JOINT REGIONAL CLIMATE AND HYDROLOGY MODELLING

Water and Energy Fluxes at the Land Surface and in the Atmosphere

Lecture hold at the International Summer School “Environmental Changes and Hydrologic processes” 25.7.-8.8.2009 Hohai University

Dr. Harald Kunstmann Institute for Meteorology and Climate Research (IMK-IFU) Karlsruhe Institute of Technology KIT Kreuzeckbahnstraße 19 D-82467 Garmisch-Partenkirchen Email: [email protected] Phone: +49 (0)8821 183208, Fax: +49 (0)8821 183243 http://www.imk-ifu.kit.edu

Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Part I: Land Surface

• Basic Aspects of Global Climate

• Global Energy and Water Fluxes

• Water in Soils

• Physics of Sensible and Latent Heat Fluxes

• Soil-Vegetation-Atmosphere-Transfer Modeling

• Concepts of Upscaling of Land Surface Parameters

2 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Basic Aspects of Global Climate

Energy budget of the Earth The sun radiates energy approximately as a blackbody with a temperature of 6000 K. its radiation spectrum extends from the ultraviolet to the infrared with a maximum in the visible range. However, gases in the earth’s atmosphere are strong absorbers of energy at selected wavelengths in this range, so the radiation striking the earth’s surface is depleted in portions of the spectrum.

Figure: Spectra of energy (a) emitted by a blackbody at 6000 K; (b) received at the Earth’s surface (global average); (c) emitted by a blackbody at 290 K; (d) emitted to space by the earth-atmosphere system (global average). Upper graph shows absorption spectrum for principal absorbing gases in the atmosphere (Dingman, 1994)

3 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Average global energy balance of the earth-atmosphere system. Numbers indicate relative energy fluxes; 100 units equals the solar constant, 1367Wm- 2.(Dingman, 1994)

Energy balance of the atmosphere: • 11 of the 15 units radiated by the surface are absorbed to heat the atmosphere. • the net effect of the exchange of radiant energy between the surface and the atmosphere is the upward transfer of 15 units of energy of which 4 are directly radiated to outer space. • Transfer of latent heat via evaporation (mostly from oceans) adds another 24 units; this is the largest source of atmospheric energy. Sensible heat transfer (conduction/convection) contributes another 7 units because the surface is, on average, warmer than the overlying air.

• energy in atmosphere: (20 +4) +(11 +7 +24)= 66 units • coming from land surface: 11+7+24 = 42 units ⇒ 42/66 = 64% • absorbed directly from solar radiation ⇒ 24/66=36% • energy due to absorption of of longwave radiation via greenhouse gases: ⇒ 11/66=17%; without the greenhouse effect the earth surface would be around -18°C

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Figure: Spectra of incoming solar and outgoing terrestrial radiation.

Stefan-Boltzmann law: All matter at a temperature above absolute zero radiates energy in the form of electromagnetic waves that travel at the speed of light:

Q = ε σ T 4

Q: rate of energy emission per unit area per unit time [Wm-2] σ: universal constant: 5.6710-8 Wm-2K-4 (Stefan-Boltzmann constant) ε: emissivity of surface [.] T: absolute temperature [K]

5 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Table: Emissivity values for different vegetations

6 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Basics of Energy Balance

SWout LWin LWout SWin

Rn = SWin − SWout + LWin − LWout

(I) = (1−α)SWin + LWin − LWout ⇒ = K + L

(K: net shortwave radiation input; L: net longwave radiation input)

: Radiation balance at the earth’s surface

α: albedo

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Table: Typical values for reflectances (albedos)

8 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Components of energy balance for a volume extending from just below the surface to the height at which the net radiation balance is determined

Rn : net incoming radiation λE : outgoing energy as evaporation H : outgoing sensible heat flux G : outgoing heat conduction into the soil P : energy absorbed by biochemical processes in the plants, typically a taken as 2% of net radiation

Ad : loss of energy associated with horizontal air movement; significant in an “oasis” situation but generally neglected otherwise. all in [W/m²] e.g. . R = λE + H + G + P + A ≈ (II) n d ≈ λE + H + G

4 I+II (1−α)SWin + LWin −σεTsfc ≈ λE + H + G

:Energy balance at land surface

9 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Exercise: Energy-Balance of Global Radiational Temperature

I. Long term average rate of input to earth-atmosphere system

i = S(1− a p )

II. Boltzmann Law: average rate at which system radiates energy to outer space:

4 q = σ Tp A

1/ 4 ⎡S (1− a p )⎤ I.=II. ⇒ Tp = ⎢ ⎥ ⎣ σ A ⎦

S = 1.74 1017 W σ = 5.78 10-8 W m-2 K-4 14 A = 4 π REarth²= 5.1 10 m² ap=0.3

⇒ Tp=253.6 K

Many naturally occurring and human induced gases strongly absorb long wave radiation (11 out of 15 units are absorbed to heat the atmosphere ⇒ greenhouse effect)

Most important greenhouse gases are 1) water vapour (accounts for 65% of the absorption), 2) carbon dioxide (33%) and methane, nitrous oxide, ozone and chlorinated fluorocarbons (2% combined).

10 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Exercise: Energy-Balance Model of Earth-Surface Temperature

Simplified two layer atmosphere: lower layer (roughly till 1.8 km) with temperature Tl: 20% of air and 50% of water vapour (main absorber of terrestrial radiation) upper layer with temperature Tu: 80% of air and 50% of water vapour

I. Total system: 4 4 S +W = a p S + σ Tu A + (1− f )σ Ts A

II. Upper atmosphere: 4 4 ku S +σ Tl A + 0.5Qe = 2σ Tu A

III. Lower atmosphere: 4 4 4 kl S + σ Tu A + fσ Ts A + 0.5Qe + Qh +W = 2σ Tl A all energy terms expressed as long term fluxes in [E T-1]

⎡(3 − 3a p − 2ku − kl )S −1.5Qe − Qh + 2W ⎤ Ts = ⎢ ⎥ ⎣ (3 − 2 f )σ A ⎦

S = 1.74 1017 W solar flux 16 Qe = 4.08 10 W latent heat flux from surface 15 Qh = 8.67 10 W sensible heat flux from surface W = 1.07 1013 W: heat generated from fossil and nuclear fuels ap = 0.3 planetary albedo ku = 0.18: fraction of solar radiation absorbed in upper layer kl = 0.075: fraction of solar radiation absorbed in lower layer f = 0.95 fraction of thermal radiation from surface that is absorbed in atmosphere

⇒ Ts=290.4 K

11 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Latitudinal energy transfer

The geometric relations of the earth’s orbit cause seasonal and latitudinal variations in the receipt of solar energy. Solar energy flux is spread out over large areas at high latitudes because the earth is a sphere. This strictly latitudinal effect is modified seasonally because the earth’s axis of rotation is tilted at an angle of 23.5° to the orbital from the sun.

Figure: Variations of solar radiation intensity [E L-2 T-1] with angel of incidence. At higher angles (higher latitudes) a given energy flux is spread over a larger area (Dingman, 2002).

Figure: Revolution of the Earth around the sun, showing that the occurrence of summer and winter in the Northern and Southern Hemispheres is determined by the 23.5° tilt of the rotational axis toward or away from the sun (Dingman, 2002)

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General circulation

The unequal latitudinal distribution of radiation and the requirements for conservation of angular momentum on the rotating earth give rise to a system of three circulation cells in the latitude bands from 0°-30°, 30° to 60°, and 60° to 90° in each hemisphere.

Figure: The general circulation of the atmosphere. “H” indicates Hadley cells. Double headed arrows in cross section indicate that the wind has a component from the east.

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Distribution of temperature

Figure: Distribution of mean temperature (°C) in a) January and b) July

14 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Distribution of precipitation

Average precipitation over the globe • Regions characterized by rising air tend to have relatively high average precipitation, those of descending air have low precipitation • The general circulation produces belts of relatively high precipitation near the equator and 60° latitude and low precipitation near 30° (where most of the world’s greats deserts occur). • Intertropical convergence zone (ITCZ) • Precipitation rates influences by topography, air temperatures, frontal activities, wind directions in relation to moisture sources

Significant deviations from general latitudinal distributions due to mountain ranges.

Figure: Latitudinal distribution of average precipitation.

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Temporal distribution of precipitation

Figure: Generalized map of seasonal precipitation regimes. Heavy dashed line encloses the monsoon region of Asia and Africa.

The reversal of circulation associated with the winter high pressure and summer low pressure over the huge landmass of Asia interacts with the topography and the migration of the ITCZ to produce a particularly strong seasonality of precipitation in much of Asia and Africa: this is the monsoon.

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Distribution of snow cover Snow has important climatic effects helping to maintain colder temperatures by reflecting much of the incoming solar energy, and, in melting, by absorbing energy that would otherwise contribute to warming the near surface environment.

During the ice ages, the reflection of solar radiation by ice and snow was an important feedback that contributed to creating and maintaining a colder climate.

Snow also acts as an insulating blanket that helps to retain heat in the soil, which is important hydrologically as well as biologically: if soil is prevented from freezing, its ability to accept infiltrating water is generally enhanced.

The principal hydrologic effect of snow is to delay the input of precipitated water into the land phase of the hydrologic cycle and thus to affect the seasonal distribution of runoff.

17 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Distribution of evapotranspiration

As expected there is a general correlation between mean annual temperature and mean annual evapotranspiration. In the oceans, the basic latitudinal patterns are distorted largely by the effects of surface currents (e.g. Gulf stream and equator ward cold currents off western South America and North America). The highest continental values are in the tropical rain forests of South America, Africa and southeast Asia While the lowest are in the Sahara desert, Antarctica, arctic North America and central Asia.

Figure: Oceanic and continental evapotranspiration in [cm yr-1]. Shaded areas: mountain zones where no data are available.

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Distribution of runoff is mostly the difference between precipitation and runoff for the continents. Virtually all the zones with the highest runoff also have the highest precipitation.

Seasonal runoff regimes In most regions that have seasonal snow cover, maximum runoff occurs in the melt season: summer in arctic and subarctic, and alpine regions, and spring at lower latitudes. A summer maximum also occurs in regions with monsoonal climates such as India and southeast Asia. Fall and winter runoff maxima are also directly related to concurrent rainfall maxima.

19 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

The Global Hydrologic Cycle

Stocks and fluxes in the global cycle

20 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Table: Global stocks and fluxes of Water

Table: Schematic diagram of stocks (km³) and annual fluxes (km³ yr-1) in the global hydrologic cycle. Inflows and outflows nay not balance for all compartments (Dingman, 1994)

Residence Times

Stock T = r Flux

⇒ TAtmosphere ≅ 12,900 km³/ 577,000 km³/year ≅ 8days ⇒ TRivers ≅ 2,120 km³/ 44,700 km³/year ≅ 17.3 days

21 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

22 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Outlook: Water Stress

Figure: Human appropriation of estimated accessible RFWS land (terrestrial renewable fresh water supply) is 30% and human appropriation of total RFWS land is 23% (Postel, 1996)

23 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Exercise: Future climate change and water resources

I. R = P − ET

II. ET = (1− w)P

R runoff ratio w ≡ P P relative change of precipitation p ≡ future Pnow ET relative change of evapotranspiration e ≡ future ETnow Relative change in runoff as a function of the present runoff ratio and the relative changes in precipitation and evapotranspiration:

R p − (1− w)e r ≡ future = Rnow w

3

e=0.7

2 p r f f 1.6 o n 1.4 ru n

i 1.2

e

g 1 1 0.8 0.7 ve chan i 0.6 lat

e 0.4 R 0

-1 0 0.10.20.30.40.50.60.70.80.9 1 Runoff ratio w

Example: Model for estimating effects of climatic changes on runoff

• Changes in runoff are everywhere more sensitive to changes in precipitation than to changes in evapotranspiration ∂r ∂r ( > ) ∂p ∂e • Relative change in runoff is always greater than relative change in precipitation (r > p) • Runoff is most sensitive to climatic changes in arid and semiarid regions where the runoff ratio w is small

24 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Water in Soils: Infiltration and Redistribution

Figure: Definitions of terms used to describe water movement in the unsaturated zone

Redistribution can involve • exfiltration (evaporation from the upper layers of the soil),

• capillary rise (movement from the saturated zone upward into the unsaturated zone due to surface tension),

• recharge (movement of percolating water from the unsaturated zone to the subjacent saturated zone),

• interflow (flow that moves downslope) and

• uptake by plant roots (transpiration)

25 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Material properties of soil

Figure: Grain size distribution curve for a soil formed on glacial till in southwestern New Hampshire. The boundaries between size classes designated clay, silt, sand and gravel are shown as vertical lines.

Figure: Soil texture triangle, showing the textural terms applied to soils with various fractions of sand, silt, and clay.

26 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Definitions

M m Particle density ρ m = Vm with Mm: mass of mineral grains Vm: Volume of mineral grains

M m M m Bulk density ρb = = Vs Va +Vw +Vm With Vs: total volume of soil sample Va: volume of air Vw: volume of liquid water Vm: volume mineral components

V +V Porosity φ = a w Vs

V Volumetric water content θ = w Vs

V θ Degree of saturation S = w = Va +Vw φ

27 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Ranges of porosities

In general, finer grained solids have higher porosities than coarser grained soils. This is in part due to the very open “House of cards” arrangement of clays, which is maintained by electrostatic forces between the roughly disk-shaped grains; In contrast, quasi-spherical sand and silt grains are in more closely packed grain-to grain architecture.

28 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Schematic diagrams of structures of aggregates of (a) clay and (b) sand. The architecture of clay particles is maintained by intergrain electrostatic forces.

29 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Soil water tension

The porous medium is made up of a matrix of particles. When the void spaces are only partially filled with water, the water is attracted to the particles surface through electrostatic forces between the water molecules’ polar bounds and the particles surfaces. This surface adhesion draws the water up around the particles surfaces, leaving the air in the center of the voids. As more water is added to the porous medium, the air exits upwards and the area of free surfaces diminishes within the medium, until the medium is saturated. The effect of soil suction can be seen if a column of dry soil is placed vertically with its bottom in a container of water – moisture will be drawn up into the dry soil to a height above the water surface at which the soil suction and gravity are equal. This height ranges from mm for a coarse sand to several meters for a clay soil.

Figure: Soil water pressure (tension) Ψ vs. degree of saturation S for soils of different textures.

30 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

31 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Explanation of surface tension

Molecules in the surface of liquid water are subjected to a net inward force due to hydrogen bonding with the molecules below the surface.

Surface tension, σ, is equal to the magnitude of that force divided by the distance over which it acts; thus its dimension are [F L-1].

Figure: Intermolecular forces acting on typical surface (S) and nonsurface (B) molecules

Water has a surface tension higher than that of most other liquids; its value at 0°C is

σ = 0.0756 Nm −1

Surface tension decreases rapidly as temperature increases:

σ ≈ 0.001⋅ (20987 − 92.613⋅T)0.4348 where T is in °C and σ in Nm-1.

32 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Sketch of phenomenon of capillary rise in a circular tube of radius r, θc is the contact angle between the meniscus and the wall. P is atmospheric pressure.

When capillary rise has ceased, the column of water is suspended from the meniscus, which is in turn attached to the walls by hydrogen bounds. Thus the water is under tension. Because of the intermolecular hydrogen bonds, the entire mass of water within the tube will also be drawn upward until the adhesive force between the molecules of the tube and those of the water is balanced by the downward forces due to the weight of the water suspended within the tube.

Upward force: Fu = σ ⋅ cos(θ c ) ⋅ 2 ⋅π ⋅ r [N]

Downward force: Fd = γ ⋅π ⋅ r² ⋅ hcr [N]

The vertical extent of water rise in the tube can be approximately calculated by (Fu=Fd)

2σ cosθ c hcr = γ wr with hcr: height of column θc: contact angle between water surface and tube r: radius of tube γw: weight density of water σ: surface tension (σ=0.0756 N m-1 at 0°C)

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The pressure in the suspended water Pm is

2 σ cos(θ ) P = −γ h = − c m cr r

Thus the pressure difference across a meniscus, like the height of capillary rise, is inversely proportional to the radius of a capillary opening.

34 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Hydraulic Properties of Soils

Infiltration and redistribution are unsaturated flows in porous media (soils) that are described by

Q d(z + p /γ w ) dh Darcy’s law Vx = = −K h = −K h Ax dx dx

Q: volumetric flow rate [L³/T] e.g. [m³/s]] Vx: specific discharge in x-direction [L/T] e.g. [m³/m²/s]] z: water table elevation above arbitrary datum [L] e.g. [m] (-> gravitational head) M γ : weight density of water (γ = g = ρ g) e.g. [N/m³] w w V w p: water pressure [F/L²] e.g. [N/m²] (i.e. gage pressure: difference between actual pressure and atmospheric pressure) Kh: hydraulic conductivity [L/T e.g. [m/s]] p h: total hydraulic head [L] ( h = z + ) γ w

p Pressure head Ψ ≡ in [L] e.g. [m] γ w

d(z + Ψ(θ )) Darcy’s law V = −K x h dx

It is conventional to measure pressure relative to atmospheric pressure: p > 0 and ψ > 0 : saturated flows

p < 0 and ψ < 0 : unsaturated flows

Negative pressure is called tension or suction.

Ψ is called tension head when p<0. The water table is the surface at which p=0.

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Infiltration: focus is on flows in vertical (z) direction.

d(z + Ψ(θ )) ⎡ d Ψ(θ )⎤ qz = −K h (θ ) = −K h (θ )⎢1+ ⎥ dz ⎣ dz ⎦

The relations between pressure and water content [ψ(θ)] and between hydraulic conductivity [Kh((θ)] are crucial determinants of unsaturated flow in soils.

36 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Hysteresis

In real soils, the value of tension at given water content is not unique, but depends on the soil’s history of wetting and drying. While this hysteresis can have significant influence on soil-moisture movement, it is difficult to model mathematically and is therefore not commonly incorporated in hydrologic models.

Figure: Hysteresis in the Ψ(θ ) −θ relation for Rubicon sandy loam. The paths with arrows trace the relation as the soil undergoes successive cycles of wetting (arrows pointing to right) and drying (arrows pointing to left).

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Analytic approximations of soil hydraulic properties

Clapp and Hornberger, 1978 ψ (S) = Ψs S −b , or ψ (θ ) = Ψs φ bθ −b

c −c c K h (S) = K h,sat S or K h (θ) = K h,satφ θ

c ≈ 2b + 3

ñ Cosby et al, 1978 2b+3 ⎛ θ ⎞ ⎜ ⎟ K h (θ ) = K h,sat ⎜ ⎟ ⎝θ sat ⎠

−b ⎛ θ ⎞ ⎜ ⎟ ψ (θ ) = Ψsat ⎜ ⎟ ⎝θ sat ⎠

Van Genuchten, 1980: 2 ⎡ 1/ m m ⎤ θ −θ ⎛ ⎛θ −θ ⎞ ⎞ K (θ ) = K res ⎢1− ⎜1− ⎜ res ⎟ ⎟ ⎥ h h,sat ⎢ ⎜ ⎜ ⎟ ⎟ ⎥ φ −θ res ⎝ φ −θ res ⎠ ⎣⎢ ⎝ ⎠ ⎦⎥

Brooks and Corey, 1966 n ⎛θ −θ ⎞ ⎜ res ⎟ K h (θ ) = K h,sat ⎜ ⎟ ⎝ φ −θ res ⎠

Baver, Gardner, and Gardner, 1972 a K (ψ ) = h ψ

Childs and Collis-George 1950 K K (ψ ) = h,sat h b +ψ n

38 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

39 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Water conditions in natural soils

Field capacity θ fc : water that can be held against gravity

Plants cannot exert suctions stronger than -15,000 cm. when the water content is reduced to the point corresponding to that value on the moisture characteristic curve, transpiration ceases and plants wilt. This water content is called

Permanent wilting point θ wp :

Available water content for plant use: θ a ≡ θ fc −θ wp

Figure: Drainage of two soils (Schematic); the arrows indicate the values of field capacity

Figure: Soil-water status as a function of pressure (tension). Natrual soils do not have tensions exceeding about -31,000 cm

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Hydrologic Horizons

Figure: Designation of hydrologic soil-profile horizons.

Ground water zone (sometimes called phreatic zone) is saturated and the pressure is positive. If there is no groundwater flow, the pressure will be hydrostatic – that is, increasingly linearly with depth according to

p(z) = γ w (z − z0 ) with p: gage pressure γw: weight density of water z: measured distance vertically downward z0: value of z at the water table Thus the water table is at atmospheric pressure; it is the level at which water would stand in a well

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Tension-saturated zone (Capillary Fringe) or vadose zone: entire zone of negative water pressures above the water table. The lowest portion of this zone is a region that is saturated or nearly saturated as a result of capillary rise. The pore spaces in a porous medium act like a capillary tube and surface tension forces draw the water into the spaces above the water table.

Water in the tension-saturated zone is under tension; pressure remains hydrostatic and head is given by

Ψ = z − z0 with z

The pressure head at the top of this zone is the air-entry tension Ψae, since at greater tensions the pores are partially filled with air. As noted, Ψae, is equal to the height of capillary rise in the soil.

The height of the tension-saturated zone may range from negligible in coarse sands to several meters in clay soils.

Intermediate Zone Water enters the intermediate zone largely by as percolation from above and leaves by gravity drainage. Thus, water content in this zone may rise above field capacity when water from rain or snow passes through the root zone, after which it gradually returns to field capacity. Tensions in this zone are usually stronger than Ψae and depend on soil texture and water content.

The intermediate zone may occupy much of the soil profile and, in arid regions, extend over many tens of meters. In other situations it may be absent at least seasonally.

Root zone The root zone (also called soil moisture zone) is the layer from which plant roots extract water during transpiration; its upper boundary is the soil surface, while its lower boundary is indefinite and irregular. Water enters by infiltration and leaves via transpiration and gravity drainage. The water content is usually above the permanent wilting point. There are many places in which the root zone extends to the water table.

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Hydraulic conductivity k [ms-1]: (Cosby et al., 1978)

θ 2b+3 k = k sat ( ) θ sat

θ sat : Volumetric water content at saturation ksat: Saturated hydraulic conductivity b: slope Parameter Mathematica exercise:

hydraulicconductivity

k 3.5 ´ 10 -12 3´ 10 -12 2.5 ´ 10 -12 2 10-7 -12 ×k 15 2´ 10 1 10-7 -12 × 1.5 ´ 10 0 10 1´ 10 -12 b 0.1 -13 5´ 10 0.2 5 0.3 θ θ 0.1 0.2 0.3 0.4 0.4

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Soil water tension Ψ(θ ) (Cosby et al., 1978)

θ ψ =ψ ( sat )b sat θ ignoring hysteresis!

Mathematica exercise:

SOIL WATER TENSION ψ

0 -200 -40ψ 0 -600 0.6 -800

0.4 θ 5

b 10 0.2

15

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Parameterizations

Soil - related parameters in the SVAT-model

- volumetric water content at saturation (θ sat ) (respectively porosity φ )

- saturation soil suction ()Ψsat

- hydraulic conductivity at saturation (K sat )

- field capacity (θ ref ), respectively (θ fc )

- wilting point ()θ w - b is an exponent in the function that relates soil water potential and water content.

Soil type −3 −1 b −3 −3 θ sat (m³m ) Ψs (m) K s (ms ) Θref (m³m ) Θ w (m³m )

Sand 0.339 0.069 1.07*E-6 2.79 0.236 0.01 Loamy sand 0.421 0.036 1.41*E-5 4.26 0.283 0.028 Sandy loam 0.434 0.141 5.23*E-6 4.74 0.312 0.047 Silt loam 0.476 0.759 2.81*E-6 5.33 0.36 0.084 Silt loam 0.476 0.759 2.81*E-6 5.33 0.36 0.084 Loamy sand 0.439 0.355 3.38*E-6 5.25 0.329 0.066 Sandy clay loam 0.404 0.135 4.45*E-6 6.66 0.314 0.067 Silt clay loam 0.464 0.617 2.04*E-6 8.72 0.387 0.12 Clay loam 0.465 0.263 2.45*E-6 8.17 0.382 0.103 Sandy clay loam 0.406 0.098 7.22*E-6 10.73 0.338 0.1 Silty clay 0.468 0.324 1.34*E-6 10.39 0.404 0.126 Clay 0.468 0.468 9.74*E-7 11.55 0.412 0.138 Organic material 0.439 0.355 3.38*E-6 5.25 0.329 0.06 Water Bedrock 0.25 7.59 9.74*E-8 11.55 0.233 0.094 Other (land - ice) 0.421 0.036 1.34*E-6 11.55 0.283 0.028

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Energy and Water Fluxes in Soils

Water content profiles

Figure: Soil moisture profiles in a light clay with φ=0.5 at various times during steady infiltration with no ponding. The number on the curve is the time since infiltration began in hours.

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Water Fluxes: Partial Differential Equation for Soil Moisture

Derivation of Richard’s-Equation

Figure: Definition sketch for terms used in deriving the Richard’s equation

d(z + Ψ(θ )) I. Darcy q ≡ V = −K z' h dz'

∂V ∂θ II. Mass balance − z' = ∂z' ∂t

∂ ⎛ ∂Ψ(θ ) ⎞ ∂K h (θ ) ∂θ I and II together yield ⎜ K h (θ ) ⎟ − = ∂z' ⎝ ∂z' ⎠ ∂z' ∂t

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Soil Water Diffusivity

∂ ⎛ ∂Ψ(θ ) ∂θ ⎞ ∂K h (θ ) ∂θ ⎜ K h (θ ) ⎟ − = ∂z' ⎝ ∂θ ∂z' ⎠ ∂z' ∂t

∂ ⎛ ∂θ ⎞ ∂K (θ ) ∂θ ⇔ ⎜ D ⎟ − h = ∂z' ⎝ ∂z' ⎠ ∂z' ∂t ∂Ψ(θ ) with D = K (θ ) soil water diffusivity [L²/T] e.g. [m²/s] h ∂θ

Mathematica exercise:

2+b ⎡ ⎛ θ ⎞ ⎤ ⎢ ⎜ ⎟ ⎥ b ⋅ k sat ⋅⎜ ⎟ ⋅ψ sat ⎢ ⎝θ sat ⎠ ⎥ D = Abs − ⎢ θ ⎥ ⎢ sat ⎥ ⎢ ⎥ ⎣ ⎦ with

Ψsat : saturated matrix potential (suction head)

hydraulic conductivity

0.2 Diff 0.1 0.8 0 0.6 6 θsat 8 0.4 10 b 12 0.2 14

48 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Excel-Exercise: Computation of soil moisture fluxes (Deep Dean, Sussex, England)

Average flux q12 between measurement points 1 and 2

h1 − h2 q12 = −K z1 − z2

Observations: Week Total head h1 at 0.8m (cm) Total head h2 at 1.8m (cm) 1 -145 -230 2 -165 -235 3 -130 -240 4 -140 -240 5 -125 -240 6 -105 -230 7 -135 -215 8 -150 -230 9 -165 -240 10 -190 -245 11 -220 -255 12 -230 -265 13 -255 -275 14 -280 -285

0 120 1234567891011121314

-0.02 100 ] y -0.04 80 da m] c m/ [ [c ce ux en r fl -0.06 60 e ff i ture s D ead Moi l -0.08 40 H Soi

-0.1 20

-0.12 0 Week Moisture flux (Baver et al) Head difference h1-h2

Assuming a=3 in Baver et al. (1972) parameterisation of K(Ψ)

49 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Mathematica Exercise: Numerical Solution of Richards Equation

Discretisation into 4 layers with thicknesses 0.1, 0.3, 0.6, 1m.

The layer – integrated form the ith soil layer is:

∂θ ⎛ ∂θ ⎞ d 1 = −D⎜ ⎟ − K z1 ∂t ∂z z1 ⎝ ⎠ z1

∂θ ⎛ ∂θ ⎞ ⎛ ∂θ ⎞ d 2 = D⎜ ⎟ − D⎜ ⎟ + K − K z2 ∂t ∂z ∂z z1 z2 ⎝ ⎠ z1 ⎝ ⎠ z2

∂θ ⎛ ∂θ ⎞ ⎛ ∂θ ⎞ d 3 = D⎜ ⎟ − D⎜ ⎟ + K − K z3 ∂t ∂z ∂z z2 z3 ⎝ ⎠ z3 ⎝ ⎠ z3

∂θ ⎛ ∂θ ⎞ d 4 = D⎜ ⎟ + K − K z4 ∂t ∂z z3 z4 ⎝ ⎠ z3

dzi: thickness of layer i [m]

50 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Energy Fluxes: Partial Differential Equation of Ground Heat Flux

Similarly to derivation of Richard’s equation: Mass balance & gradient based heat flux yields PDE for ground heat flux…:

G(z)

G(z+∆z)

Figure: Schematic of heat transfer in a vertical column of soil below a flat, horizontal surface

Through solid media and still fluids, heat is transferred primarily through conduction, which involves molecular exchanges. The rate of heat transfer (or heat flux) in a given direction is proportional to the temperature gradient:

∂T I. G = −K ∂z with G: ground heat flux [W m-2] ∂ ∂G II. (C T ) = − ∂t ∂z (law of conservation of energy)

I. and II yield

∂T ∂ ⎡ ∂T ⎤ C()θ = K(θ ) ∂t ∂z ⎣⎢ ∂z ⎦⎥

C: volumetric heat capacity [J/m3/K] K: thermal conductivity [W/m/K] T: soil temperature [K] θ : soil moisture [.] (fraction of unit soil volume occupied by water) assuming that C is not time dependent and that C and K depend on θ. 51 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Table: Molecular thermal properties of natural materials

52 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Evapotranspiration

Classification of evapotranspiration processes

The general energy balance for an evapotranspiring body during a time can be written as

∆Q LE = K + L − G − H + A − w ∆t

LE: [E L-2 T-1] z.B. Wm-2

LE: evaporation K: net shortwave radiation input L: net longwave radiation input G: net output via conduction to the ground H: net output of sensible heat exchange with the atmosphere Aw: net input associated with inflows and outflows o water (water advected energy) ∆Q: Change in amount of heat stored in the body per unit area between beginning and end of ∆t

53 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Clausius-Clapeyron Equation Condensation/evaporation and liquid-water (and also ice-particle) formation is dependent on the saturation vapour pressure. The saturation vapour pressure is the partial pressure of gas at a specific temperature when the gas is in equilibrium with a liquid or solid surface. In a gross sense, it is the maximum amount of vapour the air can hold without the vapour condensing as a liquid or depositing as ice on a surface. The temperature variation of the saturation vapour pressure of water over a liquid surface pv,s [mb] is approximated with the Clausius-Clapeyron equation

dpv,s ρv,s λe pv,s = λe = 2 dT T RvT

-3 Le : latent heat of evaporation [Jkg-1], ρ v,s [kg m ]

dpv,s 1 ⎛ Ah Bh ⎞ → = ⎜ 2 − ⎟dT pv,s Rv ⎝ T T ⎠

T = T0 = 273.15 pv,s,0 = 6.112 mb

⎡ ⎛ 1 1 ⎞ ⎛ 273.15 ⎞⎤ pv,s = 6.112 exp⎢6816 ⎜ − ⎟ + 5.1309 ln⎜ ⎟⎥ ⎣ ⎝ 273.15 T ⎠ ⎝ T ⎠⎦ with T [K], pv,s [mb].

esat mbar

120 @ D 100

80

60

40

20

ta °C -20 -10 10 20 30 40 50

Figure: Saturation vapour pressure pv,s [mb]over liquid water@ D

pv is often denoted as e and pv,s is denoted as esat.

54 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Example

T=253.15K (-20°C), pv,s = 1.26mbar

T=298.15K (25°C), pv,s = 31.60mbar

Climate change:

T=288.15K (15°C), pv,s=17.03mbar

T=289.15K (16°C), pv,s=18.16mbar ∆T=1°C -> Relative change of precipitable water by 5.5% -> impact on precipitation (e.g. intensity)

Figure: a) condensation occurs when the partial pressure of water vapour exceeds its saturation vapour pressure over liquid water, b) evaporation occurs when the saturation vapour pressure exceeds the partial pressure of water vapour (schematics not to scale).

55 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Physics of Evaporation

Figure: Schematic diagram of flux of water molecules over a water surface. The vapor pressure at the surface is es (Ts ) ; the vapor pressure of the overlying air is less than or equal to es (Ta ) . The rate of evaporation is proportional to esat (Ts ) − ea

Dry air with a temperature of Ta lies above horizontal water surface with temperature

Ts. The molecules at the surface are attracted to those in the body of the liquid by hydrogen bonds, but some of the surface molecules have sufficient energy to sever bonds and enter the air. The number of molecules with this “escape energy” increases as Ts increases. At equilibrium rate of escape and reentry are equal and the vapor pressure in the molecular layer immediately above the temperature is the saturation vapor pressure esat(Ts). The rate of evaporation is the rate at which molecules move from the saturated surface layer into the air above and that rate is proportional to the difference between vapor pressure of the surface layer and the vapor pressure of the overlying air.

E ∝ (es (Ts ) − ea )

Note: E rate of evaporation/condensation [Mass/Time] i.e. [kg/s]

λE = λv ⋅ E : rate of latent heat transfer [Energy/Time] i.e.[ J/s]

6 λv : latent heat of evaporation [Energy/Mass], i.e. [J/kg] λv ≈ 2.501 ⋅10 − 2370 ⋅Tc If E is expressed in [Length³/Length²/Time] i.e. [m³/m²/s], [m/s]

λE = ρ ⋅ λv ⋅ E [Energy/Length²/Time], i.e. [J/m²/s], [W/m²]

56 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Evaporation is a diffusive process

E = K E va (es − ea )

E: evaporation rate [L T-1] -1 -2 es, ea vapour pressure of evaporating surface and air respectively [M L T ] -1 KE: [L T² M ] exchange coefficient reflecting the efficiency of vertical transport of water vapour by turbulent eddies of wind

DWV 0.622ρ a k² K E ≡ 2 DM pρ w ⎡ ⎛ zm − zd ⎞⎤ ⎢ln⎜ ⎟⎥ ⎣ ⎝ z0 ⎠⎦

-1 DWV, DM: diffusivities of water vapour and momentum [L² T ] -3 ρa: air density [M L ] p: atmospheric pressure [M T-2 L-1] k=0.4: von Kárman constant zm: height at which air vapour pressure and wind speed are measured [L] (usually zm = 2m) zd: zero plane displacement [L] z0: roughness height [L] zd ≈ 0.7 zveg z0 ≈ 0.1 zveg

Vapor pressure relations

es = esat (Ts )

⎛ 17.3T ⎞ esat (T ) = 6.11exp⎜ ⎟ ⎝ T + 237.3 ⎠ with T in °C and e in mbar.

Vapour pressure in the air depends on relative humidity, RH:

ea = RH esat (Ta )

Latent heat exchange

LE = ρ wλv E = ρ wλv K E va (es − ea )

LE: rate of latent heat transfer [E L-2 T-1] λv: latent heat of vaporization

57 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Physics of sensible heat exchange The upward rate of sensible heat exchange by turbulent transfer, H, is given by

H = K H va (Ts − Ta )

H: sensible heat transfer [E L-2 T-1] Ts, Ta: Temperature of surface and air respectively [K] or [°C] -1 KH: [L T² M ] exchange coefficient

DH k² K H ≡ ca ρ a 2 DM ⎡ ⎛ zm − zd ⎞⎤ ⎢ln⎜ ⎟⎥ ⎣ ⎝ z0 ⎠⎦

DH: diffusivity of heat -1 -1 ca: heat capacity of air (ca=0.24 cal g °C )

The ratio of diffusivities DH/DM depends on stability conditions in the air which are determined by vertical gradients of wind speed and temperature.

DH ≈ DWV

The Bowen ratio and the psychrometric constant

H B ≡ LE

c p(T − T ) B = a s a 0.622λv (es − ea )

Thus the Bowen ratio depends on the ratio of surface-air-temperature difference to surface-air vapour-pressure difference, multiplied by a factor γ

c p γ = a 0.622λv : psychrometric constant.

⇒ K H = γρ wλv K E

58 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Derivation of the Penman Combination Equation

LE = ρ wλv E = ρ wλv K E va (es − ea )

LE: [E L-2 T-1] z.B. Wm-2

∆Q LE = K + L − G − H + A − w ∆t

K + L − H Approximation: E ≈ ρ wλv

Sensible heat transfer flux: H = K hva (Ts − Ta )

Slope of saturation-vapor versus temperature curve at air temperature

esat (Ts ) − esat (Ta ) s(Ta ) ≈ Ts − Ta

esat (Ts ) − esat (Ta ) ⇒ Ts − Ta ≈ s(Ta )

K v H = h a []e (T ) − e (T ) s(T ) sat s sat a ⇒ a K hva K hva = []esat (Ts ) − ea − []esat (Ta ) − ea s(Ta ) s(Ta )

E esat (Ts ) − ea = K E va

K hva E K hva H = − []esat (Ta ) − ea s(Ta ) K E va s(Ta )

K v K + L + h a []e (T ) − e s(T ) sat a a E = a ρ wλv + K H / K E s(Ta )

s(T )[K + L] + γK ρ λ v e (T )[1− RH] E = a E w v a sat a ρ wλv [s(Ta ) + γ ]

Penman Combination equation: mass transfer and energy balance approach combined such that no surface temperature data Ts, es are required!

59 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Assumptions revisited: • no ground heat flux • no water advected energy • no change in heat storage*

esat (Ts ) − esat (Ta ) • uses approximation s(Ta ) ≈ Ts − Ta *If it is significant, ground heat conduction term can be included by replacing K+L by K+L-G.

Essence of Penman equation can be represented as s(T ) × net radiation + γ × "mass transfer" E ∝ a s(Ta ) + γ

60 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Exercise: Evaporation from Lake Hefner, 12.7.1951

Case study Lake Hefner 12.7.1951: calculate a) net shortwave radiation, b) net longwave radiation, c) latent heat flux (neglecting all other types of heat fluxes). What is the corresponding evaporation rate in [mm/day]? Note: Latent heat of evaporation

λ [kJ/kg] can be approximated by λ = 2500 − 2.36 Tc (Tc: temperature in °C).

61 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Physics of Transpiration

Transpiration is the evaporation of water from the vascular system of plants into the atmosphere. The entire process involves absorption of soil water by plant roots, translocation in liquid form through the vascular system of the roots, stem, and branches to the leaves; and through the vascular system of the leaf to the walls of tiny stomatal cavities, where evaporation takes place. The water vapour in these cavities then moves into the ambient air through openings in the leaf surface called stomata.

Figure: Diagram of the transpiration process and enlarged views of plant celluar structure where absorption, translocation and transpiration occur.

62 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Plants live by absorbing CO2 from the air to make carbohydrates, and that gas can enter the plant only when dissolved in water. The essential function of the stomatal cavities is to provide a place where CO2 dissolution can occur and enter plant tissue; the evaporation of water is an unavoidable concomitant of that process.

Air in stomatal cavities is saturated at the temperature of the leaf, and water moves from the cavities into the air due to vapour-pressure difference, just as in open water evaporation. The major difference between transpiration and open-water evaporation is that plants can exert some physiological control over the size of the stomatal openings, and hence the ease of vapour movement, by the action of guard cells.

Figure: Photomicrograph of leaf surface showing one of the stomata and the crescent-shaped guard cells that regulate its opening.

Major factors affecting the opening and closing of guard cells are 1) light (most plants open stomata during the day and close them at night) 2) humidity (stomatal openings tend to decrease as humidity decreases below its saturation value) 3) water content of the leaf cells Iif day-time water contents become too low, stomata tend to close) Transpiration is a physical, not a metabolic process: water in the transpiration stream is pulled through the plant by potential energy gradients that originate with the movement of water vapour into the air through the stomata in response to a vapour- pressure difference. When vapour exits through the stomata, water evaporates from the walls of the stomatal cavity to replace the loss; this loss of liquid water causes a potential-energy decrease that induces the movement of replacement water up through the vascular system: this movement ultimately produces a water-content gradient between the root and the soil; and this gradient induces movement of soil water into the root. Absorption at the root surface decreases soil water content in the adjacent soil, inducing some flow of water toward the root following Darcy’s law. The great cohesive strength of water due to its intermolecular hydrogen bonds maintains the integrity of the transpiration stream to heights of the tallest trees.

63 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Modelling transpiration

Evaporation- revisited: ⎡ ⎤ ⎢ D v k² ⎥ E = K ⎢ WV a ⎥[]e − e at z − z s a ⎢ D [ln[ m d ]]²⎥ ⎢ M ⎥ ⎣ z0 ⎦

0.622ρ a with “atmospheric constant” K at = pρ w

E = Kat × Cat × [es − ea ] evaporation atmospheric × atmospheric × driving rate constant conductance gradient

v k² with C ≡ a [L T-1] at z − z [ln[ m d ]]² z0

Since evaporation from an open water surface is a “one-step” process in which water molecules pass directly from the liquid surface into the atmosphere, it can be conceptually represented by the electric-circuit analogy. In this analogy a current (water vapour) moves in response to a voltage (vapour-pressure difference) across a resistance (inverse of the atmospheric conductivity)

64 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Relation between atmospheric conductance Cat and wind speed va for vegetations of various heights zveg.

65 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Leaf conductance

Is determined by the number of stomata per unit area and the size of the stomatal opening (10,000-100,000 /cm²)

Figure: Conceptualization of (a) open water evaporation and (b) transpiration in terms of atmospheric resistance (1/Cat) and leaf resistance (1/Cleaf). ∆ev is the driving vapour-pressure difference between the evaporating surface and the atmosphere.

Plants control the size of their stomatal openings and hence leaf conductance by the response of the guard cells to 1) light intensity 2) ambient CO2 concentration 3) leaf-air vapour-pressure difference 4) leaf temperature 5) leaf water content

Cleaf = Cleaf ,max G(K in ) G(∆ρ v ) G(Ta ) G(∆θ ) with Kin: incident shortwave radiation ∆ρv: humidity deficit (difference between saturated and actual absolute humidity of the air) Ta: air temperature ∆θ: soil moisture deficit (difference between field capacity and actual water content of root zone) 66 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Maximum leaf conductances C*leaf (cm s-1) for forest tree leaves.

Stewardt’s (1988) model of leaf conductance as a function of environmental factors

67 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

68 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

69 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Canopy conductance

Ccan = LAI Cleaf with leaf area index LAI

total area of transpiring surface above ground area A LAI ≡ A

70 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

The Peman Monteith Model

Penman open water evaporation rewritten in terms of Cat and γ:

s(T )[K + L] + ρ c C e (T )[1− RH] E = a a a at sat a ρ wλv [s(Ta ) + γ ]

Monteith (1965) showed how the Penman equation can be modified to represent the evapotranspiration rate, ET, from a vegetated surface by incorporating canopy conductance:

s(T )[K + L] + ρ c C e (T )[1− RH] ET = a a a at sat a ρ wλv [s(Ta ) + γ[1+ Cat / Ccan ]] : Penman-Monteith equation

Note that for Ccan→∞ Penman-Monteith reduces to Penman.

71 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Potential evapotranspiration

Potential evapotranspiration (PET) is the rate at which evapotranspiration would occur from large area completely and uniformly covered with growing vegetation which has access to an unlimited supply of water and without advection or heat- storage effects.

Estimating actual evapotranspiration by soil moisture functions The most widely used approach for estimating actual ET makes use of meteorological data to estimate PET and then computes actual ET as

ET = f (θ ) PET where θ : water content of the root zone soil f(θ): is based on relative water content θrel as

θ −θ wp θ rel ≡ θ fc −θ wp

72 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

General forms of relations between f (θ ) = ET / PET and soil water content θ used to estimate ET. Different studies have used different functions to express the wetness of the soils. When the soil-water content variable is less than the critical value indicated by the vertical dashed line, ET is less than PET and plants are considered under water stress.

E.g. Steward (1988):

⎧ Cat ⎫ s(Ta ) + γ ⎨1+ ⎬ C [G(∆θ ) = 1]] f (θ ) = ⎩ can ⎭ ⎧ Cat ⎫ s(Ta ) + γ ⎨1+ ⎬ ⎩ Ccan [G(∆θ )]]⎭

73 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Physics of Sensible and Latent Heat Fluxes Revisited: Physics of Turbulent Transfer

Diffusion Diffusion is the process by which constituents of a fluid (such as momentum, heat content or dissolved constituent) are transferred from one position another within the fluid. Such transfers occur whenever there are differences in concentrations of the constituent in different parts of the fluid. The rate of transfer of a constituent X in the direction z is directly proportional to the gradient of X in z-direction: dC(X ) F (X ) = −D Z x dz

FZ(X): rate of transfer of X in direction z per unit area per unit time (flux of X) C(X) concentration of X

Dx: diffusivity [Length²/Time]

Figure: Conceptual diagram of the general diffusion process. Concentration of dots indicates concentration of quantity X. More of the property X moves from z1 to z2 in a given time when the diffusivity Dx is higher. 74 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Energy can be transferred between the ground surface and the air in the form of latent heat whenever there is a difference in vapor pressure.

A second mode of nonradiant heat transport occurs in the form of sensible heat that is heat energy that can be directly sensed via measurement of temperature. Sensible heat transfer occurs whenever there is a temperature difference between surface and air.

H ∝ (Ts − Ta ) H [Energy/Time/Length²]

75 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Momentum transfer Mechanisms of diffusion in the lowest layer of the atmosphere: when wind blows across a surface, the surface produces frictional drag and turbulence. The bulk vertical movement of air in turbulent eddies carries latent (in form of water vapor) and sensible heat vertically in the directions dictated by equations above.

Figure: Conceptual diagram of the process of momentum transfer by turbulent diffusion. Friction caused by surface roughness reduces average velocities (Straight arrows) near the surface and produces turbulent eddies (circular eddies), resulting in a net downward transfer of momentum. The vertical component of the eddies moves heat and water vapor upward or downward depending on the directions of temperature and vapor-pressure gradients.

Heat fluxes: quantitatively expressed by vertical transfer of momentum

pr = m ⋅ vr (Momentum p = mass times velocity, [kg m s-1])

Concentration of momentum (momentum per unit volume) equals mass density times velocity ( ρ a va )

76 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Thus, for momentum, d(ρ v ) dv F (M ) = −D a a ≈ −ρ D a Z M dz a M dz holds. M: momentum

DM: diffusivity of momentum

dv a > 0 ⇒ F (M ) < 0 dz Z

Momentum is being transferred downward from where velocities are higher to where they are lower. Turbulent eddies are the mechanism responsible for that.

F [Force /area]: Represents the horizontal shear stress due to differences of wind velocity at adjacent levels.

Because momentum is conserved, shear stress FZ(M) is conserved

2 FZ (M ) = −ρ au* which leads to

dv − ρ D a = −ρ u 2 a M dz a * 2 u* ⇔ DM = dva / dz

Assuming wind profile:

1 ⎛ z − z ⎞ ⎜ d ⎟ va = u *ln⎜ ⎟, z > zd + z0 k ⎝ z0 ⎠ with k = 0.4 von-Karman constant [.]

kvm -1 • u* = : frictional velocity [ms ], vm: wind velocity at height zm. ⎡ zm − zd ⎤ ln⎢ ⎥ ⎣ z0 ⎦

1/ 2 • The friction velocity u* describes the intensity of turbulence. It is defined as u* ≡ ()− va ' ⋅ wa ' with wa’ fluctuation of vertical wind speed ( wa = wa + wa ' ). The minus sign is required because simultaneous values of wa’ and va’ have opposite signs. 77 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: vertical distribution of wind velocity over a vegetative surface of height zveg.

zd ≈ 0.7zveg , z0 ≈ 0.1zveg

yields

dv u a = * dz k(z − zd ) and therefore

DM = k u* (z − zd )

Thus the diffusivity of momentum increases in proportion to distance above the zero- plane displacement height and in proportion to the velocity of the eddies which is characterized by u*.

78 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Latent Heat Transfer Concentration of water vapor is given by absolute humidity (mass concentration of

nv RT pv water vapor) ρv . ( ρv = = ) V RvT Since molecular weight of water vapor is 0.622 molecular weight of dry air, the relation between pv and ρv can be expressed as p p v = ρv 0.622ρ a Upward flux of water vapor [M/L²/T] equals evaporation rate ρE [M/L³][L/T] therefore

dρ ρ E = −D v w vapor dz

0.622ρ dp = −D a v vapor p dz

The upward latent heat transfer rate λE can be found directly as 0.622ρ dp λE = −D λ a v vapor v p dz

Vapor and latent heat are transported vertically by the same turbulent eddies that are involved in momentum transfer. Using u ² 1 = * DM dva / dz gives (multiplication by 1)

79 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

D 0.622ρ dp / d D 0.622ρ k 2v 2 dp λE = − vapor λ a u 2 v z = − vapor λ a m v D v p * dv / d D v p 2 dv M a z M ⎛ zm − zd ⎞ a ⎜ln ⎟ ⎝ z0 ⎠ 2 2 D 0.622ρ k v pv − pv,s ≈ − vapor λ a m (z=m) = D v p 2 v − 0 M ⎛ zm − zd ⎞ (z=m) ⎜ln ⎟ ⎝ z0 ⎠ D 0.622ρ k 2v = vapor λ a m ( p − p ) = D v p 2 v,s v (z=m) M ⎛ zm − zd ⎞ ⎜ln ⎟ ⎝ z0 ⎠

= K λE vm ( pv,s − pv (z=m) )

80 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Sensible Heat Transfer

Thermal capacity, cp: ∆H c ≡ p m ∆T relates a temperature change of a substance to a change in heat-energy content.

−1 −1 E.g. thermal capacity of water c p = 4216 J kg K The change of concentration of sensible heat h [Energy/Length³] can be expressed as

∆h = ρ a ca (Ta − Tb ) ca: heat capacity of air

Ta, Tb: air temperature Diffusion equation of sensible heat: d(ρ c T ) F (H ) = −D a a a Z H dz

DH: diffusivity of sensible heat

Similarly to latent heat: D k 2v H = H c ρ m (T − T ) = D a a 2 s (z=m) M ⎛ zm − zd ⎞ ⎜ln ⎟ ⎝ z0 ⎠

= K H vm (Ts − T(z=m) ) where KH is effectively constant for a given situation.

81 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Soil Vegetation Atmosphere Transfer Modelling

(Example: Oregon State University Land Surface Model)

(note: four instead of two layers will be used with layer thicknesses 10cm, 30cm, 60cm and 100cm)

Objective: - modeling heat fluxes (latent, sensible, ground heat flux) - modeling water fluxes (soil moisture, surface runoff) - driven only by meteorological variables Case study: - 7 days period in summer 1998, Illinois test site - comparison to observed data

82 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Energy Balance

Rn = H + E + G

Net radiation [Wm-2]

4 Rn = (1−α)SD + LD − ε ⋅σ ⋅tsfc

SD: incoming shortwave radiation [Wm-2] LD: incoming songwave radiation [Wm-2] ε : emissivity [.] σ = 5.67 10-8 Stefan Boltzmann constant [Wm-2 K-4] Latent heat of evaporation [MJ kg-1]: (Energy absorbed (released) during evaporation (condensation))

λ = 2.5 − 0.002361 (ta − 273.25) ta: air temperature [K]

Saturated water vapor pressure [kPa] (Clausius-Clapeyron equation):

1 1 273.15 6816⋅( − )+5.1309log[ ] 273.15 ta ta esat = 0.1⋅6.112⋅e

esat

16

14

12

10

8

6 ta 305 310 315 320 325 330

83 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

-1 Slope of esat [kPa K ]:

4098⋅ e Λ = sat (273015 + (ta − 273.15))²

Λ SLOPE OF ESAT 0.8

0.6

0.4

0.2

ta 300 310 320 330

Vapor pressure [kPa]: (Partial pressure exerted by water vapor)

e = rh ⋅ esat with rh: relative humidity [.]

VAPOR PRESSUR

15

e 10 0.8 5 0 0.6 290 rh 300 0.4 310 ta 320 0.2 330

84 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Psychrometric constant (factor 0.1 converts pressure from mbar into kPa)

0.0016286( p 0.1) γ = sfc λ psfc: surface pressure [kPa]

0.072 0.07γ 1050 0.068 1040 1030 290 psfc 300 1020 310 1010 ta 320 330 1000

85 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Dry air density at surface [kg/m3]:

psfc ρ dry = 100 ta ⋅ rd with rd =287 J/kg/K, gas constant for dry air

DRY AIR DENSITY AT SURFACE ρdry

1.25 ρdry1.2 330 1.15 1.1 320

1000 310 ta 1010 1020 300 psfc 1030 1040 290 1050

Moist air density at surface:

psfc ρ moist = 100 Tv ⋅ rd with Tv: virtual temperature [K]

MOIST AIR DENSITY AT SURFACE

1.2 ρmoist1.18 0.8 1.16 1.14 0.6 1000 rh 1010 0.4 1020 1030 psfc 0.2 1040 1050

86 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Virtual temperature [K] (Temperature of a sample of dry air at the same density and pressure as a sample of moist air at temperature ta.)

Tv = ta (1+ 0.608 ωv ) with ωv: mass mixing ratio of water vapor

virtual temperatur

310 Tv 300 0.8

290 rh 295 0.6 300 ta 305 310 0.4

Virtual potential temperature [K] (Is found by bringing a moist parcel to a pressure of 1000 mbar, then converting all moisture in the parcel to dry air and determining the parcel’s temperature)

1000 0.286⋅(1−0.251ωv ) θv = Tv ⋅( ) psfc

VIRTUAL POTENTIAL TEMPERATURE

306 θv304 1050 302 1040 300 1030 psfc 0.2 1020 0.4 1010 rh 0.6 0.8 1000

87 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Mass mixing ratio of water vapor [.] (kg of water vapor per kg of dry air)

0.622 ⋅ e ωv = psfc ⋅ 0.1− e

MASS MIXING RATIO OF WATER VAPOR

0.1 ωv 1000 0.05 800 0 600 psfc 0.2 400 0.4 rh 0.6 200 0.8

Saturated mass mixing ratio [.]:

0.622 ⋅ esat ωv sat = psfc ⋅ 0.1− esat

Slope of saturated mass mixing ratio:

SLOPE OF SATURATED MASS MIXING RATIO

0.125 0.1 ωvsat 1050 0.075 0.05 1040 0.025 1030 300 1020 psfc 310 1010 ta 320 330 1000

88 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Specific humidity at ta [.] (Mass of water vapor per unit mass of moist air (dry air plus water vapor))

0.622 ⋅ e qa = ( psfc ⋅ 0.1− e) + 0.622 ⋅ e

SPECIFIC HUMIDITY

0.02 0.0qa 15 1100 0.01 1080 0.005 0 1060 psfc 0.2 1040 0.4 1020 rh 0.6 0.8 1000

Saturated specific humidity [.]:

0.622 ⋅ e q = sat asat ( psfc ⋅ 0.1− e) + 0.622 ⋅ esat

SATURATED SPECIFIC HUMIDITY

0.1 qasat0.075 1100 0.05 0.025 1080 0 1060 290 psfc 300 1040 310 1020 ta 320 330 1000

89 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Slope of saturated specific humidity [K-1]:

slope of saturated specific humidity

10 ΛΛ sat 1100 5 1080 0 1060 290 psfc 300 1040 310 1020 ta 320 330 1000

90 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Hydraulic conductivity [ms-1]:

θ 2b+3 k = k sat ( ) θ sat

θ sat : Volumetric water content at saturation ksat: Saturated hydraulic conductivity b: slope Parameter

hydraulicconductivity

k 3.5 ´ 10 -12 3´ 10 -12 2.5 ´ 10 -12 2 10-7 -12 ×k 15 2´ 10 1 10-7 -12 × 1.5 ´ 10 0 10 1´ 10 -12 b 0.1 -13 5´ 10 0.2 5 0.3 θ θ 0.1 0.2 0.3 0.4 0.4

Soil water diffusivity [m2s-1] 2+b ⎡ ⎛ θ ⎞ ⎤ ⎢ ⎜ ⎟ ⎥ b ⋅ k sat ⋅⎜ ⎟ ⋅ψ sat ⎢ ⎝θ sat ⎠ ⎥ Diff = Abs − ⎢ θ ⎥ ⎢ sat ⎥ ⎢ ⎥ ⎣ ⎦ with

Ψsat : saturated matrix potential (suction head)

hydraulic conductivity

0.2 Diff 0.1 0.8 0 0.6 6 θsat 8 0.4 10 b 12 0.2 14

91 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Soil water tension [m]: θ ψ =ψ ( sat )b sat θ

SOIL WATER TENSION ψ

0 -200 -40ψ 0 -600 0.6 -800

0.4 θ 5

b 10 0.2

15

Total volumetric heat capacity [Jm-3K-1]:

Cv = θ Cv water + (1−θ sat ) Cv soil + (θ sat −θ )Cv air with 6 -3 -1 cv water= 4.2 10 [Jm K ] water volumetric heat capacity 6 -3 -1 cv soil= 1.26 10 [Jm K ] soil volumetric heat capacity -3 -1 cv air= 1004 [Jm K ] air volumetric heat capacity

92 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

-1 -1 Thermal conductivity Kt [Wm K ]:

Kt = Min[1.9, Kt,temp ]

thermal conductivity Kt

15

Kttemp 10 15 5 0 10 0.2 b 0.4 5 θsat 0.6

⎧ 0.174 If [Pf > 5.1 or Pf < 0] K t,temp = ⎨ −(2.7+P ) ⎩420 e f else with θ P = Log[10,ψ ( sat )b ] f sat θ

93 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

thermal conductivity Kt

10 7.5 -0.2 Pf 5 2.5 -0.3 0 -0.4 ψsat 0.2 -0.5 0.4 0.6 -0.6 θsat 0.8 1

Maximum infiltration [mm d-1]:

kdt⋅dts − 86400 Pd Dx(1− e ) I max = kdt⋅dts P Dx dts − d + Dx(1− e 86400 ) 86400 103 with

kdtref ⋅ k sat kdt = kref kdt ref = 3 [.] 6 kref = 2/10 [.]

Dx = dz1 + dz2 + dz3 + dz4

94 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Ground heat flux G [Wm-2]:

K ⋅ (t − t ) G = t sfc soil 1 0.5⋅ dz1

Sensible heat flux H [Wm-2]:

H = ρ moist ⋅ c p ⋅ Ch ⋅ (t sfc −θ v )

Total evaporation [Wm-2; mm d-1 : conversion factor ≈ 29]:

Etotal = Edir + Et + Ec

Potential Evaporation [mm d-1] (Penman Monteith)

Λ ⋅ (R ⋅ 0.0864) E = n + (γ (6.43 (1+ 0.536v )(e − e))) /((γ + Λ)λ) pot (Λ + γ )λ a sat

Calculation of Potential Evaporation Epot

35 Epot32.5 1.5 30 1.25 27.5 1 0.4 va 0.6 0.75 ε 0.8 0.5 1

95 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

-1 Direct soil evaporation Edir: [mm d ]

(1−σ f ) ⋅ (θ1 −θ wilt ) ⋅ E pot E dir = θ ref −θ wilt

σf: vegetation fraction Calculation of direct soil evaporation Edir

30 Edir 20 10 0.8 0 0.6 0.1 ε 0.2 0.3 θ 0.4 wilt 0.4 0.5

Canopy transpiration [mm d-1]:

⎪⎧ 0 Wc = 0 Et = ⎨ Wc n σ f ⋅ E pot ⋅ Bc (1− ( ) ) else ⎩⎪ S

Wc: canopy water content [mm] S=0.5 mm maximum allowed Wc

Wet canopy evaporation [mm d-1]:

⎪⎧ 0 Wc = 0 Ec = ⎨ Wc n σ f ⋅ E pot ( ) else ⎩⎪ S

96 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Resistance term Bc:

ΛΛ 1+ sat R Bc = r ΛΛ sat 1+ Rc ⋅Ch + Rr

resistance term Bc

1 Bc0.9 0.8 0.8 0.7 0.6 60 θref 80 0.4 100 rgl 120 0.2 140

Factor Rr:

4 R r = (4 ⋅σ ⋅ta rd ) /(100( psfc ⋅Ch ⋅ c p )) +1

Factor Rr

15 1 Rr 10 5 0.8 0 0.6 0.5 rh 0.75 0.4 1 va 1.25 0.2 1.5

97 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

-1 Canopy Resistance Rc: [sm ]

rc min R c = Lai ⋅ f1 ⋅ f 2 ⋅ f3 ⋅ f 4 with -1 rcmin: minimal stomatal resistance [sm ] Lai: Leaf area index [.]

Lai = (total area of transpiring surface above ground area A)/A

Calculation of Canopy Resistance Rc

500 Rc250 40 0 35 -250 -500 30 290 hs 300 25 310 20 ta 320 330 15

rc min + f 0 rc max f1 = 1+ f 0

Calculation of Canopy Resistance Rc

0.9 0.8 10 f10.7 0.6 8 0.5 6 10 Lai 20 4

rcmin 30 2 40 with

98 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

0.55 SD ⋅ 2 f 0 = rgl ⋅ Lai

Calculation of Canopy Resistance Rc

15 6 f0 10 5 5 0 4 50 Lai 60 3 70 80 2 rgl 90 100 1

1 f 2 = 1+ hs(ωv sat − ωv )

Calculation of Canopy Resistance Rc

0.85 1100 f20.8 0.75 1080 1060 20 psfc 1040 30 1020 hs 40 1000 50

99 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

f3 = 1- 0.0016(t ref - t a )²

f3 Calculation of Canopy Resistance Rc 1

0.8

0.6

0.4

0.2

ta 300 310 320 330

(θ1 −θ wilt ) dz1 (θ 2 −θ wilt ) dz2 (θ3 −θ wilt ) dz3 f 4 = + + (θ ref −θ wilt )(dz1 + dz2 + dz3 ) (θ ref −θ wilt )(dz1 + dz2 + dz3 ) (θ ref −θ wilt )(dz1 + dz2 + dz3 )

Calculation of Canopy Resistance Rc

1 f40.5 0 -0.5 0.4 0.1 θref 0.2 0.3 0.2 θwilt 0.4 0.5

100 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Exchange coefficient for heat [ms-1]:

C h = Abs(va ) ⋅ C n ⋅ f h

exchange coefficient for heat

0.014 Ch 1150 0.012 0.01 1100 0.06 rh 0.08 1050 0.1 psfc 0.12 0.14 1000

101 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Neutral exchange coefficient [.]

kar² C = n h h Log[ a ]* Log[ a ] z0m z0m kar =0.4 Karman constant z0m: roughness length

Cn neutral exchange coefficient 0.018

0.016

0.014

0.012

z0m 0.02 0.04 0.06 0.08 0.1 0.008

0.006

Functions modifying neutral exchange coefficient:

⎧ e −Ri ⎪ , Ri > 0 f h = ⎨ 1 0,5 1− (15Ri) /(1+ 75Cn ⋅ ( (ha * Abs[Ri])) )], R <= 0 ⎩⎪ z0m i

functions modifying neutral exchange coefficient

0.8 fh 1050 0.6 1040 0.4 1030 0.5 1020 psfc 1 1010 va 1.5 1000

102 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Bulk Richardson number (Quantifies the ratio of buoyancy to mechanical shear; Ratio: turbulence due to shear /turbulence due to buoyancy)

g ⋅ h ⋅ (ω − t ) Ri = a v sfc ωv ⋅ va ²

ha: height of met-station measurement [m] g: gravitational constant [ms-2] -1 va: wind speed at ha [ms ]

Bulk Richardson number Ri

6 Ri 4 1050 2 1040 0 1030 psfc 0.25 1020 0.5 0.75 1010 va 1 1000 1.25

103 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

VEGETATION – RELATED PARAMETERS IN THE SVAT- MODEL

- roughness length (Z0) in [m], -1 - minimal stomatal resistance (Rcmin) in [sm ]

- Rgl is the visible solar flux for which f1 is about to double its minimum value,

- hs is a parameter in f2.

Vegetation types Albedo Z 0 (m) Rc min Rgl hs Broadleaf - evergreen trees 0.11 2.653 150 30 41.69 Broadleaf - deciduous trees 0.12 0.826 100 30 54.53 Broadleaf and needleleaf trees 0.12 0.8 125 30 51.93 Needleleaf - evergreen trees 0.1 1.089 150 30 47.35 Needleleaf - deciduous trees (larch) 0.11 0.854 100 30 47.3 Broadleaf trees with groundcover 0.19 0.856 70 65 54.53 Groundcover only 0.19 0.075 40 100 36.35 Broadleaf shrubs with groundcover 0.25 0.238 300 100 42 Broadleaf shrubs with bare soil 0.25 0.065 400 100 42 Dwarf trees/shrubs with groundcover 0.16 0.05 150 100 42 (tundra) Bare soil 0.12 0.011 Cultivations 0.19 0.075 40 100 36.35 Wetland 0.12 0.04 150 100 60 Dry coastal complex 0.19 0.075 400 100 200 Water 0.19 0.01 Glacial 0.8 0.011 999 999 999

104 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Exercise: Case study Champaign, Illinois (88.37°W, 40.01°N)

Meteorological Input a) Wind

b) Temperature

c) Relative Humidity

105 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere d) Precipitation

e) Pressure

f) Shortwave radiation

106 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere g) Longwave radiation

Simulation Results a) Latent Heat Flux

600

500

400 [Wm-2]

ux 300 l

200

100 Latent Heat F 0 13.7.92 14.7.92 15.7.92 16.7.92 17.7.92 18.7.92 19.7.92 20.7.92 21.7.92 22.7.92 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 -100 Date observation simulation b) Sensible Heat Flux

200 ] 2

- 150 m [W

ux 100 Fl at

He 50 e l ib ns

e 0 S 13.7.92 14.7.92 15.7.92 16.7.92 17.7.92 18.7.92 19.7.92 20.7.92 21.7.92 22.7.92 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 -50 Date

observation simulation

107 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Part II: Meteorological Modelling and Applications to Hydrological Engineering

• Fundamentals of Atmospheric Science • Water in the Atmosphere • Continuity Equations • Momentum Equations • Global Circulation • Nonlinear Dynamics and Chaos • Numerical Solution of Partial Differential Equations using Finite Difference Schemes

108 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

EQUATIONS OF STATE

The equation of state describes the relationship among pressure, volume and absolute temperature of a real gas.

Equation of state for ideal gas

nR*T nA ⎛ R* ⎞ ⎜ ⎟ p = = ⎜ ⎟T = Nk BT : Ideal gas law (simplified equation of state) V V ⎝ A ⎠ p: Pressure (mbar) R*: Universal gas constant (R=0.083145 m3 mbar mole-1 K-1) T: Temperature (Kelvin) V: Volume (m3) n: number of moles of a gas (mole) -25 3 -1 kB: Boltzman’s constant (=1.3807 10 m mbar K ) ⎛ nA ⎞ N: number concentration of gas molecules (molecules of gas per Volume) ⎜ N = ⎟ ⎝ V ⎠ A: Avogadro’s number, A=6.02252 1023 molecules

Example Calculate the number concentration of air molecules in the atmosphere at standard sea-level pressure and temperature and at a pressure of 1 mb.

SOLUTION At standard sea level, p = 1013 mb and T = 288 K. N =2.55 1019 molecules cm-3. p = 1 mb occurs at 48 km. At this altitude and pressure, T = 270 K, N = 2.68 1016 molecules cm-3.

109 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Temperature structure of the earth’s lower atmosphere (Jacobson, 2005)

Troposphere: average rate of temperature decrease in free troposphere is about 6.5 K km-1; Thermal infrared radiation emitted by ground is absorbed by selective gases like CO2 and water vapour. These gases are most concentrated near surface. Once they absorb the radiation, they reemit half of it upwards and the other half downward, but their emission is at lower temperature than is the emission of the radiation they absorbed. Because less radiation is available for absorption with increasing altitude, temperature decreases with increasing altitude in the troposphere.

Stratosphere: O3 and O2 absorb UV radiation and emit infrared radiation heating the region.

Thermosphere: molecular oxygen and nitrogen absorb very short wavelengths of radiation in this region.

110 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Equation of state for dry air

n R *T n m R' p = d = d d T = ρ R′T d V V d nd: number of moles of dry air (mole) pd: partial pressure exerted by dry air (mbar) -3 ρd: mass density of dry air (kg m ) 3 -1 -1 R’: gas constant for dry air (= 2.8704 m mbar kg K ) =R*/md -1 md: molecular weight of dry air (=28.966 g mole )

Equation of state for water vapor

n R *T p = v = ρ R T v V v v pv: partial water pressure exerted by water vapor (mbar) -3 ρv: mass density of water vapor (kg m ) (absolute humidity) 3 -1 -1 * Rv: gas constant for water vapor (= 4.6140 m mbar kg K ) Rv=R /mv -1 mv: molecular weight of water vapour (=0.622 md [g mole ])

Mass mixing ratio of water vapour

ρv mv pv pv pv ωv = = = ε = ρ d md pd pd pa − pv (kilograms of water vapour per kilogram of dry air)

Specific humidity (moist air mass mixing ratio)

pv ρ ρ R T ε p q = v = v = v = v v ρ ρ + ρ p p p + ε p a d v d + v d v R′T RvT = mass of water vapor per unit mass of moist air ( . ) R' R *⎛ m ⎞ m with ε = = ⎜ v ⎟ = v = 0.622 Rv md ⎝ R *⎠ md

-3 ρa, ρd, ρv: density of most air, dry air and water vapor (kg m ) 111 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Example

If pv=10 mbar and pa=1010 mbar, pa=pd+pv gives pd=1000mbar; qv=0.00618 kg/kg

Figure: Global distribution of surface specific humidity [g kg-1]

Figure: Global vertical mean specific humidity [g kg-1]

112 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Equation of state for moist air : Sum of the equations of state for dry air and water vapour

ρ d + ρv Rv / R' pa = pd + pv = ρ d R'T + ρv RvT = ρ a R'T ρ a

with ε = R'/ Rv , ρ a = ρ d + ρv , ω v = ρv / ρ d it follows ρ 1+ v ρ + ρ /ε ρ ε p = ρ R'T d v = ρ R'T d and therefore a a ρ + ρ a ρ d v 1+ v ρd

1+ ωv /ε pa = ρa R'T = ρa RmT = ρa R'Tv 1+ ωv

1+ ω / ε v : gas constant for moist air Rm = R' ≈ R'(1+ 0.608ωv ) 1+ ωv

1+ ωv /ε Rm Tv = T = T ≈ T(1+ 0.608ωv ) : virtual temperature: 1+ωv R' Temperature of a sample of dry air at the same density and pressure as a sample of moist air at temperature T. ω 1+ v ε Note: ≈1+ 0.608ωv (Taylor Expansion around ωv = 0 ) 1+ ωv ∂f f (ω ) ≈ f (ω ) + (ω − 0) v v ωv =0 ωv =0 v ∂ωv

⎛ ω ⎞ ω ⎜1+ v ⎟ 1+ v ∂f ∂ ⎜ ε ⎟ 1 ε ∂f 1 with = = − 2 ; ⇒ = −1 ∂ωv ∂ωv ⎜ 1+ ωv ⎟ ε (1+ ωv ) (1+ ωv ) ∂ωv ε ⎜ ⎟ wv =0 ⎝ ⎠

1+ ωv /ε ε = 0.622 ⇒ ≈ 1+ 0.608ωv 1+ ωv Example

If pd=1013 mbar and pv=10 mbar, T=298K, calculate ωv , Rm ,Tv , ρa

−1 ω v = 0.622 ⋅10mb /1013mb = 0.00614 kg kg

3 −1 −1 Rm = 2.8704 ⋅ (1+ 0.608 ⋅ 0.00614) = 2.8809 m mb kg K

Tv = 298⋅ (1+ 0.6 ⋅ 0.00614) = 299.1 K

−3 ρ a = pa /(Rm ⋅T ) = 1023 /(2.8809 ⋅ 298) = 1.19 kg m

113 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

CHANGE IN PRESSURE WITH ALTITUDE

Hydrostatic equation (Upward gradient in air pressure is balanced by downward force of gravity multiplied by air density)

Figure: Derivation of barometric pressure equation

dpa = −ρa g dz

∂p a = −ρ g ∂z a g: gravitational acceleration (9.81 ms-2) z: altitude (m) • Hydrostatic balance: air has no vertical acceleration • Reasonable assumption when averaged over large horizontal regions • Over small regions (<2-3 km) hydrostatic assumption is not accurate (e.g. cumulonimbus)

114 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

∂pa pa,1 − pa,0 ≈ = −ρ a ,0 g ∂z z1 − z0

pa,0 pressure at altitude z0 , pa,1 pressure at altitude z1

Example -3 pa,0 =1013mbar ρ a,0 =1.23 kg m , pa,100 =? Solution:

−3 −2 pa,100 = 1013mb −1.23kgm ⋅9.81ms ⋅100m = 1000.2 mbar

(1 bar = 103 mbar = 105 N / m2 = 105 kg m / s² 1/ m² = 105 kg m−1s −2 ) ⇒ Near the surface , pressure decreases approximately 1mbar for every 10m increase in altitude

M ⋅ g dp = −ρ g dz = − p dz a a R *⋅T M M ⋅ p ( ρ = = ) V R *⋅T T: absolute temperature [K], R*=0.083145 m3 mbar mole-1 K-1 universal gas constant, M: mean mol mass of air (M=28.97g/Mol) Exponential decrease of pressure with height: ⎛ M ⋅ g ⎞ p = p0 ⋅ exp⎜− z⎟ ⎝ R *⋅T ⎠ p0: surface pressure Includes approximation: no temperature decrease with height

−1 ⎛ M ⋅ g ⎞ Scale height: z0 = ⎜ ⎟ ≈ 8 km height in which pressure is reduced to 1/e. ⎝ R *⋅T ⎠ (around 5.5 km for reduction to 1/2).

115 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Standard atmosphere

pa,s = 1013.25 mbar

Ta,s = 288 K ∂T Γ = − = +6.5 K km −1 s ∂z ∂p p a = − a g ∂ z RmT

Substituting T = Ta,s − Γs ⋅ z (where z=0km corresponds to sea level), integrating from pa,s to pa and 0 to z yields

g ⎛ T − Γ ⋅ z ⎞ ln()pa = ln⎜ a,s s ⎟ pa,s ⎜ ⎟ Γs Rm ⎝ Ta,s ⎠ Rearranging again gives altitude as a function of pressure in a standard troposphere

⎡ Γs Rm ⎤ T ⎛ p ⎞ g a,s ⎢ ⎜ a ⎟ ⎥ z = ⎢1− ⎜ ⎟ ⎥ Γs ⎝ pa,s ⎠ ⎣⎢ ⎦⎥

116 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

PHASE TRANSITIONS OF WATER

Figure: Global annual precipitation [dm/year]

Water appears in three states: gas, liquid, solid. Sources and sinks of water vapour in the atmosphere are evaporation from soils, lakes, streams, rivers, and oceans, sublimation from glaciers, sea ice, and snow packs, and transpiration from plant leaves. Approximately 85 percent o water in the atmosphere originates from ocean surface evaporation. Sinks of water vapour are condensation to liquid phase, ice deposition to the solid phase, transfer to the ocean and other surfaces. Water vapour is not only a greenhouse gas (a gas that readily absorbs thermal-infrared radiation) but also a chemical reactant and carrier of latent heat. Sources of liquid water include sea spray emission, volcanoes, combustion, condensation of water vapour, and melting of ice crystals. Sinks of liquid water include evaporation, freezing and sedimentation to the surface of aerosol particles, fog drops, drizzle, and raindrops.

117 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Ice in the atmosphere forms from freezing of liquid water and solid deposition of water vapour. Losses of ice occur by sublimation to the gas phase, melting to the liquid phase, and sedimentation to the surface.

Types of energy When water change state, it releases or absorbs energy. Kinetic energy is the energy within a body due to its motion (E=0.5 M v²). To change velocity and kinetic energy of a body, mechanical work must be done Potential energy is the energy of matter that arises due to its position, rather than its motion Internal energy is the kinetic and/or potential energy of molecules within an object. Work is the energy added to a body by the application of a force that moves the body in the direction of force Electromagnetic (radiant) energy is the energy transferred by electromagnetic waves that originate from bodies with temperatures above 0K

118 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Latent heat During condensation, freezing and deposition of a substance, energy is released. During evaporation, melting, and sublimation, energy is absorbed. The energy released or absorbed during such processes is latent heat. Latent heat absorbed (released) during evaporation (condensation) is latent heat of evaporation. It varies with temperature as

dλ e = c − c dT p,V W -1 -1 c p,V : Specific heat of water vapour at constant pressure [J kg K ]

-1 -1 cW : Specific heat of liquid water [J kg K ]

Figure: Phase changes of water. Freezing (melting) at 0°C releases (adsorbs) 333.5 Jg-1, deposition (sublimation) at 0°C releases (absorbs) 2835 Jg-1

Figure: Variation of specific heat of liquid water with temperature

119 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Since changes in c p,V and cW are small for temperatures above 0°C, these parameters are often held constant

6 -1 λe = λe,0 − (cw − c p,V )(T −T0 ) ≈ 2.501⋅10 − 2370 ⋅Tc [J kg ]

Example 0°C: 2501 J is required to evaporate 1 g of liquid water. At 100°C, 2264 J is required.

120 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Clausius-Clapeyron Equation A parameter that affects liquid-water (and also ice-particle) formation is the saturation vapour pressure. The saturation vapour pressure is the partial pressure of gas at a specific temperature when the gas is in equilibrium with a liquid or solid surface. In a gross sense, it is the maximum amount of vapour the air can hold without the vapour condensing as a liquid or depositing as ice on a surface. The temperature variation of the saturation vapour pressure of water over a liquid surface pv,s [mb] is approximated with the Clausius-Clapeyron equation

dpv,s ρv,s λe pv,s = λe = 2 dT T RvT

-3 Le : latent heat of evaporation [Jkg-1], ρ v,s [kg m ]

dpv,s 1 ⎛ Ah Bh ⎞ → = ⎜ 2 − ⎟dT pv,s Rv ⎝ T T ⎠

T = T0 = 273.15 pv,s,0 = 6.112 mb

⎡ ⎛ 1 1 ⎞ ⎛ 273.15⎞⎤ pv,s = 6.112 exp⎢6816 ⎜ − ⎟ + 5.1309 ln⎜ ⎟⎥ ⎣ ⎝ 273.15 T ⎠ ⎝ T ⎠⎦ with T [K], pv,s [mb].

esat mbar

120 @ D 100

80

60

40

20

ta °C -20 -10 10 20 30 40 50

Figure: Saturation vapor pressure pv,s [mb] over liquid water@ D

121 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Example

T=253.15K (-20°C), pv,s = 1.26mbar

T=298.15K (25°C), pv,s = 31.60mbar

Climate change:

T=288.15K (15°C), pv,s=17.03mbar

T=289.15K (16°C), pv,s=18.16mbar ∆T=1°C -> Relative change of precipitable water by 5.5% -> impact on precipitation (e.g. intensity)

Figure: a) condensation occurs when the partial pressure of water vapour exceeds its saturation vapour pressure over liquid water, b) evaporation occurs when the saturation vapour pressure exceeds the partial pressure of water vapour (schematics not to scale).

122 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Relative humidity p fr = 100% v pv,s

Figure: Global relative humidity in % at 850mbar

123 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

CONTINUITY EQUATIONS

Local and total differentiation

Total velocity in Cartesian coordinate system: vr = iu + jv + kw : i, j, k Cartesian–coordinate west-east, south-north, and vertical unit vectors dx dy dw u = , v = , w = : scalar components of velocity [ms-1] dt dt dt

r 2 2 2 r 2 2 v = u + v + w , vh = u + v

Figure: Zonally averaged west-east scalar velocities (m/s) for a) January and b) July (Jacobson, 2005)

The time rate of change of a variable such as concentration, momentum or temperature can be determined at a fixed location or in the frame of reference of the variable as it moves

dN : total derivative: time rate of change of N anywhere along the plume`s trajectory dt ∂N : partial derivative: time rate of change of N at fixed point ∂t

124 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

dN ∂N dt ∂N dx ∂N dy ∂N dz N = N(x(t),t) ⇒ = + + + = dt ∂t dt ∂x dt ∂y dt ∂z dt ∂N ∂N ∂N ∂N = + u + v + w = ∂t ∂x ∂y ∂z ∂N r = + (vr ⋅∇)N ∂t r with ∇ : Gradient Operator Lagrangian frame of reference ⇔ Eulerian frame of reference (Frame of reference that moves relative to a fixed coordinate system ⇔ frame of reference fixed relative to a fixed coordinate system)

Example Suppose the time rate of change of concentration of a gas along the path of a hot-air dN balloon travelling with the wind from east to west at u = -10 m s-1 is = 108 dt ∂N molecules cm-3s-l. If the west-east gradient in concentration is = 1010 molecules ∂x cm-3km-l (concentration increases from west to east), determine the time rate of change of concentration at a fixed point A, which the balloon passes over.

SOLUTION: ∂N u = -108 molecules cm-3s-1 : rate of change of concentration in plume resulting ∂x from west-east velocity transporting the plume. ⎛ ∂N ⎞ ⎜ ⎟ ≈ 2⋅108 molecules cm-3s-l ⎝ ∂t ⎠ A Thus, transport from the east accounts for one-half of the production rate of N at point A, and transformations along the trajectory (dN/dt) account for the other half.

125 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Derivation of continuity equations

Figure: Example of mass conservation (Jacobson, 2005)

[N]= Molecules cm-3 [∆x],[∆y],[∆z] = cm [u]= ms-1

∆N∆x∆y∆z = u1 N1∆y∆z∆t − u2 N 2 ∆y∆z∆t : Number of molecules leaving or entering control volume

∆N u N − u N = −( 2 2 1 1 ) ∆t ∆x

∂N ∂(uN) = − ∂t ∂x Continuity equation for a gas affected in one dimension

Generalization: Continuity equation for number concentration: ∂N r = −∇ ⋅ (vrN) ∂t

Similarly: Continuity equation for air density: ∂ρ r a = −∇ ⋅ (vrρ ) ∂t a

126 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

∂N ∂(uN) ∂(vN) ∂(wN) r = − − − = −∇ ⋅ (vrN) 1 ∂t ∂x ∂y ∂z

1a 2

∂N r r Aρ q = −N(∇ ⋅ vr) − (vr ⋅ ∇)N) 2 N = a molec./m³ moist air ∂t m

1b Cont. eq. for air density

∂ρ a r r ∂q r ∂ρ r r r q( + ρ (∇ ⋅ vr) + (vr ⋅ ∇)ρ ) + ρ = −ρ (vr ⋅ ∇)q a = −∇ ⋅ (vrρ ) = −ρ (∇ ⋅ vr) − (vr ⋅∇)ρ ) ∂t a a a ∂t a ∂t a a a

∂q r (1) = (vr ⋅ ∇)q ∂t Gas continuity equation for specific humidity (Advective form)

∂ρ q r (2) a + ∇ ⋅ (ρ vrq) = 0 ∂t a Humidity balance equation (Flux divergence form) kg []ρ vrq = : flux of total mass of water in a m 2 s air

1 r r r With vrN = i uN + jvN + kwN r r r 2 ∇ ⋅ (vrN) = N(∇ ⋅ vr) + (vr ⋅∇)N : identity 127 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Considering sources and sinks:

∂ρ q r a + ∇ ⋅ (ρ vrq) = Qρ ∂t a A Q: source/sink term (g/kg/s)

r kg []ρ a vq = m 2 s

128 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

TURBULENT HUMIDITY FLUXES The spatial domain in a model is divided into grid cells of finite size. In most cases atmospheric motions occur over spatial scales much smaller than the resolution of the grid cell and over temporal scales smaller than the resolution of the time step. To account for subgrid scale disturbances, each variable in the model equations can be divided into an average and perturbation component.

Figure: Precise, mean and perturbation components of scalar velocity and gas concentration (Jacobson, 2005)

For instance, actual (precise) humidity and velocity can be decomposed as

q(t) = q + q′(t);

v(t) = v + v′(t) q,v are constant in the averaging time interval q′,v′ = 0 Therefore qv = qv + q′v′

3 Additional turbulent

3 Combined temporal and spatial averaging: q = [q]+ q∗ ;v = [v]+ v∗ yields [qv] = [q][v]+ [q′v′]+ [q ∗v ∗ ] []q [v]: “standing cell”, [q ∗v ∗ ]: “standing eddies”, [q′v′]: “transient eddies”

129 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Example

q(τ ) = q + ∆qcos(τ −τ 0 ) v(τ ) = v + ∆vcos(τ ) q = 4⋅10−3 , ∆q =1⋅10−3 v =1m/ s, ∆v =10 m/ s

q ⋅v = 4⋅10−3 m/ s 1 2π q′v′ = ∆q∆v cos2τ dτ ∫ 2π 0 1 = ∆q∆v = 5⋅10−3 m/ s 2

Figure: Mechanism of turbulent humidity fluxes (Baumgartner & Liebscher, 1990)

′ Since density perturbation in the atmosphere are small, ρ a << ρ a , ρ a ≈ ρ a ; thus

∂ρ (q + q′) r a + ∇ ⋅ (ρ (vr + vr′)(q + q′)) = ρ Q ∂t a a

Taking time and grid volume average of this equation, eliminating zero valued terms, removing unnecessary overbars results in

∂q r ⎡∂ρ r ⎤ r ⎡ r ⎤ a r r′ ′ ρ a ⎢ + (v ⋅∇)q⎥ + q⎢ + ∇ ⋅ (v ρ a )⎥ + ∇ ⋅ (ρ a v q ) = ρ aQ ⎣ ∂t ⎦ ⎣ ∂t ⎦ which yields4

∂q r r 1 r r + (v ⋅∇)q + ∇ ⋅ (ρ a v′q′) = Q ∂t ρ a

∂ρ r 4 a + ∇ ⋅ (vrρ ) = 0 : continuity equation for air ∂t a 130 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Whereas models calculate spatially and temporally averaged values (e.g. q, ρ a ,u ), explicit equations for cinematic turbulent fluxes (e.g. u′q′,v′q′, w′q′ ) have not been developed. Instead these quantities are estimated with parameterizations (e.g. K- theory: gradient transport theory):

∂q ∂q ∂q u′q′ = −K ,v′q′ = −K ,w′q′ = −K xx ∂x yy ∂y zz ∂z

K: eddy diffusion coefficients (m2s-1)

In general:

∂q r r 1 r r + (v ⋅∇)q + ∇ ⋅(ρ aK∇q) = Q ∂t ρ a

⎡K xx 0 0 ⎤ where K = ⎢ 0 K 0 ⎥ : eddy diffusion tensor ⎢ yy ⎥ ⎣⎢ 0 0 K zz ⎦⎥

131 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Passing of cumulus-cloud, 1500 m vertical extension, 25m blow summit; horizontal extension: 1km. Correlation between qw and w. (Baumgartner & Liebscher, 1990)

132 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

MOMENTUM EQUATIONS … Solving for the velocities …

Spherical coordinates r r r r r r r v = iλu + jϕ v + kr w, vh = iλu + jϕ v

dx = (R cosϕ)dλ e e dy = Redϕ

Figure: Spherical coordinate symbols. Re : Earth’s radius (6371 km), ϕ : latitude, λ : r r r longitude, iλ West-East, jϕ South-Nord and kr vertical unit vectors

Forces cause objects such as air parcels to accelerate or decelerate, thereby creating or altering winds. The relationship between forces and winds is called atmospheric dynamics. The momentum equation is derived from Newtons Second Law of motion

1 r ar = F inertial m ∑

F [N or kg m s-2]: force, m [kg]: mass, a: acceleration [ms-2] Inertial acceleration is derived by considering that, to an observer in space, the absolute velocity of a body in motion near the surface is

133 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

r r r r vabs = v + Ω× Re

r -1 vabs : absolute velocity [ms ] (as seen by observer from space) vr : local velocity of body as seen by observer at earth’s surface r Ω : angular velocity vector [rad s-1] r Re : radius vector for earth [m]

r r -1 Ω× Re : rate of change in position of the body due to the earth’s rotation [ms ]

r r r r r Ω = jϕ Ωcosϕ + kr Ωsinϕ , Re = kr Re where Ω = 2π rad / 86164s = 7.292 ⋅10−5 rad s −1 magnitude of angular velocity, 86164: number of seconds for one revolution around axis (23h 56m 4s).

Inertial acceleration is defined as dvr r ar = a + Ω × v inertial dt a

r d r r r r r dvr r dR r r r r ar = (vr + Ω × R ) + Ω × (vr + Ω × R ) = + Ω × e + Ω × v + Ω × (Ω × R ) inertial dt e e dt dt e

r r dR dk With e = R r ≈ vr , it follows: dt e dt

dvr r r r r ar = + 2Ω × vr + Ω × (Ω × R ) = ar + ar + ar inertial dt e local Coriolis centripetal

dvr ar = : Local acceleration: rate of change of velocity of a parcel of air in motion local dt relative to a coordinate system fixed on earth r r r aCoriolis = 2Ω× v : Coriolis acceleration: rate of change velocity of a parcel due to rotation of a spherical earth underneath the parcel r r r r acentripetal = Ω × (Ω × Re ) : Earth’s centripetal acceleration: inward-directed rate of change of velocity of a parcel due to its motion around the earth’s axis.

134 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

1 r r r ar + ar + ar = (F + F + F ) local Coriolis centripetal m gravitation pressure viscous r Fgravitation : true gravitational force r Fpressure : pressure gradient force r Fviscous : viscous force (drag)

1 r r r r r ar = (F + F + F − F − F ) local m gravitation pressure viscous Coriolis centripetal

Whereas the centripetal and Coriolis effects are viewed as accelerations from an inertial frame of reference, they are viewed as apparent forces from a noninertial frame of reference. An apparent (or inertial) force is a fictitious force that appears to exist when an observation is made in a noninertial frame of reference. r r r Fgravitation , Fpressure Fviscous are real forces.

135 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Moment equation from a reference frame fixed on earth’s surface Atmospheric models usually require expressions for local acceleration; thus momentum equation is written most conveniently for a reference frame fixed on the surface of the earth rather than fixed outside the earth-atmosphere system. Local acceleration

dvr ∂vr r ar = = + (vr ⋅∇)vr = local dt ∂t Spherical Coordinates r ∂u u ∂u v ∂u ∂u uv tanϕ ≈ iλ ( + + + w − ) + ∂t Re cosϕ ∂λe Re ∂ϕ ∂z Re

r ∂v u ∂v v ∂v ∂v u 2 tanϕ + jϕ ( + + + w − ) + ∂t Re cosϕ ∂λe Re ∂ϕ ∂z Re r ∂w u ∂w v ∂w ∂w + kr ( + + + w ) ∂t Re cosϕ ∂λe Re ∂ϕ ∂z

(approximation: terms with uw/ Re , vw / Re removed since small for large scale motions)

Coriolis acceleration

r r r aCoriolis = 2 ⋅ Ω × v = r r r = iλ 2Ω(wcosϕ − vsinϕ) + jϕ 2Ωu sinϕ − kr 2Ωu cosϕ ≈

(1) r r ≈ jϕ 2Ωu sinϕ − kr 2Ωu cosϕ (2) r r ≈ − iλ 2Ωvsinϕ + jϕ 2Ωu sinϕ Defining the Coriolis parameter as f = 2Ωsinϕ gives another form of the Coriolis term as

r r r r r aCoriolis ≈ − iλ fv + jϕ fu = fkr × vh with vh horizontal wind speed. π Note: Coriolis force is zero at equator (sin0 = 0 ), and maximum at poles (sin =1)! 2

(1) approximation if only zonal (i.e. west-east) winds are considered (2) approximation if vertical component w and vertical components are neglected

136 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Example

−1 A mean wind speed of vh =10ms at the North pole results in a Coriolis acceleration r F magnitude of ar = Coriolis = 0.001454 ms −2 . Coriolis m

Figure: Example of Coriolis deflections. The Coriolis force deflects moving bodies to the right in the northern Hemisphere and to the left in the Southern Hemisphere.

137 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Coriolis acceleration components that results when the Coriolis force acts on a west-to-east wind travelling around the earth, parallel to the equator and into the page

Gravitational force The true gravitational force vector acts towards the center of earth: r r Fgravitation = −mkr g

GM e −2 with g = 2 = 9.81ms R e G = 6.6720 ⋅10−11 m3kg −1s−2 : universal gravitational

24 M e = 5.98 ⋅10 kg : earth’s mass

138 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Pressure-gradient force Pressure gradient force is a real force that causes air to move from regions of high pressure to regions of low pressure. The force results from pressure differences.

⎛ ∆p ∆x ⎞ Force from air pressure left side of parcel: Fp,l = ⎜ pc − ⎟∆y∆z ⎝ ∆x 2 ⎠ ⎛ ∆p ∆x ⎞ Force from air pressure right side of parcel: Fp,r = −⎜ pc + ⎟∆y∆z ⎝ ∆x 2 ⎠ Force from total air pressure in center of box: ⎛ ∆p ∆x ⎞ ⎛ ∆p ∆x ⎞ Fp,x = Fp,l + Fp,r = ⎜ pc − ⎟∆y∆z − ⎜ pc + ⎟∆y∆z ⎝ ∆x 2 ⎠ ⎝ ∆x 2 ⎠

Fp,x ∆p ⇒ = − ∆x∆y∆z ∆x

Fp,x ∆p ⇒ = − ρ a ∆x∆y∆z ρ a ∆x

Fp,x ∆p ⇒ = − m ρ a ∆x

F ∂p ∆x, ∆y, ∆z → 0: p,x = − m ρ a ∂x

∆p ∆x ∆p ∆x pc − p pc + ∆x 2 c ∆x 2

Figure: Example of pressure-gradient forces acting on both sides of a parcel of air

139 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Generalized:

r m r Cartesian Coordinates m ∂p r ∂p r ∂p F = − ∇p = − (i a + j a + k a ) = p ρ a ρ ∂x ∂y ∂z a a Spherical Coordinates m 1 ∂pa r 1 ∂pa r ∂pa = − (iλ + jϕ + kr ) ρa Re cosϕ ∂λe Re ∂ϕ ∂z

Figure: Pressure gradient force (heavy line) is perpendicular to isobars (medium lines) from high to low pressure. H and L indicate regions of high and low pressure

Example:

Two isobars (lines of constant pressure), 100km apart. Assuming ρ a = 1.2 kg / m³

−1 −2 Fp,x ∂p 1 ⎛1012 −1008mb ⎞100 kgm s −2 = − ≈ −3 ⎜ 5 ⎟ = 0.0033 ms m ρ a ∂x 1.2 kg m ⎝ 10 m ⎠ mb

140 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Viscous force Viscosity is a measure of a fluid’s ability to transmit forces laterally to the mean flow. When two layers of air slide over one another at different velocities, they exert a viscous stress, or force per unit area, on each other. The force acts parallel to the direction of motion and over the area of the plane between the two layers. The viscous force is an internal force : it is caused by molecular interactions within a parcel. Viscous force can be described in terms of wind shear and shearing stress..

∂u Wind shear ( ≠ 0 ): change of velocity with height. ∂z ∂u Shearing stress (τ =η [Nm-2]): Force per unit area, parallel to the direction of zx a ∂z motion due to wind shear. with

1.5 −5 ⎛ 416.16 ⎞⎛ T ⎞ -1 -1 ηa ≈1.8325 ⋅10 ⎜ ⎟⎜ ⎟ : dynamic viscosity of air [kg m s ] ⎝ T +120 ⎠⎝ 296.16 ⎠

Figure: Example of shearing stress in x-direction on a volume of air.

141 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

m ∂τ zx Fviscous,zx = = ρa ∂z m ∂ ⎛ ∂u ⎞ = ⎜ηa ⎟ ≈ ρa ∂z ⎝ ∂z ⎠ m ∂ 2u ≈ ηa 2 ρa ∂z

Generalised

r ηa 2 r Fviscous = m ⋅ ∇ v = ρa

Cartesian Coordinates η ⎛ r⎛ ∂ 2u ∂ 2u ∂ 2u ⎞ r⎛ ∂ 2v ∂ 2v ∂ 2v ⎞ r⎛ ∂ 2 w ∂ 2 w ∂ 2 w ⎞⎞ = m ⋅ a ⋅ ⎜i ⎜ + + ⎟ + j⎜ + + ⎟ + k⎜ + + ⎟⎟ ⎜ ⎜ 2 2 2 ⎟ ⎜ 2 2 2 ⎟ ⎜ 2 2 2 ⎟⎟ ρa ⎝ ⎝ ∂x ∂y ∂z ⎠ ⎝ ∂x ∂y ∂z ⎠ ⎝ ∂x ∂y ∂z ⎠⎠

Figure: A linear vertical wind shear results in a constant shearing stress at all heights and no net viscous force. A nonlinear wind shear results in a change of shearing stress with height and a net viscous force.

142 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Example Near surface: -1 u1=0 ms at z1=0m, -1 u2=0.4ms at z2=0.05m, -1 u3=1ms at z3=0.1m

−3 −5 −1 −1 T=288K, ρ a =1.225 kg m → ηa =1.792⋅10 kg m s

Fviscous,zx ηa 1 ⎛ u3 − u2 u2 − u1 ⎞ −3 −2 → ≈ ⎜ − ⎟ =1.17⋅10 ms m ρa (z3− z1)/ 2 ⎝ z3 − z2 z2 − z21 ⎠ Away from surface: -1 u1=10 ms at z1=1000m, -1 u2=14ms at z2=1250m, -1 u3=20ms at z3=1500m

−3 −5 −1 −1 T=280K, ρ a =1.085 kg m → η a =1.753 ⋅10 kg m s

Fviscous,zx ηa 1 ⎛ u3 − u2 u2 − u1 ⎞ −10 −2 → ≈ ⎜ − ⎟ = 5.17 ⋅10 ms m ρ a (z3 − z1) / 2 ⎝ z3 − z2 z2 − z21 ⎠

143 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Turbulent flux divergence Obstacles at the surface, such as rocks, trees, structures, and mountains, deflect air. The deflected air gives rise to wind shear and eddies downwind of or adjacent to the obstacles. Wind shear and turbulence resulting from obstacles are mechanical shear and mechanical turbulence, respectively. In atmospheric models, the effects of mechanical shear and buoyancy on the acceleration of a parcel of air are modeled as subgrid scale effects and simulated with turbulent-flux divergence terms. These terms are derived from local acceleration, the continuity equation for air Reynolds decompositions and K-theory.

∂vr r Local acceleration: (I) ar = + (vr ⋅∇)vr ⋅ ρ local ∂t a

∂ρ r r Continuity equation (II) a = − ρ (∇ ⋅ vr ) − (vr ⋅ ∇ ) ρ ) ⋅ vr ∂ t a a

r r ⎡∂v r r r⎤ r⎡∂ρ a r r ⎤ I+II: ρ a ⋅ alocal = ρ a ⎢ + (v ⋅ ∇)v⎥ + v⎢ + ∇ ⋅ (vρ a )⎥ ⎣ ∂t ⎦ ⎣ ∂t ⎦

Variables in this equation are decomposed as r r r v = v + v', ρ a = ρ a + ρ'a

Because density perturbations are generally small ( ρ'a << ρa ), the density simplifies to ρa ≈ ρa . Substituting the decomposed values into above formula gives r r r ⎧∂(v + v') r r r r r ⎫ r r ⎧∂ρa r r r ⎫ ρa ⋅ alocal = ρa ⎨ +[v + v') ⋅∇](v + v')⎬ + (v + v')⎨ + ∇ ⋅[(v + v')ρa ]⎬ ⎩ ∂t ⎭ ⎩ ∂t ⎭ Takling time and grid volume average of this equation, eliminating zero terms and removinbg unnecessary overbars gives

r ______r ⎧∂v r r r⎫ r⎧∂ρa r r ⎫ r r r r r r ρa ⋅ alocal = ρa ⎨ + (v ⋅∇)v ⎬ + v ⎨ + ∇ ⋅(vρa )⎬ + ρa (v'⋅∇)v'+v'∇ ⋅(v' ρa ) ⎩ ∂t ⎭ ⎩ ∂t ⎭

∂ρ r Remember: + ∇ ⋅ (vrρ ) = 0 ∂t a

144 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

r r r This yields alocal = alocal + a'local with

∂vr r ar = + (vr ⋅∇)vr local ∂t

r ⎡______⎤ r Fturbulent 1 r r r r r r a'local = = ⎢ ρa (v'⋅∇)v'+v'∇ ⋅(v' ρa ) ⎥ ma ρa ⎣ ⎦

r Fturbulent accounts for mechanical shear, buoancy and other eddy effects.

Expanding the turbulent flux divergence term in Cartesian altitude coordinates gives r ⎡______⎤ Fturbulent 1 r r r r r r = ⎢ ρa (v'⋅∇)v'+v'∇ ⋅(v' ρa ) ⎥ = ma ρa ⎣ ⎦ r 1 ⎡∂(ρ u'u') ∂(ρ v'u') ∂(ρ w'u')⎤ = i ⎢ a + a + a ⎥ + ρa ⎣ ∂x ∂y ∂z ⎦

r 1 ⎡∂(ρ u'v') ∂(ρ v'v') ∂(ρ w'v')⎤ + j ⎢ a + a + a ⎥ + ρa ⎣ ∂x ∂y ∂z ⎦ r 1 ⎡∂(ρ u'w') ∂(ρ v'w') ∂(ρ w'w')⎤ + k ⎢ a + a + a ⎥ ρa ⎣ ∂x ∂y ∂z ⎦

Averages such as u'u', w'v' are kinematic turbulent fluxes. Each such average must be parameterized. With K-theory, vertical kinematic turbulent fluxes are approximated with e.g. ∂u ∂v w'u' = −K , w'v' = −K m,zx ∂z m,zy ∂z where Km’s are eddy diffusion coefficients for momentum.

r Fturbulent r 1 ⎡ ∂ ⎛ ∂u ⎞ ∂ ⎛ ∂u ⎞ ∂ ⎛ ∂u ⎞⎤ = −i ⎢ ⎜ ρa Km,xx ⎟ + ⎜ ρa Km,yx ⎟ + ⎜ ρa Km,zx ⎟⎥ − ma ρa ⎣∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠⎦ r 1 ⎡ ∂ ⎛ ∂v ⎞ ∂ ⎛ ∂v ⎞ ∂ ⎛ ∂v ⎞⎤ − j ⎢ ⎜ ρa Km,xy ⎟ + ⎜ ρa Km,yy ⎟ + ⎜ ρa Km,zy ⎟⎥ − ρa ⎣∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠⎦ r 1 ⎡ ∂ ⎛ ∂w ⎞ ∂ ⎛ ∂w ⎞ ∂ ⎛ ∂w ⎞⎤ − k ⎢ ⎜ ρa Km,xz ⎟ + ⎜ ρa Km,yz ⎟ + ⎜ ρa Km,zz ⎟⎥ ρa ⎣∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠⎦

145 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Each term in this equation is an acceleration in one direction due to transport of momentum normal to that direction. The zx term is an acceleration in the x-direction due to a gradient in wind shear and transport of momentum in the z-direction Kinematic turbulent fluxes are analogous to shearing stresses in that both result when momentum is transported down a gradient of velocity. In case of viscosity, momentum is transported by molecular diffusion. In the case of turbulence, momentum is transported by eddy diffusion. In tensor (matrix) form

⎡Km,xx Km,yx Km,zx ⎤ ⎢ ⎥ Km = ⎢Km,xy Km,yy Km,zy ⎥ ⎢ ⎥ ⎣Km,xz Km,yz Km,zz ⎦ In vector and tensor notation r Fturbulence 1 r r r = − ()∇ ⋅ ρa Km∇ v ma ρa

Example 2 -1 Vertical eddy diffusion coefficient in middle of boundary layer : Km=50m s -1 u1=10 ms at z1=300m, -1 u2=12ms at z2=350m, -1 u3=15ms at z3=400m F 1 ∂ ∂u K ⎛ u − u u − u ⎞ turbulence, zx ⎛ ⎞ m,zx ⎜ 3 2 2 1 ⎟ −2 = ⎜ ρa Km,zx ⎟ ≈ ⎜ − ⎟ = 0.02ms : m ρa ∂z ⎝ ∂z ⎠ (z3 − z1) / 2 ⎝ z3 − z2 z2 − z1 ⎠ west east acceleration of wind due to transfer of westerly momentum from 400 to 300 m

146 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

SUMMARY MOMENTUM EQUATIONS Complete momentum equation

r F dv r r g 1 r ηa 2 r 1 r r r = − fk × v + − ∇pa + ∇ v + (∇ ⋅ ρa Km∇)v dt ma ρa ρa ρa

Table: Terms in the momentum equations and their horizontal and vertical magnitudes

147 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

GLOBAL CIRCULATION

Figure: Large scale global circulation

148 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

NONLINEAR DYNAMICS AND CHAOS Numerical Weather Prediction (NWP) is an initial value problem, where these initial values are based on observed weather conditions. Unfortunately these observations include instrumentation and sampling errors. How do these errors affect the long range predictability? Equations of motion are nonlinear (because they contain products of variables like uv,uw,etc ) are very sensitive to initial conditions. Such sensitivities means that substantially different weather forecasts can result from slightly different initial conditions. One famous illustration of sensitive dependence to initial conditions was suggested by Ed Lorenz: 2-D convection within a tank of water, where the bottom of the tank is heated. The vertical temperature gradient from bottom to top drives a circulation of water, with warm fluid trying to rise. The circulation can modify the temperature distribution within the tank. A very specialised, highly simplified set of equations that approximates this flow is ∂C = σ ⋅ (L − C) ∂t ∂L = r ⋅ C − L − C ⋅ M ∂t ∂M = C ⋅ L − b ⋅ M ∂t with C: circulation (positive for clockwise, greater magnitude for a more vigorous circulation) L: left-right distribution of temperature (positive for warm water on the left) M: amount of vertical mixing (0 for linear temperature gradient, positive when temperature is more uniformly mixed within the middle of the tank)

149 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Tank of fluid (shaded), showing circulation C. The vertical M and horizontal L distributions of temperature are also shown

Example Programming exercise with Excel ∂C C(t + ∆t) Solve the Lorenz equations (simple forward Euler scheme e.g. ≈ ) for ∂t ∆t the parameters and initial conditions

Parameters: σ =10, b = 8 , r = 28 3 Initial conditions: C(0)=13, L(0)=8.1, M(0)=45. Use time step ∆t = 0.01 and forecast from t=0 to t=10. Repeat the example but for slightly different initial conditions: M(0)=44

150 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Butterfly-PhaseSpace

50,00

45,00

40,00

35,00

30,00

M 25,00

20,00

15,00

10,00

5,00

0,00 -25,00 -20,00 -15,00 -10,00 -5,00 0,00 5,00 10,00 15,00 20,00 C

Butterfly-PhaseSpace

Figure: “Butterfly” showing evolution of the solution to the Lorenz equations in phase space.

151 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Lorentz Attractor

50,00

40,00

30,00

20,00 M C, 10,00

0,00 0,00 2,00 4,00 6,00 8,00 10,00 12,00 -10,00

-20,00 time

C M

Figure: Time evolution of circulation C and mixing M; M(0)=45

Lorentz Attractor

60,00

50,00

40,00

30,00

20,00 M

C, 10,00

0,00 0,00 2,00 4,00 6,00 8,00 10,00 12,00 -10,00

-20,00

-30,00 time

C M

Figure: Time evolution of circulation C and mixing M; M(0)=44

152 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

USING FINITE DIFFERENCE SCHEMES

An ordinary differential equation (ODE) is an equation with one independent variable, such as time; a Partial Differential Equation (PDE) is an equation with more than one independent variable, such as time and space. Order of a PDE,ODE: highest derivative rank of the equation Degree of a PDE,ODE: highest polynomial exponent of the highest derivative

Finite Difference Approximations to Advection–Diffusion Equations

∂q ∂q 1 ∂ ∂q + u − (ρa K xx ) = Q ∂t ∂x ρa ∂x ∂x

A finite difference approximation involves the replacement of each continuous differential operator with a discrete analog.

Figure: Discretization of a continuous west-east scalar velocity ux. (Jacobson, 2005)

153 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Figure: Derivative approximations at a point on a continuous function (Jacobson, 2005)

∂u ∆u u − u ≈ i = i+1 i−1 ∂x ∆xi xi+1 − xi−1

∂N ∂ 2 N Table: Finite difference approximations of , (Jacobson, 2005) ∂x ∂x 2

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Stability criteria for the Advection-Diffusion Equation If the eddy diffusion term is removed and no sinks/sources are present, the humidity equation reduces to the west-east advection equation

∂q ∂q + u = 0 ∂t ∂x

For certain explicit finite difference scheme approximations, stability is maintained when ∆x ∆t < min umax (Courant-Friedrich-Levy-Criterium) with umax: maximum west-east scalar velocity

∆xmin: minimum west-east grid cell-length in the domain ∆t: time step A parcel of air is not allowed to travel across a grid cell during a single time step

Example 5: -1 umax=20 ms , ∆xmin=5 km gives ∆tmax=250 s

If advection is ignored, the humidity equation reduces to the west east diffusion equation

∂q 1 ∂ ∂q − (ρa K xx ) = 0 ∂t ρa ∂x ∂x

Stability criterion:

2 ∆x min ∆t < K max

Kmax: largest eddy diffusion coefficient in the domain

Example 6: 2 -1 ∆xmin=5 km, Kmax,vertical = 50 m s gives ∆tmax=200 s

155 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

2 -1 ∆xmin=5 km, Kmax,horizontal = 2,500 m s gives ∆tmax=10,000 s Explicit solutions to the diffusion equation are more likely to become unstable in the vertical than in the horizontal direction in an atmospheric model.

Forward Euler scheme (explicit)

q − q q − q q − 2q + q i,t i,t−∆t + u i+1,t−∆t i−1,t−∆t − K i+1,t−∆t i,t−∆t i−1,t−∆t = 0 ∆t 2∆x xx ∆x 2 (K assumed to be constant) forward in time, centered in space (FTCS)

Implicit Scheme

q − q q − q q − 2q + q i,t i,t−∆t + u i+1,t i−1,t − K i+1,t i,t i−1,t = 0 ∆t 2∆x xx ∆x 2

Solution along i = 1 , ..., I is obtained by rearranging the equation as

Ai qi−1,t + Bi qi,t + Di qi+1,t = qi,t−h with ⎛ u K ⎞ Ai = −∆t⎜ + 2 ⎟ ⎝ 2∆x ∆x ⎠i−1 ⎛ 2K ⎞ Bi =1+ ∆t⎜ 2 ⎟ ⎝ ∆x ⎠i ⎛ u K ⎞ Di = ∆t⎜ − 2 ⎟ ⎝ 2∆x ∆x ⎠i+1 leading to a matrix equation

156 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

⎡B1 D1 0 0 ⎤ ⎡ q1,t ⎤ ⎡ q1,t−∆t ⎤ ⎡ A1q0,t ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢A B D 0 ⎥ q q 0 ⎢ 2 2 2 ⎥ ⎢ 2,t ⎥ ⎢ 2,t−∆t ⎥ ⎢ ⎥

⎢ 0 A3 B3 D3 ⎥ ⎢ q3,t ⎥ ⎢ q3,t−∆t ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 A B 0 ⎢ 4 4 ⎥ ⋅ ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ ⎢ ... ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ... B D 0 ⎥ ⎢q ⎥ ⎢ q ⎥ ⎢ . ⎥ ⎢ I −2 I −2 ⎥ ⎢ N −2,t ⎥ ⎢ N −2,t−∆t ⎥ ⎢ ⎥ ⎢ ... AI −1 BI −1 DI −1 ⎥ ⎢qN −1,t ⎥ ⎢ qN −1,t−∆t ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ... 0 AI BI ⎦ ⎣ qN ,t ⎦ ⎣qN ,t − ∆t⎦ ⎣DI qI +1,t ⎦ of form

M⋅ xr = sr which can be solved e.g. via

xr = M −1 ⋅ sr

Finite Differencing in Two Directions

∂q ∂q ∂q 1 ∂ ∂q 1 ∂ ∂q + u + v − (ρa K xx ) − (ρa K yy ) = Q ∂t ∂x ∂x ρa ∂x ∂x ρ a ∂y ∂y

q − q ⎛ q − q q − q ⎞ ⎛ q − 2q + q q − 2q + q ⎞ i,t i,t−∆t + ⎜u i+1, j i−1, j + v i, j+1 i, j−1 ⎟ − ⎜ K i+1, j i, j i−1, j − K i, j+1 i, j i, j−1 ⎟ = 0 t ⎜ 2 x 2 y ⎟ ⎜ xx 2 yy 2 ⎟ ∆ ⎝ ∆ ∆ ⎠t ⎝ ∆x ∆y ⎠t

(j: index in north-south direction)

The equation is linear for all i,j and can be solved implicitly in matrix form.

Disadvantage: 100x100 horizontal grid required matrix is 10,000x10,000.

Advantage: solution is unconditionally stable for all values u,v, K xx , K yy

157 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

OVERVIEW VECTORS ANALYSIS

Cartesian coordinates r r r r a = (a x , a y , a z ) = i a x + ja y + ka z

r r r a ⋅ b = a xbx + a y by + a z bz = a b cosθ : scalar or dot product

Figure: Illustration dot product

r r r c = a ×b = (a ybz − azby ,azbx − axbz ,axby − a ybx ) : cross product r cr = a b sinθ

Figure: Illustration vector product

158 Modelling Water and Energy Fluxes at the Land Surface and in the Atmosphere

Nabla operator notation r ∂ ∂ ∂ ∇ = i + j + k : Nabla operator ∂x ∂y ∂z

r ∂ ∂ ∂ ∂ ∂ ∂ ∇N = (i + j + k )N = i N + j N + k N ∂x ∂y ∂z ∂x ∂y ∂z

r ∂ ∂ ∂ ∂u ∂v ∂w ∇ ⋅vr = (i + j + k )⋅(iu + jv + kw) = + + ∂x ∂y ∂z ∂x ∂y ∂z

r ∂ ∂ ∂ ∂N ∂N ∂N (vr ⋅∇)N = (u + v + w )N = u + v + w ∂x ∂y ∂z ∂x ∂y ∂z

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