Postgraduate School

Monterey, California 93940

8. Asst Professor J. A. Gait, Code 58G1 8
Department of Oceanography

Naval Postgraduate School
Monterey, California 93940

9. LCDR Richard H. Schaus, USN 2
P.O. Box 1127

Carmel , California 93921

84

Security Classification

DOCUMENT CONTROL DATA -R&D

[Security classification ol title, body ol abstract and indexing annotation must be entered when the overall report is classified)

1 originating activity (Corporate author)

Naval Postgraduate School
Monterey, California 93940

Za. REPORT SECURITY CLASSIFICATION

Unclassified

2b. GROUP

3 REPORT TITLE

A Thermodynamic Model of a Central Arctic Open Lead

4 DESCRIPTIVE NOTES (Type ol report and, inclusive dates)

Master's Thesis: September 1971

5 Au THORISI (First name, middle initial, laat name)

Richard Harris Schaus

6. REPOR T D A TE

September 1971

7«. TOTAL NO. OF PAGES

86

7b. NO. OF REFS

25

»a. CONTRACT OR GRANT NO

b. PROJEC T NO.

9a. ORIGINATOR'S REPORT NUMBER(S)

Ob. OTHER REPORT NO(S) (Any other numbers that may be assigned
this report)

10 DISTRIBUTION STATEMENT

This document has been approved for public release and sale;
its distribution is unlimited.

11. SUPPLEMENTARY NOTES

12. SPONSO RING Ml L I T AR Y ACTIVITY

Naval Postgraduate School
Monterey, California 93940

13. ABSTRACT

A time-dependent, two-dimensional thermodynamic model of an open lead in
central arctic sea ice is presented. The model is generated by opening a lead of
finite width and infinite extent in the equilibrium sea-ice cover. From this
initial condition, the model is integrated numerically as a sea-ice cover is
reestablished over the lead. The effects of various representative advective
parameterizations, and temperature and salinity profiles in the ocean's surface
layer, on the heat flux through the lead and on the nature of ice formation are
investigated as the lead closes by thermal processes alone. Continuity equations
involving a horizontal advection term and a vertical diffusion term govern heat
and salt transport in the water. Cooling-induced convective overturn as a mechanism
for vertical heat and salt transport in the water column is treated through an
artifice of the vertical diffusion term.

DD,fr»"..1473

S/N 0101 -807-681 1

(PAGE 1)

Secur- . Classification

31408

Security Classification

KEV WOROS

Arctic

Sea Ice

Heat Budget

Air-Sea Budget

Numerical Model

Lead

Polynya

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3 2768 002 00335 2

.DUDLEY KNOX LIBRARY