GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-10-689-2017Simple process-led algorithms for simulating habitats (SPLASH v.1.0): robust indices of radiation, evapotranspiration and plant-available moistureDavisTyler W.https://orcid.org/0000-0003-4312-919XPrenticeI. Colinc.prentice@imperial.ac.ukhttps://orcid.org/0000-0002-1296-6764StockerBenjamin D.https://orcid.org/0000-0003-2697-9096ThomasRebecca T.WhitleyRhys J.WangHanhttps://orcid.org/0000-0003-2482-1818EvansBradley J.Gallego-SalaAngela V.SykesMartin T.CramerWolfganghttps://orcid.org/0000-0002-9205-5812AXA Chair of Biosphere and Climate Impacts, Grand Challenges in Ecosystems and the Environment and Grantham Institute – Climate Change
and the Environment, Department of Life Sciences, Imperial College London, Silwood Park Campus, Ascot, UKDepartment of Biological Sciences, Macquarie University, North Ryde, New South Wales, AustraliaState Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, College of Forestry, Northwest Agriculture & Forestry University, Yangling 712100, ChinaTerrestrial Ecosystem Research Network (TERN) Ecosystem Modelling and Scaling Infrastructure (eMAST), Sydney, New South Wales, AustraliaFaculty of Agriculture and Environment, Department of Environmental Sciences, The University of Sydney, Sydney, New South Wales, AustraliaDepartment of Geography, University of Exeter, Exeter, Devon, UKDepartment of Physical Geography and Ecosystem Science, Lund University, Lund, SwedenMediterranean Institute of marine and terrestrial Biodiversity and Ecology (IMBE), Aix Marseille University, CNRS, IRD, Avignon University, Aix-en-Provence, Francenow at: United States Department of Agriculture-Agricultural Research Service, Robert W. Holley Center for Agriculture and Health, Ithaca, USAI. Colin Prentice (c.prentice@imperial.ac.uk)14February201710268970829February201615April201622December201616January2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/10/689/2017/gmd-10-689-2017.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/10/689/2017/gmd-10-689-2017.pdf
Bioclimatic indices for use in studies of ecosystem function,
species distribution, and vegetation dynamics under changing climate
scenarios depend on estimates of surface fluxes and other quantities, such as
radiation, evapotranspiration and soil moisture, for which direct
observations are sparse. These quantities can be derived indirectly from
meteorological variables, such as near-surface air temperature, precipitation
and cloudiness. Here we present a consolidated set of simple process-led
algorithms for simulating habitats (SPLASH) allowing robust approximations of
key quantities at ecologically relevant timescales. We specify equations,
derivations, simplifications, and assumptions for the estimation of daily and
monthly quantities of top-of-the-atmosphere solar radiation, net surface
radiation, photosynthetic photon flux density, evapotranspiration (potential,
equilibrium, and actual), condensation, soil moisture, and runoff, based on
analysis of their relationship to fundamental climatic drivers. The climatic
drivers include a minimum of three meteorological inputs: precipitation, air
temperature, and fraction of bright sunshine hours. Indices, such as the
moisture index, the climatic water deficit, and the Priestley–Taylor
coefficient, are also defined. The SPLASH code is transcribed in C++,
FORTRAN, Python, and R. A total of 1 year of results are presented at the local and
global scales to exemplify the spatiotemporal patterns of daily and monthly
model outputs along with comparisons to other model results.
Introduction
Despite the existence of dense networks of meteorological monitoring stations
around the world, plant ecophysiology and biogeography suffer from a lack of
globally distributed observational data, especially those central to the
estimation of ecosystem-level photosynthesis, including photosynthetic photon
flux density and soil moisture. To overcome this deficiency, we present
simple process-led algorithms for simulating habitats (SPLASH) for generating
driving datasets for ecological and land-surface models (e.g., monthly carbon
and water fluxes or seasonal plant functional trait distributions) from more
readily available meteorological observations.
SPLASH is a continuation of the STASH (static shell) model, which was
originally developed for modeling the climatic controls on plant species
distributions at a regional scale
. The intention of STASH was
to provide bioclimatic indices, reflecting the environment experienced by
plants more closely than either standard summary variables such as mean
annual temperature, or such constructions as “mean precipitation of the
warmest quarter”, while requiring only standard meteorological data as
input. A key component of STASH was a simple, physically based soil-moisture
accounting scheme, first developed by , which has been used
inter alia in the original, highly cited BIOME model ,
the general forest succession model (FORSKA) described by ,
and the Simple Diagnostic Biosphere Model . Despite the
subsequent development of more complex dynamic global vegetation models
and land-surface models, the relatively simple algorithms in STASH continue to have
many applications, including to new areas such as the distribution of plant
functional traits , assessment of climate-change
impacts on specific biomes , large-scale water resource
assessments e.g.,, and simple first-principles modeling of
primary production . The continuing utility of these algorithms
owes much to their robustness, which in turn depends on the implicit
assumption that vegetation functions predictably – so that, for example,
evapotranspiration occurs at a potential rate under well-watered conditions,
and is reduced as soil water is drawn down. STASH is thus unsuitable to
answer questions like the effect of imposed vegetation changes on runoff, or
modeling vegetation–atmosphere feedbacks. Much more complex models that
dynamically couple soil, vegetation, and atmospheric boundary layer processes
exist for such applications; however, their complexity brings a burden in
terms of lack of robustness and, potentially, large inter-model differences
.
Despite their long history of use, no single publication documents the
algorithms of the STASH model. This work aims to fill that gap to allow for
the continued development and use of these algorithms. As the new incarnation
of STASH, SPLASH provides the same physically based soil-moisture accounting
scheme with updated and corrected analytical expressions for the calculation
of daily radiation, evapotranspiration, and soil moisture. Included in this
documentation are the equation derivations, variable definitions, and
information regarding model assumptions and limitations. One notable
improvement is that we have discontinued the approximation of constant
angular velocity in the orbit of Earth around the Sun. This version is thus
suitable for palaeoclimate applications, whereby orbital precession (as well
as changes in obliquity and eccentricity) influences the seasonal
distribution of insolation. SPLASH also includes explicit consideration of
elevation effects on biophysical quantities.
Key model outputs include daily insolation (incoming solar radiation at the
top of the atmosphere) and net surface radiation (Ho and HN,
respectively); daily photosynthetic photon flux density (Qn); daily
condensation, soil moisture, and runoff (Cn, Wn, and RO); and daily
equilibrium, potential, and actual evapotranspiration (Enq,
Enp, and Ena). Unlike the STASH model, SPLASH
explicitly distinguishes potential and equilibrium evapotranspiration,
recognizing that under well-watered conditions the excess of the former over
the latter is a requirement for foliage to be cooler than the surrounding
air, as has long been observed under high environmental temperatures
e.g.,.
Input values of latitude, ϕ (rad), elevation, z (m),
mean daily near-surface air temperature, Tair (∘C),
and fractional hours of bright sunshine, Sf (unitless),
are used for calculating the daily quantities of net radiation and
evapotranspiration. Daily precipitation, Pn (mmd-1), is used
for updating daily soil moisture. Tair and Pn may be derived
from various sources, including the freely available daily-averaged air
temperature and precipitation reanalysis data from the Water and Global
Change (WATCH) program's meteorological forcing dataset .
Meteorological variables are also available in the Climatic Research Unit
(CRU) gridded monthly time series datasets , which may be
downscaled to daily quantities by means of quasi-daily methods (e.g., linear
interpolation). Cloud cover fraction, for example the simulated quantities
given in the CRU TS3.21 dataset, may be used to approximate Sf.
Penman's one-complement approximation based on the cloudiness fraction is
regarded here as a sufficient estimate of Sf.
The piecewise linear method of – an adaptation of the
Doorenbos–Pruitt estimation procedure – as used in the
development of the CRU cloudiness climatology gives similar
results.
We present SPLASH comprehensively re-coded in a modular framework to be
readable, understandable, and reproducible. To facilitate varied application
requirements (including computational speed), four versions of the code (C++,
FORTRAN, Python, and R) are available in an online repository (see
Sect. ). The algorithms as presented here focus on application
to individual site locations, but a natural extension is towards spatially
distributed grid-based datasets.
In line with the intention of the original STASH algorithms, we also present
bioclimatic indices at the monthly and annual timescales to exemplify the
analytical applications of the SPLASH model outputs.
Methodology
The implementation of the soil-moisture accounting scheme follows the steps
outlined by , where daily soil moisture, Wn (mm),
is calculated based on the previous day's moisture content, Wn-1,
incremented by daily precipitation, Pn (mmd-1), and
condensation, Cn (mmd-1), and reduced by daily actual
evapotranspiration, Ena (mmd-1), and runoff, RO
(mm):
Wn=Wn-1+Pn+Cn-Ena-RO,
where Pn is a model input, Cn is estimated based on the daily
negative net radiation, Ena is the analytical integral of the
minimum of the instantaneous evaporative supply and demand rates over a
single day, and RO is the amount of soil moisture in excess of the holding
capacity. An initial condition of Wn is assumed between zero and the
maximum soil-moisture capacity, Wm (mm), for a given
location and is equilibrated over an entire year by successive model
iterations (i.e., model spin-up). Under steady-state conditions, the SPLASH
model preserves the water balance, such that ∑Pn+Cn=∑Ena+RO.
To solve the simple “bucket model” represented by Eq. (), the
following steps are taken at the daily timescale: calculate the radiation
terms, estimate the condensation, estimate the evaporative supply, estimate
the evaporative demand, calculate the actual evapotranspiration, and update
the daily soil moisture. Daily quantities may be aggregated into monthly and
annual totals and used in moisture index calculations.
RadiationTop-of-the-atmosphere solar radiation
The calculation of Cn and Ena begins with modeling the
extraterrestrial solar radiation flux, Io (Wm-2). The equation
for Io may be expressed as the product of three terms :
Io=Iscdrcosθz,
where Isc (Wm-2) is the solar constant, dr
(unitless) is the distance factor, and cosθz (unitless) is the
inclination factor. Values for Isc may be found in the literature
e.g.,; a
constant for Isc is given in Table .
Nomenclature.
Instantaneous Swevaporative supply rate, mmh-1Dpevaporative demand rate, mmh-1Eqequilibrium evapotranspiration rate, mmh-1Eppotential evapotranspiration rate, mmh-1Eaactual evapotranspiration rate, mmh-1Ioextraterrestrial solar radiation flux, Wm-2INnet radiation flux, Wm-2ISWnet shortwave solar radiation flux, Wm-2ILWnet long-wave radiation flux, Wm-2Daily Wnsoil moisture, mmPnprecipitation, mmd-1Cncondensation, mmd-1ROrunoff, mmEnqequilibrium evapotranspiration, mmd-1Enppotential evapotranspiration, mmd-1Enaactual evapotranspiration, mmd-1Hosolar irradiation, Jm-2d-1HNnet surface radiation, Jm-2d-1HN+positive net surface radiation, Jm-2d-1HN-negative net surface radiation, Jm-2d-1Qnphotosynthetically active radiation, molm-2d-1Sffraction of bright sunshine hours, unitlessTairmean air temperature, ∘CMonthly Emqequilibrium evapotranspiration, mmmo-1Emppotential evapotranspiration, mmmo-1Emaactual evapotranspiration, mmmo-1ΔEmclimatic water deficit, mmmo-1αmPriestley–Taylor coefficient, unitlessMiscellaneous cosθzinclination factor, unitlessδdeclination angle, raddrdistance factor, unitlessεobliquity, radeeccentricity, unitlessEconwater to energy conversion factor, m3J-1γpsychrometric constant, PaK-1hhour angle, radhiintersection of evaporative rates hour angle, radhnnet radiation crossover hour angle, radhssunset hour angle, radiday of month (1–31)λtrue longitude, radLvlatent heat of vaporization of water, Jkg-1νtrue anomaly, radnday of year (i.e., 1–365)Natotal number of days in a year (e.g., 365)Nmtotal number of days in a given month (e.g., 31)ω̃longitude of perihelion, radϕlatitude, radPatmatmospheric pressure, Paρwdensity of water, kgm-3rusinδsinϕ, unitlessrvcosδcosϕ, unitlessrw1-βswτIscdr, Wm-2rx3.6×1061+ωEcon, mmm2W-1h-1sslope of saturated vapor pressure–temperature curve, PaK-1τtransmittivity, unitlessτotransmittivity at mean sea level, unitlesszelevation above mean sea level, m
The distance factor, dr, accounts for additional variability in Io that
reaches the Earth. This variability is due to the relative change in distance
between Earth and the Sun caused by the eccentricity of Earth's elliptical
orbit, e (unitless), and is calculated as follows :
dr=1+ecosν1-e22,
where ν (rad) is Earth's true anomaly. True anomaly is the measure
of Earth's location around the Sun relative to its position when it is
closest to the Sun (perihelion).
The last term, cosθz, attenuates Io to account for the Sun's
height above the horizon (measured relative to the zenith angle, θz),
accounting for the off-vertical tilt of Earth's rotational axis,
ε (i.e., obliquity). The inclination factor is calculated as
follows :
cosθz=sinδsinϕ+cosδcosϕcosh,
where ϕ (rad) is the latitude, δ (rad) is the
declination angle, and h (rad) is the hour angle, measuring the
angular displacement of the Sun east or west of solar noon (-π≤h≤π). Declination is the angle between Earth's equator and the Sun at solar
noon (h=0), varying from +ε at the June solstice to
-ε at the December solstice; the changing declination is
responsible for the change in seasons. For the purposes of ecological
modeling, δ may be assumed constant throughout a single day. See,
e.g., for the precise geometric equation representing δ:
δ=arcsinsinλsinε,
where λ (rad) is Earth's true longitude (i.e., the
heliocentric longitude relative to Earth's position at the vernal equinox)
and ε (rad) is obliquity (i.e., the slowly varying tilt of
Earth's axis). Several other methods are widely used for the estimation of
δ for a given day of the year
e.g., but are not recommended because
they do not account for the change in Earth's orbital velocity with respect
to the distance between Earth and the Sun, while Eq. () does.
The relationship between true longitude, λ, and true anomaly, ν,
is by the angle of the perihelion with respect to the vernal equinox,
ω̃ (rad) :
ν=λ-ω̃.
While the three orbital parameters (i.e., e, ε, and
ω̃) exhibit long-term variability (on the order of tens of
thousands of years), they may be treated as constants for a given epoch
(e.g., e=0.0167, ε=23.44∘, and
ω̃=283.0∘ for 2000 CE), or they may be calculated
using the methods of or for palaeoclimate
studies. presents a simple algorithm to estimate λ
for a given day of the year (see Appendix ).
The daily top-of-the-atmosphere solar radiation, Ho (Jm-2),
may be calculated as twice the integral of Io measured between solar noon
and the sunset angle, hs, assuming that all angles related to
Earth on its orbit are constant over a whole day:
Ho=∫dayIo=2∫h=0hsIo=86 400πIscdrruhs+rvsinhs,
where ru=(sinδsinϕ) and rv=(cosδcosϕ), both unitless.
The sunset angle can be calculated as the hour angle when the solar radiation
flux reaches the horizon (i.e., when Io=0) and can found by substituting
Eq. () into Eq. (), setting Io equal to zero, and
solving for h:
hs=arccos-rurv.
To account for the undefined negative fluxes produced by Eq. ()
for h≥hs and h≤-hs, Io should be set
equal to zero during these nighttime hours. To account for the occurrences of
polar day (i.e., no sunset) and polar night (i.e., no sunrise),
hs should be limited to π when ru/rv≥1 and
zero when ru/rv≤-1.
Net surface radiation
The net surface radiation, HN
(Jm-2), is the integral of the net surface radiation flux
received at the land surface, IN (Wm-2), which is
classically defined as the difference between the net incoming shortwave
radiation flux, ISW (Wm-2) and the net outgoing
long-wave radiation flux, ILW (Wm-2):
IN=ISW-ILW.
The calculation of ISW is based on the reduction in Io due to
atmospheric transmittivity, τ (unitless), and surface shortwave albedo,
βsw (unitless):
ISW=1-βswτIo.
A constant value for βsw is given in Table .
Atmospheric transmittivity may be expressed as a function of elevation (to
account for attenuation caused by the mass of the atmosphere) and cloudiness
(to account for atmospheric turbidity). At higher elevations, there is less
atmosphere through which shortwave radiation must travel before reaching the
surface. To account for this, presents an equation based on
the regression of Beer's radiation extinction function at elevations below
3000 m with an average sun angle of 45∘, which can be
expressed as follows:
τ=τo1+2.67×10-5z,
where z (m) is the elevation above mean sea level and τo
(unitless) is the mean sea-level transmittivity, which can be approximated by
the Ångstrom–Prescott formula:
τo=c+dSf,
where c and d are empirical constants (unitless) and Sf is
the fraction of daily bright sunshine hours (0≤Sf≤1).
Values for c and d are given in Table .
The calculation of ILW is based on the difference between outgoing
and incoming long-wave radiation fluxes attenuated by the presence of clouds,
which may be empirically estimated by :
ILW=b+1-bSfA-Tair,
where A and b are empirical constants and Tair (∘C) is
the mean near-surface air temperature. The outgoing long-wave radiation flux
used to derive Eq. () assumes a constant ground emissivity, which
is accurate under well-watered conditions. The incoming long-wave radiation
flux is modeled based on clear-sky formulae derived by .
Values for A and b are given in Table .
HN can be decomposed into its net positive, HN+
(Jm-2), and net negative, HN- (Jm-2),
components (i.e., HN=HN++HN-). Assuming
ILW is constant throughout the day and making substitutions for
Io into Eq. (), an expression for HN+ may be
derived as twice the integral of IN between solar noon (i.e.,
h=0) and the net surface radiation flux cross-over hour angle,
hn (rad):
HN+=2∫h=0hnIN=86 400πrwru-ILWhn+rwrvsinhn,
where rw=(1-βsw)τIscdr
(Wm-2).
Here, hn is the hour angle when ISW equals
ILW and can be found by setting IN=0 in
Eq. () and solving for h, following the same substitutions as
used for hs in Eq. (), and may be expressed as follows:
hn=arccosILW-rwrurwrv.
To account for the occurrences when the net surface radiation flux
does not cross the zero datum, hn should be limited to π when
(ILW-rwru)/(rwrv)≤-1 (i.e.,
net surface radiation flux is always positive) and zero when (ILW-rwru)/(rwrv)≥1 (i.e., net surface
radiation flux is always negative).
Complementary to HN+, HN- may be calculated as
twice the integral of IN between hn and solar
midnight, defined by the piecewise function of IN between
hn and hs and -ILW between hs and
solar midnight (i.e., h=π), given as follows (note that HN- is a
negative quantity):
HN-=2∫hnhsIN-∫hsπILW=86 400πrwrvsinhs-sinhn+rwruhs-hn-ILWπ-hn.
Figure shows an example of a half-day IN curve used
in the integrals defined in Eqs. () and ().
IN, which is at its peak at solar noon, crosses zero at
hn and reaches a minimum at hs. After sunset (i.e.,
h>hs), when ISW is zero, IN is equal to
-ILW. HN+ is represented as twice the integral under
the positive net radiation curve (solid gray line), above the zero line
(solid black line), and between the vertical lines of solar noon and
hn. HN- is represented as twice the integral below
the zero line and above the negative net radiation curve (dashed gray line).
Example of the net radiation flux curve between the hours of solar
noon (i.e., h=0) and solar midnight (i.e., h=π). The IN
curve is equal to the difference between the net incoming shortwave radiation
flux, ISW (solid red line), and the net outgoing long-wave radiation
flux, ILW (dotted blue line). Positive IN, shown
decreasing from solar noon to zero at the cross-over hour angle,
hn, is denoted with a solid gray line, while negative
IN, shown decreasing from zero to -ILW between
hn and the sunset hour angle, hs, and constant
between hs and solar midnight, is denoted with a dashed gray
line. The solid black horizontal line marks the datum of zero radiation.
Photosynthetically active radiation
The daily photosynthetically active radiation in units of photon flux
density, Qn (molm-2d-1), is calculated based on the
number of quanta received (moles of photons) within the visible light
spectrum, which also corresponds to the action spectrum of photosynthesis
:
Qn=1×10-6fFEC1-βvisτHo,
where βvis (unitless) is the visible light albedo and
fFEC (µmolJ-1) is the flux-to-energy conversion
factor . This factor takes into account both the portion of
visible light within the total solar spectrum (approximately 50 %
), and the mean number of quanta in the visible light energy
band (approximately 4.6 µmolJ-1). The 1×10-6 converts the units of Qn from µmolm-2d-1 to molm-2d-1. Values for βvis
and fFEC are given in Table .
Condensation
The daily condensation, Cn, may be expressed as the water-equivalent of
the absolute value of negative net radiation, HN-:
Cn=1×103Econ|HN-|,
where Econ (m3J-1) is the water-to-energy
conversion factor that relates the energy released or required for a unit
volume of water to evaporate or condense at a given temperature and pressure,
based on a simplification of the potential
evapotranspiration for a horizontally uniform saturated surface, which may be
expressed as follows:
Econ=sLvρws+γ,
where s (PaK-1) is the slope of the saturation vapor
pressure–temperature curve, Lv (Jkg-1) is the latent
heat of vaporization of water, ρw (kgm-3) is the
density of water, and γ (PaK-1) is the psychrometric
constant. Standard values may be assumed for certain parameters (e.g.,
Lv=2.5×106Jkg-1; ρw=1×103kgm-3; γ=65PaK-1);
however, equations for the temperature dependence of s and Lve.g., and the temperature and pressure
dependence of ρw and γe.g., are available (see
Appendix ).
The barometric formula may be used to estimate the atmospheric pressure,
Patm (Pa), at a given elevation, z (m), when
observations are not available. Assuming a linear decrease in temperature
with height, which is a reasonable approximation within the troposphere
(i.e., for z<1.10×104m), the following equation may be
used :
Patm=Po1-LzTogMaRL,
where Po (Pa) is the base pressure, To (K) is the base
temperature, z (m) is the elevation above mean sea level, L
(Km-1) is the mean adiabatic lapse rate of the troposphere, g
(ms-2) is the standard gravity, Ma
(kgmol-1) is the molecular weight of dry air, and R
(Jmol-1K-1) is the universal gas constant. Values for the
constants used in Eq. () are given in
Table .
Evaporative supply
The evaporative supply rate, Sw (mmh-1), is assumed to
be constant over the day and can be estimated based on a linear proportion of
the previous day's soil moisture, Wn-1:
Sw=ScWn-1Wm,
where Sc (mmh-1) is the supply rate constant (i.e.,
maximum rate of evaporation) and Wm (mm) is the maximum
soil-moisture capacity. Constant values for Sc and Wm
are given in Table .
Evaporative demand
The evaporative demand rate, Dp (mmh-1), is set equal to the
potential evapotranspiration rate, Ep (mmh-1), as
defined by . Ep usually exceeds the
equilibrium evapotranspiration rate, Eq (mmh-1), due to the
entrainment of dry air in the convective boundary layer above an evaporating
surface . Ep is related to Eq by
the Priestley–Taylor coefficient, which may be defined as 1 plus an
entrainment factor, ω:
Dp=Ep=1+ωEq.
The constant value used for ω is given in Table .
The calculation of Eq is based on the energy–water equivalence of
IN, ignoring the soil heat flux :
Eq=3.6×106EconIN,
where 3.6×106 converts the units of Eq from ms-1 to
mmh-1. Note that Eq is defined only for positive values (i.e.,
Eq=0 for IN<0). The Priestley–Taylor potential
evapotranspiration is preferred in this context to the general
Penman–Monteith equation for actual evapotranspiration
, which requires knowledge of stomatal and
aerodynamic conductances, or to any of the “reference evapotranspiration”
formulae that specifically relate to agricultural crops.
Daily equilibrium evapotranspiration, Enq (mmd-1), is based
on the integration of Eq. () for values of positive
IN, or simply the energy–water equivalence of HN+:
Enq=1×103EconHN+,
where 1×103 converts Enq from md-1 to mmd-1.
The daily demand, which is equal to the daily potential evapotranspiration,
Enp (mmd-1), may be calculated from Enq, as in
Eq. ():
Enp=1+ωEnq.
Actual evapotranspiration
The calculation of daily actual evapotranspiration, Ena
(mmd-1), is based on the daily integration of the actual
evapotranspiration rate, Ea (mmh-1), which may be
defined as the minimum of the evaporative supply and demand rates
:
Ea=minSw,Dp,
where Sw (mmh-1) is the evaporative supply rate,
defined in Eq. (), and Dp (mmh-1) is the
evaporative demand rate, defined in Eq. ().
The analytical solution to Ena may be expressed analogous to the
methodology used for solving Ho and HN and is defined as twice
the integral of Ea between solar noon and hn, which
comprises two curves: Sw for 0≤h≤hi and Dp for
hi≤h≤hn, where hi (rad) is the hour angle
corresponding to the intersection of Sw and Dp (i.e., when
Sw=Dp):
Ena=2∫h=0hnEa=2∫0hiSw+∫hihnDp,
which may be expressed as follows:
Ena=24πSwhi+rxrwrvsinhn-sinhi+rxrwru-rxILWhn-hi,
where rx=3.6×106(1+ω)Econ
(mmm2W-1h-1). The intersection hour angle, hi, is
defined by setting Eq. () equal to Eq. () and solving
for h:
hi=arccosSwrxrwrv+ILWrwrv-rurv.
To account for the occurrences when supply is in excess of demand during the
entire day, hi should be limited to zero when coshi≥1. For
occurrences when supply limits demand during the entire day, hi should be
limited to π when coshi≤-1.
Figure shows an example of the half-day evaporative supply
and demand rate curves. Dp (dashed red line) is at a maximum at solar noon
and decreases down to zero at hn, while Sw (dotted
blue line) is constant throughout the day. The point where Sw
equals Dp is denoted by the vertical bar at hi. Ea (solid
gray line), limited by supply during most of the day, follows the
Sw line between solar noon and hi. During the time between
hi and hn, Ea is no longer limited by supply and follows
the Dp curve. After hn, both Dp and Ea are
zero. Ena is represented by twice the area above the zero line
(horizontal solid black line), below the Ea line, and between the
vertical bars of solar noon and hn.
Example of actual evapotranspiration curve between the hours of
solar noon (i.e., h=0) and solar midnight (i.e., h=π). The evaporative
demand, Dp (dashed red line), is at a maximum at solar noon and zero at
the cross-over hour angle, hn. The evaporative supply,
Sw (dotted blue line), is constant throughout the day. The point
where supply is equal to demand denotes the intersection hour angle, hi.
Actual evapotranspiration (solid gray line) is defined as the minimum of
Sw and Dp throughout the day.
Runoff
The calculation of daily runoff, RO, is based on the excess of daily soil
moisture without runoff compared to the holding capacity, Wm, and
is given by the following equation:
RO=max0,Wn∗-Wm,
where Wn∗ (mm) is the daily soil moisture without runoff
(i.e., Eq. where RO =0).
Soil moisture
With analytical expressions for Cn, Ena, and RO (i.e.,
Eqs. (), (), and (), respectively), Wn
may now be calculated by Eq. (). Once Wn is calculated, the
following limits are checked:
0≤Wn≤Wm.
The calculation of RO in Eq. () should prevent Wn from being
greater than Wm, thus satisfying the upper limit of
Eq. (). The limiting effect of Sw on
Ena, through Eqs. () and (), should, in
most cases, prevent Wn from falling below zero and satisfy the lower limit
of Eq. (); however, due to the assumption that Sw
is constant throughout the day, there is the possibility that Ena+ RO may exceed Wn-1+Pn+Cn, resulting in negative Wn. In
these rare cases, in order to maintain the mass balance of the bucket model
presented in Eq. (), Ena is reduced by an amount
equal to the magnitude of the negative soil moisture.
Bioclimatic indices
One application of the SPLASH model is for the estimation of the surface fluxes required
for the calculation of bioclimatic indices. Typically described at longer
timescales (e.g., monthly or annually), the daily SPLASH fluxes can be
integrated to monthly and annual totals:
Xm,a=∑d=1Nm,aXd,
where X is a model output parameter at a given day (Xd), month
(Xm), or year (Xa) and N is the total number of
days to sum over for a given month (Nm) or a given year
(Na).
The following sections describe three common bioclimatic indices.
Moisture index
There exists a long history that includes several variants of the moisture
index, MI, also commonly referred to as the aridity index, AI, or
moisture ratio, MR . A current definition
describes MI as the ratio of annual precipitation to annual potential
evapotranspiration , given as follows:
MI=PaEap,
where Pa (mma-1) is the annual precipitation
and Eap (mma-1) is the annual potential
evapotranspiration as calculated by Eq. (); Pa and
Eap may be substituted with their multi-year means
(i.e., Pa‾ and Eap‾) if
available. Values less than 1 are indicative of annual moisture deficit.
Climatic water deficit
The climatic water deficit, ΔE, defined as the difference between the
evaporative demand (i.e., potential evapotranspiration) and the actual
evapotranspiration, has been shown to be a biologically meaningful measure of
climate as it pertains to both the magnitude and length of drought stress
experienced by plants . At the monthly timescale, this
index is calculated as follows:
ΔEm=Emp-Ema,
where ΔEm (mmmo-1) is the monthly climatic
water deficit, Emp (mmmo-1) is the monthly
potential evapotranspiration and Ema
(mmmo-1) is the monthly actual evapotranspiration.
Emp and Ema are the monthly
totals of Enp and Ena, respectively, calculated by
Eq. (). Values of ΔE may also be computed at the annual
timescale.
Priestley–Taylor coefficient
The Priestley–Taylor coefficient, α, is the ratio of actual
evapotranspiration to equilibrium evapotranspiration, which represents the
fraction of plant-available surface moisture
. At the monthly timescale, this is
defined as follows:
αm=EmaEmq,
where αm is the monthly Priestley–Taylor coefficient,
Ema is the monthly actual evapotranspiration and
Emq (mmmo-1) is the monthly equilibrium
evapotranspiration. Due to the entrainment factor, αm may
vary between zero (i.e., no moisture) and 1+ω (i.e., unlimited
moisture). Values of α may also be computed at the annual timescale.
Results
The methodology described in Sect. was translated into
computer application code (C++, FORTRAN, Python and R). The following
sections describe the year-long SPLASH simulation results (2000 CE) at the
local and global scales along with comparisons with other model results.
Local temporal trends and bioclimatic indices
The SPLASH model was run at six locations across North America (see
Fig. ), representing six distinct climate regions across
latitudinal and elevational gradients. A total of 10 years (i.e., 1991–2000) of
monthly CRU TS3.23 data (i.e., precipitation, air temperature, and cloudiness
fraction) were extracted from the 0.5∘× 0.5∘ pixel
located over each site. Air temperature and cloudiness fraction were assumed
constant and monthly precipitation was divided equally across each day of the
month. Fractional sunshine hours were calculated as the one-complement of the
cloudiness fraction. Orbital parameters (for paleoclimatology studies) were
assumed constant and calculated for the 2000 CE epoch based on the methods
of . Model constants were assigned as per
Table .
Map of Köppen–Geiger climate regions across North America
. Six locations are selected representing the following: (a) tundra – ET (51.8∘ N, 116.5∘ W; 1383 m a.s.l.);
(b) continental with warm summers – Dfb (44.7∘ N,
73.8∘ W; 383 m.a.s.l.); (c) temperate with dry summers –
Csb (37.8∘ N, 122.4∘ W; 16 m a.s.l.); (d) hot
arid desert – BWh (32.7∘ N, 114.6∘ W; 43 m a.s.l.);
(e) equatorial monsoon – Am (26.0∘ N, 80.3∘ W;
2 m a.s.l.); (f) cold arid steppe – BSk (22.2∘ N,
101.0∘ W; 1850 m a.s.l.).
The first year of the simulation (i.e., 1991) was iterated (approximately
twice) until the daily soil moisture, initialized at zero, reached
equilibrium, after which the model was spun-up for eight years (i.e.,
1992–1999). The results, shown in Figs. and
, are for the year 2000. Accompanying the daily SPLASH
results in Fig. , shown in red, are daily surface fluxes based
on the three-layer variable infiltration capacity (VIC) model, extracted from
the 1/16∘ pixel over each of the six sites from the datasets
provided by .
Daily simulations of net radiation, HN, soil moisture,
Wn, and evapotranspiration, En, for the six climate regions defined in
Fig. : (a) tundra, (b) continental with
warm summers, (c) temperate with dry summers, (d) hot arid
desert, (e) equatorial monsoon, and (f) cold arid steppe.
Black lines represent SPLASH modeled net radiation, soil moisture, and
evapotranspiration (potential in solid black and actual in dashed black). Red
lines represent VIC three-layer modeled surface fluxes from
for net radiation, soil moisture (layer 1 in solid red, layer 2 in dashed
red, and layer 3 in dotted red), and potential evapotranspiration. Days of
the year are represented along the x axis. Data are for 1 year
(2000 CE).
Figure a shows the daily results for a tundra region over
Banff National Park in Alberta, Canada, with a mean annual temperature of
-4∘C and annual precipitation of 986 mm. The SPLASH
HN depicts a bell-shaped curve characteristic for the Northern
Hemisphere. During the spring and summer months, SPLASH HN is
higher than the VIC results, which exhibit a lower HN during the
first half of the year. The SPLASH Wn remains saturated throughout the
year at a level between the second and third layers of VIC. SPLASH and VIC
Enp are similar in magnitude throughout the year with SPLASH
Ena following Enp all year.
Figure b shows the daily results for a continental warm summer
region over the Adirondack region of New York with a mean annual temperature
of 5 ∘C and annual precipitation of 1080 mm. There is
agreement between the SPLASH and VIC HN and Enp
throughout the year. The SPLASH Wn remains saturated throughout most of
the year with a dry-down period during mid- to late summer and a recovery
period during the autumn.
Figure c shows the daily results for a temperate region with
dry summers over the Bay Area of California with a mean annual temperature of
14 ∘C and annual precipitation of 594 mm. During the
dry summer months, SPLASH HN is slightly higher than VIC, during
which time the SPLASH Wn is depleted, causing moisture-limited
Ena to occur. Before which, during the winter and early spring,
the SPLASH Wn is saturated and Ena follows Enp.
Figure d shows the daily results for a hot arid desert region
in southwestern Arizona with a mean annual temperature of 23 ∘C and annual precipitation of 39 mm. Over the entire year, SPLASH
HN is higher than VIC, with the largest differences occurring
during the summer months. The SPLASH and VIC Wn are both consistently low
throughout the year. This water limitation is expressed in the low SPLASH
Ena. During the summer months, SPLASH Enp is
slightly higher than VIC.
Figure e shows the daily results for an equatorial monsoonal
region near the southern tip of Florida with a mean annual temperature of
24 ∘C and annual precipitation of 1500 mm. There is
agreement between SPLASH and VIC HN throughout the year; however,
SPLASH Wn is higher than all three layers of VIC except for a few days
following a large rain event in October. During the drier winter, there is a
slight moisture limitation shown in the SPLASH Ena. Throughout
the year, SPLASH Enp is slightly higher than VIC.
Figure f shows the daily results for a cold arid steppe region
in San Luis Potosí, Mexico with a mean annual temperature of
18 ∘C and annual precipitation of 346 mm. During the
winter, SPLASH HN is slightly higher than VIC. The SPLASH Wn
remains low throughout the year at a level between the first and second
layers of VIC. The moisture limitation results in a lower SPLASH
Ena throughout the year. The SPLASH and VIC Enp
agree during the year.
Monthly SPLASH results of evapotranspiration, Em
(potential in solid black, actual in dashed red, and equilibrium in dotted
black), climatic water deficit, ΔEm, and Priestley–Taylor
coefficient, αm, for the six climate regions defined in
Fig. : (a) tundra, (b) continental with
warm summers, (c) temperate with dry summers, (d) hot arid
desert, (e) equatorial monsoon, and (f) cold arid steppe.
Months of the year are represented along the x axis. Results are of 1 year (2000 CE).
Figure shows the SPLASH monthly integrated
evapotranspiration results (Emp in solid black,
Emq in dotted black, and Ema in dashed
red) along with two monthly bioclimatic indices: ΔEm and
αm. For both the tundra and continental climate sites
(Figs. a and b, respectively),
Ema is equivalent to Emp, which
results in constant indices for ΔEm (i.e., 0 mm)
and αm (i.e., 1.26). At the annual timescale, ΔEa and αa for these two sites are the same as
their monthly values and MI is greater than one, suggesting that these are
water-available sites.
Figure c shows the monthly SPLASH results for the temperate region
with dry summers. Similar to the daily results (i.e.,
Fig. c), during the dry summer, Ema
falls below the Emp and Emq curves. This
results in a positive ΔEm and a drop in αm
during the summer months. At the annual timescale, ΔEa is
619 mm, which is slightly higher than the annual precipitation (i.e.,
594 mm), and both αa and MI are less than 1
(i.e., 0.633 and 0.477, respectively), suggesting that this is a water-limited
site.
The hot arid desert region presents a more extreme case as shown in
Fig. d, where Ema is constantly below
both Emp and Emq. This results in a
positive bell-shaped ΔEm curve and a shallow bowl-shaped
αm curve. At the annual timescale, ΔEa is
1450 mm, which is significantly higher than the annual precipitation
(i.e., 39 mm). Also, both αa and MI are
significantly less than 1 (i.e., 0.236 and 0.0219, respectively),
suggesting that this is a critically water-limited site.
In contrast, at the equatorial monsoonal site, shown in
Fig. e, Ema closely follows the
Emp curve, which results in an almost zero ΔEm and a nearly constant 1.26 αm. At the annual
timescale, ΔEa is 29 mm, αa is
1.24, and MI is 0.985, which all suggest that this site is not water
limited.
Similar to the hot arid desert, at the high elevation of the cold arid
steppe, shown in Fig. f, Ema is
constantly below both Emp and Emq. Unlike
the hot arid desert site, the cold arid steppe site is at a lower latitude,
which results in a flatter HN curve (as shown in
Fig. f) that leads to a more constant
Emp curve. At the annual timescale, ΔEa is 905 mm, which is greater than the annual
precipitation (i.e., 346 mm). Both αa and MI are
less than 1 (i.e., 0.482 and 0.236, respectively), which suggests that this
is a water-limited site.
Global simulation of spatial patterns
For the global simulation, 0.5∘× 0.5∘ CRU TS3.23
data were assembled for 1 year (2000 CE), including monthly precipitation
(mmmo-1), monthly mean daily air temperature (∘C), and
monthly cloudiness fraction. Monthly precipitation was converted to daily
precipitation by dividing the rainfall equally amongst the days in the month.
Fractional sunshine hours were calculated based on the one-complement of
cloudiness fraction and assumed constant over the month. Mean daily air
temperature was also assumed constant over each day of the month. Half-degree
land-surface elevation (m above mean sea level) was provided by CRU
TS3.22 . Once again, orbital parameters were assumed constant
over the year and calculated for the 2000 CE epoch based on the methods of
, and model constants were assigned as per
Table .
Global mean net downward surface radiation flux (MJm-2)
for June (left) and December (right) 2000 CE from CERES EBAF (top), from the
SPLASH model (middle), and showing the differences between SPLASH and CERES EBAF
(bottom). (a) CERES EBAF June 2000; (b) SPLASH June 2000;
(c) difference between SPLASH and CERES EBAF June 2000;
(d) CERES EBAF December 2000; (e) SPLASH December 2000; and
(f) difference between SPLASH and CERES EBAF December 2000. Color
bars on the right are linear interpolations of results in megajoules per square meter.
Differences in red indicate higher SPLASH model results.
The SPLASH simulations were driven by the data described above, one pixel at
a time, starting each pixel with an empty bucket and terminating when a
steady state of soil moisture was reached between the first and last day of
the year. Following the spin-up to equilibrate the soil moisture, the model
was driven once again to produce daily simulations of net radiation and soil
moisture.
Figure b and e show the SPLASH
results of the mean daily net surface radiation flux (MJm-2) for
the months of June and December, respectively. For comparison, the Clouds and
the Earth's Radiant Energy System (CERES) Energy Balanced and Filled (EBAF)
average all-sky surface net total flux for June and December 2000 are plotted
in Fig. a and d, respectively. The CERES EBAF net
downward radiative flux was converted from watts per square meter to
megajoules per square meter.
Overall, the SPLASH model produces a reasonable simulation of the latitudinal
gradients and seasonal shifts of net surface radiation flux. The differences
between SPLASH and CERES EBAF net downward radiative flux are highlighted in
Fig. c and f. Regions in red indicate areas where the
SPLASH model results are higher than the CERES EBAF values, while regions in
gray indicate areas where the SPLASH model results are lower. The locations
where the SPLASH model disagrees with CERES EBAF tend to occur where the
well-watered constant surface albedo assumption fails, such as in deserts and
at high-latitude ice sheets and tundra.
Figure b and e show the SPLASH results of the mean daily
relative soil moisture (%) for the months of June and December,
respectively. An ice sheet was imposed over Greenland (i.e., no soil
moisture). For comparison, the National Center for Environmental Prediction
(NCEP) Climate Prediction Center (CPC) Version 2 mean soil moisture
for June and December of the same year are plotted in
Fig. a and d, respectively. The relative soil moisture in
both datasets is computed as the ratio of millimeters of soil moisture over the
total bucket size (i.e., 760 mm in NCEP CPC and 150 mm in
SPLASH).
Global mean relative soil moisture (%) for June (left) and
December (right) 2000 CE from NCEP CPC (top), from the SPLASH model (middle), and
showing the difference between SPLASH and NCEP CPC (bottom). (a) NCEP CPC
June 2000; (b) SPLASH June 2000; (c) difference between
SPLASH and NCEP CPC June 2000; (d) NCEP CPC December 2000;
(e) SPLASH December 2000; and (f) difference between SPLASH
and NCEP CPC December 2000. Color bars on the right are linear interpolations
of results in units of relative soil moisture (%). The relative soil
moisture is based on the total bucket size (i.e., 760 mm for NCEP CPC
and 150 mm for SPLASH). Differences in red indicate higher SPLASH
model results.
Overall, the SPLASH model simulates soil-moisture patterns similar to the
NCEP CPC model results. The differences between the SPLASH and NCEP CPC model
results are highlighted in Fig. c and f. Once again,
regions in red indicate where the SPLASH model results are higher than the
NCEP CPC model results and regions in gray indicate areas where the SPLASH
model results are lower. In contrast to the NCEP CPC soil moisture, the
SPLASH model produces a relatively full bucket across wet vegetated regions.
The lower relative fullness of the NCEP CPC bucket may be contributed to its
significantly larger bucket size. Despite the differing magnitudes of soil
moisture, the spatial distributions of soil moisture show consistently drier
regions in both simulations at both time periods, especially across mid-northern latitudes (e.g., eastern North America, northern Africa, and central
Asia). Seasonal shifts in soil moisture from June to December are also
consistently shown (e.g., southern transition in Africa, eastern transition
in South America, and northern transition in Australia). There are
discrepancies in the spatial distribution of soil moisture across the high-latitude regions (especially Russia). The predominantly saturated conditions
in the SPLASH simulations across Russia for December
(Fig. e) may actually be representative of an increasing
snow pack, which could account for these differences.
Discussion
The results presented in Sect. are intended to illustrate
the dynamic patterns and trends in the SPLASH model outputs across regions
and seasons for a single year under a steady state. The SPLASH model results
are promising despite the model's simplifications and limited climatic
drivers. At the local scale, the comparison between SPLASH and VIC across
climate and elevation gradients (i.e., Fig. ) shows relatively
good agreement for Enp. There are some discrepancies between
HN, especially at the high-latitude, high-elevation tundra site
(i.e., Fig. a) and at the low-elevation hot arid desert site
(i.e., Fig. d), where the SPLASH simulations were higher than
VIC for portions to all of the year. These discrepancies are likely due to
local deviations from the globally averaged surface albedo. This is
especially true when there is snow cover, as SPLASH does not model snowpacks.
Soil moisture also showed relatively good agreement, except at the equatorial
monsoonal site (i.e., Fig. e), where the SPLASH simulation was
consistently higher than VIC. This discrepancy may be due to the assumed
constant maximum soil-moisture holding capacity. Furthermore, at the global
scale, the SPLASH model reasonably captures the latitudinal gradation of net
surface radiation flux (where surface emission assumptions are valid)
compared to the CERES EBAF results (i.e., Fig. ) and
produces similar spatial patterns of soil moisture, albeit at different
magnitudes, compared to the NCEP CPC soil-moisture results (i.e.,
Fig. ).
While the methodology presented in Sect. makes numerous
assumptions and simplifications (e.g., saturation-excess runoff generation,
invariant soil properties, and constant global parameterization), it provides
a simple and robust framework for the estimation of radiation components,
evapotranspiration, and plant-available moisture requiring only standard
meteorological measurements as input. The SPLASH model currently only
produces saturation-excess runoff. For more realistic runoff generation,
other water balance models allow runoff to occur when the bucket is less than
full, for example the empirical relationship of runoff to the weighted
relative soil moisture in the simple water balance model .
Regarding the bucket size, in principle, Wm in Eq. ()
could be formulated as a property of soil type (as was done, for example, in
the original BIOME model); there are some objections to doing so. While
Wm has a standard definition in agronomy (i.e., the difference
between field capacity and wilting point), the wilting point in reality
depends on plant properties. Also, the effective “bucket size” depends on
rooting behavior, which is highly adaptable to the soil wetness profile. The
absolute value of daily soil moisture will be influenced by the bucket size
(as shown in Fig. ) and can have an impact on the local
hydrology (e.g., Fig. e); however, plant-available moisture
indexes, such as α (i.e., the ratio of supply-limited to
non-supply-limited evapotranspiration), have commonly been found to be
relatively insensitive to the bucket size. Regarding localized effects, the
standard values presented in Table are representative of
reasonable global means; however, it is recommended that local
parameterization (e.g., shortwave albedo) be used if and when data are
available.
Over the years, a common misconception has developed regarding the
calculation of daily actual evapotranspiration (as defined by
), whereby the integration of Eq. () is
mistakenly interpreted as follows:
Ena=minS,D,
where D (mmd-1) is the total daily demand, given by
Eq. (), and S (mmd-1) is the total daily supply
over the hours of positive net radiation, which may be given by the following:
S=∫daySw=∫-hnhnSw=24πhnSw,
where hn is the net radiation cross-over angle, given
by Eq. (), and the constant coefficient converts the units of
radians to hours. As shown in Fig. , Ena is a
piecewise function consisting of two curves overlaid throughout the course of
a single day that must be accounted for simultaneously; however, even in some
recent model developments, Ena is calculated using
Eq. (), including the equilibrium terrestrial biosphere
models BIOME3 and BIOME4 and the
Lund–Potsdam–Jena dynamic global vegetation model . Only under
specific circumstances will Eq. () produce correct
results. It is the intention of this work to provide a simple analytical
solution that correctly accounts for the integration of Eq. (),
which has been provided in the form of Eq. ().
Code availability
The code, in four programming languages (C++, FORTRAN, Python, and R), is
available on an online repository under the GNU Lesser General Public License
(https://bitbucket.org/labprentice/splash). The repository includes the
present release (v1.0) and working development of the code (with makefiles
where appropriate), example data, and the user manual. All four versions of
the code underwent and passed a set of consistency checks to ensure similar
results were produced under the same input conditions. The following
describes the requirements for compiling and executing SPLASH v.1.0.
For the C++ version, the code was successfully compiled and executed using
the GNU C++ compiler (g++ v.4.8.2) provided by the GNU Compiler Collection
(Free Software Foundation, Inc., 2016). It utilizes the C numerics library
(cmath), input/output operations library (cstdio), and the standard general
utilities library (cstdlib), and it references the vector container and string
type.
For the FORTRAN version, the code was successfully compiled and executed
using the PGI Fortran compiler (pgf95 v.16.1-0) provided by The Portland
Group – PGI Compilers and Groups (NVIDIA Corporation, 2016) and the GNU
Fortran compiler (gfortran v.4.8.4) provided by the GNU Compiler Collection
(Free Software Foundation, Inc., 2016).
For the Python version, the code was successfully compiled and executed using
Python 2.7 and Python 3.5 interpreters (Python Software Foundation, 2016). It
requires the installation of third-party packages, including NumPy (v.1.10.4
by NumPy Developers, 2016) and SciPy (v.0.17.0 by SciPy Developers, 2016) and
utilizes the basic date- and time-type (datetime), logging facility
(logging), Unix-style pathname pattern extension (glob), and miscellaneous
operating system interface (os) modules.
For the R version, the code was successfully compiled and executed using
R-3.2.3 “Wooden Christmas-Tree” (The R Foundation for Statistical
Computing, 2015).
Calculating true longitude
presents a method for estimating true longitude, λ,
for a given day of the year, n, that associates uniform time (i.e., a mean
planetary orbit and constant day of the vernal equinox) to Earth's angular
position. The formula is based on classical astronomy and is suitable for
calculations in palaeoclimatology. The algorithm begins with the calculation
of the mean longitude of the vernal equinox, λm0 (rad),
assumed to fall on 21 March:
λm0=212e+18e31+βsinω̃-14e212+βsin2ω̃+18e313+βsin3ω̃,
where β=1-e2. The mean longitude, λm
(rad), is then calculated for a given day based on a daily increment
with respect to the day of the vernal equinox (i.e., day 80):
λm=λm0+2πn-80Na-1,
where Na is total number of days in the year. The mean anomaly,
νm (rad), is calculated based on the equality presented
in Eq. ():
νm=λm-ω̃,
which is then used to determine the true anomaly by the following:
ν=νm+2e-14e3sinνm+54e2sin2νm+1312e3sin3νm,
and is converted back to true longitude by the following equation:
λ=ν+ω̃.
The resulting λ should be constrained to an angle within a single
orbit (i.e., 0≤λ≤2π).
Calculating temperature and pressure dependencies
The four variables used to calculate the water-to-energy conversion factor,
Econ, given in Eq. () have temperature and/or pressure
dependencies that may be solved using the equations presented here.
The temperature-dependent equation for the slope of the saturation vapor
pressure–temperature curve, s, can be expressed as follows :
s=2.503×106exp17.27TairTair+237.3Tair+237.32,
where s ranges from about 11 to 393 PaK-1 for Tair
between -20 and 40∘C. Please be aware of the
typographical error in this formula as presented in Eq. (7) of
where 237.3 is misrepresented as 273.3.
The temperature-dependent equation for the latent heat of vaporization,
Lv, may be expressed as follows :
Lv=1.91846×106Tair+273.15Tair+273.15-33.912,
where Lv ranges from about 2.558×106 to
2.413×106JK-1 for Tair between -20 and
40∘C.
The temperature and pressure dependence of the density of water,
ρw, may be expressed as follows :
ρw=ρoKo+CAPatm∗+CBPatm∗2Ko+CAPatm∗+CBPatm∗2-Patm∗,
where ρo (kgm-3) is the density of water at 1 atm,
Ko (bar) is the bulk modulus of water at 1 atm, CA
(unitless) and CB (bar-1) are temperature-dependent
coefficients, and Patm∗ (bar) is the atmospheric pressure
(i.e., 1Pa=1×10-5bar).
The equation for ρo is based on the work of :
ρo=∑i=08CiTairi.
The equation for Ko is also based on the work of :
Ko=∑i=05CiTairi.
The equations for CA and CB are given as follows :
CA=∑i=04CiTairi,CB=∑i=04CiTairi.
The coefficients for Tair in Eqs. () through
() are given in Table .
The temperature and pressure dependence of the psychrometric constant, γ, may be expressed as follows :
γ=CpMaPatmMvLv,
where Cp (Jkg-1K-1) is the temperature-dependent specific
heat capacity of humid air; Ma (kgmol-1) and
Mv (kgmol-1) are the molecular weights of dry air and
water vapor, respectively; Lv (Jkg-1) is the latent
heat of vaporization of water; and Patm (Pa) is the
atmospheric pressure. Constants for Ma and Mv are
given in Table . The temperature dependence of Cp may
be assumed to be negligible (e.g., Cp=1.013×103Jkg-1K-1) or calculated by the following :
Cp=∑i=05CiTairi,
for Tair between 0–100 ∘C. The coefficients of
Tair are given in Table .
The Supplement related to this article is available online at doi:10.5194/gmd-10-689-2017-supplement.
I. C. Prentice, M. T. Sykes, and W. Cramer developed the original model theory
and methods. A. V. Gallego-Sala, B. J. Evans, H. Wang, and T. W. Davis contributed to model improvements.
R. T. Thomas, R. J. Whitley, B. D. Stocker, and T. W. Davis transcribed the new model code and ran simulations.
The paper was prepared with contributions from all authors.
The authors declare that they have no conflict of
interest.
Acknowledgements
This work was primarily funded by Imperial College London as a part of the
AXA Chair Programme on Biosphere and Climate Impacts. It is a contribution to
the Imperial College initiative on Grand Challenges in Ecosystems and the
Environment, and the Ecosystem Modelling And Scaling Infrastructure (eMAST)
facility of the Australian Terrestrial Ecosystem Research Network (TERN).
TERN is supported by the Australian Government through the National
Collaborative Research Infrastructure Strategy (NCRIS). BDS funded by the
Swiss National Science Foundation (SNF) and the European Commission's 7th
Framework Programme, under grant agreement number 282672, EMBRACE project. WC
contributes to the Labex OT-Med (no. ANR-11-LABX-0061) funded by the French
government through the A∗MIDEX project (no. ANR-11-IDEX-0001-02). AGS
has been supported by a Natural Environment Research Council grant (NERC
grant number NE/I012915/1). VIC simulations utilized the Janus supercomputer,
which is supported by the National Science Foundation (award number
CNS-0821794) and the University of Colorado Boulder. The Janus supercomputer
is a joint effort of the University of Colorado Boulder, the University of
Colorado Denver, and the National Center for Atmospheric Research. CERES EBAF
data were obtained from the NASA Langley Research Center Atmospheric Science
Data Center. CPC soil moisture data provided by the NOAA/OAR/ESRL PSD,
Boulder, Colorado, USA, from their website at
http://www.esrl.noaa.gov/psd/.Edited by:
D. Roche Reviewed by: two anonymous referees
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