Sensible Heat Flux Measurements

Near a Cold Surface

Yin-Chao Yen October 1995

0

0.1

0.2

0.3

012345

u2m (m/s)
 _

w' T' (m °C/s)
——

w' T' = 0.042 u2m


—— –

CRREL REPORT 95-22
Abstract

A unidirectional sonic anemometer with a fine-wire thermocouple in conjunction with a hot film anemometer were employed to measure the turbulent

fluctuating velocities of w′, u′, and the fluctuating temperature T′. Covariances were evaluated to compute the turbulent heat flux and the friction

velocity. Based on preliminary data, it can be noted that the values of

fluctuating vertical velocity and temperature, the friction velocity, and the

standard deviations of vertical and horizontal turbulent fluctuating velocities

can all be correlated rather well with a single variable, i.e., the mean wind

speed measured at a height of 2 m. In all the plots of friction velocity, vertical

and horizontal turbulent fluctuating velocities, and the fluctuating vertical

velocity and temperature vs. the mean wind speed at 2 m, the slopes are

slightly lowered as the test season progressed from early summer to the

winter. The most striking reduction can be observed in the case of the

fluctuating vertical velocity and temperature vs. mean wind speed at 2 m.

During the winter period, the slope is only about one third of that during the

spring–summer period. In other words, under unstable conditions, for the

same mean wind speed, the heat flux during the winter is only about one

third of the flux that would have occurred during the spring–summer. Under

stable conditions, the magnitude of the fluctuating vertical velocity and

temperature is much smaller, and its value shows much greater dispersion.

The values of fluctuating vertical velocity and temperature cannot be correlated in any meaningful manner, as is the case under unstable conditions, by

the mean wind speed alone. Comparisons were made with the few existing

measured data or with predictions from theoretical expressions, and they

were found to be in fairly good agreement in some cases and to have large

divergence in others.

For conversion of SI units to non-SI units of measurement consult ASTM

Standard E380-93, Standard Practice for Use of the International System

of Units, published by the American Society for Testing and Materials,

1916 Race St., Philadelphia, Pa. 19103.

Cover: Fluctuating vertical velocity and temperature ( 

w T′ ′) as a function

of the mean wind speed ( 

u2m).
CRREL Report 95-22

Sensible Heat Flux Measurements

Near a Cold Surface

Yin-Chao Yen October 1995

Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

US Army Corps


of Engineers

Cold Regions Research &


Engineering Laboratory
PREFACE

This report was prepared by Dr. Yin-Chao Yen, Research Physical Scientist, Geophysical Sciences Division, Research and Engineering Directorate, U.S. Army Cold Regions Research and

Engineering Laboratory, Hanover, New Hampshire.

Funding for this work was provided by the office of the Chief of Engineers through DA Project

4A161102AT24, Task SS, Work Unit E09, Surface Air Boundary Transfer Processes, and DA

Project 4A1623734DT08, Combat Engineering Systems.

This report was technically reviewed by Dr. Edgar L Andreas and Dr. Yoshisuke Nakano of

CRREL. The author greatly appreciates Dr. Andreas’s thorough and constructive review. He also

wishes to thank David Fisk, John Fiori, Brian Harrington, and Frank Perron for their technical

support, providing instrumentation, computer programming for data collection, and tower set-up

for the sonic measurements. The author also thanks Dr. Harold S. Boyne, former chief of the

Geophysical Sciences Branch, for his interest in and support of this new investigation.

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
CONTENTS

Page

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

Nomenclature ....................................................................................................................... v

I. General review of previous work...................................................................................... 1

II. Theoretical relations and method of analysis .................................................................. 3

A. General equation of motion ..................................................................................... 3

B. Wind structure of the surface boundary layer .......................................................... 5

C. Evaluation of φh, φw, and φm ......................................................................................................... 6

D. Critical Richardson number ..................................................................................... 7

III. Computation of turbulent fluxes .................................................................................... 8

A. Two-level measurement ........................................................................................... 8

B. One-level measurement ............................................................................................ 8

C. Empirical wind functions ......................................................................................... 11

D. Energy balance method ............................................................................................ 12

IV. Fundamentals in measuring sensible heat flux by correlation techniques .................... 13

A. General problem associated with snow-covered ground ......................................... 14

B. Coverage of frequency intervals .............................................................................. 14

C. Measurement height determination.......................................................................... 16

D. Field site selection.................................................................................................... 16

V. Experimental .................................................................................................................... 16

A. Experimental setup ................................................................................................... 16

B. Site description ......................................................................................................... 18

VI. Results and discussion ................................................................................................... 18

A. Collection and analysis of data ................................................................................ 19

B. Sensible heat flux correlation ................................................................................... 31

VII. Comparison of sensible heat flux ................................................................................. 33

VIII. Conclusions ................................................................................................................. 38

Literature cited ..................................................................................................................... 40

Abstract ................................................................................................................................ 43

ILLUSTRATIONS

Figure

1. Vertical structure of the atmosphere ........................................................................... 3

2. Ratio of bulk transfer coefficient to its neutral value from diabatic profile

theory ..................................................................................................................... 10

3. Schematic of the experimental set-up ......................................................................... 16

4. Friction velocity u* as a function of mean wind speed at u2m .................................. 24

5. Standard deviations σw′ and σu′ as a function of mean wind speed at u2m ............ 24

6. Standard deviations σu′ and σw′ as a function of friction velocity u*...................... 25

7. Variation of σ σ u w ′ ′ , and u2m with time over snow and over fluffy snow ............... 26

iii
Figure Page

8. Variation of covariance w T′ ′ with u u 2m / * ............................................................... 27

9. Variation of covariance w T′ ′ with u2m ..................................................................... 27

10. Variation of u* with u2m based on data taken in late fall, winter, and early

spring ...................................................................................................................... 28

11. Standard deviations σw′ and σu′ with u2m based on data taken in late fall,

winter, and early spring ......................................................................................... 28

12. Standard deviation σu′ as a function of u* based on data taken in late fall,

winter, and early spring ......................................................................................... 29

13. Covariance −w T′ ′ as a function of u2m during late fall, winter, and

early spring ............................................................................................................ 29

14. Covariance −w T′ ′ as a function of u2m during late fall, winter, and

early spring ............................................................................................................ 30

15. Variation of σw′/u* with stability parameter z/L ........................................................ 30

16. Relative contributions of the radiative and turbulent energy transfer over

snow as a function of season ................................................................................. 37

17. Influence of elevation on the relative contributions of radiative and turbulent

energy transfers over snow .................................................................................... 37

TABLES

Table

1. Data from test 9107101440, taken in unstable conditions ......................................... 20

2. Data from test 9107160815, taken in conditions of near-neutral stability ................. 21

3. Comparison of the ratio σw′/u* under neutral conditions ......................................... 30

4. Summary of linear correlation of computed sonic results from 1991–1992 ............. 31

iv
NOMENCLATURE

a constant in eq 52

b constant in eq 52

CO f w T′ ′ ( ) cospectrum density of fluctuating vertical velocity and temperature

ch, cw, cm bulk transfer coefficient of heat, water vapor, and momentum

ch,n, cw,n, cm,n bulk transfer coefficient of heat, water vapor, and momentum under

natural conditions

cp heat capacity

d length of sonic path

E water vapor flux

E ́, E ̃ evaporation rate

e vapor pressure

Fny Nyquist frequency

f Coriolis parameter, turbulent frequency

g gravitational acceleration

H sensible heat flux

Kh, Kw, Km turbulent transfer coefficient of heat, water vapor, and momentum

k Karman constant, thermal conductivity

L Obukhov length, latent heat

Ls latent heat of sublimation

M molecular weight

Maf snowmelt in forest canopy

n dimensionless frequency

N number of measurements

Nu Nusselt number

Pr Prandtl number

p pressure, crown density of canopy

q specific humidity

Qh sensible heat transfer

Qe latent heat transfer

Re Reynolds number

Ri Richardson number

Rif Richardson flux number

Ric critical Richardson number

Rib bulk Richardson number

St Stanton number

T, T air temperature, mean air temperature

T′ fluctuating temperature

t time

u horizontal wind speed

u mean horizontal wind speed

U wind travel (= u⋅t)

u

* friction velocity

u,v,w velocity components of the instantaneous wind vector

(,, ) u u uv v vw w w = + ′ = + ′ = + ′

uvw , , components of mean wind vector

uvw ′′ ′ , , components of the fluctuating wind vector ( ) uvw ′ = ′ = ′ = 0

x, y, z Cartesian coordinate system

v
x, y variable x and y

x y ′ ′ , fluctuating quantity in variable x and y

x y ′ ′ , covariance in fluctuations of x ́ and y ́

z vertical distance from surface

zo roughness length or height

z/L Monin-Obukhov stability parameter

— overbar indicating time average

∆θ temperature difference

φw′ ( )f spectrum density of fluctuating vertical velocity

φT′ ( )f spectrum density of fluctuating temperature

φh, φw, φm gradient functions for heat, water vapor, and momentum

ν kinematic viscosity

ρ density

σ ′ w ′ 2 , w 2 variance of fluctuating vertical velocity

σ ′ ′ T

2 , T 2 variance of fluctuating temperature

σw′ standard deviation of fluctuating vertical velocity

σu′ standard deviation of fluctuating horizontal velocity

σx′ standard deviation of x ́

σx′c calculated standard deviation of x ́

σx′t true value of standard deviation of x ́

τ Reynolds stress ( ) * τρ ρ = − uw u ′ ′ = 2

∇2 Laplacian operator

Subscripts

a air

b bulk

c critical

f flux

h heat

m momentum

n neutral

o at zero or at z = zo

w water vapor

vi
Sensible Heat Flux Measurements Near a Cold Surface

YIN-CHAO YEN

I. GENERAL REVIEW OF PREVIOUS WORK

Eddy flux can be measured directly, but the method is expensive and complex and has generally

been used only for short periods of time. For these reasons, estimates of these fluxes are generally

obtained with the use of semi-empirical formulas involving simultaneous measurements of wind,

temperature and humidity at several levels.

Field measurements have been conducted by numerous investigators, but the results usually

differ from one another. In addition, most of the work was done over grass fields, crops, open water,

ice, and glaciers. Swinbank (1951) reported the measurement of vertical transfer of heat and water

vapor by eddies in the lower atmosphere. Thorpe et al. (1973) studied the eddy correlation measurements of evaporation and sensible heat flux over Arctic sea ice and reported bulk transfer

coefficients of cn = 1.2 × 10–3 and cw = 0.55 × 10–3 with the Bowen ratio ranging from 1 to 15.

Andreas et al. (1979) reported the measurement of turbulent heat flux from Arctic leads and

suggested the sensible component of their turbulent heat flux can be predicted from bulk quantities.

They suggested the results can be either expressed in an exponential relation, i.e., Nu = 0.14

Rex

0.72, or in a linear form as Nu = 1.6 × 10–3 Rex + 1400, where Nu is the Nusselt number and Rex

is the Reynolds number based on fetch across the leads. They stated, based on the similarity theory,

that these expressions applied to the latent heat transfer as well. Over leads in winter, they found the

sensible heat flux is two to four times greater than the latent heat flux. Wesley et al. (1970)

employed a three-dimensional pressure-sphere anemometer and fast thermometer system to measure the vertical heat flux density in the atmosphere surface layer at 1–4 m above alta fescue and

snap beans. They found the system is sufficiently small and has adequate high-frequency response

and accuracy for eddy correlation measurements within 1 m of the surface.

Verma et al. (1978) reported measurements of turbulent exchange coefficient for sensible heat

and water vapor over alfalfa and soybeans under conditions of advection. The exchange coefficient

for sensible heat Kh is found to be generally greater than the exchange coefficient for water vapor

Kw, which is in contradiction to the usual assumption of equality of Kh and Kw under nonadvective

(lapse or unstable) conditions (i.e., the net transfer of both sensible heat and water vapor are

directed away from the surface). On the other hand, under advective conditions heat and water

vapor can be transferred in opposite directions. These results were confirmed by conclusions

derived from a theoretical analysis by Warhaft (1976), who stated that the greatest departure of Kh/

Kw from unity will occur where temperature and humidity gradients are of opposite signs.

Miyake et al. (1970) reported the comparison of turbulent fluxes over water determined by

profile and eddy correlation techniques. The measurements were made with limited fetch and were

under near-neutral conditions. However, the agreements were found to be well within the experimental error.

Hicks (1970) developed a general approach for the measurement of atmospheric fluxes near the

surface; a covariance computer capable of accepting analog signals representing any two atmo-
spheric variables was described. It could compute the covariance in a frequency band governed at

high frequencies by the sensor response times and at low frequencies by capacitive filters. This

instrument, using vertical wind velocity as one input, has been successfully used for measurements

of Reynolds stress and sensible and latent heat fluxes.

Though the necessity of measuring the vertical flux of heat and vapor, brought about by eddy

measurement in the lower atmosphere, has long been recognized, Swinbank (1951) was the first to

design an apparatus providing a continuous record (over a 5-minute interval) of the detailed

structure of temperature, vapor pressure, and wind speed as well as its vertical component. He

described the derivation of the vertical fluxes and presented an analysis of error estimation.

Dyer (1961) reported measurements of evaporation and heat transfer in the lower atmosphere by

an automatic eddy-correlation technique that he proved to be practical and accurate for this type of

measurement over a reasonably homogeneous surface. Under conditions of near-neutrality, the

accuracy of the technique is not limited by response time over the range of wind speed normally

encountered. With increasing atmospheric instability, the wind speed range can be further expanded and, unlike in the aerodynamic method, optimum performance is obtained at maximum instability. For measurements made at 4 m from the surface, Dyer indicated that minor surface irregularities of up to several tens of centimeters are clearly of no consequence, and if sufficient upwind fetch

is available, even greater irregularities can be alleviated by increasing the height and period of

observation.

Dyer (1967), in a study on the turbulent transport of heat and water vapor in an unstable

atmosphere, indicated that the transfer mechanism for heat and water vapor are identical for a

freely evaporating surface (i.e., φh = φw where φh and φw are Monin–Obukhov universal functions

for heat and water vapor gradients). Different expressions for φh and φw were presented dependent

on the values of the stability parameter z/L (where z is the vertical distance from the surface at

which the measurement is made, and L is the Obukhov length). He concluded that this finding is

consistent with that from shape function analysis of the same data by Swinbank and Dyer (1967).

Hicks and Martin (1972) reported experimental results on atmospheric turbulent fluxes over

snow. Their experiments were conducted under light wind along with a highly stable atmosphere

near the snow surface. Fluxatrons developed by Hicks (1970) were used to determine these eddy

fluxes. Hicks and Martin indicated that the use of eddy correlation technique to measure the fluxes

of momentum, water vapor, and sensible heat over a snow or even an ice surface has been proved to

be practical and to have a degree of accuracy satisfactory for many studies. They showed that snow

surfaces are generally extremely smooth (not different from aerodynamic smoothness) with corresponding low friction velocities, so large deviations from atmospheric neutrality may be rather

common. They claimed that direct measurement should be preferable to alternative techniques,

based on assuming either the surface roughness or the similarity of transport mechanism. Low bulk

transfer coefficients, based on data collected over Lake Mendota, Wisconsin, were expected after

considering the influence of atmospheric stability on eddy diffusivity, and they reported fluxes of 9

W/m2 for sensible heat and 22 W/m2 for latent heat.

Swinbank (1968) derived a number of relationships based on dimensional analysis that connect

the vertical fluxes of heat and horizontal momentum in the constant flux layer with other relevant

variables. Experimental data taken under unstable conditions are used to test the correctness of the

developed prediction relationship and its functional form. Swinbank found many dimensionless

groups fit the data extremely well, with correlation coefficients in the range of 0.98–0.99. He

reported that ∂θ ∂/ u is height-dependent, indicating dissimilarity in respective transfer mechanisms, and concluded that any H, u*, ∂θ ∂/ z, and ∂ ∂ u z / at one level are sufficient to prescribe the

heat transfer, the shearing stress, and the temperature and wind gradients throughout this constant

flux layer over any uniform surface. Based on the predictive relationships and experimental data of

half-hourly mean values, Swinbank concluded that up to a height of 16 m, the constant flux layer is

2
τ H E

500-1000 m

30-100 m

Free Atmosphere

(region of geostrophic balance)

Planetary Boundary Layer

(turbulent boundary layer)

Surface Boundary Layer 

(constant flux layer)

Ground

z

homogeneous in behavior throughout its depth and provides no support to the suggestion that the

layer may be divided into different regions according to stability.

Based on an exponential wind profile in a turbulent boundary layer with the air thermally

stratified, Swinbank (1964) derived the vertical gradient as

∂

∂

u

z

u

kL

z

L =− − 











 





−

* 1 1 exp

1

or in integrated form as

u u u

k

z z

L

z

L

z

L

2 1

2 1

2

1

1

1

−= − − +

−





 −

−





 −





















































* ln

exp

exp

where k is the Karman constant. He claimed that this solution is expected to be applicable in all

stabilities and to all heights below which the shearing stress and the vertical heat flux remain

constant. Comparison with experimental data shows that buoyancy modifies the transfer of heat

more strongly than does momentum, and the difference increases with height above the surface.

II. THEORETICAL RELATIONS

AND METHOD OF ANALYSIS

The vertical structure of the atmosphere may be classified into three layers, as shown in Figure

1. The free atmosphere is free from the influences of surface friction, and its motion is treated as

laminar flow. The planetary boundary layer, with considerable influence from surface friction, is

analogous to the turbulent boundary layer found on a flat plate, and its motion is treated as turbulent

flow. The surface boundary layer is the lowest 30 m or so of the planetary boundary layer, where the

motion is controlled predominantly by the presence of surface friction. This layer is also referred to

as the “constant flux layer,” i.e., the vertical fluxes of physical properties by turbulence such as

Reynolds stress and sensible heat flux are assumed to be constant to a good approximation.

A. General equation of motion

The structure of the surface boundary layer will first be explained with the aid of the fundamental principle governing the motion of the atmosphere near the surface of the ground. This principle

is expressed by the Navier–Stokes equations for a viscous, incompressible, Newtonian medium in a

uniform gravitational field and in a rotating system, which can be written in a rectangular coordinate system as

Figure 1. Vertical structure of the

atmosphere (after Kai 1982).

3
∂

∂

∂

∂

∂

∂

∂

∂ ρ

∂

∂

u

t u ux v uy w uz

p

x + + + =− + ∇ + v u fv 1 2

∂

∂

∂

∂

∂

∂

∂

∂ ρ

∂

∂

v

t u vx v vy w vz

p

y

+ + + =− + ∇ − v v fu 1 2 (1)

∂

∂

∂

∂

∂

∂

∂

∂ ρ

∂

∂

w

t u wx v wy w wz

p

z

+ + + =− + ∇ − v wg

1 2

where u, v, and w are the velocity components of the instantaneous wind vector, f is the Coriolis

parameter, x, y, and z represent the Cartesian coordinate system, p is the pressure, ρ is the air

density, g is the acceleration due to gravity, v is the kinematic viscosity, and ∇2 is the

Laplacian operator.

Equation 1 describes the instantaneous motion of the atmosphere. However, in turbulent flow it

is convenient to describe the mean rather than the instantaneous flow; this can be done by introducing components of the mean wind vector ( ) uvw , , and the components of the turbulent wind vector,

i.e.,

uuu vvv www = + ′ = + ′ ′ , , and = + (2)

With the use of eq 2 and the application of the equation of continuity and averaging, eq 1 becomes

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ρ

∂

∂

u

t u ux v uy w uz x u u y u v z u w px +++ + ′ ′ + ′ ′ + ′ ′ =− + ∇ + v u fv 1 2

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ρ

∂

∂

v

t u vx v vy w vz x v u y v v z v w py

+++ + ′ ′ + ′ ′ + ′ ′ =− + ∇ − v v fu 1 2 (3)

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ ρ

∂

∂

w

t u wx v wy w wz x w u y w v z w w pz

+++ + ′ ′ + ′ ′ + ′ ′ =− + ∇ − v wg

1 2

Considering that the vertical variation within the planetary boundary layer is usually much

greater than the horizontal variations and considering the case of steady state, eq 3 can be further

simplified as

0 1 1=− + −( ) ′ ′ +

ρ

∂

∂ ρ

∂

∂ ρ px z

u w fv

(4)

0 1 1=− + −( ) ′ ′ − ρ

∂

∂ ρ

∂

∂ ρ py z

v w fu

However, to a good approximation, near the surface the pressure gradient and the Coriolis force can

be neglected and eq 4 can be written as

∂

∂ ρz ( ) − u w′ ′ = 0 (5)

or

τρ ρ

∂

∂ = − uw K ′ ′ = = uz m Constant , (6)

in which Km is the turbulent diffusivity of momentum. Equation 6 states simply that the Reynolds

stress is a constant within the surface boundary layer and is independent of height. In a similar

4
manner, a sensible heat flux also independent of height can be derived as

H c wT c K T

z = ρ ρ ′ ′ = ∂∂ p ph . (7)

It states that the surface boundary layer is characterized essentially by the “constant flux layer.”

B. Wind structure of the surface boundary layer

In the absence of buoyancy, the wind profile is given as

∂

∂

u

z

u

kz = * , (8)

where k is the Karman constant and u* is the friction velocity defined by

u uw * = =− τ ρ/ ′ ′ . (9)

Upon integrating, the mean wind speed is

u u

k

z

z = * ln o

 . (10)

u* is employed as the reference velocity in the study of fluid flow over a rough surface. In general,

it increases with both u and the roughness of the surface zo, which is the height above the ground

(or surface) where the wind speed vanishes (this is also known as the roughness length, a characteristic of the surface).

To take into account all stability conditions other than neutral, Monin and Obukhov (1954)

modified eq 8 by introducing a universal function, φm ( ) z L , i.e.,

∂

∂ φ uz

u

kz

z

L = 





 * m (11)

in which L is the Monin–Obukhov stability length defined as

L

u cT

kgH = − *

3ρ p

3

 . (12)

For the case of neutral stability or when H = 0, we have set φm(0) = 1. If H is positive (the heat flux

is directed from the ground upward), then z/L is negative, indicating an unstable condition. On the

other hand, if H is negative, z/L is positive, indicating a stable condition.

The other two important stability parameters are the Richardson (Ri) and flux Richardson (Rif)

numbers, which are respectively defined as

Ri g

T

T

z

u

z

=

( )

























∂

∂

∂

∂

2 (13)

and

Ri g

T

w T

u w u

z

gH

Tc u

z

f

p

= ′ ′

′ ′

=

− ∂

∂ τ ∂

∂

 . (14)

The sign of Ri is determined by the gradient of mean temperature, which is by convention

negative in lapse and positive in inversion profiles. A value of Ri of zero, negative, or positive

5
indicates the conditions of neutral, unstable, or stable conditions, respectively. By a simple mathematical operation, the three parameters of Ri, Rif, and z/L are related to φm by

Ri

K

K

Ri z

L f

h

m m

= = 







1

φ . (15)

Similar expressions for mean temperature and humidity gradients can be written as

∂

∂ ρ φ Tz

H

c ku z

z

L = − 





 p *

h (16)

and

∂

∂ ρ φ qz

E

ku z

z

L = − 





 *

w , (17)

where q is the mean specific humidity, E is the water vapor flux, and φh and φw are stability

functions for heat and water vapor.

From the basic turbulent transfer equation, such as eq 8 for momentum and the following

expressions of

∂

∂ ρ

T

z

H

c K = −

p h

(18)

for heat and

∂

∂ ρ

q

z

E

K = −

w

(19)

for water vapor, we can relate the stability-related functions of φm, φh, and φw to the eddy turbulent

transfer coefficients of Kh, Km, and Kw by

K

K

K

K

w

m

m

w

h

m

m

h

= = φ

φ

φ

φ

and . (20)

The final expression for H and E in terms of these stability-related functions and mean temperature, vapor pressure, and wind speed gradients can be written as

H ckz u

z

T

z

= −ρ

∂

∂

∂

∂ φφ p

h m

2 2 1 (21)

and

E

p

k z u

z

e

z = − 0 622 1 2 2 . ρ ∂

∂

∂

∂φφw m

 , (22)

where e and p are vapor pressure in millibars. The value of 0.622 is the ratio of the molecular

weight of water vapor to that of air. Therefore, by determining the gradients of mean wind speed,

temperature, and specific humidity and the values of φh, φw, and φm, the sensible heat and water

vapor flux can be evaluated.

C. Evaluation of φh, φw, and φm

A number of researchers have used the “similarity theory” developed by Monin–Obukhov

(1954) as a framework to define the functions of φh, φw, and φm in terms of a dimensionless ratio of

6
z/L where L is invariant with height within the surface boundary layer. Experimental determinations of φh, φw, and φm are made by measuring the profiles of wind speed, temperature, humidity,

and shear stress as well as the fluxes of heat and water vapor.

A great number of determinations of φh, φw, and φm have been reported in the literature

(McVehil 1964, Dyer 1967, Dyer and Hicks 1970, Oke 1970, Webb 1970, Businger et al. 1971,

Pruitt et al. 1971). However, due to logistical problems in choosing an ideal site to ensure the

existence of the constant flux layer and other problems associated with the eddy correlation measurement technique, most of the results are not in agreement with each other. The following

relationships have been adopted by Anderson (1976) to represent the majority of the determinations. For stable conditions, he suggested that

φφ φ hwm = = =+ 





 1 5 z

L

 , (23)

which implies that Kh = Kw = Km. For unstable conditions he recommended using

φm = − 







−

1 16

1 4

z

L

/

(24)

and

φ φ h w = =− 







−

1 16

1 2

z

L

/

 . (25)

Equations 24 and 25 are commonly referred to as Businger–Dyer formulas. If the value of H and u*

are not available, the value of z/L can be related to a stability criterion that can be computed from

commonly available data, i.e., z L K K Ri / = ( ) h mmφ , then for stable conditions we have

φφ φ hwm = = =+ = − − 15 15 1 z

L ( ) Ri (26)

and for unstable conditions

φm = − ( ) − 1 16 1 4 Ri / (27)

and

φ φ h w = =− ( ) − 1 16 1 2 Ri / . (28)

At the geometric mean height of z2 and z1, the Richardson number can be evaluated from the

expression

Ri gz z T T

T Tu u = − −+ −

2 2 12 1

2 12 1

( )( )

( )( )

 . (29)

With the values of φh, φw, and φm calculated from the determined value of Ri along with the values

of wind speed, temperature, and specific humidity gradient, the sensible and water vapor flux can

be determined from eq 21 and 22.

D. Critical Richardson number

Equation 26 can be written as

φφ φ α hwm = = =− − ( ) 1 1 Ri . (30)

As z/L (see eq 26) approaches infinity, the values of φh, φw, and φm also approach infinity. Therefore

7
αRi in eq 30 approaches unity or Ri approaches α–1. A critical Ri is designated as Ric = α–1, beyond

which no turbulent condition can exist.

In his original work, Richardson (1920) reported Ric = 1.0. However, most investigators (Businger

1973) indicate the value of Ric less than 1.0 and most likely in the range of 0.15 to 0.25. Brutsaert

(1972) reported that the value of Ric increases for the case involving evaporation and radiation, but

its increase will depend on atmospheric conditions. He stated that the value of Ric is not invariant

but rather varies between 0.25, below which turbulence is very likely, and somewhat higher than

0.5, above which turbulence is not likely. In the case of a snow cover, the air temperature normally

exceeds the snow surface temperature (during daylight hours) because of the surface’s high albedo

and high emissivity and its limiting temperature of 0°C, therefore stable conditions should predominate over the snow surface.

III. COMPUTATION OF TURBULENT FLUXES

A. Two-level measurement

Under conditions of neutral stability, and with the assumption of Kh = Km, the sensible heat flux

can be obtained from the expression

H ck

u uT T

z

z

= − ( ) − ( ) −









ρ p

2 2 12 1

2

1

2

ln

 , (31)

and similarly for Km = Kw, we have

E M

pM

k u ue e

z

z

= − ( ) − ( ) −









ρ w

a

2 2 12 1

2

1

2

ln

 , (32)

where Mw and Ma are the molecular weight of water and air, respectively. Equations 21 and 22 can

be used to determine H and E fluxes under all conditions in terms of stability-related functions of

φh, φw and φm.

B. One-level measurement

In numerous cases, the measurements of wind, temperature, and humidity at two levels are not

available. The basic equations for evaluating turbulent fluxes based on one-level measurement

(essentially speaking, it is still a two-level measurement except one of the levels is located at the

surface) are generally expressed in terms of bulk transfer coefficients of heat (ch), water vapor (cw),

and momentum (cm), defined as

H ccuT T =− − ρ ph o ( ) , (33)

E

p

M =− − cue e ( ) ρ w

a w o M

 , (34)

and

τ ρρ = = u cu *

2 2 m , (35)

where e e , o are the vapor pressure at z and the snow surface (or at z = zo); and T T and o are the

mean air temperature at heights z and zo.

8
To evaluate the bulk transfer coefficients, these coefficients under neutral conditions, i.e., ch,n,

cw,n, and cm,n, would have to be evaluated. If Kh = Kw = Km is assumed, then the following expression can be obtained:

cc c k

z

z

h,n w,n m,n

o

=== 







2

2

ln

(36)

under unstable conditions. With the assumption of ch = cw, the expression of ch and cw in terms of

cm and cm,n can be written as

c c c

k

c h x

m,n

w

m,n

m

m,n m,n

1/2

cc c = = 









 −  +



 





















1 2 − 2 1

1 2 1

2

/

ln (37)

and

c c

k

x x x m

m,n

m,n

1/2

c = −  +



 



 +  +





 − +





































−

−

1 1

2

2 1

2

2

2

2

1

2

ln ln tan π , (38)

where x stands for [1–γ(z/L)]1/4 and γ is a constant determined experimentally. To compute these

ratios using observations at the surface and at height z, z/L can be related to a stability index such as

bulk Richardson number, Rib, defined as

Ri gz T T

T Tu b

o

o

= ( ) −

( ) +

2

2 , (39)

and the Obukhov length can be transformed as

L

c Tu

c kg T T =

( ) −

m

h o

3 2/ 2

 . (40)

The value of z/L can be related to Rib by

z

L

k c

c c

c

= Ri



 

h

m,n m

m,n

b

3 2 3 2 / / . (41)

Lumley and Panofsky (1964) indicated that, for stable conditions, the log-linear profile is appropriate, and thus the dimensionless wind, temperature, and specific humidity gradients can be written

respectively as

kz

u

u

z

z

L *

m

∂

∂

= +1 β ,

ku z T

z

z

L H

c

* h

p

− = +

ρ

∂

∂

1 β , and (42a,b,c)

ku z q

z

z

L E

* w − = +

ρ

∂

∂

1 β ,

where βs are treated as constant but not equal to each other even though Lumley and Panofsky have

9
set βm = βh. The bulk coefficient ratios corresponding to eq 42a,b,c can be derived and expressed as

c

c

c

c

c

c Ri m

m,n m hh,n

m

m,n

= + b

































−

1

3 2

2

β

– /

,

c

c

c

c

c

c

c

c Ri h

h,n

m

m,n h

h

h,n

m

m,n

= b











 + 



























− − 12 32 1

1

/ /

β , and (43a,b,c)

c

c

c

c

c

c

c

c Ri h

h,n

m

m,n h

h

h,n

m

m,n

= b











 + 



























− − 12 32 1

1

/ /

β .

Using eq 43a,b, a relation involving cm/cm,n with βm, βh, and Rib can be derived as

1 21 1 0

1 2

− 



 



 − − 



 















 + −





 



 = β

β

β

β

β

β

m

h

m

m,n

m

h

m

m,n

m

2

h

b 2

c

c

c

c Ri

/

 . (44)

Figure 2 shows the ratio of the transfer coefficient to its neutral value over a wide range of Rib. It

can be seen that, unlike the unstable case, the curves for the stable case calculated based on βm = 7,

βh = 11, and βw = 20 are independent of cm,n. The sudden change of slope around Rib = 0 is due to

the fact that the values of βm and βh are not equal to γ/4 and γ/2, respectively (γ = 16).

Deardorff (1968), for practical purposes, approximated the bulk transfer coefficient (over water)

for the stable case by

c

c Ri m

m,n

= −( ) m b exp 2β ,

c

c Ri h

h,n

= −+ [ ] ( ) m hb exp β β , and (45a,b,c)

c

c Ri w

w,n

= −+ [ ] ( ) m wb exp β β ,

Figure 2. Ratio of bulk transfer coefficient to its neutral value from diabatic

profile theory, over a wide range of bulk

Richardson numbers. cw /cw,n is not shown

on the unstable side for the fact that it will

coincide with ch /ch,n if Kw = Kh and with

cm /cm,n if Kw = Km (after Deardorff 1968).

3 

2 

1 

0 

– 0.5 – 0.4 – 0.3 – 0.2 – 0.1 0 0.1 0.2 

(unstable) (stable) 

Ri b 

Ratio of Transfer Coefficient

to its Neutral Value

c = 0.0016 m,n

0.0012

0.0008

0.0016

0.0012

0.0008

c / c m m,n c / c w w,n

c / c h h,n

10
and for the unstable region, the bulk coefficient ratio can be written as

c

c a aRi m

m,n

=+ − ( ) b 1 7 ln 1 and

(46a,b)

c

c b

bRi h

h,n

=+ − ( ) b 1 11 ln 1 ,

where a = 0.83 (cm,n)–0.62 and b = 0.25 (cm,n)–0.80.

Based on the eddy transfer coefficients defined in eq 8, 18, and 19 and the unstable wind,

temperature, and specific humidity profiles by

kz

u

u

z

z

L *

∂

∂ = − γ 







−

1

1 4/

 ,

kzu

H c

T

z

z

L

*

− p

= − 







−

/

/

ρ

∂

∂

1 γ

1 2

 , and (47a,b,c)

kzu

E

q

z

z

L

n

*

− = − 







−

/ ρ

∂

∂

1 γ ,

the ratio of Kh/Km for the unstable case is

K z

L

h

Km

= − 





 1

1 4

γ

/

 , (48)

and for the stable case it is

K

K

z

L

z

L

h

m

m

h

=

( ) +

( ) +

1

1

β

β

 . (49)

The ratio of Kw/Km depends on the exponent n; if n = 1/4, then Kw = Km, and if n = 1/2, then Kw/Km

= Kh/Km for the unstable case. For the stable case, the ratio of Kw/Km as for Kh/Km can be similarly

written as

K

K

z

L

z

L

w

m

m

h

=

( ) +

( ) +

1

1

β

β

 . (50)

Therefore, if βm, βh, and βw are much greater than one, then Kh/Km and Kw/Km will approach

β β h m

2 / = 11/49, corresponding to the value of Ric at which the turbulent transfers vanish.

Deardorff (1968) has indicated that Km > Kh and Kw in the stable case due to the importance of

pressure forces in diffusing momentum. If eddies behave somewhat like internal gravity waves, the

effect of mixing by molecular processes alone would be rather insignificant, thus Km > Kh and Kw.

Furthermore, the fact that the value of Kh is usually greater than Kw can probably be attributed to the

damping of thermal fluctuations by radiative transfer in all directions, and no such mechanism is

available to add the mixing of an eddy’s excessive water vapor with the surrounding unsaturated air.

C. Empirical wind functions

The turbulent vapor transfer is predominantly a function of wind speed and the vapor pressure

gradient and can be expressed in terms of Dalton’s relation as

11
E fU e e ′ = − ( )( ) o , (51)

where E ́ is the water vapor transfer expressed as a depth (mm/hr) and f U( ) is a function of the

wind speed [mm/(mb • hr)], where E ́ is positive, the direction of vapor transfer is toward the snow

cover (i.e., condensation-forest formation). When E ́ is negative, it indicates a loss from the snow

cover via the process of sublimation and evaporation. The function f U( ) is usually expressed as

f U a bU ( ) = + , (52)

where U is the product of mean wind speed and time expressed in km, b is an empirical constant

expressed in mm/(mb • km • hr), and a is a constant representing the amount of vapor transfer when

wind velocity is zero [mm/(mb • hr)]. a is found to be dependent on the time interval being used.

The values of both a and b varied a) with the heights at which the wind speed and vapor pressure are

measured, b) with the method and the accuracy of snow surface temperature measurement, and c)

with the predominating stability conditions and surface roughness of the sites.

For the evaporation of water, a finite value of a results. This reflects evaporation during calm,

unstable periods caused by radiative heating of the surface. Since stable conditions usually prevail

over a snow cover, the value of a should be zero. Values of a and b can be extrapolated to a constant

height of 1 m above the snow surface, i.e.,

u

u

e

e z z

1

z

1

= = 0 17 . (53)

where uz , u1 and ez , e1 are mean wind speed and vapor pressure at height z and 1 m above the

surface, respectively.

The latent heat transfer of evaporation or condensation can be expressed as

L E

L

E

L

fU e e s w s w s w ρ o

ρ ρ

′′ = ′ = − 10 10 ( )( ) , (54)

where E′′ is expressed in cm/hr and Ls is the latent heat of sublimation or condensation.

The ratio of sensible heat flux to latent heat flux commonly known as Bowen’s ratio, with the

assumption of cw = ch (also Kh = Kw but that does not assume Km ≠ Kh or Kw) can be written as

Q

Q

H

L E

pc T T

e e

h

e s

p o

0.622 Ls o

= = ( ) −

( ) − . (55)

Experimental evidence (Dyer 1967, Pruitt et al. 1971) suggests that the assumption of Kh = Kw is

reasonable for all stability conditions. On the other hand, in unstable conditions, Km is not equal to

Kh or Kw (or cm ≠ ch or cw). The sensible heat transfer can be written as

Q L

h fU T T s w = − o

ρ γ 10 ( )( ) , (56)

where γ = (pcp)/0.622 Ls is essentially a constant for a given location.

D. Energy balance method

Since the vertical flux of sensible heat cannot be conveniently measured over long time periods,

attempts have been made to measure the other terms in the energy balance equation. The sensible

heat flux is found by integrating the energy balance equation to height h (at which the fluxes are

measured) plus the fluxes due to radiation qR, latent heat qE, and ground conduction qG (see

Lunardini 1981), i.e.,

12
−ρ ′ ′ =− + + ( ) + ∫ ∫ ( ) ′ ′ +

∂

∂ ρ ρ

∂

∂ c wT q q q c x u T dz c u Tx dz

h h

p RGE p 0 p 0

+ + ∫ ∫ c w T

z

dz c

z

T dz h h

p 0 p 0 ρ

∂

∂

∂

∂ ( ) ρ (57)

with the use of the equation of continuity

ρ ∂ρ

∂ w ux dz z = −∫

0 .

Equation 57 becomes

− ′ ′ =− + + ( ) + − 



 



 



 

 ∫ ∫  ρ ρ

∂

∂

∂ρ

∂

∂

∂ c wT q q q c u Tz

u

x dz Tz

dz

h z

p RGE p 0 0

+ c ∫ ∫ ( ) ′ ′ + x u T dz c

z

T dz

h h

p 0 p 0

∂

∂ ρ

∂

∂ ( ) ρ . (58)

However, the integral terms are not measured in energy balance studies because the contribution

from these terms is considered to be rather small in comparison with qR or qE, but there is

uncertainty whether these terms will be small in comparison with ρc wT p ′ ′ , especially if air flow

near the test site is obstructed, creating considerable horizontal gradients. Even in the case when

there is no obstruction to the air flow, the integral terms are negligible, thus

ρc wT q q q p ′ ′ REG = ++ ( ) . (59)

The errors in qR, qE, and qG are cumulative and will be lumped together in calculating ρc wT p ′ ′ .

Since qR is usually very large, a small error in qR can thus distort the value of H ( ) = ρc wT p ′ ′

completely. Therefore, the energy balance method cannot provide the credence in evaluating the

value of ρc wT p ′ ′ .

IV. FUNDAMENTALS IN MEASURING SENSIBLE

HEAT FLUX BY CORRELATION TECHNIQUES

As indicated in eq 7, the sensible heat flux is

H c wT = ρ p ′ ′. (60)

To evaluate the heat flux, H, the covariance of w′ and T′, i.e., the fluctuating velocity and temperature,

has to be measured simultaneously. This requires the use of sophisticated measuring devices and

techniques, and the measurement is usually limited to short time spans to have a reasonable approximation of a pseudo-steady-state condition prevailing during the measurement period (this can be accomplished easily by conducting the measurement in artificially controlled conditions such as in a wind

tunnel, etc., but not in a constantly changing atmosphere). The correlation technique is the most

fundamental one, and the data results from this measurement can be used to evaluate various theories,

but we have to be aware that there is great difficulty associated with sampling frequencies, elevations,

and site suitability to derive universally applicable expressions. Such types of measurement are not

standard operations of meteorological stations, and it is unlikely that they will be in the future.

As we are aware, the turbulent exchange heat fluxes of sensible and latent heat are the most important

heat fluxes (the other is radiation heat flux) in the development of the surface energy balance. On a clear

day, the radiative flux will obviously predominate. However, during the night or on a cloudy day, the

turbulent exchange fluxes will be an essential element affecting the surface energy balance.

13
Turbulent sensible heat flux has been measured by numerous investigators since the invention of

the sonic anemometer and the advances in computer technology since 1965. However, to this

author’s knowledge, the numerous field experimental results are largely limited to terrains other

than snow-covered ground, which presents unique surface characteristics. Because the snow surface serves as the lower boundary, it is generally reported that the numerous turbulent heat flux

expressions developed from studies on terrains other than snow will not be applicable to snowcovered ground because the microphysical heat transfer process under extremely stable conditions

is still rather poorly understood.

Another phenomenon due to the presence of a snow surface as the lower boundary surface is the

validity of the widely accepted concept of the constant flux layer. Although this concept is

convenient (because the heat flux is invariant with height), variations of turbulent flux with height

have been reported. The discrepancy is attributed to the fact that a term responsible for temperature

changes due to the divergence of radiant heat transfer in the thermodynamic energy equation is not

included, even though this is considered to be a common micrometeorological practice, especially

in the first few meters above the surface. This phenomenon may become more pronounced over

snow and melting snow because of their high albedo and the upper-boundary snow temperature of

0°C. Temperature profile anomalies are introduced due to the radiation heating of the air above the

snow surface. With the upper-limit snow surface temperature at 0°C, the air over snow is heated to

above 0°C, resulting in a stable profile directing heat toward the surface. However, if the air mass is

cool, a temperature maximum has been observed in the air layer ~20–50 cm above the surface (De

La Casiniere 1974, Halberstam and Schieldge 1981) and about 5 cm above the snow surface by Yen

(1993). Above this maximum, the temperature profile is mostly unstable in the case of Halberstam

and Schieldge (1981), mostly stable in the work of De La Casiniere (1974), and nearly isothermal

for a plot T vs. ln Z (Yen 1993) as well as results from Granger (1977) in a linear plot. Under these

conditions, not only is the heat flux not a constant with height, but it undergoes a reversal in

direction at the level of the raised maximum.

Since an analytical treatment of turbulence research is out of our reach at the present time, the

following largely deals with how the field measurement system should be designed and set up to

obtain reliable and representative results of universal usage.

A. General problem associated with snow-covered ground

The most prevailing phenomenon of the atmospheric surface layer over snow-covered terrain is

the frequent occurrence of very stable conditions. Because of the low thermal conductivity of snow

and its high albedo and emissivity, the snow surface can get very cold, especially on clear nights.

The snow extracts heat from the air and thus stabilizes the surface layer. Sometimes the surface

layer becomes so stable that turbulence ceases completely and any vertical heat transfer is accomplished by much slower molecular transport processes. However, such extremely stable periods do

not persist very long because gravity waves or other disturbances distort the stable temperature

profile and provide the needed energy for renewed turbulent mixing. These processes are apparently repeated in cyclic fashion and are the key mechanism of the nocturnal boundary layer; they are

believed to be an even more intensive occurrence over snow-covered terrain. Consequently, the

widely practiced Monin–Obukhov similarity cannot handle such intermittent turbulence.

B. Coverage of frequency intervals

The study of atmospheric turbulence is not a deterministic discipline. Since the air is in random

or chaotic motion, we cannot make a single measurement and say “this is the heat flux.” In other

words, we must deal with the statistics of the turbulence. The key statistic in the study of turbulent

heat transfer over snow is the w′–T′ covariance w T′ ′ , w′ is the turbulent fluctuation in vertical

velocity, T′ is the turbulent fluctuating temperature, and the overbar indicates a time average. The

sensible heat carried to or from the snow surface is ρcp w T′ ′ , where ρ and cp are the density and the

14
specific heat of air, respectively. The simplest way to evaluate the covariance w T′ ′ is computed

from a time series measurement of w′ and T′ by

w T′ ′ = ′ ′ Nw Ti i

1 Σ , (61)

where N is the number of measurements and w′

i and T′i are the ith fluctuating vertical velocity and

fluctuating temperatures, respectively. However, this rather elementary method will not shed much

light on the intricate processes of turbulence. We obtain more information by making the same time

series measurements of w′ and T′ and computing the w′ and T′ spectra, which are defined in such a

way that they yield the variance of w′ and T′ and the covariance w T′ ′ , i.e.,

′ = =′

∞

∫ ′ w f df 2 2

0 σ φ w w ( ) ,

′ = =′

∞

∫ ′ T f df 2 2

0 σ φ T T ( ) , and (62a,b,c)

′ ′ = ∞

∫ ′ ′ w T CO f df 0 w T ( ) ,

where f is the turbulent frequency and φw′(f), φT′(f), and COw′T′(f) are the spectrum (or spectra

density) of w′ and T′ and the cospectrum of w′–T′, respectively. Equation 62 shows not only how

the variances and covariance are distributed with respect to spectra and the cospectra but also

indicate which eddies accounted for most of the variance or covariance. Kaimal et al. (1972) have

established the standard criteria of surface layer spectra. Therefore, based on the collected sonic

data we can compute the spectra and cospectra and see if the calculated spectra are in agreement

with the characteristics reported by Kaimal and his colleagues. By comparing with the given

standard characteristics, we are in a position to learn whether our instruments are operating

properly and whether our measuring techniques are correct. If the computed spectra do not possess

the well-established spectral shapes already reported, it is certain that either the turbulence measurement or the subsequent analysis or both are not operating properly. Evaluations of w T′ ′ from

eq 61 will not provide this built-in verification.

The w′–T′ spectrum viewed through eq 62c provides guidelines on how to measure w T′ ′ .

Equation 62c indicates that to measure w T′ ′ we must sample all frequencies between zero and

infinity that contribute to COw′T′. In reality, however, such sampling is neither practical nor

possible, because sampling zero frequency would require an infinitely long time series and, on the

other hand, sampling frequency to infinity would require an infinitely short sampling interval. The

fact is that not all these frequencies contribute to the integral in eq 62c. In dealing with surfacelayer turbulence spectra, it is usual to nondimensionalize frequency f by

n f z

u = , (63)

where z is the height at which the sonic anemometer is located, and u is the mean wind velocity at

height z. Kaimal et al. (1972) indicated that the peak value of COw′T′ is usually near n = 0.1 and falls

off to a value of about 1/10 of its peak value at n > 7. Therefore, we usually need only to sample

frequencies no higher than 7 u z/ . For z = 2 m and u = 2 m/s, the Nyquist frequency Fny has to be 7 Hz

and the actual sampling frequency or digitization rate must be twice Fny, or 14 Hz in this example.

To obtain satisfactory and meaningful values of w T′ ′ , Wyngaard (1973) suggested that 30 to 60

min of data collection is necessary. For 30 min of data, the lower integration limit of eq 62 is 0.0056

Hz. Even sampling w′ and T′ for a period of 30 min at a rate of 14 samples per second generates

25,200 data points for each variable. Therefore, to obtain turbulence statistics properly requires

collecting a large amount of data.

15
C. Measurement height determination

In addition to determining the frequency limits needed in the integration of eq 62, we still need

to know the height at which the sonic anemometer should be placed. Since a sonic measurement

averages w′ over a path distance between the sonic arms, it is reasonable to assume that the sonic

anemometer should not be placed too near the surface where not only are the velocity and

temperature gradients the steepest but the volume of air over which it is averaging would be far

from homogeneous; therefore, there is a limitation as to how near the surface the sonic anemometer

can be deployed effectively. Kristensen and Fitzjarrald (1984) recommended that the inhomogeneity within the sonic sampling volume is minimal if

z fd

u

fz

u

z

d

π

π

< < 1

2 or , (64)

where d is the length of the sonic path. For the example given above, i.e., for fz u/ = 7, then z must

be greater than 14 πd if we are to trust the sonic anemometer measurement. The length of d is

typically 10 or 20 cm, so the magnitude of z should be at least 4.4 or 8.8 m.

D. Field site selection

The horizontal homogeneity of the area upwind of the measurement site is an important factor in

turbulence sampling. To avoid nonrepresentative measurements introduced by internal boundary

layers, the area upwind of an instrument must be uniform for a fetch approximately 100 times the

measurement height. In the example given in section C above, the upwind fetch must be uniform

for a distance of 1400 πd.

V. EXPERIMENTAL

A. Experimental setup

Figure 3 shows the experimental setup. A Campbell Scientific CA27 one-dimensional sonic anemometer (with fine-wire thermocouple) and a hot-film anemometer are linked together and bracketed

to the tower post; they can be adjusted to any desired height but are usually set at 2 m above ground

level. The sonic heads are set 100 cm away from the tower post. The tip of the hot-film anemometer

is positioned at the middle of the sonic path but about 15 cm away from it. The sonic anemometer

has an electronic response of 40 Hz, however, the effective limitation is the loss of response to eddies

Figure 3. Schematics of experimental setup.

16
dimensionally smaller than the sonic path length. Since eddy size varies directly with z, the sonic

anemometer cannot be used below a certain height. The 10-cm sonic path length reduces the need

for the required fetch length. The zero-offset shift in the w signal caused by temperature changes on

the sonic transducer precludes the use of a CA27 sonic anemometer (SA) for absolute w measurement. For the same reason, the CA27 SA cannot be used for long-flux averaging periods (i.e., 1 hr).

It has been reported that measurements of w′ over normal averaging periods compared favorably

with values reported by Biltoft and Gaynor (1987) using other SA models. The manufacturer also

claimed that the recent model of the CA27 produced less variability in w′ than those previously

reported by Tanner et al. (1985). The CA27 has a calibration of 1 m/s/V with a range of ±4 m/s.

The tower and its attached SA are so designed that the tower axis is made vertical by adjusting

the 4 screws protruding from the corners of the concrete foundation. Ensuring that the CA27 SA is

perfectly vertical is accomplished by adjusting the ball-junction so that the air bubble-level on the

top of the CA27 SA is centered.

The temperature measurement device is a 13-μm chromel-constantan thermocouple (Model

127, Campbell Scientific, Inc.) with a response of approximately 30 Hz with its signal calibrated to

4°C/V. The absolute air temperature is not measured; instead, the measurement is referenced to the

temperature inside the base-mount of the CA27 (Fig. 3). Based on the manufacturer’s data, the

reference junction has a 20-min interval time constant, which is expected to be adequate for most

flux-averaging periods.

Care should be taken to prevent damage to the fine-wire thermocouple, which is extremely

fragile. Care also should be taken to avoid any deformation of the silver dish on the acoustical

sensors as this can introduce large effects in the wind signal. It is important to note that the

transducers employed in the CA27 must not be exposed to wet environments.

The algorithms used by the 21X Micrologger to compute statistical measures on-line require the

summation of squares, cross products, and individual values. The covariance of signals x and y is

computed from N measurements of xi and yi (where superscript i indicates instantaneous value) as

x y′ ′ = − x y

N

x y

N

Σ ΣΣ ii i i

2 ,

and the variance of x′ is given by

′ = − 





 x x

N

x

N

2

2 Σ i Σ 2

i .

In this investigation, x y′ ′ and x′

2 represent 10-min time averages. For the 10-Hz measurement

rate, 6000 samples were needed to generate a single statistical value. In conjunction with using the

21X Micrologger to compute the 20-min variances of temperature, Tanner and Green (1989)

clearly show the effect of computation error and found the error is proportional to the number of

samples in the averaging period. They reported that as the ratio of the fluctuation to the mean signal

σ′

x / x becomes smaller, the error grows more significant. The variation of relative error, i.e., the

ratio of calculated σx′

c

 to true standard deviation σx′

t

 as a function of σx′ / x , was calculated by

superimposing the sinusoidal fluctuations of the known standard deviation upon different mean

values. The results are shown below:

Ratio of σ ′

x

x Ratio of σ

σ

′

′

x

x

c

t

0.005–0.01 2–4

0.01–0.02 2–1.3

0.02–0.03 1.3–1.1

0.05 <1.05

0.10 1.01

17
The same reasoning can be applied to the errors in the covariance calculation, i.e., the computational errors become significant if the fluctuations in both signals are small with respect to their

means (i.e., x y xy ′ ′ / is small). However, in the measurement of vertical fluxes, the wind signal

oscillates around zero (i.e., w near zero), thus keeping the ratio large enough to avoid errors in 20-

min flux calculations. Based on results reported by Tanner and Green (1989), there were no

significant error ratios in fluctuations of T′ and w′ and its covariance w T′ ′ .

B. Site description

Theoretically, measurements of surface flux by the eddy-correlation method require an extended

upwind surface with reasonably homogeneous flux sources and sinks that is clear of obstacles

capable of disrupting the horizontal streamlines from parallel to the surface. These conditions were

not met in many of the field measurements reported to date, so the results or empirical formulas

developed by the various investigators varied.

In this experiment, the sonic anemometer was installed to sample the northwest prevailing wind

direction. However, during the experiment, though we did not record the change of wind direction

as a function of time, not only sharp variations in the wind speed but sudden changes in wind

direction were frequently observed. Without being equipped with a cut-off device for wind directions other than the desired one (i.e., northwest), the sonic signals gathered are bound to be rather

complex; they are composed of effects due not only to variations of wind speed and direction, but to

obstacles causing the upwind flow to be neither homogeneous nor parallel to the surface.

As indicated by Yen (1993), a considerable amount of time and effort was directed to securing a test

site that was logistically as well as economically feasible for micrometeorological measurements.

Even after a concerted effort, however, we still were unable to select a site to meet the requirements

outlined above in A, Experimental Setup. A small (about 8 m × 8 m) site located behind CRREL’s main

building was finally chosen. The site is elevated on three sides and bordered on one side with a similar

but smaller rectangular cavity. All around the site all sorts of obstacles, such as trees, buildings, a hut,

and plants, offer resistance to air flow and create extra air turbulence that contrasts with a normally

homogeneous field that has a large and smooth upwind fetch. Though we knew from the onset that

the results obtained from this non-ideal site would not provide any universal usage, it did enable the

researcher to gain some experience and insight in microphysical measurements and provided a

basis for comparison with other reported results measured under more or less similar conditions.

Along with the sonic measurement, an accompanying measurement of the air temperature (near

the sonic measurement) at various heights (up to 1.29 m) was also carried out. Its aim was to verify

the reported results that there is a temperature inversion near the surface layer. A Campbell CR7

data logger was programmed to take the eight thermocouple readings at sampling rates of 30 s, 1, or

2 min, with the selection of data averaging time ranging from 1, 10, 20, 30, or 60 min.

VI. RESULTS AND DISCUSSION

Sonic turbulent heat flux measurements commenced in May 1991 and continued for as long as

meteorological conditions permitted (it is not feasible to collect data on very windy days or when it

is raining or snowing). Over the period 1991–1992, 46 sets of flux data were collected. Due to the

lack of snowfall during the winter, only 10 sets of the data were taken over either a shallow snow

layer, a thin ice layer, or a spotty (or patchy) snow cover.

In early April 1992, due to construction activities occurring near the test site and problems with

the meteorological data-gathering facilities for wind-speed direction and net radiation data, this

series of tests was terminated. The test tower and the other associated measuring devices were

moved to the Geophysical Sciences Branch field test site in front of the Frost Effects Research

Facility in August 1992.

18
A. Collection and analysis of data

Sonic data were collected with a 21X Micrologger based on a sampling rate of 10 per second

and an averaging period of 10 minutes (mostly). Six thousand readings were used to evaluate each

data point shown in the following figures.

Instantaneous vertical wind speed, fluctuating temperatures, and the horizontal wind speed

measured by the hot-film anemometer were collected at 10 samples per second but were discarded

after the 10-minute average values were evaluated. The micrologger was programmed to receive

and compute the reference temperature; the fine thermocouple reading at the measurement height;

the average vertical wind speed w ; the mean of the fluctuating temperature T ; the mean of the

horizontal wind speed u ; the standard deviations of the vertical wind speed σ ′ = ′ 





 w w 2 , the

vertical temperature σ ′ = ′ 





T  T 2 , and the horizontal wind speed σ ′ = ′ 





 u u 2 ; and the covariances of w T′ ′ and u w′ ′.

Two typical sets of collected and computed values are shown in Table 1 and Table 2. Table 1

shows file number 9107101440, which covered a period of about 8 hr with 10-min data average

output. Table 2 shows file number 9107160815, which covered a period from 19:40 p.m. EST on 15

July, 1991, to 8:10 a.m. the next morning. The first and last 10-minute data were not used for

subsequent analysis, to ensure elimination of unsteady effects. Table 1b and Table 2b show the

friction velocity u*, the Obukhov length L, the ratio of σw′ * / u , the stability criterion z/L, and Ta,

the mean absolute air temperature. The values of T u wT , , , σw′ ′ ′, and u w′ ′ are used to define the

values of u*, L, and σw′ * / u . Looking at Table 1b, and based on the values of z/L, the atmosphere

was clearly unstable during the entire measurement period. The mean wind speed u varied from a

minimum of 1.46 to a maximum of 4.31 m/s. Since we did not have the means to determine and

record the wind direction during the measurements, the random obstacles surrounding the test site

caused a great variation of turbulence characteristics affecting the value of σ ′ = ′ 





 w w 2 , which is

a measure of the intensity of the vertical fluctuating velocity. Indirectly it also affected the sensible

heat flux (i.e., ρc wT p ′ ′ and the value of u*, which is calculated from ( ) − u w′ ′ 1 2/ .

Contrary to the results listed in Table 1b, the data shown in Table 2b are rather erratic. Since this

set of data was collected over a period from the evening to the next morning, the atmosphere was in

either a stable or a slightly unstable state. This phenomenon was evidenced by the values of w T′ ′ ,

which are mostly near zero and associated with alternate positive and negative signs (a negative

sign indicates the heat is extracted from the air and the atmosphere is stable). In cases when u w′ ′

had positive values, the data were eliminated along with the cases when the value of L was 0,

infinity, or a ratio of 0/0. It should also be noted that the values of u, , σ σ w u ′ ′ along with w T′ ′ and

u w′ ′, are much smaller in magnitude when compared with those shown in Table 1b. This is

because the atmosphere is near neutral stability, and the transports of heat and momentum are

mostly dominated by molecular processes.

In all, 47 sets of sonic heat flux measurements were made over the period from May 1991 to

April 1992. Most of the measurements were conducted from middle (or early) morning and ended

in late afternoon, depending on the weather conditions. Since the sonic anemometer cannot be used

under wet conditions, no tests were conducted on snowy, rainy, or windy days to ensure that no

damage would be done to the sonic probe and the fine-hair thermocouple.

The following figures show the variation of friction velocity u* and standard deviations 

σw′ and

 

σu′ with mean wind speed u2m , as well as the variations of σw′ and σu′ with u* (to avoid overcrowding the data points and the difficulty in using different notations for different sets of data).

Only six sets of data (91061345, 9106271405, 9107011334, 9107031408, 9107101440, and

9107301320) were plotted and only full dot notation was used. These plots were compared with

those reported by Kai (1982) in his study of statistical characteristics of turbulence and the budget

of turbulent energy in the surface boundary layer. Kai’s measurement was conducted at a site that is

circular in shape with a radius of 80 m and covered by 0.5-m-tall grass. The 30-m tower was erected

19
Table 1. Data from test 9107101440, taken in unstable conditions.

a. Measured data

Panel Air

Test Time of temperature temperature w T u2m σ w′ σ T ′ σ u′ w T′ ′ u w′ ′

period day (°C) (°C) (m/s) (°C) (m/s) (m/s) (°C) (m/s) (m °C/s) (m2/s2)

1 0850 15.93 16.83 0.124 –3.325 2.937 0.340 0.490 0.018 0.083 0.000

2 0900 16.92 17.20 0.098 –3.273 2.628 0.322 0.551 0.670 0.087 –0.025

3 0910 18.26 17.70 0.111 –3.616 1.936 0.313 0.615 0.541 0.091 –0.041

4 0920 19.92 18.07 0.075 –3.906 1.546 0.276 0.664 0.631 0.086 –0.055

5 0930 20.70 18.31 0.052 –4.160 2.016 0.287 0.587 0.624 0.083 –0.036

6 0940 20.53 19.02 0.113 –3.971 1.457 0.253 0.677 0.669 0.098 –0.038

7 0950 20.76 19.62 0.122 –4.426 1.722 0.293 0.722 0.698 0.089 –0.028

8 1000 21.06 19.96 0.052 –4.170 2.161 0.314 0.773 0.572 0.112 –0.045

9 1010 21.51 20.66 0.097 –3.819 1.795 0.295 0.679 0.643 0.099 –0.027

10 1020 22.05 21.01 0.120 –4.070 1.794 0.339 0.718 0.54 0.118 –0.017

11 1030 22.52 21.53 0.133 –3.787 1.728 0.299 0.698 0.672 0.098 –0.010

12 1040 23.10 22.08 0.069 –3.927 2.165 0.320 0.735 0.585 0.111 –0.050

13 1050 23.61 22.49 0.109 –3.860 1.638 0.326 0.844 0.684 0.134 –0.060

14 1100 23.99 22.11 0.094 –4.283 1.883 0.307 0.886 0.848 0.120 –0.068

15 1110 24.40 23.13 0.106 –3.869 1.552 0.396 0.828 0.853 0.134 –0.056

16 1120 25.11 23.27 0.144 –4.335 2.984 0.476 0.934 1.222 0.136 –0.097

17 1130 25.37 23.42 0.122 –2.521 4.306 0.714 0.584 1.418 0.143 –0.194

18 1140 25.68 23.98 0.056 –2.205 3.477 0.634 0.697 1.383 0.157 –0.176

19 1150 26.16 23.99 0.121 –2.767 3.446 0.634 0.628 1.250 0.135 –0.123

20 1200 26.58 24.54 0.077 –2.474 3.117 0.591 0.762 1.363 0.198 –0.168

21 1210 26.99 24.23 0.000 –2.876 4.056 0.699 0.693 1.450 0.193 –0.283

22 1220 27.57 24.38 0.137 –2.031 3.281 0.609 0.779 1.329 0.187 –0.098

23 1230 28.00 24.33 0.137 –2.856 3.036 0.490 0.831 1.092 0.166 –0.108

24 1240 28.29 24.26 0.128 –2.904 3.407 0.569 0.709 1.265 0.138 –0.180

25 1250 28.38 24.37 0.121 –2.050 2.617 0.474 0.840 1.05 0.155 –0.094

26 1300 28.51 24.74 0.063 –2.168 2.981 0.507 0.911 1.185 0.214 –0.147

27 1310 28.60 24.33 0.092 –2.300 3.057 0.530 0.921 1.181 0.103 –0.141

28 1320 28.66 24.70 0.069 –2.206 2.162 0.437 0.762 0.959 0.123 –0.020

29 1330 28.61 24.47 –0.035 –2.081 2.554 0.654 0.862 1.212 0.213 –0.217

30 1340 28.92 25.09 0.055 –2.342 3.455 0.614 0.922 1.288 0.233 –0.151

31 1350 29.36 24.85 0.077 –2.803 3.992 0.633 0.853 1.529 0.233 –0.249

32 1400 29.49 24.47 0.006 –2.744 3.064 0.475 0.739 1.384 0.105 –0.107

33 1410 29.80 25.33 0.097 –0.693 2.568 0.444 0.995 1.122 0.182 –0.110

34 1420 30.30 25.07 9.126 –3.566 2.411 0.428 0.851 0.966 0.147 –0.116

35 1430 30.40 24.99 0.137 –3.085 3.282 0.15 0.676 1.373 0.107 –0.070

b. Computed data.

Test u

* L Ta

period (m/s) (m)

σ w′

u*

z

L (K)

2 0.16 –3.36 2.04 –0.59 290.20

3 0.20 –6.77 1.55 –0.30 290.70

4 0.23 –11.14 1.18 –0.18 291.07

5 0.19 –6.12 1.51 –0.33 291.31

6 0.19 –5.63 1.30 –0.36 292.02

7 0.17 –3.93 1.75 –0.51 292.62

8 0.21 –6.37 1.48 –0.31 292.96

9 0.16 –3.36 1.80 –0.60 293.66

10 0.13 –1.41 2.60 –1.42 294.01

11 0.10 –0.77 2.99 –2.61 294.53

12 0.22 –7.58 1.43 –0.26 295.08

13 0.24 –8.27 1.33 –0.24 295.49

14 0.26 –11.12 1.18 –0.18 295.11

15 0.24 –7.47 1.67 –0.27 296.13

16 0.31 –16.79 1.53 –0.12 296.27

17 0.44 –45.188 1.62 –0.04 296.42

18 0.42 –35.63 1.51 –0.06 296.98

19 0.35 –24.21 1.81 –0.08 296.98

20
Table 1b (cont’d).

Test u

* L Ta

period (m/s) (m)

σ w′

u*

z

L (K)

20 0.41 –26.40 1.44 –0.08 297.54

21 0.50 –50.00 1.39 –0.04 297.23

22 0.31 –12.45 1.95 –0.16 297.38

23 0.33 –16.22 1.49 –0.12 297.33

24 0.42 –41.96 1.34 –0.05 297.26

25 0.31 –14.56 1.53 –0.14 297.37

26 0.38 –20.00 1.32 –0.10 297.74

27 0.38 –38.97 1.41 –0.05 297.22

28 0.14 –1.75 3.09 –1.15 297.70

29 0.47 –36.01 1.40 –0.06 297.47

30 0.39 –19.15 1.58 –0.10 298.09

31 0.50 –50.49 1.27 –0.04 297.09

32 0.33 –25.30 1.45 –0.08 297.47

33 0.33 –15.26 1.34 –0.13 298.33

34 0.34 –20.44 1.26 –0.10 298.07

Table 2. Data from test 9107160815, taken in conditions of near-neutral stability.

a. Measured data

 Panel Air

Test Time of temperature temperature w T u2m σ w′ σ T ′ σ u′ w T′ ′ u w′ ′

period day (°C) (°C) (m/s) (°C) (m/s) (m/s) (°C) (m/s) (m °C/s) (m2/s2)

1 1940 29.81 25.93 0.080 –4.274 0.619 0.097 0.386 0.112 0.008 –0.002

2 1950 29.40 24.90 0.090 –4.984 0.618 0.094 0.338 0.136 0.000 0.000

3 2000 29.77 23.77 0.098 –4.714 0.555 0.072 0.27 0.226 0.001 –0.003

4 2010 29.85 23.45 0.116 –4.578 0.378 0.049 0.144 0.090 0.000 0.000

5 2020 29.64 22.32 0.130 –4.180 0.373 0.045 0.435 0.112 –0.004 –0.001

6 2030 29.20 21.30 0.131 –3.193 0.450 0.051 0.344 0.168 –0.004 0.000

7 2040 28.62 20.59 0.150 –2.432 0.413 0.076 0.252 0.159 –0.002 0.000

8 2050 27.96 0.03 0.151 –1.83 0.370 0.094 0.330 0.179 0.006 0.001

9 2100 27.30 19.68 0.135 –1.074 0.434 0.064 0.409 0.202 0.001 0.000

10 2110 26.65 19.34 0.173 –0.956 0.639 0.065 0.307 0.105 0.000 0.000

11 2120 26.02 19.12 0.138 –0.452 0.734 0.066 0.204 0.111 –0.004 0.000

12 2130 25.43 18.73 0.158 –0.349 0.557 0.038 0.187 0.128 0.000 0.000

13 2140 24.88 18.29 0.160 –0.381 0.547 0.058 0.305 0.206 –0.001 0.000

14 2150 24.35 18.23 0.164 –0.222 0.623 0.098 0.349 0.147 –0.007 0.000

15 2200 23.83 17.61 0.161 –0.238 0.417 0.039 0.31 0.103 –0.001 0.000

16 2210 23.34 17.38 0.212 –0.240 0.549 0.333 0.196 0.141 –0.016 –0.003

17 2220 22.87 17.15 0.171 –0.076 0.507 0.059 0.119 0.105 0.000 0.000

18 2230 22.42 16.85 0.143 –0.124 0.534 0.048 0.135 0.173 0.000 0.000

19 2240 2.00 16.64 0.176 0.040 0.459 0.051 0.267 0.199 0.002 –0.001

20 2250 21.59 16.50 0.167 0.010 0.556 0.054 0.226 0.116 –0.002 –0.001

21 2300 21.18 16.04 0.171 –0.244 0.417 0.062 0.109 0.182 0.000 –0.001

22 2310 20.79 15.88 0.157 0.016 0.457 0.058 0.113 0.153 0.000 0.000

23 2320 20.44 15.69 0.173 0.010 0.450 0.053 0.167 0.150 –0.002 –0.002

24 2330 20.11 15.71 0.168 0.283 0.484 0.097 0.223 0.160 –0.004 –0.004

25 2340 19.77 15.59 0.158 0.082 0.480 0.073 0.129 0.153 –0.001 0.000

26 2350 19.45 15.22 0.163 0.011 0.508 0.078 0.200 0.194 –0.007 –0.001

27 0000 19.13 15.08 0.143 –0.013 0.805 0.085 0.128 0.081 0.000 –0.002

28 0010 18.84 14.93 0.167 0.013 0.568 0.095 0.226 0.200 0.001 –0.005

29 0020 18.56 14.58 0.185 0.122 0.304 0.049 0.178 0.101 0.000 0.000

30 0030 18.26 14.39 0.184 0.024 0.382 0.058 0.121 0.176 0.000 0.001

31 0040 17.97 14.20 0.195 0.171 0.290 0.036 0.137 0.078 0.001 0.000

32 0050 17.68 14.21 0.174 0.244 0.368 0.039 0.194 0.127 0.000 0.000

33 0100 17.40 13.95 0.196 0.193 0.371 0.040 0.141 0.158 0.000 0.000

34 0110 17.40 13.91 0.180 0.219 0.496 0.061 0.141 0.151 0.000 0.000

35 0120 16.90 13.84 0.184 0.183 0.405 0.054 0.192 0.118 0.002 0.000

36 0130 16.67 13.64 0.184 0.205 0.333 0.033 0.065 0.095 0.000 0.000

21
Table 2 (cont’d). Data from test 9107160815, taken in conditions of near-neutral stability.

a. Measured data (cont’d)

Panel Air

Test Time of temperature temperature w T u2m σ w′ σ T ′ σ u′ w T′ ′ u w′ ′

period day (°C) (°C) (m/s) (°C) (m/s) (m/s) (°C) (m/s) (m °C/s) (m2/s2)

37 0140 16.43 13.66 0.187 0.332 0.340 0.032 0.188 0.075 0.000 0.000

38 0150 16.21 13.44 0.208 0.206 0.503 0.051 0.134 0.122 0.000 0.000

39 0200 16.01 13.27 0.197 0.305 0.417 0.066 0.176 0.078 –0.002 0.000

40 0210 15.82 13.28 0.185 0.296 0.460 0.046 0.144 0.099 0.000 0.000

41 0220 15.64 12.96 0.188 0.279 0.237 0.036 0.173 0.078 0.000 0.000

42 0230 15.47 13.11 0.189 0.514 0.506 0.044 0.140 0.073 –0.001 0.000

43 0240 15.30 12.99 0.20 0.373 0.492 0.041 0.095 0.075 0.000 0.000

44 0250 15.13 12.96 0.180 0.581 0.496 0.090 0.266 0.176 –0.007 0.000

45 0300 14.97 12.81 0.193 0.252 0.396 0.052 0.210 0.230 –0.002 0.000

46 0310 14.81 12.53 0.172 0.478 0.036 0.036 0.136 0.094 0.000 0.000

47 0320 14.64 12.55 0.207 0.306 0.544 0.053 0.105 0.148 0.000 0.000

48 0330 14.49 12.43 0.192 0.272 0.840 0.086 0.134 0.161 0.000 –0.006

49 0340 14.38 12.53 0.191 0.400 0.718 0.089 0.099 0.087 –0.002 –0.002

50 0350 14.27 12.61 0.180 0.456 0.698 0.086 0.090 0.102 –0.002 –0.002

51 0400 14.19 12.70 0.204 0.490 0.710 0.091 0.105 0.092 –0.003 0.000

52 0410 14.13 12.59 0.181 0.406 0.599 0.079 0.076 0.157 –0.002 –0.002

53 0420 14.04 12.24 0.204 0.161 0.473 0.054 0.160 0.199 0.000 0.005

54 0430 13.91 11.94 0.192 0.242 0.278 0.043 0.203 0.087 –0.003 –0.001

55 0440 13.75 12.08 0.205 0.472 0.409 0.075 0.148 0.181 –0.002 –0.003

56 0450 13.59 11.99 0.205 0.389 0.306 0.043 0.163 0.141 0.000 –0.001

57 0500 13.47 11.72 0.209 0.353 0.603 0.065 0.135 0.179 –0.002 –0.001

58 0510 13.36 11.85 0.210 0.561 0.431 0.080 0.155 0.162 0.000 –0.003

59 0520 13.30 12.15 0.203 0.565 0.523 0.089 0.068 0.210 0.000 –0.007

60 0530 13.34 12.38 0.170 0.285 1.012 0.117 0.174 0.308 0.003 –0.007

61 0540 13.43 12.31 0.158 0.050 0.861 0.112 0.062 0.209 0.002 –0.002

62 0550 13.52 12.33 0.215 0.012 0.597 0.107 0.075 0.200 0.003 –0.006

63 0600 13.60 12.27 0.195 –0.116 0.492 0.099 0.092 0.240 0.004 –0.006

64 0610 13.67 12.20 0.161 –0.212 0.823 0.149 0.091 0.195 0.007 –0.010

65 0620 13.72 12.12 0.187 –0.226 0.860 0.141 0.112 0.213 0.007 0.000

66 0630 13.79 12.25 0.190 –0.123 0.634 0.106 0.091 0.134 0.005 –0.002

67 0640 13.85 12.38 0.172 –0.207 0.783 0.123 0.135 0.220 0.007 –0.002

68 0650 13.93 12.38 0.190 –0.341 0.682 0.109 0.145 0.315 0.008 –0.008

69 0700 14.03 12.52 0.185 –0.448 0.749 0.143 0.230 0.223 0.020 –0.009

70 0710 14.15 12.52 0.184 –0.730 0.646 0.132 0.177 0.158 0.004 –0.004

71 0720 14.30 12.70 0.164 –0.862 0.698 0.158 0.218 0.211 0.019 –0.002

72 0730 14.48 12.77 0.166 –1.179 0.745 0.142 0.232 0.245 0.013 –0.002

73 0740 14.67 12.82 0.199 –1.494 0.868 0.165 0.214 0.244 0.017 –0.005

74 0750 14.89 13.06 0.208 –1.808 1.053 0.153 0.355 0.253 0.031 –0.013

75 0800 15.17 13.23 0.201 –2.452 1.387 0.185 0.380 0.01 0.033 –0.006

76 0810 15.50 13.62 0.183 –3.089 1.022 0.168 0.579 0.407 0.044 –0.007

b. Computed data.

Test u* L Ta

period (m/s) (m)

σ w′

u*

z

L (K)

2 inconsistency: L=0/0

3 0.05 –12.44 1.31 –0.16 296.77

4 inconsistency: L=0/0

5 0.03 0.60 1.42 3.36 295.32

6 inconsistency: L=0

7 inconsistency: L=0/0

8 error: u w′ ′ is positive.

9 inconsistency: L=0

10 inconsistency: L=0

11 inconsistency: L=0/0

12 inconsistency: L=0

13 inconsistency: L=0/0

22
Table 2b (cont’d).

Test u* L Ta

period (m/s) (m)

σ w′

u*

z

L (K)

14 0.03 0.34 3.10 5.96 291.23

15 inconsistency: L=0

16 0.05 0.76 6.08 2.63 290.38

17 inconsistency: L=0/0

18 0.04 inf. 1.07 0.000 289.85

19 0.03 –1.17 1.61 –1.71 289.64

20 0.03 1.17 1.71 1.71 289.50

21 0.03 inf. 1.96 0.000 289.04

22 inconsistency: L=0/0

23 0.04 3.29 1.19 0.61 288.69

24 0.06 4.66 1.53 0.43 288.71

25 inconsistency: L=0

26 0.03 0.33 2.47 6.02 288.22

27 0.04 inf. 1.90 0.000 299.08

28 0.07 –25.97 1.34 –0.08 287.93

29 inconsistency: L=0/0

30 error: u w′ ′ is positive.

31 inconsistency: L=0

32 inconsistency: L=0/0

33 inconsistency: L=0/0

34 inconsistency: L=0/0

35 inconsistency: L=0

36 inconsistency: L=0/0

37 inconsistency: L=0/0

38 inconsistency: L=0/0

39 inconsistency: L=0

40 inconsistency: L=0/0

41 inconsistency: L=0/0

42 inconsistency: L=0

43 inconsistency: L=0/0

44 inconsistency: L=0

45 inconsistency: L=0

46 inconsistency: L=0/0

47 inconsistency: L=0/0

48 0.08 inf. 1.11 0.000 285.43

49 0.04 3.26 1.99 0.61 285.53

50 0.04 6.52 1.92 0.31 285.61

51 inconsistency: L=0

52 0.04 3.26 1.77 0.61 285.59

53 error: u w′ ′ is positive.

54 0.03 0.77 1.36 2.61 284.94

55 0.05 5.97 1.37 0.33 285.08

56 0.03 inf. 1.36 0.000 284.99

57 0.03 1.15 2.06 1.74 284.72

58 0.05 inf. 1.46 0.000 284.85

59 0.08 inf. 1.06 0.000 285.15

60 0.08 –14.21 1.40 –0.14 285.38

61 0.04 –3.25 2.50 –0.61 285.31

62 0.08 –11.28 1.38 –0.18 285.33

63 0.08 –8.46 1.28 –0.24 285.27

64 0.10 –10.39 1.49 –0.19 285.20

65 inconsistency: L=0

66 0.04 –1.30 2.37 –1.54 28.25

67 0.04 –0.93 2.75 –2.15 285.38

68 0.09 –6.51 1.22 –0.31 285.38

69 0.09 –3.11 1.51 –0.64 285.52

70 0.06 –1.32 2.09 –1.52 285.52

71 0.04 –0.34 3.53 –5.83 285.70

72 0.04 –0.50 3.18 –3.99 285.77

73 0.07 –1.52 2.33 –1.32 285.82

75 0.13 –4.48 1.46 –0.45 286.23

23
Figure 4. Friction velocity u* as a function of

mean wind speed at u2m .

0.5

0.4

0.3

0.2

0.1

0

02468

u2m (m/s)
 _

u

* (m/s)

u

*

 = 0.113 u2m


_ 

at the center with four (3-component) sonic anemometers and resistance thermometers installed at

1.6, 4.3, 12.3, and 29.5 m above the ground. The booms for the sonic anemometers and the

resistance thermometers extended 2.0 m southeast and northeast, respectively.

Figure 4 shows the relation between u* and

u2m . For six sets of data taken during the months

of June and July 1991, a simple linear relationship can be expressed as

u u * = 0 113 . 2m , (65)

which is found to be identical to the average relationship developed by Kai (1982), i.e., u u * = 0 11 .

(in Kai’s test there are four sonic anemometers,

so there is one linear relation for each level:

u u * = 0 10 . at 1.6 m, u u * = 0 12 . at 4.3 m,

u u * = 0 12 . at 12.3 m, and u u * = 0 09 . at 29.5 m.

Figure 5a shows the standard deviation σw′,

representing the turbulent fluctuation in the vertical direction, as a function of the mean wind

speed. As in Figure 4, the relationship can be

linearly represented by

σ ′ w m = 0 16 2 . u . (66)

It shows that σw′ increases with u2m . It was surprising to note that an identical relation was developed by Kai (1982) from test results gathered

from a much more homogeneous field (i.e.,

1.0

0.8

0.6

0.4

0.2

0

02468

u2m (m/s)
 –

σw' (m/s)

σw' = 0.16 u2m


_ 

Figure 5. Standard deviations σ w′ (a) and σ u′ (b) as a function of mean wind speed at u2m .

2.0

1.6

1.2

0.8

0.4

0

0246

u2m (m/s)
 _

σu' (m/s)

σu' = 0.35 u2m


_ 

a. b.

24
σ ′ w = 0 16 . u ; this relationship is also developed from measurements at four different heights, as

indicated earlier). Kai further noted that the ratio of σw′ / u at the highest level (i.e., z = 29.5 m) is

slightly lower than those at levels z = 1.6 to 12.3 m.

Figure 5b shows the relation of σu′ vs. u2m (the value of σu′ is generated not from the

unidirectional sonic anemometer but from the hot-film anemometer). A linear relationship of

σ ′ u m = 0 35 2 . u (67)

represents the data rather well, but the increase of σu′ with u2m is about 60% higher than the one

reported by Kai (i.e., σ ′ u = 0 25 . u ). This discrepancy might be due to the fact that σu′ was measured from a hot-film anemometer (which is not as sensitive as the sonic anemometer) and the test

site is much less homogeneous than the one Kai used. Kai also points out that there seems to be

smaller scatter in σw′ than in the horizontal components σu′ and σv′.

Figure 6a shows the variation of σu′ (the horizontal component of the turbulent fluctuation)

with u* and can be represented by

σ ′ u = 3 7. *u . (68)

Like the case of σu′ vs. u2m , this relation has a higher coefficient than the one presented by Kai

(i.e., σ ′ u = 2 53 . *u ). Figure 6b shows the variation of σw′ with u* and can be represented by

σ ′ w = 1 6. *u , (69)

which is nearly identical to the relation presented by Kai (i.e., σ ′ w = 1 58 . *u ) covering the measurement at four levels. As noted in Kai’s paper, in both plots of σw′

 vs. u* or u2m and σu′ vs. u* or

u2m , the horizontal components σu′

 and σv′ (in Kai’s case) have a greater scatter than the vertical

component σw′ . The ratio of σ σ u w ′ ′ / is about 2.33 in this study as compared with Kai’s of 1.60.

The higher ratio of σ σ u w ′ ′ / can be attributed to the inhomogeneity of the test site (the upwind

fetch length requirement was never met).

Figure 6. Standard deviations σ u′ (a) and σ w′ (b) as a function of friction velocity u*.

2.0

1.6

1.2

0.8

0.4

0

0 0.2 0.4 0.6 0.8

u

*

 (m/s)

σu' (m/s)

σu' = 3.7 u*

0

0 0.2 0.4 0.6

0.2

0.4

0.6

0.8

1.0

1.2

u

*

 (m/s)

σw' (m/s)

σw' = 1.6 u*

a. b.

25
Figures 7a and 7b respectively show the variation of u2m , σu′

, and σw′ as functions of time

over snow with different structures. There seemed to be no one-to-one correspondence between the

turbulence intensity of σu′ and σw′, i.e., when σu′ is at maximum σw′ is not. The variation of u2m

does resemble σu′ more than σw′. It can be noted that in both cases the peak value of σw′ was

reached before the highest value of u2m occurred. This phenomenon may be attributed to the rapid

warming of the surface air layer and its subsequent rising, which introduces added upward natural

convection. In both cases, the ratio of σu′/ σw′ varies from about 1.5 to 5.5.

Figure 8 shows the relation between the covariance w T′ ′ and the ratio of u u 2m / * , which

covered eight sets of data taken over a field of grass that was about 3 cm high. The covariance w T′ ′

can be fairly represented by

w T′ ′ = − u

u 0 212 0 013 2 . . m

*

 . (70)

It indicates that as u2m/u* increases, w T′ ′ decreases. In other words, for the same value of u2m but

with u* increasing, there will be larger values of w T′ ′ . The larger the value of u*, the more

momentum transfer there is. For a value of u2m/u* equal to ~16.3, if the above relation holds, the

value of w T′ ′ approaches zero.

Figure 9 shows the variation of w T′ ′ with u2m . This figure covers the period from May to July

1991 and consists of nine sets of data. Though the graph shows a greater scatter (which is not

unusual for micrometeorological measurements), there is a concrete trend that w T′ ′ increases with

u2m and can be fairly represented by

wT u ′ ′ = 0 042 2 . m . (71)

The lines shown in Figures 4 through 6, 8, and 9 are drawn by visual observation.

The following sets of figures are based on data taken during the late autumn, winter, and early

spring of 1992 (in all 11 sets of data). During these periods, the ground was either covered with

snow or ice, or it was bare but either partially frozen or frozen. Figure 10 shows the relation

u2m (m/s)
 – 

σu', σw' (m/s)


8 1.6

1.2

0.8

0.4

0

6

4

2

0

10 12 14 16

Time (hr)

12/19/91(over snow)

_________________


 σu'

 σw'

 u2m

 

–

u2m (m/s)
 – 

σu', σw' (m/s)


8 1.6

1.2

0.8

0.4

0

6

4

2

0

10 12 14 16

Time (hr)

12/19/91(over fluffy snow)

_____________________


 σu'

 σw'

 u2m


 

–

Figure 7. Variation of σ σ u w ′ ′ , and u2m with time (a) over snow and (b) over fluffy snow.

a. b.

26
Figure 9. Variation of covariance w T′ ′ with u2m .

between u* and u2m , and a least-squares relation was developed as

u u * = 0 079 . 2m . (72)

The dependence of u* on u2m is somewhat weaker when it is compared with that shown in

Figure 4 (i.e., u u * = 0 113 . 2m ). It can be said that over snow, ice, or partially frozen ground the

value of u* is significantly lower because there is less-intense turbulence over snow or ice than over

a grass field (~2–3 cm tall).

Figures 11 shows the relation of σw′ and σu′ with u2m . The linear least-squares correlation can

be expressed as

σ′

w m = 0 113 2 . u (73)

Figure 8. Variation of covariance

w T′ ′ with u2m /u* .

2.0 × 10–1

1.6

1.2

0.8

0.4

0

4 6 8 10 12

u2m

_ 

—––


u*

w' T' (m °C/s)
——

w' T' = 0.212 – 0.013 u2m/u*

—— –

0

0.1

0.2

0.3

012345

u2m (m/s)
 _

w' T' (m °C/s)
——

w' T' = 0.042 u2m


—— –

27
2.0

1.6

1.2

0.8

0.4

0

02468

u2m (m/s)
 _

σu' (m/s)

σu' = 0.35 u2m


–

0.5

0.4

0.3

0.2

0.1

0

02468

u2m (m/s)
 _

u

* (m/s)

u* = 0.079 u2m


–

and

σ′

u m = 0 35 2 . u . (74)

The value of 0.113 is slightly lower than that in

eq 66, but the value of 0.35 is identical to the one

given in eq 67. Figure 12 shows the variation of

σu′

 and u* for the same measurement period and

can be expressed by

σ ′ u = 3 68 . *u , (75)

which is more or less the same as the relation

shown in eq 68.

Figure 13 shows the variation of covariance

w T′ ′ with u2m . It can be represented by

wT u ′ ′ = 0 014 2 . m . (76)

The constant 0.014 is only one third of the value

given in eq 71. Neglecting the effect of variation

of ρ and cp with temperature and atmospheric pressure, the sensible heat flux is only one third of the

value during the late spring to early summer.

Though the data points in all these figures are

scattered to a great extent, the least-squares correlation coefficients of all these linear relations,

i.e., eq 71–76 are greater than 0.8313 (as high as 0.9716 for σu′ vs. u2m ). It indirectly indicates that

u2m is the predominant factor in correlating this type of data.

Figure 10. Variation of u* with u2m based on

data taken in late fall, winter, and early spring.

1.0

0.8

0.6

0.4

0.2

0

02468

u2m (m/s)
 _

σw' (m/s)

σw' = 0.133 u2m


–

Figure 11. Standard deviations σw′(a) and σu′ (b) as a function of u2m based on data taken in late

fall, winter, and early spring.

a. b.

28
2.0

1.6

1.2

0.8

0.4

0

0 0.2 0.4 0.6 0.8

u

*

 (m/s)

σu' (m/s)

σu' = 3.78 u*

Figure 12. Standard deviation σu′ as a function of u* based on data taken in late fall,

winter, and early spring.

Figure 13. Covariance w T′ ′ as a function of u2m during late fall,

winter, and early spring.

Tests under stable conditions were hard to

realize. A stable surface air layer with very calm

winds usually prevails soon after sunset. However, the magnitude of the sensible heat flux

under stable conditions is very low, and the few

data points collected during this period exhibited a much greater dispersion than in the case of

unstable conditions. Figure 14 shows the relation of −w T′ ′ with u2m (we consider the heat

flux from the air to the snow or ground surface

to be negative). The least-squares analysis only

gives a correlation coefficient of 0.0751, indicating practically no correlation between −w T′ ′

and u2m .

The ratio of σw′/u* may be expressed in terms

of a universal function of the stability parameter

z/L according to the similarity theory. Figure 15

shows the variation of σw′/u* with z/L, clearly

exhibiting the features of the ratio varying over

a range of z/L from –100 ≤ z/L ≤ 100 (in most of

the past studies, the range of z/L is from –1 ≤ z/L

≤ 10). As in other studies, there is a great dispersion of data (all the overlapping values of σw′

/

u* are omitted in the figure). No values of σw′/

u* fall into the range of –0.01 < z/L < 0.01 due to lack of near-neutral conditions. For unstable

conditions over the whole range of –z/L covered in this study, the value of σw′/u* can be fairly

represented by

σ ′ = − 











 



 w

u

z

L *

/

13 1 .

1 3

 , (77)

0

0.1

0.2

0.3

012345

u2m (m/s)
 _

w' T' (m °C/s)
——

w' T' = 0.014 u2m

—— _

29
z/L

σ w' –––


u*

0.1

1

10

–100 –10 –1 –0.1 ±0.01 0.1 1 10 100

σw' = 1.3 1 – z 

1

––– –– 

––

u

*

 

L

 

3

σw' = 1.3 1 + z 

1

––– –– 

––

u

*

 

L

 

4

Figure 15. Variation of σw′/u* with stability parameter z/L.

Figure 14. Covariance − w T′ ′ as a function of

u2m during late fall, winter, and early spring.

Table 3. Comparison of the ratio σw′

*

u under neutral conditions.

Height of

measurement

Test field σw′

u*

(m) References

Over the sea 1.23 6.4, 12.5 Natio (1978)

Over the sea 1.32 8 Pond et al. (1971)

Over the sea 1.47 1.4—4.5 Miyake et al. (1970)

Over land 1.53 1.4 McBean (1970)

Over tall grass 1.32 1.6, 4.3, 12.3, 29.5 Kai (1982)

Over snow, frozen ground, 1.30 2 This study

and partially frozen ground

0.20

0.16

0.12

0.08

0.04

0

02468

u2m (m/s)
 _

–w' T' (m °C/s)
 –––– 

30
which is different from the expression proposed by Panofsky et al. (1977) for z/L ≥ –1.0, i.e.,

σ ′ = − 











 



 w

u

z

L *

/

13 1 .

1 3

 , (78)

but Kai (1982) claimed that his data can be represented rather well by the above expression.

For stable regions (i.e., 0.01 ≤ z/L ≤ 100), even though there are less data available, a similar

expression with an exponent of 1/4 instead of 1/3 can be derived as

σ ′ = + 











 



 w

*u

z

L

13 1 3

1 4

.

/

 . (79)

Table 3 shows the comparison of the ratio σw′/u* with other results under neutral conditions. It can

be seen that the ratio of σw′/u* taken from this study is within the range reported by all the

investigators listed in Table 3. Even though the friction velocity u* varies with the homogeneity of

the upwind fetch and the surface roughness, the ratio of σw′/u* from this study (under neutral

conditions) falls nevertheless within the limits reported by other investigators. This value, along

with the relations given for u* vs. u2m (eq 65), σw′ vs. u2m (eq 66), and σw′ vs. u* (eq 69), certainly gives credence to the results obtained from this preliminary investigation. The agreement

provides confidence in carrying out the essential purpose of this investigation, i.e., the use of the

eddy correlation technique to determine the sensible heat flux.

Table 4 shows a summary of the computed sonic results. It can be seen that, with the exception

of plots of σw′ vs. u* and −w T′ ′ vs. u2m , the slope of all other plots decreases slightly as the test

runs extended from 30 May 1991 to 15 April 1992, with the lowest slope for the test runs conducted

from 19 December 1991 to 15 April 1992. It seems that in the presence of snow or ice-covered or

frozen ground, friction velocity u*, standard deviations σw′, σu′, and the covariance w T′ ′ are

slightly less dependent on u2m . In other words, for the same value of u2m , the values of u*, σw′,

σu′

, and w T′ ′ are reduced somewhat as the tests progressed from spring to winter.

B. Sensible heat flux correlation

To facilitate the practical use of the sensible heat flux data presented in Table 4, a dimensionless

analysis is needed to extend the results for other test conditions. For the test period from 19

December 1991 to 15 April 1992, the variation of w T′ ′(positive for unstable conditions or for heat

rising from the surface) with u2m was given in eq 76 as

wT u ′ ′ = 0 014 2 . m

Table 4. Summary of linear correlation of computed sonic results from 1991–1992.

9105301331 9105301331 9105301331 9105301331

Test period to to to to

9108201600 9112111530 9204151725 9104151725

Type of Correlation Correlation Correlation Correlation

plot Slope coefficient Slope coefficient Slope coefficient Slope coefficient

u u * vs. 2m 0.1034 0.9722 0.0994 0.9664 0.0893 0.9337 0.0785 0.8992

u u * vs. 2m 0.1614 0.9845 0.1572 0.9721 0.1456 0.9241 0.1330 0.8701

σu 2m ′ vs. u 0.3723 0.9816 0.3635 0.9812 0.3564 0.9765 0.3487 0.9716

σu′ vs. u* 3.4431 0.9638 3.4710 0.9634 3.5940 0.9420 3.7788 0.9182

wT u ′ ′ vs. 2m 0.0398 0.9052 0.0250 0.8649 0.0241 0.8160 0.0140 0.8313

–wT u ′ ′ vs. 2m 0.00005 0.8049 0.0003 0.1583 0.0023 0.0746 0.0045 0.0751

31
in which the constant 0.014 contains a dimension of °C. Multiply both sides of eq 76 by ρ, cp, and

z and divide by ∆T, the difference in temperature between the measurement height and the surface

(which was not measured in this study because there is no proven and reliable technique to measure

the snow surface temperature). The following equation is obtained:

Nu

T = RePr 0 014 .∆ , (80)

where Nu is the Nusselt number defined as Hz/k∆T, Re is the Reynolds number u z ρ /μ, and Pr is the

Prandtl number, cpμ/k. The essential purposes in expressing the results in terms of the dimensionless Nusselt, Reynolds, and Prandtl numbers is to include the effect of air temperature on the

variation of physical properties of air such as viscosity μ, density ρ, specific heat cp, and thermal

conductivity k. Although eq 80 is dimensionless, it is not in typical dimensionless form because it

contains the term ∆T (which was introduced to define the Nusselt number and was not measured

during the test). However, as shown in Table 4, the correlation coefficients are rather high,

demonstrating that quantities such as u*, σw′, σu′, and w T′ ′ are strongly dependent on u2m alone.

Although the tests were conducted over nearly a one-year period, the values of ∆T appear to vary

within a small range, and subsequently a more or less identical correlation will be produced if the

data ∆T is available and is used to evaluate the Nusselt number, where H is the heat flux, u is the

mean horizontal wind speed, and z is the measurement height.

Since the value of Pr for the temperature range from 240 to 300 K can be approximated by

Pr = 0.97 – 0.0009T, (81)

where T is air temperature in Kelvin (K), substituting eq 81 into eq 80 we have

Nu

T = − Re T ( ) 0 014 0 97 0 0009 . . . ∆ . (82)

As long as the mean wind speed u is known, along with the temperature at height z, even without

the data on ∆T, heat flux H can be calculated from eq 82 because ∆T is also used in defining the

Nusselt number Nu (i.e., Hz/k∆T), so we do not need to know the value of ∆T to calculate H from eq

82.

For an average temperature of 273.1 K during the winter months, eq 82 can be further

simplified as

Nu

T = Re 0 010 .∆ . (83)

On the other hand (Table 4), for the period from May to August 1991, and by assuming an average

air temperature of 293 K, an expression of

Nu

T = Re 0 028 .∆ (84)

can be derived. The value of ∆T does not come into the process to calculate heat flux H in either eq

82 or 84, so for the same value of Re values of H are about three times greater during the summer

season than during the winter. For the whole year (May 1991–April 1992) based on the data shown

in Table 4, the value of Nu can be expressed by

Nu

T = − Re T ( ) 0 024 0 97 0 0009 . . . ∆ . (85)

Therefore, as long as the mean horizontal wind speed and the air temperature at the height of the

sonic measurement are determined, the value of Re can be calculated and subsequently the heat

flux H can be determined.

32
VII. COMPARISON OF SENSIBLE HEAT FLUX

Based on boundary layer integral analysis in conjunction with a study of heat flux over Arctic

leads, Andreas (1977) presented a dimensionless relation relating the Nusselt number Nu to the

products of the Stanton number St, the Reynolds number Re, and the Prandtl number Pr, as

Nu = St • Re • Pr, (86)

in which the dimensionless groups are chosen to be defined as

Nu Hx

kT T

St H

cu T T = ( ) − =

w pw 05 05 05 . .. ( ) − , , ρ

 and Re u x

v = 0 5. ,

where Tw is the surface water temperature, and u0 5. and T0.5 are the mean wind speed and air

temperature measured at 0.5 m above the lead at fetch x, respectively. All the other properties, i.e.,

v, ρ, cp, and k, are evaluated at Tw. Andreas’s results were finally expressed either by

Nu Re = 0 18 0 71 . . (87)

or in linear form as

Nu Re =× + − 2 24 1120 . 10 3 , (88)

which was compared with the theoretical expression presented by Kays (1966) for flow over a

constant-temperature flat plate, i.e.,

St = 0.0295 Re–0.2 Pr–0.4, (89)

which is not in good agreement with the expression resulting by equating eq 87 and 86, i.e.,

St Re Pr = 0.18 Re0.71 (90)

or

St = 0.18 Re–0.29 Pr–1. (91)

For Pr = 0.714, eq 91 becomes

St = 0.2521 Re–0.29, (92)

which is quite different from Kay’s theoretical expression (eq 89). By substituting Pr = 0.714, eq 89

becomes

St = 0.0338 Re–0.2, (93)

Assuming eq 92 and 93 have a same exponent to Re, eq 92 gives about seven times larger values

than Kay’s theoretical expression (eq 93). Andreas also compared his findings with the theoretical

expression developed by Schlichting (1968) for flow over a heated plate by assuming a drag

coefficient of 1.76 × 10–3 and Pr = 1.0, i.e.,

Nu = 1.76 × 10–3 Re (94)

For a value of Re = 2.94 × 105, we have a flux value H of 36 W/m2 from eq 83. For the same

value of Re and a ∆T of 20°C, eq 88 gives an H value of ~400 W/m2, which is approximately 10

times higher than the value obtained from this study. But for the same values of Re and ∆T, eq 94

predicts a value of ~125 W/m2, which is about one third of the 400 W/m2 given by eq 88 and about

three times larger than the 36 W/m2 given by eq 83. However, it should be noted that in Andreas’s

field test not only is the temperature difference (Tw – T0.5) always about 20°C, but the measurement

33
heights are also different. On the other hand, in this study, the ∆Ts are much smaller, i.e., on the

order of 2°C (or even less), thus creating much less intense convective heat transfer.

As in this study, Tanner and Green (1989) reported an average sensible heat flux value of 49 W/

m2. His field test was conducted 14–17 August, and four sonic anemometers were used (one for

each post, but each installed at a different level). For a comparable period from May to August

(Table 4 or eq 84), and for the same Re value of 2.95 × 105, a heat flux of about 142 W/m2 can be

found, which is about three times higher than the value calculated from eq 83 for tests over snow

and frozen and partially frozen soil covered with thin ice. This value, however, is in quite good

agreement with the average heat loss over Arctic leads of about 250 W/m2 reported by Andreas

(1977). The data given by Tanner and Green were not sufficient to give a statistical analysis,

however, Andreas’s data were taken from a test with a much more homogeneous field than the one

in this study.

In a study on atmospheric turbulent flux over snow, Hicks and Martin (1972) measured eddy

fluxes of momentum, sensible heat, and water vapor over Lake Mendota, Wisconsin, and reported,

under stable atmospheric conditions, an average sensible heat flux of 9 W/m2 for a wind speed of

~2.7 m/s and ∆T from surface to air at 2 m of approximately 3°C. This average heat flux was

computed from a total of four 1-hour measurements, and flux varies from 4.2 to 11.9 W/m2. In this

study, for the period with snow or thin ice-covered surface, the heat flux was found to be on the

order of 25 W/m2, about twice the maximum value reported by Hicks and Martin. Since, as

indicated, the test site is quite inhomogeneous due to the limited fetch length and other obstacles, it

was very difficult to measure the sensible heat flux properly under stable atmospheric conditions.

Stable conditions usually prevail during a clear night with calm wind, but under these conditions

frost forms on the sonic head, inhibiting its sensitivity of measurement. Therefore, there are hardly

any reliable means to measure this low heat flux. Nevertheless, the sensible heat flux under stable

conditions for the period from May 1991 to April 1992 (Table 4) has a value of ~13 W/m2, nearly

identical to the value reported by Hicks and Martin (1972). It should be noted that even for a period

of nearly a year, there was not sufficient data to make a valid statistical analysis (note the near-zero

value of the correlation coefficient).

For unstable atmospheric conditions and for a grass-covered field, the sensible heat flux is found

to be approximtely 90 W/m2 (based on eq 84) for a wind speed of 2 m/s. Based on five sets of field

measurements for a range of 0.02 < |z/L| < 0.6, Dyer (1967) reported an average sensible heat flux

of H = 212 ∆θ3/2 where ∆θ is the potential temperature difference between heights of 1 and 4 m. In

this study, the value of ∆θ is on the order of 3°C. Therefore, based on Dyer’s expression, the

sensible flux is about ~600 W/m2, which is about seven times greater than 90 W/m2.

In a study involving measurements of evaporation and heat transfer in the lower atmosphere by

an automatic eddy-correlation technique, Dyer (1961) conducted a great number of measurements

of 5-min duration covering a 26-day period from January to March over a level pasture with wind

speeds varying from 1.96 to 7.28 m/s, with a mean wind speed of 4.2 m/s. Because of the short

response time of the sensing equipment (~0.3 s), he indicated that, for conditions of moderate and

high instability, significant deficiencies in eddy-flux measurements will occur only with wind

speeds exceeding 10 m/s. He reported an average sensible heat flux of 155 W/m2 (during his test

the wind speeds were mostly in the vicinity of 4 m/s; in only seven of the 26 tests were the wind

speeds less than 4 m/s). Using 4 m/s as the mean wind speed in eq 84 (a correlation from this

study), a heat flux of 180 W/m2 results, which is in remarkably good agreement with the value of

155 W/m2 reported by Dyer. He further stated that with measurements at a height of 4 m, minor

surface irregularities of up to several tens of centimeters are clearly of no consequence, and, if

sufficient fetch is available, great irregularities could be readily accommodated by increasing the

height and period of observation. For stable conditions, Dyer (1961) reported a heat flux value of

~14 W/m2, which is nearly identical to the value reported by Hicks and Martin (1972).

34
In a study dealing with differences between air and snow surface temperatures during snow

evaporation, Bernier and Edwards (1989) reported an average heat flux of 16 W/m2 transported

from the air stream to the snow surface under stable conditions (based on a total measurement of 14

data points from 1030 to 1930 hr), which is about 7 W/m2 higher than the value reported by Hicks

and Martin (1972) and approximately 9 W/m2 less than the value obtained from this study. Bernier

and Edwards concluded that the sensible heat flux from air to snow is the essential source of the

energy escaping from the snowpack surface through the latent heat flux. They concluded that snow

surface temperatures are generally lower than the air temperature and therefore, based on their

theoretical analysis, if the air and snow temperatures are equal and air temperatures are below 273

K, the heat flux will be overestimated. It should be noted, however, that the snow surface is rather

ill-defined because of its dynamic nature, so it is hard to measure its temperature.

Bates and Gerard (1989) discussed the various alternative means to predict snow surface

temperature. With the assumption that a measurement by an infrared radiometer was the snow

surface temperature, they compared the temperature measured by thermistor/ thermocouple, the air

temperature at 2 m, and the dew point temperature against the infrared temperature and concluded

that the radiometer would provide the most accurate estimate of the snow surface temperature,

followed in order by thermistor/thermocouple, the air temperature, and the dew point temperature.

De La Casiniere (1974) conducted studies on heat exchange over a melting snow cover with

reported means of diurnal and nocturnal balance of sensible heat flux of 0.2 MJ/m2 and 0.23 MJ/

m2, respectively, which are equivalent to 4.6 and 5.3 W/m2, respectively (without a description of

the test conditions, it is hard to compare these reported values with others).

In a comprehensive study of snow surface energy exchange, Male and Granger (1981) discussed

extensively the numerous factors affecting the turbulent energy exchange over a snow surface.

They made the following observations: a) there have been many studies involving energy transfer,

covering all areas of the world and various sites, i.e., open, forested, mountain snowpacks, isolated

snowpacks, etc.; b) most of the studies dealt with melting snow, especially on glacier snowpacks,

and virtually all were area point studies; c) very few direct measurements of turbulent energy fluxes

are available (estimates based on aerodynamic formulas do not constitute direct measurement); d)

although a number of absolute values (measured or calculated) are reported, the general trend is to

present results regarding the relative contributions of the various forms of energy transfers; and e)

only a few investigations attempted to note any diurnal or seasonal trends and to describe similarities between their findings or those of others. They concluded that the majority of investigators

estimate the turbulent sensible heat flux from the use of aerodynamic formulas derived with the

assumption of steady turbulent flow of a viscous fluid over an infinite, uniform, and hydrodynamically rough surface, implying that the transfer is only in the vertical direction and is time-independent (inferring the vertical flux is constant with height). However, the extent to which the turbulent

fluxes vary with height has been the subject of several investigations. Dyer (1968) reported a 16.8%

departure from the surface value for the turbulent sensible and latent heat fluxes at the 16-m level.

Haugen et al. (1971) reported that turbulent heat flux and shear stress are constant only to within

20% up to a height of 23 m.

The variation of heat flux with height may be more pronounced over snow and melting snow

because of their high albedo and the limiting upper-bound temperature of 0°C. De La Casiniere

(1974), Granger (1977), and Halberstam and Schieldge (1981) conducted measurements over

melting snow and showed that temperature anomalies are introduced by radiative heating of the air

above the snow surface. Because the upper-bound temperature is 0°C, the air over snow will be

heated above the snow temperature, resulting in a stable profile, but if the air mass is cool, a

temperature maximum is observed in the air layer 20–50 cm above the surface. In this situation, not

only is the heat flux not constant with height, but it undergoes a reversal in direction at the level of

the raised maximum.

35
Priestley (1959), on the other hand, points out that for an air layer being heated by radiation the

sum of the radiant and turbulent fluxes is essentially constant with height; therefore, to estimate

energy transfer over snow, one must ensure not only that the measurements of temperature and

humidity are made above the level of the raised maximum temperature, but that a simultaneous

radiation measurement is also included. Priestley suggested that we must be content with the total

surface energy transfer rather than that of the individual surface fluxes.

Large-scale parameters, such as topography, altitude, season, and air mass, were considered in

Male and Granger’s (1981) analysis of snow surface energy exchange. For brevity and comparason,

they expressed their summarized results in percentages of the total energy input, indicating not only

different values from year to year at the same site, but also making it impossible to state which

particular energy flux will dominate overwhelmingly or be negligible in any particular environment. They also indicate that not only topography but other factors such as the air mass and largescale circulation affect the process of energy transfer. Therefore, it is difficult or even dangerous to

classify sites in general as being either open, forested, or alpine, since physically similar sites with

different geographical placement will necessarily have different air mass histories. For snowcovered prairies and large open sites, theoretical approximation expressions derived after taking

into account the influences of stability and the possibility of radiative heating of the air above the

snow can be used directly to estimate the energy fluxes at the surface. In a review study of

snowmelt in a prairie environment, Male and Granger (1979) developed a number of empirical

methods of estimating energy fluxes and their accuracies and noted that the air mass exerts a

controlling influence on the relative magnitude of the various energy fluxes.

In the case of forest cover, turbulent flux measurements have received little attention either

because wind, temperature, and humidity measurements have been difficult to obtain or because

these fluxes have been ignored in the assumption that reduced wind speed results in near-negligible

energy fluxes. This may be true in the region of northern Quebec, for Hendrie and Price (1978)

pointed out that radiation alone provides an efficient predictor of the snowmelt, but farther south

turbulent exchange is found to be more significant (at least during the melting season) because

snowmelt is usually initiated by the movement of a warm air mass into the region. To estimate the

advective component of snowmelt Maf under a forest canopy compared to the corresponding part of

the Ma of an open locality, Kuz’min (1961) developed

Maf = (0.44 – 0.43 p2)Ma (95)

for coniferous and

Maf = (0.45 – 0.15 p2)Ma (96)

for deciduous, where p is the crown density of the canopy. Doty and Johnson (1969) reported 30%

and 53% reductions in evaporation rates within aspen and coniferous stands, respectively, as

compared with an open site.

Application of standard aerodynamic formulas to predict turbulent fluxes over a forest canopy

has been reported by Federer and Leonard (1971) to be inappropriate, based on the fact that in

subcanopy flow the sources of mechanical turbulence and sensible heat flux are distributed vertically through the canopy. Brutsaert (1979) formulates bulk mass and heat transfer coefficients for

canopy flow but, as pointed out by Federer and Leonard (1971), the transfer characteristics are

likely to be quite different.

In the case of an alpine environment, Obled and Harder (1979) reviewed studies of snowmelt in

a mountain environment and listed a number of factors affecting the turbulent transfer of heat and

water vapor over snow. Wind and temperature regimes in the mountain environment are complex

and make the analysis of the boundary layer on the basis of local slope winds difficult. Martin

(1975) provides a formulation for the katabatic wind profile and applies it to the calculation of the

36
turbulent fluxes over snow. In general,

however, due to the great variability of

alpine winds with respect to type,

space, and time, the type of analysis

seems to be impractical for operational purposes, and hydrologists have usually resorted to the use of standard

aerodynamic formulas in dealing with

alpine snow fields and glaciers subjected to appropriate stability correlations.

Male and Granger (1981) graphically showed the relative contributions of

radiative and turbulent energy transfers over a snow surface as functions

of season and elevation. Figure 16

shows a large scatter in the relative

contributions of the radiative and turbulent fluxes because melting of the

low-level, shallow snowpacks, which

melt quite early in the spring, depends

largely on the energy content of the air mass present. On the other hand, deeper snowpacks that

melt later in the year masked the effect of perturbations of the air masses and displayed the longterm trend of increased relative contribution of radiant energy. Figure 17 shows the effect of

elevation on the relative contribution of energy

fluxes. It can be seen that below 2000 m there is

no correlation between elevation and the relative contribution of radiant and turbulent energy fluxes, indicating the influence of other factors such as topography or air mass. For

elevations above 2000 m, there is an apparent

decrease in turbulent flux contribution as the

elevation increases. However, this trend seems

to be valid only when the effect of other factors

is negligible. In conclusion, it seems that there

is not enough data to distinguish the influence

of altitude and season on the relative contributions of radiant and turbulent energy transfers.

The influence of the air mass on the turbulent energy transfer was first reported by Sverdrup (1936) in his paper dealing with turbulent

exchange with snow. He broadly classified his

results in terms of the prevailing general weather conditions, i.e., a cold dry period is associated with a negative sensible heat transfer (heat

leaving the surface) accompanied by evaporation from the snow surface; and warm, dry and

wet periods are connected with a positive sensible heat transfer with evaporation from and condensation to the snow surface, respectively.

100

80

60

40

20

0 Mar Apr May Jun Jul Aug Sep

Relative Contribution (%)

Figure 16. Relative contributions of the radiative (dashed

line) and turbulent (solid line) energy transfer over snow

as function of season (after Male and Granger 1981).

Figure 17. Influence of elevation on the relative

contributions of radiative (open circles) and turbulent (solid circles) energy transfers over snow. Solid lines represent a range of values from a single

location and dashed lines show the relationship between elevation (above 2000 m) and the relative contribution (after Male and Granger 1981).

4000

3000

2000

1000

0 20 40 60 80 100

Relative Contribution (%)

Elevation (m)

37
The effect of energy and moisture contents of the air mass on evaporation/condensation has

generally been overlooked, until Nyberg (1965) and Rylov (1969) made gravimetric measurements

of evaporation/condensation at the snow surface and reported that the daily vapor flux is governed

by the moisture content of the air mass. Hanaford and Howard (1975) dealt with an unusual

snowmelt event due to an extreme warm upper air temperature. Granger (1977) and Granger and

Male (1978) reported that the major melt-producing flux is due to radiation, and the advancement

of the melt is governed by the energy content of the air mass. McKay and Thurtell (1978)

confirmed the generalized results reported by Sverdrup (1936) and indicated the increase in the

extent of evaporation during the transition from a warm to a cold air mass.

In general, it is fair to state that from the scatter and the variability of the results so far reported,

based on a number of investigations under a variety of environmental and site conditions, the

measurements of reliable and consistent sensible and latent heat exchanges have proven to be

difficult even at well-instrumented sites. Furthermore, techniques or methods for extrapolating

results from a small local measurement to larger areas, after taking into account the variations of

elevation, latitude, state of the air mass, or topographical characteristics, have not been successfully

developed.

Accurate information on sensible and latent heat and net radiative heat flux along with the heat

transfer processes within the snow medium would be needed to calculate the snow surface temperature. This is largely due to the fact that the snow surface itself is so ill-defined that any intrusive

measurement device, regardless of how small the sensing probe is, will disturb the surface structure, so it is uncertain whether the device is measuring the temperature of the snow grain or the air

temperature of the void space enclosed by the snow grains. Male and Granger (1981) stated that

investigations of large-scale or air mass influences on turbulent energy exchanges are likely to yield

more practical results than investigations that are confined to a few meters above the snow surfaces.

They also claim that research on air mass scale should give better insight into the factors governing

changes in sensible and latent heat flux, because such studies are aimed at the cause rather than at

an effect.

VIII. CONCLUSIONS

As indicated in section V, Experimental, the field site was chosen primarily for its convenience.

The site is small, it is elevated on the west and north, and it definitely does not meet the fetch length

requirement for eddy-correlation-type measurements. However, as indicated in section VI, Results

and Discussion, the plots u* vs. u2m (Fig. 4) and σw′ vs. u2m (Fig. 5a) are nearly identical to the

results reported by Kai (1982), who used more sophisticated and modern sonic anemometers

(three-dimensional) and a greater circular test field (160 m diameter). These surprisingly good

agreements give much credence to our sensible heat flux results.

As indicated in section VII, Comparison of Sensible Heat Flux, a number of investigators have

attempted to measure the sensible heat flux over a variety of terrains and environmental conditions

but, due to the nature of the constantly changing atmosphere, there is really no way to apply the

results obtained under a specific set of conditions to any specific combination of conditions. In

other words, it is not possible to derive formulas to predict the sensible heat flux for areas other

than the place or area where the measurement was made.

Though the sensible heat flux results obtained thus far from this study are rather preliminary and

limited, the expressions derived as shown in eq 83 and 84 can be easily used to estimate the sensible

heat flux as long as the mean wind speed and the mean air temperature at measurement height z are

given. Since the snow surface temperature (in fact, any surface temperature) can hardly be determined without the introduction of errors, there is no easy way to evaluate the temperature difference between surface and the height of the measurement. However, by looking at the definition of

38
Nu = Hz/k∆T and eq 83 and 84, it is clearly shown that we do not need to know the value of ∆T to

calculate the value of sensible heat flux H.

Table 4 clearly shows the variations of w T′ ′ with u2m over a period of one year. The slope

varies from 0.0298 for the measurements made from May to August to 0.014 for the period from

December to the next April. In the May–August period, the sensible heat flux is about three times

greater than the value calculated for the December–April period for a given Reynolds number Re.

In other words, for a same mean wind speed u2m , the friction velocity u* is lower over snowcovered ground than over grass-covered ground. This is evidenced in the variation of w T′ ′ with

the ratio of u u 2m / *

 (Fig. 8). This trend can also be seen from Table 4 in which the slopes of the

relation between u* vs. u2m , σw′ vs. u2m , and σu′ vs. u2m are slightly lowered as the test period

expanded from May to August, December and then to April. The slopes of the relation between σu ́

vs. u* (Table 4) increases slightly as the test period progressed as was expected, i.e., to have the

same value of u* during the winter period (in this case December to April). It needs a higher value

of mean wind speed u2m and thereby creates a higher value of standard deviation of fluctuating

velocity in the horizontal direction.

Though the study reported here is preliminary in nature, it has clearly established the fact that

the sensible heat flux determined during the winter months or during snow- or ice-covered ground

is about only one third of the value obtained during spring and summer months, i.e., over grasscovered ground. Therefore it is speculated that the use of expressions involving temperature and

mean velocity gradients evaluated at measurement height z to estimate sensible heat flux from

equations such as eqs 21 or 31 will definitely introduce a significant error because these equations

are derived without the consideration of snow surface as a lower-surface boundary.

For sensible heat flux determined under stable conditions (which exists mostly at night under

calm wind conditions), the data are much more scarce and showed a much greater dispersion,

discussed in section VII, Comparison of Sensible Heat Fluxes. For the case of stable atmospheric

conditions, Hicks and Martin (1972) reported an average value of 9 W/m2 (for a mean wind speed

of 2.7 m/s and ∆T of 3°C), which is about one third of the value obtained from this study under

approximately the same test conditions (i.e., 27 W/m2) (but for a period of one year, the heat flux

reduces to 13 W/m2, which is close to the value of 14 W/m2 reported by Dyer (1961) as well as of

16 W/m2 reported by Bernier and Edwards (1989).

For unstable atmospheric conditions, Dyer (1961) conducted sensible heat flux measurements

over level pasture land with a mean wind speed of 4.2 m/s and reported a heat flux value of 155

W/m2, which is nearly identical to the value of 180 W/m2 calculated from eq 84 based on the

present study. This close agreement may be accidental, because in another study Dyer (1967)

derived an expression representing the sensible heat flux by H = 212 ∆θ3/2 (W/m2). With ∆θ on the

order of 3°C, the value of H will be on the order of ~1100 W/m2, which is about seven times greater

than the value reported by Dyer (1961) as well as the value calculated based on eq 84.

In summary, based on several past studies as well as the present one, it can be concluded that not

only does sensible heat flux depend on a number of factors, but it is difficult to obtain consistent

results because of the constant changing of the atmosphere. The fact is that even during an

averaging period of 10 minutes the wind speed and its direction vary greatly. Therefore, the

fundamental assumption of constant, steady, and homogeneous upwind flow never can be realized

in the field measurement to satisfy the condition of a so-called constant flux surface layer. It is

expected that the results obtained with the eddy-correlation technique will provide a much more

reasonable estimation for the sensible heat flux than the other methods discussed in section III,

Computation of Turbulent Fluxes. Due to the constantly changing atmospheric dynamics, the

steady, homogeneous surface layer flow will never be realized in field measurements to ensure the

existence of the constant flux layer. Localized or regional measurements, which are used in this

study, may still provide a much more reliable method to find the sensible heat flux needed for

surface energy balance calculations.

39
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Sensible Heat Flux Measurements Near a Cold Surface PE: 6.11.02A

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52

Energy balance Fluctuating velocity Turbulence

Fluctuating temperature Sensible heat flux

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

A unidirectional sonic anemometer with a fine-wire thermocouple in conjunction with a hot film anemometer were

employed to measure the turbulent fluctuating velocities of w′, u′, and the fluctuating temperature T′. Covariances were

evaluated to compute the turbulent heat flux and the friction velocity. Based on preliminary data, it can be noted that the

values of fluctuating vertical velocity and temperature, the friction velocity, and the standard deviations of vertical and

horizontal turbulent fluctuating velocities can all be correlated rather well with a single variable, i.e., the mean wind speed

measured at a height of 2 m. In all the plots of friction velocity, vertical and horizontal turbulent fluctuating velocities, and

the fluctuating vertical velocity and temperature vs. the mean wind speed at 2 m, the slopes are slightly lowered as the test

season progressed from early summer to the winter. The most striking reduction can be observed in the case of the

fluctuating vertical velocity and temperature vs. mean wind speed at 2 m. During the winter period, the slope is only about

one third of that during the spring–summer period. In other words, under unstable conditions, for the same mean wind

speed, the heat flux during the winter is only about one third of the flux that would have occurred during the spring–

summer. Under stable conditions, the magnitude of the fluctuating vertical velocity and temperature is much smaller, and its

value shows much greater dispersion. The values of fluctuating vertical velocity and temperature cannot be correlated in

any meaningful manner, as is the case under unstable conditions, by the mean wind speed alone. Comparisons were made

with the few existing measured data or with predictions from theoretical expressions, and they were found to be in fairly

good agreement in some cases and to have large divergence in others.