1. Land-Climate Interactions
Water Balance at the Land Surface
Win
Wout
water storage
Control volume
Basic water balance equation:
WinDt = Dstorage + WoutDt
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2. Land-Climate Interactions
Water Balance for a Single Land Surface Slab, Without Snow (e.g., standard bucket model)
Terms on
LHS come from
the climate model.
Strongly dependent
on cloudiness, water
vapor, etc.
Terms on
RHS come are
determined by
the land surface
model.
P E R w
P = E + R + CwDw/Dt + miscellaneous
where
P = Precipitation
E = Evaporation
R = Runoff (effectively consisting of surface runoff and baseflow)
Cw = Water holding capacity of surface slab
Dw = Change in the degree of saturation of the surface slab
Dt = time step length
miscellaneous = conversion to plant sugars, human consumption, etc.
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3. Land-Climate Interactions
Usually, a combination of water balances is considered. For example:
P = Eint + Dc +
DWc Dt
Water balance associated with canopy
interception reservoir P
Eint
Eint = interception loss
Dc = drainage through canopy
(“throughfall”)
DWc = change in canopy
interception storage Wc Dc
Water balance in a snowpack
DWsnow Dt
P = Esnow + M +
P (snow)
Esnow M
Wsnow
Esnow = sublimation rate
M = snowmelt
DWsnow = change in snow
amount (“infinite”
capacity possible)
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4. Land-Climate Interactions
Water balance in a surface layer
M + Dc =
Ebs + Etr1 + Rs + Q12 + CW1DW1/Dt
M+Dc
Ebs + Etr1 Rs
Water
storage w1
Q12
Ebs = evaporation from bare soil
Etr1 = evapotranspiration from layer 1
Q12 = water transport from layer 1 to layer 2
CW1 = water holding capacity of layer 1
DW1 = change in degree of saturation of layer 1 w2 w3
Water balance in a subsurface layer (e.g., 2nd layer down)
Q12 = Q23 + Etr2 + CW2DW2/Dt
Note: some models
may include an
additional, lateral
subsurface runoff
term w2 w1 w3
Q12
Q23
water
storage
Etr2
Etr2 = evapotranspiration from layer 2
Q23 = water transport from layer 2 to layer 3
CW2 = water holding capacity of layer 2
DW2 = change in degree of saturation of layer 2
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5. Land-Climate Interactions
Water balance in the lowest layer
Qn,n-1 = QD + Etr-n + CWnDWn/Dt wn
Qn,n-1
water
storage QD
Etr-n
Etr-n = evapotranspiration from layer n, if allowed
QD = Drainage out of the soil column (baseflow)
A model may compute all of these water balances, taking care to ensure
consistency between connecting fluxes (in analogy with the energy balance
calculation). P
Eint
Esnow P Dc
Ebs
Etr1
Etr2
Etr3 Rs M W1
Q12 W3 W2
Q23 QD
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6. Land-Climate Interactions
Precipitation, P
Getting the land surface hydrology right in a climate model is difficult
largely because of the precipitation term. At least three aspects of
precipitation must be handled accurately:
a. Spatially-averaged precipitation amounts (along
with annual means and seasonal totals)
b. Subgrid distribution.
c. Temporal variability and temporal correlations.
Otherwise, even with a perfect land surface model,
Perfect land
surface model
Garbage in
Garbage
out
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7. Land-Climate Interactions
Accurate precipitation measurements are limited by availability
of rain gauges….
Each box is ~250 km
on a side
… and by inherent inaccuracies in
satellite-derived precipitation data
How Good is the Estimated SSM/I Rain Rate Climatology Data?
Over oceans, no “truth” data available for validations
Nonsystematic error includes sampling and random
Sampling error dominates
F-13 and F-14 SSM/I, with similar sampling have similar error
TMI has a slightly less, nonsystematic error
Combining F-13 & F-14 almost satisfy the TRMM 1 mm/day and 10% for heavy rain
GPM with 8 satellites will have 50% less error than combining F-13 & F-14
Technical notes for figure
Figure compliments
of Al Chang,
NASA/GSFC
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8. Land-Climate Interactions
Precipitation: subgrid variability (1)
The bottom storm is more
evenly distributed over the
catchment than the top storm.
Intuitively, the top storm will
produce more runoff, even
though the average storm
depth over the catchment
(E(Yo)) is smaller.
Key points:
-- Specifying subgrid
variability of precipitation
is critical to an accurate
modeling of surface hydrology.
-- A GCM is typically unable
to specify the spatial structure
of a given storm. The LSM
typically has to “guess” it.
From Fennessey, Eagleson,
Qinliang, and Rodriguez-Iturbe,
1986.
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9. Land-Climate Interactions
Precipitation: subgrid variability (2)
Here, the two storms have
similar spatial structure
and total precipitation
amounts. The locations of
the storms, however, are
different. If the top storm
fell on more mountainous
terrain than the bottom
storm, the top storm might
produce more runoff
Key point: A GCM is
typically unable to specify
the subgrid location of a
given storm. The LSM
typically has to “guess” it.
From Fennessey, Eagleson,
Qinliang, and Rodriguez-Iturbe,
1986.
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10. Land-Climate Interactions
Precipitation: temporal correlations
Temporal correlations are very important -- but are largely ignored --
in GCM formulations that assume subgrid precipitation distributions.
This is especially true when the time step for the land calculation is of
the order of minutes. Why are temporal correlations important?
Consider three consecutive time steps at a GCM land surface grid cell:
time step 1
time step 2
time step 3
Case 1: No temporal
correlation in storm
position -- the storm is
placed randomly with the
grid cell at each time step.
Case 2: Strong temporal
position between time steps.
Case 2 should produce, for example, stronger runoff.
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11. Land-Climate Interactions
Throughfall
Simplest approach: represent the interception reservoir as a bucket that gets filled during
precipitation events and emptied during subsequent evaporation. Throughfall occurs
when the precipitation water “spills over” the top of the bucket.
Capacity of bucket is
typically a function
of leaf area index, a
measure of how many
leaves are present.
This works, but because it ignores subgrid precipitation
variability (e.g., fractional wetting), it is overly simple.
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12. Land-Climate Interactions
Spatial precipitation variability and interception loss
SiB’s approach
(Seller’s et al, 1986)
Precipitation assumed
to fall according
to some prescribed
distribution
Area above line
is considered
throughfall
Capacity of
reservoir
Note: SiB allows
some of the
precipitation to fall
to the ground without
touching the canopy.
Original water in reservoir
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13. Land-Climate Interactions
Temporal precipitation variability and interception loss
Mosaic LSM’s approach:
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14. Land-Climate Interactions
Evaporation
See notes from energy balance lecture. Note, though, locations of
moisture sinks for bare soil evaporation and transpiration:
Bare soil evaporation water is usually taken from the top soil layer.
Transpiration water is usually taken from the soil layers that comprise the root zone. Different amounts may be taken from
different layers depending on:
-- layer thickness
-- assumed root density profile
e.g., transpiration water
taken from these layers…
but not this layer
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15. Land-Climate Interactions
Runoff
a. Overland flow:
(i) flow generated over permanently saturated zones near a river
channel system: “Dunne” runoff
(ii) flow generated because precipitation rate exceeds the infiltration
capacity of the soil (a function of soil permeability, soil water
content, etc.): “Hortonian” runoff
b. Interflow (rapid lateral subsurface flow through macropores and seepage
zones in the soil
c. Baseflow (return flow to stream system from groundwater)
Runoff (streamflow) is affected by such things as:
-- Spatial and temporal distributions of precipitation
-- Evaporation sinks
-- Infiltration characteristics of the soil
-- Watershed topography
-- Presence of lakes and reservoirs
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16. Land-Climate Interactions
Modeling runoff: basin scale
When variations in precipitation, topography, soil characteristics, etc., can
be explicitly accounted for (as in so-called “spatially distributed” hydrological
basin models), runoff can be predicted fairly accurately. The TOPMODEL
approach uses the statistics of topography to characterize the spatial
distribution of water table depth in a basin, with consequent impacts on
runoff generation.
simulated
observed
From Beven, K., “Spatially distributed
modeling; conceptual approach to
runoff prediction”, in Recent Advances
in the Modeling of Hydrologic Systems,
ed. By Bowles and O’Connell, p ,
Kluwer Academeic Pub., 1991.
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17. Land-Climate Interactions
Modeling runoff: GCM scale
Surface runoff formulations in GCMs are generally very crude, for at least
two reasons:
(i) Developers of GCM precipitation schemes have focused on producing
accurate precipitation means, not on producing accurate subgrid spatial
and temporal variability.
(ii) GCM land surface models generally represent the hydrological state of
the grid cell with grid-cell average soil moistures -- the time evolution of
subgrid soil moisture distributions is not monitored.
At best, we can expect first-order success with these
runoff formulations
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18. Land-Climate Interactions
Controls in nature
Framework of typical LSM
SCALE: HUNDREDS OF KILOMETERS
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19. Land-Climate Interactions
Note that because of the inherent inconsistency between nature and
the typical LSM’s soil layer framework, no “best” approach for modeling
runoff exists. Current LSM approaches are “all over the place”. Typically,
though, runoff is a function of the amount of moisture in the top soil layer.
Bucket model: total runoff = P + M - E if this is positive and the bucket is full.
total runoff = 0 otherwise.
GISS Model II: total runoff = max( 0.5 P Dt, excess over capacity ). W1
W1max
SiB: surface runoff =
excess over
infiltration capacity,
assuming subgrid
distribution of
throughfall.
throughfall distribution
Depth =infiltration capacity * Dt
This part of the throughfall
(above the line) runs off;
the rest infiltrates
Other approaches will be discussed later in the course.
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20. Soil Moisture Transport, Baseflow Land-Climate Interactions
First, some useful definitions:
Porosity (n): The ratio of the volume of pore space in the soil to the total volume of the soil. When a soil with a porosity of 0.5 is completely dry, it is 50% rock by volume and 50% air by volume.
Volumetric moisture content (q): The ratio of the volume of water in the soil to the total volume of soil. When the soil is fully saturated, q = n.
Degree of saturation (w): The ratio of the volume of water in the soil to the volume of water at saturation. By definition, w= q /n.
Pressure head (y): A measure of the degree to which the soil holds on to its water through tension forces. More specifically, y =p/rg, where r is the density of water, g is gravitational acceleration, and p is the fluid pressure.
Elevation head (z): The height of soil element above an arbitrary baseline.
Hydraulic head (h): The sum of the pressure head and the elevation head.
Wilting point: The soil moisture content (measured either in degree of saturation or pressure head) at which plants can no longer draw the moisture from the soil. When modeling the root zone, this is often considered to be the lowest moisture content possible.
Field capacity: The water content obtained when a saturated soil drains to the point where the surface tension on the soil particles balances the gravitational forces causing drainage.
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21. Land-Climate Interactions
Estimating water transport in the saturated zone (i.e., below water table)
Darcy’s Law states that
Q/A = flow per unit normal area = - K
where K = hydraulic conductivity
h = hydraulic head
L = separation distance
h2 - h1 L h2 h1 L
More generally, q = - K h
q = specific discharge = Q/A
Generalized Darcy’s Law:
relates flow to gravitational
and pressure forces.
(Recall: h = y + z)
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22. Land-Climate Interactions
Hydraulic conductivity, K, is related to the soil’s specific permability:
K =
krg m
Where r is the fluid’s density and m is its dynamic viscosity. K is thus a
function of soil and fluid properties.
K varies tremendously with
soil type. Small variations in
soil type, say across a field site,
could lead to orders of magnitude
difference in the ability to
transport moisture.
From Freeze and Cherry
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23. Land-Climate Interactions
Moisture transport in the unsaturated zone (e.g., in the soil near
the surface) can also be computed with Darcy’s law, if appropriate
corrections are made to pressure head and hydraulic conductivity. Z
qr = residual moisture
“specific retention”
If atmospheric
pressure defined
to be 0. qr
Recall: q = ratio of water volume
to soil volume,
n = porosity
Soil moisture
profile
p < 0
capillary
fringe
p = 0
q=n q
p > 0
Water table
Recall: w = degree of saturation,
= q/n
y(w) = ysaturated w -b
K(w) = Ksaturated w 2b+3
b = empirical coefficient
Unsaturated zone
equations (from
Clapp and Hornberger)
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24. Land-Climate Interactions
Three things that complicate moisture transport in the unsaturated zone:
1. Extreme nonlinearity.
b may have values between 4 and 10. If b=10, then K(q) = Ksaturated w 23
2. Hysteresis
Values of parameters not really a
unique function of moisture state; they
depend in part whether the soil has
previously been wet or previously
been dry -- whether the soil is
wetting up or drying down.
From Freeze and Cherry
3. Anisotropy.
Hydraulic conductivity may vary
with the direction of flow.
For a given head gradient,
flow in this direction
may be easier
than flow in
this direction
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25. Land-Climate Interactions
GCM approaches to modeling subsurface flow
Typically,
-- Assume homogeneity of soil constant Ksat, ysat
-- Ignore hysteresis
-- Concentrate on vertical transports only
-- Concentrate only on unsaturated zone and
determination of moisture drainage to water table
Discretization of Darcy’s law (e.g., SiB)
Darcy’s law for vertical flow can be written: h z
q = - K h qz = - K
= - K (z + y)
= - K ( ) z y z
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26. Land-Climate Interactions
One possible discretization of Darcy’s law (continued)
Characterize the soil as stacked layers
(d = thickness) d1 d3 d2 w1 w2 w3
recharge layer
surface layer
root layer
Compute for each layer i:
yi = ysat wi -b
Ki = Ksat wi 2b+3
Compute flow from layer i to layer i+1:
yi - y i+1
qz i,i+1 = K
+ 1 d
K = “average” K across distance = (diKi + di+1Ki+1)/(di+di+1)
d = effective depth for computing gradient = 0.5 (di+di+1)
For drainage out the bottom of the soil column (QD), one might equate it
to the hydraulic conductivity in the lowest layer. SiB, for example, goes
beyond this by also applying a “mean slope angle” term, sin x: QD = K3 sin x
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27. Land-Climate Interactions
It’s important to keep in mind that the different LSMs use very
different discretizations of the soil column -- there is no one “right way”
to do it.
Discretizations and moisture
transport paths for a wide
variety of LSMs, as outlined
by Wetzel and Chang (1996)
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28. Land-Climate Interactions
Energy balance versus water balance
Energy balance:
Implicit solution usually necessary
Results in updated temperature prognostics
Water balance:
Implicit solution usually not necessary
Results in updated water storage prognostics
How are the energy and water budgets connected?
1. Evaporation appears in both.
2. Albedo varies with soil moisture content.
3. Thermal conductivity varies with soil moisture content.
4. Thermal emissivity varies with soil moisture content.
Question: Can we address how the energy and water budgets
together control evaporation rates?
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29. Land-Climate Interactions
Budyko’s analysis of energy and water controls over evaporation
These assymptotes
act as barriers to
evaporation.
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30. Land-Climate Interactions
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31. Land-Climate Interactions
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32. Land-Climate Interactions
The equation in fact characterizes the combined energy and
water balance behavior of GCMs in general...
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33. Land-Climate Interactions
… and can thus be used to explain, in part, differences in
GCM behavior.
Each letter
corresponds to
a different GCM
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34. Land-Climate Interactions
What determines the shape of Budyko’s curve?
If only annual means mattered,
the observed curve should look
like this:
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35. Land-Climate Interactions
Seasonality, however, is important.
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36. Land-Climate Interactions
Note that if these seasonal effects
alone were considered, the
observed curve would actually
look like this:
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37. Land-Climate Interactions
This effect can bring the curve in line with the observed curve. Note, though,
that other effects also contribute to a region’s evaporation rate, including land
surface properties and the temporal variation of precipitation.
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38. Budyko’s analysis: discussion Land-Climate Interactions
1. Annual precipitation and net radiation control, to first order, annual
evaporation rates.
2. The spread of points around the Budyko curve is large, though, due to
various additional factors:
-- phasing of seasonal P and Rnet cycles
-- interseasonal storage of moisture
-- Other land surface or meteorological effects (vegetation type and
resistance, topography, rainfall statistics, …)
3. Note also:
-- Land surface processes affect the precipitation and net radiation
forcing -- there’s not truly a clean separation between land
and atmospheric effects.
-- The land’s effects on hourly, daily and monthly evaporation are
relatively much more important than they are on annual
evaporation.
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39. Land-Climate Interactions
Budyko’s equation for mean annual evaporation
Modification for interannual variability:
See Koster and Suarez, 1999
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40. Land-Climate InteractionssE P sP
/ lP
Rnet
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41. Land-Climate Interactions
Equation works well when tested with GCM data:
curve derived from
Budyko equation sE sP
/ lP
Rnet
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