CRREL REPORT 95-16

A Physically Based Model of the

Form Drag Associated with Sastrugi

Edgar L Andreas July 1995
For conversion of SI units to non-SI units of measurement consult Standard

Practice for Use of the International System of Units (SI), ASTM Standard E380-

93, published by the American Society for Testing and Materials, 1916 Race St.,

Philadelphia, Pa. 19103.

Cover: Drifting snow on Ice Station Weddell. Photograph by Keran Claffey.

Abstract

On Ice Station Weddell, some characteristics of the neutral-stability air–ice

drag coefficient at a reference height of 10 m (CDN10) were observed that had

not been documented before. The main finding was that wind-driven snow

continually alters the sea ice surface; the resulting snowdrifts determine how

large CDN10 is. In particular, this report describes three observations and attempts to explain them: 1) CDN10 is near 1.5 × 10–3 when the wind is well

aligned with the drifted snow; 2) CDN10 is near 2.5 × 10–3 when the wind

makes a large angle with the dominant orientation of the snowdrifts; 3)

CDN10 can increase by 20% if, after being well aligned with the drift patterns,

the mean wind direction shifts by as little as 20°. To investigate this behavior

of CDN10, this report adapts a model developed by Raupach that partitions the

total surface stress into contributions from form drag and skin friction. With

reasonable choices for free model parameters and with little fine-tuning, this

physically based model can reproduce the three main observations. In other

words, the model seems to include the basic physics of air–ice momentum

exchange. This modeling implies that 10-cm-high sastrugi-like roughness

elements, rather than pressure ridges, sustain most of the form drag over

compact sea ice in the western Weddell Sea. Lastly, the report speculates on

what the observations and this model say about how to parameterize CDN10

over snow-covered sea ice.
CRREL Report 95-16

A Physically Based Model of the

Form Drag Associated with Sastrugi

Edgar L Andreas July 1995

Prepared for

NATIONAL SCIENCE FOUNDATION

and

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

US Army Corps

of Engineers

Cold Regions Research &

Engineering Laboratory
PREFACE

Dr. Edgar L Andreas, Physicist, Snow and Ice Division, Research and Engineering

Directorate, U.S. Army Cold Regions Research and Engineering Laboratory, prepared

this report. He thanks Keran J. Claffey, who played a large role in collecting the data

on Ice Station Weddell, and R.B. Haehnel and Dr. S.C. Colbeck, who reviewed the

manuscript.

The Office of Polar Programs at the National Science Foundation supported this

work through grants OPP-90-24544 and OPP-93-12642, and the Department of the

Army supported it through Project 4A161102AT24.

The contents of this report are not to be used for advertising or promotional purposes. Citation of brand names does not constitute an official endorsement or approval of the use of such commercial products.

ii
iii

CONTENTS

Preface ........................................................................................................................................ ii

Introduction............................................................................................................................... 1

Raupach’s model adapted to drifted snow .......................................................................... 2

Geometry .............................................................................................................................. 2

Partitioning the stress ......................................................................................................... 4

Displacement height ........................................................................................................... 6

Results ........................................................................................................................................ 7

Discussion .................................................................................................................................. 10

Primary roughness elements ............................................................................................. 10

Parameterizing the drag coefficient .................................................................................. 10

Conclusions ............................................................................................................................... 11

Literature cited .......................................................................................................................... 11

Abstract ...................................................................................................................................... 13

ILLUSTRATIONS

Figure

1. Time series of the Ice Station Weddell CDN10 values and hourly averages of the

wind speed and direction ....................................................................................... 1

2. Snow and ice surface elevations near ISW profile tower on Julian Day 78 ............. 3

3. Geometry of the sastrugi-like snowdrifts being modeled .......................................... 3

4. Model calculation of CDN10 as a function of the wind orientation with respect

to the dominant axis of the sastrugi ..................................................................... 8

5. Stress partitioning versus the mean wind direction with respect to the dominant

axis of the sastrugi ................................................................................................... 9

6. Computed displacement height scaled by sastrugi height versus the direction of

the mean wind with respect to the sastrugi ........................................................ 9
A Physically Based Model of the

Form Drag Associated with Sastrugi

EDGAR L ANDREAS

Figure 1. Time series of the Ice Station Weddell (ISW)

CDN10 values reported in Andreas and Claffey (1995)

and hourly averages of the wind speed and direction at

5 m.

INTRODUCTION

The neutral-stability, 10-m air–ice drag coefficients (CDN10) reported by Andreas and Claffey

(1995; henceforth, AC95) clearly depend on how

wind-driven snow alters the sea ice surface. Figure 1 shows the time series of Ice Station Weddell

(ISW) CDN10 values and hourly averages of the

wind speed and direction measured at 5 m.

These CDN10 values range between 1.27×10–3 and

2.54 × 10–3 but are not randomly scattered between these limits. Rather, they generally cluster

or depend coherently on time. For example, in

AC95 we focused on the events on Julian days 64

and 65 and 79 and 80. In Figure 1, we see that, on

these days, the 5-m wind speed was typically

8 m/s or higher, the wind direction was fairly

constant, and CDN10 decreased consistently with

time.

Our explanation for this sequence was that

winds of 6–8 m/s, and higher, redistribute the

mobile snow: snowdrifts perpendicular to the

wind erode; and the new drifts, built parallel

with the mean wind, streamline the surface. Consequently, if these eroding winds persist and

their direction remains relatively constant for

just 12 hours, CDN10 will decrease by as much as

30%. If, after establishing a drift pattern, however, the wind shifts by as little as 20°, it will face

a rougher surface, and measured CDN10 values

will increase.

There are several other events in Figure 1 that

demonstrate this same smoothing in the presence of a moderate, directionally constant wind

(for example, see days 86 and 87). Thus, the

sequence is repeatable and, in fact, seems to be

the key to understanding air–ice momentum exchange in the western Weddell Sea—and probably over most other sea ice regions as well.

Form drag sustained by the snowdrifts must

explain the behavior of the CDN10 values that we

observed. Others have also recognized that form
drag can, at times, be an important means for

transferring momentum to a surface. For example, Lettau (1969) offered a heuristic formula involving obstacle geometry and silhouette area to

parameterize the roughness lengths that Kutzbach (1961) measured during his famous bushel

basket experiment on Lake Mendota. Arya (1973,

1975) assumed that pressure ridges were the

roughness elements responsible for most of the

form drag over Arctic sea ice and developed a

model to partition the total stress into skin friction and form drag on the basis of ridging intensity. Banke et al. (1976, 1980) investigated the

pressure distribution around Arctic pressure

ridges 1–3 m high in an attempt to determine the

form drag coefficient for these. But their ultimate

parameterization for CDN10 (also Banke and

Smith 1973) is more in line with our findings;

they showed that

103 CDN10 = 1.10 + 0.072 ξ (1)

where ξ (in centimeters) is the root-mean-square

surface roughness for roughness elements with

wavelengths shorter than 13 m. That is, the

smaller scale roughness dominates the momentum exchange. Jackson and Carroll (1978)

reached the same conclusion. They showed that

the angle between the mean wind and the dominant axis of the sastrugi at the South Pole determined the roughness length; they then developed a model based on Lettau’s to explain this

effect.

More recently, Raupach (1992; henceforth,

R92; also Raupach et al. 1993) developed a more

sophisticated model, based on dimensional analysis and two physically reasonable hypotheses,

to partition the total surface stress into contributions from form drag and from the stress on the

underlying surface. That model is the basis of the

analysis here. I adapt Raupach’s model to evaluate the stress partitioning for a field of sastrugilike snowdrifts. That model predicts the drag coefficient evaluated at the top of the roughness elements; I show how this quantity is related to

CDN10, which, in turn, is monotonically related to

the roughness length z0 through

CDN10 = k2 {ln[(10 – d)/z0]}–2 . (2)

Here, k (= 0.4) is the von Kármán constant, and z0

and the displacement height d must be in meters.

My objective is to see whether, with reasonable choices for snowdrift geometry and other

free model parameters, R92’s model can account

for the behavior of CDN10 at ISW that we documented in AC95. In particular, the three main observations that I hope to explain are as follows:

1. CDN10 is near 1.5 × 10–3 when the wind is

aligned with a well-developed field of drifted

snow.

2. CDN10 is near 2.5 × 10–3 when the mean wind

direction makes a large angle (say 50–140°) with

the dominant axis of the drifts.

3. CDN10 can increase by roughly 20% if, after

being aligned with a developed (or developing)

field of drifted snow, the wind turns by as little as

20°.

RAUPACH’S MODEL ADAPTED TO

DRIFTED SNOW

Geometry

Because during our deployment on ISW the

wind was directionally quite variable, the surface never developed the deep, elaborate fields

of sastrugi that Mellor (1965) and Jackson and

Carroll (1978) observed on the Antarctic continent. Rather, on ISW, the drifts were sparsely distributed and rudimentary. When the wind

reached 6–8 m/s, linear sastrugi-like drifts

would begin forming behind any protruding

obstacles. The drifts would grow as the wind

persisted; but when the wind direction changed,

these drifts would erode in winds above the 6- to

8-m/s threshold. Thus, the drifts that determined the surface roughness on ISW were generally small. The scour marks that Allen (1965) described and named scour-remnant ridges may

well be precursors to these rudimentary sastrugi.

Figure 2 shows a typical leveling profile of the

snow and sea ice surfaces in the vicinity of the

profiling mast described in AC95. We made seven such leveling surveys during our deployment

on ISW. The profiles show that snow tends to collect around obstacles in the underlying ice. The

root-mean-square (rms) roughness of the snow

surface in this particular profile is 11 cm; its icesurface roughness is 9 cm. In the seven surface

profiles that we collected, the rms snow-surface

roughness ranged from 9 to 26 cm; the ice-surface roughness ranged from 9 to 28 cm. A value

in this range is a reasonable choice for the nominal height of the snow-surface roughness elements.

Figure 3 shows the idealized geometry of the

roughness elements that I will treat with R92’s

model. These are rudimentary sastrugi with

2
3

Figure 2. Snow and ice surface elevations near our ISW profile

tower on Julian day 78 (in 1992). The sampling interval was 0.5 m.

The profiling line is not necessarily aligned with the mean wind.

Figure 3. Geometry of the sastrugi-like snowdrifts being modeled here.

per unit ground area. Suppose the fraction of the

area covered by the sastrugi-like roughness elements is γ. Thus, from Figure 3, the unit area represented by each roughness element is mnh2/2γ. The

base length of the triangular frontal area that each

roughness element presents to the wind, L(Φ), is a

function of Φ. Likewise, the silhouette area presented, A(Φ), is simply

 

A hL ( ) Φ Φ = ( ) 1

2 . (4)

Consequently, λ is also a function of Φ

 

λ γ γ Φ

Φ Φ ( ) = ( ) = 2 ( )

2

A

mnh

L

mnh . (5)

In Figure 3, we see that there are four Φ regions

for which a roughness element presents different

faces to the wind for 0 ≤ Φ ≤ 180° (symmetric for

0 ≥ Φ ≥ –180°). These and the associated values of

L, A and λ follow.

Wind

β

Φ

Top

β

Front

n h m h

h

Side

height h, width mh, and length nh. The angle Φ

describes how the mean wind is oriented with

respect to the sastrugi. When the wind has been

blowing with constant direction at speeds of

6–8 m/s, or higher, for several hours, Φ will be

zero.

From the discussion in the penultimate paragraph, we see that h is 5–20 cm. Descriptions of

drifted snow in the literature (Finney 1939, Mellor 1965, Jackson and Carroll 1978) and Plate’s

(1971, Figure 4.17) pressure distribution around

roughness elements similar to that in Figure 3

suggest that n is about 10. From Mellor’s (1965)

description of sastrugi, I assume that m is about

4. Notice, these parameters imply that, in Figure

3

β = arctan(m/2n) = 11.3°. (3)

One of the key parameters in R92’s model is λ,

the frontal area of a typical roughness element
AC95). Thus, since R92 models eq 10 for heights of

order h and higher—well above the saltation

layer—eq 10 need not include a term to account

for particle inertia. Above the saltation layer, the

concentration of blowing and drifting snow is too

low to affect the dynamics of the flow.

R92 modeled the stress on the underlying surface as

 

τλ ρ λ S Sh h

2

1

h

*

( ) = − ( )















 CU c U

u

exp . Φ (11)

Here, CSh is the drag coefficient of the underlying

surface referenced to a height h, which is where

the wind speed Uh is evaluated. Also in eq 11, ρ is

the air density; u

*

, the friction velocity, u

*

=(τ/ρ)1⁄2;

and c1, an empirical constant. The exponential

term in eq 11 accounts for the sheltering of the

underlying surface by the roughness elements.

That is, when the roughness elements are small or

sparse and λ(Φ), thus, is small, eq 11 reduces to

 

τ ρ S Sh h

2 = C U , the usual expression for the skin friction over a smooth surface. On the other hand,

when the roughness elements have a large frontal

area or are densely packed, λ(Φ) is large, and τS

approaches zero. In this case, the roughness elements completely shelter the surface; thus, skin

friction can provide none of the stress.

We can infer the value of CSh appropriate for

smooth, snow-covered surfaces from Overland’s

(1985) review and from the measurements reported by Banke et al. (1980) and Kondo and Yamazawa (1986). The lowest value for CDN10 reported by

Overland and Kondo and Yamazawa is roughly

1.1 × 10–3. Coincidentally, Figure 4 in Banke et al.

implies that the value of CDN10 for completely

smooth sea ice is 1.10 × 10–3. Call this minimum

skin friction coefficient CS10.

Converting CS10 to CSh, however, is not straightforward: We do not know the displacement height

d a priori; and near the tops of the roughness elements, the semi-logarithmic profile law (see eq 2)

is not accurate. I will explain how I deal with these

complications later; here, suffice it to say, I obtain

CSh from CS10.

R92 wrote a general expression for the force

caused by form drag on an isolated roughness element as

 

F C AU D Rh

2 = ρ (12)

where CR is the drag coefficient of the obstacle,

and A is its silhouette area. Here, I must modify

4

Region I: 0° ≤ Φ ≤ β

Here the wind sees only the front triangular

face. Thus

L(Φ) = mh cosΦ (6a)

A(Φ) = (mh2/2) cosΦ (6b)

λ(Φ) = (γ/n) cosΦ . (6c)

Region II: β ≤ Φ ≤ 90°

Here the wind sees both the front triangular

face and the side ridge. Thus

L(Φ) = nh sinΦ + (mh/2) cosΦ (7a)

A(Φ) = (h2/2)[n sinΦ + (m/2) cosΦ] (7b)

λ(Φ) = γ[m–1 sinΦ + (2n)–1 cosΦ] . (7c)

Region III: 90° ≤ Φ ≤ 180°–β

Here the wind sees only the side ridge. Thus

L(Φ) = nh sinΦ – (mh/2) cosΦ (8a)

A(Φ) = (h2/2) [n sinΦ – (m/2) cosΦ] (8b)

λ(Φ) = γ[m–1 sinΦ – (2n)–1 cosΦ] . (8c)

Region IV: 180° – β ≤ Φ ≤ 180°

Here the flow sees both side ridges as it

approaches from the rear of the drift. Thus

L(Φ) = –mh cosΦ (9a)

A(Φ) = –(mh2/2) cosΦ (9b)

λ(Φ) = –(γ/n) cosΦ . (9c)

Partitioning the stress

R92’s intent was to partition the total surface

stress (τ) into contributions from form drag (τR)

and from stress on the underlying surface (the

skin friction, τS). In general

τ = τR + τS . (10)

There is another potential sink for the momentum in eq 10 that R92 did not treat. Over erodible

surfaces, saltating particles may extract momentum from the air because of their inertia. Neither

τR nor τS reflects this momentum exchange. That

saltation layer, however, is quite thin—on the

order of a centimeter (Owen 1964, Radok 1968,
Raupach’s model, however. In my model, the obstacle’s aerodynamic properties likely change as

its orientation with the wind changes. Otherwise, the concept of streamlining would be

meaningless. Thus, CR and A must depend on Φ.

But because I could find no study in the literature

that reported the aerodynamic properties of an

obstacle like that depicted in Figure 3, I assume

that the drag contributions from the various faces are additive. Mathematically, my hypothesis is

 

F UC A

i

D h

2 Φ ΦΦ Ri i ( ) = ρ ∑ () () (13)

where CRi is the drag coefficient for a particular

face of the sastrugi-like obstacle, and Ai is the silhouette area of that face. The sum is over all faces

presented to the wind at angle Φ.

The obstacle in Figure 3 presents only three

faces. When 0° ≤ Φ ≤ 90°, the wind can see the

front triangular face; this has form drag coefficient CR1. When β ≤ Φ ≤ 180° – β, the wind can see

the side ridge; this has drag coefficient CR2.

When 180° – β ≤ Φ ≤ 180°, the wind can see only

the rear silhouette; this has drag coefficient CR3.

As I mentioned, I found no study that evaluated CR for the obstacle geometry I use here. But on

reviewing several studies that treated somewhat

similar geometries (i.e., Arie and Rouse 1956;

Arya 1973, 1975; Banke et al. 1976, 1980; Taylor

1988), I estimated what I feel are representative

values for the three faces. Here, I use CR1 = 0.10,

CR2 = 0.30, and CR3 = 0.30.

Continuing now with R92’s model, if p roughness elements cover a ground area S, the stress

they produce through form drag is

 

τ ρ R h

2 Φ ΦΦ Ri i , . p p

S

UC A V

Sh i

p

( ) = () () 













 − 



 

 ∑ 1  (14)

Here, V is the volume sheltered by an individual

roughness element; the term raised to power p,

thus, represents a mutual sheltering effect.

In my model, from eq 5

 

p

S mnh A = = ( )

( )

2

2

γ λ Φ

Φ . (15)

Using this in eq 14 gives

 

τ

λ

R h ρ 2 Φ Ri i

Φ

Φ , p Φ ΦA

UC A i

( ) = ( )

( ) () () 



 

 ∑ 

 

 

⋅ − 









 1 λ( )

( ) . ΦΦ

V

phA

p

(16)

In the limit of large p

 

1− ( )

( )











 ≈ − ( )

( )





 



 λ λ Φ

Φ

Φ

Φ

V

phA

V

hA

p

exp . (17)

Finally, following R92, for m and n greater than 1,

I estimate the sheltered volume to be

 

V cL h U

u = ( ) 1

2 2

2 Φ h

*

 (18)

where c2 is another empirical constant. Hence,

using eq 5, 17 and 18 in eq 16, we get

 

τ λρ R h

2

Ri i ΦΦ Φ

Φ

Φ ( ) = () () ( )

( )

















U C ∑ A

i A

 

 

⋅ − ( )















 exp . c

U

u

2 λ Φ h

*

(19)

In eq 19, define

 

ˆ

C C . A

i A R Ri

i Φ Φ

Φ

Φ ( ) ≡ ( ) ( )

( ) ∑ (20)

For the four Φ regions identified earlier, 

Cˆ

R follows.

Region I: 0° ≤ Φ ≤ β

 

ˆ

C C . RI R1 = (21a)

Region II: β ≤ Φ ≤ 90°

 

ˆ cos sin ( / )cos

sin ( / )cos

C . Cm C n m

n m

RII

R1 R2 = + − [ ]

+

ΦΦ Φ

Φ Φ

2

2

(21b)

Region III: 90° ≤ Φ ≤ 180° – β

 

ˆ

C C . RIII R2 = (21c)

Region IV: 180° − β ≤ Φ ≤ 180°

 

ˆ

C C . RIV R3 = (21d)

Combining eq 11, 19 and 20 in eq 10, we get

 

τρ λ ( ) Φ Φ = − ( )



























UC c U

u

h

2

Sh

h

*

exp 1

 

 

+ () () − ( )



























λ λ ΦΦ Φ ˆC c exp .

U

u

R

h

*

2 (22)

5
6

To make solving eq 22 easier, R92 assumed that

c1 = c2 = c; for c he used 0.25, which provided the

best fit to the data that he tested his model

against. With this assumption, eq 22 reduces to

 

τρ λ λ () () ˆ Φ ΦΦ Φ = + ( ) exp ( ) . [ ] −















 UC C c U

u

h

2

Sh R

h

*

(23)

Notice

 

τ

ρ

( ) Φ =≡ ≡ U

u

U

C

r h

2

h

2 Dh *

2

2

1 (24)

is the drag coefficient for the total surface stress

referenced to height h. On inserting this definition and rearranging terms in eq 23, we get

 

cr cr λ λ ( ) Φ Φ − ( )



 



 2 2 exp

 

= ( )[ ] + () () c −

C C λ λ ΦΦ Φ

2

1 2

Sh Rˆ . /

(25)

This has the form

X e–X = a (26)

where a is a constant for a given value of Φ.

I solved eq 26 for X using Newton’s method;

the solution required only two to three iterations.

There are multiple solutions to eq 26, however;

we must select the right one. As R92 explained,

eq 26 has the following properties:

For 0 < a < e–1, eq 26 has two solutions, X1 and

X2, where X1 < 1 and X2 > 1.

For a = e–1, X = 1.

For a > e–1, there is no solution to eq 26.

The only nontrivial case is the first of these; and

X1 is the only physical solution, since X must approach zero as either λ(Φ) or a approaches zero.

Thus, to start Newton’s method, my first guess

was X = a.

Solving eq 26 yields X, which, in turn, is related to the total drag coefficient we seek

 

C X

c

Dh

–1/2 = ( )

2

λ Φ . (27)

Notice, to check this solution, we can make an

asymptotic approximation in eq 23. When λ(Φ) is

near zero

 

CC C Dh Sh R = + λΦ Φ () () ˆ . (28)

Displacement height

To convert the total drag coefficient CDh to its

10-m value, we need to know the wind speed

profile law. When a surface is covered with

roughness elements that are much larger than z0,

it is often necessary to include a displacement

height d (Thom 1971) in the semi-logarithmic

wind profile. That is

 

U z

u k

z d

z

( ) ln

*

=  −



 



 1

0

 (29)

where U(z) is the average wind speed at height z.

But near the tops of the roughness elements,

eq 29 cannot be accurate because of the enhanced

turbulence there created by flow around these

obstacles. R92 therefore hypothesized that, for

h ≤ z ≤ zw

 

U z

u k

z d

z

z d

z d

( ) ln

* w

=  −



 



 + −

−





 





















1

0

ψ (30)

where ψ is a profile influence function similar to

the stability corrections required by MoninObukhov similarity. At zw and above, ψ is zero.

R92 went on to show that

 

ψ

z d

z d

z z

z d

z d

z d

−

−





 



 = −

− + −

−





 



 w

w

w

w ln (31)

and that

zw – d = cw(h – d) (32)

where cw is a constant. He also defined

 

ψ ψ h

w

w w

–1 ≡ −

−





 



 = ( ) + −

h d

z d

ln 1 c c (33)

which will be useful shortly. Notice, there are errors in two of R92’s equations (eq 29 and 31). In

an unpublished correction to his paper dated 23

October 1992, Raupach explained that, in light of

these errors, cw is approximately 4 rather than

1.5, the value he originally gave. Hence,

ψh = 0.64, a value independent of surface properties.

Equation 32 implies that

 

z ch c d ww w = +− ( ) 1 . (34)

With cw = 4 and the assumption that zw > h, eq 34

implies that d < h: The displacement height is always less than the height of the roughness elements.
To obtain CDN10 from CDh, we must match the

profile laws, eq 29 and 30, at zw. From eq 29 and

32, we have the following results for z ≥ zw

 

C U

u k

d

z

DN10

*

– / ( ) ln 1 2 10 1 10

0

≡ =  −



 



 (35)

and

 

C U z

u k

h d

z

Dz c w

w w

*

– / ( ) ln ln 1 2 1

0

≡ =  −



 



 + ( ) 















 (36)

where h, d, z0 and zw must be in meters. Subtracting eq 35 from 36, we get

 

C C

k

h d

d Dz DN10 wc w

–/ –/ ln ln . 12 12 1

10

= + −

−





 



 + ( ) 



 



 (37)

From eq 30–33, for h ≤ z ≤ zw

 

C U

u k

h d

z

Dh

h

h

*

– / ln 1 2 1

0

≡ =  −



 



 +















 ψ (38)

and

 

C U z

u k

h d

z

Dz c w

w w

*

– / ( ) ln ln . 1 2 1

0

= =  −



 



 + ( ) 













 (39)

Subtracting eq 38 from 39 yields

 

C C

k Dz Dh w h c w

–/ –/ ln . 12 12 1

= + [ ] ( ) − ψ (40)

On equating eq 37 and 40, we finally can relate

CDN10 to CDh

 

C C

k

d

h d DN10 Dh h

–/ –/ ln . 12 12 1 10

= + −

−





 



 − 



 

 ψ  (41)

We still do not know the displacement height d,

however.

Thom (1971) identified d as the effective level

at which the roughness elements absorb the momentum being transferred to the surface. R92

used this definition to derive d. That is, if dR is the

centroid of the form drag on the roughness elements and if dS is the centroid at which the skin

friction acts

 

dd d d ττ τ τ = += RR SS RR (42)

because dS = 0 by definition. Thus

 

d d = R R ( ) τ τ / . (43)

7

* Personal communication (Arctic and Antarctic Research

Institute, St. Petersburg, Russia).

Notice, from eq 19, 20 and 23

 

τ

τ

λ

λ

R R

Sh R

= () ()

+ () ()

Φ Φ

Φ Φ

ˆ

ˆ . C

C C (44)

R92 then related dR to h. Following his arguments but applying them to the specific geometry of my model, I get

 

d h cC A R d Dh = − [ ] () () – / / 1 2 1 2 / Φ Φ λ (45)

in which cd is a constant that R92 took to be 0.6.

Inserting eq 5 in this and combining it with eq 43

yields

 

d h c C mn = ( ) ττ γ R d Dh / /. [ ] − ( ) – / / 1 2 1 2 1 2 (46)

Thus, it is easy to find τR/τ from eq 44; and once

my model yields CDh, it is simple to compute d.

Using this value in eq 41 finally gives the quantity we seek, CDN10.

I pointed out in the previous section that we

need to convert CS10 to CSh before we can begin

computations. Equation 41 also makes this conversion. Unfortunately, we need CSh before we

have obtained d. I could handle this problem by

iterating the entire set of equations on d. But, as I

will show in the next section, d/h ≈ 0.3, a result

also consistent with eq 46. At this step in the

computations, this simple approximation is reasonable—especially in light of uncertainties in

the other model parameters—because CDN10 is

not very sensitive to CS10 for the range of values

that this parameter can realistically assume.

RESULTS

I have not yet discussed the value of γ, the

fractional coverage of sastrugi-like roughness elements. Vladimir Churun* tried to quantify the

roughness of the ISW floe using the radar on the

Akademik Fedorov at the time the station was deployed. His survey suggested that hummock

coverage varied over the floe from 10 to 30%.

While hummock coverage is not the same as sastrugi coverage, my personal experience on ISW

suggests that 10–30% sastrugi coverage is also

about right. Notice, because of the geometry of

the sastrugi that I am modeling (Fig. 3), the tightest possible packing of roughness elements will

yield a γ value of only 0.5. Thus, my first guess,
γ = 0.15, represents a conservative but not insignificantly small sastrugi coverage.

Figure 4 shows two sets of predictions for

CDN10 as a function of Φ based on the model

developed in the preceding section. Figure 4a

shows computations of CDN10 for γ = 0.15 and for

sastrugi heights of 5, 10 and 20 cm, typical heights

represented in our snow-surface roughness surveys (e.g., Fig. 2). The 10-cm case fits our observations quite well. Here, CDN10 is 1.43–1.45 × 10–3

when the flow is head-on or within ±12° of headon (that is, when Φ <

~ β). These values agree well

with the nominal minimum value for CDN10 of

1.5 × 10–3 that we reported in AC95.

When Φ gets larger than 11.3° (see eq 3), the

wind can see the side ridge of the obstacle. In effect, it sees a larger silhouette area and a higher

value of CR; CDN10 must thus increase. Figure 4

shows that if the wind rotates just 20° from headon, CDN10 will increase by 15–24%, in good agreement with our observations in AC95.

As Φ increases and the wind thus has an increasing component perpendicular to the axis of

the sastrugi, CDN10 continues increasing. CDN10 has

its maximum at Φ = 101.3°, where it is 2.73 × 10–3

for the γ = 0.15 and h = 10 cm case. Again, m and n

set the location for this maximum; this is where

A(Φ) has its maximum. The CDN10 value there is in

fair agreement with the maximum value that we

observed on ISW, 2.54 × 10–3 (AC95).

As Φ increases toward 180°, the model is entirely speculative, since we have few data with which

to compare such a reversal in the wind and because CR3 is the least certain of the form drag coefficients. For now, the decrease as Φ approaches

180° seems reasonable because the silhouette area

is decreasing rapidly. But the absolute values for

CDN10 in this region are certainly open to debate.

In Figure 4a, the calculations with h values of 5

and 20 cm are less realistic than for h = 10 cm. For

h = 5 cm, the maximum CDN10 value calculated is

8% less than our observed maximum; for h = 20

cm, the calculated maximum is 27% higher than

what we observed. For all three heights, however,

8

b. h = 10 cm and γ = 0.10, 0.15, 0.20, or

0.25. Notice, the γ = 0.15 line is the same

as in Figure 4a

a. γ = 0.15 (the sastrugi cover 15% of the

surface) and h = 5, 10, and 20 cm.

Figure 4. Model calculation of CDN10 as a function of the wind orientation (Φ) with respect to the

dominant axis of the sastrugi. Model parameters are m = 4, n = 10, CS10 = 1.10 × 10–3, CR1 = 0.10,

CR2 = 0.30, CR3 = 0.30, c = 0.25, cw = 4 and cd = 0.6.

γ = 0.15

h = 20 cm

10 cm

5 cm

0 40 60 80 100 120 140 160 180 20

5

4

3

2

1

0

Wind Direction, Φ (°)

103 CDN10

h = 10 cm

γ = 0.25

0.20

0.10

0.15

0 40 60 80 100 120 140 160 180 20

5

4

3

2

1

0

Wind Direction, Φ (°)

103 CDN10
CDN10 values for head-on flow are in a reasonable

range, 1.36–1.51 × 10–3.

Figure 4b investigates the sensitivity of the

model to γ when h = 10 cm. For this value of h,

γ = 0.10 yields CDN10 values for head-on flow of

1.36 × 10–3. Although we did measure four CDN10

values in this range, the more likely lower limit is

about 1.5 × 10–3. Also, as Φ increases, the predicted CDN10 values reach a maximum that is 6% below our observed maximum. When γ is 0.20 and

0.25, the predicted head-on values for CDN10 are

roughly correct, 1.50–1.55 × 10–3. But for these γ

values, the maximum predicted values are, respectively, 17 and 26% above our observed maximum. Notice, γ = 0.25 means that the sastrugi

cover half the area that they would if they were

packed as tightly as possible. This value is, therefore, probably an unrealistically large sastrugi

coverage for ISW.

For the γ = 0.15 and h = 10 cm case—the one

that fits the data best—I also ran the model with

CS10 = 1.00 × 10–3 (not shown), roughly the smallest value compatible with the surveys I cited earlier. Resulting predictions for CDN10 were 4–6%

lower for 0° ≤ Φ ≤ 20° and for 160° ≤ Φ ≤ 180° than

for calculations based on CS10 = 1.10 × 10–3. For

30° ≤ Φ ≤ 160°, the differences were only 2–3%.

Consequently, the choice of CS10 does not have a

big effect on the results, as long as the value is a

reasonable one. Secondly, my handling of the displacement height in converting from CS10 to CSh is

appropriate, given this weak sensitivity to CS10.

Figure 5 shows some examples of how the

stress is partitioned between form drag and skin

friction. I plot τR/τ; clearly, τS/τ=1 – τR/τ. In the

figure, the (γ = 0.10, h = 10 cm) and the (γ = 0.20,

h = 20 cm) lines, respectively, show the minimum

and maximum values of τR/τ that I obtained for

the range of model parameters considered. My

computations suggest that, even for a very rough

surface, γ = 0.20 and h = 20 cm, skin friction produces more than half of the total stress when the

flow is approximately head-on. In other words,

the streamlined shape of the sastrugi is quite effective at reducing form drag. When the flow is at

right angles to the sastrugi, however, form drag

dominates the total stress because the wind has

so much more surface to push against. This dominance of form drag for such wind orientations

explains why the drifts erode so quickly (AC95).

Equation 46 yields the displacement height;

Figure 6 shows several computations of d/h. For

9

Figure 6. Computed displacement height d scaled by

sastrugi height h versus the direction of the mean

wind with respect to the sastrugi. Here, γ is the fractional sastrugi coverage, and h is in centimeters.

Figure 5. Stress partitioning, τR/τ, versus the mean

wind direction with respect to the dominant axis of

the sastrugi. Realize that τS/τ = 1–τR/τ. Here, γ is the

fractional sastrugi coverage, and h is the sastrugi height

in centimeters.

(γ, h) = (0.20, 20)

(0.15, 10)

(0.10, 10)

0 40 60 80 100 120 140 160 180 20

1.0

0.8

0.6

0.4

0.2

0

Wind Direction, Φ (°)

τ R –––

τ

(γ, h) = (0.20, 20)

(0.20, 10)

(0.15, 20)

(0.15, 10)

(0.15, 5)

0 40 60 80 100 120 140 160 180 20

0.5

0.4

0.3

0.2

0.1

0

Wind Direction, Φ (°)

d

––

h
densely packed roughness elements, such as

crops, trees or other vegetation, d/h is typically

reported to be about 0.7 (Stanhill 1969, Thom

1971, Fazu and Schwerdtfeger 1989). For the

more widely spaced sastrugi that I am considering, however, d/h is 0.15–0.35. This is also roughly what R92 (his Figure 5) computed for λ values

in the range relevant to this problem, 0.01–0.06.

These plots justify my use of d/h = 0.3 when I

had to convert CS10 to CSh in the section on Partitioning the Stress. They also show that, for sastrugi typically 10 cm high, d is usually less than 3

cm. We were thus safe to ignore d in analyzing

our wind speed profiles in AC95. For the lowest

profiling level used in AC95, 0.5 m, including a

displacement height of 3 cm in a computation

based on eq 29 would yield a 

u

*

value roughly

1% larger than a computation with d = 0. For the

higher levels, the difference would be even less.

Since the nominal uncertainty in our wind speed

measurements was ±5 cm/s, this small bias in

our computations is negligible.

Figure 6 shows that d has some interesting

structure—it is not a constant for all wind directions. Equation 46 explains this behavior: d/h responds mainly to two variables, τR/τ and CDh;

d/h increases linearly with τR/τ but decreases

linearly with 

CDh

1/2 . The peak in d/h near 20–30°

corresponds to the rapid increase in τR/τ in this

range (Fig. 5), while the broad minimum

centered near 100° is where CDh has its maximum (Fig. 4). Another curious feature in Figure 6

is that the d values in this broad minimum are

usually less than the values for head-on flow.

Clearly, d does not change in concert with z0 or

CDh—parameters that quantify the aerodynamic

roughness of the surface. This is an important

conclusion because others have claimed that d is

a linear function of z0 (e.g., Brutsaert 1982, p. 113;

Sugita and Brutsaert 1990).

In summary, the Raupach’s Model Adapted to

Drifting Snow section explains that my adaptation of R92’s model has 11 adjustable parameters:

h, m, n, γ, CS10, CR1, CR2, CR3, c, cw and cd. With

reasonable choices for these and with little finetuning, I have demonstrated that that model fits

our ISW observations quite well. I do not mean

to imply here, however, that the model is infallibly accurate—only that it seems to capture the

essential physics of how wind transfers momentum to snow-covered sea ice.

DISCUSSION

Primary roughness elements

The obvious (and, perhaps, surprising) conclusion from the calculations described in the last

section is that roughness elements with heights of

only 10 cm can have a dominant effect on CDN10.

This is essentially the same conclusion that Jackson and Carroll (1978) reached. Arya’s (1973,

1975) model, in contrast, predicted that pressure

ridges 1 to 4 m high accounted for most of the

form drag that determined the large-scale roughness and drag coefficient over Arctic sea ice.

My calculations provide a theoretical basis for

the empirical result that Banke et al. (1976, 1980)

obtained, eq 1. Their roughness parameter ξ reflects the small-scale roughness; they reported ξ

values of 3–13 cm, where ξ came from integrating

the roughness spectrum over all wavelengths

shorter than 13 m (also see Andreas et al. 1993).

Such wavelengths are far smaller than either Arctic or Antarctic ridge spacings (Lytle and Ackley

1991). Thus, eq 1 and my calculations point to the

same conclusion: Pressure ridges are relatively

unimportant in setting the local drag coefficient;

CDN10 responds primarily to sastrugi-size roughness elements. Pressure ridges, however, will

probably be important in establishing an areally

averaged or effective roughness length, Z0. Fiedler and Panofsky (1972), Arya (1975), Overland

(1985), and Claussen (1991), among others, have

offered some thoughts on inferring Z0 from z0.

But pursuing that connection is beyond my scope

here.

Parameterizing the drag coefficient

These model results and the behavior of CDN10

over snow-covered sea ice that we documented in

AC95 suggest what might be necessary in a

scheme for parameterizing CDN10. The snow surface is not in general isotropic, especially during,

or for some time after, high wind events. The

wind builds sastrugi and therefore streamlines

the surface in the mean wind direction. Figure 4

shows how CDN10 behaves for various wind directions once the surface has been streamlined.

Thus, the key is to estimate the dominant direction of the sastrugi without the benefit of in

situ observations. Maybe in the future, satellites

will provide real-time information on ice-surface

roughness, including any preferential alignment

of the roughness elements. But for now, all we can

10
do is estimate the orientation of roughness elements by tracking the history of surface winds that

have been inferred from analyzed geostrophic

winds. According to AC95, if the surface wind

blows at 8 m/s or higher with constant direction, it

will build sastrugi parallel to it and, thus, streamline the surface. If such conditions persist for 12

hours, CDN10 will fall from its initial value to about

1.5 × 10–3 and will remain here as long as the wind

remains aligned with the sastrugi.

If subsequent surface winds fall below 6 m⁄s,

Figure 4 shows how CDN10 will vary as the wind

turns with respect to the axis of the sastrugi.

The difficult part of the parameterization will

be treating directionally variable winds of 6–8 m⁄s,

or higher. These are strong enough to begin reworking the surface—eroding the snowdrifts

present—but, because of their variability, will not

necessarily build new drifts and thereby streamline the surface in another direction. Neither the

data in AC95 nor the model developed here provides a clear suggestion on how to evaluate CDN10

for such winds.

In summary, except for the complication just

mentioned, it seems possible to now estimate

CDN10 over snow-covered sea ice by tracking the

history of the surface wind computed from analyzed pressure fields. I have described the algorithm in words; coding and testing it is the next

step.

CONCLUSIONS

I have adapted R92’s model for the form drag

on a field of interacting roughness elements and

applied it to rudimentary sastrugi-like snowdrifts. The key step was treating the complex geometry of the sastrugi. In the absence of laboratory values for the form drag on the various faces of

the sastrugi, I assumed that the form drag reflects

additive contributions from the various faces of

the individual sastrugi (see eq 13).

The resulting model does quite well in reproducing the values of CDN10 that we observed in

AC95. For sastrugi of height 10 cm and with an

areal coverage of 15%, the model predicts that

CDN10 = 1.43 × 10–3 when Φ is less than 12°. In

AC95, the nominal head-on value of CDN10 that

we observed was 1.5 × 10–3. In the model, as Φ

gets larger than 12°, CDN10 increases rapidly. In

this example, CDN10 increases by 17% as Φ turns

from 12 to 20°. As Φ continues increasing, CDN10

eventually reaches a maximum of 2.73 × 10–3 at

Φ = 101.3°. This increase with Φ also mirrors the

directional sensitivity we reported in AC95,

though the predicted maximum is about 7%

higher than our measured maximum for CDN10,

2.54 × 10–3.

The upshot of this modeling is that the small,

10-cm-high roughness elements associated with

drifting snow—rather than pressure ridges or

other mesoscale features—determine the local

drag coefficient over snow-covered sea ice.

Though some have stated this conclusion explicitly (e.g., Joffre 1982) or implicitly (e.g., Banke et

al. 1976, 1980), my model is the first to provide a

theoretical foundation for it.

These model predictions and the observations

in AC95 suggest how we should attempt to parameterize CDN10 in terms of what we can measure.

Because CDN10 depends crucially on how the

mean wind is oriented with respect to the sastrugi, the key is predicting sastrugi orientation.

Our observations showed that this orientation

depends on the history of the surface wind. By

inferring surface winds from analyzed fields

of the geostrophic wind, we should be able to deduce this orientation. The event to look for is a

directionally constant surface wind of at least

8 m/s that persists for 12 hours. At the end of

such an event, CDN10 will be roughly 1.5 × 10–3.

Figure 4 then shows how to estimate CDN10 when

the wind direction subsequently changes.

LITERATURE CITED

Allen, J.R.L. (1965) Scour marks in snow. Journal of

Sedimentary Petrology, 35: 331–338.

Andreas, E.L and K.J. Claffey (1995) Air–ice drag

coefficients in the western Weddell Sea: 1. Values

deduced from profile measurements. Journal of

Geophysical Research, 100: 4821–4831.

Andreas, E.L, M.A. Lange, S.F. Ackley and P.

Wadhams (1993) Roughness of Weddell Sea ice

and estimates of the air–ice drag coefficient. Journal of Geophysical Research, 98: 12,439–12,452.

Arie, M. and H. Rouse (1956) Experiments on twodimensional flow over a normal wall. Journal of

Fluid Mechanics, 1: 129–141.

Arya, S.P.S. (1973) Contribution of form drag on

pressure ridges to the air stress on Arctic ice. Journal of Geophysical Research, 78: 7092–7099.

Arya, S.P.S. (1975) A drag partition theory for determining the large-scale roughness parameter

and wind stress on the Arctic pack ice. Journal of

Geophysical Research, 80: 3447–3454.

11
Lettau, H. (1969) Note on aerodynamic roughness-parameter estimation on the basis of roughness-element description. Journal of Applied

Meteorology, 8: 828–832.

Lytle, V.I. and S.F. Ackley (1991) Sea ice ridging in

the eastern Weddell Sea. Journal of Geophysical Research, 96: 18,411–18,416.

Mellor, M. (1965) Blowing snow. USA Cold Regions Research and Engineering Laboratory,

Cold Regions Science and Engineering Monograph, Part III, Section A3c.

Overland, J.E. (1985) Atmospheric boundary layer structure and drag coefficients over sea ice.

Journal of Geophysical Research, 90: 9029–9049.

Owen, P.R. (1964) Saltation of uniform grains in

air. Journal of Fluid Mechanics, 20: 225–242.

Plate, E.J. (1971) Aerodynamic Characteristics of Atmospheric Boundary Layers. Oak Ridge, Tennessee:

U.S. Atomic Energy Commission.

Radok, U. (1968) Deposition and erosion of snow

by the wind. USA Cold Regions Research and Engineering Laboratory, Research Report 230.

Raupach, M.R. (1992) Drag and drag partition on

rough surfaces. Boundary-Layer Meteorology, 60:

375–395.

Raupach, M.R., D.A. Gillette and J.F. Leys (1993)

The effect of roughness elements on wind erosion

threshold. Journal of Geophysical Research, 98:

3023–3029.

Stanhill, G. (1969) A simple instrument for the

field measurement of turbulent diffusion flux.

Journal of Applied Meteorology, 8: 509–513.

Sugita, M. and W. Brutsaert (1990) Wind velocity

measurements in the neutral boundary layer

above hilly prairie. Journal of Geophysical Research,

95: 7617–7624.

Taylor, P.A. (1988) Turbulent wakes in the atmospheric boundary layer. In Flow and Transport in

the Natural Environment: Advances and Applications

(W.L. Steffen and O.T. Denmead, Eds.). Berlin:

Springer-Verlag, p. 270–292.

Thom, A.S. (1971) Momentum absorption by vegetation. Quarterly Journal of the Royal Meteorological Society, 97: 414–428.

12

Banke, E.G. and S.D. Smith (1973) Wind stress on

Arctic sea ice. Journal of Geophysical Research, 78:

7871–7883.

Banke, E.G., S.D. Smith and R.J. Anderson (1976)

Recent measurements of wind stress on Arctic sea

ice. Journal of the Fisheries Research Board of Canada,

33: 2307–2317.

Banke, E.G., S.D. Smith and R.J. Anderson (1980)

Drag coefficients at AIDJEX from sonic anemometer measurements. In Sea Ice Processes and Models

(R.S. Pritchard, Ed.). Seattle: University of Washington Press, p. 430–442.

Brutsaert, W.H. (1982) Evaporation into the Atmosphere: Theory, History and Applications. Dordrecht: D. Reidel.

Claussen, M. (1991) Estimation of areally-averaged surface fluxes. Boundary-Layer Meteorology,

54: 387–410.

Fazu, C. and P. Schwerdtfeger (1989) Flux-gradient relationships for momentum and heat over a

rough natural surface. Quarterly Journal of the Royal Meteorological Society, 115: 335–352.

Fiedler, F. and H.A. Panofsky (1972) The geostrophic drag coefficient and the ‘effective’

roughness length. Quarterly Journal of the Royal

Meteorological Society, 98: 213–220.

Finney, E.A. (1939) Snow drift control by highway

design. Engineering Experiment Station, Michigan State College, East Lansing, Bulletin 86.

Jackson, B.S. and J.J. Carroll (1978) Aerodynamic

roughness as a function of wind direction over

asymmetric surface elements. Boundary-Layer Meteorology, 14: 323–330.

Joffre, S.M. (1982) Momentum and heat transfers

in the surface layer over a frozen sea. BoundaryLayer Meteorology, 24: 211–229.

Kondo, J. and H. Yamazawa (1986) Bulk transfer

coefficient over a snow surface. Boundary-Layer

Meteorology, 34: 123–135.

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surface roughness. In Annual Report 1961, Department of Meteorology, University of Wisconsin, Madison, p. 71–113.
14

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

July 1995

A Physically Based Model of the Form Drag OPP-90-24544

Associated with Sastrugi OPP-93-12642

PR: 4A161102AT24

Edgar L Andreas

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road CRREL Report 95-16

Hanover, New Hampshire 03755-1290

Office of Polar Programs Office of the Chief of Engineers

National Science Foundation Washington, DC 20314-1000

Arlington, Virginia

Approved for public release; distribution is unlimited.

Available from NTIS, Springfield, Virginia 22161.

On Ice Station Weddell, some characteristics of the neutral-stability air–ice drag coefficient at a reference

height of 10 m (CDN10) were observed that had not been documented before. The main finding was that winddriven snow continually alters the sea ice surface; the resulting snowdrifts determine how large CDN10 is. In

particular, this report describes three observations and attempts to explain them: 1) CDN10 is near 1.5 × 10–3

when the wind is well aligned with the drifted snow; 2) CDN10 is near 2.5 × 10–3 when the wind makes a large

angle with the dominant orientation of the snowdrifts; 3) CDN10 can increase by 20% if, after being well aligned

with the drift patterns, the mean wind direction shifts by as little as 20°. To investigate this behavior of CDN10,

this report adapts a model developed by Raupach that partitions the total surface stress into contributions

from form drag and skin friction. With reasonable choices for free model parameters and with little fine-tuning, this physically based model can reproduce the three main observations. In other words, the model seems

to include the basic physics of air–ice momentum exchange. This modeling implies that 10-cm-high sastrugilike roughness elements, rather than pressure ridges, sustain most of the form drag over compact sea ice in the

western Weddell Sea. Lastly, the report speculates on what the observations and this model say about how to

parameterize CDN10 over snow-covered sea ice.

Air–sea–ice interaction Form drag

Drag coefficient Snow drifts

Drifting snow Weddell Sea

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