Characteristics of the Heat Flux in the Unstable Atmospheric Surface Layer

MAITHILI SHARAN AND PIYUSH SRIVASTAVA

Centre for Atmospheric Sciences, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India

(Manuscript received 22 September 2015, in final form 11 June 2016)

ABSTRACT

The behavior of the heat fluxH with respect to the stability parameter z (5z/L, where z is the height above

the ground, and L is the Obukhov length) in the unstable atmospheric surface layer is analyzed within the

framework of Monin–Obukhov similarity (MOS) theory. Using MOS equations,H is expressed as a function

of z and vertical surface-layer potential temperature gradient ›u/›z. A mathematical analysis is carried out to

analyze the theoretical nature of heat flux with the stability parameter by considering the vertical potential

temperature gradient as (i) a constant and (ii) a power-law function of heat flux. For a given value ofH, two

values of z associated with different stability regimes are found to occur in both the conditions, suggesting the

nonuniqueness of MOS equations.

Turbulent data over three different sites—(i) Ranchi, India; (ii) the Met Office’s Cardington, United

Kingdom, monitoring facility; and (iii) 1999 Cooperative Atmosphere–Surface Exchange Study (CASES-99;

United States—are analyzed to compare the observed nature ofH with that predicted by MOS. The analysis

of observational data over these three sites reveals that the observed variation of H with z is consistent with

that obtained theoretically from MOS equations when considering the vertical temperature gradient as a

power-law function of heat flux having the exponent larger than 2/3. The existence of two different values of

the stability parameter for a given value of heat flux suggests that the application of heat flux as a boundary

condition involves some intricacies, and it should be applied with caution in convective conditions.

1. Introduction

Monin and Obukhov similarity (MOS) theory (Monin

andObukhov 1954) is widely used to estimate the stability

parameter z (5z/L, where z is the height above the ground,

and L is the Obukhov length) and surface fluxes in atmo-

spheric models for weather forecasting as well as for

air quality and climate modeling (Arya 1988; Beljaars and

Holtslag 1991;Garratt 1994;Oleson et al. 2008; Skamarock

et al. 2008; Jimenez et al. 2012; Giorgi et al. 2012; Pielke

2013). These surface fluxes are of crucial importance not

only because they influence the steady state of the atmo-

sphere, but also because they determine the mean pro-

files in the atmospheric boundary layer (Holtslag and

Nieuwstadt 1986; Beljaars and Holtslag 1991). The input

parameters, such as exchange coefficients, boundary layer

height, required in short-range forecasts and air pollution

modeling are dependent on the surface fluxes. Further,

in single-column and large-eddy simulation models for

the study of the atmospheric boundary layer, surface layer

sensible heat flux is frequently used as a boundary condi-

tion (Basu et al. 2008; Gibbs et al. 2015).

A number of studies (Taylor 1971; De Bruin 1994;

Malhi 1995; Derbyshire 1999; van de Wiel et al. 2007;

Basu et al. 2008; Wang and Bras 2010; van de Wiel et al.

2011; Gibbs et al. 2015) suggest that the MOS equations

have multiple solutions in the stable atmospheric condi-

tion, and thus it is not legitimate to use sensible heat flux

as a boundary condition in numerical models in stable

conditions. Basu et al. (2008) have pointed out the

shortcomings of using sensible heat flux as a surface

boundary condition in numerical models and suggested

that, in order to capture the very stable regime in nu-

merical models, one should prescribe surface tempera-

ture as a boundary condition in place of sensible heat flux.

Wang and Bras (2010) have pointed out that the

nonuniqueness of MOS equations is a numerical artifact

Supplemental information related to this paper is available at

the Journals Online website: http://dx.doi.org/10.1175/JAS-D-15-

0291.s1.

Corresponding author address: Prof. Maithili Sharan, Centre for

Atmospheric Sciences, Indian Institute of Technology Delhi, Hauz

Khas, New Delhi 110016, India.

E-mail: mathilis@cas.iitd.ernet.in

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and have postulated an extremum hypothesis of turbulent

transport in the atmospheric surface layer to derive a

unique solution for theMOS equations. The hypothesis of

Wang and Bras (2010, 2011) leads to possible simplifica-

tion of MOS equations by replacing the universal simi-

larity functions with some empirical constants in both

stable and unstable atmospheric conditions. The inter-

pretations of Wang and Bras (2010, 2011) are contradic-

tory to those reported in the literature by a number of

researchers in the stable atmospheric conditions. Conse-

quently, van de Wiel et al. (2011) have argued that the

nonuniqueness of MOS equations is not a mathematical

artifact and have provided a physical justification for the

heat flux maxima and dual nature of heat flux resulting in

nonexistence of any preferred stability state. However, the

study of van de Wiel et al. (2011) is limited to the stable

conditions, and, to the authors’ knowledge, the detailed

theoretical and observational analyses of the behavior of

sensible heat flux in daytime convective conditions have

not yet been carried out to assess the possible non-

uniqueness of MOS equations and its implications in nu-

mericalmodels.Hence, an attempt ismade here to analyze

the nature of sensible heat flux in an unstable atmospheric

surface layer within the framework of MOS theory.

2. Methodology

According to MOS theory, nondimensional gradients

of the mean wind speed U and virtual potential tem-

perature u in a homogeneous surface layer are universal

functions of the stability parameter z:

kz

u*

›U

›z5u

m(z) , (1)

kz

u*

›u

›z5u

h(z) , (2)

where k is the von Kármán constant, z is the height

above the ground, u* and u* are the friction velocity and

temperature scales, and um and uh are, respectively, the

similarity functions describing dimensionless wind and

temperature profiles in terms of z. Here, z is defined as

z5kzg

u

(2H/rCp)

u3

*, (3)

in which u is the mean potential temperature in the

layer, g is the acceleration due to gravity, r is density of

the dry air, Cp is specific heat capacity at constant

pressure, and H is the heat flux, defined as

H52rCpu*u*. (4a)

Substituting the value of u* from Eq. (4a) in Eq. (3), one

gets

H

rCp

!2

5kzg

u

u3*z. (4b)

We assume that z is finite (0,2z,‘) and nonzero

(z 6¼ 0). In the unstable atmospheric conditions, there

is a positive upward heat flux and negative temperature

gradient ›u/›z. Hence, the temperature scale u* and

stability parameter z are negative in unstable conditions.

As a result of this, the expression on the right-hand side

of Eq. (4b) is positive.

Substituting the value of u* from Eq. (2) in Eq. (4b),

one gets

H

rCp

!2

5 (kz)4g

u

(›u/›z)3

u3h(z)z

. (5)

A number of expressions for the universal similarity

functions um and uh have been proposed in the litera-

ture by various researchers (Kazansky and Monin 1956;

Ellison 1957; Yamamoto 1959; Panofsky 1961; Sellers

1962; Businger et al. 1971; Dyer 1974; Högström 1996;

De Bruin 1999; Wilson 2001). However, the Businger–

Dyer similarity functions are commonly used in atmo-

spheric models (WRF; Skamarock et al. 2008). Thus, in

the subsequent analysis, an expression for the function

uh in unstable conditions is taken as

uh(z)5Pr

t(12 g

hz)21/2 , (6)

in which Prt 5 (uh/um)z50 is the turbulent Prandtl number

for neutral stability and gh is a constant. The functions um

anduh differ on the values of Prt, gm, and gh. The constant

gm appears in the expression of um. The values of Prt, gm

and gh are Prt 5 0:74, gm 5 16, and gh 5 9 (Businger

et al. 1971); Prt 5 1 and gm 5 gh 5 16 (Dyer 1974); and

Prt 5 0:95, gm 5 19, and gh 5 11:6 (Högström 1996).

Substituting the expression ofuh fromEq. (6) into Eq.

(5), we obtain

H

rCp

!2

5 (kz)4g

u

(›u/›z)3

Pr3t (12 ghz)23/2

z. (7)

As ›u/›z, 0 and z, 0 in unstable conditions, without loss

of generality,we replace ›u/›zwith2›un/›z and zwith2h.

Accordingly, Eq. (7) can be rewritten as

H

rCp

!2

5 (kz)4g

u

(›un/›z)3

Pr3t (11 ghh)23/2

h. (8)

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Taking the square root of both the sides and retaining

the positive sign, as H is positive in unstable conditions,

one obtains

H*5 (kz)2ffiffiffig

u

r(›u

n/›z)3/2

Pr3/2t (11 ghh)23/4(h)1/2

, (9)

in which H*5H/rCp.

The behavior ofH* with h depends on the value of the

vertical potential temperature gradient (›un/›z) in sur-

face layer or the lapse rate. In the subsequent analysis,

the nature of H* with h is analyzed for two assumed

forms of vertical potential temperature gradient: (i) as a

constant and (ii) as a power-law function of heat flux.

a. Case I: Constant vertical temperature gradient

For a given constant d. 0 such that ›un/›z5 d, Eq. (9)

can be rewritten as

H*5 (kz)2ffiffiffig

u

rd3/2

Pr3/2t (11 ghh)23/4(h)1/2

. (10)

For determining maximum and minimum values, the van-

ishing of the first-order derivative of the function leads to

the critical points, and then the positive (negative) value of

the second-order derivative at the critical point ensures the

point is minimum (maximum). For this purpose, differen-

tiating Eq. (10) with respect to h, one gets

dH*

dh5

"(kz)2

ffiffiffig

u

r �d

Prt

�3/2#h23/2(11g

hh)21/4

2

3

�2(11 g

hh)1

3

2hg

h

�. (11)

For the critical points of H*, dH*/dh5 0, which leads

to a critical point h5hc 5 2/gh. The value of the second-

order derivative with respect to h at the critical point

hc 5 2/gh turns out to be

d 2H*

dh2

����h5hc

5

"(kz)2

ffiffiffig

u

r �d

Prt

�3/2#"

(2/gh)23/2(3)21/4

gh

4

#,

which is positive, as gh . 0 and d. 0. Thus,H* attains its

minimum value at hc. This suggests that, for a given

value of ›un/›z, H* decreases with h until it attains its

minimum value at hc and then starts increasing with

increasing instability. The corresponding minimum

value of Hmin is obtained from Eq. (10) as

Hmin

rCp

5

�27g2

h

4

�1/4

(kz)2ffiffiffig

u

r �d

Prt

�3/2

. (12)

In nondimensional form, Eq. (10) is written as

H

Hmin

5

�4

27g2h

�1/41

(11 ghh)23/4(h)1/2

. (13)

Thus, for a given temperature gradient, there exist two

values of z(52h) corresponding to the same magnitude

of heat flux (Fig. 1a).

b. Case II: Varied vertical temperature gradient

The vertical potential temperature gradient is

considered to vary as a power-law function of heat

flux: that is,

�›u

n

›z

�5b(H*)a , (14)

in which a and b are constants.

Using Eq. (14), Eq. (9) can be written as

(H*)(123a/2) 5 (kz)2ffiffiffig

u

rb3/2

Pr3/2t

1

(11 ghh)23/4(h)1/2

. (15)

To obtain the critical points, differentiating Eq. (15)

with respect to h, one gets

�12

3a

2

�(H*)(23a/2)dH*

dh

5

"(kz)2

ffiffiffig

u

r �b

Prt

�3/2#h23/2(11 g

hh)21/4

2

3

�2(11 g

hh)1

3

2hg

h

�. (16a)

Thus, in this case H* has a critical point at

h5hc 5 2/gh provided a 6¼ 2/3. The value of second-

order derivative at the critical point hc 5 2/gh turns

out to be

d 2H*

dh2

����h5hc

51

(12 3a/2)

1

H*(23a/2)

"(kz)2

ffiffiffig

u

r �b

Prt

�3/2#

3

"(2/g

h)23/2(3)21/4

gh

4

#.

(16b)

As b. 0 and gh . 0, d2H*/dh2 . 0, provided a, 2/3

and d2H*/dh2 , 0 for a. 2/3 [Eq. (16b)]. This suggests

that, for a. 2/3, H* first increases with h until it

attains a maximum value at hc 5 2/gh and then starts

decreasing with increasing value of h. The corre-

sponding maximum value of Hmax is obtained from

Eq. (15) as

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Hmax

rCp

5

"�27

4g2h

�1/4

(kz)2ffiffiffig

u

r �b

Prt

�3/2#1/(123a/2)

, a. 2/3.

(17)

In nondimensional form, Eq. (15) is written as

H

Hmax

5

"�4

27g2h

�1/41

(11 ghh)23/4(h)1/2

#1/(123a/2)

. (18)

The same critical point (i.e., h5 2/gh) is a point of min-

imum for a, 2/3 (Fig. 1c). Thus, there exist two values of

stability parameter z(52h), corresponding to the same

magnitude of heat flux (Figs. 1b,c), provided a 6¼ 2/3.

For a5 2/3, Eq. (15) leads to an identity

(kz)2ffiffiffig

u

r �b

Prt

�3/21

(11 ghh)23/4(h)1/2

5 1, (19)

and heat flux no longer appears in this equation.

3. Field observations

Data from three experimental programs carried out

under different topographical conditions are used in the

present study: (i) Ranchi, India (http://odis.incois.gov.

in/index.php/project-datasets/ctcz-programme/data);

(ii) the Met Office’s Cardington monitoring facility

(http://browse.ceda.ac.uk/browse/badc); and (iii) 1999

Cooperative Atmosphere–Surface Exchange Study

(CASES-99) (Poulos et al. 2002; http://www.eol.ucar.

edu/projects/cases99/).

a. Ranchi (India) dataset

The data used in the present study are obtained

from a fast response sensor [Campbell Scientific 3D

sonic anemometer CSAT3)] installed at 10-m height to

measure the three components of wind and tempera-

ture at a frequency of 10Hz, mounted over a 32-m

micrometeorological tower deployed in a remote and

grassland area of the Birla Institute of Technology,

Mesra, Ranchi (23.4128N, 85.4408E), India, with an

average elevation 609m above mean sea level in a

tropical region (Dwivedi et al. 2015). This study ana-

lyzes turbulent measurements for the year 2009 taken

at 10-m height with a sonic anemometer. The hourly

fluxes are calculated using an eddy correlation tech-

nique. The whole dataset is divided based on the wind

speed and the stability regimes (Srivastava and Sharan

2015). As the focus of the study is on unstable

FIG. 1. Variation of nondimensional heat flux with the stability parameter in unstable conditions.

(a) Case I, in which heat flux H is nondimensionalized by its minimum value Hmin; (b) the case

(›un/›z)5b(H*)a witha5 1, inwhichH is nondimensionalized by itsmaximumvalueHmax; (c) the

case (›un/›z)5b(H*)a with a5 0:5, in whichH is nondimensionalized by its minimum value.

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conditions, data during the daytime conditions with

z, 0 are used. The data that correspond to a wind

speed of less than 0.1m s21, the transition period be-

tween day and night, or before and after rainfall are

excluded. A total of 2865 hourly data points with z, 0

are obtained with 0.1 , U , 10m s21.

b. Met Office’s Cardington (United Kingdom)monitoring facility dataset

Another dataset used in this study is taken from the

meteorological tower installed at Cardington in Bed-

fordshire (52.068N, 0.258W) in theUnited Kingdomwith

average height of 29m above mean sea level. The site

is a large grassy field and relatively flat (Luhar et al.

2009). The dataset contains the high-frequency wind and

temperature observations recorded from the ultrasonic

anemometers at 10, 25, and 50m above the ground

surface and slow-response temperature measurements

at 1.2, 10, 25, and 50m obtained from the platinum re-

sistance thermometer. In the present study, 30-min-

averaged data at the 10-m level for the months of June,

July, and August of years 2007, 2008, and 2009 are used

to calculate the hourly averaged turbulent quantities.

After the analysis, data corresponding to daytime un-

stable conditions (i.e., z, 0) are chosen for the study. A

total of 2382 hourly data points with z, 0 and 0.1,U,10ms21 are selected for the study.

c. CASES-99 (United States) dataset

The CASES-99 experiment was performed near

Leon (37.388N, 96.148W) in southeastern Kansas in the

United States during the month of October 1999. The

turbulence measurements were taken on a 60-m tower

at 1.5 (0.5)-, 5-, 10-, 20-, 30-, 40-, 50-, and 55-m levels by

eight sonic anemometers. The experiment was focused

on the stable atmospheric boundary layer; however, it

provides good observations for the study of surface

layer characteristics during free convection (Poulos

et al. 2002; Steeneveld et al. 2005). The terrain of the

main experimental site was relatively homogeneously

flat and lacked obstacles in the surroundings. The

turbulence measurements taken from a sonic ane-

mometer at the 10-m level on a 60-m tower (Poulos

et al. 2002) are used in the present study. Detailed

information on the experiment is given in Poulos et al.

(2002). The turbulence at a sampling rate of 20Hz was

used in the analysis to calculate the fluxes. Hourly

turbulent fluxes at the 10-m level are calculated from

the observations by using an eddy correlation tech-

nique. After the analysis of turbulence data at the 10-m

level, 249 hourly averaged data points corresponding

to unstable conditions (i.e., z , 0) are selected for

the study.

4. Results and discussion

a. Theoretical analysis

An analysis is carried out to analyze the behavior of

heat flux with stability in unstable conditions within the

framework of MOS theory. The sensible heat flux ob-

tained from MOS equations in gradient form is

expressed as a function of stability parameter z and the

vertical potential temperature gradient ›u/›z. The be-

havior of the heat flux is analyzed by considering the

vertical surface layer potential temperature gradient as

(i) a constant and (ii) a power-law function of heat flux.

For a prescribed value of temperature gradient, heat flux

attains its minimum value at z522/gh. The non-

dimensional heat flux H/Hmin is observed to decrease

until it attains a minimum value at z522/gh and then

starts increasing with increasing instability (Fig. 1a).

This suggests that, for a given value of heat flux, there

exist two different values of stability parameter. Out of

these two values, one corresponds to weakly unstable

conditions, and the other one characterizes a relatively

stronger instability.

By considering the power-law profile for vertical

temperature as2›u/›z5b(H/rCp)a, the analysis shows

that, if a. 2/3, H attains its maximum value at

z522/gh. The magnitude of nondimensional heat flux

H/Hmax is small in near-neutral conditions. This in-

creases with increasing instability until it attains the

maximum value at z522/gh and then starts decreasing

with further increasing instability (Fig. 1b). The theo-

retical behavior of heat flux for 2›u/›z5b(H/rCp)a

remains similar to that for 2›u/›z5 d if a, 2/3 (Fig. 1c).

It is interesting to note that the extremum point remains

the same in all three conditions (i.e., z522/gh). The

extremum point implies the value of z at which heat flux

takes a maximum or minimum value.

The theoretical behavior of heat flux H with stability

appears to be consistent with that observed if one takes

2›u/›z5b(H/rCp)a with a. 2/3. This behavior of the

heat flux with stability may be justified as follows. In the

weakly unstable conditions, the magnitude of heat flux is

small because of a weak temperature gradient. In this

condition, the mechanical production of turbulence is

dominant. As the instability increases, the role of me-

chanical as well as thermal turbulence is significant, and

heat flux attains a maximum value at a critical point

z522/gh. However, beyond this point of maximum

heat flux, the thermal turbulence is more significant than

the mechanical turbulence, resulting in a large value of

2z but comparatively small value of H. Here, we have

analyzed the nature ofH as a function of z and ›u/›z and

found that existence of multiple values of z for a given

value of heat flux persists in the case of unstable

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conditions in a manner similar to that in stable condi-

tions. In the next section, we have analyzed three dif-

ferent datasets to support the theoretical behavior of H

in unstable conditions with the observed behavior with

respect to z.

b. Observational analysis

Figures 2a–e show the variation of heat flux with

stability in unstable conditions for the Ranchi, Car-

dington, and CASES-99 datasets. The variation of H

with the stability parameter z is bounded by a curve.

This curve first shows increasing behavior with2z until

it attains a peak at z’20:12 and then decreases further

with increasing instability. In the near-neutral condi-

tion, the value of the heat flux is small because of a

weak temperature gradient. It starts increasing with

instability and attains a maximum value around

z’20:12. Beyond this value, it again starts decreasing

with increasing instability. This shows that a given

value of heat flux characterizes two different

FIG. 2. Variation of heat flux H with respect to the stability parameter z in unstable conditions for (a) Ranchi,

India; (c) Cardington, United Kingdom; and (e) the CASES-99 datasets. (b),(d),(f) Five unstable sublayers similar

to those of Srivastava and Sharan (2015), as suggested byKader andYaglom (1990) andBernardes andDias (2010),

are shown for each of the datasets.

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atmospheric stability conditions. Of these two values,

one is associated with a weakly unstable condition,

whereas the other corresponds to a relatively stronger

instability. This observed nature ofH with z appears to

be similar to the stable conditions, where the same

value of H corresponds to near-neutral and very stable

atmospheric conditions (Taylor 1971; De Bruin 1994;

Malhi 1995; Derbyshire 1999; van de Wiel et al. 2007;

Basu et al. 2008;Wang and Bras 2010; van deWiel et al.

2011; Gibbs et al. 2015). The observations show that the

value of heat flux remains small in both near-neutral

conditions and very unstable conditions. It attains a

maximum value in between these two stability regimes

at approximately z’20:12.

The nature of H is further analyzed in the five

unstable sublayers, similar to Srivastava and Sharan

(2015), as suggested by Kader and Yaglom (1990) and

Bernardes and Dias (2010). These sublayers are dy-

namic (DNS), dynamic–dynamic convective transition

(DNS–DCS transition), dynamic convective (DCS), dy-

namic convective–free convective transition (DNS–FCS

transition), and free convective (FCS) based on the strict

ranges of the stability parameter z in each sublayer. The

quantitative description of the data in each sublayer is

given in Table 1. The data lying in the DNS and DNS–

DCS transition layers characterize weakly to moderately

unstable conditions, while data belonging to the DCS,

DCS–FCS transition, and FCS layers represent moder-

ately to strongly convective conditions. The heat fluxH

shows an increasing behavior with2z from DNS to the

DNS–DCS transition layer until it attains a peak at

z’20:12 (Figs. 2b,d,f). However, a decreasing trend of

H is observed from its peak value with increasing in-

stability. Table 2 shows the observed average value of

H in each sublayer. It is clear from Table 2 that the

average value of H increases from DNS to DNS–DCS

transition layers. However, the values decrease from

the DCS to the FCS sublayers suggesting that H in-

creases with increasing instability until it reaches its

maximum at the transition point of the DNS–DCS and

DCS sublayers and then decreases with increasing

instability.

The behavior appears to be similar for all three

datasets chosen in the present study (Figs. 2a,e). How-

ever, the relationship is relatively more pronounced in

Ranchi and Cardington data as compared to CASES-99

data. There is a variation in the point of extremum over

the different datasets. In the Cardington dataset, the

extremum point is observed to lie in the DCS sublayer.

However, the heat flux attains its maximum value in the

DNS–DCS transition sublayer over the Ranchi and

CASES-99 datasets. There might be a number of rea-

sons for the differences in the point of maximum heat

flux in unstable conditions. These include frequency of

occurrence of low-wind conditions, roughness lengths of

heat and momentum of the sites, and the orographic

features of the surrounding area. For example, signifi-

cant differences are observed in the values of the sta-

bility parameter for the same wind and temperature

values with different values of momentum roughness

length (Sharan and Srivastava 2014). However, a precise

physical reason for the differences in the point of max-

imum heat flux among the different datasets is not clear

to us at the moment.

The theoretical value of z corresponding to maximum

heat flux is found to be;20.22,20.125, and20.172 for

the values of gh fromBusinger et al. (1971), Dyer (1974),

and Högström (1996). The difference in the theoreti-

cal value arises as a result of the choice of the constant

gh in the functional form of uh. Also, the observed

TABLE 1. The quantitative description of the data in each of the unstable sublayers according to Bernardes and Dias (2010). Columns 1

and 2 represent the sublayer and the corresponding range of z, while columns 3–8 represent the number of data points for Ranchi, India;

Cardington; and CASES-99 datasets for different wind speeds U (m s21).

Sublayers Range of z

Number of hours

Ranchi, India Cardington CASES-99

U# 2 U. 2 U# 2 U. 2 U# 2 U. 2

DNS 20:04# z# 0 65 164 0 485 6 66

DNS–DCS transition 20:12# z,20:04 152 685 3 775 3 62

DCS 21:20# z,20:12 720 840 127 853 28 52

DCS–FCS transition 22:00# z,21:20 58 26 28 31 16 1

FCS z,22:0 119 38 52 32 11 4

TABLE 2. Average value of heat flux in each of the sublayers.

Sublayers

Heat flux (Wm22)

Ranchi, India Cardington CASES-99

DNS 35.41 42.13 93.32

DNS–DCS transition 120.50 89.40 145.60

DCS 111.54 106.19 117.40

DCS–FCS transition 87.90 103.67 66.50

FCS 77.84 95.87 54.00

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behavior of H with z from all the datasets is found to

be consistent with that obtained theoretically from

MOS equations when considering the vertical tem-

perature gradient as a power-law function of heat flux

having an exponent larger than 2/3. This is also in line

with the value taken by Rao and Narasimha (2006) in

their study.

Wang and Bras (2010) have analyzed the behavior of

H* and ›u/›z with respect to u* and observed that a

given value of ›u/›z corresponds to two values of u*unless ›u/›z reaches a minimum value, which results

in a unique solution of u*. They argued that the non-

uniqueness of theMOS equations in unstable conditions

is a mathematical artifact, and it persists as long as the

similarity functions are dependent on the stability pa-

rameter z. To overcome this problem, they have pro-

posed an extremumhypothesis to have a unique solution

for MOS equations, which leads to a possible simplifi-

cation of MOS equations by replacing the universal

similarity functions with some empirical constants.

However, the present study suggests that nonuniqueness

of MOS equations in the case of convective conditions

does not appear to be a mathematical artifact and is

consistent with the observations. Analysis of the obser-

vational data over three different sites suggests that the

value z522/gh, corresponding to the extremum value

of heat flux, which is interpreted by Wang and Bras

(2010, 2011) as the only physically consistent value of z,

appears to be one out of many possible values in un-

stable conditions similar to the stable conditions, as

pointed out by van de Wiel et al. (2011). The present

study supports the three-sublayer model of Kader and

Yaglom (1990) rather than the existence of any pre-

ferred stability state, as speculated by Wang and Bras

(2010, 2011).

5. Issues and limitations

In this study, the behavior of heat flux is analyzed with

respect to the stability parameter in the unstable surface

layer. Some underlying issues and the limitations asso-

ciated with the analysis are discussed.

a. Functional form of similarity functions

The conclusions drawn from the present study rely

on the nature of the similarity function uh(z) and the

assumptions on the form of the negative potential

temperature gradient. According to Monin–Obukhov

similarity theory, the nondimensional wind and tem-

perature profiles are universal functions of stability

parameter z. Detailed forms of these functions are not

given by the theory but must be determined using field

experiments. However, based on the theory, certain

asymptotic predictions can be made for the nature of

these similarity functions (Högström 1996). In the

neutral limit, um,h(z) must be a constant. For strong

instability, when buoyancy dominates the local tur-

bulence production, the dimensional analysis (Monin

and Yaglom 1971) suggests that um,h(z) should vary

as a21/3 power of the stability parameter. However,

the various field experiments have not always veri-

fied these theoretical limits (Högström 1996). In

view of these asymptotic limits, various expressions

for the similarity functions are proposed in the litera-

ture by researchers (Kazansky andMonin 1956; Ellison

1957; Yamamoto 1959; Panofsky 1961; Sellers 1962;

Businger et al. 1971; Dyer 1974; Högström 1996; De

Bruin 1999; Wilson 2001). The historical Kazansky–

Ellison–Yamamoto–Panofsky–Sellers (KEYPS) formula

(Lumley and Panofsky 1964) for the similarity func-

tion for momentum um (Kazansky and Monin 1956;

Ellison 1957; Yamamoto 1959; Panofsky 1961; Sellers

1962) is given by

u4m(z)2 gzu3

m(z)5 1,

in which g is a constant. This can be interpreted as an

interpolation function that recovers the theoretical

scaling for near-neutral conditions while capturing

the 21/3 power law in free convection as predicted by

MOS theory. However, theoretical justification of the

KEYPS equation remained heuristic and assumed a

constant heat-to-momentum eddy diffusivity, an as-

sumption that was negated by the Kansas experiment,

which prevented the widespread acceptance of the

KEYPS equation (Katul et al. 2011). Based on the

simple mixing-length model (Fleagle and Businger

1980), Businger derived the relationships um(z)5(12 gz)21/4 and uh(z)5Prt(12 gz)21/2 (Businger et al.

1971; Businger 1988), which are commonly referred to in

the literature as the Businger–Dyer (BD) relationships.

The derivation of the KEYPS formula requires the ratio

of heat to momentum eddy diffusivity (Kh/Km) to be a

constant, while the Businger–Dyer formula does not

assume a constant ratio of heat to momentum eddy

diffusivity (Businger 1988). The original KEYPS for-

mula can be easily recovered from the BD profiles for a

constant ratio of heat to momentum eddy diffusivity

(Businger 1988). The BD functions have been success-

fully evaluated using the famous Kansas dataset. A

major limitation of Businger–Dyer similarity functions

is their failure to recover the well-known classical free-

convection limit. However, Fleagle and Businger (1980)

pointed out that this might be attributed to the fact that,

in free convection, large eddies are associated with

varying vertical shear, and, therefore, there is shear

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production of turbulent kinetic energy near the surface

even though the shear averaged over many eddies van-

ishes (Fleagle and Businger 1980). Businger–Dyer sim-

ilarity functions are still commonly used in all the

climate, atmospheric, hydrological, and ecological ap-

plications or models of land surface processes when land

surface fluxes are to be coupled to the state of the at-

mosphere (Parlange et al. 1995; Katul et al. 2011).

De Bruin (1999) revised the mixing-length model

(Fleagle and Businger 1980) and proposed the expression

for similarity functions as uh 5 (12 3z)2/3(12 10z)21.

The expression proposed by De Bruin (1999) follows the

classical 21/3 power law in free convection [i.e.,

uh } (2z)21/3 for2z � 1]. Based on the reanalysis of the

Kansas dataset, Wilson (2001) proposed the expression

uh 5Prt[11 g(jzj)2/3]21/2 with g as a positive constant,

satisfying the classical free convection behavior.

We have carried out a similar analysis for these func-

tional forms of similarity functions. In the case of the

similarity function of De Bruin (1999), for a given tem-

perature gradient, heat flux decreases with increasing

instability until z521/11 and then slightly increases and

attains a constant value (see supplementary material file

Hflux_1.pdf). By considering the power-law profile for

vertical temperature as 2›u/›z5b(H/rCp)a, the analy-

sis shows that, if a. 2/3, H attains its maximum value at

z521/11 and then slightly decreases and attains a con-

stant value with further increasing instability. For the

similarity functions of Wilson (2001), no critical points

are found, which suggests the unique solution of MOS

equations (see supplementary material file Hflux_1.pdf).

b. Assumption on vertical potential temperaturegradient

The assumption that heat flux varies as a power-law

function of temperature gradient or vice versa is our hy-

pothesis, which can be partially addressed on the basis of

the classical flux gradient equation in which the non-

linearity is through the diffusivity parameter that is a

nonlinear function of heat flux (Wang and Bras 2010).

Also, this is justifiable from the observations (see sup-

plementary material file Hflux_2.pdf). Hence, for the

analysis, we have taken a simple hypothesis that the

temperature gradient can be expressed as a power-law

function of heat flux. However, the hypothesis that

(›un/›z)5b(H*)a leads to an identity for a5 2/3 given

as

(kz)2ffiffiffig

u

rb3/2

Pr3/2t

1

(11 ghh)23/4(h)1/2

5 1

Since the heat flux no longer appears in the expression, it

is the limitation of our hypothesis in this case. However,

in view of studies available in the literature, the fol-

lowing interpretation may be applicable. When the

ground is heated under low-wind conditions (i.e., wind

shear is negligible), Priestley (1955) pointed out that the

heat flux H follows the relation

H

rCp

ffiffiffiffiffiffiffig/u

pj›u/›zj3/2z2

5HN, (20)

where HN is a constant. This constant HN is termed the

nondimensional heat flux by Priestley (1955) and is

subsequently studied by Taylor (1956) and Dyer (1965,

1967). In the case of free convection, when the non-

dimensional heat flux is observed to be independent of

the stability of the convection-controlled layer, the heat

flux follows the 3/2 power law of the negative potential

temperature gradient. In the case of forced convection,

when the contribution of both the buoyancy and shear to

produce turbulent kinetic energy is significant,HN must

be a function of the stability of the atmosphere: that is,

HN5 f (z) . (21)

Thus, although our hypothesis for a5 2/3 leads to an

identity, it has an apparent significance in unstable

conditions.

6. Conclusions

A systematic mathematical analysis of the MOS

equations is carried out to analyze the theoretical be-

havior of heat flux H with respect to the stability pa-

rameter z in unstable conditions. The heat flux

H is expressed as a function of z and temperature

gradient ›u/›z, and two cases—(i) 2›u/›z5 d,

(ii)2›u/›z5b(H/rCp)a—are considered for the analysis.

For a given constant ›u/›z, the heat flux is observed to

decrease until it attains a minimum value at z522/gh

and then starts increasing with increasing instability.

However, for2›u/›z5b(H/rCp)a, with a. 2/3, the heat

flux increases with increasing instability until z522/gh

and then starts decreasing with further increasing in-

stability. From the analysis, it is found that multiple

values of z exist for a given value of H in both the con-

ditions with an identical point of extremum (i.e.,

z522/gh). Turbulent data over three different sites—

(i) Ranchi, India; (ii) the Met Office’s Cardington

monitoring facility; and (iii) CASES-99—are analyzed

to compare the observed nature of H with its theoretical

nature as predicted by MOS theory. The analysis of ob-

servational data over the three different sites reveals that

the behavior of H with z is consistent with that obtained

theoretically from MOS equations by considering the

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vertical temperature gradient as a power-law function of

heat flux having an exponent larger than 2/3. The data do

not suggest any preferred stability state in unstable con-

ditions, as speculated byWang andBras (2010, 2011). The

value z522/gh, corresponding to the extremum value of

heat flux, appears to be one out of many possible states in

unstable conditions, similar to the stable conditions as

argued by van de Wiel et al. (2011). The present study

suggests that the existence of multiple values of z for a

given value of H persists in the case of convective con-

ditions, similar to the case of stable atmospheric condi-

tions that was not reported in the earlier studies.

Acknowledgments. The authors wish to thank

Dr. Manoj Kumar for providing observational data for

Ranchi, the Met Office for use of Cardington measure-

ments, and the National Center for Atmospheric Re-

search (NCAR) for CASES-99 observations. This work

is partially supported by the Ministry of Earth Sciences,

Government of India, under the CTCZ program

(MoES/CTCZ/16/28/10); JC Bose Fellowship to MS

from DST-SERB, Government of India (SB/S2/JCB-79/

2014); and SRF to PS from University Grants Commis-

sion. The authors thank the reviewers for their valuable

comments and suggestions.

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