3 February 2004 Lecture 3 - Koster CLIM 714 Land-Climate Interactions W in W out water storage...

3 February 2004 Lecture 3 - KosterCLIM 714

Land-Climate Interactions

Win Woutwater storage

Control volume

Basic water balance equation: Wint = storage + Woutt

Water Balance at the Land Surface



Water Balance for a Single Land Surface Slab, Without Snow(e.g., standard bucket model)

P = E + R + Cww/t + miscellaneous

where P = Precipitation E = Evaporation R = Runoff (effectively consisting of surface runoff and baseflow) Cw = Water holding capacity of surface slab w = Change in the degree of saturation of the surface slab t = time step lengthmiscellaneous = conversion to plant sugars, human consumption, etc.

P E

w

Terms on LHS come fromthe climate model.Strongly dependenton cloudiness, watervapor, etc.

Terms on RHS come aredetermined by the land surfacemodel.

R



Usually, a combination of water balances is considered. For example:

Water balance associated with canopyinterception reservoir P = Eint + Dc +

Wc

t

Eint = interception lossDc = drainage through canopy (“throughfall”)Wc = change in canopy interception storage

P Eint

Dc

Wc

Water balance in a snowpack

P (snow) Esnow

M Wsnow

P = Esnow + M + Wsnow

t

Esnow = sublimation rateM = snowmeltWsnow = change in snow amount (“infinite” capacity possible)



Water balance in a subsurface layer (e.g., 2nd layer down)

Water balance in a surface layer

w2

w1

w3

Q12Ebs = evaporation from bare soilEtr1 = evapotranspiration from layer 1Q12 = water transport from layer 1 to layer 2CW1 = water holding capacity of layer 1W1 = change in degree of saturation of layer 1

Waterstorage

M+Dc Ebs + Etr1 Rs

M + Dc = Ebs + Etr1 + Rs + Q12 + CW1W1/t

Q12 = Q23 + Etr2 + CW2W2/t

w2

w1

w3

Q12

Q23

waterstorage

Etr2

Etr2 = evapotranspiration from layer 2Q23 = water transport from layer 2 to layer 3CW2 = water holding capacity of layer 2W2 = change in degree of saturation of layer 2

Note: some modelsmay include anadditional, lateralsubsurface runoffterm



Water balance in the lowest layer

wn

Qn,n-1

waterstorage QD

Etr-n

Qn,n-1 = QD + Etr-n + CWnWn/t

Etr-n = evapotranspiration from layer n, if allowedQD = Drainage out of the soil column (baseflow)

A model may compute all of these water balances, taking care to ensureconsistency between connecting fluxes (in analogy with the energy balance calculation).

W2

W1

W3

Q12

Q23

RsMEbs

QD

P Eint

Dc

PEsnow

Etr1 Etr2 Etr3



Precipitation, P

Getting the land surface hydrology right in a climate model is difficultlargely 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 landsurface model

Garbagein

Garbageout



Each box is ~250 kmon a side

Accurate precipitation measurements are limited by availabilityof rain gauges….

How Good is the Estimated SSM/I Rain Rate Climatology Data?

Estimated Nonsystematic Error

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14

Rain Rate (mm/day)

Percen

t E

rro

r (

%)

F-14 F-13 TMI F-13+F-14 GPM F-13 AM

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

… and by inherent inaccuracies insatellite-derived precipitation data

Figure complimentsof Al Chang, NASA/GSFC

Technical notes for figure



The bottom storm is more evenly distributed over thecatchment than the top storm.Intuitively, the top storm willproduce more runoff, eventhough the average stormdepth over the catchment (E(Yo)) is smaller.

Key points: -- Specifying subgridvariability of precipitationis critical to an accuratemodeling of surface hydrology.-- A GCM is typically unableto specify the spatial structureof a given storm. The LSMtypically has to “guess” it.

From Fennessey, Eagleson,Qinliang, and Rodriguez-Iturbe,1986.

Precipitation: subgrid variability (1)



Here, the two storms havesimilar spatial structureand total precipitationamounts. The locations ofthe storms, however, aredifferent. If the top stormfell on more mountainousterrain than the bottomstorm, the top storm mightproduce more runoff

Key point: A GCM is typically unable to specifythe subgrid location of agiven storm. The LSM typically has to “guess” it.

From Fennessey, Eagleson,Qinliang, and Rodriguez-Iturbe,1986.

Precipitation: subgrid variability (2)



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 ofthe 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 3Case 1: No temporalcorrelation in storm position -- the storm isplaced randomly with thegrid cell at each time step.

Case 2: Strong temporalcorrelation in storm position between time steps.

Case 2 should produce, for example, stronger runoff.



Throughfall

Simplest approach: represent the interception reservoir as a bucket that gets filled duringprecipitation events and emptied during subsequent evaporation. Throughfall occurs when the precipitation water “spills over” the top of the bucket.

This works, but because it ignores subgrid precipitationvariability (e.g., fractional wetting), it is overly simple.

Capacity of bucket istypically a functionof leaf area index, ameasure of how manyleaves are present.



Spatial precipitation variability and interception loss

SiB’s approach (Seller’s et al, 1986)

Original water in reservoir

Precipitation assumed to fall accordingto some prescribeddistributionArea above line

is consideredthroughfall

Capacity of reservoir

Note: SiB allowssome of the precipitation to fallto the ground withouttouching the canopy.



Temporal precipitation variability and interception loss

Mosaic LSM’s approach:



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 fromdifferent layers depending on: -- layer thickness -- assumed root density profile

e.g., transpiration watertaken from these layers…

but not this layer



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



Modeling runoff: basin scaleWhen variations in precipitation, topography, soil characteristics, etc., canbe explicitly accounted for (as in so-called “spatially distributed” hydrologicalbasin models), runoff can be predicted fairly accurately. The TOPMODELapproach uses the statistics of topography to characterize the spatialdistribution of water table depth in a basin, with consequent impacts on runoff generation.

simulated

observed From Beven, K., “Spatially distributedmodeling; conceptual approach torunoff prediction”, in Recent Advancesin the Modeling of Hydrologic Systems,ed. By Bowles and O’Connell, p. 373-387,Kluwer Academeic Pub., 1991.



Modeling runoff: GCM scaleSurface runoff formulations in GCMs are generally very crude, for at leasttwo 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



Controls in nature Framework of typical LSM

SCALE: HUNDREDS OF KILOMETERS



Note that because of the inherent inconsistency between nature and the typical LSM’s soil layer framework, no “best” approach for modelingrunoff 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 t, excess over capacity ).W1

W1max

SiB: surface runoff = excess over infiltration capacity, assuming subgrid distribution of throughfall.

throughfall distribution

Depth =infiltration capacity * t

This part of the throughfall(above the line) runs off;the rest infiltrates

Other approaches will be discussed later in the course.



Soil Moisture Transport, Baseflow

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 (): The ratio of the volume of water in the soil to the total volume of soil. When the soil is fully saturated, = 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= /n.

Pressure head (): A measure of the degree to which the soil holds on to its water through tension forces. More specifically, =p/g, where 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.



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 = + z)



Hydraulic conductivity, K, is related to the soil’s specific permability:

K = kg

Where is the fluid’s density and is its dynamic viscosity. K is thus a function of soil and fluid properties.

K varies tremendously withsoil type. Small variations insoil type, say across a field site,could lead to orders of magnitude difference in the ability totransport moisture.

From Freeze and Cherry



Moisture transport in the unsaturated zone (e.g., in the soil nearthe surface) can also be computed with Darcy’s law, if appropriatecorrections are made to pressure head and hydraulic conductivity.

Water table

Z

=n

Soil moisture profile

capillary fringe

r

r = residual moisture

“specific retention”

Recall: = ratio of water volume to soil volume, n = porosity

p < 0

p = 0

p > 0

If atmospheric pressure defined to be 0.

(w) = saturated w -b

K(w) = Ksaturated w 2b+3

b = empirical coefficient

Unsaturated zoneequations (from

Clapp and Hornberger)

Recall: w= degree of saturation, = /n



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() = Ksaturated w23

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 directionmay be easier than flow in this direction



GCM approaches to modeling subsurface flow

Typically, -- Assume homogeneity of soil constant Ksat, sat

-- 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:

q = - K h qz = - K

= - K (z + )

= - K ( + 1)

h

z

z

z



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 layerCompute for each layer i:i = sat wi -b

Ki = Ksat wi

2b+3

Compute flow from layer i to layer i+1:

qz i,i+1 = K i - i+1

d+ 1

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



It’s important to keep in mind that the different LSMs use verydifferent 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)



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 budgetstogether control evaporation rates?

Energy balance versus water balance

Energy balance:Implicit solution usually necessaryResults in updated temperature prognostics

Water balance:Implicit solution usually not necessaryResults in updated water storage prognostics



Budyko’s analysis of energy and water controls over evaporation

These assymptotesact as barriers to evaporation.







The equation in fact characterizes the combined energy andwater balance behavior of GCMs in general...



… and can thus be used to explain, in part, differences in GCM behavior.

Each letter corresponds to a different GCM



What determines the shape of Budyko’s curve?

If only annual means mattered,the observed curve should look like this:



Seasonality, however, is important.



Note that if these seasonal effects alone were considered, theobserved curve would actually look like this:



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 landsurface properties and the temporal variation of precipitation.



Budyko’s analysis: discussion

1. Annual precipitation and net radiation control, to first order, annualevaporation rates.

2. The spread of points around the Budyko curve is large, though, due tovarious 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.



Budyko’s equation for mean annual evaporation

Modification for interannual variability:

See Koster and Suarez, 1999



PRnet

EP

P

E



Equation works well when tested with GCM data:

E

P

PRnet

curve derived fromBudyko equation

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3 February 2004 Lecture 3 - Koster CLIM 714 Land-Climate Interactions W in W out water storage...

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