Table 1: List of commonly used variables for sea-ice studies.

Symbol Descriptionc sea-ice concentrationf thermodynamic growth rateg probability density function of ice thicknessh thicknessi unit vector in x directionj unit vector in y directionk unit vector in z directionk thermal conductivitym massq specific humidityt timeu velocity in x directionv velocity in y directionz depthA areaCd drag coe!cientCp specific heat capacityH sea-surface height fieldI penetrating solar radiationLf latent heat of fusionQ energy fluxS salinityT temperatureF force vectorV velocity vector! albedo/surface reflectance" emissivity# strain rate$ density! stress tensor" traction vector% boundary layer turning angle" degree day# mechanical redistribution function

1

Table 2: List of commonly used subscripts for sea-ice variables (Briegleb et al.2004)

Symbol Description0 surface value

1...n category/level indicatora atmospheref freezingi sea icen last value in an arrayo pure ices snoww sea waterz depth

2

Table 3: List of commonly used physical constants for sea-ice studies (Briegleb etal. 2004)

Symbol Description Valueag gravitational acceleration 9.80616 m s!1

ko thermal conductivity of pure ice 2.034 W m!1 K!1

ks thermal conductivity of snow 0.31 W m!1 K!1

Cpo specific heat of pure ice 2054 J kg!1 K!1

Cps specific heat of snow 0 J kg!1 K!1

Cpw specific heat of sea water 4218 J kg!1 K!1

Lfi latent heat of fusion of ice 3.337 × 105 J kg!1

Si sea ice reference salinity 4 pptSo solar constant 1373 W m!2

Sw ocean reference salinity 34.7 pptTff w freezing temperature of fresh water 273.15 KTfsw freezing temperature of sea water 271.35 K!w ocean albedo 0.06&i thermal conductivity constant 0.13 W m!1 K!1

"i ice emissivity 0.95"s snow emissivity 0.95$i density of sea ice 917 kg m!3

$s density of snow 330 kg m!3

$w density of sea water 1026 kg m!3

' Stefan-Boltzmann constant 5.67 × 10!8 W m!2 K!4

$ angular speed of the earth 7.3 × 10!5 rad s!1

3

Chapter 1

INTRODUCTION

Sea ice covers approximately 7 % of the earths oceans at any time (Wadhams

2000), greatly a%ecting the exchange of energy, momentum, and mass between the

ocean and the atmosphere. Heat transfer within the sea-ice zone is primarily an

open water phenomenon and the distributions of thin ice, leads, and polynyas have

important implications for the regional energy balance (Maykut 1978, King and

Turner 1997). Although sea ice greatly a%ects the surface energy balance within the

polar regions, it remains poorly represented in global circulation and climate system

models (Hanna 1996, Allison 1997, Drinkwater and Liu 2000, Holland et al. 2001)

and no comprehensive climatology of the Antarctic sea-ice thickness distribution

exists to evaluate model output (Worby and Ackley 2000, Geiger et al. 2000).

Currently the best source of information on the distribution of Antarctic

sea-ice thickness for use in model evaluation studies is an extensive record of in

situ sea-ice observations. In situ sea-ice observations provide high-quality, detailed

information on sea-ice conditions but, these data are limited seasonally and geo-

graphically and do not provide the continuous, circumpolar information on sea-ice

conditions needed to evaluate a simulated sea-ice thickness distribution (Worby

1999, Timmermann et al. 2004). Recent studies suggested that thickness estimates

derived from remotely-sensed imagery have the potential to provide continental-

scale thickness information that could be used to validate model output (Geiger et

al. 2000, Schellenberg et al. 2002, DeLiberty and Geiger 2005). The purpose of

4

this study was to evaluate the utility of remotely-sensed Antarctic sea-ice thickness

estimates for use in model evaluation and to compare these observations with the

Antarctic ice-thickness distribution produced by a stand-alone sea-ice model.

1.1 Antarctic Sea-Ice Environment

Antarctic sea ice is highly variable, with large seasonal and complex inter-

annual variations in extent, concentration, thickness, and volume (Hanna 1996).

Antarctic sea ice has a maximum extent of approximately 19× 106 km2 in Septem-

ber with a minimum extent of about 4×106 km2 in February (Allison 1997, Wadhams

2000). Comprised of mostly thin, first-year ice, Antarctic sea ice extends from about

60 − 80" S latitude, except within the Weddell Sea sector where it may extend fur-

ther northward (King and Turner 1997, Wadhams 2000). At its maximum extent,

Antarctic sea ice covers 20 % more area than its Arctic counterpart and has a much

larger seasonal variation, approximately 80 % or 16 × 106 km2, than that observed

in the Arctic (King and Turner 1997).

The Antarctic sea-ice zone is uninhibited by land to the north, and divergent

processes within the ice zone result in ice drift away from the continent, leaving

behind areas of open water (Nihashi and Ohshima 2001). Approximately 21 %

(4.0× 106 km2) of the sea ice zone is open water during the season of maximum sea

ice extent and about 37 % (1.5×106 km2) when Antarctic sea ice is at its minimum

extent (Allison 1997, Wadhams 2000). The exchange of energy over areas of open

water and thin ice is almost equal to that of the open ocean, ranging between 100-

1000 Wm!2, and may be reduced by over 95 % in areas covered by > 1 m (Hanna

1996). An ice-free ocean has an albedo below 10 % while the albedo of sea ice varies

from about 50 % to 90 % depending on the age of the ice, the presence of snow, and

sun angle (Hanna 1996, Wadhams 2000).

5

1.2 Importance of Sea Ice in the Climate System

Sea ice is an important feature of the global climate system because it is

sensitive to and also indicative of changes in climate, playing a complex role in

global climatic and oceanographic processes (Remund et al. 2000, Worby and Ackley

2000, Geiger et al. 2000). Arctic sea-ice extent has decreased (Parkinson et al.

1999), which has been partially attributed to greenhouse warming (Vinnikov et al.

1999). Many simulations indicated that Antarctic sea-ice cover should also decrease

in response to greenhouse warming (Gordon and OFerrell 1997, Wu et al. 1999).

However, the negative trend in Northern Hemispheric ice cover is in contrast with

the slight positive trend (1.0± 0.4% per decade) in Southern Hemispheric ice cover

(Zwally et al. 2002, Parkinson 2002). The significant decrease in Arctic sea-ice cover

and the slight increase in Antarctic sea-ice cover coincide with an approximately 0.2

K per decade increase in average global temperature (Hansen et al. 1999).

The degree to which sea ice a%ects global atmospheric and oceanic processes

is primarily a function of its thickness (Timmermann et al. 2004). Due to the

insulating e%ect of sea ice on the polar surface energy balance, it a%ects the exchange

of energy between the ocean and the atmosphere (Simmonds and Jacka 1995, King

and Turner 1997, Wadhams 2000, Timmermann et al. 2004). It is also a major

component in a number of climate feedbacks, some of which could strengthen in

response to an increase in global temperature (Ebert and Curry 1993, Watkins and

Simmonds 2000).

Sea ice is also a driving force in global thermohaline circulation, maintaining

the vertical structure of the ocean and therefore global oceanic circulation. The

main processes involve brine rejection from forming sea ice, which creates unstable

heavy water at the surface, initiating vertical convection. This is believed to be the

primary form of Antarctic bottom water formation (Wahdams, 2000).

Although much of the work on sea ice has focused on the Arctic, Antarctic sea

6

ice plays a complex role in regional and global climatic and oceanographic processes

as noted above. A decrease in Antarctic sea-ice concentration in response to an

increase in global temperature could a%ect the formation and movement of mesoscale

and synoptic-scale cyclones throughout the Southern Ocean (Simmonds and Wu

1993) and midlatitudes of the Southern Hemisphere (Simmonds and Jacka 1995).

Simmonds and Wu (1993) suggested that a decrease in sea-ice concentration will

result in an increase in cyclogenesis at about 61 "S, however, this may only have local

significance (Hanna 1996). A decrease in sea-ice concentration could also inhibit the

northward movement of cold air leading to a decrease in the equatorward transport

of sensible and latent heat. The resulting decrease in the amount of energy within the

midlatitudes of the Southern Hemisphere could impede the formation and decrease

the severity of midlatitude weather systems (Hanna 1996).

1.3 Previous Studies

Several studies suggested that Antarctic sea ice will exhibit changes in extent,

thickness, and compactness in response to long-term changes in global climate, as

well as short-term climate variations (Ledley 1991, Rind et al. 1995, Jacobs and

Comiso 1997, Geiger et al. 1997). Due to the lack of reliable information on the

distribution of sea-ice thickness within the Southern Ocean, sea-ice concentration

(percent of ice-covered area within the sea-ice region), has been the most commonly

used sea-ice variable in large-scale climate studies. Remotely-sensed information

on sea-ice concentration was used to represent the Antarctic sea-ice environment

in several modeling studies (Bromwich et al. 2001, Guo et al. 2003, Bromwich et

al. 2004), resulting in a reasonable approximation of some atmospheric variables.

Due to a lack of long-term and large-scale thickness observations within the Antarc-

tic sea-ice zone, these studies used a constant sea-ice thickness distribution, thus

misrepresenting sea ice in the simulated climate system (Worby and Ackley 2000).

7

Although many climate simulations indicated a decline in northern and south-

ern hemispheric ice extent in response to atmospheric warming (Hanna 1996, Cav-

alieri et al. 1997), changes in sea-ice thickness may provide more information on

the sea-ice response to climate change (King and Turner 1997). Most general cir-

culation models (GCM) produced conflicting results regarding the impact on the

e%ect global warming may have on the distribution of Antarctic sea-ice thickness

(Hanna 1996). Such conflicts may be indicative of poor model representation of the

sea-ice environment (Worby and Ackley 2000), biases in model physics, and the lack

of reliable information for model evaluation (Geiger et al. 2000).

High-quality information on sea-ice thickness can be obtained from electro-

magnetic induction sounding (EMI), upward-looking sonar (ULS), and ship-based

observations. However, such data are limited temporally and spatially and do not

provide circumpolar sea-ice thickness information at regular time scales (Worby et

al. 1999, Worby et al. 2001). Although sea-ice thickness cannot be directly obtained

from satellite imagery (Worby et al. 2001), satellite-derived proxy information, such

as ice type and stage of development, may be used to infer ice thickness (Dedrick et

al. 2001, Schellenberg 2002, Schellenberg et al. 2002).

The National Ice Center (NIC) has produced weekly, hemispheric ice charts

containing information on ice conditions obtained from remotely-sensed imagery

since 1973. These charts were originally intended for navigation but information on

stage-of-development, a proxy for sea-ice thickness, could potentially resolve some

variability in sea-ice thickness on a regional- to basin-scale for the entire Antarctic

sea-ice pack (Dedrick et al. 2001, Schellenberg et al. 2002, DeLiberty et al. 2004,

DeLiberty and Geiger 2005).

A recent study comparing thickness estimates between NIC ice charts and

ship-based observations within the Ross Sea sector showed an encouraging correla-

tion between remotely-sensed and in situ estimates (Schellenberg 2002, Schellenberg

8

et al. 2002). Further analysis indicated that thickness estimates obtained using

the NIC ice charts also corresponded with the annual sea-ice cycle (DeLiberty et

al. 2004). They concluded that the thickness estimates obtained using the NIC

ice charts provided a reasonable approximation of the sea-ice thickness distribution

within the Ross Sea and have the potential to provide a basin-scale record of ice con-

ditions that can be used to evaluate model output (DeLiberty et al. 2004, DeLiberty

and Geiger 2005).

1.4 Research Objectives

Ship observations are valuable resources for information on observed Antarc-

tic sea-ice thickness (Worby 1999), but these data are limited seasonally and spa-

tially. Remotely-sensed imagery provides the spatial coverage necessary for model

evaluation, but the e%ect of spatial resolution on in situ and remotely-sensed sea-ice

observations had yet to be evaluated. In order to assess the utility of sea-ice thick-

ness estimates derived from remotely-sensed imagery for model evaluation, errors

resulting from the spatial distribution of data points were evaluated for both in

situ and NIC thickness estimates following Willmott and Johnson (2005). Results

from the spatial resolution error assessment supported the use of the NIC sea-ice

thickness estimates for model evaluation.

Once it was determined that the NIC sea-ice thickness estimates were ap-

propriate for use in model evaluation studies they were used to evaluate the ice-

thickness distribution produced by the stand-alone version of the sea-ice model

component of the National Center for Atmospheric Research (NCAR) Community

Climate System Model, Version 3.0 (CCSM3), the Community Sea Ice Model, Ver-

sion 5.0 (CSIM5). Model output was evaluated using statistical methods of mean

absolute error (MAE) (Willmott et al. 1985, Willmott and Matsuura 2005) and

the index of agreement (d) (Willmott et al. 1985, Legates and McCabe 1999).

9

Chapter 2

SEA ICE PHYSICS

Sea ice, which is a discontinuous solid, exists along the boundary between two

continuous fluids - the ocean and the atmosphere. The distribution of sea ice and

its thickness is in direct response to the exchange of energy, mass, and momentum

along this boundary. The sea-ice environment is a dynamic environment in which

the distribution of sea-ice thickness is governed by the direction and magnitude of

heat energy through the ice pack and the movement of the surrounding air and

water (Washington and Parkinson 2005).

2.1 Sea Ice Dynamics

Sea-ice dynamics describe the movement of sea ice, which is primarily driven

by surface atmospheric and oceanic circulation, and the deformation of sea ice, re-

sulting from ice interaction. The mathematical representation of sea-ice dynamics

is based on the conservation of momentum in which the velocity of sea ice is deter-

mined by the sum of the forces acting upon it (Hibler 1986, Flato 2003, Washinton

and Parkinson 2005). Wind is the primary force driving sea ice motion over short

timescales, while the ocean currents and ice interactions play a greater role in ice

dynamics over longer timescales (Thorndike 1986, Hibler 2003).

10

2.1.1 Conservation of Momentum for Sea-Ice Motion

In general, sea-ice drift follows the Euler equations of motion and can be

expressed as

$ihidVi

dt= # • ! + X (2.1)

where $i is the density of sea ice, hi represents sea-ice thickness, Vi corresponds to ice

velocity, ! is the second order stress tensor due to ice interaction, and X represents

the force acting on the ice. Sea-ice density and thickness together represent the

mass (mi) of sea ice per unit area and ! corresponds to a nonlinear function of the

strain rate (Hibler 2003, Flato 2003).

Expanding equation 2.1 for X, the equation for sea-ice motion becomes a

momentum balance equation in which

midVi

dt= " a + "w + FD + FG + FI (2.2)

(Hibler 2003, Flato 2003, Washington and Parkinson 2005). In equation 2.2, the

left-hand side of the equation, which represents sea-ice acceleration, is equal to the

sum of the five major dynamic stresses, or forces, that drive sea-ice motion. These

terms include stress, or traction, from the air above the ice (" a) and the water

below ("w), stress created by earths rotation, or Coriolis force (FD), stress resulting

from the surface pressure due to sea-surface tilt, or dynamic topography (FG), and

stress from interstitial ice pressure (FI ; Hibler 2003, Flato 2003, Washington and

Parkinson 2005).

The stresses due to wind, water, and ice interaction have the greatest e%ect

on sea-ice motion and are an order of magnitude larger than the Coriolis parameter

and sea-surface height field (Hibler 2003). The acceleration term is often simplified

by assuming a steady-state velocity field over which midVidt = 0 (Washington and

Parkinson 2005). This is a reasonable assumption given that the magnitude of the

acceleration term is three times less than that of the air and water stress terms

(Rothrock 1973, Steele et al. 1997).

11

2.1.2 Mathematical Representation of Sea-Ice Motion

Given the di!culty in numerically representing sea-ice dynamics, the forces

that drive sea-ice motion are often parameterized in various ways depending on the

application (Table A.3). Wind stress can be represented using nonlinear boundary

layer functions in the form of:

" a = $aCda | Vg | (Vg cos%a + k ×Vg sin%a) (2.3)

where $a is the density of air, ca is the atmospheric drag coe!cient, Vg is the

geostrophic wind velocity, %a is the atmospheric boundary layer turning angle, and

k is the unit vector normal to the surface. Here, the wind stress is simplified by

using Vg instead of the vector di%erence between Vg and Vi, (Vg − Vi) , which

is physically reasonable given that Vg is much greater than Vi (Washington and

Parkinson 2005).

Similarly, water stress, "w can be expressed as:

"w = $wCdw | Vd | (Vd cos%w + k × Vd sin%w) (2.4)

where $w is the density of sea water, Cdw is the oceanic drag coe!cient, Vd is the

vector di%erence between the ocean geostrophic velocity Vo and ice velocity Vi, and

%w is the ocean boundary layer turning angle. The formulation for the wind and

water stresses can be simplified by assuming the drag coe!cients and turning angles

are constant (Washington and Parkinson 2005).

The numerical representation of the Coriolis force, FD, includes only known

parameters and can be expressed as:

FD = $ihifc(k × Vi) (2.5)

fc = 2$ sin( (2.6)

12

where fc is the Coriolis parameter, $ is the angular speed of the Earth, and (

is latitude. The dynamic topography, FG, can also be determined using known

parameters and is represented by:

FG = −$ihiag # H (2.7)

where ag is the acceleration due to gravity and #H is the gradient of the sea

surface height field H , (Washington and Parkinson 2005). The stress due to internal

ice pressure, FI , is a function of ice thickness, salinity, and air content, and is

proportional to dynamic topography, such that:

FI = # • ! (2.8)

(Hibler 2003, Washington and Parkinson 2005).

Due to the di!culty in expressing each term mathematically, certain terms

may be parameterized or are assumed insignificant and ignored. However, all terms

should be included for a complete model representation of a dynamic sea-ice envi-

ronment under all scenarios (Washington and Parkinson 2005).

2.1.3 Sea-Ice Rheologies

Sea ice exhibits both viscous and plastic behaviors and is not uniform in its

response to compression. Due to the numerous uncertainties involved in representing

internal ice stress, the relationship between stress (!) and the strain rate (#), or

rheology, is represented within sea-ice models using several di%erent methods (Flato

2003, Washington and Parkinson 2005).

Free drift is the simplest rheology and represents divergent conditions within

an open ice pack in which there is no stress acting upon the ice (FI = 0) as shown

in Figure 2.1a. A linear-viscous rheology assumes a linear relationship between con-

verging stress and strain rate (Figure 2.1b), but this solution produces similar results

for both the central and near-shore regions of the ice pack. Therefore, linear rhe-

ologies are limited to areas of homogenous sea-ice conditions away from boundaries

13

where as only nonlinear rheologies are appropriate for the entire ice pack (Hibler

2003, Flato 2003, Washington and Parkinson 2005).

The development of nonlinear rheologies evolved from the study of plastic

theory, leading to the formulation of plastic failure models with various representa-

tions of ice mechanics that characterize the behavior of pack ice as a whole (Hibler

2003). The simplest nonlinear rheology is the ideal plastic rheology, in which stress

is assumed constant and independent of the strain rate as shown in Figure 2.1c

(Washington and Parkinson 2005). The viscous-plastic rheology (VP) is commonly

used in coupled ice-atmosphere-ocean models and is a combination of the linear-

viscous and ideal plastic rheologies. In the viscous-plastic rheology (Figure 2.1c),

stress and the strain rate are proportional when the strain rate is small while stress

is constant when the strain rate is large (Hibler 1979, 2003).

The relationship between stress and the strain rate is more dynamic in the

collision-induced rheology where the strain rate increases as convergence increases

up to a limiting strain rate value (Figure 2.1d) (Washington and Parkinson 2005).

In the elastic-viscous-plastic rheology (EVP), an elastic component is added to VP

that accounts for sea-ice velocity and is more sensitive to synoptic weather systems.

Although the ice-thickness distribution produced by the elastic-viscous-plastic rheol-

ogy is similar to that produced by viscous-plastic rheology, is more computationally

e!cient and better for model parallelization (Washington and Parkinson 2005).

2.2 Sea Ice Thermodynamics

Thermodynamic sea-ice growth and decay is primarily driven by the net flux

of energy between the ocean and the atmosphere through the ice pack. Energy flux

calculations are used to determine sea-ice concentration (c) and thickness (h) for

both ice (i) and snow (s) thickness as well as the temperature profile and brine

content of the ice pack (Washington and Parkinson 2005).

14

(A)

(C) (D)

(B)

Collision InducedRheology

Linear ViscosityFree Drift

Ideal PlasticViscous Plastic

Figure 2.1: Schematic representation of the most commonly used rheologies includ-ing (A) free drift, (B) linear viscosity, (C) ideal and viscous plastic,and (D) collision induced. Modified from Washington and Parkinson(2005; Figure 3.24)

15

2.2.1 Sea-Ice Energy Balance

The energy flow through the ice pack includes several forms of long and short

wave energy (Figure 2.2). The primary energy fluxes that influence thermodynamic

ice growth and decay include incoming (QSW #) and reflected (QSW $) solar radiation

as well as emitted (QLW $) and counter-radiated longwave radiation (QLW #). The

previously mentioned energy fluxes per unit area may be written as:

QSW # = So

!d

d

"2

cos )s (2.9)

QSW $ = QSW #(1 − !0) (2.10)

QLW = "'T 4 (2.11)

where So is the solar constant,#

dd

$2is the eccentricity, cos)s is the solar zenith

angle, !0 is the surface reflectance, or albedo, ' is the Stefan-Boltzmann constant,

and " and T represent the emissivity and temperature of the surface for QLW $

and the atmosphere and QLW # (Hartmann 1994). Sea-ice thermodynamics are also

influenced by sensible (QH) and latent heat (QL) fluxes and conductive fluxes (QG)

through ice and snow,

QG0 = k

!dT

dz

"

0

(2.12)

where QG0 is the value of the upward conductive flux at the surface z = 0, k is

the thermal conductivity, and dTdz is the surface value of the temperature gradient.

Finally, the energy fluxes associated with snow and ice melt can be expressed as

QM = Lfdh

dt(2.13)

where QM represents ice or snow melt and Lf is the latent heat of fusion for snow

or ice (Wadhams 2000, Washington and Parkinson 2005).

2.2.2 Conservation of Energy for Sea-Ice Growth/Melt

In general, sea-ice thermodynamics may be expressed numerically using the

steady-state equation for the conservation of energy in which the exchange of energy

16

Figure 2.2: Schematic representation of the energy balance vertically through anice pack. Modified from Washington and Parkinson (2005; Figure3.21).

is balanced along the air/snow, air/ice, snow/ice, and ice/ocean interfaces. The

steady-state equation for the conservation of energy at the surface of ice covered

water follows:

QH +QL +QLW # +(1−!0)QSW #−I0−QLW $ +QG0 =

%&

'0 if T0 < Tf

QM if T0 = Tf

(2.14)

where I0 is the amount of solar radiation that penetrates the snow/ice column, and

Tf is the salinity dependent freezing point. The surface energy balance for the sea-

ice zone will be equal to zero for surface temperatures below freezing (T0 < Tf ),

otherwise melt will occur (Wadhams 2000, Washington and Parkinson 2005). It

should be noted that for sea ice, the sensible and latent heat fluxes are positive

downward (↓) (Washington and Parkinson 2005).

The steady-state equation for the conservation of energy along the air/snow

interface follows equation 2.14 for the snow surface and the values for emissivity,

17

albedo, and the conductive flux are specific to the snow surface ("s, !s, and QGs).

Snowmelt is dependent on surface temperature, which is that of the snow surface,

and equals 0 for surface temperature below freezing. Once the T0 = Tf , snow will

melt until the snow thickness, hs, equals 0, at which time the ice surface will begin

to melt (Wadhams 2000, Washington and Parkinson 2005).

In the case of bare ice, there is no snow cover to insulate the ice surface,

therefore, warming will result in direct surface ice melt rather than snow melt. The

energy balance for the air/ice interface can be expressed as

QH + QL + QLW # + (1 − !i)QSW # − Ii − QLW $ + QGi − Mi = 0 (2.15)

where "i and !i represent sea-ice emissivity and surface reflectance respectively,

and (FG)i is the surface value of the upward conductive flux through the ice pack

(Washington and Parkinson 2005).

The conservation of energy along the snow/ice interface depends on the bal-

ance between the conductive fluxes of snow and ice (QGs = QGi) such that:

ks

!*Ts

*z

"

hs

= ki

!*Ti

*z

"

hs

(2.16)

In this case, hs indicates the depth to the snow/ice interface, which is equivalent to

the snow thickness (Wadhams 2000, Hibler 2003, Washington and Parkinson 2005).

The conservation of energy along the ice/water interface depends on the balance

between the ice melt/growth term (QMi) and the di%erence between the ocean long-

wave radiation flux (QLWw) and the conductive flux through the ice column (QGi).

The relative magnitudes of the ocean and conductive heat fluxes at the ice/water

interface determines whether bottom, or basal, ice melt (+QMi) or growth (−QMi)

will occur following:

−Lf

!*hi

*t

"

hs+hi

= QLW #0 − ki

!*Ti

*z

"

hs+hi

(2.17)

In this case, hs + hi indicates the depth to the ice/water interface, which is equal

to the sum of the snow and ice thickness. If the ocean flux is positive and larger

18

in magnitude than the conductive flux, then the right-hand side of the equation is

positive and sea-ice melt will occur (−*hi/*t). If the ocean flux is negative or smaller

in magnitude than the conductive flux, the flux di%erence is negative resulting in

ice growth (*hi/*t) (Wadhams 2000, Washington and Parkinson 2005).

2.2.3 Thermal Structure of Sea Ice

The transfer of energy through a snow-covered ice pack may also be used

to calculate the temperature profile of ice and snow layers. Using a numerical

approximation for heat conduction through ice and snow, the temperature profile

for each can be estimated by:

$c*T

*t= k

*2T

*z2+ +I0 exp !!z (2.18)

where + is the bulk extinction coe!cient for snow or ice (Wadhams 2000, Hibler

2003, Washington and Parkinson 2005). The second term on the righthand side of

equation 2.17 allows solar radiation to penetrate the snow and ice layers using + to

approximate Beers extinction law (Maykut and Untersteiner 1971, Washington and

Parkinson 2005).

Although the specific heat and thermal conductivity of ice are often parame-

terized and assumed constant in many cases, they are both functions of temperature

and salinity (Wadhams 2000). The ice thermal conductivity can be approximated

by:

ki = ko + &Si

Ti(2.19)

where ko is the thermal conductivity of pure ice such that:

ko = 9.282 exp!0.0057Ti (2.20)

19

(Yen 1981), Si is the salinity of the ice in practical salinity units (psi), Ti is the

ice temperature in "C, and & = 0.13Wm!1. Similarily, the specific heat can be

approximated by:

ci = co + aTi + bSi

T 2i

(2.21)

where Cpo is the specific heat of pure ice, a = 7.53 J kg!1 "C!2, and b = 0.018 MJ "C kg!1.

In addition to approximating ki and Cpi, $iCpi can be approximated by:

$iCpi = ($Cp)o + ,Si

T 2i

(2.22)

where ($c)p is the product of the density and specific heat for pure ice and , = 17.15 MJ kg!1 K

(Wadhams 2000).

2.2.4 Ice Thickness Distribution

An ice thickness distribution is the numerical approximation of the distribu-

tion of sea-ice thickness within an ice pack and is a quantifiable way to define the

character and state of sea-ice. Information on the distribution of sea-ice thickness

is essential for understanding the exchange of energy between the ocean and the at-

mosphere, degree of ice deformation, and strength of the sea ice. The ice thickness

distribution may also be used to determine average thickness, which together with

the ice velocity, is used to determine the mass flux (rate of transport) of sea ice

(Wadhams 2000).

A typical ice thickness distribution is shown in Figure 2.3a resulting from de-

formation processes governed by sea-ice dynamics as well as thermodynamic growth

and decay (Thorndike 1992, Wadhams 2000, Hibler 2003). Sea-ice thermodynamics

lead to thinner ice by the ablation of thick ice and ridges, and thicker ice through

thermodynamic growth of thin ice (Figure 2.3b; Hibler 2003). Sea-ice deformation

causes ice to converge (Figure 2.3c) creating thicker ice through pressure ridging,

and diverge (Figure 2.3d) producing areas of open water in which new ice forms.

By creating areas of open water, where new ice forms, and forming ridges, sea-ice

20

dynamics a%ect the amount of ice that falls within the thinnest and thickest ice

thickness categories, while sea-ice thermodynamics a%ect the amount of ice within

the middle of the ice thickness distribution (Hibler 2003).

)

Figure 2.3: Schematic representation of the sea-ice thickness distribution producedby (A) thermodynamic and dynamic processes, (B) only thermody-namic processes, (C) divergence, and (D) mechanical redistribution.Modified from Wadhams (2000, Figure 5.2)

Numerical approximation of the ice thickness distribution involves solving

a thickness distribution function for a given area using equations that represent

both dynamic and thermodynamic processes (Thorndike et al. 1975). The ice

thickness distribution is represented by an areal probability density function (g) of

ice thickness (hi), which represents the proportion, or area (A), of ice within region

21

R(x, y, t) at time t with an ice thickness between h + (h + dh) given by:

g(h)dh =dA(h,h + dh)

R(2.23)

(Thorndike 1992, Wadhams 2000). The right-hand side of equation 2.23 represents

dynamic and thermodynamic processes and can be expanded such that:

dg

dt= −g # •V − *

*h(fg) +# (2.24)

where V is the horizontal velocity vector, f is the vertical growth rate, and # is

the mechanical redistribution function. The first term in equation 2.24 accounts for

divergence within the ice pack, the second represents thermodynamic growth, and

the redistribution function accounts for mechanical processes such as ridging and the

formation of leads (Thorndike 1992). The thermodynamic term allows ice thickness

to be redistributed between ice-thickness categories such that the areal distribution

of ice within each thickness category varies as ice grows or melts (Hibler 2003).

The representation of sea-ice thermodynamics in equation 2.24 only accounts

for vertical growth of sea ice, however, the lateral growth and decay of sea ice also

a%ects the distribution of sea-ice thickness. Therefore equation 2.24 expands to

include an additional term, L(h,g), to account for the lateral growth and decay of

sea ice (Hibler 1980). Equation 2.24 is now expressed as:

dg

dt= −g # •v( )* +

1

− *

*h(fg)

( )* +2

−L(h,g)( )* +3

+ #()*+4

(2.25)

in which term 1 represents the dynamic redistribution of ice thickness due to di-

vergence, terms 2 and 3 represent the vertical and horizontal redistribution of ice

thickness due to thermodynamic growth and decay, and the fourth and final term

represents the mechanical redistribution of sea-ice thickness due to ridging and raft-

ing (Hibler 2003, Briegleb 2004).

Horizontal divergence within the ice pack, as represented by the first term in

equation 2.25, accounts for ice divergence and the creation of leads. Term 1 therefore

22

increases the proportion of open water within the ice pack by decreasing the fraction

of ice with thickness h > 0. Thermodynamic growth and melt, represented by terms

two and three, allow thin ice to grow and thick ice to melt, changing the areal fraction

of ice within each thickness category. The mechanical redistribution function allows

for the formation of thick ice due to ridging and rafting while also leading to the

formation of open water (Wadhams 2000).

The mechanical redistribution function di%ers from terms 1 - 3 in that it does

not create nor destroy ice, but a%ects the proportion of ice within the thickness

distribution that is involved in ridging. Mass is conserved as the areal fraction of

ice involved in ridging is redistributed throughout thicker ice categories following

, %

0

h#dh = 0 (2.26)

taking into account the areal fraction of ridged ice and open water created by di-

vergence, convergence, and shearing (Hibler 2003). The mechanical redistribution

function is the most di!cult term to approximate mathematically and is often pa-

rameterized or determined empirically (Wadhams 2000, Briegleb 2004).

2.3 Summary

The distribution of sea-ice thickness is determined by the direction and mag-

nitude of dynamic and thermodynamic forces. The dynamics influence the move-

ment of sea ice, creating thicker ice by convergent processes and open water by

divergent processes. Thermodynamic processes lead to the vertical and horizontal

growth and decay of sea ice in response to the flow of heat energy through the sea

ice (Washington and Parkinson 2005). Together, ice dynamics and thermodynam-

ics account for the amount of sea ice at any given location within the sea-ice zone

(Wadhams 2000).

23

Although the dynamic forces acting on the ice and the flow of energy through

the sea ice can be described mathematically, individual terms are di!cult to com-

pute. Therefore, it is di!cult to create a representative sea-ice thickness distribution

numerically. In the past, sea-ice models addressed either dynamic or thermody-

namic processes and although the current generation of sea-ice models include both

dynamic and thermodynamic processes, many variables, such as the relationship

between stress and strain rate as well as oceanic and atmospheric components, are

parameterized or ignored (Washington and Parkinson 2005).

24

Chapter 3

MODELING SEA ICE

Due to the complexity of sea ice, the main purpose of sea-ice models in global

climate simulations is to determine the presence or absence of ice, its thickness, and

concentration within each model grid cell at every time-step. Although the physical

characteristics of sea ice are of secondary importance in modeling, they must be

known to determine the distribution of sea ice and the e%ect it has on atmospheric

and oceanic model components (Washington and Parkinson 2005).

The distribution of sea-ice thickness produced by a sea-ice model is deter-

mined by solving a set of nonlinear equations concerning the dynamic and thermo-

dynamic characteristics of the ice pack. The temperature structure and thickness

of sea-ice are described by thermodynamics, which are determined using an energy

balance to resolve net ice growth and melt. Sea-ice dynamics describe the motions

of sea ice using the momentum equation, which is then solved for ice velocity (Flato

2003, Washinton and Parkinson 2005). The conservation of area and mass are also

important in modeling sea-ice thickness and represent the redistribution of sea-ice

thickness to thinner or thicker ice-thickness categories within the ice pack (Flato

2003).

The first sea-ice model, a simple thermodynamic model, was developed in the

late nineteenth century, which led to the development of empirical thermodynamic

models during the first half of the twentieth century (Stefan 1890, Wadhams 2000).

25

Thermodynamic sea-ice models improved as the understanding of sea-ice thermody-

namics improved, leading to the development of multi-level numerical models (Wad-

hams 2000, Washington and Parkinson 2005). Similarly, dynamic models improved

with the understanding of sea-ice dynamics leading to the development of one-

dimensional plastic models and more complex rheologies. Finally, thermodynamic

and dynamic model were coupled leading to the development of the ice-thickness

distribution models in use today (Hibler 2003, Washington and Parkinson 2005).

3.1 Thermodynamic Models

The first thermodynamic sea-ice model, and the first attempt at modeling

sea ice in general, was developed by Stefan (1890). Stefans model was based on the

relationship between sea-ice growth and air temperature (Ta) in which the thickness

(hi) of young ice follows:dhi

dt∝ (Ta − TB)

hi(3.1)

where TB is the temperature at the bottom of the ice and t is time. This early model

accounts for the main properties of sea-ice thermodynamics including the decrease

in ice growth as ice thickens and, under thermodynamics alone, tries to reach a

uniform thickness (Wadhams 2000).

Assuming steady-state boundary conditions and a linear temperature profile,

sea-ice growth can be approximated using the Stefan number (Sn) such that:

Sn =Lfi

Cpi(Tf − TB)(3.2)

where Lfi is the latent heat of fusion for sea ice, Cpi is the specific heat capacity of

sea ice, Tf is the melting point of ice, and TB is the temperature at the bottom of

the ice. Sea-ice growth can then be approximated by:

$iLfi h ≈ ki(Tf − TB)

hi(3.3)

26

Figure 3.1: Schematic diagram of a linear temperature profile through a singlelayer of snow-covered ice following the Stefan approximation. Modifiedfrom Hibler (2003; Figure 7.23).

where $i represents the density of sea ice, ki is the thermal conductivity of sea ice,

and h is the growth function. Equation 3.3 can be solved for ice thickness (hi)

following:

hi ≈!

2kit

Sn

" 12

(3.4)

assuming Sn is small (Figure 3.1) (Worster 1999, Hibler 2003).

However, Sn is usually large for sea ice and the Stefan model does not prop-

erly account for the exchange of energy at the top and bottom of the ice column.

Although the Stefan approach to the numerical approximation of sea-ice thermo-

dynamics is limited, it has provided the framework for the development of more

complex thermodynamic sea-ice models (Wadhams 2000, Hibler 2003).

27

3.1.1 Degree-Day Thermodynamic Models

The development of thermodynamic numerical sea-ice models continued dur-

ing the first half of the twentieth century with the development of simple, freezing

degree-days thermodynamic models (Wadhams 2000). These early models of sea-ice

growth were based on the empirical relationship between ice growth and air tem-

perature, treating sea-ice thickness as a function of the number of days in which the

air temperature remained below freezing (Barnes 1928, Lebedev 1938, Zubov 1945).

A degree-day ("), in units of "C day!1, is the sum of the number of days in

which the air temperature (Ta) above or below a reference temperature, in this case

the freezing point of sea water (Tfsw),

"(t) =

, t

0

(Tfsw − Ta)dt (3.5)

(Wadhams 2000, Hibler 2003). The empirical relationship between ice thickness (hi)

and degree-days has been represented in a variety ways beginning with Lebedev

(1938) defining thickness as:

hi = 1.33"0.58 (3.6)

Zubov (1945) as:

h2i + 50 hi = 8" (3.7)

and Anderson (1961) as:

h2i + 5.1 hi = 6.7" (3.8)

(Wadhams 2000, Hibler 2003). Similarly, Bilello (1961) developed a relationship

between sea-ice decay and cumulative degree-days above freezing ("), representing

the change in sea ice ('hi) in summer by:

'hi = 0.55"& (3.9)

(Wadhams 2000).

28

Combining elements of the Stefan approximation and degree-day models, the

simple degree-day models previously described were expanded to account for snow-

covered ice. Based on equations 3.3 and 3.8, this extended degree-day approximation

of ice thickness includes snow thickness (hs) and parameterized heat flux and thermal

conductivity coe!cients for ice and snow which, when solved, become:

hs+i + (13.1hs + 16.8)hs+i = 12.9" (3.10)

(Hibler 2003).

As with the Stefan approximation, ice growth in degree-day models slows

as ice thickness increases and thus provides an adequate approximation of sea-ice

thickness (Hibler 2003). However, such models parameterize or ignore important

components of the surface energy balance, such as internal heat capacity and the

presence of brine pockets, and assume either no snow cover or snow of constant

thickness (Wadhams 2000).

3.1.2 Multi-Level Thermodynamic Models

Early, empirical models treated sea ice as one, snow-free layer of constant

thickness with later adjustments for snow cover (Wadhams 2000). However, the

physical properties of sea ice vary vertically through a column of ice and snow.

Variability in the physical properties of snow-covered ice a%ects the vertical temper-

ature profile of the ice column. Therefore, a single, linear temperature profile does

not accurately reflect the vertical distribution of temperature within a column of ice

(Washington and Parkinson 2005).

Multi-level, thermodynamic sea-ice models were developed out of the need

to improve the vertical resolution of thermodynamic ice models by accounting for

changes in the temperature profile throughout a column of ice. The vertical reso-

lution is improved by dividing the ice pack into more than one layer of snow and

29

ice. Each layer of thickness (hi) is assumed to have a linear temperature (Ti) profile

such that: !*Ti

*z

"

hs

=

!*Ti

*z

"

hi+hs

=(Ti)hi+hs − (Ti)hs

hi(3.11)

for each layer of sea ice (i) and!*Ts

*z

"

0

=

!*Ts

*z

"

hs

=(Ts)hs − (Ts)0

hs(3.12)

for each layer of snow (s). In this case, the temperature at the bottom of the ice

column, (Ti)hi+hs, is equal to the freezing point of the underlying ocean (Figure 3.2;

Washington and Parkinson 2005).

Although ice models vary, snow melt (QMs) is typically set equal to zero

at the beginning of each model time step and equations 2.11, 2.12, 2.14, and 2.16

are solved for the surface temperature at the air/snow (Ts)0 and snow/ice interfaces

(Ti)0. If the snow surface temperature is above freezing, (Ts)0 is set equal to freezing,

the temperature dependent flux terms in equation 2.14 are recalculated for melting

snow (QMs (= 0), and equation 2.13 is solved for snow thickness. The change in sea

ice thickness due to ablation or accretion at each grid cell is then determined using

finite di%erence solution to equation 2.17 (Washington and Parkinson 2005).

3.2 Dynamic Models

The development of dynamic sea-ice models began in the second half of the

20th century as part of the Arctic Ice Dynamics Joint Experiment (AIDJEX) and

work published by Campbell (1964), Coon (1980), Hibler (1979), and Parkinson

and Washington (1979). These early calculations were later improved through the

works of Flato and Hibler (1992), Steele et al. 1997, Zhang et al. (1998), and

Hunke and Zhang (1999) leading to the modern dynamic sea-ice models in use

today (Washington and Parkinson 2005).

Dynamic sea-ice models determine sea-ice velocity by solving a set of equa-

tions regarding the forces acting upon the ice (equation 2.2). Simulating the stresses

30

Figure 3.2: Schematic diagram of the temperature profile through snow-covered icein a multi-level thermodynamic ice model. Modified from Washingtonand Parkinson (2005; Figure 3.22).

due to air (" a), water ("w), the Coriolis force (FD), and sea-surface tilt (FG)is rel-

atively straightforward (equations 2.3 - 2.7), however, the internal ice stress (FI)

(equation 2.8) is di!cult to simulate and is treated in a variety of ways depending

on the model. Internal ice stress is a function of the relationship between stress and

strain, or rheology, ice strength and mass, all of which must be accounted for by a

dynamic ice model (Wadhams 2000, Washington and Parkinson 2005).

31

Figure 3.3: Schematic diagram of free drift ice motion in response to wind stresswithout friction. Modified from Wadhams (2000; Figure 4.1a).

3.2.1 Free-Drift Solution to Internal Ice Stress

The free drift solution for internal ice stress assumes that the ice cover is a

single flow with no external interaction (Wadhams 2000). The free drift solution to

FI assumes there is no collision with other flows, the internal ice pressure (P ) is

equal to zero, and the change in velocity over a certain distance can be greater than

zero#"v"y ≥ 0

$(Geiger 1996, Wadhams 2000).

For example, the free drift solution to the movement of a single ice flow acted

on by wind would result in ice movement in the direction of the force, or wind

direction. The velocity will increase to the point at which the Coriolis force deflects

the ice to the left in the Antarctic and to the right in the Arctic (Figure 3.3). When a

drag force due to ocean circulation is added, ice motion will continue in the direction

of the wind while being deflected by the Coriolis force, but at a slower rate (Figure

3.4). With time, this motion eventually becomes uniform at an angle of 45 to the

right or left or the wind vector, depending on the hemisphere (Wadhams 2000).

Although free drift has been used to simulate sea-ice motion, sea ice rarely

behaves as a single unit and convergence is not negligible. Free drift is therefore

only appropriate for individual flows within and open ice pack and areas where ice

32

Figure 3.4: Schematic diagram of free drift ice motion (ui) in response to forcesgenerated by wind (F ), the Coriolis force (FD) and water stress ("w).Modified from Wadhams (2000; Figure 4.1b).

33

motion is primarily divergent. The internal ice stress for a typical sea ice pack is

better approximated assuming FI ≥ 0 (Wadhams 2000, Washington and Parkinson

2005).

3.2.2 One-Dimensional Plastic Models

The development of plastic deformation models began in the early 1980s in

order to improve the model representation of FI . Plastic models assume sea ice

behaves as a plastic material in which failure occurs only when the stress reaches a

critical value and stress is independent of the rate of deformation (Coon 1980, Wad-

hams 2000, Washington and Parkinson 2005). This is a more reasonable approach

to simulating FI given the non-linear properties of ice flow (Wadhams 2000, Hibler

2003).

The simplest example of a plastic sea-ice model is a one-dimensional, hori-

zontal representation of the momentum balance assuming a linear ocean drag and

constant wind stress. This one-dimensional, plastic model is expressed as:

Cdui −*'i

*x= -a (3.13)

where Cd is a linear drag coe!cient, ui represents the ice velocity in the x direction

(vi represents velocity in the y direction), 'i represents a one-dimensional ice stress,

and -a is a constant wind stress in the x plane. In the case of-"ui"x

.> 0, or free drift,

assume 'i = 0, while 'i = −P , where −P represents pressure exerted by the ice, can

be assumed for-"ui"x

.< 0, the condition in which yielding, or convergence, occurs.

In between free drift and yielding, the ice is rigid, undeformed, and characterized

by incompressible flow in which-"ui"x

.= 0 and −P ≤ 'i ≤ 0 (Geiger 1996, Hibler

2003).

Internal ice stress can be represented by a simplified plastic model, the cavi-

tating fluid model, developed by Flato and Hibler (1990), in which shear strength is

34

Figure 3.5: Latitudinal transect through a theoretical ice flow

ignored and the velocity is adjusted to allow convergence (Wadhams 2000). Apply-

ing this model to a case in which a hypothetical, latitudinal transect through an ice

pack (0 ≤ x ≤ L ≤ N), where L is the location of the ice edge along the transect,

bounded at each end (xo = 0, xn = N) by a wall and solving for P , equation 3.13

becomes:dP

dx= -aCdui (3.14)

(Geiger 1996, Hibler 2003). In this case, ice of constant thickness exists at the

base of the wall located at xo, extending from 0 ≤ x ≤ L with open water from

L < x ≤ N (Figure 3.5) (Geiger 1996).

The cavitating fluid constitutive rheology includes considerations for free drift-"ui"x ≤ 0, P = 0

., incompressible flow

-"ui"x ≤ 0, 0 ≤ P ≤ Pmax

., and isotropic yield-

ing-"ui"x ≤ 0, P = Pmax

.resulting in three solutions for P along xo ≤ x ≤ xn (Geiger

1996). In the case of constant P , dPdx = 0 and equation 3.14 becomes:

ui =-aCd

(3.15)

where P = 0 for free drift, and P = Pmax for isotropic yielding. In the case of

incompressible flow, the velocity gradiant is equal to zero due to a constant ui and

35

P is variable such that equation 3.14 becomes:

P − Pref

x − xref= -a − Cdui (3.16)

where Pref and xref refer to the value of P at a particular point x along the transect

(Geiger 1996).

Solving for P and ui for the entire transect yields (Table 3.1):

P =

%///&

///'

MIN(−-aL, Pmax) at x = 0

PxoL (L − x) at 0 < x ≤ L

0 at L ≤ x ≤ xn

(3.17)

ui =

%//////&

//////'

0 at x = 0

#aCd

+ PxoCdL at 0 < x ≤ L

ui0<x<L+uiL<x<xn2 at x = L

#aCd

at L ≤ x ≤ xn

(3.18)

The pressure exerted by the wall located at x0 is equal to Pmax, which causes the

ice to deform at x0. If the reaction pressure at the wall (P0 = −-aL) exceeds Pmax,

the excess pressure (P0Pmax) causes the ice to move as it deforms. Isotropic yielding

occurs when the magnitude of -a is greater than the resistive force per unit length

exerted on the ice (Pmax/L). Velocity at x0 is zero and the sign of *ui/*x indicates

the direction the ice motion, resulting in convergence at x0 for *ui/*x < 0 and

divergence at x0 for *ui/*x > 0. If the wind velocity is constant, isotropic yielding

only occurs at the wall and the ice pack follows incompressible flow to the ice edge

(x0 < x ≤ L) and free drift past the ice edge (L < x < xn) (Figure 3.6) (Geiger

1996).

Comparisons between simulated and observed pressure and velocity values

indicate that the plastic representation of ice dynamics is more realistic than the

free drift solution. For high wind velocity, the free drift and rigid solutions for

ice velocity are similar, however, there is a large di%erence in ice velocity between

36

Figure 3.6: Gridded, latitudinal transect through a theoretical ice flow in a cavi-tating fluid model.

Table 3.1: Results of a cavitating fluid model for a transect through the theoreticalice flow depicted in Figure 3.6. In this example, 'x = 4.44 × 105 m,-a = −0.1256 kg m!1 s!2, Cd = 0.6524 kg m!1 s!2, Pmax = 55000 Pa m,and P0 = 474014.37 Pa m.

Location Latitude Pressure (Pa m) Velocity (m s2) Condition0 -80 55000.00 0.00 Isotropic Yielding1 -76 48529.41 -0.17 Incompressible Flow2 -72 42058.82 -0.17 Incompressible Flow3 -68 35588.24 -0.17 Incompressible Flow4 -64 29117.65 -0.17 Incompressible Flow5 -60 22647.06 -0.17 Incompressible Flow6 -56 16176.47 -0.17 Incompressible Flow7 -52 9705.88 -0.17 Incompressible Flow8 -48 3235.29 -0.18 Incompressible Flow9 -44 0.00 -0.19 Free Drift10 -40 0.00 -0.19 Free Drift

37

free drift and rigid plastic solutions for low wind velocity. Although the simulated

values for internal ice pressure tend to be high, plastic failure models are able to

represent the relationship between wind and ice velocity by allowing ice to slow, and

eventually stop, under low wind speeds (Hibler 2003).

3.3 Ice Thickness Distribution Models

Given the complexity of the sea-ice environment, it cannot be described by

dynamics or thermodynamics alone (Hibler 2003). Numerical approximations of

both the dynamic and thermodynamic processes are necessary to account for changes

in ice volume due to ablation and accretion, as well as changes in ice area due to

convergence and divergence (Flato 2003). The need for a numerical sea-ice model

to account for dynamic and thermodynamic ice sources and sinks lead to the de-

velopment of ice thickness distribution (ITD) models in which sea-ice thickness is

distributed between more than one ice thickness category (Hibler 2003).

3.3.1 Two-Category Ice Thickness Distribution Model

Changes in ice volume, usually evaluated in terms of ice thickness (hi), and

area (Ai) can be expressed as:

*hi

*t= −# (uihi) + Shi (3.19)

*Ai

*t= −# (uiAi) + SAi (3.20)

where −# (uihi) and − # (uiAi) account for ice motion and Shi and SAi are the

thermodynamic source terms for sea-ice thickness and area (Flato 2003). Equations

3.19 and 3.20 have been expanded to represent a two-category ITD model in which

the ice pack is divided into one category of thick ice with area, Ai, and a thin ice

category of area, 1 − Ai. For simplicity, the ice within category 1 − Ai is assumed

to have a thickness of zero and represents open water (hi = 0) and thin ice up to a

38

thickness of hi0 . The thickness of sea ice within category Ai is uniformly distributed

between hi = 0 and hi = 2 hi/Ai (Hibler 1979, 2003, Flato 2003).

In Hiblers (1979) two-category thickness distribution model, the distribution

of mean ice thickness (hi) and compactness (Ai) is expressed using the continuity

equation for sea-ice mass such that:

*hi

*x= −*(uihi)

*x− *(vihi)

*y+ Shi

(3.21)

*Ai

*x= −*(uiAi)

*x− *(viAi)

*y+ SAi (3.22)

Equation 3.22 accounts for ridging, which occurs when Ai = 1 and any increases in

Ai, resulting in Ai > 1, are removed by ridging. Ice strength is then parameterized

and represented by:

P = P $hi exp[−C(1 − Ai)] (3.23)

where P $ = 2.75 × 103 N m!2 and C = 20 (Flato 2003, Hibler 2003).

Although the two-level ITD model is an improvement upon stand-alone dy-

namic and thermodynamic models, it still lacks the complexity required to ade-

quately simulate the sea-ice thickness distribution. It does not take into account

the variability of sea-ice thickness over small spatial scales, which has a non-linear

e%ect on the thermodynamic terms in the two-category ITD model. The model also

poorly represents ridging and parameterizes ice pressure based on ice thickness and

concentration (Flato 2003). Therefore, the need to improve the treatment of thin

ice, the distribution of ice thickness within thicker ice, and ridging became the basis

for the development of multi-category ITD models (Hibler 2003).

3.3.2 Multi-Category Ice Thickness Distribution Model

Multi-category ITD models were developed based on the concept of the ice-

thickness distribution function, g(hi). The ice-thickness distribution function was

redefined and treated as a probability density for ice thickness defined by equation

39

2.25 (Wadhams 2000, Flato 2003). The probability density function for sea-ice

thickness is generally expressed as:

dg

dt= −g # •v − *

*h(fg) − L(h,g) +# (3.24)

where the distribution of sea-ice thickness is a function of ice advection, horizontal

and vertical thermodynamic growth, and redistribution (Thorndike et al. 1975,

Thorndike 1992, Flato 2003).

The thickness distribution function represents the proportion, or fractional

area, of ice of a given thickness that falls within a thickness category, hi to hi + dhi

(Wadhams 2000, Lipscomb 2001). The number of categories (M) in g(hi) can vary,

depending on the model and the experiment, but studies indicate that M = 5 for

hi > 0, plus an additional category for open water, is a physically reasonable ITD for

use in coupled climate system models (Bitz et al. 2001, Washington and Parkinson

2005).

Ice mechanics are determined by the ice advection and redistribution terms

in equation 3.24. The ice advection term, −g # •v, is determined using a two-

dimensional distribution approach and accounts for the creation of leads in response

to divergent wind stress (Lipscomb 2001). The ice advection term adds area to

the open water category at the expense of the other thickness categories (Wadhams

2000).

The redistribution function, #, is determined using a ridging model, and

accounts for the redistribution of sea-ice thickness and area due to convergence,

divergence, and shearing (Lipscomb 2001). The redistribution term moves ice from

thinner to thicker categories through ridge building, increasing the probability that

ice area will fall within thicker ice categories (Wadhams 2000). The inclusion of a

ridging model within an ITD model improves the parameterization of ice strength,

a%ecting the amount of energy lost by ridging (Flato 2003).

40

The thermodynamic growth/decay terms are solved using a transport mech-

anism that distributes ice between thickness categories as the ice grows and melts.

The horizontal growth/decay term, − ""hi

(fg), can be solved as a finite di%erence

problem or using a remapping scheme in the form of:

d

dt

, hi

hin!1

gdhi = fn

, hi

hin!1

gdhi (3.25)

Similarly, volume can be expressed as:

d

dt

, hi

hin!1

higdhi = 0 (3.26)

where f ≡ dhndt is the thermodynamic growth rate for sea ice(Lipscomb 2001).

Although there are numerous remapping schemes for ITD models (Hibler 1980,

Dukowicz and Baumgardner 2000, Williamson and Laprise 2000, Bitz et al. 2001)

most modern climate system models use Lagrangian or semi-Lagrangian methods

for remapping (Washington and Parkinson 2005).

3.4 Summary

Although some sea-ice models represent only the dynamic or thermodynamic

properties of sea-ice, sea-ice models designed for use with general circulation and

climate system models generally address both dynamic and thermodynamic sea-

ice processes (Flato 2003, Washington and Parkinson 2005). Sea-ice models have

been developed as stand-alone models, representing certain aspects of the sea-ice

environment as a whole, or to represent the sea-ice environment within a climate

system model. Regardless of model complexity, numerical approximation of ice

behavior within both stand-alone or coupled sea-ice models results in errors within

the simulated sea-ice environment. However, sea-ice models can provide valuable

information on the role of the sea-ice cover in the climate system (Flato 2003).

41

Chapter 4

SEA ICE OBSERVATIONS

The interactions between sea ice, ocean circulation, and global climate have

been the subject of several modeling studies (Fichefet and Morales Maqueda 1997,

Goosse and Fichefet 1999, Ogura at al. 2004, Stossel and Markus 2004, Holland et

al. 2006). However, sea ice remains poorly represented in even the most complex

climate system model (Wu et al. 1999, Turner et al. 2001, Worby and Ackley 2000,

Bitz et al. 2001). The variability in sea-ice area, observed in terms of sea-ice extent

and concentration, within the Southern Ocean has been well documented since the

late 1970s using Scanning Multichannel Microwave Radiometer (SMMR) and Special

Sensor Microwave/Imager (SSM/I) data (Stammerjohn and Smith 1997, Gloersen

et al. 1999, Watkins and Simmonds 2000, Parkinson 1994, 1998, 2002, Zwally et

al. 2002), although the seasonal and interannual variability in sea-ice thickness

is poorly understood (Geiger et al. 2000). Remotely sensed information on sea-

ice concentration has been used to represent the Antarctic sea-ice environment in

several modeling studies (Bromwich et al. 2001, Guo et al. 2003, Bromwich et al.

2004) but such studies may be limited due to a lack of incorporating long-term,

large-scale thickness observations of the Antarctic sea-ice zone (Worby and Ackley

2000).

Ship-based observations provide an extensive record of sea-ice thickness es-

timates for the Southern Ocean and currently the primary source of large-scale

thickness information used to evaluate sea-ice model output (Holland et al. 2006).

42

However, it is di!cult to obtain continental-scale in situ observations and as a re-

sult, a simulated ice-thickness distribution for the entire Antarctic region may only

be partially evaluated due to temporal and spatial gaps in the current record of

ship-based Antarctic sea-ice thickness estimates (Figure 4.1) (Timmermann et al.

2004). Sea-ice thickness cannot be directly obtained from satellite imagery (Worby

et al. 2001) but satellite-derived proxy information, such as ice type and stage of

development, may be used to infer an ice thickness estimate (Figure 4.2) (Dedrick

et al. 2001, Schellenberg 2002, Schellenberg et al. 2002).

4.1 Ship-Based Antarctic Sea-Ice Thickness Estimates

Ship-based observations are some of the most valuable resources for informa-

tion on Antarctic sea ice, most notably sea-ice thickness. The Scientific Committee

on Antarctic Research (SCAR) developed the Antarctic Sea-Ice Processes and Cli-

mate (ASPeCt) program to compile and standardize an extensive archive of ship

observations. Prior to the inception of the ASPeCt program, observations made

during separate voyages were not consistent, making it di!cult to compare data

collected from di%erent voyages. To date, the ASPeCt program has identified in-

formation collected during 81 voyages sponsored by the United States, Australia,

Russia, and the United Kingdom, that contain quantifiable information on sea-ice

concentration and thickness (Worby 1999, Worby et al. 2008). Currently, the AS-

PeCt data archive includes 21,710 individual records of Antarctic sea ice collected

between 1981 and 2005 (Worby 1999, Schellenberg et al. 2002, Timmermann et al.

2004, Worby et al. 2008).

In situ records include information on sea-ice concentration, thickness, form,

and snow cover, and are based on observations of sea-ice characteristics made within

a 1 km radius of the ship over the length of the ships path through the ice pack.

Individual voyages represent sea-ice conditions at sub-synoptic scales (Worby 1999)

making it di!cult to obtain complete coverage of the entire Antarctic ice pack with

43

-45

-135

-90

135

-75

45

A 0 990 1,980495 Kilometers

-45

-135

-90

135

-75

45

D

-45

-135

-90

135

-75

45

C

-45

-135

-90

135

-75

45

E

-45

-135

-90

135

-75

45

B

Figure 4.1: Distribution of (A) December-January-Febuary (summer), (B) March-April-May (fall), (C) June-July-August (winter), and (D) September-October-November (spring) in situ sea-ice thickness observations forthe Southern Ocean from 1981 to 2005 from the ASPeCt ship-dataarchive (ASPeCt 2008).

44

cm

0 1233 km

Figure 4.2: Distribution of seasonally averaged, remotely-sensed sea-ice thick-ness estimates derived from stage-of-development proxy informationfor (A) December-January-Febuary (summer), (B) March-April-May(fall), (C) June-July-August (winter), and (D) September-October-November (spring) from 1995 to 1998 (NIC 1995-1998).

45

great frequency. Therefore, not all sectors of the Southern Ocean are represented

seasonally or even annually throughout the period of record (Figure 4.1 and 4.3)

(Timmermann et al. 2004).

0

200

400

600

800

1000

1200

1400

1600

Summer (DJF) Autumn (MAM) Winter (JJA) Spring (SON)

Indian Ocean Western Pacific Ocean

Ross Sea Admundsen/Bellingshausen

Weddell Sea

Num

ber o

f Obs

erva

tions

(198

1-20

01)

Figure 4.3: The number of seasonal in situ sea-ice thickness observations for thefive sectors of Southern Ocean from 1981 to 2005 (ASPeCt 2008).

Sea-ice thicknesses are reported as a range of thicknesses based on sea-ice

type or a measured thickness estimate for individual point locations (Table 4.1).

Thickness observations, which are accurate within ±5 cm, are made by trained ice

observers who estimate ice thickness, in centimeters, as the ice moves past a buoy

of known diameter or rule attached to the side of the ship (Worby 1999, Worby et

al. 2008). Thickness estimates made by observers may have errors or biases that

reflect the observers level of training and experience. Such biases can be mitigated

by cross validating observations made by two groups of two observers, each making

observations simultaneously. Navigational biases may result in the underestimation

46

Table 4.1: Classification of sea-ice thickness and ice type used by the ASPeCtprogram (Worby 1999).

Average Thickness Ice Ice TypeThickness Range Thickness Classification

(cm) (± cm) Code0.0 0 0 No Ice5.0 5 10 Frazil5.0 5 11 Shuga5.0 5 12 Grease5.0 5 20 Nilas10.0 5 30 Pancakes12.5 2.5 40 Young Grey Ice (10-15 cm)22.5 7.5 50 Young Grey-White Ice (15-30 cm)50.0 20 60 First-Year Ice (30-70 cm)95.0 25 70 First-Year Ice (70-120 cm)160.0 40 80 First-Year Ice (>120 cm)160.0 40 85 Multi-Year Ice100.0 100 95 Fast Ice999 999 999 No Data

of thicker ice since ships can have trouble navigating through thick, ridged, and

rafted ice (Worby 1999).

In situ ice-thickness estimates (zi), in centimeters, were calculated from voy-

age data using observed estimates of sea-ice concentration (ci) and thickness (zi) for

the thickest (i=1), middle (i=2), and thinnest (i=3) partial ice categories at each

observation point.

ci(.,(, t) = ci ± c&i (4.1)

zi(.,(, t) = zi ± z&i (4.2)

Each observation point has a specific location in terms of latitude (() and longitude

(.) and each observation represents sea-ice conditions at that location at a specific

point in time (t). The estimated sea-ice thickness within each partial ice category

47

is the sum of the thickness of each ice form, level ice (u), ridged ice (r), and snow

(s), observed at each data point (Schellenberg 2002).

zi = zui + zri + zsi (4.3)

Sea-ice concentration and thickness estimates include an empirically determined

level of uncertainty (c&i, z&i) based on thickness and form (Worby et al. 2008).

c&i = ±0.1 (4.4)

z&ui=

%///&

///'

±0.5 at 0 ≤ z&li < 10 cm

±0.3 at 10 ≤ z&li < 30 cm

±0.2 at z&li ≥ 30 cm

(4.5)

z&ri= ±0.5 (4.6)

z&si= ±0.5 (4.7)

Thickness and error calculations for each ice category (i = 1...3) can be described

as:

z =30

i=1

cizi (4.8)

z& =

130

i=1

2!c&ici

"2

+

!z&izi

"23

(cizi)2

41/2

(4.9)

following Schellenberg (2002: 34).

Equation (4.8) is a good approximation for level ice and snow thickness, but

adjustments need to be made to account for the disproportionate volume of ice per

unit area contained within ridged ice compared to level ice. Estimates of ridged-ice

thickness should reflect the three-dimensional characteristics of the ridge as well

as the thickness of the level ice that formed the ridge (Worby et al. 2008). The

following equation is an empirically derived model for ridge ice thickness where:

zri = 2.7RiSi + zui (4.10)

48

R is the fractional area of ridged ice, S is the ridge sail height and 2.7RiSi represents

the cross-sectional area of the sail (0.5RiSi) and keel (2.2RiSi) (Figure 4.4a). Level

ice (zui) is estimated following:

zui = Rizui + (1 − Ri)zui (4.11)

Substituting zui in equation 4.10 with equation 4.11 and conserving mass by moving

half of the level ice located between the sail and the keel (Rizui) to the level ice

(Figure 4.4b), ridged ice thickness becomes:

zri =Ri

2(5.4Si + zui) +

!1 − Ri

2

"zui (4.12)

(Worby et al. 2008).

4.2 Remotely-Sensed Antarctic Sea-Ice Thickness Estimates

The National Ice Center (NIC), a joint organization sponsored by the U.S.

Navy, the National Oceanic and Atmospheric Administration (NOAA), and the U.S.

Coast Guard, has produced weekly ice charts for the Southern Ocean since the early

1970s. The sea-ice charts are produced by trained sea-ice analysts, who discern sea-

ice concentration and stage-of-development information, more commonly referred

to as ice type, using aircraft reconnaissance, visible and infrared Advanced Very

High Resolution Radiometer (AVHRR), Operational Line Scanner (OLS) imagery,

passive microwave Electrically Scanning Microwave Radiometer (ESMR), SMMR,

and SSM/I data, Synthetic Aperature Radar (SAR) data, and freezing degree-day

models (NIC et al. 1996, Schellenberg et al. 2002). The NIC charts are produced

weekly, reporting average ice conditions integrated from data usually collected over

a 3-5 day period. Sea-ice charts represent areas of homogenous sea-ice conditions as

discrete polygons with information on sea-ice concentration, ice type, and form as

mandated by the World Meteorological Organization (Figure 4.5) (WMO 1970).

Symbols representing sea-ice information for each polygon of homogenous ice

conditions are displayed on the chart within an egg code. An egg code contains

49

Figure 4.4: Schematic diagram of (A) sea-ice ridge structure defined by equation4.10 and (B) a numerical approximation of sea-ice ridge structure.Modified from Worby et al. (2008; Figure 3).

50

Figure 4.5: Ross Sea section of an operational sea-ice chart for the week of Septem-ber 19, 1997 (NIC 1997).

51

Figure 4.6: Schematic diagram of the type of sea-ice information provided by and”eggcode”. A typical eggcode always contains information on totalconcentration (C), usually contains information on one or more of thepartial concentrations (Ca,Cb, and Cc) and corresponding ice type orstages of development (Sa,Sb, and Sc) along with the floe character-istics of the partial ice category (Fa). The remaining information isoccasionally (So and Sd) or rarely (Fb and Fc) included. Modified fromDedrick et al. (2001)

information on the total concentration of sea ice within individual polygons as well

as the partial concentrations, stages of development, and any notable ice forms

associated with thickest (i = a), middle (i = b), and thinnest (i = c) ice within

each polygon (Figure 4.6). In addition to displaying important information about

the within each sea-ice polygon, the information within an egg code is the basis for

creating digital ice charts used to calculate sea-ice thickness (Dedrick et al. 2001).

NIC operational sea-ice charts in Sea Ice Data in Digital Form (SIGRID)

52

Table 4.2: WMO and SIGRID classification of sea-ice thickness and stage-of-development used by the NIC.

Egg Code Ice SIGRID Ice Stage ofThickness Thickness Code Development

0 00 Ice Free1 81 New Ice (<10 cm: Slush, Shuga)2 82 Nilas (<10 cm)4 84 Grey Ice (10-15 cm)3 83 Young Ice (10-30 cm)5 85 Grey-White Ice (15-30 cm)6 86 First-Year Ice (30-200 cm)7 87 White Ice (30-70 cm)1. 91 First-Year Ice (70-120 cm)4. 93 First-Year Ice (>120 cm)7. 95 Old Ice (survived one summer)F8 08 Land-Fast Ice

80, 98, 99, -1 No Data

are available for the Southern Ocean from 1973 to 1994 and contain information on

total sea-ice concentration with a special designation for fast ice (NIC 1996). Be-

cause these data do not contain stage-of-development information, sea-ice thickness

estimates cannot be obtained for this time period. Charts reporting ice type were

introduced in 1995 and contain additional information on first-year ice, dividing

the first-year ice categories into thin, medium, and thick ice, from which reliable

thickness estimates were obtained (Table 4.2). In late 1999, the three first-year

ice categories were combined into one first-year ice category, ranging in thickness

from 70-120 cm, and are unsuitable for thickness estimates (Dedrick et al. 2001,

Schellenberg 2002, Geiger et al. 2000).

The detail and accuracy of individual ice charts vary from year to year de-

pending on the quality, resolution, availability of remotely-sensed data, and carto-

graphic measures used by the analyst. A change in cartographic techniques resulted

53

in a marked decrease in the area of individual sea-ice polygons between the pa-

per charts produced prior to October 1997 and the more recent, digital ice charts

(Schellenberg 2002). Other sources of error include ice analyst expertise and the

subjective nature of image interpretation, ice-type determination, and delineating

boundaries (Dedrick et al. 2001, Schellenberg 2002). Given the resolution of the

imagery used to produce the NIC ice charts, areas smaller than 20 to 40 km2 cannot

be resolved and polygon boundaries are typically accurate within ±10 km (Enomoto

and Ohmura 1990).

Weekly, digitized or scanned paper sea-ice charts for 1995 through 1998 were

processed using ESRI’s ArcGIS resulting in a four-year, digital record of weekly

sea-ice conditions for the Southern Ocean, in a vector format, from which sea-ice

thickness estimates were obtained (Schellenberg 2002, Schellenberg et al. 2002,

DeLiberty et al. 2004, DeLiberty and Geiger 2005). Thickness estimates, in cen-

timeters, were calculated from weekly NIC ice charts for 1995 through 1998 using

sea-ice concentration (Ci) and ice type (Si) observations for the thickest (i=1), mid-

dle (i=2), and thinnest (i=3) ice within each sea-ice polygon. Stage-of-development

is a quantifiable indicator of ice type determined from remotely sensed data. Spe-

cific ice types are associated with a range of thicknesses and the midpoint of each

range serves as a proxy for sea-ice thickness used here to estimate total ice thickness

(ZNIC) (Schellenberg 2002).

ZNIC =30

i=1

Ci

10Si (4.13)

The weekly vector maps containing sea-ice thickness estimates were converted to a

raster data format using ESRI’s ArcGIS. Sea-ice thickness maps in a raster format

generated from vector polygon maps contain 50km2 equal-area grid cells.

Concentration and thickness estimates derived from NIC sea-ice charts have

an associated level of uncertainty following equations (4.1) through (4.6). The error

54

associated with the total thickness estimate for each NIC sea-ice polygon is defined

as:

Z &NIC =

130

i=1

2!Ci

c&i/10

"2

+

!z&ilSi

"23 !

Ci

10Si

"241/2

(4.14)

where sea ice is assumed to be level and c&i and z&i are defined by equations (4.4)

and (4.5) respectively (Schellenberg 2002). The assumption that the sea ice is level

leads to an underestimation of ridged and rafted ice within the NIC dataset.

4.3 Evaluation of Sea-Ice Thickness Data

In order to assess the utility of the NIC sea-ice thickness estimates for model

evaluation, the remotely-sensed estimates were compared to the ASPeCt in situ

thickness estimates to determine the accuracy of the NIC estimates and the signifi-

cance of the spatial distribution of observations. To test the accuracy of the thick-

ness estimates derived from remotely-sensed sea-ice observations, the NIC thickness

estimates were correlated against in situ total thickness estimates following Schellen-

berg (2002). The significance of the spatial distribution of observations, evaluated

in terms of resolution error, was determined for remotely-sensed and in situ datasets

following Willmott and Johnson (2005).

4.3.1 Data Comparison

Assessment of the potential utility of thickness estimates calculated using

NIC ice charts was conducted using five weeks of coincident in situ and satellite

data for the Ross Sea sector. Remotely-sensed and in situ estimates for correspond-

ing time periods were reprojected, and spatially and temporally merged (Figure

4.7). Because ship thickness estimates represent point observations, an average

of the point thickness estimates within each individual polygon were compared to

the remotely-sensed thickness estimate for the corresponding polygon (Schellenberg

2002, DeLiberty et al. 2004).

55

NoData0 100 30 70 120 200 cm

0 210 420 630 840Kilometers

180∞160∞E 160∞W

75∞S

0 60 120Kilometers

65∞S

175∞W

65∞S

NoData0 100 30 70 120 200 cm

0 210 420 630 840Kilometers

180∞160∞E 160∞W

75∞S

0 60 120Kilometers

65∞S

175∞W

NoData0 100 30 70 120 200 cm

0 210 420 630 840Kilometers

180∞160∞E 160∞W

75∞S

NoData0 100 30 70 120 200 cm

NoData0 100 30 70 120 200 cm

0 210 420 630 840Kilometers

180∞160∞E 160∞W

75∞S

0 210 420 630 840Kilometers

180∞160∞E 160∞W

75∞S

0 210 420 630 840Kilometers

180∞160∞E 160∞W

75∞S

0 60 120Kilometers

65∞S

175∞W

0 60 120Kilometers

0 60 120Kilometers

0 60 120Kilometers

65∞S

175∞W

65∞S

Figure 4.7: Comparison of thickness estimates derived from NIC remotely-sensedsea-ice observations and in situ total thickness estimates calculatedusing ASPeCt voyage data within the Ross Sea sector for May, June,and August of 1995 and May and June of 1998. Modified from Schel-lenberg (2002), figure 5.3, p. 45.

56

Results of the data comparison indicate that there is an encouraging correla-

tion between in situ and remotely-sensed thickness estimates for total ice thickness

(Figure 4.8). The NIC estimates reasonably approximate sea-ice thickness for thin-

ner ice but tend to underestimate thicker ice, resulting in a higher mean absolute

error (MAE = 26.1 cm) and lower index of agreement (d1 = 0.57). The underesti-

mation of thicker ice within remotely-sensed thickness estimates reflect the inability

of the aereal image to determine the location of ridges, which contain the thickest

ice within an ice pack (Worby 1999). Remotely-sensed estimates of thicker ice would

be improved by the implementation of an algorithm that would account for ridged

ice.

4.3.2 Resolution Error

Flaws in climate data may reflect biases resulting from both the variability

within the data and grid-cell size. The errors associated with grid-cell size, or spatial

resolution errors, are a commonly overlooked source of inaccuracies. Climate data

and model output are often spatially averaged to coarse resolutions, which decreases

the spatial variability within and the variability about the mean value of the dataset.

As a result, bias errors (EB) usually decrease as resolution decreases, which may

lead to misinterpretation of the quality of a dataset (Willmott and Johnson 2005).

Therefore, errors attributed to spatial resolution, or resolution error (ER), must be

considered when evaluating the total error (ET ) associated with a set of gridded

spatial data such that:

ET = ER + EB (4.16)

(Willmott and Johnson 2005).

The e%ect of spatial resolution on gridded sea-ice thickness observations,

evaluated in terms of resolution error, was determined for remotely-sensed and in

situ thickness estimates following Willmott and Johnson (2005). The resolution

57

error associated with gridded sea-ice data accounts for sea-ice processes that occur

at sub-grid-cell resolutions. Average, seasonal sea-ice thickness distributions were

produced from weekly NIC thickness estimates by calculating the average thickness

value at the centroid of each raster grid cell for all weeks within each season. The

seasonal distributions of ASpeCt sea-ice thickness estimates were generated using

all sea-ice thickness observations within each season.

Seasonally averaged thickness estimates for the Southern Ocean, south of

−60" latitude, derived from ASPeCt in situ observations and NIC operational sea-

ice charts were interpolated to spherical 0.5" resolution using the inverse distance

weighting algorithm (I) such that:

zk = I[/znk,.,(] =

5nki=1 wizi5nki=1 wi

(4.17)

where zk is the interpolated average thickness estimate at each k grid-cell centroid,

and /znkis a vector containing nk=15 neighboring thickness values that influence

zk, and wi accounts for the spherical distance between the grid point and and the

observation (Willmott and Matsuura 1995).

The gridded in situ and remotely-sensed thicknesses were spatially aggregated

to coarse resolutions (R) following:

ZRi =

53j=1 wijZ0.5ij5m"

j=1 wij

(4.18)

where:

m' =

!R

0.5

"2

(4.19)

for R = 1, ..., 5 and ZRi represents a reliable spatial average of the 0.5" thickness

values aggregated to 1", 2", 3", 4", and 5" resolutions.

Results of the resolution error assessment indicate that there is an inverse

relationship between the amount of sub-grid-scale variability and both the spatial

resolution and the number of observations (Figure 4.9). The highest resolution er-

rors are associated with coarser, or lower, resolutions as well as fewer observations.

58

The e%ect of the number of observations on the resolution error is clearly shown

given the much higher resolution errors associated with in situ estimates compared

with remotely-sensed. The resolution error associated with in situ estimates is much

greater than those associated with remotely-sensed observations and is highest dur-

ing the seasons with the fewest number of observations. Resolution error also in-

creases as the areal extent of sea-ice coverage increases during the winter and fall

seasons, regardless of the number of observations. This indicates a relationship be-

tween resolution error and the number of non-zero grid cells, which is greater for

seasons with a greater sea-ice extent.

4.4 Summary

The evaluation of remotely-sensed sea-ice thickness estimates, in terms of

accuracy and spatial coverage, indicates that although in situ thickness estimates

are a more accurate representation of the ice thickness at the observed locations, the

continuous spatial extent NIC ice chart estimates provide a better representation

of the hemispheric distribution of sea-ice thickness. Previous studies also indicate

that thickness estimates obtained using the NIC ice charts correspond with the

annual sea-ice cycle, thus providing a reasonable approximation of the annual cycle

of ice growth and decay (DeLiberty et al. 2004, DeLiberty and Geiger 2005). The

largest errors within the NIC thickness record occur within the estimates for thicker

ice, which tend to be underestimated due to the fact that the dataset does not

account for ridged ice (Dedrick et al. 2001). However, the spatial resolution errors

associated with in situ thickness estimates are much greater than those associated

with remotely-sensed thickness estimates due to gaps in the spatial coverage of in

situ observations compared with the hemispheric spatial coverage of remotely-sensed

observations. Therefore, the hemispheric record of ice conditions, which includes

sea-ice thickness, provided by the NIC operational sea-ice charts are appropriate for

use in model evaluation.

59

Figure 4.8: The correlation between coincident NIC and ASPeCt sea-ice thicknessestimates.

60

0

10

20

30

40

50

60

70

80

90

1 2 3 4 5

Resolution (degree)

Figure 4.9: Resolution errors associated with ASpeCt (solid) and NIC (open) sea-ice thickness estimates for summer (red), fall (orange), winter (blue),and spring (green) at 1, 2, 3, 4, and 5 degree grid-cell resolutions.

61

Chapter 5

MODEL EVALUATION

The National Center for Atmospheric Research (NCAR) Community Climate

System Model, version 3.0 (CCSM3) is a coupled climate system model with dy-

namic and data-driven versions of the atmosphere, land surface, ocean, and sea-ice

components. The dynamic versions of each model component interact with a central

coupler, sending data to and receiving forcing information from the coupler. Data

is sent but no forcing information is received from the coupler for the data-driven

versions of each model component.

The Community Sea Ice Model, version 5.0 (CSIM5) is the fifth version of

the sea-ice component of the CCSM3 and can be used as part of a coupled CCSM3

simulation or as a stand-alone sea-ice model (Collins et al. 2005). The model

was used as a stand-alone model in this study to isolate errors due to ice-model

physics from those produced by coupled model components. Identification of errors

directly related to ice physics provides the basis for understanding how the ice model

interacts with other model components.

5.1 Model Description

The sea-ice component is a dynamic-thermodynamic, sea-ice thickness dis-

tribution model in which energy and ice volume are conserved by using an elastic-

viscous-plastic rheology, horizontal and vertical redistribution of ice mass, and mul-

tiple ice-thickness categories (Briegleb er al. 2004). Thermodynamic and dynamic

62

sea-ice processes are represented by a sub-grid-scale ice-thickness distribution fol-

lowing Thorndike et al. (1975),

*g

*t=

*

*hi(fg) + L(g) −# • (vg) + &(h, g,v) (5.1)

where g(h)dh is the fraction of ice within ice thickness catagory h to h+dh and the

cumulative distribution function is represented by,

G(h) =

, h

0

g(h)dh (5.2)

The first term computes the change in sea ice thickness due to stationary melt and

thaw, the second accounts for lateral melting and ice formation, the third represents

the horizontal divergence, and the fourth computes changes in ice thickness due to

rafting and ridging (Thorndike et al. 1975, Holland et al. 2005).

Equation 5.1 is solved in stages beginning with changes in ice thickness due

to thermodynamic growth/melt, followed by dynamic transport and the mechanical

redistribution of sea ice. The thermodynamic growth rate (f) determines the initial

fractional ice area within each thickness category (term one), which is then redis-

tributed by lateral, thermodynamic ice growth/melt (term two). Ice velocity is then

computed, transporting ice horizontally (term three), followed by the mechanical

redistribution of ice between ice-thickness categories using a parameterized ridging

and rafting scheme (Holland et al. 2005).

In addition to being solved in stages, equation 5.1 is divided into Mc number

of thickness categories bound by h'mc

, mc = 0, 1, 2...Mc, plus an additional category

for open water, h'0 (Briegleb et al. 2005, Holland et al. 2005). Bitz et al. (2001)

investigated the e%ect of the number of thickness categories (Mc) on simulated ice

thickness distribution. Setting Mc = 1, 3, 5, 10, and 15, with an additional category

for open water, it was determined that the area consisting of open water and thin

ice responds little to Mc > 5.

63

The fractional ice area within each thickness category, mc, is solved using a

discrete set of equations following,

Amc =

, h"mc

h"mc!1

g(h)dh (5.3)

where the total area covered by ice h > 0 is A =5Mc

mc=1 Amc (Briegleb et al. 2004).

Ice volume (V ) is also determined for each category mc following,

Vmc =

, h"mc

h"mc!1

hg(h)dh (5.4)

Fractional ice area (Amc) and volume (Vmc) within each thickness category are eval-

uated in terms of thermodynamic (ST ) and mechanical (SM) sources/sinks such

that*Amc

*t= STAmc

−# • (uAmc) + SMAmc(mc = 1, 2, ...Mc) (5.5)

and*Vmc

*t= STVmc

−# • (uVmc) + SMVmc(mc = 1, 2, ...Mc) (5.6)

from which sea-ice thickness (hmc = Vmc/Amc) is derived (Briegleb et al. 2005).

Linear remapping transports ice between neighboring thickness categories

in response to vertical thermodynamic processes (Briegleb et al. 2004, Holland et

al. 2005). When ice grows beyond the upper boundary of the thickness category

(h'mc

), it is added to the category directly above (mc+1). Conversely, ice that melts

below the lower bound (h'mc!1

) is added to the category directly below (mc!1). Ice

that melts completely is assumed to have a thickness equal to the lower bound

of category one (h'0). Overall, the redistribution of ice between multiple thickness

categories preserves not only area but volume and energy as well (Briegleb et al.

2004).

5.2 Model Configuration

The model was configured using a dipole, 1" nominal horizontal grid (gx1v3)

of 320 longitudes and 384 latitudes (Vertenstein et al. 2004a, Collins et al. in

64

press), Mc = 5 thickness categories, distributed to reflect an Arctic ice thickness

distribution, plus open water (Table 5.1), and four vertical ice layers plus one snow

layer (Briegleb et al. 2004). Vertical melt or growth within the ice column is

determined by temperature dependent fluxes computed for the boundaries between

each layer of ice and snow (Holland et al. 2005). A slab ocean mixed layer model

distributed with CSIM5 was used to calculate ocean surface energy and momentum

transfers to the ice pack (Schramm et al. 2004). Atmospheric flux variables were

calculated using an atmospheric boundary layer model that uses stability based flux

calculations to determine the transfer of energy and momentum from the atmosphere

to the ice surface (Hunke csim code).

Table 5.1: The sea-ice thickness distribution (cm) used by the CSIM. The rangeincludes all thicknesses within each thickness category, Mc.

M Range0 01 0+ − 0.652 0.65 − 1.393 1.39 − 2.474 2.47 − 4.605 > 4.60

The oceanic heat flux is calculated using sea surface temperature and salinity

provided by Parallel Ocean Program (POP) model output (Schramm et al. 2004).

Potential heat flux from the ocean determines the rate of frazil ice growth within

open water (positive) and basal and lateral heat transfer (negative; Holland et al.

2005). The atmospheric heat contribution is a function of parameterized solar and

terrestrial energy inputs and atmospheric moisture content. Ice motion is driven by

variable surface ocean currents from POP model output and constant wind speed

and direction set within the atmospheric boundary layer model (Hunke 2004).

65

5.3 Atmospheric Boundary Layer Calculations

Sea ice exists at the boundary between the atmosphere and ocean and re-

sponds to the transfer of energy between the atmosphere and the ocean by thermo-

dynamic growth or melt (Eq. 2.14). Ice also responds to the transfer of momentum

from the overlying atmosphere and the underlying ocean by ice motion, expressed in

terms of ice velocity (Eq. 2.2). Therefore, a reasonable representation of constantly

fluctuating atmospheric and oceanic fluxes is necessary to produce a reasonable

ice-thickness distribution with respect to time and space.

Oceanic thermodynamic and dynamic flux variables are provided by ocean

model output, which varies with each time step. Variations in oceanic flux variables

represent both daily and annual cycles in the exchange of energy and momentum be-

tween the ocean and overlying ice pack. Atmospheric thermodynamic and dynamic

fluxes are also calculated every time step however, many important atmospheric

flux components are parameterized (Hunke 2004). Parameterization of atmospheric

state variables, such as incoming solar energy and wind speed, do not represent daily

and annual cycles in energy and momentum fluxes from the atmosphere (Table 5.2).

Table 5.2: Parameterized values of commonly used atmospheric terms set withinthe CSIM5 atmospheric model (Hunke 2004).

Variable Term Value Description Unitsua uatm 5 Horizontal Wind Component ms!1

va vatm 5 Vertical Wind Component ms!1

QSW# Fsw 0 Incoming Shortwave Radiation Wm!2

... swvdr 0 Shortwave Visible Direct Wm!2

... swvdf 0 Shortwave Visible Di%use Wm!2

... swidr 0 Shortwave Infrared Direct Wm!2

... swidf 0 Shortwave Infrared Di%use Wm!2

Ta Tair 273 Air Temperature Kqa Qa 0.014 Specific Humidity kgkg!1

QLW # Flw 280 Incoming Longwave Wm!2

66

Parmeterization of the atmospheric energy contribution to the ice pack limits

the ability of the thermodynamic model to represent seasonal variations in the energy

balance and therefore seasonal ice growth and melt. Given the paramaterization of

both shortwave and longwave energy terms within the model, Eq. 2.14 can be

rewritten as:

QH + QL + 280.0 − QLW$ + QG0 =

%&

'0 if T0 < Tf

QM if T0 = Tf

(5.7)

where QLW# = 280 Wm!2 and all shortwave terms are omitted. The transfer of

sensible heat between the atmosphere and the surface can not vary seasonally due

to the parameterization of air temperature. Latent heat transfer is highly influenced

by the parameterized specific humidity value, which does not allow humidity to vary

with wind direction.

Model simulations generated using the parameterized atmosphere included in

the original model distribution result in an atmosphere that is too warm to support

sea ice (Ta, qa, and QLW #). Thin ice within the initial ice pack, set by POP model

output, melts during the summer months of January and February, leaving only

very thick, multi-year and landfast ice along coast. Ice melt continued into the fall

months of March, April, and May until even the thickest ice melted, resulting in

an ice pack of zero thickness by the middle of June. Given the constant ice melt

produced by the ice model, a more realistic representation of energy flux variables

was added to the model.

5.3.1 Incoming Shortwave Radiation

Incoming shortwave radiation (QSW#) varies both temporally and spatially

from 0Wm!2 during the winter months of June, July, and August to over 1000Wm!2

67

at 50 "S latitude around the winter solstice (Figure 5.1). Incoming shortwave radi-

ation was estimating for all latitudes at each hourly time step such that:

QSW# = So

!d

d

"2

cos"s (5.8)

where So is the solar constant,#

dd

$2

corrects for earth’s eccentricity, and cos"s is

the solar zenith angle (Sellers 1965, Washington and Parkinson 2005).Incoming Shortwave Flux (Wm-2)

0

200

400

600

800

1000

1200

1400

0 50 100 150 200 250 300 350

Julian Day

Figure 5.1: Annual distribution of incoming shortwave radiation for 50 "S (lightblue), 60 "S (medium blue), 70 "S (dark blue), 80 "S (purple) latitudefollowing equation 5.8. A Julian day of 0 represents January 1st.

The eccentricity correction, derived from Spencer’s (1971) fourier series, ex-

pands to:!

d

d

"2

= 1.00011 + 0.34221 cos"d + 0.00128 sin"d

+0.000719 cos 2"d + 0.000077 sin 2"d

68

(5.9)

where the time of year ("d) was determined using the day number (dn) from n = 0

to 364.

"d =20dn

365(5.10)

The solar zenith angle accounts for lat ((), declination (*), formulated as a function

of Julian day, and the hour angle (H).

cos"s = sin ( sin * + cos( cos * cos H (5.11)

* = 0.006918− 0.399912 cos"d + 0.070257 sin"d

−0.006758 cos 2"d + 0.000907 sin 2"d

−0.002697 cos 3"d + 0.00148 sin 3"d

(5.12)

The hour angle increases in magnitude by pi/12 or 15" for every hour before or after

solar noon, when it is equal to zero (Sellers 1965, Washington and Parkinson 2005).

Accounting for hourly, daily, and latitudinal variations in solar input results

in a more reasonable representation of the surface energy balance. However, it also

increases the energy input into the ice pack, leading to further ice melt. Therefore,

additional calculations were necessary to correct for parameterized energy balance

components.

5.3.2 Incoming Longwave Radiation

Incoming longwave radiation (QLW#) varies depending on the amount of en-

ergy emitted by the atmosphere toward the surface. The amount of energy emitted

by the atmosphere may be treated as a function of air temperature (Ta) such that

QLW# = "'T 4a (5.13)

69

assuming an atmospheric emissivity (") of one and ' is the Stefan-Boltzmann con-

stant. Longwave emission toward the surface is also a function of cloud cover fraction

(cc), which is not considered in the model. Clouds absorb upwelling longwave radi-

ation and reradiate longwave energy toward the surface. High cloud cover fractions

result in greater longwave flux toward the surface than clear sky conditions. There-

fore, average monthly cloud cover fraction estimates, which vary by latitude (Figure

5.2), were input into the model and a cloud cover approximation was added to the

Stefan-Boltzmann equation (Geiger et al. 1997).

QLW# = "'T 4a

61.0 − 0.261 exp [−0.000777(273.0− Ta)

27× (1.0 + 0.275cc) (5.13)

Using a cloud cover approximation to estimate incoming longwave radiation

for air temperatures below freezing results in longwave values below the parameter-

ized value of 280Wm!2 set in the model (Figure 5.3). Creating a variable approxi-

mation of incoming longwave radiation within the model lowers the surface energy

available for ice melt. However, this does not lower the surface energy enough to

produce a realistic sea-ice distribution.

5.3.3 Specific Humidity

The distribution of sea ice within CSIM5 is particularly sensitive to atmo-

spheric moisture due to the influence of latent heat on ice growth and melt. High

humidity values result in a greater latent heat flux toward the ice surface, which

promotes ice melt. It also leads to a decrease in ice area by melting new ice as it

forms. Specific humidity (q) is a measure of the moisture content of the atmosphere.

Relative humidity is a function of the specific humidity and temperature such that

RH =q

qs100 (5.14)

where qs represents the maximum amount of moisture the atmosphere can hold at

a given temperature. Colder air has a lower capacity for moisture than warm air.

70

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

-80 -70 -60 -50 -40

Latitude

Clou

d Co

ver

Frac

tion

Figure 5.2: Monthly average cloud cover fraction by latitude. Cloud cover is low-est in July (purple), steadily increasing during the months of Au-gust (blue), September (Green), October (yellow), November (lightorange), and December (orange) to the period of highest cloud coverin January (red). Cloud cover then decreases again from Januarythrough February (orange), March (light orange), April (yellow), May(green), and June (blue). Modified from Parkinson and Washington(1979, figure 9).

71

Temperature (oC)

Downwelling Longwave Flux Wm-2

210

220

230

240

250

260

270

280

-10 -8 -6 -4 -2 0

Figure 5.3: Longwave flux estimates for air temperatures below freezing assuminga maximum cloud cover of 85 % (red) and a minimum cloud cover of40 % (blue) following equation 5.13.

Polar air temperatures typically range from near to well-below freezing, reducing

the amount of moisture necessary to reach saturation (Washington and Parkinson

2005).

The value for the specific humidity originally used by the model (q = 14gkg!1)

results in relative humidity values well above saturation for temperatures just below

freezing (Figure 5.4). This results in a latent heat flux toward the ice surface over

400 Wm!2. Typically, polar latent heat flux values should be low in magnitude

(< 100Wm!2) and primarily negative (away from the surface), depending on the

amount of ice growth. Assuming an average relative humidity value of 50 % for the

Southern Ocean, a specific humidity value of 2 gkg!1 results in a more reasonable

72

latent heat flux for the Antarctic sea-ice environment (Figure 5.5).

370

380

390

400

410

420

430

440

450

-2 -1.5 -1 -0.5 0

Temperature (oC)

Figure 5.4: Relative humidity estimates over ice (gray) and water (black) for airtemperatures below freezing for q = 14 gkg!1 following equation 5.14.

Humidity varies greatly with wind direction. Air flow from the continent is

dry, with relative humidity values below 50 %. Air flow originating over open water

is more humid, with relative humidity values possibly exceeding 50 %. A parame-

terized specific humidity value assuming a RH = 50% will result in a higher specific

humidity, causing higher latent heat flux values, for areas dominated by continental

air flow, which tends to be drier. Assuming a RH = 50 % for areas along the ice

edge, which are dominated by marine air flow, will result in an underestimation

of atmospheric humidity and lower latent heat values (Figure 5.6) (Geiger pers.

communication).

73

-200

-150

-100

-50

0

50

100

150

200

250

300

350

400

450

500

-2 -1.5 -1 -0.5 0

Temperature (oC)

Late

nt H

eat

Flux

(Wm

-2)

Figure 5.5: Latent heat flux estimates for air temperatures below freezing for dif-ferent values of specific humidity. Specific humidity values representedinclude 2 gkg!1 (blue), 3 gkg!1 (green), 4 gkg!1 (yellow), 5 gkg!1 (or-ange), and 14 gkg!1 (red).

Due to the influence of wind direction on humidity and the latent heat flux,

the value of the specific humidity was changed from a parameter assuming 50 %

relative humidity to a variable as a function of wind direction and air temperature.

Continental and marine influences were simulated using ECMWF wind direction

data from 1991. The relative humidity within grid cells with a southerly wind com-

ponent was set at 40 % to represent the drier, continental influence. A relative

humidity of 60 % was used for grid cells with a northerly wind component to repre-

sent a more humid marine influence (Geiger pers. communication). The maximum

74

Spec

ific

Hum

idit

y (g

kg-1)

Temperature (oC)

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

-2 -1.5 -1 -0.5 0

Figure 5.6: Specific humidity values for temperatures below freezing assuming arelative humidity of 60 % (light blue), 50 % (medium blue), and 40 %(dark blue) following equation 5.18.

specific humidity (qs) was determined using the saturation vapor pressure as a func-

tion of average monthly air temperature (Ta), derived from ECMWF data, following

Bolton (1980)

es = 611.2 ∗ exp17.67 ∗ Ta

243.5 + Ta(5.15)

where air temperature is in degrees Celsius and

qs =0.622 ∗ es

P − es(5.16)

Air pressure (P ) was determined by

P = $a ∗ R ∗ Ta (5.17)

75

where R = 287.0 Jkg!1K!1 is the universal gas constant and $a = 1.3 kgm!3 is the

density of the atmosphere. The specific humidity was then determined by

q =RH ∗ qs

100(5.18)

(Washington and Parkinson 2005).

Accounting for continental and marine influences on humidity lead to in-

creased ice thickness near the coast and slightly reduced the overall ice extent.

However, erroneously higher than normal latent heat flux values near the coast,

where air flow is strongly influenced by the continent, reduced ice growth and new

ice formation, especially during the fall months of March, April, and May. Erro-

neously lower humidity, and therefore lower than normal latent heat flux values,

along the ice margin, where there is a stronger marine influence, resulted in dimin-

ished ice melt (Figures 5.7-10).

The resulting ice distribution included thinner than normal ice along the coast

with large polynas developing within the Weddell and Ross Seas, and a greater than

normal sea-ice extent. A more sophisticated treatment of humidity within the Ross

and Weddell Seas, where katabatic winds have a strong influence on latent heat

transfer, is necessary to improve the simulated sea-ice distribution. There was also

a lag in the start of the growth season due to higher than normal latent heat values

near the coast, which inhibited frazil ice growth. Improvements also must be made

in the transfer of latent heat through the ocean-atmosphere interface during the fall

growth period.

5.4 Model Evaluation

The simulated sea ice thickness distribution for the Southern Hemisphere pro-

duced by the CSIM5 was evaluated using sea-ice thickness estimates derived from the

National Ice Center operational sea-ice charts for Antarctica. Model performance

was first evaluated using the distribution of model prediction errors throughout the

76

cm

0 854 km

Figure 5.7: The sea-ice thickness distribution produced by the CSIM5 for thesummer months of December, January, and February.

77

cm

0 854 km

Figure 5.8: The sea-ice thickness distribution produced by the CSIM5 for the fallmonths of March, April, and May.

78

cm

0 854 km

Figure 5.9: The sea-ice thickness distribution produced by the CSIM5 for wintermonths of June, July, and August.

79

cm

0 854 km

Figure 5.10: The sea-ice thickness distribution produced by the CSIM5 for thespring months of September, October, and November.

80

Southern Ocean (Willmott and Matsuura 2005). Average model performance was

then assessed in terms of the average error associated with seasonally averaged simu-

lated gridded thickness estimates as well as the degree to which the model predicted

the distribution of gridded thickness estimates (Willmott 1981, Legates and McCabe

1999, Willmott and Matsuura 2005).

5.4.1 Model Evaluation Statistics

Model prediction errors (ei) associated with each grid cell were assessed for

seasonally averaged simulated ice-thickness distributions. Seasonly-averaged, sim-

ulated sea-ice thickness distributions were determined by calculating the average

sea-ice thickness value for each grid cell from the monthly distributions within each

season. The di%erence between simulated, or predicted (Pi), ice-thickness estimates

and NIC, or observed (Oi), thickness estimates were determined for each correspond-

ing grid cell.

ei = Pi − Oi (5.19)

Errors associated with each grid cell were mapped to determine areas where the

model over-predicted (ei > 0) sea-ice thickness with respect to observations and

areas where the model under-predicted (ei < 0) sea-ice thickness (Willmott and

Matsuura 2005).

The mean absolute error (MAE) represents the magnitude of the average

error within a set of predicted values, of size N , when compared to observations.

Average errors are weighted by a latitudinal dependent area weight (wi).

MAE =

5Ni=1 wi |Pi − Oi|5N

i=1 wi

(5.20)

wi = | sin((u − (l)| (5.21)

The MAE is the upper limit of the average or mean bias error (MBE) and the lower

limit of the root-mean-square error (RMSE) such that MBE ≤ MAE ≤ RMSE.

81

Although RMSE is often used in model evaluation studies, it is not a useful statistic

in that it reflects not only the average error, but the variance about the average error

and the sample size (Willmott et al. 1985, Willmott and Matsuura 2005).

The index of agreement (d1) statistic is a dimensionless measure of error

between 0 and 1 that accounts for di%erences between individual observed and pre-

dicted values as well as the di%erence from the average observed (O) value (Willmott

et al. 1985, Willmott pers. communication).

d1 = 1.0 −5N

i=1 wi(Oi − Pi)5Ni=1 wi

88Pi − O88 +

88Oi − O88 (5.22)

The index of agreement is a measure of how well the model predicts the distribution

of sea-ice thickness within the simulated ice pack. A d1 value of 1 indicates a perfect

agreement between the model and the data while a value of 0 indicates that there

is no agreement between predicted and observed ice estimates. The interpretation

of d1 is similar to the coe!cient of determination (r2), where higher values indicate

better agreement. However, d1 is less sensitive to outliers than r2, resulting in a

more informative measure of model performance (Legates and McCabe 1999).

5.4.2 Model Performance

Evaluation of grid-cell model prediction errors indicates that the model gen-

erally underpredicts sea-ice thickness in all seasons (Figures 5.11-14). The largest

errors occur along the coast and within the Weddell Sea due to the sensitivity of

the model to erroneously high latent heat values near the coast. The greatest area

of underpredicted sea-ice thicknesses occurs during the fall months in which new

ice formation is delayed by the greater than normal latent heat flux predicted by

the model. The model tends to overpredict sea-ice thickness along the ice margin,

particularly during the winter. The overestimation of ice thickness along the outer

edge of the ice pack can also be explained by erroneous latent heat, which reflect an

underestimation of the marine influence on humidity and the latent heat flux.

82

No Data -1.0 -0.5 0.0 0.0 0.5 1.0 m

0 854 km

Figure 5.11: The di%erence between the sea-ice thickness distribution producedby the CSIM5 (Pi) and that estimated using NIC operational sea-icecharts (Oi) for summer.

83

No Data -1.0 -0.5 0.0 0.0 0.5 1.0 m

0 854 km

Figure 5.12: The di%erence between the sea-ice thickness distribution producedby the CSIM5 (Pi) and that estimated using NIC operational sea-icecharts (Oi) for fall.

84

No Data -1.0 -0.5 0.0 0.0 0.5 1.0 m

0 854 km

Figure 5.13: The di%erence between the sea-ice thickness distribution producedby the CSIM5 (Pi) and that estimated using NIC operational sea-icecharts (Oi) for winter.

85

No Data -1.0 -0.5 0.0 0.0 0.5 1.0 m

0 854 km

Figure 5.14: The di%erence between the sea-ice thickness distribution producedby the CSIM5 (Pi) and that estimated using NIC operational sea-icecharts (Oi) for spring.

86

Results of the model evaluation statistics indicate that the model performs

reasonably well given the low MAE values (annual average MAE value of 0.10 m)

and d1 values above 0.75 (Table 5.3). The model best simulates the observed sea-

ice environment during the summer months, during which ice melt is the dominant

thermodynamic process. The highest MAE and lowest d1 values occur during the

period of greatest sea-ice extent in which the influence of the latent heat flux on ice

growth is greatest.

Table 5.3: Model evaluation statistics for seasonally averaged simulated ice thick-ness distributions produced by CSIM5.

Season MAE d1

Summer (DJF) 0.075 m 0.78Fall (MAM) 0.068 m 0.76Winter (JJA) 0.108 m 0.77Spring (SON) 0.154 m 0.75

Although the model generally underpredicts sea-ice thickness, it predicts an

appropriate seasonal cycle of ice growth and melt and a reasonable ice-pack extent.

The model also simulates a reasonable ice thickness distribution in which ice is

thickest within the interior of the ice pack and thins toward the ice margin. Errors

within the simulated ice pack are the result of poor model treatment of atmospheric

humidity, which strongly influences the surface latent heat flux and the total energy

available for thermodynamic sea-ice processes.

87

Chapter 6

SUMMARY AND CONCLUSIONS

The Antarctic sea-ice environment is incredibly complex and the exchange

of energy and momentum through this environment is very di!cult to numerically

approximate. One of the main hurdles associated with modeling the Antarctic sea-

ice environment is the lack of quality, large-scale observations with which to evaluate

numerical model output. Previously, information on sea-ice concentration, extent,

and area were the only variables routinely observed over the entire Southern Ocean

(Stammerjohn and Smith 1997, Gloersen et al. 1999, Watkins and Simmonds 2000,

Zwally et al. 2002, Parkinson 1994, 1998, 2002). Observational information on sea-

ice thickness, derived primarily from ship observations, contain spatial and temporal

gaps, limiting the area that can be evaluated(Worby et al. 1999, Worby et al. 2001).

Sea-ice thickness estimates derived from remotely-sensed observations have

only recently become available for analysis and model evaluation (Schellenberg 2002,

Schellenberg et al. 2002, DeLiberty et al. 2004). This study is the first to use

sea-ice thickness estimates derived from remotely-sensed observations to evaluate

a simulated Antarctic sea-ice thickness distribution. The improved resolution and

observation frequency provided by remotely-sensed observations allow sea-ice mod-

elers to evaluate sea-ice model performance over the entire Southern Ocean over a

variety of time-scales.

88

6.1 Model Evaluation

A simulated sea-ice thickness distribution for the Southern Ocean was gen-

erated using CSIM5, the fifth generation sea-ice model produced by NCAR. The

model, developed as part of the CCSM3 coupled climate system model, is designed

for coupled model runs but was run independently for this analysis with some mod-

ifications to better approximate the atmospheric state variables within the model.

The model was run as a stand-alone model to evaluate the sea-ice physics within

the model without the influence of errors generated by other model components.

Errors in the sea-ice model output indicate sensitivities in the treatment of sea-ice

physics within the model rather than errors output from coupled model components.

Identification of sensitivities specific to the sea-ice model is essential for evaluating

coupled model output.

Evaluation of model output using continental-scale, ice-thickness estimates

derived from NIC operational sea-ice charts indicated that although the distribution

was a reasonable approximation of the observed distribution, the model generally

underpredicted sea-ice thickness. Thinner ice indicates errors within sea-ice model

thermodynamics, specifically, an erroneously high energy input into the ice pack.

Further investigation indicated that the model overestimated the magnitude and

direction of the latent heat flux due to an oversimplified approximation of atmo-

spheric moisture. Without continental-scale thickness observations, identification of

large-scale patterns in model prediction errors, and interpretation of those errors, is

more di!cult and may lead to misinterpretations of model performance.

6.2 Future Research

Evaluation of the sea-ice thickness distribution produced by CSIM5 using

remotely-sensed thickness estimates provided a better understanding of the factors

that influence ice growth, melt, and redistribution within the model. This infor-

mation is the basis for further research into how the di%erent thermodynamic and

89

dynamic components approximated by the model influence model output. Future

research will include

• further investigation of the apparent model sensitivity to latent heat and spe-

cific humidity.

• improving the representation of specific humidity within CSIM5.

• investigating the sea-ice response to changes in atmospheric state variables

using CSIM5 as a stand-alone model.

• investigating the sea-ice response to atmospheric state variables using CSIM5

coupled with the Community Atmosphere Model (CAM3).

This research will attempt to not only produce a better simulated sea-ice distribu-

tion, but provide a better understanding of the role of sea ice within the climate

system.

90

REFERENCES

Allison I., Physical processes determining the Antarctic sea ice environment, Aus-tralian Journal of Physics, 50, 759-771, 1997.

Anderson, D. L., Growth rate of sea ice, Journal of Glaciology, 3 (30), 1170-1172,1961.

Barnes, H. T., Ice Engineering, Renouf, Montrial, p. 364, 1928.

Bilello, M. A., Formation, growth and decay of sea ice. Arctic, 14 (1), 3-24, 1961.

Bitz, C. M., M. M. Holland, A. J. Weaver, and M. Eby, Simulating the ice-thicknessdistribution in a coupled climate model, Journal of Geophysical Research,106, C2, 2441-2463, 2001.

Bolton, D., The computation of equivalent potential temperature, Monthly WeatherReview, 108, 1046-1053, 1980.

Briegleb, B. P., C. M. Bitz, E. C. Hunke, W. H. Lipscomb, M. M. Holland, J. L.Schramm, and R. E. Moritz, Scientific Description of the Sea Ice Componentin the Community Climate System Model, Version Three, Technical ReportNCAR/TN-436+STR, National Center for Atmospheric Research, Boulder,CO, 70 p., 2004.

Bromwich, D.H., J.J. Cassano, T. Klein, G. Heinemann, K.M. Hines, K. Ste%en,and J.E. Box, Mesoscale modeling of katabatic winds over Greenland withthe Polar MM5. Monthly Weather Review, 129, 2290-2309, 2001.

Campbell, W. J., On the Steady-State Flow of Sea ice, University of Washington,Seattle, Washington, 167 p., 1964.

Cavalieri, D. J., P. Gloersen, C. L. Parkinson, L. C. Comiso, and H. J. Zwally,Observed hemispheric asymmetry in global sea ice changes, Science, 278,1104-1106, 1997.

91

Coon, M. D., A review of AIDJEX modeling. In Pritchard, R. S., ed., Sea IceProcesses and Models, University of Washington Press, Seattle, 12-17, 1980.

Dedrick, K. R., K. Partington, M. Van Woert, D. A. Bertoia, and D. Benner, U.S. National/Naval Ice Center digital sea ice data and climatology, CanadianJournal of Remote Sensing, 27, 457-475, 2001.

DeLiberty, T. A., C. A. Geiger, and M. D. Lemcke, Quantifying sea ice in theSouthern Ocean using ArcGIS, Proceedings of the ESRI International UserConference, San Diego, CA, August 9-13, 2004.

Deliberty, T. A. and C. A. Geiger, Temporal and regional variations of sea ice thick-ness in the Ross Sea during 1995 and 1998, Proceedings of the 8th Conferenceon Polar Meteorology and Oceanography, AMS. San Diego, CA. 9-13 January2005.

Drinkwater, M. R., and X. Liu, Seasonal to interannual variability in Antarctic sea-ice surface melt, IEEE Transactions on Geoscience and Remote sensing, 38,4, 1827-1842, 2000

Dukowicz, J. K. and J. R. Baumgardner, Incremental remapping as a transport/advectionalgorithm, Journal of Computational Physics, 160, 318-335, 2000.

Ebert, E. E., and J. A. Curry, An intermediate one-dimensional thermodynamicsea ice model for investigation of ice-atmospheric interactions, Journal ofGeophysical Research, 98, 10,085-10,109, 1993.

Enomoto, H., and A. Ohmura, The influences of atmospheric half-yearly cycle onthe sea ice extent in the Antarctic, Journal of Geophysical Research, 95,9497-9511, 1990.

Fichefet, T. and M. A. Morales Maqueda, Sensitivity of a global sea ice model tothe treatment of ice thermodynamics and dynamics, Journal of GeophysicalResearch, 102, 12,609-12,646, 1997.

Flato, G. M., Sea-ice modelling, In Bamber, J. L., and A. J. Payne, eds., MassBalance of the Cryosphere: Observations and Modelling of Contemporaryand Future Changes, Cambridge University Press, 367-390, 2003.

Flato, G. M. and W. D. Hibler, III., On a simple sea-ice dynamics model for climatestudies. Annals of Glaciology, 14, 72-77, 1990.

92

Flato, G. M. and W. D. Hibler, III., Modeling pack ice as a cavitating fluid. Journalof Physical Oceanography, 22,626-651, 1992.

Geiger, C. A., Investigation of dynamic sea ice processes in the Weddell Sea during1992. Thayer School of Engineering Doctor of Philosophy Thesis, p. 384,1996.

Geiger, C.A., S.F. Ackley, W.D. Hibler, III, Year-round pack ice in the WeddellSea, Antarctica: response and sensitivity to atmospheric and oceanic forcing,Annals of Glaciology, 25, 269-275, 1997.

Geiger, C. A., A. P. Worby, T. DeLiberty, M. VanWoert, and S. F. Ackley, Inves-tigation of Interannual Sea Ice Thickness variability in the Southern Ocean,NSF Proposal, 17 p., 2000.

Gloersen, P., D. L. Parkinson, D. J Cavalieri, J. C. Comiso, and H. J. Zwally, Spatialdistribution of trends and seasonality in the hemispheric sea ice covers: 19781996, Journal of Geophysical Research, 104, C9, 20827-20835, 1999.

Gossse, H. and T. Fichefet, Importance of ice-ocean interactions for the global oceancirculation: A model study, Journal of Geophysical Research-Oceans, 104,C10, 23337-23355, 1999.

Gordon, H. B. and S. P. OFarrell, Transient climate change in the CSIRO coupledmodel with dynamic sea ice, Monthly Weather Review, 125, 875-907, 1997.

Guo, Z., D.H. Bromwich, and J.J. Cassano, Evaluation of Polar MM5 simulationsof Antarctic atmospheric circulation. Monthly Weather Review, 131, 384-411,2003.

Hanna, E., The role of Antarctic sea ice in global climate change, Progress in PhysicalGeography, 4, 371-401, 1996.

Hansen, J., R. Ruedy, J. Glascoe, and M. Sato, GIS analysis of surface temperaturechange, Journal of Geophysical Research, 104, (D24), 30,997-31,022, 1999.

Hartmann, D. L., Global Physical Climatology, Academic Press, San Diego, Califor-nia, 411 p,. 1994.

Hibler, W. D. III., A dynamic thermodynamic sea ice model, Journal of PhysicalOceanography, 9, 4, 815-846, 1979.

93

Hibler, W. D. III., Modeling pack ice as a viscous-plastic continuum: some prelimi-nary results, In Pritchard, R.S., ed., Sea Ice Processes and Models, Universityof Washington Press, 163-176, 1980.

Hibler, W. D. III., Ice dynamics, In Untersteiner, N., ed., The Geophysics of SeaIce, Plenum Press, New York, New York, 577-640, 1986.

Hibler, W. D. III., Modelling the dynamic response of sea ice, In Bamber, J. L.,and A. J. Payne, eds., Mass Balance of the Cryosphere: Observations andModelling of Contemporary and Future Changes, Cambridge University Press,227-333, 2003.

Holland, M. M., C. M. Bitz, and A. J. Weaver, The influence of sea ice physicson simulations of climate change, Journal of Geophysical Research, 106, C9,19,639-19,655, 2001.

Holland, M. M., C. E Bitz, E. C. Hunke, W. H. Lipscomb, and J. L. Schramm,Influence of the sea ice thickness distribution on polar climate in CCSM3,Journal of Climate, 19, 2398-2414, 2006.

Hunke, E. C., and Y. Zhang, A comparison of sea ice dynamics models at highresolution, Monthly Weather Review, 127, 396-408, 1999.

King, J. C. and J. Turner, Antarctic Meteorology and Climatology, Cambridge Uni-versity Press, New York, New York, 1997.

Lebedev, V. V., Rost ldo v arkticheskekh, rekakh i moriakh v zavisimosti ot otrit-satelnykh tempertur vozdukha. Problemy Arktiki, 5, 9-25, 1938.

Legates, D. R., and G. J. McCabe, Evaluating the use of goodness-of-fit measures inhydrologic and hydroclimatic model validation, Water Resources Research,35, 233-241, 1999.

Lipscomb, W. H., Remapping the thickness distribution in sea ice models, Journalof Geophysical Research, 106 (C7), 13,989-14, 2001.

Maykut, G. A., Energy exchange over young sea ice in the central Arctic, Journalof Geophysical Research, 83, 3646-3658, 1978.

94

Maykut, G. A., and N. Untersteiner, Some results from a time-dependent thermo-dynamic model of sea ice, Journal of Geophysical Research, 76, 1550-1575,1971.

National Ice Center (NIC), 1972-1994 Arctic and Antarctic Sea Ice Data, A CD-ROM produced by the NIC, Fleet Numerical Meteorology and OceanographyDetachment, and the National Climatic Data Center (NCDC), Washington,D.C., 1996.

Nihashi, S. and K. I. Ohshima, Relationship between ice decay and solar heatingthrough open water in the Antarctic sea ice zone, Journal of GeophysicalResearch, 106, C8, 16767-16782, 2001.

Ogura, T., A. Abe-Ouchi, and H. Hasumi, E%ects of sea ice dynamics on the Antarc-tic sea ice distribution in a couple ocean atmosphere model, Journal of Geo-physical Research, 109, 20-22, 2004.

Parkinson, C. L. Spatial patterns in the length of the sea ice season in the SouthernOcean, 1979-1986, Journal of Geophysical Research, 99, 16, 327-339, 1994.

Parkinson, C. L. Spatial patterns in the length of the sea ice season in the SouthernOcean, 1979-1986, Journal of Geophysical Research, 99, 16, 327-339, 1994.

Parkinson, C. L., Trends in the length of the Southern Ocean sea-ice season, 1979-99,Annals of Glaciology, 34, 435-440, 2002.

Parkinson, C. L., and W. M. Washington, A large-scale numerical model of sea ice,Journal of Geophysical Research, 84, 311-337, 1979.

Remund, Q. P., D. G. Long, and M. R. Drinkwater, An interative approach to mul-tisensor sea ice classification, IEEE Transactions on Geoscience and RemoteSensing, 38, 4, 1843-1856, 2000.

Rothrock, D. A., The steady drift of an incompressible Arctic ice cover, AIDJEXBulletin, 21, 49-77, 1973.

Schellenberg, B. A., Investigation of sea-ice thickness variability in the Ross Sea,Masters Thesis, University of Delaware, 133 p., 2002.

95

Schellenberg, B. A., T. L. DeLiberty, C. A. Geiger, J. Silberman, and A. P. Worby,Investigation of seasonal sea-ice thickness variability in the Ross Sea, Pro-ceedings of the 13th Global Change and Climate Variation, American Mete-orological Society, Orlando, Florida, January 13-17, 2002, 130-132, 2002.

Sellers, W.D., Physical Climatology, University of Chicago Press, Chicago, 272 p.,1965.

Simmonds, I. and T. H. Jacka, Relationships between the interannual variabilityof Antarctic sea ice and the southern oscillation, Journal of Climate, 8, 3,637-647, 1995.

Simmonds, I. and X. Wu, Cyclone behavior response to changes in winter southernhemisphere sea-ice concentration, Quarterly Journal of the Royal Meteoro-logical Society, 119, 1121-1148, 1993.

Spencer, J.W., Fourier series representation of the position of the Sun, Search, 2,5,172, 1971.

Stammerjohn, S. E. and R. C. Smith, Opposing Southern Ocean climate patterns asrevealed by trends in regional sea ice coverage, Climate Change, 37, 617-639,1997.

Steele, M., J. Zhang, D. Rothrock, and H. Stern, The force balance of sea ice in anumerical model of the Arctic Ocean, Journal of Geophysical Research, 102,21,061-21,079, 1997.

Stefan, J., ’Uber die Theorie der Eisbildung, insbesondere ’uber die Eisbildung imPolarmeere, Sitzber. Akad. Wiss. Wien. p. 98, 1890.

Stossel A., and T. Markus, Using satellite-derived ice concentration to representAntarctic coastal polynyas in ocean climate models, Journal of GeophysicalResearch, 109, 17-79, 2004.

Thorndike, A. S., D. A. Rothrock, G. A. Maykut, and R. Colony, The thicknessdistribution of sea ice, Journal of Geophysical Research, 80, 4501-4513, 1975.

Thorndike, A. S., Kinematics of sea ice, In Untersteiner, N., ed., The Geophysics ofSea Ice, Plenum Press, New York, New York, 489-550, 1986.

96

Thorndike, A. S., Estimates of sea ice thickness distribution using observations andtheory, Journal of Geophysical Research, 97, C8, 12,601-12.605, 1992.

Timmermann, R., A. Worby, H. Goosse, and T. Fichefet, Utilizing the ASPeCtsea ice thickness data set to evaluate a global coupled sea ice-ocean model,Journal of Geophysical Research, 109, 2004.

Turner, J., W. Connolley, D. Cresswell, and S. Harangozo, The simulation of Antarc-tic sea ice in the Hadley Centre Climate Model (HadCM3), Annals of Glaciol-ogy, 33, 585-591, 2001.

Vinnikov, K. Y., A. Robock, R. J. Stou%er, J. E. Walsh, C. L. Parkinson, D. J.Cavalieri, J. F. B. Mitchell, D. Garrett, and V. F. Zakharov, Global warmingand Northern Hemisphere sea ice extent, Science, 286, 1934-1937, 1999.

Wadhams, P., Ice in the Ocean, Gordon and Breach, Australia, p., 2000.

Washington, W. M., and C. L. Parkinson, An Introduction to Three-DimensionalClimate Modeling, 2nd edition, University Science Books, Sausalito, Califor-nia, 353 p., 2005.

Watkins, A. B. and I. Simmonds, Current trends in Antarctic sea ice: The 1990simpact on a short climatology, Journal of Climate, 13, 4441-4451, 2000.

Williamson, D. L. and R. Laprise, Numerical approximations for global atmosphericgeneral circulation models, In Mote, P. and A. ONeill, eds., Numerical Mod-elling of the Global Atmosphere in the Climate System, NATO Science Series,p. 127, 2000.

Willmott, C.J., On the validation of models, Physical Geography, 2, 184-194, 1981.

Willmott, C. J. and M. L. Johnson, Resolution errors associated with gridded pre-cipitation fields, International Journal of Climatology, 25, 1957-1963, 2005.

Willmott, C. J. and K. Matsuura, Smart interpolation of annually averaged airtemperature in the United States, Journal of Applied Meteorology, 34, 2577-2586, 1995.

Wooster, M. G., Solidification of fluids. In Batchelor, G. K., H. K. Mo%att, andM. G. Wooster, eds., Perspectives in Fluid Dynamics. Cambridge UniversityPress, 1999.

97

Worby, A. P., Observing Antarctic Sea Ice: A practical guide for conducting sea iceobservations from vessels operating in the Antarctic pack ice. A CD-ROMproduced for the Antarctic Sea Ice Processes and Climate (ASPeCt) pro-gram of the Scientific Committee for the Antarctic Research (SCAR) GlobalChange and the Antarctic (GLOCHANT) program, Hobart, Australia, 1999.

Worby, A. P. and S. F. Ackley, Antarctic research yields circumpolar sea ice thicknessdata, EOS, Transactions, AGU, 81 (17), 181, 184-185, 2000.

Worby, A. P., G. M. Bush, and I. Allison, Seasonal development of the sea-ice thick-ness distribution in East Antarctica: Measurements from upward-lookingsonar, Annals of Glaciology, 33, 177-180, 2001.

Worby, A. P., P. W. Gri!n, V. I. Lytle, and R. A. Massom, On the use of electro-magnetic induction sounding to determine winter and spring sea ice thicknessin the Antarctic, Cold Regions Science and Technology, 29, 49-58, 1999.

Worby, A. P., C. A. Geiger, M. J. Paget, M. L. Van Woert, S. F. Ackley, and T. L.DeLiberty, Thickness distribution of Antarctic sea ice, Journal of GeophysicalResearch, 113, 2008.

World Meteorological Organization (WMO), WMO Sea-Ice Nomenclature, Volume1: Terminology and Codes, World Meteorological Organization, report 259,Geneva, Switzerland, 1970.

Wu, X. R., W. F. Budd, and T. H. Jacka, Simulations of Southern Hemispherewarming and Antarctic sea-ice changes using global climate models, Annalsof Glaciology, 29, 61-65, 1999.

Yen, Y. C., Review of thermal properties of snow, ice and sea ice, US Army ColdRegions Research and Engineering Laboratory, Hanover, New Hampshire,Research Report, 81-110, 1981.

Zhang, G. J., W. D. Hibler, III, M. Steele, and D. A. Rothrock, Arctic ice-oceanmodeling with and without climate restoring, Journal of Geophysical Re-search, 28, 191-217, 1998.

Zwally, H. J., J. C. Comiso, C. L. Parkinson, D. J. Cavalieri, and P. Gloersen,Variability of Antarctic sea ice 1979-1998, Journal of Geophysical Research-Oceans, 107, C5,2002.

98

Zubov, N. N., Arctic Ice. Izdatelstvo Glavsermorputi, Moscow, 1945. (TranslationAD426972, National Technical Information Service, Springfield, VA).

99