GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-10-2875-2017The Analytical Objective Hysteresis Model (AnOHM v1.0):
methodology to determine bulk storage heat flux coefficientsSunTingting.sun@reading.ac.ukhttps://orcid.org/0000-0002-2486-6146WangZhi-Huahttps://orcid.org/0000-0001-9155-8605OechelWalter C.https://orcid.org/0000-0002-3504-026XGrimmondSuec.s.grimmond@reading.ac.ukhttps://orcid.org/0000-0002-3166-9415Department of Meteorology, University of Reading, Reading, RG6 6BB, UKDepartment of Hydraulic Engineering, Tsinghua University, Beijing
100084, ChinaState Key Laboratory of Hydro-Science and Engineering, Tsinghua
University, Beijing 100084, ChinaSchool of Sustainable Engineering and the Built Environment, Arizona
State University, Tempe, AZ 85287, USAGlobal Change Research Group, Department of Biology, San Diego State
University, San Diego, CA 92182, USADepartment of Environment, Earth and Ecosystems, The Open University,
Milton Keynes, MK7 6AA, UKTing Sun (ting.sun@reading.ac.uk) and Sue Grimmond (c.s.grimmond@reading.ac.uk)27July2017107287528908December201610January201718June201719June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/10/2875/2017/gmd-10-2875-2017.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/10/2875/2017/gmd-10-2875-2017.pdf
The net storage heat flux (ΔQS) is important in the
urban surface energy balance (SEB) but its determination remains a
significant challenge. The hysteresis pattern of the diurnal relation between
the ΔQS and net all-wave radiation (Q∗) has been
captured in the Objective Hysteresis Model (OHM) parameterization of ΔQS. Although successfully used in urban areas, the limited
availability of coefficients for OHM hampers its application. To facilitate
use, and enhance physical interpretations of the OHM coefficients, an
analytical solution of the one-dimensional advection–diffusion equation of
coupled heat and liquid water transport in conjunction with the SEB is
conducted, allowing development of AnOHM (Analytical Objective Hysteresis
Model). A sensitivity test of AnOHM to surface properties and
hydrometeorological forcing is presented using a stochastic approach (subset simulation). The sensitivity test suggests that the albedo, Bowen
ratio and bulk transfer coefficient, solar radiation and wind speed are most
critical. AnOHM, driven by local meteorological conditions at five sites with
different land use, is shown to simulate the ΔQS flux well
(RMSE values of ∼ 30 W m-2). The intra-annual dynamics of OHM
coefficients are explored. AnOHM offers significant potential to enhance
modelling of the surface energy balance over a wider range of conditions and
land covers.
Introduction
The essential role of an integrated land surface model is to physically
predict the land–atmosphere interactions by resolving the transfer of energy,
water, and trace gases (Katul et al., 2012; Liang et al., 1994; Sellers et
al., 1997). Such land–atmospheric interactions are strongly modulated by the
partitioning of solar energy at the land surface (Chen and Dudhia, 2001;
McCumber and Pielke, 1981; Yang and Wang, 2014) which can be considered
through the surface energy balance (SEB) equation (Oke, 1988):
Q∗-ΔQS=QH+QE,
where Q∗, ΔQS, QH, and QE are the net
all-wave radiation, net storage, turbulent sensible, and latent heat fluxes,
respectively. Equation (1) distinguishes the available energy at the land
surface (left-hand side) from the heat transfer through turbulent transport
(right-hand side).
The turbulent and radiative fluxes (Q∗, QH, and QE) are
more readily measured using standard techniques (e.g. eddy-covariance
instruments, radiometry) than ΔQS
(Offerle et al., 2005; Pauwels and
Daly, 2016; Roberts et al., 2006; Wang, 2012). For ΔQS the
net energy stored or released by changes in sensible heat within the canopy
air layer, roughness elements (e.g. vegetation, buildings in an urban
environment), and the ground all have to be considered. The volume of interest
extends from the top of the roughness sub-layer to the depth in the ground
where the daily averaged vertical net heat conduction is zero (see Fig. 2 in
Masson et al., 2002); this presents very significant challenges of spatial
sampling.
Knowledge of ΔQS is crucial to a wide range of
processes and applications: from modelling turbulent heat transfer and
boundary layer development to predicting soil thermal fields. In rural sites,
or simple bare soil sites, the flux may be a small fraction of the net all-wave radiation (Oliphant et al., 2004). However, in areas where there is more
mass, such as cities, the term becomes much more significant
(Kotthaus and Grimmond, 2014a) and a key element of the SEB and well-known effects such as the urban heat island.
In urban systems a wide range of techniques have been used to estimate
ΔQS (Grimmond et al., 1991; Roberts et al., 2006). These
include the following:
Heat conduction approach: the weighted average of heat flows through all
urban materials and surfaces by solving heat conduction equations – e.g.
buildings, streets, vegetated lands
(Offerle et al., 2005; Wang et al., 2012; Yang
et al., 2014).
Thermal mass scheme: the storage heat is inferred from the changes in
thermal mass of all components of the urban system (Kerschgens and Kraus,
1990).
Heat flux plates: combined measurements from grass and paved surfaces
(Kerschgens and Drauschke, 1986; Kerschgens and Hacker, 1985).
Parameterization as a function of Q∗: either as a linear
function (Oke et al., 1981), hyperbolic (cotangent, secant) function
(Doll et al., 1985), or hysteresis relation (Camuffo and Bernardi, 1982).
The last of these is used in the Objective Hysteresis Model (OHM) (Grimmond et al.,
1991).
Residual: practical difficulties of direct measurement of ΔQS in urban areas, result in the SEB residual (i.e. Q∗+QF-QH+QE) frequently being the “preferred”
observations (Ao et al., 2016; Ching et al., 1983; Doll et al., 1985; Li et
al., 2015; Oke and Cleugh, 1987) (where QF is the anthropogenic heat
flux).
The focus here is on the OHM approach, which is forced by Q∗ and
accounts for the diversity of the surface materials (sub-facets i) in the
measurement source area of interest with weightings (f) for their two- or
three-dimensional extent (Grimmond et al., 1991):
ΔQS=∑ifia1,iQi∗+a2,i∂Qi∗∂t+a3,i,
where the a1, a2, and a3 coefficients are for individual facets
determined by least-square regression between ΔQS and
Q∗ using results from observations (e.g. asphalt road (Anandakumar,
1999), wetlands (Souch et al., 1998), forests
(Oliphant et al., 2004), or numerical modelling – e.g. urban
canyons (Arnfield and Grimmond, 1998) and roofs (Meyn and Oke, 2009). These
coefficients capture the net behaviour of a facet type in a typical setting,
rather than being required to identify the component materials within a facet
(e.g. multiple materials making up a roof, wall, with varying thermal
connectivity and individual properties). As such, OHM is one of the less
demanding parameterizations, yet does capture a more realistic understanding
of the relation between ΔQS by Q∗ compared with
other approaches. Despite the shortage of OHM coefficients for the wide range of
facet types found in cities, OHM captures the urban ΔQS
overall generally well (Grimmond and Oke, 1999; Järvi et al., 2011, 2014;
Karsisto et al., 2015; Roth and Oke, 1995).
OHM is a cornerstone in the urban land surface models SUEWS (Surface Urban Energy
And Water Balance Scheme; Järvi et al., 2011, 2014; Ward et al.,
2016) and LUMPS (Local-Scale Urban Meteorological Parameterization Scheme; Grimmond and Oke, 2002), and plays an essential role in determining
the initial energy partitioning at each time step of the models' simulations.
Previous modelling studies (Arnfield and Grimmond, 1998; Meyn and Oke, 2009)
have led to better understanding of the OHM coefficients. Solution of the
one-dimensional advection–diffusion equation of coupled heat and liquid water
transport by Gao et al. (2003, 2008) was used to explore the physical
relation of OHM coefficients a1 and a2 to the phase lag between
ΔQS and Q∗. However, insight into a3 remain
unclear
(Sun et al., 2013).
In this paper, the solutions of the one-dimensional advection–diffusion
equation of coupled heat and liquid water transport (Gao et al., 2003, 2008)
are employed with the SEB (Eq. 1) to investigate more fully the three OHM
coefficients, the outcomes of which lead to development of the Analytical
Objective Hysteresis Model (AnOHM) (Sect. 2). The Monte Carlo-based subset
simulation (Au and Beck, 2001) approach is then used to undertake a
sensitivity analysis of AnOHM to surface properties and hydrometeorological
conditions (Sect. 3). An offline evaluation of AnOHM's performance for five
sites with different land covers (Sect. 4) provides evidence that this is an
alternative approach to obtain OHM coefficients. Given that this allows
applications across a much wider range of environments and meteorological
conditions, we conclude that AnOHM has important implications for land
surface modelling (urban and non-urban).
Model developmentParameterization of storage heat flux ΔQS for a
land surface
For a given land surface (e.g. bare soil), the governing heat
conduction–advection equation can be written (Gao et al., 2003, 2010) as
∂T∂t=λ∂2Tdz2+W∂T∂z,
where T is the temperature at a reference depth z (positive downward),
t is time, λ is the thermal diffusivity, and W=∂λ/∂z-CW/Cgwφ is the soil water
flux density (Ren et al., 2000), with CW the volumetric heat
capacity of water, Cg the volumetric heat capacity of soil, w the pore
water velocity, and φ the volumetric soil water content.
The steady-periodic solution of Eq. (3) corresponding to the principal
Earth rotation frequency (ω=2π2π2424, in radh-1), with boundary condition
TS=ATSsinωt-γ+T‾S
is given by (Gao et al., 2003, 2010)
Tz,t=ATSexp-z/Msinωt-z/N-γ+T‾S,
where M=2λΔ+W, N=Δω,
and Δ=W2+W4+16λ2ω22; with T‾S, ATS, and γ
denoting the daily mean value, amplitude, and initial phase of surface
temperature, respectively, which need to be determined by the boundary
conditions imposed by the SEB.
From Fourier's law, the soil heat flux is then given by
Gz,t≡-k∂T∂z=kATSM2+N2MNexp-zMsinωt-zN-γ+δ,
where δ=arctanMN=arctan2λω(Δ+W)Δ and k is the thermal
conductivity. In particular, at the surface z=0, the ground heat flux
G0 is given by
G0t=kATSM2+N2MNsinωt-γ+δ,
and a simple written form of ΔQS (if only one surface) can
be given as
ΔQS=G0=cηsinωt+η,
where η=δ-γ and cη=kATSM2+N2MN .
Although the above derivation only considers the land surface made of a
single material type, the derived ΔQS (Eq. 8) can be
adapted for surfaces made of composite materials or volumes given appropriate
bulk/ensemble properties.
Parameterization of net all-wave radiation Q∗ for a land
surface
Given the parameterizations of incoming longwave radiation L↓,
outgoing longwave radiation L↑, sensible heat flux QH,
latent heat flux QE, and storage heat flux ΔQS as
follows:
L↓=εaσTa4,L↑=εsσTS4︸1+(1-εs)L↓︸2QH=ChUTS-Ta,QE=QH/β,ΔQS=G0.
The boundary condition imposed by
the SEB relation can be rewritten as
1-αK↓+εaσTa4-εsσTS4=ChU1+β-1TS-Ta+G0,
where the turbulent fluxes QH and QE are parameterized as functions
of temperature gradient TS-Ta with albedo α,
bulk transfer coefficient Ch, wind speed U, and Bowen ratio (β=QH/QE). Theoretically, the second part of Eq. (10) (i.e. 1-εsL↓) should be accounted for in the
estimation of L↑ (Oke, 1987); however, given that it is usually less
than ∼5 % of the first part of the equation (see full discussion in
Appendix A) for most land covers (Oke, 1987), here it is omitted from
consideration and in the development of AnOHM.
By assuming that the incoming solar radiation K↓ and air
temperature Ta follow sinusoidal forms through a day as function
of the mean value for the day (e.g. K‾↓) (Sun et
al., 2013),
K↓=AKsinωt+K‾↓,Ta=ATsinωt-τ+T‾a,
and introducing the solar radiation scale,
AK∗=1-αAK,
and longwave radiation scale (assuming εa≈εs≈ε as a first-order estimate (as AnOHM is
insensitive to this parameter; see Sect. 3.2); see clear sky of ∼ 0.85 (Staley and Jurica, 1972) and urban surfaces of ∼ 0.95 (Kotthaus et al., 2014):
AT∗=4εσT‾a3+1+β-1ChUAT=(fL+fT)AT=fAT,
where τ denotes phase differences between Ta and
K↓, the f=fL+fT consists of the longwave energy
redistribution factor: fL=4εσT‾a3 and a turbulent energy redistribution factor: fT=1+β-1ChU. Linearizing the fourth-order longwave
expressions of temperature at mean daily air temperature Ta‾ (Sun et al., 2013), the values of TS‾
and ATS are obtained:
T¯S=1-αfK‾↓+T‾aATS=fMNsinτNfM+ksinγ-kMcosγAT=1M∗2+N∗2sinτsinγ-ζAT=χγAT,
where ζ=arctanN∗/M∗, γ=ζ+arctansinτcosτ+AK∗/AT∗, M∗=1+k/(fM),
N∗=k/(fN) and χγ=1M∗2+N∗2sinτsinγ-ζ.
The net all-wave radiation Q∗ is parameterized as
Q∗=1-αK↓+εσTa4-εσTS4=1-αAKsinωt+K‾↓+fLTa-TS=cφsinωt+φ+fLf1-αK‾↓,
where φ=arctanχγsinγ-sinτfAK∗/fLAT∗-χγcosγ-cosτ and
cφ=fAK∗2fLAT∗2-χγcosγ-cosτ2+βγsinγ-sinτ2.
Derivation of AnOHM coefficients
Based on the above parameterizations of Q∗ (Eq. 21) and ΔQS (Eq. 8), together with OHM for a specific surface:
ΔQS=a1Q∗+a2∂Q∗∂t+a3,
the coefficients can be readily derived from the parameterization in Sect. 2.2, as
a1=cηcφcosη-φ,a2=cηωcφsinη-φ,a3=-cηcφcosη-φ⋅fTf1-αK‾↓=-a1⋅fTf1-αK‾↓.
In the densest parts of cities, the anthropogenic heat (QF) often has a
large influence on the SEB and it needs to be accounted for
(Allen et al., 2011; Chow et al.,
2014; Nie et al., 2014; Sailor, 2011). This requires the governing SEB
relation (Eq. 14) to be rewritten:
1-αK↓+εσTa4-εσTS4+QF=ChU1+β-1TS-Ta+G0.
Assuming QF is diurnally invariant (as a first-order estimate – e.g. Best
and Grimmond, 2016), the derivation (Sect. 2.2) can be extended to
include a first-order estimate of QF to obtain
a3F=-cηcφcosη-φ⋅fTf1-αK‾↓-QF=-a1⋅fTf1-αK‾↓-QF,
where a3F (subscript “F” indicates the inclusion of QF). The other
two coefficients remain unchanged.
Physical interpretations of AnOHM coefficients
Based on the parameterizations of AnOHM coefficients (Eqs. 23, 24, 25/27),
physical interpretations can be more fully described compared with OHM:
a1 characterizes the ratio of ΔQS and Q∗ and
depends on the energy scales (i.e. cη and cφ) and their
phase difference (i.e. η-φ). The energy scales, representing
daily amplitudes of ΔQS and Q∗, determine the
overall magnitude, while the phase difference moderates the ratio value.
a2 accounts for the temporal changes in ΔQS and
Q∗ by including the principal Earth rotation frequency ω,
in addition to the same determinants of a1 (i.e. cη,
cφ, and η-φ). The complementary sinusoidal functions,
with phase difference (i.e. sinη-φ and cosη-φ), in the formulations of a1 and a2
are inversely related with a stronger lag effect from a2, and less
contribution to ΔQS by Q∗ (i.e. smaller a1).
a3 (or a3F) indicates the baseline ΔQS determined
by energy redistribution factors (i.e. fT and f) and energy inputs
(i.e. K‾↓, and QF if anthropogenic heat is
considered) as well as a1. It can be inferred from Eq. (2) that the
nocturnal ΔQS is largely determined by a3 when the
absolute values and variability of Q∗ are small at night. A larger
daytime energy input (i.e. K‾↓, and QF if
anthropogenic heat is considered) suggests more heat released at night.
Sensitivity analysis
Given the complex dependence of AnOHM coefficients on surface properties
and meteorological forcing (Sect. 2.3), the impacts of these coefficients
are assessed further by a sensitivity analysis.
Subset simulation
To improve the computational efficiency of undertaking Monte Carlo
sensitivity analyses, subset simulation is used (Au and Beck, 2001).
This is an adaptive stochastic simulation procedure with particular
efficiency in analysing the short-tail of a distribution probability (while
also adaptable to long-tail scenarios) (Wang et al., 2011).
If the probability that a critical response Y exceeds a threshold y,P(Y>y), a range of exceedance regions can be
specified and sampled using Markov chains. Initially a direct Monte Carlo
method is used to choose possible values for the parameter of interest in the
anticipated range with a specified distribution (or probability distribution
function, PDF) of the uncertainty. From this (level 0), the first exceedance
level probability is determined, F1 at which PY>y1.
Then a Markov chain Monte Carlo (MCMC) procedure is used to generate samples
of a given conditional probability p0, leading to the exceedance of
y1 in the earlier simulations. This procedure is repeated, for
exceedance events Fi at which PY>yi=p0i,
i=1,2,3,…, until simulations reach a target
exceedance probability, e.g. associated with rare events or risk analysis.
Further details of this subset simulation process are provided in Wang et al. (2011).
Subset simulation efficiently generates conditional samples with Metropolis
algorithms (Hastings, 1970; Metropolis et al., 1953). This is the basis of
MCMC. To generate samples that successively approach a certain conditional
probability, a specific Markov chain is designed with the target PDF as its
limiting stationary distribution trend as its length increases. The selection
of a distribution is key as this controls the next sample
generated from the current one. Ideally, the distribution selection would
be automatic but this has an efficiency cost relative to the robustness
benefit. For the surface parameters (Table 1a) and hydrometeorological
forcing (Table 1b) analyses a normal distribution PDF is used (Au
and Beck, 2003; Au et al., 2007), with three conditional levels
(Nlevel=3) and a conditional probability of p0=0.1 – i.e. at each
level the highest 10 % of the outputs are considered to exceed the
intermediate threshold. As such, the three-level simulation can effectively
capture a rare event with the target exceedance probability of 10-4
(i.e. the probability of occurrence is less than 1 in 10 000) and generate
appropriate samples of different conditional probabilities.
Range of values used as basis for the sensitivity analysis: (a)
surface parameters and (b) hydrometeorological variables. All are assumed to have
normal PDF. Values of surface parameters are based on values reported in
Stull (1988).
Parameter/variable UnitMinMaxMeanStandard deviation(a) Surface Thermal conductivitykW m-1 K-1031.20.1Bulk material heat capacityCpMJ m-3 K-1042.00.04Albedoα–010.270.07Emissivityε–0.81.00.930.025Midday* mean Bowen ratio (inverse)β-1–0200.050.05Bulk transfer coefficientChJ m-3 K-10840.5(b) Hydrometeorological Amplitude or range of the daily incoming shortwave radiationAKW m-201200800200Mean daytime incoming shortwave radiationK‾↓W m-2050020050Amplitude or range of the daily air temperatureAT∘C01582Mean daily air temperatureT‾a∘C040307.5Phase lag between radiation and air temperatureτrad0π/2π/4π/10Mean daytime wind speedUm s-10420.5Mean daily water flux densityW10-7 m3 s-1 m-20100105
* midday period: 1000–1400 local standard time.
Characteristics of the flux towers at the study
sites.
SiteUK-LdnUS-WlrCA-NS5US-SRMUS-SO4Location51.50∘ N, 0.12∘ W37.52∘ N, 96.86∘ W55.86∘ N, 98.49∘ W31.82∘ N, 110.87∘ W33.38∘ N, 116.64∘ WLand cover classificationUrban/built-upGrasslandEvergreen needleleaf forestWoody savannasClosed shrublandsLand cover codeURBGRAENFWSACSHStudy year20112003200420042005ReferenceKotthaus and Grimmond (2014a, b)Klazura et al. (2006), Coulter et al. (2006)Goulden et al. (2006)Scott et al. (2009)Luo et al. (2007)
The metric S (in %), used to indicate the sensitivity of the model
output Y to a specific uncertainty parameter X (Wang et al., 2011), is
S=1Nlevel∑i=1NlevelEX|Y>yi-EXEX×100,
where i=1,2,…,Nlevel is the index of
conditional sampling level, EX is the expectation that the
unconditional distribution of a specific uncertainty parameter X, while
EX|Y>yi is the expectation of X at conditional level
i. A positive (negative) S indicates an increase will lead to increase
(decrease) in simulated value. Hence the sign of S indicates the impact of a
change in parameter uncertainty. The absolute magnitude of S indicates the
sensitivity.
This assessment does not consider if the simulated values have low
probability. Later analyses (Sect. 4) consider the simulation results
relative to observed fluxes.
Impacts of surface properties
Following the sensitivity analysis of AnOHM coefficients to the surface
properties, the distributions of conditional samples for thermal
conductivity k, bulk heat capacity Cp, and emissivity ε are
similar to the original proposal distributions (Fig. 1), implying weak
dependence of a1, a2, and a3 on these properties. However, for
albedo (α) both a2 and a3 are sensitive, but a1is
not; changes in inverse Bowen ratio (β-1) impact all three
coefficients; and the bulk transfer coefficient Ch impacts a1 and
a2, but has little effect on a3.
Histograms of conditional samples at different conditional levels
for surface property parameters (rows from top: thermal conductivity k in W m-1 K-1, heat capacity Cp in MJ m-3 K-1, albedo
α, emissivity ε, inverse Bowen ratio
β-1, and bulk transfer coefficient Ch in J m-3 K-1) with AnOHM coefficients as the model output (columns from left:
a1, a2 and a3). Each subplot x axis is the parameter value and
y axis is the PDF value. The original proposal distribution (dashed line)
and simulation levels (different colours) are shown.
Using S (Eq. 28) to quantify this, it is found that the surface properties
(k, Cp, and ε) have less sensitivity, with less skewed
conditional samples between levels, so S values close to 0 (Fig. 2). The S of
k is the largest of the three. From the S results for the α sensitivity
analysis (Fig. 2), it is apparent that an increase in α will
increase a1 while decreasing a2 and a3, whereas the reverse
occurs for β-1 and Ch (i.e. their decreases leads to larger
a2 and a3 values but smaller a1).
Relative variation in sensitivity (S, %, Eq. 28) to surface
parameters. See Fig. 1 for further details.
From this, the links between the key surface parameters and the storage heat
flux can be considered. With an increase in α, there is reduced solar
energy in the SEB. This reduces the temporal change in ΔQS
(smaller a2) and decreases the baseline value of ΔQS
(smaller a3); larger β-1 indicates that more available energy is
dissipated by QE than by QH, leading to decreased Ts and
ΔQS (smaller a1); a smaller portion of Q∗ will be dissipated by ΔQS (smaller a1) as the
increased Ch can facilitate the turbulent convection and thus increase
the total turbulent fluxes.
Impacts of hydrometeorological conditions
Similarly, the sensitivity of AnOHM to hydrometeorological variables is
explored (Fig. 3). The air temperature (range, mean) and water flux density
related variables (i.e. AT, T‾a, and W) have
minimal influence on the skewness of the conditional samples. In contrast,
the incoming shortwave (solar) radiation (range, mean) and wind-related
variables (i.e. AK, K‾↓, and U) and the phase
lag τ between K↓ and Ta have large impacts. In
terms of the greatest impact on the coefficients (a1, a2, and
a3): AK and U influences a1, τ impacts a2, and
a3 responds more to AK and K‾↓ than the
other variables.
Histograms of conditional samples at different conditional levels
for ambient forcing parameters (rows from top: incoming solar radiation
amplitude AK in W m-2 and its daytime mean K‾↓ in W m-2, air temperature amplitude AT
in ∘C and its daily mean T‾a in
∘C, the phase lag τ in rad between
K↓ and Ta, wind speed U in m s-1,
and water flux density W in m s-1) with AnOHM coefficients as the
model output (columns from left: a1, a2 and a3). As Fig. 1.
Variables that strongly modulate the interactions between ΔQS and Q∗ can be informed by the S results (Fig. 4).
For instance, a greater range in K↓ (i.e. larger AK) will
occur with larger energy input from solar radiation, leading to stronger
heating of the near-surface atmosphere and a smaller portion to ΔQS (smaller a1) but higher baseline ΔQS
(larger a3). This is consistent with a reduction in K↓¯ having a decrease in a3. The temporal change in ΔQS is highly correlated with the change in τ, an increase in
which implies a slower response of the surface to solar radiation and an
overall decrease in ΔQS (smaller a1, a2, and a3).
The greater sensitivity to τ of a2 is a key part of the original
hysteresis nature of the heating/cooling of a surface. The sensitivity
responses of a1, a2, and a3 to U are very consistent with
those to Ch, suggesting the similar pathway that turbulent fluxes (i.e.
QH and QE) modulate ΔQS. As W mostly influences
the heat conduction–diffusion in the underlying surface as thermal properties
(i.e. Cp and k), less dependence is observed on it. This is similar
with Cp and k.
Relative variation in sensitivity (S, %, Eq. 28) to forcing
parameters. See Fig. 3 for further details.
Model evaluation
In this section, the actual ability of AnOHM to determine the storage heat
flux relative to observations is evaluated using 30 min observations from
five sites of different land use/covers (Table 2). The measurements include
turbulent sensible and latent fluxes, along with incoming and outgoing
shortwave and longwave radiation and basic meteorological variables (see
Kotthaus and Grimmond, 2014a, b; Klazura et al., 2006; Coulter et al., 2006; Goulden et al., 2006; Scott et al., 2009; Luo et al.,
2007,
for details). Anthropogenic heat flux
QF at the urban site (i.e. UK-Ldn) is estimated using the GreaterQF
model (Iamarino et al., 2011); the heat storage flux ΔQS is thus estimated as the modified residual of urban energy
balance as ΔQS=Q∗+0.75QF-1.2QH+QE (Kotthaus and Grimmond, 2014a, b), which is then used in
this evaluation. A similar approach for estimating ΔQS
(i.e. residual of surface energy balance, ΔQS=Q∗+QF-QH+QE) is applied at the other (non-urban)
sites but with QF=0.
Surface properties used in AnOHM simulation for the study sites
based on calibration. The values of α and β are monthly
climatology from January to December and are used when observations are not
available (see Table 1 for notation definition).
AnOHM is first calibrated with observations under sunny conditions, when the
assumptions of AnOHM are best satisfied (i.e. diurnal cycles of
K↓ and Ta follow sinusoidal forms), to obtain
surface properties required by AnOHM (Table 3). As the Bowen ratio β varies daily and monthly (Kotthaus and Grimmond, 2014a, b),
β is either determined as the daily value if available, or based
on the observation-based monthly climatology (Table 3). The seasonality in
albedo α is accounted for also by using its monthly climatology
(Table 3). AnOHM is driven by atmospheric forcing (i.e. K↓,
Ta, and U) and/or their derived scales (AK, K‾↓, AT, T‾a, and τ) to
generate the OHM coefficients (i.e. a1, a2, and a3, see Fig. 5), from which the net heat storage flux ΔQS can be
predicted (Fig. 6) using the observed Q∗ with Eq. (2).
Intra-annual variations of OHM coefficients: (a)a1, (b)a2, and (c)a3. LOESS fits (solid lines) through the daily values
predicted by AnOHM and daily values (squares) measured at an asphalt road
site (Anandakumar, 1999) are shown. The LOESS (Cleveland and Devlin, 2012)
fitting is a locally weighted polynomial regression approach.
Monthly median (line) diurnal cycles and interquartile range (shaded)
values of ΔQS for AnOHM predictions (blue), OHM predictions
(orange) and observations (green) at (a) UK-Ldn (URB), (b) US-Wlr (GRA), (c) CA-NS5 (ENF), (d) US-SRM (WSA), and (e) US-SO4 (CSH) (see Table 2 for site
information). Statistics include average bias and RMSE (W m-2). The OHM
coefficients a1, a2, and a3 used for different land covers are:
0.553, 0.303, and -37.6 at the urban site (UK-Ldn) (Ward et al., 2016),
0.32, 0.54, and -27.4 at the grass-covered sites (US-Wlr and US-SRM)
(Grimmond and Oke, 1999), and 0.11, 0.11, and -12.3 at the forest-covered sites (CA-NS5 and US-SO4) (Grimmond and Oke, 1999).
To examine the seasonality of the OHM coefficients, rather than the daily
variations in hydrometeorological forcing, LOESS (LOcally wEighted
Scatter-plot Smoother; Cleveland and Devlin, 1988) curves are obtained to
filter out day-to-day variations in the OHM coefficients (see Appendix B for
a direct comparison of these coefficients by different modelling and
observational regression approaches). Intra-annual variations are found in
all the three OHM coefficients (Fig. 5), indicating the strong impact of
seasonality of meteorological conditions. These controls, as indicated by
Eqs. (23)–(25/27), are complex and will vary with local conditions. For
instance, comparison of OHM coefficients between the AnOHM predictions (LOESS
fitted solid lines in Fig. 5) and observations at an asphalt road site in
Alland, Austria, reported in Anandakumar (1999) (empty squares in Fig. 5) demonstrates differences in a1 (Fig. 5a) and a2 (Fig. 5b)
but general similarity in a3 (Fig. 5c). Compared to a1 and
a2, it is noteworthy that, in addition to the S results (see Fig. 4)
given the more explicit mechanism by which the atmospheric conditions
moderate a3 (see Eqs. 25 and 27), such seasonality in a3 is
predicted by AnOHM, and evident in the observations (Fig. 5c, also Ward et
al., 2013). Larger K‾↓ in warm seasons
(May–September) will lead to smaller a3 (Eqs. 25, 27) and
vice versa.
The AnOHM simulated and observed ΔQS agree well at the five
different land cover sites, with RMSE values of ∼ 30 W m-2. For
comparison purposes, it is noted that the urban land surface model comparison
(Best and Grimmond, 2015; Grimmond et al., 2011)
found ΔQS to be the most poorly represented among all the
SEB components with the best RMSE values of 53 W m-2 (Lipson et al.,
2017). Although the much smaller ΔQS RMSE obtained by AnOHM
uses a prescribed Bowen ratio in the offline evaluation, such improvement
indicates the ability of AnOHM to simulate a more consistent ΔQS with observations. Compared with OHM predictions (orange lines
in Fig. 6), AnOHM (blue lines in Fig. 6) better reproduces the
seasonality in ΔQS but gives larger bias at two sites with
natural land covers (i.e. US-SRM and US-SO4). This can be attributed to the
overestimates of nocturnal ΔQS by AnOHM. Overall, the
evaluation demonstrates good performance of AnOHM in predicting the long-term
ΔQS with clear seasonality reproduced across a wide range
of surface types.
Discussion and concluding remarks
In this study, the Analytical Objective Hysteresis Model (AnOHM) is developed
to obtain OHM coefficients across a wide range of surface and meteorological
conditions and to improve physical understanding of the interactions between
ΔQS and Q∗. The sensitivity of AnOHM to surface
properties and hydrometeorological conditions is analysed through Monte Carlo-based subset simulations (Au and Beck, 2001). The results
highlight the importance of the albedo, the Bowen ratio, and the bulk transfer
coefficient, and the importance of solar radiation and wind speed in
regulating the heat storage. The importance of albedo in modulating the heat
storage was also found by Wang et al. (2011), who also used the same subset
simulation approach with the single-layer urban canopy model (SLUCM; for details
see Kusaka et al., 2001). This demonstrates the consistency in heat
storage modelling between AnOHM and SLUCM. From the sensitivity results,
variations in OHM coefficients of a similar size may arise from either
surface property parameters or hydrometeorological forcing that are
associated with the same physical processes (see bulk transfer coefficient
Ch in Fig. 2 and wind speed U in Fig. 4). This supports the
ability of AnOHM in representing physical processes. An offline evaluation of
AnOHM using flux observations from five sites with different land covers
demonstrates its ability to predict the intra-annual dynamics of OHM
coefficients and shows good agreement between simulated and observed storage
heat fluxes. In particular, the seasonality in the OHM coefficient a3
observed in a previous study (Anandakumar, 1999) is well predicted by
AnOHM.
The limitations of AnOHM are important to consider. First, given the
assumption that the incoming solar radiation K↓ and air
temperature Ta diurnal cycles are sinusoidal, optimal performance
of AnOHM occurs under clear-sky conditions. The current parameterizations of
K↓ and Ta within AnOHM only consider the harmonics
of principal frequencies for formulation simplicity. More frequencies may
potentially resolve more realistic diurnal variations in K↓
and Ta. As the reflected part of L↓ (i.e. 1-εsL↓) is assumed negligible, and
similar emissivity values are assumed for sky and land surface (i.e.
εs≈εa≈ε), the
outgoing longwave radiation is underestimated. These simplifications greatly
facilitate the AnOHM formulation without qualitatively changing the final
results as the sensitivity analyses (see the minimal S values for ε in Fig. 2) demonstrate. The inclusion of water flux density W equips
AnOHM with an ability to investigate the hydrological impacts of the
underlying surface on land–atmosphere interactions. However, estimation of
W remains challenging (Wang, 2014) and the resulting uncertainty in the
final results warrants caution in conducting simulations over land covers
with strong soil moisture dynamics (e.g. grassland with high soil moisture
under clear-sky condition).
Despite these limitations, AnOHM does permit improved modelling of the
surface energy balance through its physically based parameterization scheme
for storage heat flux ΔQS. Compared to OHM, AnOHM has the
benefit of allowing ΔQS to be simulated for land covers for
which coefficients are not available and to allow for seasonal variability to
be accounted for. As AnOHM shares similar hydrometeorological forcing inputs
(i.e. K↓, Ta and U) to other land surface
models (LSMs), it can potentially be used within in LSMs to estimate ΔQS, or if turbulent fluxes are included to be a complete LSM. The
overall improvements from adopting AnOHM in modelling land surface processes
will be presented in forthcoming work in the SUEWS–AnOHM framework.
The Fortran source code for AnOHM can be obtained from the corresponding
authors upon request.
Rationale for a simplified formulation of outgoing longwave
radiation
In the formulation of outgoing longwave radiation L↑, a
simplified form (i.e. εsσTs4) is used for AnOHM
by ignoring part 2 of Eq. (10) (i.e. 1-εsL↓). The rationale for such simplification is that given
εs is usually larger than 0.9, 1-εsL↓ contributes a relatively small portion to the total
longwave component (Oke, 1987) and omission of this part is well accepted in
the parameterization of outgoing longwave radiation for land surface modelling
across various land covers (Bateni and Entekhabi, 2012; Lee et al., 2011;
Stensrud, 2007).
Using the parameterization of incoming longwave radiation in the AnOHM
framework (i.e. L↓=εaσTa4≈εsσTa4), we conduct
a sensitivity analysis of the ratio between the ignored part (i.e. 1-εsL↓) and total outgoing longwave
radiation (i.e. εsσTs4+1-εsL↓) at a constant air temperature of 20 ∘C and
find this ratio is generally less than 5 % given εs ranges
between 0.90 and 0.99 (Fig. A1).
Moreover, if 1-εsL↓ is included in
the net longwave radiation, the induced effect can be incorporated into a
modified sky emissivity
εa′=εsεa as
follows:
Lnet=L↓-L↑=L↓-εsσTs4+1-εsL↓=εsL↓-εsσTs4=εsεaσTa4-εsσTs4=εa′σTa4-εsσTs4.
Then by assuming ε≈εa′≈εs, the derivation following Eq. (18) still holds. The
sensitivity analysis suggests that the derived coefficients are insensitive
to ε (see S for ε in Fig. 2).
As such, we deem the omission of 1-εsL↓ will not qualitatively change the results of this
work.
Ratio between the second part of Eq. (10) (i.e. 1-εsL↓) and total outgoing
longwave radiation (i.e. εsσTs4+1-εsL↓) at a constant air
temperature of 20 ∘C.
Comparison in OHM coefficients between different modelling
approaches and observation regression
The comparison in OHM coefficients by different modelling and observational
regression approaches (Fig. B1) indicate AnOHM generally follows the
results by observation regression, whereas the typical coefficient values
adopted by OHM do not.
Comparison of OHM coefficients (left, central and right columns for
a1,
a2 and
a3, respectively) between different
modelling approaches and observation regression at five sites: UK-Ldn (a, b, c), US-Wlr (d, e, f), CA-NS5 (g, h, i), US-SRM (j, k, l) and US-SO4 (m, n, o). The blue dots denote the paired values between AnOHM and observation
regression. The orange lines represent the reference value used in OHM
simulations for land covers of grass and tree (Grimmond and Oke, 1999),
whereas the green lines show median values derived from results by
observation regression at corresponding sites.
The authors declare that they have no conflict of interest.
Acknowledgements
Funding is acknowledged from Met Office/Newton Fund CSSP- China (SG),
National Science Foundation of China (51679119, TS), and U.S. National
Science Foundation (CBET-1435881, ZHW). The authors thank Ivan Au
(University of Liverpool) for providing the Subset Simulation package. The
authors acknowledge the large number of people who have contributed to the
data collection, the agencies that have provided sites and the agencies that
funded the research at the individual sites. The US Department of
Energy's Office of Science funded AmeriFlux data (ameriflux-data.lbl.gov)
are from US-Wlr (PIs: David Cook and Richard L. Coulter), CA-NS5 (PI:
Mike Goulden), US-SRM (PI: Russell Scott) and US-SO4 (PI: Walt
Oechel, funded by San Diego State University and SDSU Field Stations
Program). The London data are supported by NERC ClearfLo (NE/H003231/1),
NERC/Belmont TRUC (NE/L008971/1), EUf7 BRIDGE (211345), H2020 UrbanFluxes
(637519), King's College London and University of Reading. In particular,
the authors thank Simone Kotthaus (University of Reading) for her
detailed preparation of the UK-Ldn site data. For access to the UK-Ldn site
data, please contact
c.s.grimmond@reading.ac.uk.Edited by: Chiel van Heerwaarden
Reviewed by: three anonymous referees
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