A Note on the Flux-Variance Similarity Relationships for Heat and Water Vapour in the Unstable Atmospheric Surface Layer

Abstract

Atmospheric surface layer (ASL) experiments over the past 10 years demonstrate that the flux-variance similarity functions for water vapour are consistently larger in magnitude than their temperature counterpart. In addition, latent heat flux calculations using the flux-variance method do not compare as favorably to eddy- correlation measurements when compared to their sensible heat counterpart. These two findings, in concert with measured heat to water vapour transport efficiencies in excess of unity, are commonly used as evidence of dissimilarity between heat and water vapour transport in the unstable atmospheric surface layer. In this note, it is demonstrated that even if near equality in flux-profile similarity functions for heat and water vapour is satisfied, the flux-variance similarity functions for water vapour are larger in magnitude than temperature for a planar, homogeneous, unstably-stratified turbulent boundary-layer flow.

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Page 1

A NOTE ON THE FLUX-VARIANCE SIMILARITY RELATIONSHIPS
FOR HEAT AND WATER VAPOUR IN THE UNSTABLE ATMOSPHERIC
SURFACE LAYER
Research Note
GABRIEL G. KATUL1,2and CHENG-I HSIEH1,2
1School of the Environment, Duke University, Durham, NC 27708-0328, U.S.A.;2Center for
Hydrologic Sciences, Duke University, Durham, NC 27708, U.S.A.
(Received in final form 27 April 1998)
Abstract. Atmospheric surface layer (ASL) experiments over the past 10 years demonstrate that
the flux-variance similarity functions for water vapour are consistently larger in magnitude than their
temperature counterpart. In addition, latent heat flux calculations using the flux-variance method do
not compare as favorably to eddy-correlation measurements when compared to their sensible heat
counterpart. These two findings, in concert with measured heat to water vapour transport efficien-
cies in excess of unity, are commonly used as evidence of dissimilarity between heat and water
vapour transport in the unstable atmospheric surface layer. In this note, it is demonstrated that even
if near equality in flux-profile similarity functions for heat and water vapour is satisfied, the flux-
variance similarity functions for water vapour are larger in magnitude than temperature for a planar,
homogeneous, unstably-stratified, turbulent boundary-layer flow.
Keywords: Atmospheric surface layer, Closure models, Flux variance method, Scalar transport,
Scalar variance similarity functions.
1. Introduction
Estimating heat and water vapour fluxes using the so-called flux-variance method
(FVM), first introduced by Tillman (1972), is gaining rapid popularity in surface
hydrology and micrometeorology (Wesely, 1988; Weaver, 1990; Lloyd et al., 1991;
De Bruin et al., 1991, 1993; De Bruin, 1994; Padro, 1993; Kustas et al., 1994;
Albertson et al., 1995; Paw U et al., 1995; Katul et al., 1995, 1996; Hsieh et
al., 1996; Bink, 1996; Bink and Meesters, 1997; Asanuma and Brutsaert, 1998;
Andreas et al., 1998). The method is based on surface-layer similarity theory for
scalar variances in the atmospheric surface layer (ASL) and is given by:
σc
c∗
= χc(ξ);
ξ =
−z
Lmo,
Lmo=
−ρu3
H
¯Tcp
∗
kg
?
+ 0.61E
?,
Boundary-Layer Meteorology 90: 327–338, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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GABRIEL G. KATUL AND CHENG-I HSIEH
where σc(= (c?2)1/2) is the standard deviation of the concentration fluctuation of
a scalar entity (C), overbar is time averaging, primed quantities indicate departure
from the mean state by turbulence, z is the height above the zero-plane displace-
ment height, Lmois the Obukhov length, E (= w?q?) is the evaporation rate, u∗
is the friction velocity, χc(ξ) is the flux-variance similarity for a given scalar C
(χT and χqare the flux-variance similarity functions for heat and water vapour,
respectively), c∗is a scalar concentration defined by
c∗=w?c?
u∗
,
k = 0.4 is the von Karman constant, H = ρcpw?T?is the sensible heat flux, cpis
the specific heat capacity of dry air at constant pressure, ρ is the mean air density,
w?, T?, and q?are the vertical velocity, temperature, and water vapour concentration
fluctuations, respectively,¯T is the mean air temperature, and g is the gravitational
acceleration.
Field experiments over the past 10 years demonstrated that:
(1) The flux-variance method reproduces eddy-correlation sensible heat flux well
(Lloyd et al., 1991; Albertson et al., 1995; Katul et al., 1995; Hsieh et al.,
1996).
(2) The latent heat flux comparisons between flux-variance predictions and eddy
correlation measurements are inferior to those of sensible heat for the same site
(Weaver, 1990; Padro, 1993; Katul et al., 1995; Roth and Oke, 1995; Asanuma
and Brustsaert, 1998).
(3) For unstable atmospheric conditions, heat is transported more efficiently than
water vapour (Katul et al., 1995; Roth and Oke, 1995; Bink, 1996; Asanuma
and Brutsaert, 1998).
(4) When χcis expressed as χc= Cc(ξ)−1/3, the similarity constant Ccfor heat
(hereafter referred to as CT) is smaller than that for water vapour (hereafter re-
ferred to as Cq). Many ASL field experiments (e.g., Wyngaard,1971; Sorbjan,
1989; Kader and Yaglom, 1990; Albertson et al., 1995; Katul et al., 1995; Roth
and Oke, 1995; Asanuma and Brutsaert, 1998) determined a CT ≈ 0.93–0.98
while others (e.g., Hogstrom and Smedman-Hogstrom, 1974; Ohtaki, 1985;
De Bruin et al., 1991; Katul et al., 1995; Roth and Oke, 1995; Asanuma and
Brutsaert, 1998) determined a Cq ≈ 1.1–1.5. We note that these ranges for
CT and Cqwere determined using different sensor types (e.g., for humidity,
these sensors include Lyman-α sensors, krypton hygrometers, and infrared gas
analyzers; for temperature, these sensors are mainly fine wire thermocouples
and triaxial sonic anemometer temperature). Hence, the fact that CT < Cq
cannot be due to systematic instrument biases. In addition, the measurements

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FLUX-VARIANCE SIMILARITY RELATIONSHIPS FOR HEAT AND WATER VAPOUR
329
of σq/q∗while scaling well with ξ−1/3are more scattered when compared to
σT/T∗, with a bias
σq
q∗
This inequality was used as a diagnostic for the lack of similarity between heat
and water vapour (e.g., Roth and Oke, 1995; Andreas et al., 1998; Asanuma
and Brutsaert, 1998). In fact, flux-profile similarity theory investigations re-
ported in Brutsaert and Parlange (1992) conclude good agreement between
eddy-correlation measured and predicted sensible and latent heat fluxes de-
rived from measured profiles of¯T and ¯ q. Asanuma and Brutsaert (1998) com-
pared flux-variance calculations for heat and water vapour fluxes with eddy-
correlation measurements and reported good agreement for heat but not water
vapour. In addition, Asanuma and Brutsaert (1998) reported that Cqdecreases
with stronger correlation between temperature and water vapour fluctuations.
In short, ASL field experiments over the past 10 years suggest that flux-profile
similarity functions for heat (=φh(ξ)) and water vapour (=φq(ξ)) appear to
be in closer agreement with each other when compared to their flux-variance
similarity function counterparts (i.e., χq(ξ) ?= χT(ξ)) for planar homoge-
neous flow (e.g., Brutsaert, 1982; Ohtaki, 1985; Stull, 1988; Garratt, 1992;
Kaimal and Finnigan, 1994). We note many other studies have demonstrated
that φh(ξ) ?= φq(ξ) even for a planar homogeneous flow (e.g., Warhaft, 1976;
Bink, 1996).
>σT
T∗.
The objective of this note is to demonstrate that even if φh(ξ) ≈ φq(ξ), χT(ξ)
must be smaller than χq(ξ) for a planar-homogeneous unstable ASL in the absence
of subsidence, advective transport, and flux-divergence. For advective conditions,
it is known that φh(ξ) ?= φq(ξ) as demonstrated by Lang et al. (1983) and Mc-
Naughton and Laubach (1998) and that Cq?= CTas demonstrated by de Bruin et
al. (1991), Katul et al. (1995), and Bink (1996). The proposed analytical approach
is based on second-order closure principles applied to the flux-budget equations for
heat and water vapour.
2. Theory
The steady-state one-dimensional equations for the budgets of w?T?and w?q?, in
the absence of flux-divergence terms but including the buoyancy production terms,
are:
∂w?T?
∂t
= 0 = −w?2∂¯T
= 0 = −w?2∂ ¯ q
∂z− T?∂p?
∂z− q?∂p?
∂z+ gβT?2,
∂w?q?
∂t ∂z+ gβT?q?,
(1)

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GABRIEL G. KATUL AND CHENG-I HSIEH
where β = 1/¯T, p is the static pressure, and w?2(=σ2
variance. In order to obtain expressions that are strictly dependent on surface-layer
similarity flow variables, the classical Rotta type parameterization for the scalar-
pressure covariances is used (Mellor, 1973; Mellor and Yamada, 1974; Launder et
al., 1975; Lumley, 1978; Moeng and Wyngaard, 1986). These parameterizations
are:
w) is the vertical velocity
c?∂p?
∂z
=Cs
τw?c?,
(2)
where c?isa scalar concentration fluctuation (e.g., T?orq?), Csis aclosure constant
(Lumley, 1978), τ = Ttke/¯ ? is a relaxation time scale for dissipating the turbulent
kinetic energy (Ttke), and ¯ ? is the mean turbulent kinetic energy dissipation rate.
Upon replacing this closure model in the flux-budget equations, we obtain:
T?2=
1
gβ
?
w?2∂¯T
?
∂z+ Csw?T?
τ
?
?
,
T?q?=
1
gβ
w?2∂ ¯ q
∂z+ Csw?q?
τ
.
(3)
The flux-profile similarity formulations for¯T and ¯ q is:
∂¯T
kz
∂ ¯ q
kz
∂z=−φh(ξ)T∗
∂z=−φq(ξ)q∗
,
,
(4)
where T∗= w?T?/u∗, q∗= w?q?/u∗, φh(ξ) and φq(ξ) are, as before, the stability
correction functions for heat and water vapour flux-profile relations, respectively.
Upon replacing these expressions in the flux-budget equations, we obtain:
T?2=
1
gβ
?
−w?2φh(ξ)T∗
?
kz
+ CsT∗u∗
τ
?
,
T?q?=
1
gβ
−w?2φq(ξ)q∗
kz
+ Csq∗u∗
τ
?
.
(5)
By noting that T?q?is RT,qσTσq, σqcan be replaced by the identity
σq=
1
RT,q
σT
T?2T?q?.
(6)

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FLUX-VARIANCE SIMILARITY RELATIONSHIPS FOR HEAT AND WATER VAPOUR
331
Upon replacing the formulations in (5) for T?q?and T?2in (6), we obtain:
?
1
gβ
kz
σq=
1
RT,qσT
1
gβ
−w?2φq(ξ)q∗
−w?2φh(ξ)T∗
kz
+ Csq∗u∗
+ CsT∗u∗
τ
?
?.
?
τ
(7)
Upon dividing both sides by q∗and rearranging, (7) simplifies to:
?
?
Equation (8) explicitly relates σq/q∗to σT/T∗for a planar homogeneous unsta-
ble ASL in the absence of scalar flux divergences. Simplifications to (8) in the
context of the flux-variance similarity functions and heat/water vapour transport
efficiencies are discussed next.
σq
q∗
=
1
RT,q
σT
T∗
−w?2φq(ξ)
−w?2φh(ξ)
kz
+ Csu∗
+ Csu∗
τ
?
?.
kz
τ
(8)
3. Results and Discussions
3.1. FLUX-VARIANCE SIMILARITY FUNCTIONS FOR HEAT AND WATER
VAPOUR
If φh(ξ) = φq(ξ), then (8) reduces to:
σq
q∗
RT,q
and
=
1
σT
T∗,
(9a)
χq(ξ) =
1
RT,qχT(ξ).
(9b)
Since the correlation coefficient RT,qis less than unity (as demonstrated by almost
all ASL field experiments over homogeneous and heterogeneous surfaces), then
χq(ξ) > χT(ξ).
(10)
Note that (10) assumes similarity in mean temperature and water vapour profiles
(i.e., φh(ξ) = φq(ξ)). That is, inequality in the flux-variance similarity relation-
ships does not necessarily imply dissimilarity in the flux-profile relationships.