Land-surface atmosphere interaction Author: Dr. Ferenc Ács Eötvös Loránd University Institute of Geography and Earth Sciences Department of Meteorology Financed from the financial support ELTE won from the Higher Education Restructuring Fund of the Hungarian Government
knowledge on the phenomenology of the atmospheric transport processes in the vicinity of the land-surface, Land-surface atmosphere interaction Goals: TO PROVIDE BASIC PHYSICS KNOWLEDGE, MORE PRECISELY knowledge on the phenomenology of radiation transfer above the land-surface, knowledge on the phenomenology of heat and water transport processes in the soil,
Land-surface atmosphere interaction Goals: TO PROVIDE BASIC PHYSICS KNOWLEDGE, MORE PRECISELY knowledge in detail about Monin- Obukhov s similarity theory, knowledge on the water transfer processes in the soil-vegetation system, knowledge on the energy transfer processes in the soil-vegetation system.
Introduction (Gaia and the vegetation) Characteristics of the soil-vegetationatmosphere system: central element: the vegetation (photosynthesis: the most important and ancient process on the Earth (Gaia)), Physical, chemical and biological phenomena and processes. Weather: physical processes. Climate: phyisical, chemical and biological processes.
Introduction (Gaia and the vegetation) Monteith et al. (1975)
Introduction (Gaia and the vegetation) Water: Flux densities and reservoirs. Soil is the largest water reservoir! Therefore meteorology cannot disregard the soil. Monteith et al. (1975)
Introduction (Gaia and the vegetation) Resistances: stomatal resistance is the largest. Therefore meteorology cannot disregard vegetation. Rose (1966) Ψ potential; r t - soil resistance; r gy root resistance; r x xylem vessel resistance; r s stomatal resistance; r cu cuticular resistance; r a aerodynamic resistances in the boundary (lower) and turbulent (upper) atmospheric layers; légkör = atmosphere; vízkészlet = water amount in the soil
Radiation Vegetation canopy: Radiation features of the leaf (r (reflection), tr (transmision) and a (absorption) spectra, water content), radiation features of the vegetation canopy (r and tr spectra), albedo (solar elevation angle), radiation balance.
Radiation Bare soil: Radiation features of the soil particles (r spectra), radiation features of the soil types (r spectra, humus and iron oxides), albedo (solar elevation angle, soil moisture content, roughness), radiation balance.
Radiation - vegetation Radiation (optical) properties of a "typical" leaf Jones (1983)
Radiation - vegetation radiation properties of the leaf Jones (1983)
Radiation - vegetation canopy radiation properties of the vegetation canopy, Jones (1983)
Radiation - vegetation canopy radiation properties of the vegetation canopy Braden (1985)
Radiation - vegetation canopy radiation properties of the vegetation canopy, Braden (1985)
Radiation - vegetated surface Albedo solar elevation When the irradiation is "low" the albedo is "high" and its changes are great. When the irradiation is "high" the albedo is "low" and its changes are "small". Sellers and Dorman (1987)
Radiation - vegetated surface Radiation balance: 4 4 4 2 ) 1 ( c c g g a a v v v T T T tr R R and if tr v = 0 4 4 4 2 ) 1 ( c c g g a a v v T T T R R (rough approach and the simplest form)
Radiation - bare soil surface radiation properties of the soil particles, Szász and Zilinyi (1994)
Radiation - bare soil surface radiation properties of the soil types, Jones (1983)
roughness: it has the smallest effect of the three parameters. Radiation - bare soil surface albedo (solar elevation, soil moisture content, roughness) solar elevation: the same dependence as in the case of vegetation, soil moisture content: dry soil higher albedo; moist soil lower albedo; the transition is non-linear,
Radiation - bare soil surface Radiation balance: R b R(1 b ) a T 4 a g T 4 g. (rough approach and the simplest form)
Soil definition Soil is a medium consisting of organic and inorganic materials, where the transfer of matter and energy occur continuously via physical, chemical and biological processes. Therefore soil possesses various horizons, so it has a stratified structure. Soil deviates from its bedrock source material by having such a layered structure. This layered structure is its important feature, and characterises it.
Soil - profiles Soil has a layered structure. The distribution of the horizons according to depth is called the soil profile. Each profile is composed of horizons A, B and C. The surface horizon A is the most weathered soil layer with the highest humus content. The sub-surface horizon B has a lower humus content than the surface horizon A. Horizon C is the least weathered soil layer and has the smallest humus content of the soil horizons.
Soil texture This notion expresses how large the soil particles are. The largest soil particles (50 2000 μm) are called sand. Sand s water conduction is high, consequently its water retention is low. Sandy soils have a very low CEC (Cation Exchange Capacity).
Soil texture Medium size soil particles (2 50 μm) are called silt. This possesses moderately high (neither good nor bad) water conduction and moderately low water retention. Its ion holding capacity is moderate.
Soil texture Soil particles with a diameter smaller than 2 μm are known as clay. Clay possesses low water conduction and a high water retention capacity. Its ion holding capacity is high.
Soil texture Stefanovits, Filep, Füleky (1999) Soil textural triangle: schematic diagram for representing soil particle composition (sand, silt and clay fractions expressed in per cent). (Remark: designations in the triangle represent soil texture classes)
Soil texture: classification according to soil particle composition Cosby et al. (1984)
Soil types Do not mistake soil type for soil texture! Soil type refers to soils formed under similar environmental conditions, in a similar state of development, possessing similar process associations.
Physical properties of soil Soil is made up of solids, liquids and gases. It is useful to define several variables which describe the physical condition of the three-phase soil system. M t = total mass, M s = mass of solids, M l = mass of liquid, M g = mass of gases, V t = total volume, V s = volume of solids, V l = volume of liquids, V g = volume of gases and V f = V l + V g = volume of fluid (sum of V l + V g ).
Physical properties of soils Particle density Dry bulk density Total porosity Void ratio e s b f V V M V f s s M V V t V. s s f t,,,
Physical properties of soil.,, S f l l b l b s l b s l l t l s l V V S w M M M M V V M M w Mass wetness or mass based water content Volume wetness or volumetric water content Degree of saturation
Particle size distribution in soils: particle size distribution curve A particle size distribution curve is a plot of the number of particles having a given diameter versus diameter. Particle size distribution in soil is approximately lognormal (a plot of number of particles vs. log diameter would approximate a Gaussian distribution function).
Soil heat flow Heat flow in the soil occures from particle to particle. The relationship between heat flux density and temperature is described by the Fourier law (first formulated by Fourier in 1822). The highest heat flow is in the vertical direction, since the temperature gradient is the highest in the vertical. Therefore a 1- dimensional treatment is common.
Fourier law Fourier law: This is an empirical formula, i.e. a parameterization. The negative sign regulates the direction of f h. λ is thermal conductivity (Wm -1 K -1 ) f h ( z, t) ( z) T z.
Differential equation of heat conduction Heat flux density f h is not constant over depth! Where z f h 0 (divergence) the temperature has to decrease, and vice versa, where f h 0 (convergence) z the temperature has to increase. Combining the Fourier law with the continuity equation f h z C h T t.
Differential equation for heat flow The equation can only be physically interpreted by using a minus sign on the left side of the equation! Namely, in the case of divergence of f h temperature T has to decrease over time [(δt/δt) < 0)], while in the case of the convergence of f h T has to increase over time [(δt/δt) > 0)]. C h is the volumetric specific heat of the soil. It is equal to the product of soil density (kgm -3 ) and specific heat (Jkg -1 K -1 ).
Differential equation for heat flow If λ and C h are independent of z, the equation could be written as where k=λ/c h is thermal diffusivity. t T z C z T z z h ) ( ] ) [( t T z T k 2 2
Thermal properties of soil materials The thermal properties of soil materials deviate markedly. Campbell (1985)
Parameterization of volumetric specific heat The volumetric specific heat of soil is the weighted sum of the specific heats of all soil constituents: C h C m m C w C a a C o o. Φ is the volume fraction of the components (m, w, a and o indicate mineral, water, air and organic constituents).
Parameterization of volumetric specific heat Since C a is too small and Φ o can be neglected (2-4% on average), C h of mineral soil becomes C h C m ( 1 ) C. f w
Thermal conductivity of soil It depends upon many factors f ( b,, q, o) Campbell (1985)
Thermal conductivity of soil different parameterizations Thermal conductivity change versus relative soil moisture content for fine and coarse soil textures (Johansen model) Ács et al. (2012) 1.8 1.6 Johansen - Coarse Johansen - Fine 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.01 0.11 0.21 0.3 0.4 0.5 0.6 0.7 0.79 0.89
Thermal conductivity of soil different parameterizations Thermal conductivity change versus relative soil moisture content for fine, coarse and very coarse soil textures of mineral soils and of organic soils (Côté Konrad model) Ács et al. (2012) 1.8 1.6 1.4 1.2 C - vcoarse - minerso 1 0.8 C - coarse - minerso C - fine - minerso C - organic - minerso 0.6 0.4 0.2 0 0.01 0.11 0.21 0.3 0.4 0.5 0.6 0.7 0.79 0.89
Thermal conductivity of soil different parameterizations Thermal conductivity change versus relative soil moisture content for coarse, mineral soils using different parameterizations Hővezető képesség (W m-1 K-1) Ács et al. (2012) 1.8 1.6 1.4 1.2 1 J - coarse C - coarse 0.8 N - coarse 0.6 0.4 0.2 0 0.01 0.11 0.21 0.3 0.4 0.5 0.6 0.7 0.79 0.89 Relatív talajnedvesség-tartalom
Analitical solution to the heat flow equation The heat flow equation can be analitically solved using the boundary conditions as follows: Heat flux density at the soil surface: f h ( 0, t) fh0 fh0 sin( t ), 4 2. T At an infinite depth: f (, t) h 0.
Analitical solution to the heat flow equation Using former boundary conditions T( z, t) T Te z ds sin( t z d S ) where d S 2k 2. C At z=d S, the amplitude ΔT is e -1 =0.37 times its value at the surface. This is the so called damping depth T z d S T e 0,37T.
Analitical solution to the heat flow equation According to the solution the amplitude of the temperature wave decreases exponentially over depth, the phase of the temperature wave is linearly displaced over depth z.
The shape of f h (z,t) Combining the equations T(z,t) and f h (z,t), f h (z,t) can be written as f h ( z, t) f h0 e z / d sin( t ). 4 In doing so, we also used the following equation: sin x cosx S z d 2 sin( x S 4 ).
The shape of f h (z,t) f h (z,t) can also be written as This equation will be used for discussing the so called force-restore method, which serves for predicting soil surface temperature.. 2 2 2 ) 2 ( ], ), ( ), ( 1 [ ), ( 0 2 / 1 S S h h d C d T f C where T t z T t t z T t z f
Water flow in the soil Water flow in the soil is similar to diffusion, leakage. This is caused by the tortuosity of the soil via the effect of capillary and gravitational forces. Gravitational force is imlicitly directed downwards. The direction of capillary forces is variable, it is the same with direction of water potential gradient. If the water potential gradient is directed upwards and the capillary force is larger than the gravitational force, the water flows upwards.
Water flow in the soil Gravitational force governs water flow in the macropores where the water is free (not bound to soil particles). This water is the so-called gravitational water. Capillary force governs water flow in the micropores where the water is bound by soil particles. This water is the so called capillary water. Capillary water is held by cohesion (attraction of water molecules to each other) and adhesion (attraction of water molecules to the soil particles).
Water flow in the soil Water flux density (f w ) is determined by both capillary and gravitational forces. This joint effect could be written as f w f f, wk wg where f wk K z and f wg K or Kg depending on the units used. Ψ is the water potential and K is the hydraulic conductivity. The formula for flux density f wk is the Darcy law, it is empirically based. Before discussing Ψ and K, let s get to know their units!
The unit of Ψ If the volume of water is considered, Ψ s unit is Jm -3, that is Nm -2 =Pa. Instead of Pa, water column height could also be used as the unit. The relationship: 1 hpa = 1 cm of water column height. If the mass of water is considered, Ψ s unit is Jkg -1.
The unit of K If the unit of Ψ is water column height and the unit of flux density f w is ms -1 (this comes from m 3 m -2 s -1 because water volume is considered), then the unit of K is also ms -1. In this case, flux density f wg = K. If the unit of Ψ is Jkg -1 and the unit of flux density f w is kgm -2 s -1, then the unit of K is kg s m -3. In this case, flux density f wg = K g.
Differential equation for water flow Flux density f w is not constant over depth! Where z f w 0 (divergence) the soil moisture content (θ) has to decrease and vice versa, where f w 0 (convergence) z the soil moisture content has to increase. Combining the flux density equation with the continuity equation one can obtain the so called Richards equation. f w z w. t
Differential equation for water flow If the water flow is mostly governed by capillary forces, that is when f w = f wk (this is the simpler case), then w [ K t z K K Dw. C ] z [ D z where D w is the water diffusivity. C represents the change of soil moisture content for a unit change of Ψ. The most important assumption for this transformation is that Ψ is a function of θ and, vice versa, θ is a function of Ψ. This is true only for capillary and osmotic potentials. w ], z
Differential equation for water flow In the former equation, the unknown variable is θ. Nevertheless, the equation could also be expressed as a function of Ψ. Then, since Ψ = Ψ m w C m. m t m w C t m z [ K z m ], where Ψ is the total water potential, while Ψ m is the matric or capillary potential. The relationship between Ψ and Ψ m as well as their dependence on θ will be discussed later.
Differential equation for water flow If the water flow is governed not only by matric but also by gravitational potential (this is the most general case), i.e. when f w = f wk +f wg, then w C t z [ K ]. z Since m gz, w C t m z [ K z m Kg].
Differential equation for water flow The latter equation approaches reality closely, since it takes into account both the capillary and gravitational effects. To be able to consider the equation, we have to know more about Ψ and K. Note that Ψ is a state variable, while K is a parameter! Let s first take a look at Ψ!
Water potential Water potential is the potential energy per unit mass (or volume) of water in a system, compared to that of pure, free water. According to the convention, the potential energy of free water is zero. So, the potential energy of bound water possesses negative values. The more the water is bound by soil particles, the more negative Ψ is, i.e. the higher the absolute value of Ψ is.
Water potential In the definition, Ψ is referring to both mass and volume. If it is reffering to volume, Ψ s unit is Nm -2, that is Pa. The negative Ψ can be interpreted as suction, the magnitude of which is equal to the pressure and, implicitly, it is directed opposite to it. It was also mentioned that Pa could also be replaced by water column height. 1 hpa = 1 cm water column height. Considering water mass, the unit of Ψ is Jkg -1.
Water potential Campbell (1985)
Water potential Water potential is not only determined by capillary and gravitational forces. In the vicinity of plant roots, water flow is also influenced by osmotic potential (Ψ o ). Osmotic potential is equivalent to the work required to transport water reversibly and isothermally from a solution to a reference pool of pure water at the same elevation. If the water column is continous, hydrostatic pressure could also act as an external force. This is characterized by a pressure potential Ψ p.
Water potential Total water potential (Ψ) is the sum of the water potential components, i.e. m g o p.
Water potential Among the water potential components, the matric (the result of the attraction between water and soil particles) and osmotic potentials depend on soil moisture content. Ψ is also function of θ via Ψ m and Ψ o. The Ψ m (θ) relationship is of basic importance, it is called the soil moisture characteristic or moisture release curve. The Ψ m (θ) relationship (in most cases this is the same as Ψ(θ)) is called the pf curve, when Ψ is represented as the logarithm of the water column height expressed in cm (y axis) versus relative soil moisture content (θ/θ S ) (x axis).
Water potential source= internet S= sand, L= loam, T= clay, WP= wilting point, FK= field capacity
Water potential The function Ψ(θ) can be estimated using statistical evaluations applied to soil sample data. Campbell s (1974) parameterization is based on the assumtion that the relationship between lnψ and ln[θ/θ S ] is linear (this is the simplest approach). S ( S ) b.
Water potential S ( ) S b. b is the porosity index, Ψ S is Ψ at saturation and analogously θ S is θ at saturation. Their values were determined by Clapp and Hornberger (Clapp and Hornberger, 1978) using data from USA soil samples. Clapp-Hornberger s data set (Clapp and Hornberger, 1978) is widely used in meteorological models.
Water potential Ács (1989)
Water potential There are also more complex parameterizations, van Genuchten s is one such parameterization (van Genuchten, 1980). This parameterization is widely used in soil science.
Hydraulic conductivity K changes similarly to Ψ in a broad range. In the large pores, where the gravitational effect is dominant, K is a function of Ψ S. K for saturated soil can be expressed after theoretical considerations as follows: K S 2 w 2 2 S (2b 2 S 1)(2b, 2) where σ is the surface tension of water, ν is the viscosity of water, θ S is the saturated soil moisture content, Ψ S is the saturated water potential, ρ w is the water density and b is the porosity index.
Hydraulic conductivity The former equation can also be written as K S S K is obviously proportional to K S and it is inversely related to Ψ S2. Ψ S can be interpreted as characteristic microscopic length. The characteristic length for a soil can be taken as the radius of the largest pores. 2 const.
Hydraulic conductivity Function K(θ) as the function Ψ(θ) could be estimated using statistical evaluations applied to data referring to soil samples. As it was mentioned, one of the simplest relations for K(θ) is obtained by Campbell (Campbell, 1974). ( ) 2b3 The values of K S, θ S and b are determined by Clapp and Hornberger (Clapp and Hornberger, 1978). Campbell s parameterization with values of K S, θ S and b obtained for USA are widely used in meteorological applications. K K S S.
Wetness characteristics and soil texture Water flow in the soil is regulated by pores, more precisely by their magnitude and size distribution. These two factors depend indirectly on the features of soil particles (magnitude, form, material composition). So, wetness characteristics as Ψ S, K S, θ S and b depend indirectly on soil texture. How? Is there any rule or relationship? Yes, relationships can be observed, in short, they are as follows.
Ψ S and the soil texture The magnitude of Ψ S increases going from coarser (sand) to finer (clay) soil textural classes. This increase could be quantified as it is done in the ISBA (Interaction Soil Biospere Atmosphere) biophysical scheme (Meteo France), nevertheless such quantification is not common in meteorological applications. The observed increase can be easily explained. At saturation, water retention in smaller pores is higher than water retention in larger pores.
K S and soil texture The magnitude of K S decreases going from coarser (sand) to finer (clay) soil textural classes. K S is extremely sensitive to the magnitude of the large pores since water runs out first from the largest pores when the water content decreases. It is logical but it has to be said: water runs out of the smaller pores only after it has run out of the larger pores.
θ S and the soil texture Concerning porosity (total pore volume) the basic question is as follows: How large is the porosity of many small pores with respect to the porosity of much fewer large pores? Observations show that porosity increases going from coarser (sand) to finer (clay) soil textural classes. Since θ S is practically equal to porosity, the same change can also be observed for θ S.
b and the soil texture b is the slope of the best-fit line between lnψ and ln[θ/θ S ]. Therefore b represents the change of lnψ for a unit change of ln[θ/θ S ]. If we construct these straight lines for all soil textural classes (on the basis of soil sample data), we shall see that the slope of the lines increases going from coarser to finer soil textures. More precisely: the lower b is the lower the porosity (light soils) and vice versa, the larger b is the larger the porosity (heavy soils).
Wetness characteristics of different soil textures for USA and Hungarian soils Ács et al. (2010)
Infiltration and redistribution Water flow in the soil is also determined by soil surface conditions. Precipitation flux density splits into surface run off (liquid water does not enter the soil) and infiltration (liquid water enters the soil). This partitioning depends upon relief and soil characteristics, primarily upon the soil texture and the soil moisture conditions. Hydrologists are interested in run off, while meteorologists and pedologists in infiltration. Let s find out more about the most important features of the infiltration!
Infiltration Infiltration rate f i (t) depends strongly on soil moisture content. It is higher for dry and lower for moist soil. Campbell (1985) Infiltration rate is initially high, but decreases over time to a constant value.
Infiltration When water enters soil, it develops a transmission zone from the soil surface to the wetting front (boundary between wet and dry soil). This sharp front is a result of the sharp decrease in K. In the transmission zone, K is high because θ is high. Below it K is low because θ is much lower than in the transmission zone. Cambell (1985)
Infiltration The observed infiltration rate f i (t) can also be theoretically deduced. Let x f be the depth of the transmission zone. Ψ f and Ψ i are the water potential at the wetting front and at the soil surface. Let [K] be the average hydraulic conductivity in the transmission zone. Then, the average infiltration rate is f i [ K] f x f i.
Infiltration During infiltration the observable wetting front moves through the soil with a velocity dx f /dt, threby increasing the water content in the transmission zone by Δθ. Δθ can be expressed as i 2 f 0, where θ 0 = soil water content before infiltration, θ i = soil water content at the inflow and θ f = soil water content at the wetting front.
Infiltration On the basis of the continuity equation [ K] i x f f dx Integration gives x f as a function of time. x f is directly proportional to the square root of time. dt f. x f 2[ K]( i f ) t.
Infiltration Combining x f and f i (t), f i ( t) [ K]( 2t i f ). The infiltration rate [f i (t)] is directly proportional to Δθ 1/2 and inversely proportional to t 1/2.
Infiltration Integrating f i (t) over time one can obtain cumulative infiltration. I I I 0 f ( t) dt [ K]( i { 2 t i 2 [ K]( t 0 [ K]( i { 2t ) t. dt, Cumulative infiltration is proportional to t 1/2. i f ) } 1/ 2 f t 0 t 1/ 2 f ) } 1/ 2 dt,
Soil water transport equations in the biophysical scheme SURFMOD The movement of water in the soil is represented in SURFMOD by Richards equation: w t f z In this equation, the so called source-sink term (for instance water uptake by roots) is not represented. By implementing it one gets w f w t z w. SST.
Soil water transport equations in the biophysical scheme SURFMOD Integrating the former equation between an upper level a and a lower level b and assuming that θ and SST are constant within the layer thickness D ab, one gets the following equation: w D ab t ( f wb f wa ) D ab SST, where D ab z b z a.
Prediction of θ in the top soil layer In the SURFMOD, this layer is denoted by D 1. So, D ab = D 1 [see Figures 2.3 and 2.5 in Ács et al. (2000)] f f D wa wb ab inf Q P SST run1 By substituting these terms, one gets an equation which agrees with equation (3.9) in Ács et al. (2000). Now let s look at Q 1! R1 Q Q R0 Q 1 E 0 and S, p.
Prediction of θ in the top soil layer Q 1 is constituted by both capillary and gravitational terms. Therefore Q K ( 1 z 1 1 w 1 z (δψ/δz) at z 1 refers to z b being equal to level D 1. Expressing levels via layer thicknesses and using finite difference approximation, one can simply obtain Q 1 as Q wk 1 1 (1 2 D 1 1 ). D 2 2 ),
Prediction of θ in the top soil layer where D 2 is the thickness of the intermediate soil layer [see Figure 2.3 in Ács et al. (2000)] The obtained Q 1 agrees with equation (5.19) for i=1 in Ács et al. (2000).
Prediction of θ in the intermediate soil layer In the SURFMOD, this layer is denoted by D 2. So, D ab = D 2 [see Figures 2.3 and 2.5 in Ács et al. (2000)]. Furthermore f wa Q 1 Q R1, f wb Q 2 and D ab SST Q run2. By substituting these terms one gets an equation which agrees with equation (3.12) in Ács et al. (2000).
Prediction of θ in the intermediate soil layer Furthermore Q wk 2 2 (1 2 D 2 2 D 3 3 ), The obtained Q 2 agrees with equation (5.19) for i=2 in Ács et al. (2000). Note that Figures 2.3 and 2.5 in Ács et al. (2000) can help in understanding how the equations are obtained.
Prediction of θ in the bottom soil layer In the SURFMOD, this layer is denoted by D 3. So, D ab = D 3 [see Figures 2.3 and 2.5 in Ács et al. (2000)]. Furthermore, there are no roots in this layer. So f f D wa wb ab Q Q By substituting these terms one gets an equation which agrees with equation (3.13) in Ács et al. (2000). 2 3 SST, and Q run3.
Phenomenology of the atmospheric transport processes in the vicinity of land-surface Structure and features of the near surface atmosphere (Foken, 2002)
Phenomenology of the atmospheric transports in the vicinity of landsurface Bonan (2002) What is transferred to where? Why and how?
Phenomenology of the atmospheric transports in the vicinity of land-surface What is the relationship between the flux densities [E (evaporation), H (heat) and τ (momentum)] and state variables [q (specific humidity), T (temperature), u (wind speed)]? In common practice, the state variables (q, T, u) are measured (routinely only at one level), while flux densities (except precipitation and radiation) are calculated! One important goal in micrometeorological education is to present the most important methods for calculating vertical flux densities, for instance, evapotranspiration.
Flow types Laminar flow (molecular diffusion; feature of the medium; it is near the surface) Turbulent flow [eddy (diffusion-like transfer) transfer; feature of the flow; it is far above the surface]
Turbulent flow domains Microscale turbulence ƒ=h/l (l=u τ) h = height above ground l = horizontal size of the eddy viscous subgroup, ƒ >> 1 inertial subgroup ƒ 1 micrometeorological domain mechanical turbulence 1 > ƒ 0.3 mechanical and thermal turbulence ƒ 0.3
Turbulent flow coefficients Eddy diffusivity (K) K flux density of the quantity concentration gradientof the quantity aerodinamic resistance (r) concentration differenceof the quantity r flux density of the quantity
Turbulent flow coefficients K refers to the level, while r to the layer! The relationship between them is as follows: r z 2 z 1 1 dz. K( z) This is derived from their definitions!
Mechanical turbulence ground surface turbulence caused by wind shear (wind speed change with the height), neutral stratification (vertical temperature gradient is equal to zero), mass transfer is possible, but heat transfer is not. Monteith et al. (1975) Logarithmic wind profile u( z) u k * ln( z z 0 )
Mechanical turbulence above the vegetation canopy Roughness length (z 0 ) [wind speed becomes zero not at the surface (this could be called the geometrical level ), but somewhat above the surface (it could be called the aerodynamic level )], Zero plane displacement height (d) [there is a shift between aerodynamic levels above vegetation and bare soil. Vegetation acts as a protective wall of height d against wind, though it is a porous medium.] u( z) Monteith et al. (1975) u k * z ln( z 0 d )
Mechanical turbulence above the vegetation canopy τ parameterizations, r and K calculations 2 u * 2 ) u( z C am r am 1 u( z) C am u( z) u 2 * K M k u* ( z d) lu* ahol lk( zd)
Thermal and mechanical turbulence ground surface Turbulence caused by both wind shear and surface heating, stable (the vertical temperature gradient is positive) and unstable (the vertical temperature gradient is negative) stratifications, wind profile: near to the logarithmic (but not logarithmic) Bonan (2002)
Thermal and mechanical turbulence ground surface There is heat transport beside momentum and mass transport. [all three profiles (wind, humidity, temperature) have to be considered] Land-surface: vegetation (d+z 0 ), bare soil (z 0 ).
Aerodynamic method Let stratification be neutral! Then, ) ( * d z ku E z q z q E K M. ) ( * d z ku z u z u K M
Aerodynamic method Let stratification be stable or unstable instead of neutral! Then, ) *( q q M Est d z ku E K E K E z q, ) *( d z ku c H K c H K c H z p M p Hst p. ) ( * m m M Mst d z k u K K z u
Aerodynamic method According to similarity theory, the functions φ(ς) are dimensionless so-called universal functions, where z L mon, L mon 3 * u g H k T c L mon is that height where the turbulent kinetic energy generated by wind shear and thermal stratification is equal. p.
Aerodynamic method We are interested to know the integral form of the equations (Brutsaert, 1982) since the measurements are at discrete levels, so q u E q ( 2) q ( 1), 1 q2 ku* H ) ( ), 12 ( 2 1 ku* c p u k ) ( ). * 2 u1 m( 2 m 1
Aerodynamic method where, ) ( 2 1 d q q, ) ( 2 1 d. ) ( 2 1 d m m
Aerodynamic method We are also interested in the relationship between the stable and unstable on the one hand and the neutral stratifications on the other. This could be characterized by introducing the so called stability function (ψ) (Brutsaert, 1982). ) ( ) ( ) ln( )) ( (1 1 1 1 2 1 2 2 1 d ) ( ) ( ) ln( )) ( (1 1 1 1 2 1 2 2 1 m m m m d ) ( ) ( ) ln( ) ( 1 )) ( (1 1 1 1 2 1 2 2 1 2 1 2 1 q q q q q d d d
Aerodynamic method where So. ) ( 1 ) ( 2 1 d, ) ( ) ( ) ln( 1 2 1 2 * 2 1 q q ku E q q, ) ( ) ( ) ln( 1 2 1 2 * 2 1 c p ku H. ) ( ) ( ) ln( 1 2 1 2 * 1 2 m m k u u u
Aerodynamic method Brutsaert (1982) Stability function
Aerodynamic method according to similarity theory h z0 ( planetaryboundarylayer height) 2, 1 L mon. The lower level is not in the atmosphere, instead at the land-surface because of the lack of the measurements! q 1 qs u1 0, 1 s and q2 q, u2 u,, 2.
Aerodynamic method, ) ( ln 0 * q q s z d z ku E q q, ) ( ln 0 * z d z c ku H p s. ) ( ln 0 * m m z d z k u u
Aerodynamic method In order to integrate ψ we need to know φ. Many functions of φ are suggested. Here, the functions suggested by Dyer and Hicks (1970) will be used. For unstable stratification q (1 16 ) 1/ 2, (1 16 ) 1/ 2, (1 16 m ) 1/ 4.
Aerodynamic method For stable stratification 15 0 1 q m 6 1.
Aerodynamic method Universal functions Brutsaert (1982) Brutsaert (1982)
Aerodynamic method Businger et al. (1971) Universal functions Foken (2002)
Aerodynamic method, 1 1 2ln ) ( 2 0 2 q q x x, 1 1 2ln ) ( 2 0 2 x x ), ( 2 ) ( 2 ) (1 ) (1 ) (1 ) (1 ln ) ( 0 2 0 2 0 2 2 m m m m x arctg x arctg x x x x., ) 16 (1, ) 16 1 ( 0 0 4 1/ 0 0 4 1/ mon L mon z és L d z x x For unstable stratification:
Aerodynamic method, 2 1 2ln ) ( 2 x, 2 1 2ln ) ( 2 x q. 2 ) 2arctan( 2 1 ln 2 1 2ln ) ( 2 x x x m For unstable stratification:
Aerodynamic method For stable stratification: q ( ) ( ) m ( ) 5 5 z d L mon.
Aerodynamic method We could see that flux densities E, H and τ depend upon L mon, and, vice versa, L mon depends upon E, H and friction velocity (u * ). When there is such an interdependence the iterative procedure has to be applied!
Energy balance of the vegetation canopy Beside roughness, the energy balance (available energy flux density) of the surface is also an important factor. Let s take a look at the energy balance of an air column! The air column is within the Prandtl layer. Oke (1978)
1. radiation balance at the top of the air column (R n ), 2. heat flux density across the soil surface (G), 3. turbulent heat flux densities (sensible heat flux density (H) and the latent heat flux density (λ E)) in the air column (we suppose that they are constant over height) Energy balance of the vegetation canopy What are flux densities? Vertical flux densities:
Energy balance of the vegetation canopy What are flux densities? Horizontal flux densities (advection (D)), Heat storage: 1. Heat storage in the column of the vegetation canopy (air, leaves, stems, thin soil surface layer) (J), 2. Radiation energy used by photosynthesis (μ A). μ is the fixation energy of CO 2 (1,15 10 4 J g -1 ), A is the assimilation rate (g m -2 s -1 )
Energy balance of the vegetation canopy Adding input and output flux densities referring to the air column one obtains the energy balance equation for the vegetation canopy: R n G D J A H E 0. The terms D, J and μ A could be neglected with respect to R n -G, so: R n G A e H E.
Energy balance of the vegetation canopy A e is the available energy flux density at the surface (note: A e is energy flux density (unit: W m -2 ) and not energy (unit: J)). Atmosphere gets the A e (in the form of H+λ E), therefore it is important for us. The partitioning of A e between H and λ E is regulated by the water availability of the surface.
Energy balance of the vegetation canopy How large are the flux densities? How do they change during the day? Monteith et al. (1975)
Energy balance of the vegetation canopy A P R. P= photosynthesis (mg m -2 s -1 ), Baldocchi (1994) R= respiration (mg m -2 s -1 ).
Bowen method Input data: air temperature (T), water vapour pressure (e) at least at two levels and the available energy flux density at the surface (A e ). A e is a new important term! Output quantities: sensible (H) and latent heat (λ E) flux densities. There are fewer input data (there is A e, but there is no u(z)) as compared to the aerodynamic method and the energy balance is fulfilled.
Bowen method. sin, 1, 1 1 E H ce A H A E e e E H E H E p H p K K because and K K e T z e K c z T K c E H β is the Bowen ratio. It can be estimated on the basis of the so called gradient measurements.. e T
Bowen method The accuracy of β depends on how well the best-fit straight line T(e) is estimated. Ács (1989) Gradient measurement: location - Rimski Sancevi (in Hungarian Római Sáncok), date 1982, 19 th May, local time - 14 hours, land-surface type bare soil
Bowen method Applicability: The method may be applied well when A e is large and is less applicable when A e is small (about zero).
Penman-Monteith s equation Combining the energy balance approach and the aerodynamic treatment one gets Penman-Monteith s equation. This is possible if water balance information is also available and used. Input data: air temperature (T), partial water vapor pressure (e) and wind velocity (u) at one level (the levels must not be at the same height), the available energy flux density of the surface and information referring to the availability of water on the surface. Output quantities: sensible (H) and latent heat (λ E) flux densities.
Penman-Monteith s equation Usually more input data are used than in the Bowen method since the so called surface resistance of the land-surface also has to be estimated. It takes into account the atmospheric stratification. The Bowen method does not. It is one of the most widespread equations in environmental meteorology.
Penman-Monteith s equation How is it derived? Here are the basic equations! A H E, r r r ah ae st e c c c p p p T (0) T ( z), H e(0) e( z) és E e T[(0)] e(0). E 4 equations, 4 unknowns. The unknowns are: H, λe, T(0) and e(0). S
Penman-Monteith s equation Now let s sum the last and the next to last equations! r ae r st c p e S [ T(0)] E e( z). Herewith e(0) is eliminated.
Penman-Monteith s equation How can T(0) be eliminated? e S [ T (0)] e S [ T ( z)] [ T (0) T ( z)], where es ( T ) T and T (0) T ( z) Ae E c p r ah.
Penman-Monteith s equation Substituting these Redistributing according to λe. ) ( ] [ )] ( [ E z e r c E A z T e c r r ah p e S p st ae. )] ( ) ( [ ) ( ah e S p ah st ae r A z e z T e c E r r r E
Penman-Monteith s equation Multiplying by γ and dividing by r ah E A e c p { e S ( r [ T ae ( z)] e( z)}/ r st ) / r ah r ah. Since r ah =r ae =r a and δe=e S [T(z)]-e(z) E A e c p (1 e r r st a / r ) a.
Priestley-Taylor s equation Input data: A e and the air temperaure (T) at one level. Output quantities: sensible (H) and latent heat (λe) flux densities. Contrary to Penman-Monteith s equation (PM equation) Priestly-Taylor s equation does not take into account the stratification effect. Priestley-Taylor s equation (PT equation) is more popular since satellite measurements of radiation became available.
Priestley-Taylor s equation How to derive it? Let s start from the PM equation! The PM equation can be divided into two terms. The first characterises the surface (term ΔA e ), while the second the evaporative demand of the atmosphere (term δe).
Priestley-Taylor s equation First supposition: the second term is usually less than the first. Therefore the second term can be expressed as a part of the first term. Second supposition: the surface is wet, therefore its surface resistance is small, i.e. r st 0. If the two suppositions are valid, then E PT A e where PT 1.25
Synthesis of the methods Four methods are presented for estimating H and λe: the aerodynamic method, the Bowen method, Penman-Monteith s equation and Priestly-Taylor s equation. Now let s compare the methods!
Synthesis of the methods Aerodynamic method Bowen method (T 2, e 2, u 2 ) (T 2, e 2 ) (T 1, e 1, u 1 ) (T 1, e 1 ) and A e estimation of no estimation of stratification stratification PM equation PT equation (T 1, e 1, u 1 ) A e and Θ T 1 and A e estimation of no estimation of stratification of stratification
Evaporation fraction and the Bowen ratio The Bowen ratio has already been introduced. The evaporation fraction α is defined as E A e Both α and β depend on the available water and energy of the surface, so indirectly on weather and climate. Neverthless, their changes can be analyzed in a simpler way. How?.
Evaporation fraction and the Bowen ratio Now let s take the resistances! So far two resistances were introduced: r a (aerodynamic resistance) and r st (stomatal resistance). Now let s see the so called climatic resistance! r i c p A e e. r i depends both on the state of the surface and on the state of the atmosphere.
Evaporation fraction and the Bowen ratio α and β can be easily expressed as function of r a, r st and r i. So, they can be also analyzed in terms of resistances (Jones, 1983)!. i a i a st st a a i a r r r r r and r r r r r
Soil and vegetation as water reservoirs Water transfer in the soil-vegetation system will be considered from the meteorological point of view. From the meteorological point of view soil and vegetation are primarily water reservoirs. Vegetation stores water not only in its body, but also on its surface.
Soil and vegetation as water reservoirs Soil can store the largest amount of water. This amount is much larger than the amount stored by vegetation. At the same time the amount of water storable in the body of vegetation is much larger than the amount of water storable on its surface. The ratio of water storable in the soil, in the body of vegetation and on the surface of vegetation is roughly equal to the ratio 100 : 10 : 1.
Soil and vegetation as water reservoirs It is important to say that soil s water storage capacity is comparable to annual flux densities entering and leaving it. It has to be underlined that soil is not only a great water reservoir but also a great carbon reservoir. But annual carbon flux densities which enter and leave it cannot be compared (they are much less) to its carbon storage capacity.
Soil and vegetation as water reservoirs Water amount is changing in both reservoirs, nevertheless they are not independent. They are connected via transpiration and root water uptake. Monteith et al. (1975)
Water flux densities in the soilvegetation system Which water flux densities are the largest? Precipitation, evapotranspiration and run-off. Evapotranspiration is composed of three components: transpiration, soil evaporation and evaporation of the intercepted water. As we see, the last two terms are both evaporation. Meteorologists are interested in precipitation and evapotranspiration, while hydrologist in run-off.
Water flux densities in the soilvegetation system One important contribution of the science of the biophysical modeling in meteorology is that it recognized and quantified the role and impact of transpiration in the formation of weather and climate.
(Water storage on leaves is called interception.) Water flux densities in the soilvegetation system Which water flux densities are moderately large (not small, not large) but important? The interception and evaporation of the intercepted water. Why? Since this water comes back into the atmosphere without entering the soil. With this, the water cycle becomes faster and the forcing of local convective weather events is stronger. This phenomenon is the strongest and therefore the most important in tropical regions.
Water flux densities in the soilvegetation system Now let s take a look at transpiration and root water uptake! These two water flux densities are different (root water uptake is always a little bit greater than transpiration), but they can be treated as equal from the meteorological point of view. This fact is important since the estimation of transpiration in meteorological models is based on this fact. The details related to this topic will be explained later.
Water storage in the soil-vegetation system Now let s also consider the characteristics of soil and vegetation as water reservoirs! The smallest water reservoir is the vegetation surface. Its average maximum value in meteorological models is 0.2 mm/lai, where LAI is the leaf area index. This means that a maximum of 2 dl water can be kept on a leaf surface of 1 m 2 without run-off. So, 1 l water can be stored on a leaf surface of 5 m 2. This value is an average value. This varies depending on vegetation type, but in meteorological models this is usually not taken into account.
Vegetation water content The ratio between the stored water and dry mass is an important vegetation characteristic. For herbaceous plants this ratio is 6:1. This ratio could also be used as a guideline for other vegetation types.
Vegetation water content According to the previous consideration as a first estimation vegetation water content is six times the dry mass contained in a unit of LAI. Water content possesses a daily course and it varies during the growing season. Daily maximum is at dawn, while the minimum is during the midday hours. In the growing season, it increases with the increase of biomass. At the end of the growing season, the water content of grasses is about 10 mm. This can be interpreted that 10 mm is accumulated over the course of about 100 days, so, the average accumulation rate is 0.1 mm/day.
The accumulation rate and transpiration Let s compare the accumulation rate and transpiration! We saw that the daily accumulation rate is 0.1 mm/day. The daily sum of transpiration is 1-4 mm/day. We can see that the accumulation rate is one order of magnitude or more smaller than transpiration. This means that water practically flows through the vegetation, its storage is minimal. Vegetation is simply a channel between the soil and atmosphere. This flow through the channel is independent from the stored water.
Soil water content What is the maximum storable water in the soil? When the storage is maximum, the pores are completely full with water, then θ=θ S. For 1 m 3 soil this is roughly 0.5 m 3, or 500 litres of water. Can vegetation gain access to this water? Not completely, only partly. Vegetation can take up water only from the θ f θ w soil moisture content zone, which is called the plant available water holding capacity.
Soil water content Plant available water holding capacity is less than θ S and greater than 0. The amount θ f θ w also depends upon soil texture. For sand it is the least (about 0.1 m 3, that is 100 litres of water), while for loam it is much greater (about 0.2-0.3 m 3, that is 200-300 litres of water). These facts also show the reason why loam is one of the best and sand is the least appropriate soil texture class for crop production.
Soil water content The meaning of θ f and θ w has still not been explained! θ f is the field capacity soil moisture content, while θ w is the wilting point soil moisture content. θ f is that minimum soil moisture content for which the force of gravity is still greater than the capillary force for holding the water. Consequently, the soil column is not able to hold the water in it for all cases when θ θ f. θ w is that soil moisture content value below which the plant is not able to take up water. In other words, the moisture content of soil after the plants have removed all the water they can.
Soil water holding capacity θ f and θ w values: Clapp and Hornberger (1978)
Soil water content We can see that plants are not only able to take up water in extreme dry (θ θ w ) but also in extreme wet (θ θ f ) conditions. This fact shows that plants need some soil air in order to take up water. Of course, this is regulated and solved by plants living in the water in a certain way (for instance, by building an aerenchyma system).
Soil water content There is a certain degree of uncertainity in the definition of θ f and θ w. Namely, gravitational drainage or the process of wilting cannot be observed unequivocally on the basis of the use of a number of criteria. Consequently, their values are uncertain. Different criteria are used to determine their values. Without discussing this issue, let s underline that in the biophysical modelling of the land-surface their numerical values are uncertain though they are important.
Water transport in the soil-plant system Let s take a look at water transport in the soil-plant system! We showed on the diurnal scale that the amount of stored water in vegetation (accumulation rate) could be neglected with respect to transpiration. Transpiration can be succesfully modelled taking into account the above fact. The use of the aforementioned assumption is widespread in meteorological applications. Therefore, the basic equations of this approach will be briefly presented.
Transpiration model: basic equations Root water uptake (Q R ) is the input water flow. This water flow will be simulated using an analogy to Ohm s law (current is the ratio of potential difference and resistance). Voltage is the difference between the leaf water potential (Ψ leaf ) and soil water potential (Ψ soil ). Ψ leaf refers to the average leaf of the canopy which is represented by one big leaf located on the level d+z 0. Ψ soil reflects an average potential reffering to the 1-m deep soil moisture content profile in the root zone.
Transpiration model: basic equations Soil puts up a resistance r S, while vegetation a resistance r P to the water flow in the soil-plant system. r S is greater the drier the soil is, and vice versa, r S is smaller the wetter the soil is. Note that r S is comparable to stomatal resistance when the soil is dry! r P is mainly caused by xylem vessels. It is taken as a constant.
Resistances in the soil-plant system Rose (1966) Ψ potential; r t - soil resistance; r gy root resistance; r x xylem vessel resistance; r s stomatal resistance; r cu cuticular resistance; r a aerodynamic resistances in the boundary (lower) and turbulent (upper) atmospheric layers; légkör = atmosphere; vízkészlet=water amount in the soil
Transpiration model: basic equations Root water uptake can be expressed as Q R soil r S r P leaf. (1)
Transpiration model: basic equations Soil water potential and leaf water potential are given in unit of water column height [m H 2 O]. Resistances are given in seconds, though such a resistance unit is very unusual. This is true because Q R is parameterized after Ohm s law. Such a parameterization can be done since the water flow in the soil-plant system is almost a steady state (quasi steadystate). The unit in seconds can be interpreted as follows: if the resistance is high, the water flows slowly, consequently the transport needs more time. So, a long time is equivalent to great resistance, and vice versa, a short time corresponds to a low resistance value.
Transpiration model: basic equations We have already mentioned that root water uptake (input water flux, Q R ) is practically equal to transpiration (output water vapor flux, E T ), i.e. Q E R T. (2)
Transpiration model: basic equations Transpiration can be calculated either by Penman-Monteith s formula or by the gradient method, as presented below: ) (3. ) ( ) (3, ) (1 / b r r e T e c E L a r r r e c R E L c a r vg S p T a c a p n T
Transpiration model: basic equations One of the most important terms in eq. (3a) is r c. In the meteorological land-surface modelling community, r c is commonly parameterized by Jarvis (1976) formula: r c r st min F ad LAI GLF F ma. Jarvis (1976) formula consists of the product of different environmental functions, often called stress functions contained in F ad and in F ma.
Transpiration model: basic equations Beside such multiplicative formula, there are also such formulae where the whole effect is expressed by the addition of environmental functions (see, for instance, Federer, 1979). Symbols: r stmin is the minimum stomatal resistance at optimum environmental conditions, LAI is the leaf area index, GLF is the green leaf fraction, F ad is the function for representing the atmospheric demand effect upon stomatal functioning and F ma is the function for representing soil moisture availability effect upon stomatal functioning.
Transpiration model: basic equations The function F ma can be expressed via Ψ leaf since it depends upon soil water availability. Taking these facts into account, F leaf cr ma, soil, S cr that is r c r LAI F GLF st min ad F ma f ( leaf ). (4)
Transpiration model: basic equations Input data: state variables and fluxes: S, T, e, U, P; parameters: ρ, c p, γ, L, LAI, GLF, r stmin, Ψ cr, Ψ soil,s. Quantities to be calculated: Δ, R n,δe. Parametrizations: r S, r P, Ψ soil, r a, F ad. Symbols: see Table 2.1 in Ács et al. (2000, page 22, 23)
Transpiration model: basic equations We have four unknowns in four equations. The unknowns are: Ψ leaf, r c, E T and Q R. Ψ leaf could be expressed by combining equations. Of course, E T could also be estimated on the basis of Ψ leaf.
Transpiration model: basic equations The form of the equation for Ψ leaf depends upon how the function F ma is parameterized. If the F ma /Ψ leaf relationship is linear, the equation for Ψ leaf is a quadratic equation. Only the positive signed square root solution is the real, physically based solution.
Transpiration model: basic equations Model results: Monteith et al. (1975)
Transpiratiom model: applications in the SURFMOD The derivation of the equation for estimating Ψ leaf based on the use of equation (3a) for calculating LE T can be found in Ács et al. (2000) on page 59. The same, but for equation (3b) can be found in Ács et al. (2000) on page 58.
Vegetation canopy surface resistance Rose (1966) As already mentioned, one of the most important parameters in Penman-Monteith s equation is vegetation canopy resistance r c.
The functioning of stomata Since r c is an important parameter in calculating transpiration, the functioning of stomata (opening, closing) has to be described as fully as possible in meteorology too. source: internet
Stomata Large area density small area density Chaloner (2003) Chaloner (2003)