MONOGRAPH 96-2

The Atmospheric Boundary Layer Over

Polar Marine Surfaces

Edgar L Andreas June 1996
Cover: Schematic depictions of unstable (or convective) and stable atmospheric boundary layers. The unstable boundary layer is characterized

by large eddies, convective plumes, a capping version, and well mixed

(i.e., constant) profiles of wind speed and potential temperature. The

stable boundary layer is shallower and has smaller eddies and steeper

vertical gradients in wind speed and potential temperature. The density

gradient through the inversion may allow gravity waves to exist at the

top of the boundary layer; there also may be a low-level jet resulting from

inertial oscillations and intermittent turbulence near the inversion. [Both

figures adapted from Wyngaard (1992).]

Abstract

The Atmospheric Boundary Layer (ABL) over polar marine surfaces is, in ways,

simpler and, in other ways, more complex than ABLs in other environments. It

is simpler because topographic effects are rarely a concern, the surface is fairly

homogeneous, and roughness lengths over sea ice and the ocean are much

smaller than they are over land. It is complex because the stratification is usually

stable, and stable ABLs have not yielded to quantification as readily as

convective ABLs have. This report reviews some of these characteristics of ABLs

over polar marine surfaces. The ABL, by definition, is the turbulent layer between

the Earth’s surface and the (generally) nonturbulent free atmosphere. Hence, the

emphasis is on turbulence processes—in particular, the turbulent transfer of

momentum and sensible and latent heat over sea ice. As such, this report

reviews both the theoretical and observational bases for our understanding of

the mean structure of the ABL. Understanding this structure then allows

predicting the turbulent surface fluxes of momentum and sensible and latent

heat.
Monograph 96-2

The Atmospheric Boundary Layer Over

Polar Marine Surfaces

Edgar L Andreas June 1996

Prepared for

NATIONAL SCIENCE FOUNDATION

Approved for public release; distribution is unlimited.

US Army Corps


of Engineers

Cold Regions Research &


Engineering Laboratory
PREFACE

This report was prepared by Dr. Edgar L Andreas, Research Physicist, Snow and Ice Division, Research and Engineering Directorate, U.S. Army Cold Regions Research and Engineering Laboratory.

The author thanks Kerry Claffey for help with the Ice Station Weddell (ISW) observations;

his Ice Station Weddell colleagues Aleksandr Makshtas and Boris Ivanov for their conscientious airborne snow observations, which yielded Figure 14; Alfred Blackadar for allowing

reproduction of Figure 24; Rachel Jordan and Peter Guest, who offered helpful comments on

the manuscript; Mark Hardenberg for editorial assistance; the U.S. National Science Foundation, which supported this work through grants OPP-90-24544 and OPP-93-12642; the Office

of Naval Research, which supported it with contract N0001496MP30005; and the U.S. Department of the Army, which 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

Basic equations of an ice floe .............................................................................................. 1

The Navier-Stokes equation ................................................................................................ 2

Monin-Obukhov similarity .................................................................................................. 6

Surface-layer profiles .......................................................................................................... 10

Neutral stratification ....................................................................................................... 10

Including stratification effects ........................................................................................ 11

Bulk transfer coefficients for heat and momentum over sea ice.......................................... 14

Mathematical background .............................................................................................. 14

Drag coefficient .............................................................................................................. 16

Scalar bulk transfer coefficients...................................................................................... 21

The Ekman layer.................................................................................................................. 23

Thermal wind....................................................................................................................... 26

Rossby number similarity .................................................................................................... 27

Summary.............................................................................................................................. 34

Literature cited..................................................................................................................... 34

Abstract................................................................................................................................ 39

ILLUSTRATIONS

Figure

1. Demonstration of Monin-Obukhov similarity theory ................................................ 9

2. Semi-logarithmic wind speed profile ......................................................................... 10

3. Hourly averaged, semi-logarithmic wind speed profiles observed on Ice Station

Weddell ............................................................................................................... 10

4. Nondimensional wind speed and scalar gradients as functions of stability for

unstable conditions ............................................................................................. 11

5. Various suggestions for the functional form of the nondimensional wind speed and

scalar gradients for stable conditions .................................................................. 12

6. Sample wind speed profiles ....................................................................................... 15

7. Sample potential temperature profiles ....................................................................... 15

8. The 10-m, neutral-stability drag coefficient over Arctic sea ice ................................ 17

9. A 300-m-long snow-surface and ice-surface elevation profile measured on Ice

Station Weddell ................................................................................................... 17

10. Snow-surface and ice-surface roughness spectra computed for the profiles in

Figure 9 ............................................................................................................... 17

11. Neutral-stability, 10-m air–ice drag coefficients from the winter Weddell Gyre

study ........................................................................................................................... 18

12. Time series of the neutral-stability, 10-m drag coefficient measured on a fixed

mast on Ice Station Weddell ............................................................................... 18
iv

Figure

13. Two long events on Ice Station Weddell characterized by a relatively constant

wind direction ..................................................................................................... 19

14. Ice Station Weddell observations showing the percentage of the time with winddriven snow for a given 5-m wind speed ............................................................ 19

15. CDN10 values in Figure 12, averaged over 1-m/s bins of the 10-m wind speed......... 20

16. Geometry of the sastrugi-like roughness elements envisioned to dictate the drag

coefficient over snow-covered sea ice ................................................................ 20

17. Model calculations of the neutral-stability, 10-m drag coefficient as a function of

the wind orientation ............................................................................................ 21

18. Schematic depiction of the microscale roughness of an aerodynamically smooth

and an aerodynamically rough surface ............................................................... 21

19. Model predictions of zT/z0 and zQ/z0 over snow-covered surfaces ............................ 22

20. Model predictions of CHN10 and CEN10 over snow-covered surfaces as a function

of the surface roughness parameter and the 10-m wind speed ........................... 23

21. Hodographs of the wind vector in Ekman layers in the Northern and Southern

Hemispheres ....................................................................................................... 25

22. Nondimensional profiles of the longitudinal and transverse velocity components

for Ekman layers in the Northern and Southern Hemispheres ........................... 25

23. The 00 GMT radiosounding from Ice Station Weddell on 15 May 1992 .................. 26

24. Northern Hemisphere hodographs of Ekman spirals in the presence of a constant

thermal wind ....................................................................................................... 27

25. Observations of the height of the core of the low-level jet and of the corresponding

height of the inversion on Ice Station Weddell ................................................... 29

26. Observations of the height of the jet core versus the estimated height of the

turbulent layer from a Richardson number criterion .......................................... 29

27. Evaluations of FU and FV for very unstable, weakly stable and very stable conditions ................................................................................................................. 30

28. Evaluations of FΘ for unstable, weakly stable and moderately stable conditions ..... 31

29. Yamada’s resistance laws for the longitudinal and transverse velocity components

and for the potential temperature as functions of stability ................................. 32

30. Geostrophic drag coefficient and the turning angle as functions of stability ............. 33

TABLES

Table

1. Predicted behavior of the Deacon and Richardson numbers in very stable

conditions ............................................................................................................ 13

2. Values of the coefficients in the polynomial that predicts zT/z0 and zQ/z0................. 22
INTRODUCTION

The Atmospheric Boundary Layer (ABL) is the lowest few hundred meters of the atmosphere, where the

Earth’s surface most directly influences atmospheric

processes. The vast majority of Earth’s human inhabitants are never outside the ABL.

I cannot hope, in a few pages here, to compete with

the many good books recently published on the ABL

(Arya 1988, Stull 1988, Sorbjan 1989, Garratt 1992,

Kaimal and Finnigan 1994)—nor do I need to. My purpose here is to present some of the basic concepts of

boundary-layer meteorology, to define some of the jargon a relative novice might encounter in the above (and

other) texts, and to describe a few of the unique problems associated with the ABL over polar marine surfaces.

This report divides into two logical parts. In the first

part, I focus on microscale processes in the so-called

Atmospheric Surface Layer (ASL). The height scale

relevant in the ASL is generally 10–30 m; relevant horizontal scales are a few hundred meters. In the second

part, I tie surface-layer processes to the structure of the

entire ABL. Here, the relevant height scale is the height

of the ABL—typically a few hundred meters over polar marine surfaces. The relevant horizontal scale is on

the order of kilometers.

BASIC EQUATIONS OF AN ICE FLOE

The momentum balance of a floating ice floe is (e.g.,

Hibler 1979)

 

m D

Dt mf mg H

v v v v v u r v =− + × ∇+ τ τw k u– I. (1)

Here, m is the mass of the floe; 

v

u is the velocity vector

of the floe; D/Dt is the material derivative

 

D

Dt t

= + ⋅∇ ∂

∂

v v u (2)

where 

v

τ = vectoral stress exerted on the top of the floe

by the wind

 

v

τw = vectoral stress exerted on the underside of

the floe by the water

f = Coriolis parameter

 

v

k = vertical unit vector

 

v

∇H = gradient in sea surface height

 

v

I = internal ice forces vector.

 

v

τ and 

v

τw are the only turbulence terms in eq 1. Thus,

the main concern boundary-layer meteorologists have

with eq 1 is evaluating 

v

τ , the surface stress on the top

side of the ice. The desire to know 

v

τ motivates much of

what I will write in this monograph.

The energy budget at the surface of an ice floe is

(e.g., Maykut 1978, Parkinson and Washington 1979,

Makshtas 1991)

BQ Q Q Q H H C =− + − − − + s ss L L s L α ↓ ↑ (3a)

 = melting/freezing + storage/release . (3b)

Here Qs = incoming shortwave radiation

αs = shortwave albedo

αsQs = reflected shortwave radiation

QL↓ = incoming longwave radiation

QL↑ = emitted longwave radiation

C = conduction to the ice surface from below

Hs = turbulent sensible heat flux (the flux driven

by a difference in temperature between the

ice surface and the air)

HL = turbulent latent heat flux (the flux driven

by a difference in water vapor density between the surface and the air).

In eq 3, my convention is that positive terms add heat

to the ice surface; negative terms carry heat away. Thus,

for example, when Hs and HL are positive, turbulence

is carrying heat from the surface into the air.

If the seven terms on the right-hand side of eq 3a do

The Atmospheric Boundary Layer Over Polar Marine Surfaces

EDGAR L ANDREAS
not add to zero, the energy balance B is nonzero. As

a result, this energy imbalance must reflect phase

changes (freezing or melting) or warming or cooling

(storage or release of heat) of the ice within the floe.

As a boundary-layer meteorologist, I could justifiably study any but the C term on the right-hand side of

eq 3a. But since my main interest is turbulence in the

ABL, I will confine my attention here to Hs and HL.

These are the only two turbulence terms in eq 3. As I

said for 

v

τ , the desire to know Hs and HL is the basis for

most of what I will write here.

THE NAVIER-STOKES EQUATION

All geophysical fluid dynamics starts with the

Navier-Stokes equation (e.g., Busch 1973, Businger

1982)

(4)

In my notation, a wavy overbar indicates an instantaneous value; 

u ̃i , for example, is the i

th component of

the instantaneous fluid velocity vector. Thus, eq 4 shows

the processes that affect this component.

In eq 4, term I is the time rate of change (the acceleration) of the i

th component. Term II really contains

three terms since the repeated j index indicates a sum.

It shows the advective effects on the i

th velocity component. In term III, 

ρ ̃ is fluid density and p ̃ is the pressure. Term III, thus, shows the effects of a pressure gradient on the fluid—fluid tends to accelerate down the

gradient. In term IV, g is the acceleration of gravity,

and δij is the Kronecker delta, where δij = 1 if i = j, and

δij = 0 if i ≠ j. Term IV therefore represents the gravitational force and affects only the 

u ̃ 3 component of the

velocity (the vertical component). Term V shows the

Coriolis effects on the fluid motion. Here Ω is the Earth’s

rotation rate, 2π radians per day (7.27×10–5 s–1), and ηj

is the unit vector of the Earth’s rotational axis

 

v

η λλ = ( , cos , sin ) 0 (5)

where λ is the latitude. Notice, the quantity f in eq 1 is

f = 2Ω sinλ . (6)

Finally, term VI quantifies viscous effects on the flow,

where ν is the kinematic viscosity of the fluid.

We can write an equation analogous to eq 4 for the

conservation of any conservative scalar—potential temperature 

 ̃

θ, for example (Busch 1973, Businger 1982)

Here, term I is the time rate of change of the instantaneous temperature. Term II quantifies advective effects

on the temperature. Term III contains the molecular effects on the temperature, where D is the thermal diffusivity of heat in the fluid.

In the atmosphere, the instantaneous specific humidity q ̃ is usually another conservative scalar. We can write

an equation for it with exactly the same form as eq 7

∂

∂

∂

∂

∂

∂ ∂

 ̃  ̃ q  ̃  ̃

t

u q

x

D q

x x

+ = j

j

w

j j

2

(8)

where Dw is the molecular diffusivity of water vapor in

air.

Equation 4, especially, is too complex to treat as it

stands; we need to simplify it. Our first simplification is

to recognize that each instantaneous quantity in eq 4, 7

and 8 can be decomposed into an average and a turbulent fluctuation about that average. That is,

uUu  ̃i ii = + (9a)

 

 ̃

θ θ = + Θ (9b)

qQq  ̃ = + (9c)

 

ρρ ρ  ̃ = +0 (9d)

pPp  ̃ = + . (9e)

In eq 9, a capital letter indicates an averaged quantity,

except in eq 9d where ρ0 indicates the average air density. A lower case letter without the wavy overbar is a

zero-mean turbulent fluctuation about that average.

Let me demonstrate how to manipulate these decomposed quantities. Let a straight overbar indicate averaging. Then, with u ̃i for example

u ̃i ≡ Ui (10)

since the average of ui is zero by definition ( u ̃i ≡ 0 ). Similarly

uu U u U u  ̃  ̃ ( )( ) ij i i j j =+ + (11a)

 = +++ UU uU uU uu i j i j j i ij (11b)

 = + UU uu i j ij (11c)

since Ui and Uj are already averaged quantities.

2

∂θ

∂

∂θ

∂

∂ θ

∂ ∂

 ̃  ̃  ̃  ̃ . t u x D x x + = j

j jj

 I II III

2

(7)

∂

∂

∂

∂ ρ

∂

∂ δ εη ν ∂∂ ∂

 ̃  ̃  ̃

 ̃

 ̃  ̃  ̃ . u

t u ux

p

x g u ux x

i

j

i

j i

i3 ijk j k

i

j j

= – 1 – –2

I II III IV V VI

+ + Ω

2
I could digress here and spend 10–20 pages discussing what the averaging denoted by that straight overbar means. Ideally, it denotes an ensemble average

(Lumley and Panofsky 1964, p. 6)—the average formed

from data collected during repeated, identical experiments. In the atmospheric sciences, however, we do

not have the luxury of ensemble averaging. To create

averages, we must make an ergodic hypothesis (Lumley and Panofsky 1964, p. 35)—that conditions are near

enough to steady state that a time average is equivalent

to an ensemble average. Lumley and Panofsky (1964,

p. 35 ff.), Haugen et al. (1971), Wyngaard (1973) and

Andreas (1988), among others, have considered what

the appropriate averaging time is for ABL statistics. In

light of these studies, henceforth in this report I will

use upper-case variables to mean averages obtained

from an hour’s worth of instantaneous values. Likewise, the straight overbar, in u ui j for example, will

mean a turbulence statistic obtained by averaging for 1

hour. Notice that, because these are time averages not

volume averages, an averaged quantity may still depend on position within the fluid.

To write eq 4, 7 and 8, we have already made some

approximations. To simplify these further, we will make

additional approximations. These are the Boussinesq

approximations (Busch 1973; Businger 1982; Garratt

1992, p. 20 f.), which I summarize here.

Now back to the Navier-Stokes equation, eq 4. Substitute in it eq 9a, 9d and 9e, then average. The result is

The second and third terms on the left-hand side derive

from the assumption that the atmosphere is incompressible

3

The Boussinesq Approximations

1. The dynamic viscosity, μ = ρ0ν; the thermal conductivity of air, kT = ρ0cpD, where

cp is the specific heat of air at constant pressure; and Dw are constants throughout the fluid.

2. The flow speeds are low enough that the air behaves as an incompressible fluid.

3. Turbulent fluctuations in fluid properties are much smaller than the corresponding

averages; that is θ/Θ<<1, q/Q<<1, ρ/ρ0<<1 and p/P<<1.

4. p/P can be neglected in comparison to θ/Θ, q/Q and ρ/ρ0.

5. The heat generated by viscous stresses can be neglected. In other words, term VI

in eq 4 does not need to appear as a source term in eq 7.

6. Turbulent density fluctuations, ρ, are significant only when they multiply g.

(12)

∂

∂

U

x

j

j

= 0 (13a)

∂

∂

u

x

j

j

= 0. (13b)

The first term on the right-hand side of eq 12 is a consequence of another of the Boussinesq approximations—that fluctuations in fluid density are much

less than the mean density and, therefore, only important when they multiply g.

In horizontally homogeneous conditions, the x (or x1)

and y (or x2) derivatives in eq 12 produce zeroes, except

in ∂P/∂x and ∂P/∂y, which are imposed synoptic-scale

forcings. Near an impermeable surface, such as sea ice

or the ocean, the average vertical velocity, W (or U3),

must be zero. Lastly, the viscous term in eq 12 is

always small except within a few millimeters of the

surface. Hence, in an ABL above a horizontally homogeneous surface, the three equations implicit in eq 12

become

U equation

∂

∂

∂

∂ ρ

∂

∂

U

t

uw

z

P

x

=− − + f V 1

0

(14a)

V equation

∂

∂

∂

∂ ρ

∂

∂

V

t

vw

z

P

y =− − − f U 1

0

(14b)

W equation

g

P

z = − 1

ρ0

∂

∂ . (14c)

The W equation, which results from the assumption that

vertical accelerations are much less than pressure gradient forces (Pielke 1984, p. 30 ff.), shows that, on

average, the ABL is in hydrostatic balance.

∂

∂

∂

∂

∂

∂ ρ

∂

∂

δ εη ν ∂

∂ ∂

U

t

U U

x x u u Px

g U U

x x

i

j

i

j j

i j

0 i

i3 ijk j k

i

j j

++ =

−− +

1

2

2

Ω .
In eq 14a and b, it is common to substitute the geostrophic wind components Ug and Vg, defined as

fV P

x g ≡ 1

ρ0

∂

∂ (15a)

− ≡ fU P

y g

1

ρ0

∂

∂ . (15b)

Thus, eq 14a and b become

∂

∂

∂

∂

U

t

uw

z

=− + − fV V ( ) g (16a)

∂

∂

∂

∂

V

t

vw

z =− − − fU U ( ) g . (16b)

The ∂ ∂ uw vw and terms in eq 16 really contain the

essence of what we mean by an atmospheric boundary

layer. Without these terms, and assuming for the moment steady-state conditions, eq 16 would reduce to a

simple geostrophic balance characterized by twodimensional flow. Such conditions are common in the

free atmosphere, but the ABL is by definition a turbulent layer, and turbulence by definition is threedimensional. The ∂ ∂ uw vw and terms in eq 16, thus,

give the ABL its character.

For the atmosphere, the ideal gas law is the appropriate equation of state (Lumley and Panofsky 1964, p.

214)

 ̃  ̃  ̃ p Rt = ρ . v (17)

Here R (= 287.056 J kg–1 K–1) is the universal gas constant for dry air, and  ̃tv is the instantaneous virtual temperature, a temperature that reflects the fact that the

presence of water vapor affects air density.

As usual

 ̃t Tt . v vv = + (18)

Lumley and Panofsky (1964, p. 213 f.) showed that

t t  ̃  ̃  ̃ M

M q v a a

w

=+ − 



 





















1 1 (19a)

 = + ( )  ̃t q .  ̃ a 1 0 61 (19b)

where Ma (= 28.9644 × 10–3 kg/mol) is the molecular

weight of dry air, and Mw (= 18.0160 × 10–3 kg/mol) is

the molecular weight of water. Here, also,  ̃ta is the instantaneous air temperature

 ̃t T a = +θ (20)

where T is the average air temperature. In eq 20, we

also use the fact that the turbulent fluctuations in potential temperature and in air temperature are virtually

identical. From eq 19b, 18, 9c and 20, we see that, to a

good approximation

TT Q v = + ( ) 1 0 61 . (21a)

t Q Tq v = + θ( ) 1 0 61 0 61 . .. + (21b)

Expanding eq 17, we get

PpRT t

T

t

T

+= + + + 



 

 ρ  ρ

ρ

ρ

ρ

0

0 0

v 1 v

v

v

v

. (22)

On ignoring the fourth term on the right-hand side of

eq 22, because it is much smaller than the other terms,

then averaging, we get the mean equation of state

PRT = ρ0 v. (23)

On subtracting this from eq 22, we find

p

R T

p

P

t

ρ T

ρ

0 0 v ρ

v

v

== + . (24)

According to the Boussinesq approximations, however,

the left side of this equation is negligible in comparison to the right side. Consequently, the equation of state

for the fluctuating quantities is

ρ

ρ0

= − t

T

v

v

. (25)

In eq 4, let us look at term III with this equation of

state in mind. We have

 

1 1

0  ̃

 ̃ . ρ

∂

∂ ρρ

∂

∂

p

x

P p

i i x = +

( ) +

(26)

After some simple manipulations, this becomes

11 1

0 0 0

2

0

2  ̃

 ̃ . ρ

∂

∂ ρ

∂

∂ ρ

∂

∂

ρ

ρ

∂

∂

ρ

ρ

∂

∂

p

x

P

x

p

x

P

x

p

x i iiii

=+−− (27)

Again, the fourth term on the right is much smaller than

the others; we ignore it. According to the Boussinesq

approximation, the density fluctuation in the third term

on the right is important only when it multiplies the

acceleration of gravity. Thus, from eq 14c and 25, this

term is

− = =− ρ

ρ

∂

∂

ρ

ρ

δ δ

0

2

0

P

x

g t g

i T i3 vv

i3. (28)

4
Consequently, term III in eq 4 is

11 1

0 0  ̃

 ̃ . ρ

∂

∂ ρ

∂

∂ ρ

∂

∂

δ p

x

P

x

p

x

t g

iii T

v

v

=+− i3 (29)

With eq 29 substituted into eq 4, we subtract eq 12 to

get an equation for the turbulent velocity component ui

∂

∂

∂

∂

ρ

∂

∂ δ εη ν ∂∂ ∂

u

t x

Uu uU uu uu

p

x

t g

T

u

u

x x

i

i

ij i j ij ij

i

v

v

i3 ijk j k

i

j j

+ + +− ( )

=− + − +

1 2

0

2

Ω .

Multiply this equation by ui

, then average. After some

manipulations, we get

Here

e uu u v w

2 22 2 1

2

1

2

≡ = ++ i i ( ) (32)

and e2 is the average Turbulent Kinetic Energy (TKE).

In eq 31, we henceforth ignore the last term on the

right-hand side, the viscous transport term, because it is

small except very near the surface. The Coriolis term in

eq 31 is identically zero because εijkuiuk = uiuk – ukui

for all j.

Again invoking horizontal homogeneity in an atmospheric surface layer (where W = 0) and assuming that

the mean wind is in the x direction (i.e., V = 0), we finally derive the turbulent kinetic energy equation from

eq 31

∂

∂

∂

∂

∂

∂ ρ

ε e

t uw UT

wt

z we wp

2

2

0

=− + 



 



 − z + g –

I II III IVa IVb V

v

v .

Here, term I is the time rate of change of TKE. Term II

represents mechanical production of TKE; the Reynolds

stresses (e.g., uw ) extract energy from the mean wind

speed gradient. Term III is the buoyancy production of

TKE. Term IV shows that TKE changes because of turbulent transport. Term IVa represents the vertical turbulent advection of TKE; term IVb represents a transport

resulting from a correlation between the vertical velocity fluctuations and the pressure fluctuations. Term V,

which is

ε ν

∂

∂

∂

∂ ≡ ux

u

x

i

j

i

j

(34)

is called the dissipation rate of TKE. Because in eq 34

∂ui/∂xj is a squared quantity, ε is always positive. But

since ε appears with a minus sign in eq 33, term V is

always a sink for TKE. It represents the dissipation of

the turbulence to heat because of viscous effects.

In summary, eq 33 shows the processes that are important in maintaining the turbulence in an atmospheric

surface layer. For example, for steady-state conditions,

the turbulent transport terms in eq 33 (IV) are often small.

Consequently, mechanical (II) and buoyant (III) production nearly balance viscous dissipation (V). Production

equals dissipation (Panofsky and Dutton 1984, p. 92 ff.;

Fairall and Larsen 1986).

Starting with the scalar conservation equations, eq 7

and 8, we can follow a procedure similar to that above to

derive equations for scalar means and variances. With

potential temperature as an example, insert eq 9a and b

into eq 7 and average. The result is

∂

∂

∂

∂

∂ θ

∂

∂

∂ ∂

ΘΘ Θ

t

U

x

u

x

D

x x

+ += j

j

j

j jj

2

. (35)

Now subtract eq 35 from eq 7 to get a conservation

equation for the turbulent temperature fluctuations

∂θ

∂

∂

∂

∂θ

∂

∂ θ

∂

∂ θ

∂

∂ θ

t ∂ ∂

u

x

U

x

u

x

u

x

D

x x

+ + +−= j

j

j

j

j

j

j

j jj

Θ 2

. (36)

Multiply this equation by θ and average

∂θ

∂

θ ∂

∂

∂θ

∂

∂ θ

∂

∂θ

∂

∂θ

∂

∂ θ

∂ ∂

2 2 2 2

2 2

2

2

t u x U x

u

x

D x x D x x

=− − −

− +

j

j

j

j

j

j

j j jj

Θ

. (37)

As in eq 31, we can ignore the molecular transport term,

the last term in eq 37. Again assume horizontal homogeneity, that we are within a few tens of meters of the surface (so W = 0), and that x is in the direction of the mean

wind vector (i.e., V = 0). With these simplifications, the

mean equation for potential temperature becomes

∂

∂

∂ θ

∂

∂

∂

Θ Θ

t

w

z

D

z

+ =

2

2 (38)

and the conservation equation for temperature variance

is

5

(30)

(33)

∂

∂

∂

∂

∂

∂

∂

∂ ρ

∂

∂

εη ν

∂

∂

∂

∂

ν

∂

∂ ∂

e

t u u Ux U ex

e u

x

u p

x

wt g

T

u u u

x

u

x

e

x x

2 2 2

0

2 2

1

2

=− − − − +

− −+

i j

i

j

j

j

j

j

i

j

v

v

ijk j i k

i

j

i

j jj

Ω . (31)
∂θ

∂

θ ∂

∂

∂ θ

∂ θ

2 2

2

t w z

w

z =− − − N Θ .

I II III IV

(39)

In eq 39, term I is the time rate of change of temperature variance. Term II is the production of temperature

variance through the interaction of the vertical temperature flux (wθ) with the mean temperature gradient.

Term III is the turbulent transport of temperature variance by the vertical velocity fluctuations. Term IV, which

by definition is

N D

x x

θ

∂θ

∂

∂θ

∂ ≡ 2

j j

(40)

is always positive. Since it appears with a minus sign in

eq 39, in analogy with ε, it represents the rate at which

molecular processes dissipate the temperature variance.

From the conservation equation for specific humidity, eq 8, we can follow exactly the same procedure as

above for potential temperature to derive conservation

equations for the mean specific humidity

∂

∂

∂

∂

∂

∂

Q

t

wq

z

D Q

z

+ = w

2

2 (41)

and for the humidity variance

∂

∂

∂

∂

∂

∂

q

t

wq

Q

z

wq

z

N

2 2

=− − − 2 q (42)

where

N D q

x

q

x q w

j j

≡ 2 ∂

∂

∂

∂ (43)

is the dissipation rate of specific humidity variance.

As in the TKE equation, eq 33, it is common in eq 39

and 42 to assume steady state and that the turbulent transport terms are small. Production then again equals dissipation (Large and Pond 1982; Panofsky and Dutton

1984, p. 94).

MONIN-OBUKHOV SIMILARITY

The equations for the mean velocity components, eq

16, the mean potential temperature, eq 38, and the mean

specific humidity, eq 41, provide some important insights

into processes in the atmospheric surface layer.

Let me define nondimensional versions (denoted by

+) of the variables in eq 16 that are appropriate in an

ASL

U UG + = / (44a)

V VG + = / (44b)

U UG g g + = / (44c)

V VG g g + = / (44d)

t ft + = (44e)

z zz + = / . 0 (44f)

Here

GU V = + ( ) g

2

g

2 1 2/

(45)

is the magnitude of the geostrophic wind, and z0 is the

aerodynamic roughness length of the surface. In addition, I will make uw and vw nondimensional by dividing by u

*

2, where u

* is the friction velocity, a scale I

will say much more about later.

Following Tennekes and Lumley (1972, p. 168 ff.),

I make eq 16 nondimensional

1 1

2

C Ro

U

t

uw u

z C Ro

V V

g g

g

*

/ *

*

∂

∂

∂

∂

+

+ +

+ + = − ( ) + − ( )

(46a)

1 1 2

C Ro

V

t

vw u

z C Ro

U V

g g

g

*

/ *

*

. ∂

∂

∂

∂

+

+ +

+ + = − ( ) − − ( )

(46b)

Here

C

u

G g ≡ * (47)

is the geostrophic drag coefficient, which, as I will show

later, is typically 0.03. And

Ro

u

f z * ≡ *

0

(48)

called the friction Rossby number (Tennekes and Lumley 1972, p. 170), is typically of order 106 over sea ice.

If our scaling is accurate in eq 46, the terms containing the nondimensional variables should all be of order

one. When the first and last terms in each of these two

equations, however, are divided by C Ro g * , terms of

order 10–4 result. Consequently, in the ASL, all that

remains of eq 46 are

∂

∂

uw u

z

/ *

2

0 ( ) ≅

+

(49a)

∂

∂

vw u

z

/ . *

2

0 ( ) ≅

+

(49b)

6
Thus, uw and vw are basically constant with height in

the ASL. We usually align the x axis with the mean wind

so vw = 0 and then define

u uw * . 2 = − (50)

Thus, u* is a fundamental velocity scale in the surface

layer because it is constant with height.

We can apply the same scaling procedure to eq 38.

Here, in addition to eq 44, however, we also need

Θ Θ + = / t

* (51)

where t

* is a temperature scale that I will explain shortly.

In eq 38, I make wθ nondimensional with u t * *. The nondimensional form of eq 38 is thus

1 1 2

Ro t

w ut

+ + z PrR z

* *

+ *

+

+

2

∂

∂

∂ θ

∂

∂

∂

Θ Θ

+ ( ) = /

(52)

where Pr (≡ ν/D) is the Prandtl number and

R

u z

* * 0 ≡ ν

(53)

is the roughness Reynolds number.

Again, as with eq 46, ∂ ∂ Θ+ + / t should be of order

one if our scaling is accurate. But in eq 52, ∂ ∂ Θ+ + / t is

divided by 106. In the atmosphere, Pr is about 0.7; and

over sea ice R* is rarely smaller than 10, except in very

light winds (Andreas and Claffey 1995). Thus, the third

term in eq 52 is also small. Consequently, again the turbulence term is all that remains of eq 52 in the ASL

∂ θ

∂

w ut

z

/ . * *

+

( ) ≅ 0 (54)

As a result, wθ, the kinematic sensible heat flux, is basically constant with height in the ASL. We define its value as

ut w * * ≡− θ. (55)

Thus, because it is independent of height, t

* is the fundamental surface-layer temperature scale.

The mean humidity equation, eq 41, yields to the same

scaling arguments that the mean temperature equation

did. We thus see that wq , the kinematic latent heat flux,

is also independent of height in the ASL. We define its

value as

u q wq * * ≡ − (56)

which therefore introduces the fundamental surface-layer

humidity scale q

*

, another quantity that is constant with

height.

Following arguments similar to these, Monin and

Obukhov (1954) recognized u t * and * as fundamental

flux scales in the ASL. In light of eq 33, they also took

g/Tv to be an important parameter. Lastly, they knew

that the height of the observation, z, was a fundamental

length scale in the surface layer. Consequently, they

hypothesized that all surface-layer statistics should scale

with combinations of these four quantities.

We now know, however, that for scaling properties

that depend on air density, rather than t

* alone, we need

a scale that also includes q

* (Zilitinkevich 1966, Busch

1973). This is the virtual temperature flux scale, t

v*,

defined from eq 21b as

t t Q Tq v* * * = + ( ) 1 0 61 0 61 . . + (57)

where Q T and must be layer-averaged mean values.

Because t

* and q* are constant with height in the surface layer, t

v* is too.

From u t gT * *v and v , / , it is possible to define a fundamental length scale L that is also a constant in the

ASL

1 0 61

1 0 61 2 2 L

kgt

T u

k g

Tu

t

T

Q

≡= + q

+





 



 v

v

*

* * * *

.

. (58)

where k (= 0.4) is the von Kármán constant. Many call

L the Monin-Obukhov length. I prefer, however, to call

it the Obukhov length, since Obukhov defined it in print

eight years before the Monin-Obukhov (1954) paper

appeared (Businger and Yaglom 1971, Obukhov 1971).

On recognizing the dynamical significance of the surface layer scales u t * * , , z and L (we have since added

t

v*and q

*

), Monin and Obukhov (1954) speculated that

all surface-layer turbulence statistics should behave

similarly when properly expressed in terms of these

scales (see also Businger 1973, Wyngaard 1973). In particular, Monin-Obukhov similarity quantifies stability

effects in the ASL with the nondimensional parameter

z/L ≡ ζ. Shortly, when we see that ∂ ∂ U z u kz / / ≅ * , it

will be evident that ζ is the ratio of the buoyant production to the mechanical production (term III to term II) in

the turbulent kinetic energy equation, eq 33. Thus, roughly

when ζ > 1, buoyancy effects dominate mechanical processes in the surface layer; when ζ < 1, mechanical effects dominate. When ζ < 0, the surface is heating and,

thus, destabilizing the air in the ASL through the turbulent exchange of sensible and latent heat. This process

creates unstable stratification. When ζ > 0, the surface

is extracting heat from the surface layer and thereby

cooling it from below. This results in a stably stratified

surface layer.

7
As a demonstration of the power of Monin-Obukhov

Similarity Theory (MOST), let me consider the variances

of vertical velocity, w2 2 ≡ σw , and temperature, θ σ 2 2 ≡ t .

Theory predicts that when these statistics are properly

scaled, they can be functions only of ζ. That is

σw

*u = fz L (/ ) (59)

σt

*

t = gz L (/ ) (60)

since u

* is our primary velocity scale in the ASL and t* is our primary temperature scale.

While MOST predicts that f(ζ) and g(ζ) should exist,

it does not predict their functional forms. Ultimately,

these must be evaluated experimentally. MOST does,

however, provide some guidance as to the asymptotic

behavior of f(ζ) and g(ζ). Let me elaborate.

In the so-called free-convection limit, the buoyancy

production term in eq 33 swamps the mechanical production term; thus, u

* loses its significance as the fundamental velocity scale. From the remaining fundamental parameters, wt t g T v v v or * ( ), / and z, however, it is

possible to define a new velocity scale appropriate for

the free-convection limit (e.g., Hess 1992)

u z g wt

T f v

v

≡ 



 





1 3/

. (61)

As in eq 55, we can, in turn, define a new temperature

scale for the free-convection limit

t

wt

u

T wt

z g wt f

f

v

v

≡ = 











3 1 3/

. (62)

If these are the proper scales in the free-convection

limit, making σw and σt nondimensional with them

should yield universal functions. But now, without u

*,

we have used all the scales available to us. In other words,

without u

*, L cannot be defined; therefore, σw⁄uf

 and

σt/tf can depend on no other variables. The only conclusion is that they are constants in the free-convection limit

σf

f

constant

u = (63)

σt

f

constant.

t = (64)

Since u

* must lose its dynamical significance gradually as ζ decreases, a matching region must exist where

both eq 59 and 63 and where both eq 60 and 64 are approximately true. Using the definitions eq 58, 61 and 62

thus implies that asymptotically in the free-convection

limit

σ ζ w

*u ∝ −( )1 3/

(65)

and

σ ζ t

*

t ∝ −( )1 3/ . (66)

Again, although MOST was the basis for our deriving

these asymptotic expressions, it tells us nothing about

the values of the implied proportionality constants. These

must be found experimentally.

The opposite extreme from free convection is very

stable stratification—in which the vertical density gradient is actually steep enough to suppress vertical exchange. Thus, the eddies reaching a sensor placed at

height z are compacted in their vertical dimension and

may never have been in contact with the surface. In other

words, contrary to near-neutral or unstable conditions, in

very stable conditions, z is no longer a meaningful scale;

a turbulence sensor placed at height z can make no measurement that tells it where the surface is. Thus, z drops

out of our list of surface-layer scales.

Without z, we cannot form the stability parameter

z/L. Therefore, the nondimensional standard deviations

in vertical velocity and temperature must be independent of stability in very stable conditions

(67)

(68)

As usual, MOST does not tell us what these constants

should be—only that they should exist.

Figure 1 demonstrates the veracity of MOST. Figure

1a shows σw * / u as a function of stability from measurements I made with three-axis and vertical sonic anemometers at the Sevilleta Long-Term Ecological Refuge near Socorro, New Mexico, in August 1991. Figure

1b shows σ σ t q * * / / t q and , where σq is the standard

deviation in specific humidity, measured during the same

experiment. Some of the σt values came from the temperature fluctuations measured by the three-axis sonic

anemometer–thermometer; some came from a 76-μm

chromel-constantan thermocouple. The humidity data

came from a krypton hygrometer.

In Figure 1a, σw * / u goes as (–ζ)

1/3 for ζ < –0.4, as

eq 65 predicts. For ζ >0, σw * / u is almost independent

of ζ, as eq 67 suggests.

The data in Figure 1b are more scattered than those

in Figure 1a because t

* and q* also reflect the uncer8

σ

σ

w

t

*

*

constant

constant.

u

t

=

=
9

– 10 – 1 – 0.1 ± 0.01 0.1 1

z/L

0.1

1

10

–

From 3-Axis Sonic


 Best Data


 Questionable Data


From Eddy-Correlation


 Vertical Sonic

σw/u*

Figure 1. Demonstration of Monin-Obukhov similarity theory using

data collected over a semi-arid grassland in New Mexico in August

1991. One anemometer was a three-axis sonic anemometer–thermometer;

the other was a single-axis sonic that was part of an eddy-correlation system manufactured by Campbell Scientific. The three-axis sonic yielded u* for nondimensionalizing both sets of σw measurements. Questionable data

were collected in a directionally variable wind. σ σ t q t

* q

* /, /

a. Standard deviation of vertical velocity, σw. The line is σw/u* =

1.20(0.70 – 3.0ζ)1/3 for –4 < ζ ≤ –0.1, 1.20 for –0.1 ≤ ζ ≤ 0, and

1.20(1 + 0.2ζ) for 0 ≤ ζ < 1.

b. Nondimensional standard deviations of the scalars temperature

(σt) and humidity (σq). The line is σ ζ s * / .( . ) / s = − − 3 4 1 28 4 1 3 for

–4 < ζ ≤ 0, and 3.4 for 0 ≤ ζ < 1.

tainty in the u* measurement (see eq 55 and 56). Nevertheless, both the temperature and the humidity data do

seem to collapse to a single functional form and, thus,

to confirm the predictions of MOST. In particular, for

ζ < –0.1, σt * / t and σq/q* go as (–ζ)–1/3, as eq 66 predicts. And for ζ > 0, the nondimensional standard deviations seem to be independent of ζ.

There have been many other confirmations of

Monin-Obukhov similarity theory in the last 30 years

(see Haugen [1973] and Panofsky and Dutton [1984] for

reviews), and it is now the foundation for our understanding of processes in the atmospheric boundary layer.
SURFACE-LAYER PROFILES

Neutral stratification

Long before the advent of Monin-Obukhov similarity theory, turbulence researchers used scaling arguments

to model the wind speed profile in neutral stratification.

As I explained, u* is the fundamental velocity scale in

the ASL, and z is a fundamental length scale. In neutral

stratification (i.e., when L is infinite or wtv is zero), these

are the only scales available to us. Consequently, the

vertical gradient in wind speed must obey

dU

dz

u

k z = * (69)

where k, the von Kármán constant, assures the equality.

The no-slip boundary condition means that U(z) is

zero at the surface. Hence, we can integrate eq 69 easily

to obtain

U z

u

z

( ) ln = + * z b (70)

where b is an integration constant. Because of the

logarithm on the right side of eq 70, we cannot write

U(z = 0) = 0. Rather, we define a new length scale, the

roughness length z0, where U(z = z0) = 0. Thus

b u

k = − z * ln . 0 (71)

Consequently

U z

u

k ( ) ln / = ( ) z z * 0 (72)

the familiar semi-logarithmic form of the wind speed

profile in neutral stratification.

Figure 2 is a schematic interpretation of eq 72. If we

measure U at several heights and plot these data in a

coordinate system that is logarithmic in z, the data

should lie on a straight line. The slope of this line is

u k * / ; it intersects the U = 0 axis at ln z0.

Equation 72, however, is not just a theoretical construct. Semi-logarithmic wind speed profiles are fairly

common in nature. Figure 3 shows 10 of the 197 such

profiles that Andreas and Claffey (1995) observed over

sea ice in the western Weddell Sea.

If wtv is small enough that conditions are still near

neutral though t

* (i.e., the sensible heat flux) is nonzero, the potential temperature profile obeys the same

scaling as in eq 69. Because t

* is the appropriate temperature scale

d

dz

t

k z

Θ = * . (73)

There is no reason to assume a priori that the same

multiplicative coefficient k–1 should appear in both eq

69 and 73. In fact, the Kansas results (Businger et al.

1971) showed not only that k = 0.35—rather than the

more common value of 0.40—but also that there should

be an additional multiplicative constant of value 0.74

10

Figure 2. Schematic representation of the semilogarithmic wind speed profile.

Wind Speed (m/s)

Height (m)

10

1

0.1

0 2 4 6 8 10 12 14 16 18 20 22 24

8.51


64


22

8.47


64


23

8.84


65


0

8.31


65


1

8.36


65


2

8.89


65


3

9.14


65


4

8.85


65


5

8.86


65


6

7.33


65


7

Ice Station Weddell

Figure 3. Hourly averaged, semi-logarithmic wind speed profiles observed on Ice Station Weddell (Andreas and Claffey

1995). The lowest level of the left-most profile is assigned a value

of 2 m/s; subsequent lowest levels are offset by 2 m/s. Thus, these

profiles reflect relative rather than absolute values. The lines

are least-squares fits according to eq 72. Under each profile, the

upper number is the extrapolated value of the wind speed (in

m/s) at 10 m; the middle number is the Julian day in 1992; the

lower number is GMT.

U

Inz

Inz 0

Slope = ––– u

*

k

Semi- Logarithmic Profile

0
on the right side of eq 73. Wieringa (1980), however,

reexamined the Kansas data, concluded that k = 0.41,

nearly the traditional value of 0.40, and found no need

for an additional multiplicative constant in eq 73.

Högström’s (1988) review suggested k = 0.40 ±0.01

and, again, that eq 69 and 73 represent correct scaling

in neutral stratification. Henceforth, I eschew the popular Kansas results and take the more traditional but

observationally defensible position that k = 0.40 and

that the von Kármán constant is the only factor necessary in both eq 69 and 73.

As we did with eq 69, we can integrate eq 73 from

the surface to height z. The result is

Θ( ) ln z T

t

k

=+ +z c s * (74)

where Ts is the surface temperature, and c is a constant

of integration. As in eq 71, we find c by requiring Θ to

be Ts at the surface; thus

Θ( ) ln / z T

t

k

= + ( ) z z s T * (75)

where zT is a new length scale, the roughness length

for temperature.

Exactly the same arguments that we used to predict

the potential temperature gradient also apply to specific humidity. Thus, in near-neutral stratification

dQ

dz

q

k z = * . (76)

Integrating this yields

Qz Q q

k ( ) ln( / ) = + z z s Q * (77)

where Qs is the surface value of the specific humidity,

and zQ is yet another length scale, the roughness length

for humidity. z0, zT and zQ are not necessarily equal, as

I will explain later.

Including stratification effects

Because atmospheric stratification is rarely near

neutral, it often affects the shape of surface-layer profiles. Thus, eq 69, 73 and 76 are not strictly accurate in

diabatic conditions. We can extend these, however, on

the basis of Monin-Obukhov similarity theory. These

diabatic profile corrections, in fact, are at the core of

Monin and Obukhov’s (1954) work. They simply tried

to retain the basic form of eq 69, 73 and 76 by multiplying these by functions that depend only on z/L

dU

dz

u

k z = ( ) * φ ζ m (78a)

d

dz

t

k z

Θ = ( ) * φ ζ h (78b)

dQ

dz

q

k z = ( ) * φ ζ h . (78c)

As in our earlier discussion, these φ functions—

though presumably universal—are empirical: They

must be found experimentally. Notice, I use the same φ

function in both the temperature and humidity equations because there is no good theoretical reason why

these should be different (Hill 1989). A definitive experiment that verifies this assumption, however, has not

been done, though several experiments support it (e.g.,

Dyer 1974, Dyer and Bradley 1982).

For unstable conditions, φm and φh are fairly well

known (e.g., Paulson 1970)

φζ ζ m( ) = − ( )− 1 16 1 4/ (79a)

φζ ζ h ( ) = − ( )− 1 16 1 2/ . (79b)

These are commonly called the Businger-Dyer relations

because Joost Businger (1966) and Arch Dyer derived

them independently in the mid-1960s (Businger 1988).

Though the constant multiplying ζ in eq 79 may vary

somewhat among the various experimental evaluations

of φm and φh (e.g., Dyer and Hicks 1970, Businger et

al. 1971, Wieringa 1980, Dyer and Bradley 1982,

Högström 1988), the same basic functional form comes

through.

Figure 4 shows eq 79 as functions of stability.

Notice in the figure and in eq 79, φm = φh = 1 at ζ = 0

(i.e., for neutral stratification) as eq 69, 73, and 76 require. Figure 4 also shows that as –ζ gets larger—as

conditions become more unstable—both φm and φh

11

Unstable 1.0

0.8

0.6

0.4

0.2

0

–10 –8 –6 –4 –2 0

ζ = z/L

φm = (1 – 16 ζ)–1/4

φh = (1 – 16 ζ)–1/2

Figure 4. Nondimensional wind speed and scalar gradients, φm and φh, as functions of stability for unstable

conditions.
decrease monotonically. This means, according to eq

78, that the vertical gradients of wind speed, temperature and humidity get weaker as the instability

increases. The vertical exchange that the increasing

buoyancy fosters homogenizes the profiles.

In contrast to the situation for unstable stratification,

for an ASL that is stably stratified there are a host of

suggestions as to the forms of φm and φh. Figure 5 shows

some of these functions.

Often the wind speed and scalar profiles are assumed

to be log-linear in stable conditions (Webb 1970). That

is, the profiles are basically semi-logarithmic as in eq

72, 75 and 77, but each has an additional additive term

that depends on ζ. φm and φh would, thus, have the form

φ ζ φ ζ γζ mh s ( ) = ( ) = +1 . (80)

The constant γs is generally reported to be in the range

from 5 (Webb 1970, Dyer 1974, Large and Pond 1981)

to 7 (Wieringa 1980, Large and Pond 1982, Högström

1988).

From wind speed and temperature profiles at the South

12

Stable

φm = 1 + 7 ζ

= (1 + 4.5 ζ)3/4

= 1 + 0.7 ζ + 0.75 ζ (6 – 0.35 ζ) exp (–0.35 ζ)

= 1 + 5 ζ

0

0

2

4

6

8

10

12

14

0.5 1.0 1.5 2.0

ζ = z/L

φm

Figure 5. Various suggestions for the functional form

of the nondimensional wind speed (φm) and scalar (φh)

gradients for stable conditions.
Pole, Lettau (1979) deduced the nondimensional gradient functions

φζ ζ m( ) = + ( .) / 1 45 3 4 (81)

and

φζ φ ζ ζ h m ( ) = =+ () ( . ) 2 32/ 1 45 (82)

which are also depicted in Figure 5. Lettau’s φh function predicts a very steep scalar gradient with increasing ζ that has not been confirmed by independent observations.

Although Lettau (1979) developed his functions especially to treat very stable conditions, the log-linear

form (eq 80) is also often applied in very stable conditions, though it has been tested only for 0 ≤ ζ < 1 (e.g.,

Dyer 1974, Hicks 1976, Yaglom 1977). As an alternative to the log-linear form for 0 ≤ ζ ≤ 10, Holtslag and

de Bruin (1988) and Beljaars and Holtslag (1991) proposed the function

φ ζ φζ

ζζ ζ ζ

m h ( ) =

=+ + − −

( )

1 0 7 0 75 6 0 35 0 35 . . ( . ) exp( . ).

(83)

I will refer to this as the Dutch formulation.

In Figure 5, we see that there is little difference

among the four functions suggested for φm and φh when

0 ≤ ζ ≤ 0.5. Thus, because the log-linear form is much

simpler mathematically than Lettau’s (1979) or the

Dutch formulation, it might be preferable for weakly

stable conditions. Some other considerations, however,

might help us decide which of the four formulations is

best in more stable conditions.

The gradient Richardson number is

Ri g

T

dz

U dz =

v ( )

∂

∂

Θ /

/ . 2 (84)

From eq 78 and the definition of ζ (eq 58) we can write

this as

Ri g

T

t kz

u kz

= ( ) ( )

( ) ( ) = ( )

v ( )

h

m

2

h

m

2 *

*

/

/

. φ ζ

φ ζ

ζφ ζ

φ ζ 2 (85)

In stable conditions, turbulence ceases when the

Richardson number exceeds a critical value, Ricr. Thus,

we should expect accurate profile functions to yield this

critical value through eq 85; that is

lim . ζ→∞

Ri Ri = cr (86)

Traditionally, Ricr is assumed to be 0.20–0.25 (Busch

1973, Businger 1973). But Mahrt (1981) and Heinemann and Rose (1990) have reported that a larger value

is sometimes indicated. Woods (1969) may have explained this range of values by demonstrating that there

is a hysteresis in Ricr. He concluded that a turbulent

flow becomes laminar when Ri exceeds 1, but a laminar flow does not become turbulent until Ri falls below

0.25 (see also Plate 1971, p. 76). In his observations at

South Pole, however, Lettau (1979) frequently found

turbulence to exist even when Ri exceeded unity; he

thus concluded that, in stable conditions, there was no

critical Richardson number. Monin and Yaglom (1971,

p. 440 f.) also argued that there seems to be no critical

Richardson number in stable conditions. In light of this

controversy, I show in Table 1 what the four formulations for φm and φh depicted in Figure 5 predict as the

behavior of Ri in very stable conditions.

Lettau (1979; see also Viswanadham [1979, 1982])

described two other parameters that characterize

surface-layer profiles, the Deacon numbers for wind

speed and potential temperature, Dm and Dh, respectively. These quantify profile curvature. For wind speed

D z d U dz

dU dz m ≡ − ( /)/ .

2 2

(87)

With eq 78a, it is easy to show that

D d

d m

m

m ( ) ( )

( ) ζ . ζ

φ ζ

φ ζ

ζ = −1 (88)

Similarly, for potential temperature

D z d dz

d dz h ≡ − ( /)/

2 2 Θ

Θ (89)

which, with eq 78b, becomes

D d

d h

h

h ( ) ( )

( ) ζ . ζ

φ ζ

φ ζ

ζ = −1 (90)

13

Table 1. Predicted behavior of the Deacon and

Richardson numbers in very stable conditions according to the four gradient formulation shown in Figures

4 and 5.

limit ζ → ∞

φm, φh Dm Dh Ri

1 + 5ζ 0 0 1/5 = 0.20

1 + 7ζ 0 0 1/7 = 0.14

φm = (1 + 4.5ζ)3/4

φh = (1 + 4.5ζ)3/2

1 + 0.7 ζ + 0.75ζ(6 – 0.35ζ) exp(–0.35ζ) 0 0 1.4

1/4 –1/2 ζ
Table 1 also shows the Deacon numbers that the four

φm and φh formulations in Figure 5 predict in very

stable conditions.

Table 1 yields contradictory suggestions as to which

of the φm and φh formulations is best in stable conditions. The Webb (1970) and Wieringa (1980) functions

predict that Ricr is 0.20 and 0.14, respectively—roughly

the traditional range for Ricr (Lumley and Panofsky 1964,

p. 117). The Dutch formulation predicts Ricr = 1.4, a

value in line with more recent appraisals. The critical

Richardson number based on Lettau’s (1979) formulation is unbounded, a result that Monin and Yaglom

(1971, p. 441) defended theoretically and that Lettau

believed was possible in light of his South Pole observations.

In very stable conditions, turbulence is suppressed;

the vertical exchange of heat and momentum must thus

be by molecular processes alone. In such conditions, the

vertical profiles of wind speed and temperature depend

linearly on height, and their vertical gradients are, thus,

constants with height; Dm and Dh would then be zero.

Viswanadham’s (1982) analysis also suggested that zero

is the limiting value, at least for Dm, in very stable conditions. (He did not consider Dh.) In Table 1, three of

the formulations predict that zero is the limit for the

Deacon numbers in strong stability. But, again from his

measurements at South Pole, Lettau (1979) found that

Dm = 1/4 and Dh = –1/2 in very stable conditions and,

therefore, tuned his φm and φh functions to produce these

limits.

In stable stratification, wave phenomena can supplant

forced convection and molecular diffusion as the dominant mechanisms for transferring heat and momentum.

These processes are necessarily intermittent. The stratification builds and builds, until a wave breaks and introduces new turbulence that episodically homogenizes

the wind speed and temperature profiles. The overriding stratification eventually damps this turbulence until

the cycle repeats. Thus, it seems that all the limiting

Deacon and Richardson numbers listed in Table 1 may

be appropriate at times. The very steep temperature gradient that Lettau (1979) reported would be consistent

with a relatively quiet, strongly stratified surface layer.

But the weaker gradient of the Dutch formulation seems

plausible, too, in a surface layer frequently mixed by

breaking gravity waves. Thus, the stable atmospheric

surface layer is still rife with interesting questions about

turbulence processes.

BULK TRANSFER COEFFICIENTS FOR HEAT

AND MOMENTUM OVER SEA ICE

Mathematical background

For estimating turbulent surface fluxes from field

data, in computer models, and for some analytical studies, it is often convenient to know the so-called bulk

transfer coefficients. These relate a turbulent flux to

more readily available quantities.

Here, I will discuss the bulk transfer coefficients for

momentum (usually called the drag coefficient) and

for sensible and latent heat. These are defined, respectively, as

τρ ρ ρ =− = = uw u C U * Dr r

2 2 (91)

H c wt c u t C c U T sp p * * = =− = − ρρ ρ Hr p r s r ( ) Θ (92)

H L wq L u q C L U Q Q Lv v * * = =− = − ρρ ρ Er v r s r ( ). (93)

In these cp = specific heat of air at constant pressure

Lv = latent heat of vaporization or sublimation

Ts, Qs = surface values

Ur, Θr, Qr = values at some reference height, usually

10 m

CDr, CHr = bulk transfer coefficients. Since these are

and CEr height dependent, I explicitly include a

height subscript in their symbols.

In the last section, I introduced the gradient functions φm(ζ) and φh(ζ) in eq 78. These let us quantify Ur,

Ts – Θr and Qs – Qr in eq 91–93 and thereby provide a

mathematical framework for handling the bulk transfer coefficients. Take wind speed as an example. From

eq 78a

dU

dz

u

k z = z L * φm( / ). (94)

Panofsky (1963) and Paulson (1970) showed how to

integrate this. The trick is to write

dU

dz

u

k z

= −+ [ ] * 1 1 φ ζ m ( ) (95)

then the integration becomes

dU

u

k

dz

z

d

U z

U z

z

z

′ = ′

− − ′

′ ′



















 ∫ ∫ ∫

( )

( ) ( ) .

0 0

1

0

* φ ζ m

ζ ζ

ζ

(96)

Hence, because U(z0) = 0

U z

u

k ( ) ln( / ) ) . = − [ ] z z * m ( 0 ψ ζ (97)

Thus, the trick in eq 95 leads again to a semilogarithmic profile with an additive stability correction.

That stability correction in eq 97, ψm, is defined as

14
ψ ζ φ ζ

ζ ζ

ζ

m m ( ) ( ) = . − ′

′ ′ ∫

1

0

d (98)

We can follow exactly the same procedure with the

potential temperature and specific humidity gradients

in eq 78b and 78c to obtain the profiles of these variables

Θ( ) ln( / ) ) z T

t

k

=+ − [ ] z z s Th * ψ ζ( (99)

Qz Q q

k ( ) ln( / ) ) =+ − [ ] z z s Qh * ψ ζ( (100)

where

ψ ζ φ ζ

ζ ζ

ζ

h

h ( ) ( ) = . − ′

′ ′ ∫

1

0

d (101)

To see the ψm and ψh functions that result from integrating the φm and φh functions that I described in the

last sections, see Paulson (1970), Lettau (1979),

Holtslag and de Bruin (1988), Arya (1988, p. 166 f.),

or Launiainen and Vihma (1990).

Figures 6 and 7 show sample wind speed and potential temperature profiles computed from eq 97 and eq 99.

Note the curvature of the unstable wind speed and temperature profiles. The vertical exchange characteristic

of unstable stratification tends to homogenize the profiles; both wind speed and temperature bend more towards the vertical with increasing height. In stable conditions, on the other hand, the stratification suppresses

vertical exchange. Consequently, steeper gradients are

possible.

From eq 91–93, 97, 99 and 100, we see that the bulk

transfer coefficients are clearly stability dependent (e.g.,

Andreas and Murphy 1986)

C k

rz rL Dr

m

= −

2

0

2 [ln( / ) ( / )] ψ

(102)

C k

rz rL rz rL Hr

m Th

= − −

2

0 [ln( / ) ( / )] [ln( / ) ( / )] ψ ψ

(103)

C k

rz rL rz rL Er

m Qh

= − −

2

0 [ln( / ) ( / )] [ln( / ) ( / )] . ψ ψ

(104)

At neutral stability (r/L = 0) both ψm and ψh are zero

by definition. Such conditions define the neutralstability bulk transfer coefficients. From eq 102–104,

these are

C k

r z

DNr =

2

0

2 [ln( / )] (105)

(106)

15

0

0.01

0.1

1

10

20

2 4 6 8 10 12 14

Wind Speed, U (m/s)

, z


Unstable


(L = –2 m)

Stable


(L = 2 m)

Neutral

u

*

 = 0.20 m/s


z0 = 1.0 × 10–3 m

Potential Temperature, θ (°C)

–20 –15 –10 –5 0

0.01

0.1

1

10

20

t, z


Unstable
 Neutral

(L = –2 m,


t

*

 = –0.5°C)

Stable


(L = 2 m,


t

*

 = 0.1°C)

Height, z (m)

Figure 6. Sample wind speed profiles. Conditions are as labeled. The ψm in eq 97 is based

on the Businger-Dyer form (eq 79 and Figure

4) for the unstable case and on the Dutch formulation (eq 83 and Figure 5) for the stable case.

Figure 7. Sample potential temperature profiles. Conditions are

as labeled. The ψm in eq 97 is based on the Businger-Dyer form

(eq 79 and Figure 4) for the unstable case and on the Dutch formulation (eq 83 and Figure 5) for the stable case.

Height, z (m)

 

C k

rz rz

k C

r z

C

kC z z

HNr

T

DNr

1/2

T

DNr

–1

DNr

1/2

T

ln( /

1–

= =

=

2

0

0

[ln( / )][ln( / )] )

ln( / )
(107)

Thus, knowing z0, zT and zQ is equivalent to knowing

the neutral-stability bulk transfer coefficients for any

reference height r.

We also see from eq 106 and 107 that equality of z0,

zT and zQ implies that CDNr = CHNr = CENr. Although

some measurements have suggested that CHNr = CENr,

none have confirmed that CDNr is, in general, equal to

either of these, as I will explain shortly. Thus, equality

between z0 and zT or zQ is not true either.

From eq 102–104, it is clear that if we know the

neutral-stability transfer coefficients or z0 and zT and

zQ, we know the stability-dependent transfer coefficients

needed in eq 91–93 to compute the turbulent fluxes of

interest. Deardorff (1968) was one of the first to show

plots of the bulk transfer coefficients as functions of stability. More recently, Smith (1988) presented updated

plots using modern expressions for the necessary similarity functions. In a nutshell, for constant wind speed,

CD, CH and CE all increase as conditions become more

unstable; and, again for a constant wind speed, all decrease as stability increases. These results follow intuitively from our discussion of how vertical exchange

is enhanced in unstable conditions and suppressed in

stable conditions.

Since we can obtain the bulk transfer coefficients at

any height and for any stratification from the roughness

lengths or from the neutral-stability coefficients, we can

confine the rest of our discussion to the values of these

neutral-stability coefficients. I will also settle on the standard reference height of 10 m and, henceforth, confine

my discussion to the neutral-stability transfer coefficients

at 10 m, CDN10, CHN10 and CEN10. For completeness,

the following equations show how to compute CDr, CHr

and CEr from these neutral-stability values (Andreas and

Murphy 1986)

 

C C

k C r rL

Dr

DN10

–1

DN10

1/2 m

=

{ } 1 10 + −

2 [ln( / ) ( / )] ψ

(108)

C C CC

k C C r rL Hr

DN10 Dr DN10

–1

HN10 DN10

–1/2

h

= + −

(/ )

[ln( / ) ( / )]

1 2/

1 10 ψ

(109)

C C CC

k C C r rL Er

EN10 Dr DN10

–1

EN10 DN10

–1/2

h

= + −

(/ )

[ln( / ) ( / )]

1 2/

1 10 ψ

(110)

where r must be expressed in meters.

Drag coefficient

Overland (1985) reviewed 45 evaluations of the drag

coefficient over sea ice published through 1984. Since

then, Anderson (1987), Fairall and Markson (1987),

Guest and Davidson (1987, 1991), Martinson and

Wamser (1990), Wamser and Martinson (1993),

Andreas et al. (1993b), and Andreas and Claffey (1995)

have added to this information pool.

But I still feel that one of the most important works

on this subject is that by Banke et al. (1980). They

showed that CDN10 is related to a measurable surface

roughness parameter ξ through (Fig. 8)

10 1 10 0 072 3 CDN10 = + . . ξ (111)

where ξ must be in centimeters. Here, ξ obtains from an

integration under the snow-surface roughness spectrum,

Φs

ξ κκ

κ

2

0

=

∞

∫ Φs( )d (112)

where κ is the wavenumber and κ0 = 0.5 rad/m, which

corresponds to a maximum wavelength of about 13 m.

In turn, the snow-surface roughness spectrum, Φs(κ),

is the spectral density (Andreas et al. 1993b) obtained

from a Fourier analysis of a snow-surface elevation profile. Figure 9 shows such a snow-surface elevation profile and the actual ice-surface elevation profile measured

on Ice Station Weddell (ISW) in 1992 (Anonymous 1992,

ISW Group 1993). Figure 10 shows the resulting (nondimensional) snow-surface (Φs) and ice-surface (Φi

)

roughness spectra implied by these profiles (see Andreas

et al. [1993b] for computational guidelines). According

to eq 112, ξ is obtained from the Φs line in Figure 10

by integrating from κ = 0.5 rad/m to infinity. That upper

integration limit, however, is not actually infinity but rather

the Nyquist wavenumber 2π/2∆, where ∆ is the sampling

interval. Since Banke et al. (1980) used a sampling interval of 1 m, for their ξ values the upper integration limit

was π rad/m. Because the sampling interval for the ISW

profiles was 0.5 m, however, the Nyquist wavenumber for

these is 2π rad/m (Fig. 10).

Figure 8 and eq 111 suggest that form drag is important in fostering momentum exchange. As the surface gets measurably rougher, there are more vertical

surfaces for the wind to push against; CDN10 consequently increases. Measurements in the 1980s in the

marginal ice zone—arguably the roughest oceanic surface—corroborated this hypothesis that as the surface

gets visibly rougher the drag coefficient increases

(Andreas et al. 1984; Brown 1986, 1990; Anderson 1987;

Guest and Davidson 1987, 1991).

16

 

C k

rz rz

k C

r z

C

kC z z

ENr

Q

DNr

1/2

Q

DNr

–1

DNr

1/2

Q

ln( /

1–

= =

=

2

0

0

[ln( / )][ln( / )] )

ln( / ) .
17

2.0

2.4

1.6

1.2

0.8 0.001

0.002

0.005

0.01

0.02

0.05 (

0.1

0.2

0 2 4 6 8 10 12 14

ξ (cm)

103 CDN10

z0 (cm)

Figure 9. A 300-m-long snow-surface and ice-surface elevation profile measured on Ice Station Weddell.

Figure 8. The 10-m, neutral-stability drag coefficient (CDN10) over

Arctic sea ice is related to a parameter that characterizes surface

roughness (ξ) (after Banke et al. 1980).

0.01

0.00

0.02

0.03

0.01 0.1 1 10

Snow Surface


Ice Surface

Wavenumber, κ (rad/m)

κΦ(κ) (m2)

Figure 10. Snow-surface and ice-surface roughness spectra computed

for the profiles in Figure 9. For the snow-surface spectrum, ξ = 3.7 cm.
The power of eq 111 is its implicit suggestion that it

may be possible to determine the drag coefficient remotely from instruments sensitive to surface roughness.

In this light, eq 111 warranted further investigation. The

Winter Weddell Gyre Study (WWGS; Augstein et al.

1991) on the Polarstern gave us an opportunity to try

this inverse use of eq 111. During a 1989 transect of

the Weddell Sea, WWGS scientists collected 47 snowsurface profiles like that in Figure 9, except most were

only about 100 m long. We computed Φs from these,

then ξ from eq 112, and finally CDN10 from eq 111 (Andreas et al. 1993b). Figure 11 shows a histogram of the

resulting values.

Lange and Eichen (1991) had developed a classification scheme for sea ice in the Weddell Sea. They identified four ice classes, as follows:

Class I: deformed first-year ice

Class II: undeformed first-year ice

Class III: undeformed second-year (multiyear) ice

Class IV: deformed second-year (multiyear) ice

In Figure 11, I distinguish the drag coefficients com18

Figure 11. Neutral-stability, 10-m air–

ice drag coefficients computed from

eq 111 using snow-surface elevation

profiles collected during the Winter

Weddell Gyre Study (Andreas et al.

1993b).

ce Stat o edde

3.0

2.5

2.0

1.5

1.0

62 67 72 77 82 87 92

Julian Day

103 CDN10

Figure 12. Time series of the neutral-stability, 10-m

drag coefficient measured on a fixed mast on Ice

Station Weddell (see Andreas and Claffey [1995] or

Andreas [1995a]).

puted for each ice class. Because WWGS scientists had

little opportunity to sample second-year floes, the statistics on classes III and IV in the figure are uncertain.

But for classes I and II, where the sample size is larger,

Figure 11 corroborates our earlier speculation: When

the ice is identifiably rougher (i.e., deformed), the drag

coefficient is generally larger.

Much of the appeal of eq 111 is the unification it

provides; it confirms our intuition about the importance

of roughness on drag with hard data. Unfortunately, no

one has found independent confirmation of eq 111 (e.g.,

Shirasawa 1981). The actual form of eq 111 may be a

moot point, however. Our recent observations on Ice

Station Weddell in the western Weddell Sea imply that

eq 111 oversimplifies air–ice coupling.

Figure 12 is a time series of ISW CDN10 values that

Andreas and Claffey (1995) deduced from four-level

wind speed profiles, including those in Figure 3. Though
these data came from a single mast that did not move

with respect to the surrounding ice for the duration of the

experiment, the CDN10 values range from 1.27 × 10–3 to

2.54 × 10–3. And these values are not randomly scattered but rather behave coherently with time. Figure 13

shows two long storm events extracted from Figure 12. In each case, CDN10

decreased as the wind persisted from a

fairly constant direction and blew with

speeds of about 8 m/s or higher.

19

140

160

180

6

8

10

12

2.5

2.0

1.5

120

140

160

6

8

10

12

2.5

2.0

1.5

64 65 66 79 80 81

0

Julian Day

103 CDN10 Wind Direction (°) U5 (m/s)

Figure 13. Two long events on Ice Station Weddell

that were characterized by a relatively constant wind

direction (hourly average). U5 is the hourly averaged

wind speed at a height of 5 m; the CDN10 values came

from Figure 12.

Percent of Time with Wind-Driven Snow

at Given Wind Speed

Figure 14. Ice Station Weddell observations showing the percentage

of the time with wind-driven snow

for a given 5-m wind speed. The

terminology follows National

Weather Service (1991) guidelines.

Above each histogram bar is the

number of hourly averaged wind

speeds within the indicated range.

On ISW we observed that snow generally began

drifting when the wind at 5 m reached 6 m/s. When the

wind reached 8–10 m/s, blowing and drifting snow were

virtually guaranteed. Figure 14 summarizes these observations.
We thus inferred that, for most of the hourly values

depicted in Figure 13, wind-driven snow was present

(Andreas and Claffey 1995). As a result, the decrease in

the CDN10 values during the two events suggests that

snowdrifts perpendicular to the wind were eroding, and

this snow was depositing in drifts that tended to streamline the surface in the mean wind direction. This streamlining lowered the aerodynamic roughness and therefore

the drag coefficient.

Visual inspection of Figure 13 may suggest that the

behavior of CDN10 is simply a wind speed effect: Lower

CDN10 values seem to be associated with lower wind

speeds. But looking at the entire ISW set of drag coefficients (Fig. 15) does not corroborate this speculation.

CDN10 does not have any obvious dependence on the

wind speed for speeds between 4 and 14 m/s. Likewise,

Shirasawa (1981) found no wind speed dependence in

drag coefficients measured over snow-covered lake ice

for winds up to 11 m/s, and Inoue (1989) also found no

wind speed dependence at Mizuho Station on the Antarctic continent for winds up to about 15 m/s.

To investigate further this hypothesis that streamlining

of the surface by wind-driven snow has a major effect

on the air–ice drag coefficient, I (Andreas 1995a, 1995b)

developed a physically based form drag model following Raupach’s (1992) guidelines. Figure 16 depicts the

rudimentary sastrugi-like roughness elements that I treat

with the model. Although we made no formal observations of sastrugi or surface texture on ISW, we often

saw that high wind sculpted the snow surface into linear

drifts that looked like Figure 16. Figure 9 supports these

casual observations; it shows that the snow collected

around protrusions in the sea ice surface but that the resulting drifts were generally small. Hence, I assume that

the typical sastrugi height, h in Figure 16, was roughly

10–15 cm on ISW.

Figure 17 shows model predictions of how CDN10

depends on the wind’s orientation with respect to the

long axis of the sastrugi (Φ in Fig. 16). In the figure, γ is

the fractional surface coverage of the roughness elements

depicted in Figure 16. Notice, because of the geometry

of the modeled sastrugi, the maximum possible value of

γ is 0.50.

Figure 17 reproduces some of the observations that I

have been describing. When the wind is well aligned

(within ±15°) with the sastrugi, CDN10 has a minimum.

As the wind turns to blow more perpendicularly to the

sastrugi (without being high enough to erode them),

CDN10 increases. For h = 15 cm and γ = 0.15, the model

predicts CDN10 values in line with what we observed on

ISW. The model yields CDN10= 1.43 × 10–3 when the

wind is within ±12° of head-on to the sastrugi, while on

20

0 3 6 9 12 15

1.0

1.5

2.0

2.5

3.0

U10 (m/s)

17

32

23

23

45

23 17

10 7

103 CDN10

Figure 15. CDN10 values in Figure 12, averaged

over 1-m/s bins of the 10-m wind speed; the filled

circle denotes this average. Next to each circle is

the number of CDN10 values in each bin. The vertical bar indicates the 90% confidence interval for

the population within each bin, assuming a beta

distribution with lower limit 1.2 × 10–3 and upper

limit 2.60 × 10–3 (Harr 1977, p. 495 f.).

Wind

Front

n h m h

h

β

Φ

Top

Side

β

Figure 16. Geometry of the sastrugi-like roughness elements

envisioned to dictate the drag coefficient over snow-covered

sea ice. If the wind has been blowing above 6–8 m/s for

10–12 hours, the sastrugi will be aligned with the mean

wind direction; Φ will then be 0°.
ISW the minimum CDN10 value was about 1.5 ×10–3.

When the wind is roughly at a right angle to the sastrugi,

the model predicts a maximum in CDN10 of 2.70 × 10–3,

while our observed maximum was 2.54 × 10–3.

Our ISW observations and this modeling have, thus,

shown some deficiencies in eq 111. Because drifting

snow makes a sea ice surface anisotropic, eq 111 can be

true only if ξ is obtained from a snow-surface profile

aligned with the mean wind. But our observations and

modeling also reiterate one of the conclusions implicit

in eq 111: Small roughness elements (heights of order

10–15 cm) with fairly short wavelengths (order

of meters) dominate the local form drag over snowcovered surfaces. Jackson and Carroll (1978) had reached

the same conclusion. Because the horizontal scales involved are much smaller than either Arctic or Antarctic ridge spacings (Lytle and Ackley 1991), pressure

ridges seem to be less important in local processes.

Nevertheless, pressure ridges may still be important in

setting the large-scale roughness and drag coefficient

over sea ice (Arya 1973, 1975).

Scalar bulk transfer coefficients

The Reynolds analogy (Schlichting 1968, p. 268 f.

and 662 f.; Hinze 1975, p. 746) is the hypothesis that

momentum and scalars, such as temperature or humidity, are transferred similarly. An immediate conclusion

of the Reynolds analogy is that CDN10 = CHN10 = CEN10.

In discussing eq 105–107, however, I noted that, in geophysical flows at least, CDN10 has never been proven to

equal CHN10 or CEN10. The reason is that, over natural

surfaces, momentum and scalars are transferred by different processes.

Figure 18 is a depiction of the microscale of a natural surface such as snow. The surface has millimetersize roughness elements that sometimes are embedded

in the viscous sublayer and sometimes protrude above

the viscous sublayer. The roughness Reynolds number,

eq 53, parameterizes the aerodynamic character of a

surface. When R

*≤ 0.135, the surface is aerodynamically smooth; when R

*≥2.5, the surface is aerodynamically rough; when 0.135 < R

*< 2.5, the surface is in

transition (Businger 1973).

21

103 CDN10

Figure 17. Model calculations of the neutral-stability, 10-m drag coefficient,

CDN10, as a function of the wind orientation, Φ, with respect to the dominant

axis of the sastrugi depicted in Figure 16 (from Andreas 1995a,b). Here, the

sastrugi height, h, is 15 cm; and γ is the fraction of the surface that the sastrugi

cover. Also, m = 4 and n = 10 (see Fig. 16).

























Aerodynamically Smooth, R* ≤ 0.135

Aerodynamically Rough, 2.5 ≤ R*

Figure 18. Schematic depiction of the microscale roughness of an aerodynamically smooth and an aerodynamically rough surface. The shading shows the viscous sublayer, where momentum transfer to the surface must be by

molecular processes.

5

4

3

2

1

0

0 20 40 60 80 100 120 140 160 180

Wind Direction, Φ (° )

h = 15 cm

γ = 0.20

0.15

0.10

0.05
 10–2 10–1 100 101 102 103 10–5

10–4

10–3

10–2

10–1

 100

 101

R*

Smooth Transition Rough

Water Vapor

Temperature

Physically, what these distinctions mean are that

when the surface is aerodynamically smooth, the roughness elements are embedded in the viscous sublayer;

as a result, momentum is transferred to the surface

strictly by molecular processes. When the surface is

aerodynamically rough, the roughness elements protrude through the viscous sublayer; turbulent eddies can

therefore transfer momentum through pressure forces

on the roughness elements as well as through molecular processes. In eq 33, term IVb reflects this explicit

dependence of the TKE budget on transfers resulting

from the covariance of velocity and pressure fluctuations. We find no similar pressure transport terms

in either the temperature variance budget, eq 39, or the

humidity variance budget, eq 42. Scalars cannot be

transported by pressure forces; scalar transfer at a

surface is always molecular. Consequently, over aerodynamically rough surfaces, at least, the Reynolds

analogy is inherently inappropriate (cf., Joffre 1982).

Measurements have not yielded conclusive results

on the behavior of the scalar transfer coefficients over

snow-covered surfaces. Hicks and Martin (1972) and

Thorpe et al. (1973) measured both CHN10 and CEN10

over lake ice and sea ice, respectively. Joffre (1982)

measured CHN10 over snow-covered sea ice. Kondo and

Yamazawa (1986) measured CHN1 (1-m value) and

Barry and Munn (1967) measured CEN0.3 (30-cm

value) over snow-covered ground. The inferred CHN10

and CEN10 values are quite scattered because, over snow,

the potential temperature and humidity gradients necessary to compute the transfer coefficients (see eq 92

and 93) are generally small and, thus, difficult to measure precisely. The CHN10 values from the measurements

are typically about 1×10–3, but CEN10 measurements

do not suggest a consensus value. Andreas (1987) reviewed some of these observations.

In the absence of definitive experimental work, I devised a model to predict CHN10 and CEN10 (Andreas

1987). My work built on Brutsaert’s (1975a, b) surfacerenewal model. Its main result is to predict the scalar

roughness, zs, as a function of z0 and R*

ln( / ) ln (ln zz b b R b R ) s * * 0 01 2 2 = + + (113)

where Table 2 gives the polynomial coefficients when

zs is zT or zQ. Figure 19 shows this equation.

We see from eq 106 and 107 that knowing zT/z0 and

zQ/z0 is essential for predicting CHNr and CENr. But we

also see from these equations that both scalar transfer

coefficients also depend on CDNr. Hence, to use eq 113

in eq 106 and 107 to predict CHN10 and CEN10, I used

eq 111 to model CDN10 (Andreas 1987). As a result,

both scalar transfer coefficients depend on wind speed

(because zs/z0 depends on R*) and ξ. Figure 20 shows

model predictions of CHN10 and CEN10.

Figure 20 shows not only CHN10 and CEN10 but also

CDN10 computed from eq 111, which is independent of

wind speed. Only at very low wind speeds, when the

flow is aerodynamically smooth or in transition, are

CHN10 and CEN10 near CDN10—the Reynolds analogy.

For higher winds, CHN10 and CEN10 are clearly less

than CDN10. Figure 19 shows the same behavior; zT

and zQ can be orders of magnitude less than z0 as the

flow gets more aerodynamically rough.

Munro (1989) verified some of the predictions of

this model with data collected over a Canadian glacier.

Kondo and Yamazawa’s (1986) data collected over

snow-covered ground provide another test of the

model’s prediction of CHN10 (Fig. 20a). Because Kondo

and Yamazawa reported only CHN1 and CDN1 (both at

22

Table 2. Values of the coefficients in the polynomial (eq 113) that predicts zT/z0 and zQ/z0.

R* ≤ 0.135 0.135 < R* < 2.5 2.5 ≤ R* ≤ 1000

Temperature (zT/z0)

b0 1.250 0.149 0.317

b1 — –0.550 –0.565

b2 — — –0.183

Water vapor (zQ/z0)

b0 1.610 0.351 0.396

b1 — –0.628 –0.512

b2 — — –0.180

zs/z0 Figure 19. Model predictions of zT/z0 and zQ/z0 over

snow-covered surfaces (from Andreas 1987). The three

regions label where the flow is aerodynamically smooth,

aerodynamically rough, or in transition.
1 m), I had to obtain their raw data* to compute the

points plotted in Figure 20a. In addition, because they

had not measured ξ, I inferred the ξ values associated

with the CHN10 values in Figure 20a by using eq 111 to

estimate ξ from their measured drag coefficients.

Although there are some discrepancies between the

model predictions and the data in Figure 20a, the comparison is quite encouraging. The data support the model

prediction that CHN10 increases with increasing CDN10

at a given wind speed and that CHN10 decreases with

increasing wind speed. Unfortunately, no high quality

measurements are yet available for testing the model’s

prediction for CEN10, though several have accepted its

predictions for both CHN10 and CEN10 (Morris 1989,

Munro 1989, Launiainen and Vihma 1990).

THE EKMAN LAYER

Polar meteorology has, of course, relied on the discoveries of meteorologists working at lower latitudes.

But, in turn, meteorologists and oceanographers everywhere owe a debt to a polar scientist, Fridtjof Nansen.

During the drift of the Fram across the Arctic Ocean in

the mid-1890s (Nansen 1897), Nansen noticed that his

ship and the sea ice generally drifted to the right of the

wind. Ekman (1905) developed a mathematical explanation for these observations and thereby provided meteorology and oceanography with the first rigorous planetary

boundary layer model. Though the spiral in the wind

vector that Ekman’s model predicts is rarely seen, his

model is still a basis for current ABL modeling. Hence,

I will describe it in some detail here.

Start with eq 16, which are equations for the mean

horizontal wind vector. Assuming steady-state simplifies these, but the turbulence terms still make analytic

solution difficult. We, thus, need a closure assumption.

The Ekman solution is to assume that the turbulent transport can be modeled as a turbulent diffusion process

with a constant turbulent diffusivity K. That is, the surface stresses in the x and y directions, τx and τy, are

modeled as

τ ρ

∂

∂ x / =− = uw K Uz

(114a)

τ ρ

∂

∂ y / . =− = vw K Vz

(114b)

With these and the assumption of steady-state, eq 16

yield (e.g., Stull 1988, p. 210 f.; Garratt 1992, p. 42)

−−= fV V K U

z

( ) g

∂

∂

2

2 (115a)

fU U K V

z

( ) − = g

∂

∂

2

2 (115b)

23

103 CHN10

103 CEN10

Figure 20. Model predictions (from Andreas 1987) of CHN10 andCEN10 over snow-covered surfaces as a function of

the surface roughness parameter ξ (in cm) and the 10-m wind speed, U10. The arrows on the right show CDN10 for the

indicated ξ value.

a. CHN10. The data points are from measurements by

Kondo and Yamazawa (1986).

b. CEN10.

* Personal communication with J. Kondo, Professor of Meteorology, Geophysical Institute, Tohoku University, Sendai, Japan, 1986.
or

−−= − fV V K

U U

z

( ) ( )

g

g ∂

∂

2

2 (116a)

fU U K

V V

z

( ) ( ) − = −

g

g ∂

∂

2

2 (116b)

since Ug and Vg are independent of height.

If we define

S U U iV V ≡− + − ( )( ) g g (117)

we can combine eq 116 a and b into one equation in S

ifS

K

S

z = ∂∂

2

2 . (118)

The solution of this is easiest if we align the x axis with

the geostrophic wind, that is, Vg = 0. Then the boundary conditions at z = 0 are

U(0) = V(0) = 0 (119)

so

S(0) = – U. (120)

As z approaches infinity, the boundary conditions are

lim ( )

z

Uz U →∞= g (121a)

lim ( )

z

V z →∞= 0 (121b)

so

lim ( ) .

z

S z →∞= 0 (122)

With these boundary conditions, the solution of eq

118 is fairly straightforward. It is (e.g., Businger 1982;

Stull 1988, p. 211; Garratt 1992, p. 42)

Uz U z z ( ) [ exp( / )cos( / )] = −− g 1 δ δ (123a)

V z sgnf U z z ( ) exp( / )sin( / ). = − g δ δ (123b)

Here, sgnf is the sign of the Coriolis parameter: plus in

the Northern Hemisphere, minus in the Southern Hemisphere. And δ is a height scale

δ = 











2

2

K

f . (124)

Let us look at what eq 123 implies about processes

24

.

at the surface. From eq 114 and 123, we see that

τ ρ δ x

g / = K U

(125a)

τ ρ δ y

g / . = −K U sgnf (125b)

That is, the surface stress is to the right of the geostrophic wind in the Northern Hemisphere, as Nansen

observed; in the Southern Hemisphere, it is to the left.

Since by definition

ττ τ ρ = + ( ) x = 2

y

2 *

2 1 2/ u (126)

we also find from eq 125 another relation between the

constants

u

K U

*

2 g = 21 2/

. δ (127)

Using this in eq 124, we see that

δ = 21 2/. u

f U *

2

g

(128)

While solving the Ekman equations in the geostrophic wind frame is easiest, observers at the surface

know the direction of the surface stress (remember, in

an Ekman layer the surface wind vector is zero) better

than they know the direction of the geostrophic wind.

Hence, for me at least, it is easier to visualize the

Ekman winds in a coordinate system aligned with the

surface stress vector than in a coordinate system aligned

with the geostrophic wind. The following equations

accomplish this transformation of velocity components

in the geostrophic wind frame [denoted as (U,V)] to

components in the surface stress frame [denoted as

(U′,V′)]:

′ = + − U z U z sgnf V z () [ () / 2 1 2 ( )] (129a)

′ = − − V z sgnf U z V z () [ / 2 1 2 ( ) + ( )]. (129b)

Thus, substituting eq 123 into eq 129 yields the

Ekman solution in a reference frame aligned with the

surface stress

′ =

{ } −− − [ ]

− Uz G

zz z

( )

exp( / ) cos( / ) sin( / )

/ 2

1

1 2

δδ δ

(130a)

 

′ = −

{ } −− + [ ]

V z G sgnf −

zz z

( )

exp( / ) cos( / ) sin( / )

/ 2

1

1 2

δδ δ

(130b)

×

×
Southern

Hemisphere

Northern

Hemisphere

–0.8

–0.6

–0.4

–0.2

0

0.2 0.6 0.8 1.0

0.2

0.4

0.6

0.8

0.4

U'/G —V'

G

π

π

2

—π

3

2

—π

3

—π

2

—π

2

—π

3

—π

3

—π

4

—π

4

—π

6

—π

6

—π

12

—π

12

—π

18

—π

18

—π

36

—

π

36

45°

45°

Figure 21. Hodographs of the wind vector in

Ekman layers in the Northern and Southern Hemispheres. The dots with numbers nearby mark nondimensional heights, z/δ. The straight line at 45° in

each hemisphere shows the geostrophic wind.

Here, G is the magnitude of the geostrophic wind; and,

in this frame, the y component of the geostrophic wind

is no longer zero. Thus

GU U V = = ( ) ′ + ′ g g

2

g

2 1 2/ . (131)

Figure 21 shows plots of eq 130 in both the Northern

and Southern Hemispheres. The obvious feature in these

hodographs is the so-called Ekman spiral. The height of

the Ekman layer, hE, is commonly taken as

hE = π δ (132)

since π δis the lowest height at which the velocity vector has the same direction as the geostrophic wind (see

Fig. 21). From eq 128, we can estimate that hE is on the

order of 300 m.

We see in Figure 21 that the Ekman winds turn with

increasing height. In a Cartesian coordinate system with

positive angles counterclockwise from the x axis, the

turning angleα between the surface stress and the geostrophic wind is given by

tan lim ( )

( )

α = ′

′ = − z→∞

V z

U z sgnf (133)

or

α = –45° sgnf. (134)

The geostrophic wind makes a 45° angle with the surface stress in an Ekman layer.

That 45° turning angle is easy to remember; remembering the direction of the turning, however, is harder

since some of us deal with ABLs in both hemispheres.

Thus, I find it helpful to bring some of the sophistication of physics to boundary-layer meteorology through

the following right-hand rule.

Figure 22 presents another view of the Ekman solution. Notice, the longitudinal velocity component is dis25

–1.0 –0.5 0 0.5 1.0 0

5π

––


2

2π

3π

––


2

π

π

—


2

5π

––


2

2π

3π

––


2

z


—
 δ

z


—
 δ

π

π

—


2

0.5 1.0

V'/G U'/G

Northern


Hemisphere

Southern


Hemisphere

Figure 22. Nondimensional profiles of the longitudinal and

transverse velocity components for Ekman layers in the Northern and Southern Hemispheres. The x (or U′) axis is aligned

with the surface stress.

A Right-Hand Rule for the Ekman Layer

1. Point the fingers of the right hand in the direction of the surface stress.

2. Curl the fingers in the direction of f. (f is up in the Northern Hemisphere; down

in the Southern Hemisphere.)

3. The right thumb then points in the direction that the wind will turn with increasing height in an Ekman layer.
tinctly supergeostrophic; it has a bulge near z/δ = π/2

where U U ′ > ′

g . The transverse velocity component is

also a bit supergeostrophic just below z/δ = π. Notice,

too, for z > πδ = hE, the wind vectors are approximately

geostrophic.

The question now becomes: How realistic is the

Ekman solution; is the Ekman spiral, for example, ever

observed in nature? We made a lot of radiosounding profiles on Ice Station Weddell (Claffey et al. 1994); it would

be reasonable to look through these for evidence of an

Ekman layer. But remember, the Ekman layer is derived

assuming neutral stratification. On ISW, stable stratification was the norm. Figure 23 does, however, show one

ISW radiosounding that found a nearly neutral layer

200 m deep. In this layer, the wind clearly turns counterclockwise with height, as Figure 21 suggests it should in

the Southern Hemisphere. But since the upper-level wind

components have different magnitudes, the turning

angle is not the 45° predicted for an Ekman layer; it is

actually closer to 30°. Nevertheless, there are enough

similarities between Figure 23 and the Ekman model—

the turning, the height over which the turning occurs—

for us to see that the essence of the model must be correct. For an even better demonstration of the veracity of

the Ekman solution, see McPhee and Martinson (1994).

They observed a fairly well behaved Ekman spiral in

the oceanic boundary layer under the sea ice at ISW.

THERMAL WIND

Many phenomena are present in the atmosphere that

void the assumptions on which Ekman based his model.

Thus, the paucity of observations of a true atmospheric

Ekman layer is not surprising. The thermal wind is one

such phenomenon that complicates the analytical description of the ABL.

26

–2 

–12 –10 –8 –6 –4 

0

0

200

400

600

800

2 4 6 8 10

Temperature (°C)

Wind Vector (m/s)

U Component
 IS051500

V Component


Temperature

Height (m)

Figure 23. The 00 GMT radiosounding from

Ice Station Weddell on 15 May 1992. The coordinate system is such that the surface wind has

no transverse component (i.e., the coordinate

system is aligned with the surface stress).

Remember, eq 15 defined the geostrophic wind components. Suppose, rather than being constant as Ekman

assumed, the geostrophic wind varies with height. On

using the ideal gas law, eq 5, and taking the z derivative

in eq 15, we can show (e.g., Arya and Wyngaard 1975;

Sorbjan 1989, p. 177 f.)

∂

∂

∂

∂

∂

∂

U g

f T

T

y

g

f

T

y

g

P P z = − = − ln (135a)

∂

∂

∂

∂

∂

∂

V g

f T

T

x

g

f

T

x

g

P P z = = ln , (135b)

where the subscript P indicates that the derivatives are

along surfaces of constant pressure. If these terms are

not zero, meteorologists say that there is a thermal wind

or geostrophic shear; oceanographers prefer to describe

this effect as baroclinicity. The terms are all equivalent

(Arya and Wyngaard 1975).

For the terms in eq 135 to be nonzero, we see that a

horizontal temperature gradient must exist. In other

words, a surface that is not horizontally homogeneous

in temperature will induce thermal wind effects. How

do these effects manifest?

By integrating eq 135 from the surface to height z,

we get the following

Uz U g

f

T

y

dz

z

g g () () ln = − ′ ∫ 0

0

∂

∂ (136a)

Vz V g

f

T

x

dz

z

g g () () ln

= + ′ ∫ 0

0

∂

∂ (136b)
27

where, for convenience, I have dropped the subscript P

that appeared in eq 135. For illustrative purposes, suppose

∂

∂

ln T

y = m (137a)

∂

∂

ln T

x = n (137b)

where m and n are constants (e.g., Sorbjan 1989, p. 179).

Then eq 136 simply become

Ug(z) = Ug(0) – mgz/f (138a)

Vg(z) = Vg(0) + ngz/f . (138b)

That is, in this example, the baroclinicity causes the

geostrophic wind components to increase or decrease

with height depending on the direction of the gradient

in surface temperature.

Again, it is hard for me to remember the direction in

which the geostrophic wind will change in response to

a surface temperature gradient. I, thus, developed another right-hand rule.

For practice with this rule, suppose you are over sea

ice in the Weddell Sea (i.e., in the Southern Hemisphere). There is open water to the east; thus, the positive temperature gradient points east. Suppose the surface geostrophic wind is blowing due north. According

to the above rule, Vg must increase with height to the

south. That is, Vg(z) will get smaller with height and

may even turn negative (i.e., the wind may blow

toward the south at some height).

Sorbjan (1989, p. 179 f.) showed how to incorporate simple thermal wind parameterizations like eq 138

into the Ekman solutions. The result is a distorted

Ekman spiral whose shape depends on the relative magnitudes and directions of the surface geostrophic wind

and the horizontal temperature gradient. Figure 24

shows some similarly modified Ekman spirals from

Blackadar (1963). Clearly, the presence of a thermal

wind can hide the Ekman spiral.

ROSSBY NUMBER SIMILARITY

As I have hinted, the Ekman solution has shortcomings. In particular, it predicts a low-speed layer at the

surface that is fairly thick (see Fig. 22). In reality, the

wind speed increases rapidly above the surface; at a

height of only 1 m, it is already an appreciable fraction

of the geostrophic wind speed (e.g., see Fig. 6). Conversely, the surface-layer profiles I described in earlier

sections are not accurate for heights above, roughly,

1

/

10 the height of the ABL. Blackadar and Tennekes

(1968), however, demonstrated that, by matching

surface-layer and Ekman-layer solutions in a region

where they overlap, it is possible to mathematically

describe the entire ABL with an extended Ekman solution that also treats stability effects.

Figure 24. Northern Hemisphere hodographs of Ekman

spirals in the presence of a constant thermal wind. The

thermal wind effect is modeled as in eq 138; the magnitude

of the plotted thermal wind vector is roughly mgz/f (or

ngz/f) for z of 800–900 m. The surface-level geostrophic

wind is labeled Ug(0). The numbers near the dots indicate

the height in meters (after Blackadar 1963; see also Brown

1974).

 Ug (0)

 Ug (0)

 Ug (0)

 Ug (0)

Thermal Wind

Thermal Wind

Thermal Wind

Thermal Wind

18.4°

26.3°

26.8°

17.7°

1.2

2.4

5.2

11

23 49

104 134172

220

282

363 466

769 598

1267 987

1.2 2.4

5.2

11

23 49 104 134171220

283

363

466

599

769

987

1268

1.2

1.2

2.4

2.4

5.2

5.2

11

11

23

23

49

49

104

104

134

134

172

172

220

220

599

599

283

282

363

363

466

466

769

769

A Right-Hand Rule for the Thermal Wind

1. Point the fingers of the right hand in the direction of f. (f is up in the Northern

Hemisphere; down in the Southern Hemisphere.)

2. Curl the fingers in the direction of the positive temperature gradient.

3. The right thumb then points in the direction in which the geostrophic wind

increases with height.
We can write the surface-layer solutions eq 97, 99

and 100 as

U z

u k

z z

( ) ln /

*

= [ ] ( ) − m( ) 1

0 ψ ζ (139a)

V z

u

( )

*

= 0 (139b)

Θ( ) ln / z T

t k

z z − = [ ] ( ) − ( ) s

*

T h

1 ψ ζ (140)

Qz Q

q k

z z

( ) ln / . − = [ ] ( ) − ( ) s

*

Q h

1 ψ ζ (141)

Equation 139b is true because Monin-Obukhov similarity aligns the x axis with the mean wind (the U component) and assumes no turning of the wind in the surface

layer.

Likewise, on seeing eq 116 and considering the scales

involved, we can show that the profiles in the Ekman

layer should obey defect laws (Blackadar and Tennekes

1968; Yamada 1976; Tennekes 1982; Brutsaert 1982, p.

72 ff.) such that

Uz U

u

F z hh L ( ) ˆ (/, / ) − =

*

U (142a)

Vz V

u

sgnf F z h h L ( ) ˆ (/, / ) − =

*

V (142b)

Θ Θ

Θ

( ) ˆ (/, / ) z

t

F z hh L − =

*

(143)

Qz Q

q

F z hh L ( ) ˆ ( / , / ). − =

*

Q (144)

In these, ˆ, ˆ, UV Q Θˆ ˆ and are, respectively, longitudinal

velocity, transverse velocity, potential temperature and

specific humidity scales that I will explain shortly.

If the scaling is accurate, the functions FU, FV, FΘ

and FQ should depend only on the dimensionless ratios

z/h and h/L, where h is now the height of the ABL and L

is again the Obukhov length. In particular, these functions should not depend on the surface Rossby number

G fz / 0 (Blackadar and Tennekes 1968, Tennekes 1973,

Hess 1992). Thus, these and subsequent arguments

are generally referred to as Rossby number similarity

(Blackadar and Tennekes 1968, Hess 1973, Tennekes

1973).

Although to continue the similarity arguments we do

not need a formal definition of h, this is a good time to

digress on its meaning. As I explained, the presence of

turbulence is essential to the definition of the ABL. Consequently, h should denote the surface that separates the

turbulent ABL from the (generally) nonturbulent upper

air. Rarely, however, do we have profiles of turbulence

intensities through the ABL, let alone into the

upper air. Thus, it is common to use a surrogate, such as

the height of the temperature inversion zi, as an estimate

for h (e.g., Kaimal et al. 1976). While zi is a good estimate for h in the convective boundary layer, in the stable boundary layer often found over polar marine surfaces, zi can overestimate h (e.g., Mahrt et al. 1979).

Hanna (1969) reviewed a host of empirical methods

for estimating h from surrogate information in both

stable and unstable conditions. Mahrt et al. (1979) concentrated on estimating h in the stable ABL. Their main

conclusion was that the height of the core of the lowlevel jet (zj) that is common in stable boundary layers is

a good estimator of h. In turn, the height of the turbulent

layer predicted by a Richardson number criterion, zRi, is

well correlated with zj. Here, a bulk gradient Richardson

number is computed from

Ri z g z

z

z T

U Vz ( (z)

s

2 ) ( )

( )

( ) = −

Θ +

Θ

2 (145)

and zRi is defined as the height at which

Ri(zRi) ≥ 0.4 (146)

where 0.4 is the critical Richardson number (see also

Heinemann and Rose 1990).

On Ice Station Weddell, our radiosondes often found

low-level jets (Andreas et al. 1993a, Claffey et al. 1994,

Andreas et al. 1995). If we assume, as Mahrt et al. (1979)

did, that zj is an estimate for h, Figure 25 shows that the

inversion height is usually a poor estimate of h in the

stable ABL. In defining zi in this figure, we used Kahl’s

(1990) definition. As such, zi could be called the top of

the inversion layer. In unstable conditions, on the other

hand, zi is often taken as the base of the inversion (e.g.,

Kaimal et al. 1976). The conflicting definitions arise

because in a stable ABL the inversion base is often at the

surface (Kahl 1990, Serreze et al. 1992, Claffey et al.

1994).

Figure 26, in contrast, suggests that zRi, which

approximates the height at which turbulence ceases, is a

fair estimate of h—if we assume that h corresponds to

zj, the height of the jet core. Thus, we see that over sea

ice in late fall and winter, the ABL is generally quite

shallow; in Figure 26, zj is usually between 50 and 300 m.

Overland and Guest (1991) and Guest and Davidson (1994)

offered additional insight into how the radiation budget,

especially during the winter, dictates the thermal and turbulent structure of the ABL and, therefore, its height.

28
Now back to Rossby number similarity.

Since eq 139–141 and eq 142–144 are two sets of

expressions for the same quantities, there should exist a

height interval in which both sets are true. That is, as

z z/ 0 → ∞ (and zz zz / ,/ T Q →∞ →∞ ) and as

z h/ → 0, the two formal descriptions of the atmospheric

profiles should be simultaneously accurate (Blackadar

and Tennekes 1968, Tennekes 1982, Hess 1992). In these

asymptotic limits, eq 139–141 and eq 142–144 require

that

U z

u k

zz zL

F z hh L U

u

( ) ln / /

/,/ ˆ

*

m

U

*

= [ ] ( ) − ( )

= ( ) +

1

0 ψ

(147a)

V z

u

sgnf F z h h L V

u

( ) /,/ ˆ

*

V

*

= = 0 ( ) +

Θ

Θ

Θ

( ) ln / /

/,/ ˆ

z

t k

zz zL T

t

F z hh L

t

*

T h s

*

*

= [ ] ( ) − ( ) +

= ( ) +

1

ψ

(148)

Q z

t k

zz zL Q

q

F z hh L Q

q

( ) ln / /

/,/ ˆ.

*

Q h s

*

Q

*

= [ ] ( ) − ( ) +

= ( ) +

1

ψ

(149)

It is straightforward to rearrange and rewrite eq 147–

149 as

A hz k U

u

kF z h z h

( ) ln / ˆ

/ , ln /

μ

μ ψμ

≡ ( ) −

= ( ) + ( ) − ( )

0

*

U m

(150a)

B k V

u

( ) sgnf k F z h

ˆ μ μ ≡ / , − = ( ) *

V (150b)

C hz

k T

t

kF z h z h

( ) ln /

ˆ

/ , ln /

μ

μ ψμ

≡ ( ) + ( ) −

= ( ) + ( ) − ( )

T

s

*

h

Θ

Θ

(151)

 

D hz kQ Q

q

kF z h z h

( ) ln /

ˆ

/ , ln / .

μ

μ ψμ

≡ ( ) + ( ) −

= ( ) + ( ) − ( )

Q

s

*

Q h

(152)

In each of these, the middle term does not depend

on z; consequently, although there is a z in the right

term, this term must be independent of z also. Likewise, because the right term depends explicitly on the

new stability parameter μ = h/L (first introduced by

Kazanski and Monin 1960), the middle term must also.

Therefore, Rossby number similarity lets us define the

new profile functions A, B, C and D that depend only

on μ. These functions are often called resistance laws

because they show how the bulk boundary layer

parameters ˆ, ˆ, ˆ, , ˆ UV T QQ θ , s s and h are related to the surface properties u*, t

*, q*, z0, zT, zQ and L. In a way, A,

B, C and D are thus comparable to the bulk transfer

coefficients CDr, CHr and CEr.

29

0

200

400

600

800

Height of Jet Core, zj (m)

0 200 400 600 800 1000 1200

1:1

Figure 25. Observations of the height of the core of the low-level

jet (zj) and of the corresponding height of the inversion (zi) on Ice

Station Weddell (Claffey et al. 1994, Andreas et al. 1995).

0

500

400

300

200

100

0

100 200 300 400 500

zRi (m)

1:1

zj (m)

Figure 26. Observations of the height of the

jet core (zj), a surrogate for h, versus the estimated height of the turbulent layer from a

Richardson number criterion (zRi) (see eq 145

and 146) (Andreas et al. 1995).

(147b)

Height of Inversion, zi (m)
30

–40 –20 0

0

0.2

0.4

0.6

08.

1.0

20 40 –40 –20 0 20 40

—


z


h

 ^ U – U


———
 u

*

 ^ V – V


———
 u

*

–150 ≤ h/L ≤ –120

–40 –20 0

0

0.2

0.4

0.6

08.

1.0

20 40 –40 –20 0 20 40

—


z


h

 ^ U – U


———
 u

*

 ^ V – V


———
 u

*

0 ≤ h/L ≤ 30

–40 –20 0

0

0.2

0.4

0.6

08.

1.0

20 40 –40 –20 0 20 40

—


z


h

 ^ U – U


———
 u

*

 ^ V – V


———
 u

*

180 ≤ h/L ≤ 210

Figure 27. Evaluations of FU and FV for very unstable (–150 ≤ μ ≤ –120), weakly stable (0 ≤ μ ≤ 30), and very stable

(180 ≤ μ ≤ 210) conditions (after Yamada 1976).
As with the φm and φh or ψm and ψh functions of

Monin-Obukhov similarity, Rossby number similarity

essentially leads us on a quest to find the nondimensional, universal functions A, B, C and D. Part of this

quest requires defining the scales ˆ, ˆ, UV Q Θˆ ˆ and . In

analogy with the Ekman equations (i.e., eq 116), we

might expect ˆ , ˆ , ˆ ( ) ˆ U U V V h Q Qh = == = ( ) g g Θ Θ and .

But in light of frequent baroclinicity (a thermal wind),

Arya and Wyngaard (1975) found that more robust values for Uˆ and Vˆ derive from averaging the geostrophic

wind from the surface to h. Likewise, Yamada (1976)

found that Θˆ is best taken as the height-averaged potential temperature. By extension, I define Qˆ analogously.

Mathematically, we define

χ χ = ∫

1

0

h

z dz

h

( ) (153)

where χ is U(z), V(z), Ug(z), Vg(z), Θ(z) or Q(z). Thus

(Arya and Wyngaard 1975, Yamada 1976)

UU U ˆ = = g (154a)

VV V ˆ * u

f h = =− g

2

(154b)

Θ Θ ˆ = (155)

ˆ

Q Q = . (156)

After defining Uˆ , Vˆ and Θˆ as above, Yamada

(1976) made the classic attempt to find FU, FV and FΘ

and, thus, A, B and C. Figures 27 and 28 show the FU,

FV and FΘ functions that he deduced from plots of data

from the Wangara experiment in Hay, Australia. (Brutsaert [1982] offered one of the few discussions of FQ

and D.)

Figures 27 and 28 emphasize why it is appropriate

to refer to eq 150–152 as resistance laws. These figures

show that in very unstable conditions FU, FV and FΘ

are all near zero. This means that the ABL is well mixed;

the wind vector and temperature at h are very near the

values at the surface. That is, there is efficient coupling

between the upper air and the surface. In very stable

conditions, in contrast, U(z), V(z) and Θ(z) vary strongly

with height. In other words, in stable conditions the

transfer of properties from the surface to the top of the

ABL, or vice versa, is inefficient.

From his analysis, Yamada (1976) was able to estimate A, B and C. Many others since Yamada have

suggested alternate formulations (e.g., Arya 1977,

Zilitinkevich 1989a, b). I do not have room here to review all of these; thus, since Yamada’s functions are

the most frequently cited, I will base the remainder of

this section on them. He found

A(μ) = 10.0 – 8.145 (1 – 0.008376 μ)–1/3

for μ ≤ 0 (157a)

= 1.855 – 0.380 μ

for 0 ≤ μ ≤ 35 (157b)

= –2.94 (μ – 19.94)1/2

for 35 ≤ μ (157c)

B(μ) = 3.020 (1 – 3.290 μ)–1/3

for μ ≤ 0 (158a)

31

Figure 28. Evaluations of FΘ for unstable (–60 ≤ μ ≤ –30), weakly stable (0 ≤ μ ≤ 30),

and moderately stable (60 ≤ μ ≤ 90) conditions (after Yamada 1976).

–40 –20 0 20 40 –40 –20 0 20 40 –40 –20 0 20 40

0

0.2

0.4

0.6

0.8

1.0

z


—

h

θ – θ ^

–––– t

*

–60 ≤ h/L ≤ –30 0 ≤ h/L ≤ 30 60 ≤ h/L ≤ 90
= 3.020 + 0.300 μ

for 0 ≤ μ ≤ 35 (158b)

= 2.85 (μ – 12.47)1/2

for 35 ≤ μ (158c)

 C(μ) = 12.0 – 8.335 (1.0 – 0.03106 μ)

–1/3

 for μ ≤ 0 (159a)

= 3.665 – 0.829 μ

for 0 ≤ μ ≤ 18 (159b)

= –4.32 (μ – 11.21)1/2

for 18 ≤ μ . (159c)

Figure 29 shows plots of these.

Again, we can interpret the behavior of the curves

in Figure 29 as an indication of the resistance the ABL

offers to the vertical transfer of properties. In unstable

conditions (μ < 0), A, B and C are relatively small. From

eq 150 and 151 we thus infer that the ABL is well mixed.

Equation 150b shows that there is little turning in the

wind between the top of the surface layer and the top

of the ABL. And eq 150a and 151 show a nearly se

mi-logarithmic relationship between height and Uˆ

nd Θˆ —the same basic relation found in the surface lay

r. Thus, again, conditions extant at the top of the surf

ce layer continue, with little change, high into the ABL.

In the stable half of Figure 29 (μ > 0), however, we

see the opposite behavior. A, B and C deviate increasingly from zero as the stability increases. Equation

150b thus shows that V becomes significant. In

other words, the wind vector turns. Likewise, eq 150a

and 151 suggest that we cannot easily infer the profiles

of U(z) and Θ(z) higher in the ABL from the values at

the top of the surface layer.

Equation 47 defined the geostrophic drag coefficient

Cg as u*/G. To be compatible with the preferred

Ekman-layer scales, rather than defining G as in eq 131,

here I use

GU V = + ( ) ˆ ˆ . / 2 2 1 2(160)

Thus, using eq 150, we can express Cg in terms of the A

and B functions

 

C u

G

k

hz A B

g ≡ =

[ ] ( ) − ( ) + ( ) 











*

/

ln /

.

0

2 2 1 2

μ μ

(161)

Likewise, eq 133 defined the turning angle α. Our

Rossby number similarity solutions, however, provide

another definition of α that depends on stability

tan ˆ

ˆ α =

V

U

(162)

where α is now the turning angle between the surfacelayer wind and the vertically integrated geostrophic

wind (see eq 154). Again, from eq 150, we can express

162 as

tan ( )

ln( / ) ( )

α . μ

μ = −−

sgnf B

hz A 0

(163)

Figure 30 shows plots of Cg and α as functions of

stability when Yamada’s (1976) functions are used for

32

40

20

0

–150 –100 –50 0 50 100 150

–20

–40

–60

B

A

C

μ = h/L

After Yamada (1976)

Figure 29. Yamada’s (1976) resistance laws for the longitudinal (A) and transverse (B) velocity components and for the

potential temperature (C) as functions of the stability parameter μ = h/L (see eq 157–159).
A and B. This figure again points out the resistance-law

interpretation of A and B. In unstable conditions, Cg is

large and α is near zero. As a result, the surface stress

responds strongly to geostrophic forcing. And because

of this strong coupling, the ABL is well mixed so the

turning of the wind vector with height is small. On the

other hand, in very stable conditions, Cg is smaller and

α is larger in magnitude. That is, because the stratification suppresses the vertical turbulent exchange, the surface stress is less responsive to the geostrophic forcing.

Also with less efficient vertical coupling, the wind vector can rotate more with height because the upper-level

winds have little opportunity to mix their longitudinal

momentum down into the ABL.

Brown (1981, 1990), Overland and Davidson (1992)

and Overland and Colony (1994) showed some alternative formulations for Cg and α over sea ice and also

presented Cg and α data that came from Arctic sea ice

regions. Although their data and the computed parameters in Figure 30 may differ in detail, the trends and

the general magnitudes of the quantities agree.

As we did in the atmospheric surface layer, we can

also write an expression for the geostrophic drag coefficient at neutral stability. From eq 161, this is

C k

hz A B

gN =

{ } [ ] ln / ( ) − ( ) + ( )

. /

0

2 2 1 2

0 0

(164)

Yamada’s (1976) functions (eq 157–158) suggest that

A(0) = 1.855 and B(0) = 3.020. For comparison,

Zilitinkevich (1989b) recommended A(0) = 1.7 and

B(0) = 4.5 after an extensive review.

Earlier, I showed that the surface-layer neutralstability drag coefficient at 10 m (CDN10) is monotonically related to z0. We can use that relation, eq 105,

here to relate CgN and CDN10. Equation 164 thus becomes

C k

kC h A B

gN

DN10

–1/2

=

[ ] + ( ) − ( ) + ( ) 









 ln /

/

10 0 0

2 2

1 2

(165)

where h must be in meters.

We see two effects here. As CDN10 increases, so does

CgN. But as the ABL gets deeper (i.e., as h increases),

CgN decreases if CDN10 remains constant. This latter

effect is analogous to what we see in the surface-layer

bulk-transfer coefficients. Equations 105–107 show that

CDNr, CHNr and CENr all decrease as the reference height

r increases. The reason is that the forcing quantities

Ur – 0, Ts – Θr and Qs – Qr (see eq 91–93) all increase

in magnitude with r; thus, the corresponding transfer

coefficient can decrease proportionately.

From eq 164 we can also solve for z0 in terms of

CgN

z h A kC B 0

2 2 1 2

0 0 ef = −+ − ( ) gN





 















 exp ( ) ( / ) ( ) .

/

(166)

Since this roughness length derives from Ekman layer

parameters, it is the effective roughness length (Fiedler

and Panofsky 1972, Arya 1975, Mason 1988, Claussen

1990) in contrast to the local roughness length, which

33

Figure 30. Geostrophic drag coefficient (Cg) and the turning angle (α) as functions of

stability when Yamada’s (1976) A and B functions (eq 157–158) are used in eq 161 and

163, respectively. To make this calculation, I used h/z0 = 5.0 × 105. The α values are for the

Northern Hemisphere; Southern Hemisphere values have the opposite sign.

0

1.0

2.0

3.0

4.0

5.0

–150 –100 –50 50 100 150 0

μ = h/L

100 × Cg Northern


Hemisphere

–50

–40

–30

–20

–10

0

α (°)

h/z0 = 5.0 × 105
is the correct surface-layer interpretation of z0. Thus,

we see that—as in the surface layer—finding the geostrophic drag coefficient at neutral stability is tantamount

to finding the effective roughness length, and vice

versa.

SUMMARY

Several good problems in the atmospheric boundary

layer over sea ice still exist. I have highlighted a few of

these. One is to investigate turbulent exchange and the

turbulence statistics in the atmospheric surface layer in

stable conditions. A related study is to narrow the uncertainty in the nondimensional gradient functions, φm and

φh, for stable conditions (see Fig. 5). Efforts here would

benefit not only polar scientists but boundary-layer

meteorologists everywhere. The stable boundary layer

is seldom in quasi-steady-state at lower latitudes; consequently, obtaining robust statistics in stable stratification is more difficult here than it is in the polar regions.

Another good problem is to devise a new parameterization for CDN10 in terms of some measurable surface

properties that—preferably—can be sensed remotely.

The relation from Banke et al. (1980) (eq 111) is quite

suggestive but evidently impractical in light of my recent work on how drifting snow can rapidly alter CDN10.

A related issue is obtaining good measurements of

CHN10 and CEN10 over sea ice. These are conspicuous

by their absence from the literature. The primary reason

for the scanty number of observations is that the forcing

terms Ts – Θr and Qs – Qr necessary to compute CHNr

and CENr are small over sea ice and notoriously difficult to measure. It may, thus, be time to revise our typical

measurement practices or to rethink the entire method of

parameterizing sensible and latent heat fluxes over sea ice

in terms of bulk transfer coefficients.

Because the geostrophic drag coefficient is related

monotonically to the effective roughness length z0ef , it

should be possible to bootstrap information about geostrophic drag from the many observations of z0 and

CDN10 over sea ice. The key step is inferring z0ef from

z0. Several have thought about this problem, but, to my

knowledge, only Arya (1975) has attempted to include

the unique topographic features of sea ice in the transformation. In light of the recent flurry of papers on area

averaging (e.g., Mason 1988; Claussen 1990, 1991;

Vihma and Savijärvi 1991; Blyth et al. 1993; Moore et

al. 1993), this problem might be ripe for a productive

revisit.

LITERATURE CITED

Anderson, R.J. (1987) Wind stress measurements over

rough ice during the 1984 Marginal Ice Zone Experiment. Journal of Geophysical Research, 92: 6933–6941.

Andreas, E.L (1987) A theory for the scalar roughness

and the scalar transfer coefficients over snow and sea

ice. Boundary-Layer Meteorology, 38: 159–184.

Andreas, E.L(1988) Estimating averaging times for point

and path-averaged measurements of turbulence spectra.

Journal of Applied Meteorology, 27: 295–304.

Andreas, E.L (1995a) Air-ice drag coefficients in the

western Weddell Sea: 2. A model based on form drag

and drifting snow. Journal of Geophysical Research, 100:

4833–4843.

Andreas, E.L (1995b) A physically based model of the

form drag associated with sastrugi. USA Cold Regions

Research and Engineering Laboratory, CRREL Report

95-16.

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 and B. Murphy (1986) Bulk transfer coefficients for heat and momentum over leads and polynyas. Journal of Physical Oceanography, 16: 1875–1883.

Andreas, E.L, W.B. Tucker III and S.F. Ackley (1984)

Atmospheric boundary-layer modification, drag coefficient and surface heat flux in the Antarctic marginal ice

zone. Journal of Geophysical Research, 89: 649–661.

Andreas, E.L, K.J. Claffey and A.P. Makshtas (1993a)

Low-level jets and inversions on Ice Station Weddell 1.

Antarctic Journal of the United States, 28(5): 274–276.

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

(1993b) Roughness of Weddell Sea ice and estimates of

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

Andreas, E.L, K.J. Claffey and A.P. Makshtas (1995)

Low-level atmospheric jets over the western Weddell Sea.

In Preprint Volume, Fourth Conference on Polar Meteorology and Oceanography, Dallas, Texas, 15–20 January 1995. American Meteorological Society, Boston,

p. 252–257.

Anonymous (1992) U.S. and Russian scientists complete

historic Weddell Sea investigation. Antarctic Journal of

the United States, 27(4): 8–11.

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.

Arya, S.P.S. (1977) Suggested revisions to certain boundary layer parameterization schemes used in atmospheric

circulation models. Monthly Weather Review, 105: 215–

227.

Arya, S.P.S. (1988) Introduction to Micrometeorology.

San Diego: Academic Press.

34
Arya, S.P.S. and J.C. Wyngaard (1975) Effect of baroclinicity on wind profiles and the geostrophic drag law

for the convective planetary boundary layer. Journal of

the Atmospheric Sciences, 32: 767–778.

Augstein, E., N. Bagriantsev and H.W. Schenke (1991)

The expedition ANTARKTIS VIII/1–2, 1989 with the

Winter Weddell Gyre Study of the research vessels

“Polarstern” and “Akademic Fedorov.” Berichte zur

Polarforschung, 84: 1–129.

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.

Barry, P.J. and R.E. Munn (1967) Use of radioactive

tracers in studying mass transfer in the atmospheric

boundary layer. Physics of Fluids, 10: 5263–5266.

Beljaars, A.C.M. and A.A.M. Holtslag (1991) Flux

parameterization over land surfaces for atmospheric

models. Journal of Applied Meteorology, 30: 327–341.

Blackadar, A.K.(1963) The vertical distribution of wind

in a baroclinic adiabatic atmospheric boundary layer.

Presented at 211th national meeting of the American

Meteorological Society, 21–24 January 1963, New York,

N.Y. (Bulletin of the American Meteorological Society,

43: 661–674.)

Blackadar, A.K. and H. Tennekes (1968) Asymptotic

similarity in neutral barotropic planetary boundary

layers. Journal of the Atmospheric Sciences, 25: 1015–

1020.

Blyth, E.M., A.J. Dolman and N. Wood (1993) Effective resistance to sensible- and latent-heat flux in heterogeneous terrain. Quarterly Journal of the Royal Meteorological Society, 119: 423–442.

Brown, R.A. (1974) Analytical Methods in Planetary

Boundary Layer Modelling. New York: John Wiley and

Sons.

Brown, R.A. (1981) Modeling the geostrophic drag coefficient for AIDJEX. Journal of Geophysical Research,

86: 1989–1994.

Brown, R.A. (1986) The planetary boundary layer in the

marginal ice zone. MIZEX Bulletin VII, p. 41–54. (USA

Cold Regions Research and Engineering Laboratory,

Special Report 86-3.)

Brown, R.A. (1990) Meteorology. In Polar Oceanography (W.O. Smith, Jr., Ed.). San Diego: Academic Press,

p. 1–46.

Brutsaert, W. (1975a) A theory for local evaporation

(or heat transfer) from rough and smooth surfaces at

ground level. Water Resources Research, 11: 543–550.

Brutsaert, W. (1975b) The roughness length for water

vapor, sensible heat and other scalars. Journal of the

Atmospheric Sciences, 32: 2028–2031.

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

D. Reidel.

Busch, N.E. (1973) On the mechanics of atmospheric

turbulence. In Workshop on Micrometeorology (D.A.

Haugen, Ed.). Boston: American Meteorological Society, p. 1–65.

Businger, J.A. (1966) Transfer of momentum and heat in

the planetary boundary layer. In Proceedings of the Symposium on the Arctic Heat Budget and Atmospheric Circulation (J.O. Fletcher, Ed.). Santa Monica, California:

Rand Corporation, RM–5233-NSF, p. 305–332.

Businger, J.A. (1973) Turbulent transfer in the atmospheric surface layer. In Workshop on Micrometeorology

(D.A. Haugen, Ed.). Boston: American Meteorological

Society, p. 67–100.

Businger, J.A. (1982) Equations and concepts. In Atmospheric Turbulence and Air Pollution Modelling (F.T.M.

Nieuwstadt and H. van Dop, Ed.). Dordrecht: D. Reidel,

p. 1–36.

Businger, J.A. (1988) A note on the Businger-Dyer profiles. Boundary-Layer Meteorology, 42: 145–151.

Businger, J.A. and A.M. Yaglom (1971) Introduction

to Obukhov’s paper on “Turbulence in an atmosphere with

a non-uniform temperature.” Boundary-Layer Meteorology,

2: 3–6.

Businger, J.A., J.C. Wyngaard, Y. Izumi and E.F.

Bradley (1971) Flux-profile relationships in the atmospheric surface layer. Journal of the Atmospheric Sciences,

28: 181–189.

Claffey, K.J., E.L Andreas and A.P. Makshtas (1994)

Upper-air data collected on Ice Station Weddell. USA

Cold Regions Research and Engineering Laboratory,

Special Report 94-25.

Claussen, M. (1990) Area-averaging of surface fluxes in

a neutrally stratified, horizontally inhomogeneous atmospheric boundary layer. Atmospheric Environment, 24A:

1349–1360.

Claussen, M. (1991) Estimation of areally-averaged surface fluxes. Boundary-Layer Meteorology, 54: 387–410.

Deardorff, J.W. (1968) Dependence of air-sea transfer

coefficients on bulk stability. Journal of Geophysical

Research, 73: 2549–2557.

Dyer, A.J. (1974) A review of flux-profile relationships.

Boundary-Layer Meteorology, 7: 363–372.

Dyer, A.J. and E.F. Bradley (1982) An alternative analysis of flux-gradient relationships at the 1976 ITCE.Boundary-Layer Meteorology, 22: 3–19.

Dyer, A.J. and B.B. Hicks (1970) Flux-gradient relationships in the constant flux layer. Quarterly Journal of the

Royal Meteorological Society, 96: 715–721.

Ekman, V.W. (1905) On the influence of the Earth’s rotation on ocean-currents. Arkiv für Matematik, Astronomi

och Fysik, 2: 1–53.

Fairall, C.W. and S.E. Larsen (1986) Inertial-dissipation

35
methods and turbulent fluxes at the air/ocean interface.

Boundary-Layer Meteorology, 34: 287–301.

Fairall, C.W. and R. Markson (1987) Mesoscale variations in surface stress, heat fluxes and drag coefficient in

the marginal ice zone during the 1983 Marginal Ice Zone

Experiment. Journal of Geophysical Research, 92: 6921–

6932.

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.

Garratt, J.R. (1992) The Atmospheric Boundary Layer.

Cambridge, England: Cambridge University Press.

Guest, P.S. and K.L. Davidson (1987) The effect of observed ice conditions on the drag coefficient in the summer East Greenland Sea marginal ice zone. Journal of

Geophysical Research, 92: 6943–6954.

Guest, P.S. and K.L. Davidson (1991) The aerodynamic

roughness of different types of sea ice. Journal of Geophysical Research, 96: 4709–4721.

Guest, P.S. and K.L. Davidson (1994) Factors affecting

variations of snow surface temperature and air temperature over sea ice in winter. In The Polar Oceans and

Their Role in Shaping the Global Environment (O.M.

Johannessen, R.D. Muench and J.E. Overland, Ed.). Washington, D.C.: American Geophysical Union, p. 435–442.

Hanna, S.R.(1969) The thickness of the planetary boundary layer. Atmospheric Environment, 3: 519–536.

Harr, M.E. (1977) Mechanics of Particulate Media: A

Probabilistic Approach. New York: McGraw-Hill.

Haugen, D.A. (Ed.) (1973) Workshop on Micrometeorology. Boston: American Meteorological Society.

Haugen, D.A., J.C. Kaimal and E.F. Bradley (1971) An

experimental study of Reynolds stress and heat flux in

the atmospheric surface layer. Quarterly Journal of the

Royal Meteorological Society, 97: 168–180.

Heinemann, G. and L. Rose (1990) Surface energy balance, parameterizations of boundary-layer heights and

the application of resistance laws near an Antarctic ice

shelf front. Boundary-Layer Meteorology, 51: 123–158.

Hess, G.D. (1973) On Rossby-number similarity theory

for a baroclinic planetary boundary layer. Journal of the

Atmospheric Sciences, 30: 1722–1723.

Hess, G.D. (1992) Observations and scaling of the atmospheric boundary layer. Australian Meteorological Magazine, 41: 79–99.

Hibler, W.D. III (1979) A dynamic thermodynamic sea

ice model. Journal of Physical Oceanography, 9: 815–

846.

Hicks, B.B. (1976) Wind profile relationships from the

‘Wangara’ experiment. Quarterly Journal of the Royal

Meteorological Society, 102: 535–551.

Hicks, B.B. and H.C. Martin (1972) Atmospheric turbulent fluxes over snow. Boundary-Layer Meteorology, 2:

496–502.

Hill, R.J. (1989) Implications of Monin-Obukhov similarity theory for scalar quantities. Journal of the Atmospheric Sciences, 46: 2236–2244.

Hinze, J.O. (1975) Turbulence. New York: McGraw-Hill,

2nd edition.

Högström, U. (1988) Non-dimensional wind and temperature profiles in the atmospheric surface layer: A reevaluation. Boundary-Layer Meteorology, 42: 55–78.

Holtslag, A.A.M. and H.A.R. de Bruin (1988) Applied

modeling of the nighttime surface energy balance over

land. Journal of Applied Meteorology, 27: 689–704.

Inoue, J. (1989) Surface drag over the snow surface of

the Antarctic Plateau: 1. Factors controlling surface drag

over the katabatic wind region. Journal of Geophysical

Research, 94: 2207–2217.

ISW Group (1993) Weddell Sea exploration from ice

station. Eos, Transactions of the American Geophysical

Union, 74: 121–126.

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. Boundary-Layer Meteorology, 24: 211–229.

Kahl, J.D. (1990) Characteristics of the low-level temperature inversion along the Alaskan coast. International

Journal of Climatology, 10: 537–548.

Kaimal, J.C. and J.J. Finnigan (1994) Atmospheric

Boundary Layer Flows: Their Structure and Measurement. New York: Oxford University Press.

Kaimal, J.C., J.C. Wyngaard, D.A. Haugen, O.R. Coté,

Y. Izumi, S.J. Caughey and C.J. Readings (1976) Turbulence structure in the convective boundary layer. Journal of the Atmospheric Sciences, 33: 2152–2169.

Kazanski, A.B. and A.S. Monin (1960) A turbulent regime above the ground atmospheric layer. Izvestiya,

Geophysical Series, no. 1, p. 110–112.

Kondo, J. and H. Yamazawa (1986) Bulk transfer coefficient over a snow surface. Boundary-Layer Meteorology, 34: 123–135.

Lange, M.A. and H. Eichen (1991) The sea ice thickness distribution in the northwestern Weddell Sea. Journal of Geophysical Research, 96: 4821–4837.

Large, W.G. and S. Pond (1981) Open ocean momentum flux measurements in moderate to strong winds.

Journal of Physical Oceanography, 11: 324–336.

Large, W.G. and S. Pond (1982) Sensible and latent heat

flux measurements over the ocean. Journal of Physical

Oceanography, 12: 464–482.

Launiainen, J. and T. Vihma (1990) Derivation of turbulent surface fluxes—An iterative flux-profile method

allowing arbitrary observing heights. Environmental Software, 5: 113–124.

Lettau, H.H. (1979) Wind and temperature profile pre36
diction for diabatic surface layers including strong inversion cases. Boundary-Layer Meteorology, 17: 443–464.

Lumley, J.L. and H.A. Panofsky (1964) The Structure

of Atmospheric Turbulence. New York: Interscience.

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.

Mahrt, L. (1981) Modeling the depth of the stable

boundary-layer. Boundary-Layer Meteorology, 21:3–19.

Mahrt, L., R.C. Heald, D.H. Lenschow, B.B. Stankov

and I. Troen (1979) An observational study of the structure of the nocturnal boundary layer. Boundary-Layer

Meteorology, 17: 247–264.

Makshtas, A.P. (1991) The Heat Budget of Arctic Ice in

the Winter. Cambridge, England: International Glaciological Society.

Martinson, D.G. and C. Wamser (1990) Ice drift and

momentum exchange in winter Antarctic pack ice. Journal of Geophysical Research, 95: 1741–1755.

Mason, P.J. (1988) The formation of areally-averaged

roughness lengths. Quarterly Journal of the Royal

Meteorological Society, 114: 399–420.

Maykut, G.A. (1978) Energy exchange over young sea

ice in the central Arctic. Journal of Geophysical Research,

83: 3646–3658.

McPhee, M.G. and D.G. Martinson (1994) Turbulent

mixing under drifting pack ice in the Weddell Sea. Science, 263: 218–221.

Monin, A.S. and A.M. Obukhov (1954) Basic laws of

turbulent mixing in the ground layer of the atmosphere

(in Russian). Trudy, Akademiia Nauk SSSR, Geofizicheskii Instituta, 24: 163–187.

Monin, A.S. and A.M. Yaglom (1971) Statistical Fluid

Mechanics: Mechanics of Turbulence. Cambridge, Massachusetts: MIT Press, vol. 1.

Moore, K.E., D.R. Fitzjarrald and J.A. Ritter (1993)

How well can regional fluxes be derived from smallerscale estimates? Journal of Geophysical Research, 98:

7187–7198.

Morris, E.M. (1989) Turbulent transfer over snow and

ice. Journal of Hydrology, 105: 205–223.

Munro, D.S. (1989) Surface roughness and bulk heat

transfer on a glacier: Comparison with eddy correlation.

Journal of Glaciology, 35: 343–348.

Nansen, F. (1897) Farthest North. New York: Harper

and Brothers, vol. I and II.

National Weather Service (1991) Observing Handbook

No. 1: Marine Surface Weather Observations. Silver

Spring, Maryland: U.S. Department of Commerce,

National Oceanic and Atmospheric Administration.

Obukhov, A.M. (1971) Turbulence in an atmosphere

with a non-uniform temperature. Boundary-Layer

Meteorology, 2: 7–29.

Overland, J.E. (1985) Atmospheric boundary layer

structure and drag coefficients over sea ice. Journal of

Geophysical Research, 90: 9029–9049.

Overland, J.E. and R.L. Colony (1994) Geostrophic drag

coefficients for the central Arctic derived from Soviet

drifting station data. Tellus, 46A: 75–85.

Overland, J.E. and K.L. Davidson (1992) Geostrophic

drag coefficients over sea ice. Tellus, 44: 54–66.

Overland, J.E. and P.S. Guest (1991) The Arctic snow

and air temperature budget over sea ice during winter.

Journal of Geophysical Research, 96: 4651–4662.

Panofsky, H.A. (1963) Determination of stress from wind

and temperature measurements. Quarterly Journal of the

Royal Meteorological Society, 89: 85–94.

Panofsky, H.A. and J.A. Dutton (1984) Atmospheric Turbulence: Models and Methods for Engineering Applications. New York: John Wiley and Sons.

Parkinson, C.L. and W.M. Washington (1979) A largescale numerical model of sea ice. Journal of Geophysical Research, 84: 311–337.

Paulson, C.A. (1970) The mathematical representation

of wind speed and temperature profiles in the unstable

atmospheric surface layer. Journal of Applied Meteorology, 9: 857–861.

Pielke, R.A. (1984) Mesoscale Meteorological Modeling. Orlando, Florida: Academic Press.

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

Atomic Energy Commission.

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

surfaces. Boundary-Layer Meteorology, 60: 375–395.

Schlichting, H. (1968) Boundary-Layer Theory. New

York: McGraw-Hill, 6th edition.

Serreze, M.C., J.D. Kahl and R.C. Schnell (1992) Lowlevel temperature inversions of the Eurasian Arctic and

comparisons with Soviet drifting station data. Journal of

Climate, 5: 615–629.

Shirasawa, K. (1981) Studies on wind stress on sea ice

(In Japanese, English summary). Low Temperature Science, Ser. A, 40: 101–118.

Smith, S.D. (1988) Coefficients for sea surface wind

stress, heat flux, and wind profiles as a function of wind

speed and temperature. Journal of Geophysical Research,

93: 15,467–15,472.

Sorbjan, Z. (1989) Structure of the Atmospheric Boundary Layer. Englewood Cliffs, New Jersey: Prentice Hall.

Stull, R.B. (1988) An Introduction to Boundary Layer

Meteorology. Dordrecht: Kluwer.

Tennekes, H. (1973) Similarity laws and scale relations

in planetary boundary layers. In Workshop on Micrometeorology (D.A. Haugen, Ed.). Boston: American

Meteorological Society, p. 177–216.

Tennekes, H. (1982) Similarity relations, scaling laws

and spectral dynamics. In Turbulence and Air Pollution

Modelling (F.T.M. Nieuwstadt and H. van Dop, Ed.).

Dordrecht: D. Reidel, p. 37–68.

Tennekes, H. and J.L. Lumley (1972) A First Course in

37
terion for laminar-turbulent-laminar transition in the

ocean and atmosphere. Radio Science, 4: 1289–1298.

Wyngaard, J.C. (1973) On surface-layer turbulence. In

Workshop on Micrometeorology (D.A. Haugen, Ed.).

Boston: American Meteorological Society, p. 101–149.

Wyngaard, J.C. (1992) Atmospheric turbulence.

Annual Review of Fluid Mechanics, 24: 205–233.

Yaglom, A.M. (1977) Comments on wind and temperature flux-profile relationships. Boundary-Layer Meteorology, 11: 89–102.

Yamada, T. (1976) On the similarity functions A, B and

C of the planetary boundary layer. Journal of the

Atmospheric Sciences, 33: 781–793.

Zilitinkevich, S.S. (1966) Effect of humidity stratification on hydrostatic stability. Akademiia Nauk SSSR,

Atmospheric and Oceanic Physics, 2: 655–658.

Zilitinkevich, S.S. (1989a) Velocity profiles, the resistance law and the dissipation rate of mean flow kinetic

energy in a neutrally and stably stratified planetary

boundary layer. Boundary-Layer Meteorology, 46: 367–

387.

Zilitinkevich, S.S. (1989b) The temperature profile and

heat transfer law in a neutrally and stably stratified planetary boundary layer. Boundary-Layer Meteorology, 49:

1–5.

38

Turbulence. Cambridge, Massachusetts: MIT Press.

Thorpe, M.R., E.G. Banke and S.D. Smith (1973) Eddy

correlation measurements of evaporation and sensible

heat flux over Arctic sea ice. Journal of Geophysical

Research, 78: 3573–3584.

Vihma, T. and H. Savijärvi(1991) On the effective roughness length for heterogeneous terrain. Quarterly Journal

of the Royal Meteorological Society, 117: 399–407.

Viswanadham, Y. (1979) Relation of Richardson number to the curvature of the wind profile. BoundaryLayer Meteorology, 17: 537–544.

Viswanadham, Y. (1982) Examination of the empirical

flux-profile models in the atmospheric surface boundary layer. Boundary-Layer Meteorology, 22: 61–77.

Wamser, C. and D.G. Martinson (1993) Drag coefficients for winter Antarctic pack ice. Journal of

Geophysical Research, 98: 12,431–12,437.

Webb, E.K. (1970) Profile relationships: The log-linear

range and extension to strong stability. Quarterly Journal of the Royal Meteorological Society, 96: 67–90.

Wieringa, J. (1980) A revaluation of the Kansas mast

influence on measurements of stress and cup anemometer overspeeding. Boundary-Layer Meteorology, 18:

411–430.

Woods, J.D. (1969) On Richardson’s number as a cri-
1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

6. AUTHORS

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION

 REPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING

 AGENCY REPORT NUMBER

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

14. SUBJECT TERMS 15. NUMBER OF PAGES

16. PRICE CODE

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

 OF REPORT OF THIS PAGE OF ABSTRACT

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and

maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,

including suggestion for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington,

VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

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.

June 1996

The Atmospheric Boundary Layer Over Polar Marine Surfaces

Edgar L Andreas

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road Monograph 96-2

Hanover, New Hampshire 03755-1290

National Science Foundation and Office of the Chief of Engineers

Arlington, Virginia 22230 Washington, DC 20314-1000

Approved for public release; distribution is unlimited.

Available from NTIS, Springfield, Virginia 22161.

45

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

The Atmospheric Boundary Layer (ABL) over polar marine surfaces is, in ways, simpler and, in other ways, more complex than ABLs in other environments. It is simpler because topographic effects are rarely a concern, the surface is fairly

homogeneous, and roughness lengths over sea ice and the ocean are much smaller than they are over land. It is complex

because the stratification is usually stable, and stable ABLs have not yielded to quantification as readily as convective

ABLs have. This report reviews some of these characteristics of ABLs over polar marine surfaces. The ABL, by definition, is the turbulent layer between the Earth’s surface and the (generally) nonturbulent free atmosphere. Hence, the emphasis is on turbulence processes—in particular, the turbulent transfer of momentum and sensible and latent heat over sea

ice. As such, this report reviews both the theoretical and observational bases for our understanding of the mean structure

of the ABL. Understanding this structure then allows predicting the turbulent surface fluxes of momentum and sensible

and latent heat.

Air–sea–ice interactions Cold regions Polar meteorology

Atmospheric boundary layer Polar marine surfaces Turbulence