# Surface Energy Fluxes

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1 Chapter 7 Surface Energy Fluxes CHAPTER 7 SURFACE ENERGY FLUXES INTRODUCTION SURFACE ENERGY BUDGET LEAF TEMPERATURE AND FLUXES SURFACE TEMPERATURE AND FLUXES VEGETATED CANOPIES SURFACE CLIMATE TABLES FIGURE LEGENDS... 30

5 Chapter 7 Surface Energy Fluxes the diffusion of materials is related to the concentration difference divided by a resistance to diffusion. For example, the exchange of water vapor and CO 2 between a leaf and the surrounding air depends on two resistances connected in series: a stomatal resistance from inside the leaf to the leaf surface and a boundary layer resistance from the leaf surface to the air. The overall leaf resistance is the sum of these two resistances. If stomata are located on both sides of the leaf, the upper and lower resistances acting in parallel determine the overall leaf resistance. Convection is the transport of heat through air movements. A common example is the warmth felt as warm air rises from a radiator or the cooling of a breeze on a hot summer day. This heat exchange is called sensible heat. Sensible heat flux is represented through a resistance network in which heat flux is directly proportional to the temperature difference between an object and the surrounding air and inversely proportional to a transfer resistance ( T T ) a s H = ρc p r H where T s and T a are the surface and air temperatures ( C), respectively. An object loses energy if its temperature is warmer than the air (a positive flux); it gains energy when it is colder than the air (a negative flux). The density of air (ρ, kg m -3 ) varies with temperature and humidity but ρ = 1.2 kg m -3 is typical. The term C p is the heat capacity of air and is approximately 1010 J kg -1 C -1. The transfer resistance r H (s m -1 ) depends on wind speed and surface characteristics. The larger the temperature difference, the larger the heat flux (Figure 7.2). Larger transfer resistances result in less heat flux for a given temperature gradient. Heat is also lost from an object by evapotranspiration. Considerable energy is required to change water from liquid to vapor. Evapotranspiration, therefore, involves a transfer of mass and energy to the atmosphere. Transfer of mass is seen as wet clothes dry on a clothesline. Heat loss is why a person may feel cold on a hot summer day when wet but hot after being dried with a towel. When water changes from liquid to gas (vapor), energy is absorbed from the evaporating surface without a rise in temperature. This latent heat of vaporization varies with temperature, but is J kg -1 at 30 C (Table 5.2). A typical summertime rate of evapotranspiration is 5 mm of water per day, which with a density of water of 1000 kg 5

6 Ecological Climatology m -3 is equivalent to 5 kg water per square meter (5 kg m -2 /1000 kg m -3 = m). With a latent heat of vaporization of J kg -1, a water loss of 5 kg m -2 day -1 is equivalent to a heat loss of 141 W m -2 (i.e., kg 1 day J J m 2 day s kg = sm 2 ) For comparison, the solar radiation over the same 24-hour period is about 200 W m -2 for a cloud-free sky. This heat is transferred from the evaporating surface to the air, where it is stored in water vapor as latent heat. It is released when water vapor condenses back to liquid. Evaporation occurs when a moist surface is exposed to drier air. Water evaporates from the surface, increasing the amount of water vapor in the surrounding air. When the air is saturated with water vapor, evaporation ceases. Evaporation, therefore, is related to the vapor pressure deficit of air the difference between the actual amount of water in air and the maximum possible when saturated. Saturation vapor pressure increases exponentially with warmer temperature (Figure 7.4). The actual amount of water vapor in air is related to relative humidity. Relative humidity is the ratio of actual vapor pressure (e a ) to saturated vapor pressure evaluated at the air temperature (e * [T a ]) expressed as a percent [ ] RH = 100 ea e Ta For a constant vapor pressure, relative humidity decreases as temperature and therefore saturated vapor pressure increases. Specific humidity is another measure of atmospheric moisture. This is the mass of water divided by the total mass of air. Specific humidity (q a, kg kg -1 ) is related to vapor pressure (e a, Pa) as q a = 0.622ea P 0.378e a where P is atmospheric pressure (Pa). At sea level, P = Pa and because P >> e a, specific humidity can be approximated as q a = e / P. The right axis in Figure 7.4 shows corresponding specific a humidity at saturation for various temperatures. One kilogram of air can hold about 6 g of water when saturated at 7 C, 18 g at 23.5 C, and 36 g at 35.5 C. Latent heat flux is represented by a resistance network similar to sensible heat in which ( [ ]) ρcp ea e Ts λe = γ rw 6

7 Chapter 7 Surface Energy Fluxes The term e * [T s ] represents the saturation vapor pressure (Pa) evaluated at the surface temperature T s, and the term e a is the vapor pressure of air (Pa). The difference between these, ( e Ts e ), is the vapor a pressure deficit between the evaporating surface, which is saturated with moisture, and air. The term r W is a resistance (s m -1 ) analogous to r H. This resistance increases as the surface becomes drier (i.e., less saturated) so that a wet site has a higher latent heat flux than a dry site, all other factors being equal. The term γ is the psychrometric constant, which depends on heat capacity (C p, J kg -1 C -1 ), atmospheric pressure (P, Pa), and latent heat of vaporization (λ, J kg -1 ) as γ = ( C P) /(0.622 λ). A typical value is 66.5 pascals p per degree Celsius (Pa C -1. ). This equation is analogous to that for sensible heat flux, and a positive flux means loss of heat and water to the atmosphere (Figure 7.2). The fifth way in which objects gain or lose energy is conduction. Conduction is the transfer of heat along a temperature gradient from high temperature to low temperature but in contrast to convection due to direct contact rather than movement of air. The heat felt when touching a steaming mug of coffee is an example of conduction. The rate at which an object gains or loses heat via conduction depends on the temperature gradient and thermal conductivity as G = k( T / z) where k is thermal conductivity (W m -1 C -1 ) and T / z is the temperature gradient between the two objects ( C m -1 ) (Figure 7.2). Thermal conductivity is a measure of an object s ability to conduct heat. Differences in thermal conductivity can create perceptions of hot or cold when touching an object. A metal spoon in a cup of hot soup feels warmer than a wooden spoon because it conducts heat from the soup to a person s hand much more rapidly than a wooden spoon. The radiation that impinges on a surface or object must be balanced by the reflected and emitted radiation and by energy lost or gained through sensible heat, latent heat, and conduction. This balance is maintained by changing surface temperature. As an example, consider the temperature and energy balance of an object on a summer day in which S = 700 W m -2. With an albedo of r = 0.2, the object absorbs 560 W m -2 solar radiation. A typical downward longwave flux from the atmosphere is L = 340 W m -2, which gives a radiative forcing of Q a = = 900 W m -2. If the surface temperature is T s = 31.8 C, the 7

8 Ecological Climatology upward longwave flux is L = 490 W m -2 with ε = 1. Air temperature is T a = 29.4 C, relative humidity is 50% (so that e a = Pa), r H = 25 s m -1, and r W = 100 s m -1. The sensible heat flux is H = 111 W m -2 and the latent heat flux, with e Ts = Pa, is λe = 461 W m -2 (in this example, ρ = 1.15 kg m -3, C p = 1005 J kg -1 C -1, γ = 66.5 Pa C -1 ). For convenience, assume there is no heat loss by conduction (i.e., G = 0). In this case, the surface energy imbalance is Qa L H λe= = 162 W m -2 : i.e., the object loses 162 W m -2 more energy than it receives. Its temperature must decrease to restore the energy balance. If surface temperature drops to T s = 29.0 C, with e Ts = Pa, the upward longwave, sensible heat, and latent heat fluxes are L = 473, H = -18, and λe = 340 W m -2, respectively. The object gains 105 W m -2 more energy than it loses ( ( 18) 340= 105). The true temperature of this object under these conditions is somewhere between 29.0 C and 31.8 C. 7.3 Leaf temperature and fluxes The manner in which energy fluxes determine microclimates can be illustrated by the energy budget and temperature of a leaf. The leaf energy budget is Q a 4 = εσ ( Tl ) + ( ) ρc ( e [ T ] e ) Tl Ta p l a ρc + p r /2 γ r b s + r b Leaf temperature (T l ) is the temperature that balances the energy budget. The boundary layer resistance (r b ) governs heat flow from the leaf surface to the air around the leaf (Figure 7.5). Stomatal resistance (r s ) acting in series with boundary layer resistance regulates transpiration. For a leaf, heat and moisture transfer are governed by a leaf boundary layer resistance, which depends on leaf size (d, meters) and wind speed (u, m s -1 ) and is approximated per unit leaf area (onesided) as r 200 d/ b = u This resistance has units of seconds per meter (s m -1 ). Plant physiologists often use m 2 s mol -1 instead of s m -1. At sea level and 20 C, 1 m 2 s mol -1 = 41 s m -1 (Jones 1992, p. 357). The boundary layer resistance represents the resistance to heat and moisture transfer between the leaf surface and free air above the leaf 8

9 Chapter 7 Surface Energy Fluxes surface. Wind flowing across a leaf is slowed near the leaf surface and increases with distance from the surface (Figure 7.6). Full wind flow occurs only at some distance from the leaf surface. This transition zone, in which wind speed increases with distance from the surface, is known as the leaf boundary layer. It is typically 1 to 10 mm thick. The boundary layer is also a region of temperature and moisture transition from a typically hot, moist leaf surface with bright sunlight to cooler, drier air away from the surface. The boundary layer regulates heat and moisture exchange between a leaf and the air. A thin boundary layer produces a small resistance to heat and moisture transfer. The leaf is closely coupled to the air and has a temperature similar to that of air. A thick boundary layer produces a large resistance to heat and moisture transfer. Conditions at the leaf surface are decoupled from the surrounding air and the leaf is several degrees warmer than air. Boundary layer resistance increases with leaf size and decreases with wind speed (Figure 7.7). This expression for bounday layer resistance is derived for a fluid moving smoothly across a surface a condition known as laminar forced convection (Gates 1980, pp ; Monteith and Unsworth 1990, pp ; Campbell and Norman 1998, pp ). For a flat plate of length d (meters), the resistance to heat transfer from one side of the plate is r = ( ρc d)/( knu) b p where ρ is the density of the fluid (kg m -3 ), C p is the specific heat of the fluid (J kg -1 C -1 ), k is the thermal conductivity of the fluid (W m -1 C -1 ), and Nu is the dimensionless Nusselt number. For a flat plate with laminar flow, this is given by Nu = 0.66 Re Pr where Pr is the dimensionless Prandtl number and Re is the dimensionless Reynolds number. The Reynolds number depends on fluid velocity (u, m s -1 ) and kinematic viscosity (ν, m 2 s -1 ) Re = ( ud) / ν Combining equations, ( ρc p ν ) d d r = = a b (0.66k Pr 0.33 ) u u 9

10 Ecological Climatology For air at 20 C, ρ = kg m -3, C p = 1010 J kg -1 C -1, ν = m 2 s -1, k = W m -1 C -1, and P r = 0.72 so that a = 312 s 1/2 m -1 for a flat plate. Values of Nu for leaves are generally higher, so that r b is lower, than that of a flat plate. The boundary layer resistance for flat plates must be divided by 1.4 for leaves in the field (Campbell and Norman 1998, p. 224), giving a = 223 s 1/2 m -1. Gates (1980, pp ) recommended a value of 174 s 1/2 m -1 for leaves. An approximate value for leaves is a = 200 s 1/2 m -1. This is the resistance for heat exchange from one side of a leaf. Sensible heat is exchanged from both sides of a leaf so that heat exchange is regulated by two resistances (each defined by r b ) in parallel (Figure 7.3) and the effective resistance for heat transfer is r b /2(Figure 7.5). Gates (1980, pp. 27, 30, 351) used a value of a = 133 s 1/2 m -1 for sensible heat flux from a leaf. Campbell (1977, pp ) reduced r b by one-half for sensible heat. Jones (1992, p. 63) also distinguished between one-sided and two-sided heat transfer. Transpiration occurs when stomata open to allow a leaf to absorb CO 2 during photosynthesis (Figure 7.6). At the same time, water diffuses out of the saturated cavities within the foliage to the drier air surrounding the leaf. The resistance for latent heat exchange, therefore, includes two terms: a stomatal resistance (r s ), which governs the flow of water from inside the leaf to the leaf surface, and the boundary layer resistance, which governs the flow of water from the leaf surface to surrounding air. The total resistance is the sum of these two resistances (Figure 7.5). Stomata open and close in response to a variety of conditions (Chapter 9): they open with higher light levels; they close with temperatures colder or hotter than some optimum; they close as the soil dries; they close if the surrounding air is too dry; and they vary with atmospheric CO 2 concentration. Stomatal resistance is a measure of how open the pores are and varies from about 100 s m -1 when stomata are open to greater than 5000 s m -1 when stomata are closed. For transpiration, the boundary layer resistance, which is formulated based on heat exchange from one side of a leaf, is not reduced by one-half because stomata are typically, but not always, located on only one side of a leaf whereas sensible heat exchange occurs on both sides. For example, Gates (1980, pp. 27, 30, 32, 351) used a = 133 s 1/2 m -1 for sensible heat and a = 200 s 1/2 m -1 for latent heat. Campbell (1977, pp ) reduced r b by one-half for sensible heat but not latent heat when stomata are on one side of the leaf. If stomata are on both sides of the leaf, the resistances are in parallel (Figure 7.3) and the total resistance is ( r + r )/2. s b 10

12 Ecological Climatology temperature of 91 C with longwave radiation only to a more comfortable temperature of 31 C. Cooling by transpiration is greatest with large radiative forcings and decreases as radiation decreases. It is largest for calm conditions and decreases as wind increases. Figure 7.8 shows the cooling effect of transpiration in more detail over a range of air temperature and relative humidity. Latent heat flux decreases and leaf temperature increases as relative humidity increases. For example, with an air temperature of 45 C and a relative humidity of 20%, the leaf temperature is about 37.5 C. This is 7.5 C colder than the air. However, at 90% relative humidity, when the latent heat flux is much smaller, the leaf temperature is approximately equal to the air temperature of 45 C. The same is true for all air temperatures: as relative humidity increases, transpirational cooling decreases, and leaf temperature increases. For air temperature greater than 21 C, the relative humidity at which the leaf is warmer than air increases with warmer air temperature. In a hot environment, the leaf is cooler than air for all but the most humid conditions. In a cool environment, the leaf is warmer than air for all but the most arid conditions. The cooling effect of evaporation is why we sweat, and it is why a person may feel comfortable in dry climates, where low relative humidity results in rapid evaporation of sweat, but hot and uncomfortable in humid climates, where evaporation is not as efficient. In Chapter 5, Thornthwaite s equation was introduced as a means to calculate evapotranspiration. However, this method is not based on the surface energy budget and instead uses air temperature as a surrogate for the energy available to evaporate water. The Penman-Monteith equation is an alternative formulation of evapotranspiration based on the energy budget. It combines the surface energy balance with physiological and aerodynamic controls of heat transfer. The Penman-Monteith equation is derived from the energy balance and the expressions for sensible and latent heat. In this equation, the saturation vapor pressure at the leaf temperature ( * es ), which is a non-linear function of temperature (Figure 7.4), is approximated by e * s = e * a+ s( Ts T a) where * ea is the saturation vapor pressure evaluated at the air temperature (Ta) and s= de * / dt is the slope of the saturation vapor pressure versus temperature evaluated at T a. Substituting this expression for * es into λe gives 12

13 Chapter 7 Surface Energy Fluxes λe = * ( e + s T T ea) ρc ( ) p a s a γ r W Then, from the definition of the surface energy budget and sensible heat flux, s a H p n ( λ ) T T = ( r / ρc ) R E G Substituting this expression for Ts Ta into the latent heat flux gives, after algebraic manipulation, λe = ρc s R G p e * e rh s+ γ ( r / r ) ( n ) + ( a a) W H For a leaf, G = 0, r r /2, and so that H = b r = W r + s r b ρc sr p * n + e e rb /2 λe = rb + r s s + γ rb /2 ( a a) This equation relates transpiration to net radiation (R n ), the vapor pressure deficit of air ( e * a ea ), and the boundary layer and stomatal resistances. Evapotranspiration increases as more net radiation is available to evaporate water and as the atmospheric demand (i.e., vapor pressure deficit) increases. The role of stomata in regulating leaf transpiration can be seen in the two limiting cases when leaf boundary layer resistance is very large or very small (Jarvis and McNaughton 1986). If the leaf boundary layer resistance becomes very large, so that the leaf is decoupled from the surrounding air by a thick boundary layer, λe = sr /( s+ γ ) n which is known as the equilibrium evaporation rate. In this case, transpiration is independent of stomatal resistance and depends chiefly on the net radiation available to evaporate water. If the boundary layer resistance is small, so that there is strong coupling between conditions at the leaf surface and outside the leaf boundary layer, transpiration is at a rate imposed by stomatal resistance * ( ) λ E = ρ C e e /( p a a γr s ) 13

14 Ecological Climatology In this case, an increase or decrease in stomatal resistance causes a proportional decrease or increase in transpiration. In between these two extremes of equilibrium and imposed transpiration, intermediate degrees of stomatal control prevail. The degree of coupling between a leaf and surrounding air depends on leaf size and wind speed (Figure 7.7). Small leaves, with their low boundary layer resistance, approach strong coupling. Large leaves, with the high boundary layer resistance, are weakly coupled. Leaves in still air are decoupled from the surrounding air while moving air results in strong coupling. 7.4 Surface temperature and fluxes The same principles that determine the temperature and energy fluxes of a leaf also determine the temperature and energy fluxes of land surfaces. Consider, for example, the energy balance of a forest (Figure 7.9). The forest absorbs and reflects solar radiation, absorbs and emits longwave radiation, and exchanges latent and sensible heat with the atmosphere. Sensible heat flux is directly proportional to the difference in air temperature (T a ) and the effective surface temperature (T s ) and varies inversely in relation to an overall resistance. Latent heat flux is directly proportional to the difference in vapor pressure of air (e a ) and the saturated vapor pressure of the surface (e * [T s ]) and is inversely related to an overall resistance. The exchanges of sensible and latent heat between land and atmosphere occur because of turbulent mixing of air and resultant heat and moisture transport. The movement of air can be represented as discrete parcels of air, each with its own temperature, humidity, and momentum (mass times velocity), moving vertically and horizontally. As the parcels of air move, they carry with them their heat, moisture, and momentum. Wind mixes air and transports heat and water vapor in relation to the temperature and moisture of the parcels of air being mixed. The importance of wind in mixing air in the atmospheric boundary layer near the surface is illustrated with two simple examples. Over the course of a summer day, productive vegetation can absorb enough CO 2 during photosynthesis to deplete all the CO 2 in the air to a height of 30 m (Monteith and Unsworth 1990, p. 146). In practice, however, CO 2 is not depleted because wind mixes the air and replenishes the absorbed CO 2. Likewise, productive vegetation can transpire 10 kg m -2 (10 mm) of water on a hot summer day (Campbell and Norman 1998, p. 63). This would increase the moisture in the first 100 m of the atmosphere by 100 g m -3 if it were not transported away. This is much 14

15 Chapter 7 Surface Energy Fluxes more water than the atmosphere can hold when saturated (about 17 g m -3 at 20 C). Instead, wind replaces the moist air with drier air. Turbulence is generated when wind blows over Earth s surface (Figure 7.10). The ground, trees, grasses, and other objects exert a retarding force on the fluid motion of air. The frictional drag imparted on air as it encounters rough surfaces slows the flow of air near the ground. The reduction in wind speed transfers momentum from the atmosphere to the surface, creating turbulence that mixes the air and transports heat and water from the surface into the lower atmosphere. With greater height above the surface, eddies are larger so that transport of momentum, heat, and moisture is more efficient with height above the surface. The derivation of surface fluxes can be understood in a one-resistor formulation such as for bare ground (Figure 7.11). Momentum (τ, kg m -1 s -2 ), sensible heat (H, W m -2 ), and water vapor (E, kg m -2 s -1 ) fluxes between the atmosphere at some height z with horizontal wind (u a, m s -1 ), temperature (T a, C), and specific humidity (q a, kg kg -1 ) and the ground are τ = ρ( u u )/ r a s am H = ρc ( T T )/ r p a s ah E = ρ( q q )/ r a s aw where u s, T s, and q s are corresponding surface values. The resistances r am, r ah, and r aw are aerodynamic resistances (s m -1 ) to momentum, heat, and moisture, respectively, and represent turbulent processes. (This derivation of evapotranspiration is in terms of water vapor (E, kg m -2 s -1 ) not heat (λe, J m -2 s -1 ) and specific humidity nor vapor pressure. Substitution of q a = e / P and multiplying both sides of the a equation by latent heat of vaporization (λ) gives the previously defined latent heat flux.) These fluxes are nearly constant with height in the layer of the atmosphere near the surface (lowest 50 m or so). Monin-Obukhov similarity theory (Brutsaert 1982; Arya 1988) relates turbulent fluxes in the surface or constant flux layer to mean vertical gradients of horizontal wind (u), temperature (T), and specific humidity (q) as 15

16 Ecological Climatology τ H E ku ( z d) u z * = ρ φm( ζ) ρc ku ( z d) T z * = p φh( ζ) ku ( z d) q ρ z * = φw( ζ) where k = 0.4 is the von Karman constant, z is height in the surface layer (m), and d is the displacement height (m). The functions φ m (ζ), φ h (ζ), and φ w (ζ) are universal similarity functions that relate the constant fluxes of momentum, sensible heat, and water vapor to the mean vertical gradients of wind, temperature, and moisture in the surface layer. The friction velocity (u *, m s -1 ) is given by 2 τ = ρu * The terms in brackets in each equation are turbulent transfer coefficients for momentum, heat, and moisture. These coefficients increase with height above the surface (z) because of larger eddies. To maintain a constant flux with respect to height, increases in transport with greater height must be balanced by corresponding decreases in the vertical gradient ( u/ z, T / z, q/ z). Indeed, observations show that in the surface layer, mean vertical gradients in horizontal wind, temperature, and specific humidity decrease with height so that greater turbulent transfer is balanced by decreased vertical gradient and fluxes are constant (e.g., Figure 7.10). Turbulent mixing also increases with greater wind speed, and hence the coefficients increase with greater wind speed. This is represented by the friction velocity (u * ). Greater turbulence results in a weaker vertical gradient. In addition to surface friction, buoyancy generates turbulence. In daytime, the surface is typically warmer than the atmosphere and strong solar heating of the land provides a source of buoyant energy. Warm air is less dense than cold air, and rising air enhances mixing and the transport of heat and moisture away from the surface. In this unstable atmosphere, sensible heat flux is positive, temperature decreases with height, and vertical transport increases with strong surface heating. At night, longwave emission generally cools the surface more rapidly than the air above. The lowest levels of the atmosphere become stable, with cold, dense air trapped near the surface and warmer air above. Sensible heat flux is negative (i.e., toward the surface). Under these conditions, vertical motions are suppressed, and transport is reduced. 16

17 Chapter 7 Surface Energy Fluxes The effect of buoyancy on turbulence is represented by the similarity functions φ m (ζ), φ h (ζ), and φ w (ζ), where ζ = ( z d)/ L and L (m) is the Obukhov length scale (a measure of atmospheric stability). These functions have values of one when the atmosphere is neutral, <1 when the atmosphere is unstable, and >1 when the atmosphere is stable. As a result, the vertical profile is weaker in unstable conditions than in stable conditions (Figure 7.12). u/ z z2 > z 1 Integrating between two arbitrary heights ( ) gives u u * ln z d z u d z ψ d = m + ψm k z1 d L L 2 1 The surface is defined as u 1 = 0 at 1 0m z d z = z + d and ψ m = L 1 0 so that wind speed at height z is u z u z d ln ψm( ζ) k z0m * = The height z0m+d is known as the apparent sink of momentum. The roughness length (z0m, m) is the theoretical height at which wind speed is zero. A typical value is 5 cm or less for bare ground and as large as 1 m for tall forests. The displacement height (d, m) is the vertical displacement caused by surface elements. This displacement height is zero for bare ground and greater than zero for vegetation. Similarly, T / z and q/ z can be integrated between height z and the surface as H z d Tz Ts = ln ψh( ζ) kρcpu* z0h q E z d q = ln ψ ( ζ) z s w kρu* z0w For temperature and specific humidity, surface values are defined as T s at height z = z0w + d z z0h d = + and qs at. Similar to momentum, the effective heights at which heat and moisture are exchanged with the atmosphere are defined by the heights z0h + d and z0w + d, where z 0h and z 0w are the roughness 17

18 Ecological Climatology lengths for heat and moisture, respectively. These heights are known as the apparent sources of heat and moisture. The functions ψ m (ζ), ψ h (ζ), and ψ w (ζ) account for the influence of atmospheric stability on turbulent fluxes. They have negative values when the atmosphere is stable, positive values when the atmosphere is unstable, and equal zero under neutral conditions. These equations describe logarithmic vertical profiles of mean wind, temperature, and moisture in the atmosphere. Figure 7.13 illustrates wind profiles under neutral atmospheric conditions (i.e., ψ ( ζ ) = 0) that might be found over grassland and forest. In the grassland, wind speed decreases m sharply close to the ground surface, attaining a value of zero at z = z0m + d = 40 cm. For forest, wind speed attains a value of zero at z = z0m + d = 8 m. At any height above the vegetation, wind is stronger over the smooth grassland surface than over the rougher forest, and in general wind speed at a given height decreases as roughness length increases. The logarithmic wind profile is valid only for heights z > z0m + d z = z0m + d and does not describe wind within plant canopies or close to the surface. The height is the height where the wind profile extrapolates to zero. In reality, however, wind does blow within plant canopies. Also, it only applies to an extensive uniform surface of 1 km 2 or more. Similar logarithmic profiles can be constructed for temperature and moisture. When sensible heat and evapotranspiration are positive, so that heat and water vapor are exchanged into the atmosphere, temperature and specific humidity decrease with height. This typically occurs during the day. At night, when sensible heat is negative (i.e., towards the surface), temperature increases with height. Similar to wind, these profiles only apply to extensive homogenous terrain. The aerodynamic resistances to momentum, heat, and moisture transfer between height z and the surface are found by relating the profile equations to the flux equations to give 1 z d z d r = ln ψ ( ) ln ( ) am 2 m ζ ψ m ζ k ua z0m z0m 1 z d z d r = ln ψ ( ) ln ( ) ah 2 m ζ ψh ζ k ua z0m z0h 1 z d z d r = ln ψ ( ) ln ( ) aw 2 m ζ ψw ζ k ua z0m z0w 18

19 Chapter 7 Surface Energy Fluxes where u a is wind speed (m s -1 ) at height z. These resistances are less than the neutral value when the atmosphere is unstable, resulting in a large flux for a given vertical gradient, and more than the neutral value when the atmosphere is stable, resulting in a smaller flux for the same vertical gradient. 7.5 Vegetated canopies At the scale of an individual leaf, stomatal control of transpiration is quantified by stomatal resistance. At the scale of a canopy of leaves, an aggregated measure of surface conductance is required to account for evaporation from soil and transpiration from foliage. In particular, the stomatal physiology of individual leaves, which is directly measurable in terms of the response of stomata to light, water, temperature, and other environmental factors, must be scaled over all leaves to the canopy, where photosynthesis and transpiration can only be measured for the aggregate canopy (Figure 7.14, color plate). The relationship between the bulk surface resistance obtained from canopy-scale micrometeorological measurements to leaf resistance, a biological parameter obtained directly from leaf gas exchange, is the subject of considerable research. Whereas stomatal resistance describes the transpiration resistance of an individual leaf, canopy or surface resistance describes the aggregate resistance of a canopy of leaves. This can be done as a single bulk representation of the effective surface for moisture exchange or by partitioning evapotranspiration into ground and vegetation, represented as one layer of leaves or multiple layers (Figure 7.11). In a bulk representation of a vegetated canopy, evapotranspiration is controlled by two resistances acting in series: a surface resistance that represents the combined foliage transpiration and soil evaporation and an aerodynamic resistance that represents turbulent processes. Alternatively, evapotranspiration can be partitioned into soil evaporation and transpiration. Soil evaporation is regulated by aerodynamic processes within the plant canopy. Transpiration is regulated by a canopy resistance that is an integration of leaf resistance over all the leaves in a canopy. For very dense canopies, soil evaporation is negligible and the distinction between surface resistance and canopy resistance is not important. One means to estimate canopy resistance is to represent the canopy as a big leaf with a resistance representative for all the leaves in the canopy (Figure 7.5). Canopy resistance is then the leaf 19

20 Ecological Climatology resistance divided by leaf area index. If r /2 b and (r + r ) are the leaf resistances for sensible and latent s b heat, respectively, the corresponding canopy resistances are r /(2 L) and ( r + r )/ Lwhere L is leaf area b s b index. Leaf area index is the total one-sided or projected area of leaves per unit ground area. A square centimeter of ground covered by a leaf with an area of 1 cm 2 has a leaf area index of one. The same area covered by two leaves, each with an area of 1 cm 2, has a leaf area index of two. Canopy exchange is perhaps more obvious in terms of conductance. Canopy conductance per unit ground area is the leaf conductance per unit leaf area times leaf area index. A typical leaf area index for a productive forest is 4-6 m 2 m -2. Greater leaf area index increases latent heat exchange with the atmosphere by increasing the surface area from which moisture is lost. The same is true for sensible heat flux. A one-layer formulation of latent heat from a plant canopy illustrates the processes controlling canopy fluxes (Figure 7.11). Total latent heat is the sum of three fluxes. Water vapor is exchanged between the leaves and air in the canopy. This flux is λe v ρcp ( ev eac) = γ r c where e ac is the vapor pressure of canopy air, e v is the saturated vapor pressure of the vegetation, and r c is the canopy resistance. Moisture is also lost from the ground as λe g ρcp ( eg eac) = γ r ac where e g is the vapor pressure at the ground and r ac is an aerodynamic resistance within the canopy. Water vapor is also exchanged between the air within the canopy and air above the canopy. This flux is λe a ρcp ( eac ea ) = γ r a where e a is the vapor pressure of air above the canopy and r a is the aerodynamic resistance to moisture transfer from within the canopy to above the canopy (Section 7.4). If there is no storage of water vapor in the canopy air, these three fluxes are related as λe = λe + λe a g v 20

21 Chapter 7 Surface Energy Fluxes In a very dense canopy, soil evaporation is minimal so that total latent heat is the sum of the vegetation and canopy air fluxes ρcp ( ev ea) λe = λe + λe = v a γ rc+ ra That is, latent heat exchange between a canopy of leaves and the atmosphere is the sum of two resistances acting in series: a canopy resistance that regulates transpiration and an aerodynamic resistance (Figure 7.5). Water vapor must first diffuse out of stomata to the leaf surface, governed by the resistance r / L, then s from the leaf surface to canopy air surrounding the leaf ( r / L), and then to air above the canopy (r b a). This additional canopy resistance is seen in derivations of bulk vegetation parameters obtained from measurements of sensible and latent heat fluxes over vegetated canopies. For example, roughness length can be derived from measured fluxes if all other terms in the flux equations are known. Such derivations show that the roughness lengths for heat and moisture in a bulk aerodynamic formulation of vegetated surfaces are typically 20% that for momentum (Campbell and Norman 1998, p. 96). That is, the apparent sources of heat and moisture in general occur at a lower height in the canopy than the apparent sink of momentum, and the aerodynamic resistances for sensible heat and water vapor are greater than that for momentum. This is most obvious under neutral conditions, when the ψ functions have a value of zero. In this case, the aerodynamic resistance for heat is [ z d z ] [ z d z ] [ z z ] ln ( ) / 0h ln ( ) / 0m ln 0m/ 0h * r = = + = r + r ah ku* ku* ku* am b * where u = u k / ln[( z d) / z0. The term is the additional boundary layer resistance for sensible * m ] r a b heat exchange from the canopy (Monteith and Unsworth 1990, p. 248). The Penman-Monteith equation applied to a canopy of leaves gives insights to surface and canopy resistances and their relationship with leaf resistance (Kelliher et al. 1995). For a canopy, ρc s R G e e a λe = = rc + r a s + γ ra p * ( n ) + ( a a) * r s( Rn G) + ρcpga( ea ea) ( 1 / ) s+ γ + g g a c 21

22 Ecological Climatology where r c is canopy or surface resistance, r a is the aerodynamic resistance for water vapor, and g c and g a are the corresponding conductances. By measuring evapotranspiration from a canopy and if all other terms are known, the Penman-Monteith equation can be solved for canopy or surface resistance. Theory suggests that canopy conductance increases linearly with leaf area index at low leaf area (Figure 7.15). Canopy conductance is relatively constant at a value largely determined by leaf conductance at high leaf area. Canopy conductance becomes constant with high leaf area because low light levels on leaves deep in the canopy close stomata. Surface conductance approaches canopy conductance as leaf area increases because soil evaporation becomes negligible in a dense canopy. Surface conductance significantly exceeds canopy conductance only for low leaf area index less than about 3 m 2 m -2, where soil evaporation is more important. For a given leaf conductance, surface conductance is nearly independent of leaf area because the tendency for canopy conductance to decrease with low leaf area is counteracted by increasing proportion of soil evaporation. Across a variety of forests, grasslands, and croplands, surface and canopy conductances derived from the Penman-Monteith equation show relationships similar to that predicted from theory (Figure 7.15). Maximum surface conductance in relation to leaf area index is within the bounds determined by three values of leaf conductance representative of woody plants, natural herbaceous plants, and agricultural crops (Table 5.3). Maximum surface conductance increases linearly with maximum leaf conductance with a slope of approximately 3. This is consistent with theory, which predicts that for any leaf conductance surface conductance is constrained within limits set by leaf area and that surface conductance is largely determined by leaf conductance at high leaf area. As with an individual leaf, vegetation has degrees of coupling to the atmosphere determined by the magnitude of the aerodynamic resistance (Jarvis and McNaughton 1986). Tall forest vegetation has a low aerodynamic resistance and is closely coupled to the atmosphere. Evapotranspiration approaches a rate imposed by canopy resistance λe = ρc e e γr ) * p( a a)/( c 22

23 Chapter 7 Surface Energy Fluxes Short vegetation such as grassland has a large aerodynamic resistance and is decoupled from the atmosphere. Evapotranspiration approaches an equilibrium rate dependent on available energy with little canopy influence λe = s( R G)/( s+ γ ) n This coupling and decoupling with the atmosphere can be seen in leaf temperature. By influencing aerodynamic resistance, the height of plants has a large influence on leaf temperature. Tall vegetation is aerodynamically rough. Heat is readily exchanged with the atmosphere, and the temperature of leaves is closely coupled to that of the air. In contrast, short vegetation is aerodynamically smooth and dissipates heat less effectively. The leaves of short vegetation experience warmer temperatures than that of air. In cold climates, short stature may convey an advantage by warming leaf temperature (Wilson et al. 1987; Grace 1988; Grace et al. 1989). Figure 7.16 illustrates this for alpine forest and shrub vegetation. For both types of vegetation the leaf-to-air temperature difference increases with greater net radiation. In the tall forest, the slope of this relationship is near zero ( C per W m -2 ). Leaf temperatures are similar to air temperature, with a maximum excess of 5 C. In the dwarf shrub vegetation, the slope is larger ( C per W m -2 ) and declines with increasing wind speed. The maximum temperature excess is 19 C in bright sunshine and low wind speed. These results are explained by the effect of wind and vegetation height on the efficiency with which heat is carried away from leaves. The aerodynamic resistance increases with shorter vegetation and weaker winds (Figure 7.7). The effect of wind speed to decrease resistance is greater for short vegetation than for tall vegetation. 7.6 Surface climate Several atmospheric variables affect surface fluxes (Table 7.2). Incoming solar and longwave radiation determine net radiation. Air temperature and atmospheric humidity determine the diffusion gradients for sensible and latent heat, respectively. Wind speed affects sensible heat, latent heat, and momentum fluxes through turbulence. Precipitation determines soil water, and through this, latent heat. Snow on the ground affects surface albedo. Soil temperature affects the amount of heat flowing into the soil. In addition, solar radiation, air temperature, atmospheric humidity, atmospheric CO 2, and soil water 23

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