An analysis of the shapes of sea ice ridges

cold regions science and technology ELSEVIER

Cold Regions Science and Technology 25 (1997) 65-77

An analysis of the shapes of sea ice ridges G.W. T i m c o a, R.P. Burden b a Canadian Hydraulics Centre, National Research Council, Ottawa, Ont. KIA OR6 Canada Memorial Universi~, St. John's, Nfld. A I B 3X5 Canada

Received 1 May 1996; accepted 5 July 1996

Abstract

An analysis has been made of the salient features of 112 first-year and 64 multi-year sea ice ridges. Based on this information, the important characteristics of the ridges have been related through simple equations. In particular, the ratio of the keel-depth to sail-height was found to be 4.4 for first-year ridges, and 3.3 for multi-year ridges; the ratio of the keel-area to sail-area was 8.0 for first-year ridges and 8.8 for multi-year ridges. Also, for first-year ridges, the ratio of the keel-width to sail-height was approximately 15, and the ratio of the keel-width to keel-depth was 3.9. An analysis of the sail and keel angles indicates a distribution of values with an average sail angle of 21 ° for temperate ridges, and 33 ° for ridges in the Beaufort Sea. In this paper, the results of this analysis are described, and the important ridge characteristics are discussed. Keywords: Sea ice; Ridges; Shape; Morphology; Sail; Keel

1. Introduction

A characteristic of ice-covered waters is the presence of ice ridges. These ridges form by the breaking and deformation of the ice cover which is being driven under the action o f winds, currents and Coriolis forces. These pressure ridges can be formidable. They can inhibit navigation and generate high loads during an interaction with an offshore structure. For these reasons, the characteristics of sea ice ridges have been extensively studied. Sea ice ridges are complex objects with a wide variability of sizes and shapes. This has made their characterization difficult. Numerous studies have been made t9 measure the properties of ridges, and based on these studies, some general information on ridge properties has emerged. Ridges have been categorized by their mode of formation (either compression or shear), but this method is not unambiguous,

since it is not always clear how the ridge formed. A more common method to categorize ridges is based on the a g e of the ridge - - either first-year or multi-year ridges. In general, ridges are linear features with large amounts of ice below the waterline (the keel), and smaller amounts of ice above the waterline (the sail). Tucker (1989) has suggested that the keel-to-sail elevation ratios are on the order of 4.5 : 1 for a first-year ridge and 3.2 : 1 for a multi-year ridge. Analysis of laser and sonar profiles have suggested a negative exponential distribution fits well for the distribution o f both sail heights (Tucker et al., 1979; Weeks et al., 1980) and keel spacing (Wadhams et al., 1985). More recently, Davis and W a d h a m s (1995) have suggested that a log-normal distribution better describes the ridge spacing relationship than a negative exponential distribution. Information of this type on the characteristics and relationships of the various parameters in an ice

0165-232X/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0165-232X(96)00017- 1

66

G. W. Timco, R.P. Burden/Cold Regions Science and Technology 25 (1997) 65 77

ridge is very important. The information on the ridge spacing has important implications for ship routing in ice-covered waters, and the information on the ridge size has important implications in terms of both shipping, and the loads that a ridge could exert on an offshore structure. In this regard, the development of functional relationships to describe the characteristics of an ice ridge has increasing importance in probabilistic numerical models for calculating ice loads. Information on the size and shape of a ridge have been obtained in a number of different ways. All of the techniques, however, can be subdivided into 2 basic categories - - continuous scanning of the ice surface, or discrete measurements of specific ice features. Both techniques have advantages and disadvantages. With continuous scanning, information can be obtained about the ridge spacing and orientations, as well as the sail or keel shapes. In this case, a laser or upward-looking sonar is fixed in one location and used to record the surface elevation of the ice as the ice moves past it. With this technique, a large amount of information can be obtained with relatively little effort. However, usually only one side of the ice surface is profiled (either the top or bottom), and direct correlation of the top and underside surfaces is not possible. This restricts information on the sail/keel properties of individual ridges. With a discrete measurement, information can be obtained on the overall size and shape of an individual ridge at one particular location, both above and below the waterline. In this case, several holes are drilled in the ice, and the thickness is measured using either a tape measure or a sonar transducer probe that is lowered into the drill holes. This approach can provide detailed quantitative information on both the sail and keel size and shape, but does not supply any information on ridge spacing. Also, the discrete measurement technique is significantly more labour-intensive than continuous profiling. Recently, Davis and Wadhams (1995) have summarized the findings obtained by profiling ridges using laser-profiles and upward-looking sonar. This analysis has provided valuable insight into the ridge spacing and orientation characteristics for a wide number of ridges in several geographical locations. In this paper, the results of a number of different investigations of discrete measurements of sea ice

ridges are summarized. The information reported here has been obtained from a large number of published profiles of individual sea ice ridges. An emphasis was placed on ridges in which information on both the sail and keel was available. Detailed information on the measured sail and keel properties has been summarized, and the salient characteristics have been determined.

2. D a t a sources

Information on the shape of sea ice ridges was obtained from 22 different sources, covering a wide geographic area. In total, the details on 176 ridges were obtained. This included full cross-section profiles of individual ridges, with information on a number of different ridge properties. The 176 ridges were initially classified according to their age (firstyear or multi-year), and then they were further subdivided according to their location, Fig. 1 provides a flow diagram illustrating the category breakdown, and the number of ridges in each category. The information on first-year ridges was obtained from the following sources: • Banke and Lowings (1983) provides details of 7 ridges in McKinley Bay in the Beaufort Sea region; • Kovacs and Mellor (1971) provides information on 1 ridge in the Beaufort Sea; • Gladwell (1975) provides information on 8 ridges in the Beaufort Sea; • McGonigal (1978) provides information on 13 ridges in the Canadian Beaufort Sea; • Vaudrey (1980) profiled 14 different ridges in the

AGE? I I FIRST1 -YEAR

I MULT~4YEAR1

LOCATION?

LOCATION?

I

i

BEAUFORT 53

QE ISLANDS I

Fig. 1. Flow diagram illustrating the category breakdown for the ridges, and the number of ridges in each category.

G.W. Timco, R.P. Burden/Cold Regions Science and Technology 25 (1997) 65-77

Beaufort Sea, with 7 of the ridges profiled at 2 locations; • Voelker et al. (1981) provides information on 40 ridges in the Bering Sea; • Voelker et al. (1982) provides profiled information on 3 ridges in the Beaufort Sea; • Evers (1986) provides details of 3 profiles each across 4 different ridges in the Labrador Sea; • Kankanp~i~i (1989) provides information on 8 different ridges from the Gulf of Bothnia and the Gulf of Finland; • Veitch et al. (1991a) provides profiles of 2 ridges in the Gulf of Bothnia in the Baltic Sea region; • Veitch et al. (1991b) studied a first-year ridge at two different times of the year (February and March) in the Baltic Sea region; • Lepp~iranta and Hakala (1991) provide details of 6 ridges in the Bay of Bothnia in the Baltic Sea region; • Williams and Kirby (1994) provides information on 5 ridges in Northumberland Strait in Canada. These 112 ridges were subdivided according to their location, with 46 from the Beaufort Sea region, and 66 ridges from " t e m p e r a t e " regions, including the Labrador Sea, Baltic Sea, Northumberland Strait, and the south Bering Sea. The information on multi-year ridges was obtained from the following sources: • Kovacs and Mellor (1971) provides profiles of 4 ridges in the Beaufort Sea; • Kovacs et al. (1973) provides profiles of 1 ridge at 3 different locations; • Kovacs and Mellor (1974) discuss multi-year ridges and provide information on 1 ridge in the Beaufort Sea; • Kovacs et al. (1975) provides information on 17 ridges in the Beaufort Sea; • Kovacs (1976) provides information on 3 ridges in the Beaufort sea off the coast of Alaska; • Kovacs (1977) provides details of 2 ridges; • Cox (1972) provides information on 1 ridge during the AIDJEX 1972 experiment; • Harrington (1978) provides profiles of 3 different ridges, with 3 profiles on one of the ridges; • Dickens (1981) provides information on 11 different ridges with a few multiple profiles on different ridges in the Queen Elizabeth Island region;

Ws SAIL Ps S A I L

67

W I D T H :,

Hs

POROSI

SAIL H E I G H T

I

SEA LEVEL

-ii-oi, :

.

p. ,EEL.O"OS' /

c

Wk

KEEL W I D T H

.

.

.

.

.

.

.

hk

;,

Fig. 2. Schematic illustration of a first-year ridge showing the definition of terms used in the analysis.

• Kovacs (1983) provides 3 profiles and information on 11 ridges in Prudhoe Bay; • Voelker et al. (1982) provides information on 10 ridges during the trials of the USCG Polar Sea. These ridges were subdivided according to geographic location into 2 main categories - - the Beaufort Sea region (53 ridges), and the Queen Elizabeth Island region (11 ridges). There was a wide range of information supplied on the characteristics of the ridges. In many cases, detailed profiles were given, but in some cases, the information on the ridge was simply summarized. For the present work, as much of the available information as possible was extracted from the descriptions and profiles of the ridges. Fig. 2 shows a schematic illustration of a typical first-year ridge, and it provides the definition of terms used in the analysis. All of the information on the ridges was put in tabular form, including details of the sail and keel (width, thickness, area and angles), the consolidated region (minimum, average and maximum thickness), the snow layer (thickness and distribution), and the strength and porosity of the ridge. This data, including a large number of the detailed ridge profiles have been published in Burden and Timco (1995). There was considerable variation in the level of detail supplied on the ridges by the individual authors, and full information was not supplied for any reported ridge. Nevertheless, there was enough information to obtain some definite details on the shape and characteristics of sea ice ridges.

68

G. W. Timco, R.P. B u r d e n / C o l d

R e g i o n s S c i e n c e a n d T e c h n o l o g y 25 (1997) 6 5 - 7 7

Table 1 Summary of parametric relationships Equation

n

r2

a

b

value

st. error

t-value

95% confidence rain

Hk = a l l b Wk = a l l b Wk = aH~b

A k = aA b

Hk.m = aH~bm Ak,m = aAb•m

97 65 75 33 47 10

st. error

t-value

95% confidence rain

max

. 0.05

.

. . 18.17

0.79

0.98

. . 12.05

0.72

1.01

. . 12.34

0.65

0.90

0.70

0.94

max

0.783 0.793

3.95 4.60

0.12 0.31

34.09 15.07

3.72 4.00

4.18 5.2 !

0.88

0.739 0.746

3.91 5.67

0.16 1.13

24.53 5.02

3.59 3.41

4.22 7.92

. 0.07

.

0.87

0.675 0.713

14.85 20.75

0.63 1.98

23.48 10.51

13.59 16.81

16.11 24.69

1.00 0.78

. 0.06

.

0.871 0.896

7.96 17.46

0.33 4.28

24.32 4.08

7.29 8.73

8.63 26.19

0.82

0.873 0.878

3.17 3.66

0.08 0.30

41.83 12.37

3.02 3.06

3.33 4.26

1.00

-

--

0.91

0.05

19.23

0.82

1.01

0.931 0.921

8.81 8.82

0.44 4.42

19.81 1.99

7.80 -- 1.42

9.82 19.06

. . 9.75

0.76

1.24

A large number of different properties were plott e d to i n v e s t i g a t e

value

any general trends that exist for

1.00

1.00

1.00

1.00

t.00

-

-

0.06

. 0.10

-

14.43

.

-

had no apparent physical basis -b e s t c u r v e - f i t to a l i m i t e d n u m b e r

it w a s s i m p l y t h e of data points.

ridges• In many cases, there was a good correlation

T h u s , it w a s d e c i d e d to c h a r a c t e r i z e t h e r e l a t i o n s h i p s

found between

using

d i f f e r e n t r i d g e c h a r a c t e r i s t i c s , b u t in

2 types

of curves

--

one,

a simple

linear

o t h e r s , t h e r e w a s little o r n o c o r r e l a t i o n . A n u m b e r

relationship (forced through the origin), and two, the

of techniques

were

best-fit power relationship. In most cases, there was

relationships.

Usually,

tween 2 parameters

u s e d to q u a n t i f y

the important

the best-fit relationship

was a complex

be-

relationship that

v e r y little d i f f e r e n c e

between

the two

approaches,

w i t h t y p i c a l c o r r e l a t i o n ( r 2) c o e f f i c i e n t s o n t h e o r d e r

Table 2 Summary of distributions Relationship

Type

n

Mean

St. dev.

X2

p-value

D.O.F.

keel/sail ratio - - first-year ice

normal log-normal

97 97

4.46 1.41

1.85 0.41

36.11 10.05

0.01 0.69

13 13

sail angle - - temperate

normal log-normal

40 40

20.68 2.87

I1.45 0.61

16.80 16.80

0.21 0.21

13 13

sail angle - - Beaufort

normal log-normal

40 40

32.90 3.45

9.16 0.29

13.60 14.40

0.40 0.35

13 13

keel angle - - first-year ice

normal log-normal

70 70

26.56 3.17

13.39 0.47

30.11 16.40

0.00 0.23

13 13

keel/sail ratio - - multi-year ice

normal log-normal

47 47

3.34 1.12

0.85 0.20

37.77 18.70

0.00 0.13

13 13

sail angle - - multi-year ice

normal log-normal

73 73

19.50 2.81

8.50 0.68

11.38 47.33

0.58 0.00

13 t3

69

G.W. Timco, R, P. Burden/Cold Regions Science and Technology 25 (1997) 65-77

of 0.7 to 0.9, This level of correlation should be considered to be quite good, given the wide natural variability of the ridges. The best-fit linear and power relationships are summarized in Table 1, along with the information on the number of data points (n), and the statistical correlation ( r 2) between the parameters. In addition, the value of the coefficients along with their standard error and 95% confidence limits are presented. An analysis was also performed to investigate the distribution of the measured sail and keel angles of the ridges. For this analysis, the reported angles for both the sail and keel of the different ridges were tabulated, and a statistical function was fit to the data. Unfortunately, in many cases, there was not sufficient data to be able to unambiguously define the best functional relationship. Therefore, the data was fit to either a simple normal or log-normal distribution, depending upon the general shape of the distribution. Table 2 lists the information on the mean and standard deviation for the normal distribution, and the log(mean) and log(standard deviation) for the log-normal distribution. In addition, the goodness of fit for each distribution is listed as determined using a A"2 test. The results of the analysis is presented in the following sections, for both first-year and multi-year ridges.

3. First-year ridges There was a wide variation in the shapes of the first-year sea ice ridges. The schematic illustration of Fig. 2 presents a representative picture of a ridge, but in reality, their shape can vary significantly from this general shape. Although there were a few ridges that showed this classic shape, in general, their shapes are much more complex. In many cases, there were large asymmetries in the ridge, with non-symmetrical sails and keels, and significant non-alignment of the centre of the sail to the keel. On the other hand, the general characteristics shown in Fig. 2 were evident for all ridges, and, as such, it provides a reasonable representation. The analysis of the ridge information provides a key to the general relationship of the important properties of sea ice ridges. Fig. 3 shows a plot of the keel depth versus the

3O

~10 v

0 2

4 SAIL

6

HEIGHT

(m)

Fig. 3. Keel depth versus sail height for first-yearridges.

sail height for first-year ridges. The data have been differentiated to show both the Beaufort and temperate ridges. Although there is scatter, there is a general trend of increasing keel depth with increasing sail height, with no particular distinction between the Beaufort and temperate ridges. The best-fit linear relationship and the best-fit power relationship were: Hk = 3.95H~ or H k = 4 . 6 0 H °88

(1)

where H k is the keel depth and H, is the sail height. As seen in Fig. 3, there is little difference between these curves, except for very large ridges. The data in the figure indicates that, although there is a clear trend, there is a large scatter associated with the possible keel-to-sail ratio for first-year ridges. To quantify this characteristic, the keel-to-sail ratio was fitted to a log-normal distribution. Fig. 4 shows the distribution for data from 97 ridges. The mean of the

1.4i ,, FIR

i

i

i

i

i

6

7

i i i MEAN = 4 . 4 6

T YEAR ICE

r

/

1.2

v

i

~0.8 ~0,6 ~

0.4

°

0.2 0 0

1

2

3

4 KEEL

Fig.

4.

Probability

density

first-year ice ridges.

fi -

SAIL

function

fl

9

10

11

I2

RATIO for

the

keel-to-sail

ratio

for

70

G. W. Timco, R.P. Burden/Cold Regions Science and Technology 25 (1997) 65-77

I 150

]

1

F(. BEA'O'T

l,,fl

v

TEMPERATE

BEAUFORT

]

W k = 2 0 . 7 5 I I °'va,

150

v

W"k = 1 ~ . 8

.~ 1~5

_

.

~, = ,.E,~:°'

! b- 1 0 0 v 75 .1 :~

50

,

50

*v v,

v v~

v

v

v

M 25

10 KEEL

20 DEPTH

2

30

4

6

8

10

SAIL HEIGHT ( m )

(m)

Fig. 5. Keel width versus keel depth for first-year ridges.

Fig. 6. Keel width versus sail height for first-year ridges.

distribution is 4.46, and the standard deviation is 1.85. The mean value of 4.46 lies between the values of 3.95 and 4.60 of Eq. (1), for the linear and power fit. This difference reflects the differences in the various approaches used to characterize the data. The ratio of keel depth to sail height of 4.46 from Fig. 4 is very similar to the value of 4.5 quoted by Tucker (1989). Fig. 5 shows a plot of the keel width versus the keel depth for both temperate and Beaufort first-year ridges. There is a general trend of increasing keel width with increasing keel depth. The best-fit linear relationship and power law relationship for this data is

Published ridge profiles that contained complete profiles were digitized and, using a commercial software package, the area of the keel, sail and snow were determined. This was done for 33 first-year ridges. Based on this information, the relationship between the area of the sail and keel could be determined. Fig. 7 shows a plot of the keel area versus the sail area for first-year ridges, for both temperate and Beaufort regions. Although there is not a lot of data, the existing data shows a general trend of a linear relationship between the keel and sail area. The best-fit relationships are

Wk= 5.67H °'sv

where A k is the area of the keel, and A~ is the area of the sail. In addition to investigating the relationships between the heights and areas of the different components of the ridge, the angles of both the sail ( % )

Wk = 3 . 9 1 H k or

(2)

where Wk is the width of the keel. It is not possible to comment on the differences between temperate and Beaufort ridges, since their data ranges are different. In general, however, Eq. (2) well describes both types of first-year ridges. Since the keel depth is related to both the sail height (Eq. (1)), and the keel width (Eq. (2)), the keel width should also be related to the sail height. Fig. 6 shows this relationship. Although there is scatter, there is a general trend of increasing keel width with sail height. The best-fit relationship for this data is W k = 14.85H~ or Wk = 2 0 . 7 5 H °78

A k ~ 7.96A~ or A k = 17.46A~ "82

t2oo]

.

.

.

~

BEAUFORT

-<

! u~ 4001

/ / ~ / ~ "

•. ~ < -

(3)

The curves representing these 2 relationships are shown on the figure. In general, there is little difference between the 2 curves over most of the range of reported ridges.

. v

(4)

1

0

¢,~.v)~/

I'x

0

--A~

"

v

= 17,oA:"' "

°

J~

~5

50 SAIL

t

75 AREA(m

)00

[25

2)

Fig. 7. Keel area versus sail area for first-year ridges.

_~ 150

G. W. Timco, R.P. Burden/Cold Regions Science and Technology 25 (1997) 65-77 i

1

FIRST

i

i

i

d

r

i

i

r

1.3

--

E RI.-T FERATE

FIRST

i

~

YEAR

T

71 T

T

ICE

~

S.D.

=

r 13.4

°

=

v hi0 o

0.75

.d

.~

-I

= 26.6 °

MEAN

i

~

0.5

0,8

~0.6 m

\

0.4

0.25 .o

o 0.2

o P., 0

0

zo

30 SAIL

40

50

ANGLE

eo

70

80

90

10

30

(deg.)

30

40

KEEL

50

ANGLE

60

70

80

90

(deg.)

Fig. 8. Probability density function for the sail angle for first-year temperate ice ridges.

Fig. 10. Probability density function for the keel angle for first-year ice ridges.

and keel ( a k) were determined, based on the published profiles for the ridges. Figs. 8 and 9 are plots of the probability density distributions showing a log-normal distribution for the sail angles for both the temperate ridges (Fig. 8), and Beaufort ridges (Fig. 9). In both cases, there was little difference between a normal or log-normal fit for this data. However, there is a significant difference in the characteristics of the distributions. For the temperate ridges, the mean sail angle is 20.7 °, with a standard deviation of 11.5 ° (based on 40 measurements). For the Beaufort ridges, the mean sail angle is higher at 32.9 °, with a standard deviation of 9.2 ° (based on 40 measurements). With regard to the keel angles, there was no apparent difference between the 2 types of ridges. Fig. 10 shows the probability density distribution for the keel angles for first-year ridges. It fits well to a log-normal distribution with a mean of

26.6 ° and a standard deviation of 13.4 ° (based on 70 measurements). There was some information available on the thickness of the refrozen (consolidated) layer for a number of different ridges. Information on the variation of thickness of this layer is especially important in ice engineering applications, since the consolidated layer often exerts the highest forces on offshore structures during a ridge/structure interaction. Presentation of the information on the consolidated layer thickness was not straightforward, since there was a wide range of thickness for the ridges. To get some insight into the variability of the thickness of the consolidated layer, the maximum, minimum and average thickness values were determined for 25 different ridges. With this information, the ratio of

CONSOLIDATED MEAN 2.4

i FIRST

l YEAR

ICE

i -

BEAU

i --

~

ORT

i

MEAN S.D.

I

2

LAYER MEAN

0.51

= 1.58

i

= 32.9 ° =

9.2"

I

v

N=40

o O m

,,

A o

~ 0.8 i

i

o0.4 0

i 10

, r 20

30 SAIL

40

ANGLE

50

60

70

B0

90

(deg.)

Fig. 9. Probability density function for the sail angle for first-year Beaufort ice ridges.

0.5 MINIMUM/AVERAGE THICKNESS

I.

2

2.5

MAXIMUM/AVERAGE THICKNESS

Fig. 11. Probability density functions for the ratios of the minimum-to-average thickness, and the maximum-to-average thickness in the consolidated layer of first-year ridges.

72

G. W. Timco, R.P. Burden/Cold Regions Science and Technology 25 (1997) 65-77

the minimum-to-average thickness and the maximum-to-average thickness were determined for each ridge. These ratios are shown as 2 different probability density plots in Fig. 11. The mean of the minimum-to-average thickness ratio was 0.51 with a standard deviation of 0.2. The mean of the maximum-to-average thickness ratio was 1.68 with a standard deviation of 0.37. These ratios show that there is a large amount of variability in the thickness of the consolidated layer in ice ridges. An analysis was performed to determine the porosity of the sail and keel regions, based on an assumption of iso-static equilibrium for the ridge. The sail and keel porosity of an ice ridge are related by

A~n p~,, +A~(I (l

-

- p,) pico(l - ~/)

=

(5) Pw -

where Pk = keel porosity, p, = sail porosity, A~, = snow area, A~ = sail area, p~, = snow density (assumed to be 0.300 Mg m-3), Pi~e = density of keel ice (assumed to be 0.91 Mg m-3), "q = correction factor to account for sail ice density being less than keel ice density (this is assumed to be 0.05), A k = keel area, and Pw = density of sea water (1.027 Mg m

3).

This equation contains two unknowns terms (Pk and p,), and therefore it cannot be solved uniquely. It is, however, possible to determine the relationship between the 2 unknowns. To solve the equation, a series of sail porosity values were assumed, and the corresponding keel porosity was calculated using Eq. (5). This was only done for 33 ridges which had complete profiles. The ridge profiles were digitized, and the sail, keel and snow areas were determined. Using this approach, the best-fit equation to describe the sail and keel porosity for first-year ridges is Pk = 0.14 + 0.73 p,

(6)

There was little difference between the data for the Beaufort and temperate ridges, and the above equation is a reasonable representation for both regions. It should be noted that this equation predicts that the keel porosity is non-zero, even with a zero sail porosity. Eq. (6) can be used to estimate the porosity of the keel, if the sail porosity is measured. A number of other properties were investigated

for first-year ridges, but they did not show any pronounced trends. In particular, there was n o apparent correlation between the following properties of first-year ridges: sail width and sail height sail height and the thickness of the level ice maximum consolidated depth and the keel depth keel depth and the thickness of the level ice average keel angle and the keel depth average sail angle and the sail height

4. Multi-year ridges There was not as much information available on the detailed profile properties of multi-year ridges compared to first-year ridges. There was information on 64 multi-year ridges, with the majority (53) from the Beaufort region. In addition, there was some information on 11 ridges from higher latitudes in the Canadian Queen Elizabeth Islands. Similar to the first-year ridges, there was a wide range of shapes and sizes for the multi-year ridges. In general, the keels of the multi-year ridges tended to be much broader, and more "rectangular" in shape, compared to the more triangular keel shapes of first-year ridges. Many different properties were plotted to investigate the general trends for multi-year ridges. In most cases, there was not sufficient information to determine a large number of valid correlations. There were, however, a number of important trends obtained from the data. Similar to the analysis for the first-year data, linear and simple power-law relationships were determined. The best-fit relationships are summarized in Table 1, along with the information on the number of data points (n), and the statistical correlation (r 2) between the parameters for both the linear and power relationships. In addition, the value of the coefficients of the equation are presented, along with estimates of the standard error and 95% confidence limits. Table 2 lists the relationships determined for the statistical distributions for the sail and keel angles for the multi-year ridges. Fig. 12 shows a plot of the keel depth versus the sail height for the multi-year ridges from both the Beaufort and Queen Elizabeth Island regions. There is a strong relationship between these properties for

G. W. Timco, R.P. Burden/Cold Regions Science and Technology 25 (1997) 65-77 4D

i

i

i

r

i

73

2000

]

~AUVO~T

•

~zAvroav

QZ ISLANDS DATA

o

Qz ~SLAWDS SArA

~'-'1500

~30

,m

~20 •

[~

.

~

\Hk.m

=

3"66H°'~t

i

1000

= 3.17H., m 500

vl0 •

~

o

~

LMULT] ~ = ' R ICEI

0 0

2

r

[

4

6

SAIL

--

HEIGHT

0

I

50

8

1O

12

I

I

100

150

SAIL

AREA(m

200

250

2)

(m)

Fig. 14. Keel area versus sail area for multi-year ridges•

Fig. 12. Keel depth versus sail height for multi-year ridges.

the Beaufort data, but considerable scatter for the ridges from the QE Island region, A best-fit relationship was determined for only the Beaufort data as

Hk, m = 3 . 1 7 H ~ , m o r Hk.,n = 3 . 6 6 H , ° • '

(7)

where Hk, r, is the depth of the keel, and H,,~ is the height of the sail. When compared to first-year ice ridges (see Fig. 3), it is clear that the sail-to-keel ratio is lower for multi-year ice, and that there is much less scatter in the data. Fig. 13 shows a plot of the distribution of the recorded keel-to-sail ratios for multi-year ice. A log-normal curve has been fit to the Beaufort data which exhibits a mean value of 3.34 and a standard deviation of 0.85 (based on 47 ridges). The keel-to-sail ratio for the ridges in the QE Islands region is higher, with a mean of 4.7 and a standard deviation of 1.5 (based on 12 ridges). The reason for this is not known. The mean value of the keel-to-sail

ratio for the Beaufort ridges of 3.34 is very similar to the value of 3.2 quoted by Tucker (1989) for multiyear ridges. Fig. 14 shows a plot of the keel area versus the sail area for a few multi-year ridges. There is a good correlation between them, with a linear relationship Ak, m = 8.81 As, m

where Ak, na is the area of the keel of a multi-year keel and A~,m is the area of the sail of a multi-year ridge. This ratio of 8.81 is slightly higher than the value of 7.96 determined for first-year ridges (see Fig. 7). Fig. 15 shows a plot of the probability density of the sail angle for multi-year ridges. In contrast to the log-normal distribution observed for first-year ridges, the sail angles appear to follow a normal distribution with a mean value of 19.5 °, and a standard deviation of 8.5 ° . This mean value is similar to that observed

=-4i=UL ..... i=,~,

0.08

MEAN = 3 . 3 4 S.D. = 0 8 5 N = 47

7

d MULTI

S

1

~ YEAR

~

i

~

~

ICE

MEAN S.D.

i = =

T - -

19.5 ° 8.5 °

0.06

v

~2.5 .,.a

(8)

°

¢~ a) 0 . 0 4

¢v 1. 5 ¢= o

0.02

,.~ 1 o ~" 0 . 5 0

__1

0 O

1

2

3 KEEL

4

5

-

SAIL

6

7

t

9

!0

RATIO

Fig. 13. Probability density function for the keel-to-sail ratio for multi-year ice ridges in the Beaufort Sea.

10

20

30 SAIL

40 ANGLE

R 50

~ 60

i 70

_L 80

_ 90

(deg.)

Fig. 15. Probability density function for the sail angle for multiyear ice ridges.

74

G.W. Timco, R.P. Burden / Cold Regions Science and Technology 25 (1997) 65-77

for first-year temperate ridges (Fig. 8), but it is lower than that observed for first-year Beaufort ridges (Fig. 9). Due to a lack of data, an evaluation of the values and distribution of keel angles was not possible for multi-year ridges. An examination of the profiles of the keels showed a wide variation in angles, ranging from virtually 0 ° (i.e. a long flat-bottomed ridge) to approximately 50 ° . An analysis of the ridge porosity was performed, similar to that discussed above for the first-year ridges. For the multi-year ice, the best-fit relationship between the keel porosity (ek.m) and sail porosity (P~,m) was Pk,m = 0.06 + 0.91P,,m

(9)

This equation can be used to estimate the keel porosity if the sail porosity is measured.

5. Discussion The collection of the data on ridge shapes and sizes and its analysis has led to some interesting trends. It is clear that although there is a very wide variety of shapes and sizes for sea ice ridges, there are some very definite relationships between several ridge properties. The present analysis details these relationships and provides information on the variation of the important ridge properties. With this information, it is possible to define the conditions of an " a v e r a g e " sea ice ridge. Figs. 16 and 17 show composite sketches of an " a v e r a g e " first-year ridge and multi-year ridge respectively. The figures illustrate the important relationships, and point to some differences between first-year and multi-year ridges. It is interesting to compare these features to the C(s

20.7° (temperate) ~32.9 ° (Beaufort)

= 0 14

<

~ W k

¢

Hk [ Hs = 4.4

INk/

Wk/Hk

Ak/As=

= 3.9

*

Hs = 15.1 80

Fig. 16. Sketch showing the features of an "average" first-year ridge.

.............

i;_

up ,o 5° °

Hk [

.,F..............

IH

Hs = 3.3

Ak/As=

,,:

/---

8.8

Fig. 17. Sketch showing the features of an "average" multi-year

ridge.

general ridge shape predicted by the 2 main theories for the ridge-building process. Both theories are developed in terms of the deformation of ice floes and ice blocks with constant thickness. Limiting sail heights and keel depths are calculated for ice sheets with a range of ice thickness. The data outlined in this paper, however, cannot be used directly to validate the models, since the thickness of the ice at the time of ridge formation is not known. The correlation between the sail height and ice block size has been investigated (Tucker et al., 1984), and this type of information is more useful for model validation. However, the information presented here can be used to compare with calculated values of the equilibrium profile of the ridges. Parmerter and Coon (1972) constructed a 2-dimensional kinematic model which used 2 ice floes of equal thickness separated by a lead filled with ice rubble to a depth corresponding to the thickness of the floes. As the 2 floes converged, the rubble was displaced both above and below the ice floes. An assumed angle of repose shaped the rubble piles. The rubble between the floes was assumed to be in hydrostatic equilibrium, and the ice floes were allowed to bend and break in response to the accumulated rubble on and beneath them. The model predicted that there is a limiting height of the ice rubble which depends upon the thickness and strength of the ice floes. Parmerter and Coon's simulations showed a wide variety of ridge shapes, and these appear to be in good agreement with the general shape of the ridges summarized in the present paper. They often found ridge asymmetry which was a very common feature observed for ice ridges in the field. Parmerter and Coon used values for the angle of repose of 45 ° for the sail and this gave a sail angle of 25 ° . This is in good agreement with the field observed values of 20.7 ° for temperate ridges and 32.9 °

G.W. Timco, R.P. Burden/Cold Regions Science and Technology 25 (1997) 65-77

for Beaufort ridges. A keel slope of 35 ° was predicted by their model, which is in reasonable agreement with the field observations which give an average keel angle of 26.6 ° . Their model predicts limit heights for the ridges, based on the ice floe thickness and strength. It is not possible to directly compare their predictions to the field data, since the ridges profiled in the field were not necessarily generated with the ice thickness that surrounded the ridge during the profile process. That is, the ice ridges were usually formed much earlier when the ice was thinner. In general, however, the predicted sail heights and keel depths are certainly in-line with the range of values measured for sea ice ridges. More recently, Hopkins, in a series of papers (Hopkins and Hibler, 1989; Hopkins et al., 1991; Hopkins, 1994) has developed a physics-based, 2-dimensional particle simulation approach to model the ridge-building process. In this sophisticated model, the instantaneous shape, position, orientation and velocity of each particle in an array are stored in a computer program. The contact and body forces on each particle are calculated at each time-step, and each particle is allowed to move to new locations with new conditions which depend upon the resultant of the forces. By starting with a similar situation to that used by Parmerter and Coon, the model clearly shows the evolution of a ridge with time. To date, Hopkins has concentrated on developing this complex model, and investigating the energetics of the ridge-building process. However, he has used the model to predict the general ridge shape for a few cases. He reports that with the model and average sail heights of 1 to 2 m, the average sail angle is approximately 18° + 6 °, and keel angles of 46 ° + 10°. These are in reasonable agreement with the measured values.

6. Summary This paper has presented a summary of some of the key characteristics of both first-year and multiyear sea ice ridges. The characteristics were based on an analysis of the details and profiles of 176 ridges, with 112 first-year ridges, and 64 multi-year ridges. Relationships between several important ridge char-

75

acteristics have been established using best-fit linear and power law behaviour. In virtually all cases, however, the simple linear equation appears to be adequate for describing the relationship. For first-year ridges, the following characteristics were determined: • The overall shape of ridges can be quite variable and non-symmetrical, but they are usually triangular in shape. • The ratio of the keel depth to sail height follows a log-normal distribution with a mean value of 4.4 and a standard deviation of 1.9. The ratio of the keel width to keel depth is approximately 3.9. The ratio of the keel width to sail height is approximately 15.1. The ratio of the keel area to sail area is approximately 8.0. The sail angles can be described by a log-normal distribution with a mean value of 20.7 ° for temperate ridges, and 32.9 ° for Beaufort ridges. The keel angle can be described by a log-normal distribution with a mean value of 26.6 °. There is considerable variability in the thickness of the consolidated layer of the ridge, with thickness ratios of 0.51 for the minimum-to-average thickness, and a ratio of 1.68 for the maximumto-average thickness. Thus, there can be variability by a factor of over 3 in the thickness of the consolidated layer. The keel and sail porosity can be related by a linear equation with a non-zero intercept of 0.14, and a slope of 0.73. For multi-year ridges, the following information was determined: The overall sizes of multi-year ridges can be significantly larger than first-year ridges, and they are more rectangular in shape. The ratio of the keel depth to sail height is approximately 3.3 for Beaufort Sea multi-year ridges. The ratio of the keel area to sail area is approximately 8.8. The sail angle can be described by a normal distribution with a mean value of 19.5 °. The keel and sail porosity can be related by a linear equation with a non-zero intercept of 0.06, and a slope of 0.91.

76

G. W. Timco, R.P. Burden / Cold Regions Science and Technology 25 (1997) 65 77

Acknowledgements This work was partially sponsored by the Panel on Energy Research and Development (PERD) and the National Energy Board of Canada, and the authors would like to thank them, and Dr. I. Konuk of the NEB for their support. This work is a contribution from the PERD Project (6A5009) - - NRC Centre for Ice/Structure Interaction. References Banke, E.G. and Lowings, M.G., 1983. Consolidation of Pressure Ridges in First Year Landfast Ice Eastern Beaufort Sea, Winter 1981-1982. Martec Report to Gulf Resources Canada Ltd., Calgary, Alta. Burden, R.P. and Timco, G.W., 1995. A Catalogue of Sea Ice Ridges. National Research Council of Canada Report TR1995-27, Ottawa, Ont. Cox, G.F., 1972. Variation of Salinity in the Multiyear Ice at the AIDJEX 1972 Camp. Masters Thesis, Dartmouth College, Hanover, NH. Davis, N.R. and Wadhams, P., 1995. A Statistical analysis of Arctic pressure ridge morphology. J. Geophys. Res., 100: 10915-10925. Dickens, D., 1981. Multi-year pressure ridge study Queen Elizabeth Islands. In: Proceedings of Port and Ocean Engineering under Arctic Conditions, POAC'81, Quebec, Vol. 2, pp. 765775. Evers, K.-U., 1986. Erste Eisbrechtechnische Expedition mit F.S. Polarstern. Schlussbericht, Band III, Kap. 13: Geometrie und Konsolidierungsgrad der yon "Polarstern" durchbrochenen Presseisriicken. Hamburg. Harrington, A.G., 1978. Geometry and Strength of Multi-Year Pressure Ridges in the Alaskan Beaufort Sea. Gulf Research and Development Company Report, Houston Technical Services Centre, Houston, TX. Hopkins, M.A., 1994, On the ridging of an intact ice sheet. J. Geophys. Res., 99(C8): 16351 - 16360. Hopkins, M.A., Hibler, W.D., Ill and Flato, G.M., 1991. On the numerical simulation of the sea ice ridging process. J. Geophys. Res., 9(C3): 4809-4820. Hopkins, M.A. and Hibler, W.D., III, 1989. On modelling the energetics of the ridging process. In: Proceedings OMAE'89, The Hague, Vol. 4, pp. 175-178. Gladwell, R.W., 1975. Field Studies of Eight First Year Pressure Ridges in the Southern Beaufort Sea. Arctic Petroleum Operators' Association Report # 75 (APOA 75), Calgary, Alta. Kankanpaii, P,, 1989. Structure of first year pressure ridges in the Baltic Sea. In: Proceedings of Port and Ocean Engineering under Arctic Conditions, POAC'89, Lulea, Vol. 1, pp. 87-102. Kovacs, A., 1976. Grounded Ice in the Fast Ice Zone Along the Beaufort Sea Coast of Alaska. Cold Regions Research and Engineering Laboratory (CRREL) Report 76-32. Hanover, NH.

Kovacs, A., 1977. Sea ice thickness profiling and under ice oil entrapment. In: Ninth Offshore Technology Conference, Houston, TX, Vol. 3, pp. 547-554. Kovacs, A., 1983. Characteristics of multi-year pressure ridges. In: Proceedings of Port and Ocean Engineering under Arctic Conditions, POAC'83, Helsinki, Vol. 3, pp. 173-182. Kovacs, A. and Mellor, M., 1971. Sea Ice Pressure Ridges and Ice Islands. Creare Inc. lor Arctic Petroleum Operators Association, Technical Note 122. Calgary, Alta. Kovacs, A. and Mellor, M., 1974. Sea ice morphology and ice as a geologic agent in the southern Beaufort Sea. In: J.C. Reed and J.C. Sater (Editors), Symposium on the Coast and Shelf of the Beaufort Sea. Arctic Institute of North America, Arlington, VA, pp. 113-164. Kovacs, A., Weeks, W.F., Ackley, S.F., and Hibler, W.D., III, 1973. Structure of a multiyear pressure ridge. Arct. J. Arct. Inst. N. Am., 26(1): 23-31. Kovacs, A. Dickens, D. and Wright, B., 1975. An Investigation of Multi-Year Pressure Ridges and Shore Pile-Ups. Arctic Petroleum Operators Association No. 89, Calgary, Alberta, Canada. Lepp~iranta, M. and Hakala, R. 1991. The structure and strength of first year ice ridges in the Baltic Sea. Cold Reg. Sci. Technol., 20:291-311. McGonigal, D., 1978. First Year Pressure Ridge Study Beaufort Sea, April-May 1978. Frontier Research Division, Gulf Canada, Calgary, Alta. Parmerter, R.R. and Coon, M.D., 1972. Model of pressure ridge formation in sea ice. J. Geophys. Res., 77(33): 6565-6575. Tucker, W.B., III, 1989. An overview of the physical properties of sea ice. In: Proceedings of the Workshop on Ice Properties. NRC Technical Memo 144, NRCC 30358, Ottawa, Ont., pp. 71-89. Tucker, W.B.. Ill, Sodhi, D,S. and Govoni, J.W., 1984. Structure of first-year pressure ridge sails in the Prudhoe Bay region. In: P.W. Barnes, D.M. Schell and E. Reimnitz (Editors). The Alaskan Beaufort Sea: Ecosystems and Environraents. Academic Press, New York, NY, pp. 115-135. Tucker, W.B., IIl, Weeks, W.F. and Frank, M., 1979. Sea ice ridging over the Alaskan Continental Shelf. J. Geophys. Res., 84(C8): 4885-4897. Vaudrey, K.D., 1980. Beaufort Sea first-year ice features survey - - 1979 Volume 1: Field Investigation. Gulf Research and Development Company Report, Houston Technical Services Centre. Houston, TX. Veitch, B., Lensu, M., Riska, K., Kosloff, P., Kelley, P. and Kujala, P., 1991a. Field observations of ridges in the northern Baltic Sea. In: Proceeding of Port and Ocean Engineering under Arctic Conditions. POAC'91, St. John's, Nfld., Vol. 1, pp. 381-400. Veitch, B., Kujala, P., Kosloff, P. and LeppRranta, M., 1991b. Field Measurements of the Thermodynamics of an Ice Ridge. Helsinki University of Technology, Laboratory of Naval Architecture and Marine Engineering, Report No. M-I14, Helsinki. Voelker, R.P., DeBord, F.W., Nelka, J.J., Jacobi, J.W. and Coburn, J.L., 1981. Assessment of Ice Conditions in the South Bering

G.W. Timco, R.P. Burden / Cold Regions Science and Technology 25 (1997) 65-77 Sea Based on April 1980 USCG Polar Class Trafficability Test Data. Vols. 1 and 2. U.S. Maritime Administration Artec Inc., Springfield, VA. Voelker, R.P., DeBord, F.W., Geisel, F.A., Coburn, J.L. and Dane, K.E., 1982. Winter 1981 Trafficability Tests of the USCGC Polar Sea. Vols. 1 and 2. U.S. Maritime Administration Artec Inc., Springfield, VA. Wadhams, P., McLaren, A.S. and Weintraub, R., 1985. Ice thickness distribution in Davis Strait in February from submarine sonar profiles. J. Geophys. Res., 90(C1): 1069-1077.

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Weeks, W.F., Tucker, W.B., Frank, M. and Fungcharoen, S., 1980. Characterization of the surface roughness and floe geometry of the sea ice over the continental shelves of the Beaufort and Chukchi Seas. In: R.S. Pritchard (Editor), Sea Ice Processes and Models. University of Washington Press, Seattle, WA, pp. 300-312. Williams, F.M, and Kirby, C.S., 1994. Ice Features in Northumberland Strait. N R C / I M D Report TR-1994-08, St. John's, Nfld.