Transcript Slide 1
Energy Balance at the Land Surface
Control volume
IE
Ein
Eout
(internal energy)
Basic energy balance equation:
EinDt = DIE + EoutDt
Energy Balance for a Single Land Surface Slab, Without Snow
Terms on
LHS come from
the climate model.
Strongly dependent
on cloudiness, water
vapor, etc.
Sw
Sw
Lw
Lw
H
T
lE
Terms on
RHS come are
determined by
the land surface
model.
Swi + Lwi = Swh + Lwh + H +lE + CpDT + miscellaneous
where
Swi = Incoming shortwave radiation
Lwi = Downward longwave radiation
Swh = Reflected shortwave radiation
Lwh = Upward longwave radiation
H = Sensible heat flux
l = latent heat of vaporization
E = Evaporation rate
Cp = Heat capacity of surface slab
DT = Change in slab’s temperature, over the time step
miscellaneous = energy associated with soil water freezing, plant
chemical energy, heat content of precipitation, etc.
Other energy balances can also be considered. For example:
Energy balance of a vegetation canopy
Energy balance in a surface layer
Sw
Note: same
symbols are
used, but
values will
be different.
Sw
Sw
Lw
Lw
H
lE
Sw
Lw
Internal
energy
T1
Tc
Sw Sw
Lw Lw
H lE
Lw
G12
T2
T3
H H
G12 = heat flux between soil layers 1 and 2
Energy balance in a subsurface layer
Energy balance in snowpack
Sw
T1
Internal
energy
T2
T3
G12
Sw
Internal
energy
Lw
Lw
Tsnow
H l sE
GS1
lmM
T1
G23
lm = latent heat of melting
ls = latent heat of sublimation
M = snowmelt rate
GS1 = heat flux between bottom of pack and soil layer 1
In practice, several energy balance calculations
may be combined into a single “model”
Sw
Sw
Lw
Lw
H
lE
Tc
Sw Sw
Lw Lw
Sw
lE
H H
Sw
Lw
Lw
Tsnow
T1
G12
T2
G23
H l sE
GS1
lmM
T3
The trick is to keep the fluxes between the “control volumes” consistent. If the energy balance calculation
for the snowpack includes a flux GS1 from the bottom of the pack to the ground, then the energy balance for
the top soil layer must include an input flux of G S1.
In the above example, a total of five energy balances are computed: one for the canopy, one for the snowpack,
and one for each of three soil layers. Note that some models may include additional soil layers or may
divide the snowpack itself into layers, each with its own energy balance.
Reflected Shortwave Radiation
S
# bands
Assume: Sw =
b=1
+ S Sw
# bands
Sw
direct, band b
Compute: Sw =
S
b=1
diffuse, band b
reflectance for
spectral band
# bands
b=1
Sw
+ S b=1 Sw
direct, band b
a direct, band b
# bands
Simplest description: consider
only one band (the whole
spectrum) and don’t
differentiate between diffuse
and direct components:
albedo
Sw = Sw a
diffuse, band b
a diffuse, band b
Typical albedoes (from Houghton):
sand
.18-.28
grassland .16-.20
green crops .15-.25
forests
.14-.20
dense forest .05-.10
fresh snow .75-.95
old snow
.40-.60
urban
.14-.18
July Total Net Flux (W/m**2)
Color range: blue - red - white, light
green = 0 W/m**2, Values: -50 250W/m**2
Global mean = 111W/m**2, Minimum =
-65W/m**2, Maximum = 249W/m**2
July Surface Albedo
Color range: blue - red - white, gray:
undefined, Values: 0 - 1
Global mean = 0.16, Minimum = 0.05,
Maximum = 0.90
NASA Langley Atmospheric Sciences Data Center
http://srb-swlw.larc.nasa.gov/DataSets/sample.html
Upward Longwave Radiation
Stefan-Boltzmann law: Lw = e s T4
where e = surface emissivity
s = Stefan-Boltzmann constant = 5.67 x 10-8 W/(m2K4)
T = surface temperature (K)
Emissivities of natural
surfaces tend to be slightly
less than 1, and they vary
with water content. For
simplicity, many models
assume e = 1 exactly.
http://www-surf.larc.nasa.gov/surf/pages/ems_bb.html
Sensible heat flux (H)
Spatial transfer of the “jiggly-ness” of molecules, as represented by temperature
Equation commonly used in climate models: H = r cp CH |V| (Ts - Tr),
where
r = mean air density
cp = specific heat of air, constant pressure
CH = exchange coefficient for heat
|V| = wind speed at reference level
Ts = surface temperature
Tr = air temperature at reference level (e.g., lowest GCM grid box)
For convenience, we can write this in terms of the aerodynamic resistance, ra:
H=
r cp (Ts - Tr)
ra
where ra = 1/ (CH |V|)
H=
r cp (Ts - Tr)
ra
Why is this form convenient? Because it allows the use of the Ohm’s law analogy:
Tr
V1
R
Electric
Current
Sensible
heat flux
Ts
V2
Current = Voltage difference / Resistance
I = (V2 – V1) / R
ra
H=
r cp (Ts - Tr)
ra
The aerodynamic resistance, ra, represents the difficulty with which heat (jiggliness of
molecules) can be transferred through the near surface air. This difference is strongly
dependent on wind speed, roughness length, and buoyancy, which itself varies with
temperature difference:
10000
H=
1000
ra (s/m)
r cp (Ts - Tr)
ra
100
10
1
-10
0
10
Ts - Tr
Idealized picture
LATENT HEAT FLUX: The energy used to tranform
liquid (or solid) water into water vapor.
Latent heat flux from a liquid surface: lvE, where
E = evaporation rate (flux of water molecules away from surface)
lv = latent heat of vaporization
= (approximately) (2.501 - .002361T)106 J/kg
Latent heat flux from an ice surface: lsE, where
ls = latent heat of sublimation
= lv + l m
lm = latent heat of melting = 3.34 x 105 J/kg
For the purpose of this class, lv and lv will both be assumed constant.
We can then discuss the latent heat flux calculation in terms of the
evaporation calculation.
Now, some definitions.
es(T) = saturation vapor pressure: the
vapor pressure at which the condensation
vapor onto a surface is equal to the upward
flux of vapor from the surface.
Clausius-Clapeyron equation:
l
es(T) varies as exp(-0.622
)
RdT
Useful approximate equation:
es(T) = exp(21.18123 – 5418/T)/0.622,
where T is the temperature in oK.
Specific humidity, q: Mass of vapor per mass of air
qr = 0.622 er /p
(p = surface pressure, er = vapor pressure)
Dewpoint temperature, Tdew: temperature to which air must be reduced to begin
condensation.
Relative humidity, h: The ratio of the amount of water vapor in the atmosphere
to the maximum amount the atmosphere can hold at that temperature.
Note: h = er/ es(Tr) = es(Tdew)/ es(Tr) = qs(Tdew)/ qs(Tr)
Potential Evaporation, Ep: The evaporative flux from an idealized, extensive
free water surface under existing atmospheric conditions. “The evaporative
demand”.
Four evaporation components
Transpiration: The flux of moisture drawn out of the soil and then released
into the atmosphere by plants.
Bare soil evaporation: Evaporation of soil moisture without help from plants.
Interception loss: Evaporation of rainwater that sits on leaves and ground
litter without ever entering the soil
Snow evaporation: sublimation from the surface of the snowpack
Evaporation from a fully wetted surface (=Ep)
Here’s the famous Penman equation:
“equilibrium evaporation for
a saturated air mass passing
over a wet surface”
Epenman =
D = d(es)/dT
(Rnet - G)D + (rcp/ra) (es(Tr) - er)
g = cpp/(0.622l)
Contains terms that are
relatively easy to measure
contribution due to
subsaturated air
D+g
G = heat flux into Rnet = net
ground
radiation
The Penman equation can be shown to be equivalent to
the following equation, which lies at
vapor pressure at
the heart of the potential evaporation
reference level
calculation used in many climate
=es(Tdew)
models:
er
Evaporative
flux
ra
Ep =
0.622r es(Ts) - er
p
ra
es(Ts)
Note: the ra used here is that same as that
used in the sensible heat equation. Does that
make sense?
Here’s one rationale:
The air is full of eddies. Buoyancy tends
to make warm pockets of air rise and cool
ones sink.
Land surface
Rising warm
pockets bring
both warm air
and moist air up
with them
Descending
cool pockets
bring both cool
air and dry air
down with them
In other words, the
same process contributes
to both sensible heat
flux and evaporation
flux. Thus, the same
“resistance” applies.
Simplest model for transpiration:
E =
0.622r es(Ts) - er
p
r a + rs
Equation that
lies at the heart
of standard land
surface models!
er
ra
Evaporative
flux
stomatal resistance
rs
leaf surface
es(Ts)
Note: above equation
is equivalent to the
famous Penman-Monteith
evaporation equation
-- Assumes saturated conditions
within plant stomata
-- Assumes the plant/soil system
determines rs, the stomatal
resistance.
-- Employs “Ohm’s Law” analogy,
placing stomatal and aerodynamic
resistances in series.
stomate
Epenman-monteith =
(Rnet - G) D + (rcp/ra) (es(Tr) - er)
D + g (1 + rs/ra)
Stomatal resistance is not easy to quantify.
rs varies with:
-- plant type and age
-- photosynthetically active radiation (PAR)
-- soil moisture (w)
-- ambient temperature (Ta)
-- vapor pressure deficit (VPD)
-- ambient carbon dioxide concentrations
Effective rs for a full canopy (i.e., rc) varies with leaf density, greenness fraction,
leaf distribution, etc. rc is essentially a spatially integrated version of rs .
Modeling stomatal resistance:
“Jarvis-type” models: rs = rs-unstressed(PAR) f1(w)f2(Ta)f3(VPD)
Many newer models: rs = f(photosynthesis physics)
Key point: Because plants close their stomata during times of
environmental stress, rs is modeled so that it increases during
times of environmental stress.
Optimal
temperature
range (K)
Minimum Stomatal Resistance [sec m-1}]
(from BATS and CLSM, via LDAS)
1. Evergreen Needleleaf Forest
175
2. Evergreen Broadleaf Forest
150
3. Deciduous Needleleaf Forest
175
4. Deciduous Broadleaf Forest
175
5. Mixed Cover
175
6. Woodland
173.
7. Wooded Grassland
169.
8. Closed Shrubland
175
9. Open Shrubland
178.
10. Grassland
165
11. Cropland
117
12. Bare Ground
175
13. Urban and Built-Up
154.
From SiB, as
used in Mosaic
Wilting
point matric
potential (m)
From SiB, as
used in Mosaic
268-313
273-318
-250.
-500
273-318
-250.
283-323
-400
283-328
-230.
Typical approaches to modeling latent heat flux (summary)
Transpiration
lv E =
0.622lr es(Ts) - er
p
ra + r s
Evaporation from bare soil
lv E =
0.622lvr es(Ts) - er
p
ra + rsurface
Interception loss
lv E =
0.622lvr es(Ts) - er
p
ra
ls E =
0.622lsr es(Ts) - er
p
ra
Snow evaporation
Resistance to
evaporation
imposed by soil
Note: more complicated
forms are possible, e.g.,
inclusion (in series) of
a subcanopy aerodynamic
resistance.
Bowen Ratio, B: The ratio of sensible heat flux to latent heat flux.
Evaporative Fraction, EF: The ratio of the latent heat flux to the
net radiative energy.
Over long averaging periods, for which the net heating of the ground is
approximately zero, these two fractions are simply related: EF = 1/(1+B).
Maximum B is infinity (deserts).
Minimum EF is 0 (deserts).
Minimum B could be close to zero, maximum EF could be close
to 1 (rain forests).
HEAT FLUX INTO THE SOIL
One layer soil model: Let G be the residual energy flux at the
land surface, i.e.,
G = Sw + Lw
- Sw
- Lw
- H - lE
Then the temperature of the soil, Ts, must change by DTs so that
G = CpDTs/dt
where
Cp is the heat capacity
dt is the time step length (s)
The choice of the heat capacity can have a major impact on
the surface energy balance.
Low heat capacity case
time of day
High heat capacity case
time of day
-- Heat capacity might, for example, be chosen so that it represents the
depth to which the diurnal temperature wave is felt in the soil.
-- Note that heat capacity increases with water content. Incorporating this
effect correctly can complicate your energy balance calculations.
Heat Flux Between Soil Layers
One simple approach:
G12 = L (T1 - T2) / Dz
Internal
energy
T1
G12
T2
G23
T3
Dz
where
L = thermal conductivity
Dz = distance between
centers of soil layers.
temperature
-- Using multiple layers rather than a single layer allows
the temperature of the surface layer (which controls
fluxes) to be more accurate.
-- Like heat capacity, thermal conductivity increases with
water content. Accounting for this is comparatively easy.
depth
Another approach sometimes used to update soil temperature:
“force-restore”
T
2T
t
z
The full diffusion equation for soil heat transport is c ---- = D ----.
2
Under a pure sinusoidal energy forcing (“G”) at the ground surface, the
full diffusion equation has the same solution as the following linear
ordinary differential equation:
“force”
dTs
dt
=
2
dc
“restore”
G + w(T - Ts)
This equation is
implemented directly
in some land surface
schemes
Where d = depth of diurnal temperature wave
w = frequency of diurnal wave (=1/86400 s)
c = volumetric heat capacity
T = deep soil temperature (unaffected by diurnal wave)
Snow modeling: Plenty of “If statements”
Albedo is high when
the snow is fresh, but
it decreases as the
snow ages.
Energy balance in snowpack
Sw
Internal energy
a function of
snow amount,
snow temperature,
and liquid water
retention
Solid
fraction
Snowmelt occurs only
when snow temperature
reaches 273.16oK.
Sw
Internal
energy
Lw
Tsnow
H l sE
GS1
lmM
T1
Thermal conductivity
within snow pack
varies with snow age.
It increases with snow
density (compaction
over time) and with
liquid water retention.
1
0
Temperature
Lw
273.16
Critical property of snow: Low thermal conductivity
strong insulation
To capture such properties,
the snow can be modeled
as a series of layers, each
with its own temperature.
250oK
260oK
Temperature
profile
snow
soil
270oK
272oK
snow
soil
How do we deal with snowmelt?
One way:
1. Solve the energy balance for a layer, and determine
the updated temperature.
2. If the new temperature is less than or equal to 273.16oK,
then you’re done.
3. If the new temperature is greater than 273.16oK, then
recompute the energy balance assuming the new
temperature is exactly 273.16oK. The excess energy
flux obtained should be used to melt snow:
Excess energy flux = lmM
where lm = latent heat of melting
M = snowmelt rate
SOLVING THE ENERGY BALANCE EQUATION
Sw
Sw
Lw
Lw
H
lE
T
Sw + Lw
= Sw
+ Lw
+ H + lE + CpDT/dt
The key to solving the energy balance equation is to notice
that all fluxes on the right hand side of the equation (except Sw )
are functions of the temperature, Ts.
Simplest calculation: Assume heat capacity of surface is zero.
Sw + Lw
= Sw + f(Ts)
Solve for Ts
Implicit calculation (i.e., don’t assume heat capacity is zero):
determine change in (D Ts) that allows a perfect energy balance.
Step 1: Express each flux in terms of the value it would take at the initial
temperature Told (at the beginning of the time step) and its derivative with
respect to temperature:
Lw = Lw |T-old +
Lw
T
DTs
H = H|T-old +
H
T
DTs
lE = lE|T-old + l
E
DTs
T
Step 2: Rewrite energy balance equation:
Sw + Lw
= Sw + (Lw + H + LE ) | T-old
+
T
(Lw + H + LE ) +
Cp
Dt
DTs
Step 3: Solve for DTs.
Told + DTs is the temperature
that both ensures a proper energy
balance and properly updates the
soil temperature.
Note: for a multi-layer soil moisture, G12 can be similarly
linearized in terms of DTs, and the additional terms can be
incorporated into the implicit equation.
When solving the energy balance equation, it is essential to update the
boundary layer properties (ra) implicitly when solving for DTs. (Recall
that ra is itself varies strongly with surface temperature. Otherwise, you
get oscillations.
ra small
boundary layer
unstable
lE, H large
surface warms
surface cools
lE, H small
Time step
Time step
boundary layer
stable
ra large
What are the
relative magnitudes
of the energy
balance terms?
Sellers
What are the
relative magnitudes
of the energy
balance terms over
a diurnal cycle?
Final Word: Surface energy balance
in the context of larger system.
Traditional view:
Surface energy balance determined by imposed meteorology
More “holistic” view:
Surface energy balance, boundary layer structure are entwined and
inseparable.
Soil moisture versus cloud-base
height, from AMS Horton
lecture by Alan Betts