Surface Energy Fluxes

CHAPTER 7 SURFACE ENERGY FLUXES ... 1

7.1 INTRODUCTION... 2

7.2 SURFACE ENERGY BUDGET... 2

7.3 LEAF TEMPERATURE AND FLUXES... 8

7.4 SURFACE TEMPERATURE AND FLUXES... 14

7.5 VEGETATED CANOPIES... 19

7.6 SURFACE CLIMATE... 23

7.7 TABLES... 28

7.8 FIGURE LEGENDS... 30

7.1 Introduction

Energy is exchanged between the atmosphere and Earth surface. Solar radiation and atmospheric longwave radiation warm the surface and provide energy to drive weather and climate. Some of this energy is stored in the ground or oceans. Some of it is returned to the atmosphere, warming the air. The rest is used to evaporate water. These surface energy fluxes are an important component of Earth’s global mean energy budget. At the global scale, Earth’s energy budget shows the atmosphere has a deficit of energy while the surface has a surplus (Figure 2.1). The atmosphere absorbs 67 W m^-2 of solar radiation and 350 W m^-2 of longwave radiation from the surface; it emits 195 W m^-2 of longwave radiation to space and 324 W m^-2 onto the surface. The excess loss of radiation compared with absorption is -102 W m^-2. The surface, in contrast, gains 168 W m^-2 of solar radiation and 324 W m^-2 of longwave radiation from the atmosphere while emitting 390 W m^-2 of longwave radiation. This gives the surface a net surplus of 102 W m^-2. This surplus energy is returned to the atmosphere as sensible heat (24 W m^-2) and latent heat (78 W m^-2). These heat fluxes arise as winds carry heat (sensible heat) and moisture (latent heat) away from the surface. Over land, sensible and latent heat are important determinants of microclimates. Microclimates are the climate near the ground and represent local climates due to topography, vegetation, soils, landforms, and structures. This chapter introduces the energy fluxes needed to understand microclimates. The next chapter examines spatial patterns in landscapes and resulting microclimates.

7.2 Surface energy budget

Energy continually flows through the climate system. As solar radiation passes through the atmosphere, some is absorbed, primarily by water vapor and clouds, and some is scattered, both upwards to space and downwards onto the surface, by clouds, air molecules, and particles suspended in the air. This downward scattered radiation, known as diffuse radiation, emanates from all directions of the sky. In contrast, direct beam radiation is not scattered and originates from the Sun’s position in the sky. The Sun is the principal source of radiant energy, but all objects with a temperature warmer than –273.15 °C (0 Kelvin) emit longwave radiation in proportion to their temperature raised to the fourth power.

These radiative sources of energy are depicted in Figure 7.1 for a person standing next to a wall.

In this case, direct beam radiation is received based on the Sun’s angle above the horizon and its location with respect to compass direction. Diffuse radiation is received in all directions from solar radiation reflected by the sky, surrounding ground, and the wall. Likewise, the sky, ground, and wall emit longwave radiation that is absorbed by the person. Incident solar radiation varies with time of year, time of day, cloudiness, and atmospheric turbidity. A typical midday summertime value for clear, dry sky at tropical latitudes is as much as 1000 W m^-2. With cloudy skies or when in shade (so that direct beam radiation is blocked), this may be as little as 100 W m^-2. Diffuse radiation is most important when scattering is high.

On overcast days, all the radiation is diffuse. The fraction of total radiation that is diffuse is also high when the low solar altitude angle leads to a longer path through the atmosphere, such as in mornings, evenings, winter, and at high latitudes. When the Sun is directly overhead, solar radiation travels through less of the atmosphere than when it is at an angle and hence less is scattered. Consequently, diffuse radiation may account for 25-50% of the total radiation when the Sun is low on the horizon, but only 10-20% when the Sun is high in the sky. The proportion of the total radiation that is diffuse increases, often by a factor of two, for cloudy, overcast, or polluted skies. Incident longwave radiation varies with temperature, humidity, and cloudiness. Typical clear sky fluxes are 400 W m^-2 at 30 °C and 200 W m^-2 at 0 °C. Under cloudy conditions, these fluxes can increase by 20% or so. During the day, solar radiation is the dominant flux under clear, dry skies. Solar and longwave fluxes are both important with cloudy skies. At night, longwave radiation is the sole source of radiant energy to the surface.

An object has five means to dissipate the radiant energy impinging on it (Figure 7.1). Some of the incident solar radiation is reflected away from the object; the remainder is absorbed. Longwave radiation is emitted in proportion to temperature to the fourth power. Heat is also transferred by direct contact with another object (conduction), movement of air that carries heat away from the object (convection or sensible heat), and latent heat exchange in which heat is dissipated through a change in water from liquid to gas.

The radiation that impinges on a surface or object must be balanced by the energy radiated back to the atmosphere, energy lost or gained by sensible and latent heat, and heat storage. The energy balance is

(1−r S) ↓ + ↓= ↑ + +L L H λE+G

The left-hand side of this equation constitutes the radiative forcing (Qa), which is the sum of absorbed solar radiation ( ) and longwave radiation (L↓), respectively. The right-hand side of the equation consists of the emitted longwave radiation (L↑), sensible heat (H), latent heat (λE), and heat exchange by conduction (G). Expressed in a different manner, the net radiation absorbed by an object is

(1−r S) ↓

(1 ) ( )

R r S L L H E G

n= − ↓ + ↓ − ↑ = +λ +

Net radiation (Rn) is balanced by sensible, latent, and conduction heat fluxes. The effect of different surfaces on microclimates can be understood by considering the various terms in the energy budget (Gates 1980; Grace 1983; Monteith and Unsworth 1990; Campbell and Norman 1998).

All surfaces except for the blackest reflect sunlight. This amount is equal to , where S↓ is the incident radiation onto the surface and r is the albedo, defined as the fraction of S↓ that is reflected by the surface. The remainder, , is the solar radiation absorbed by the surface. A perfect mirror has an albedo of one (i.e., all the incoming light is reflected). An object that completely absorbs all radiation incident upon its surface, known as a blackbody, has an albedo of zero (i.e., all light is absorbed). The albedo of natural landscapes ranges from greater than 0.9 for fresh snow to less than 0.1 for some types of vegetation.

rS↓

(1−r S) ↓

Emission of longwave radiation is a second means of radiative cooling. Terrestrial objects emit electromagnetic radiation in the infrared band at long wavelengths between 3 µm and 100µm. This emission is proportional to temperature raised to the fourth power. For temperatures in degrees Celsius,

(

)

L T

εσ s

↑= +

where Ts +273.15is absolute temperature in Kelvin (K) and σ = 5.67 × 10^-8 W m^-2 K^-4 is the Stefan- Boltzmann constant (Figure 7.2). The emissivity (ε) of an object generally ranges from 0.95 to 1.0. A blackbody has an emissivity equal to one. Most natural surfaces are ‘grey bodies’, with emissivity less than one.

Movement of air carries heat and moisture away from an object. These exchanges are represented as a diffusion process analogous to electrical networks (Figure 7.3). The electrical current between two points on a conducting wire is equal to the voltage difference divided by the electrical resistance. Similarly,

the diffusion of materials is related to the concentration difference divided by a resistance to diffusion. For example, the exchange of water vapor and CO2 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

(^Ta ^Ts)

H C

p rH

ρ ⁻

= −

where Ts and Ta 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 Cp is the heat capacity of air and is approximately 1010 J kg^-1 °C^-1. The transfer resistance

rH (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 2.43 × 10⁶ 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

m^-3 is equivalent to 5 kg water per square meter (5 kg m^-2/1000 kg m^-3 = 0.005 m). With a latent heat of vaporization of 2.43 × 10⁶ 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., 5 ^{kg 1 day} 2 430 000 ^J 141 ^J

2 86 400 s kg 2

m day× × = sm )

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 (ea) to saturated vapor pressure evaluated at the air temperature (e*[Ta]) expressed as a percent

[ ]

100

RH = ea e T∗ a

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 (qa, kg kg^-1) is related to vapor pressure (ea, Pa) as

0.622 0.378

a a

a

q e

P e

= −

where P is atmospheric pressure (Pa). At sea level, P = 101 325 Pa and because , specific humidity

can be approximated as . The right axis in Figure 7.4 shows corresponding specific 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.

P e

>> a

0.622 /

q e

a = a P

Latent heat flux is represented by a resistance network similar to sensible heat in which

( [ ])

Cp ea e Ts

E rW

λ ^{= −}ργ ^{− ∗}

The term e*[Ts] represents the saturation vapor pressure (Pa) evaluated at the surface temperature Ts, and the term ea is the vapor pressure of air (Pa). The difference between these, ( ), is the vapor

pressure deficit between the evaporating surface, which is saturated with moisture, and air. The term is a resistance (s m

e∗⎡ ⎤ −⎣ ⎦Ts ea

rW -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 (C P) /(0.622 )

γ = p λ . A typical value is 66.5 pascals 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 Qa = 560 + 340 = 900 W m^-2. If the surface temperature is Ts = 31.8 °C, the

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 ea = 2049.6 Pa), rH = 25 s m^-1, and rW = 100 s m^-1. The sensible heat flux is H = 111 W m^-2 and the latent heat flux, with e∗⎡ ⎤ =⎣ ⎦Ts 4701.6Pa, is λE = 461 W m^-2 (in this example, ρ = 1.15 kg m^-3, Cp =

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=900 490 111 461− − − =−162W 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 Ts = 29.0 °C, with Pa, the upward longwave, sensible heat, and latent heat fluxes are L↑ = 473, H = -18, and λE = 340 W m

4005.5 e∗⎡ ⎤ =⎣ ⎦Ts

-2, respectively. The object gains 105 W m^-2 more energy than it loses (900 473 ( 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

( ) ( [ ] )

4

/ 2

( 273.15)

a l

C e T e

T Tl a p l a

Cp rb rs b

Q =εσ T + +ρ ⁻ + ^ργ ^∗ +⁻ r

Leaf temperature (Tl) is the temperature that balances the energy budget. The boundary layer resistance (rb) governs heat flow from the leaf surface to the air around the leaf (Figure 7.5). Stomatal resistance (rs) 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 (one- sided) as

200 /

r d

b = u

This resistance has units of seconds per meter (s m^-1). Plant physiologists often use m² s mol^-1 instead of s m^-1. At sea level and 20 °C, 1 m² 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

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. 268-303; Monteith and Unsworth 1990, pp. 121-131; Campbell and Norman 1998, pp. 99-101). For a flat plate of length d (meters), the resistance to heat transfer from one side of the plate is

( ) /( N

r C d k

b = ρ p u)

where ρ is the density of the fluid (kg m^-3), Cp 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

0.5 0.33 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, m² s^-1)

Re=(ud) /ν

Combining equations,

( )

(0.66 Pr0.33)

C p d d

r a

b k u u

ρ ν

= =

For air at 20 °C, ρ = 1.204 kg m^-3, Cp = 1010 J kg^-1 °C^-1, ν = 15.5 × 10^-6 m² s^-1, k = 0.026 W m^-1 °C^-1, and Pr

= 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 rb 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. 297-303) 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 rb) in parallel (Figure 7.3) and the effective resistance for heat transfer is (Figure 7.5). Gates (1980, pp. 27, 30, 351) used a value of a = 133 s

b/ 2 r

1/2 m^-1 for sensible heat flux from a leaf. Campbell (1977, pp. 119-123) reduced rb 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 CO2 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 (rs), 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 CO2 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. 119-123) reduced rb 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

. (rs+rb)/ 2

Table 7.1 shows the importance of sensible and latent heat in cooling leaf temperature under a variety of radiative forcings and wind speeds for a summer day.The leaf has a radiative forcing of 1000, 700, and 400 W m^-2, which is representative of values for a clear sky at midday, a cloudy sky at midday, and night, when solar radiation is zero and the leaf receives only longwave radiation. If longwave radiation is the only means to dissipate this energy, the leaf has temperatures of 91 °C, 60 °C, 17 °C with high, moderate, and low radiative forcings.

Heat loss by convection – sensible heat – cools the leaf (Table 7.1). Under calm conditions, with a wind speed of 0.1 m s^-1, sensible heat loss decreases leaf temperature by 38 °C (to a temperature of 53 °C) with the high radiative forcing and by 20 °C (to a temperature of 40 °C) with the moderate forcing. Higher wind speeds lead to even cooler temperatures. At 4.5 m s^-1, the temperature of the leaf exposed to the high radiative forcing has been reduced from 91 °C to 34 °C. At low radiative forcing, convection warms the leaf because it is colder than the surrounding air and heat is transferred from the air to the leaf. This example illustrates the powerful effect wind has in transporting heat away from an object, thereby cooling the object.

Greenhouse microclimates are an example of the warm temperatures that can arise in the absence of convective heat exchange (Avissar and Mahrer 1982; Mahrer et al. 1987; Oke 1987). It is generally thought that greenhouses provide a warm environment to grow plants because glass or other translucent coverings allow solar radiation to penetrate and warm the interior of the greenhouse while longwave radiation emitted by these warm interior surfaces is trapped by the glass. Indeed, this is the analogy that spawned the term ‘greenhouse effect’ by which increasing concentrations of CO2 in the atmosphere warm Earth’s climate. While this can happen, the daytime warmth in greenhouses is largely a result of negligible convective heat exchange with the outside environment. The sensible heat lost from the warm interior surfaces is trapped within the greenhouse, warming the interior air.

Latent heat exchange also cools the leaf (Table 7.1). Under calm conditions (0.1 m s^-1) and high radiative forcing, transpiration cools the leaf an additional 14 °C, from a temperature of 53 °C with longwave radiation and convection to a temperature of 39 °C. Higher winds result in even lower temperatures. With a wind speed of 4.5 m s^-1, the leaf temperature has been reduced from a lethal

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 T( s−T )a where is the saturation vapor pressure evaluated at the air temperature (T

ea*

a) and s de= ^*/dT is the slope of the saturation vapor pressure versus temperature evaluated at Ta. Substituting this expression for _es^*into λE gives

(

)

p ( )

C a s a

W

e s T T e

E r

λ ρ

γ

+ − −

= ^a

Then, from the definition of the surface energy budget and sensible heat flux,

( )

( / C )p

s a H n

T −T = r ρ R −λE G−

Substituting this expression for Ts−Tainto the latent heat flux gives, after algebraic manipulation,

( )

(

)

( / )

n a

H

W H

s R G e e

E r

s r r

a

ρ

λ γ

− + −

= +

For a leaf, G = 0, r r / 2, and so that

H = b r r r

W = s + b

( )

p *

C / 2

/ 2

n a

b

b s

b

sR e e

E r

r r

s r

a

ρ λ

γ

+ −

= ⎛ + ⎞

+ ⎜ ⎟

⎝ ⎠

This equation relates transpiration to net radiation (Rn), the vapor pressure deficit of air ( ), 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.

e*a−ea

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,

/( ) E sR s λ = n +γ

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

(

)

C /(

E p e e

a a rs)

λ =ρ − γ

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 (Ta) and the effective surface temperature (Ts) and varies inversely in relation to an overall resistance. Latent heat flux is directly proportional to the difference in vapor pressure of air (ea) and the saturated vapor pressure of the surface (e*[Ts]) 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 CO2 during photosynthesis to deplete all the CO2 in the air to a height of 30 m (Monteith and Unsworth 1990, p. 146). In practice, however, CO2 is not depleted because wind mixes the air and replenishes the absorbed CO2. 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

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 (ua, m s^-1), temperature (Ta, °C), and specific humidity (qa, kg kg^-1) and the ground are

( ) /

( ) /

( ) /

a s am

p a s ah

a s aw

u u r

H C T T r

E q q r

τ ρ ρ ρ

= −

= − −

= − −

where us, Ts , and qs are corresponding surface values. The resistances ram, rah, and raw 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 and multiplying both sides of the equation by latent heat of vaporization (λ) gives the previously defined latent heat flux.)

0.622 /

q e

a = a P

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

*

*

*

( )

( )

( )

( )

( )

( )

m

p h

w

ku z d u z ku z d T

H C

z ku z d q

E z

τ ρ φ ζ

ρ φ ζ

ρ φ 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

u*

τ ρ=

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.

The effect of buoyancy on turbulence is represented by the similarity functions φ_m(ζ), φ_h(ζ), and φ_w(ζ), where

(z d) /L ζ = −

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).

Integrating ∂ ∂u/ zbetween two arbitrary heights (z₂ >z₁) gives

* 2 2 1

2 1

1

ln _{m m}

u z d z d z d

u u

k ^{⎡ ⎛} z ⁻d ^⎞ ψ ^⎛ L^{− ⎞} ψ ^⎛ L ^⎤

− = − + − ⎞

⎢ ⎜⎝ − ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠⎥

⎣ ⎦

The surface is defined as u₁=0at z₁=z_0m+d and _m z¹ d 0

ψ ^⎛⎜⎝ L⁻ ⎞ =⎟⎠ so that wind speed at height z is

*

0

ln ( )

z m

m

u z d

u = k ^⎡⎢ ^⎛⎜ z^{− ⎞}⎟−ψ ζ ^⎤⎥

⎢ ⎝ ⎠ ⎥

⎣ ⎦

The height _{z m+d0} 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,

and can be integrated between height z and the surface as

/ T z

∂ ∂ ∂ ∂q/ z

* 0

* 0

ln ( )

ln ( )

z s h

p h

z s w

w

H z d

T T

k C u z

E z d

q q

k u z

ρ ψ ζ

ρ ψ ζ

⎡ ⎛ ⎞ ⎤

− −

− = ⎢ ⎜ ⎟− ⎥

⎢ ⎝ ⎠ ⎥

⎣ ⎦

⎡ ⎛ ⎞ ⎤

− −

− = ⎢ ⎜ ⎟− ⎥

⎢ ⎝ ⎠ ⎥

⎣ ⎦

For temperature and specific humidity, surface values are defined as Ts at height z=z_0h+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

z=z0w+d

z0h +dand z_0w+d, where z0h and z0w are the roughness

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., ) that might be found over grassland and forest. In the grassland, wind speed decreases

sharply close to the ground surface, attaining a value of zero at ( ) 0

ψm ζ =

z=z0m+d= 40 cm. For forest, wind speed attains a value of zero at = 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

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

z=z0m+d

z>z0m +d z=z0m +d

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 ln ( ) ln ( )

2 0 0

1 ln ( ) ln ( )

2 0 0

1 ln ( ) ln ( )

2 0 0

z d z d

ram k ua z m m z m m

z d z d

rah k ua z m m z h h

z d z d

raw k ua z m m z w w

ψ ζ ψ ζ

ψ ζ ψ ζ

ψ ζ ψ ζ

⎡ ⎛ − ⎞ ⎤ ⎡ ⎛ − ⎞ ⎤

⎢ ⎥ ⎢ ⎥

= ⎜ ⎟− ⎜ ⎟−

⎢ ⎝ ⎠ ⎥ ⎢ ⎝ ⎠ ⎥

⎣ ⎦ ⎣ ⎦

⎡ ⎛ − ⎞ ⎤ ⎡ ⎛ − ⎞

⎢ ⎥ ⎢

= ⎜ ⎟− ⎜ ⎟−

⎢ ⎝ ⎠ ⎥ ⎢ ⎝ ⎠

⎣ ⎦ ⎣

⎡ ⎛ − ⎞ ⎤ ⎡ ⎛ − ⎞

⎢ ⎥ ⎢

= ⎜ ⎟− ⎜ ⎟−

⎢ ⎝ ⎠ ⎥ ⎢ ⎝ ⎠

⎣ ⎦ ⎣

⎤⎥

⎥⎦

⎤⎥

⎥⎦

where ua 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