Wet-bulb temperature (symbol Tw) and psychrometer. 33. 2.6 Exercises ...... be integrated from reference level (denoted with index o) to level z. Next, we denote.
ATMOSPHERE-LAND INTERACTIONS THEORETICAL BACKGROUND AND METHODS OF OBSERVATION
H.A.R. de Bruin
Meteorology and Air Quality Group, Wageningen University
February 2004
2
CONTENT Chapter 1 An Introductory Survey 1.2 A calm sunny summer day at Cabauw 1.2.1 Air temperature 1.2.2 Specific Humidity 1.2.3 Wind speed 1.2.4 The energy balance of the earth's surface 1.2.4.1 Radiation 1.2.4.2 Sensible, latent and soil heat fluxes 1.2.4.3 Soil temperature and heat flux 1.3 Summary 1.4. First interpretation and course outline 1.4.1 Simple picture 1.5 Vertical heat transfer into the atmosphere 1.6 Some other examples 1.7 Course outline Exercise MATHCAD module Chapter 2 Properties of air and water vapor 2.1 Composition of air 2.2 Properties of perfect gas 2.3 Hydrostatic equilibrium 2.4 Atmospheric thermodynamics 2.5 Water vapor 2.5.1 Thermal effects of phase changes 2.5.2 Water vapor variables a. Absolute humidity (symbol v units kg m -3). b. Water vapor pressure (symbol ea units N m-2, hPA or mbar) c. Mixing ratio and specific humidity (symbol r and q, dimensionless) d. Virtual temperature (Tv units K) e. Saturation water vapor pressure (es same units as ea) f. Relative humidity (symbol RH) h. Dew point (symbol Td units either 0C or K) g. Wet-bulb temperature (symbol Tw) and psychrometer 2.6 Exercises Chapter 3 Radiation 3.1. Introduction 3.2 Brief review of radiation laws 3.3 Radiation balance at the surface 3.4 Radiation in wavelength interval 3.4.1 Apparent Stefan-Boltzmann laws for a wavelength interval 3.5 Short- and longwave radiation 3.6 The natural greenhouse effect 3.7 Shortwave radiation in the atmosphere 3.7.1 Absorption 3.7.2 Scattering Law of Rayleigh: > d Law of Mie; ≈ d Laws of geometric optics: < d
6 6 6 8 10 11 11 13 14 16 17 17 18 21 22 23 23 24 24 24 25 26 30 30 30 30 30 30 32 32 32 33 33 34 37 37 38 40 40 41 46 47 48 48 49 49 49 49
3 3.8 Incoming shortwave radiation 3.8.1 Solar zenith angle and extraterrestrial radiation 3.8.2 Daily mean values 3.9 Outgoing shortwave radiation 3.10 Incoming longwave radiation 3.10.1 Empirical expressions for L0 and Lc . 3.11 Outgoing longwave radiation 3.12 Net longwave radiation 3.13 Exercises 3.14 Mathcad modules 3.14.1 Module on wavelength interval 3.14.2 Module on sub-pixel fire detection with NOAA-3 3.14.3 Module on the greenhouse effect 3.14.4 Module on extraterrestrial radiation and day length Chapter 4 Soil Temperature and Heat Flux 4.1 Introduction 4.2 Bare soil 4.3 Theory 4.3.1 Soil Temperature 4.3.2 Semi-infinite homogeneous soil with sine-wave at the surface 4.4 Soil heat flux density G 4.4.1 The force-restore method 4.5 Fully vegetated surfaces 4.6 Semi-infinite homogeneous soil with sudden step change 4.7 Semi-infinite homogeneous soil with constant heat flux at surface 4.8 Numerical mathematical solutions 4.8.1 Introduction 4.8.2 Forward-difference approximation 4.8.3 Logarithmic z-spacing** 4.9 Radiation forcing at the surface 4.10 Radiation forcing at the surface and 'closed-box' atmosphere 4.11 Exercises Chapter 5 Turbulence 5.1. Introduction 5.2 Statistical description 5.3 Pressure, temperature and density fluctuations 5.4 Vertical fluxes and eddy correlation method 5.5 'Webb-correction' 5.6 K-theory 5.7 Some scales and definitions 5.8 Richardson numbers and turbulent kinetic energy 5.8.1 The Obukhov length 5.9 Exercises Chapter 6 Fluxes in the surface layer and similarity theory 6.1. Introduction 6.2. Surface layer or constant-flux layer 6.3. Free convection 6.3.1 Free convection flux-profile relationship 6.3.2 Free-convection relationship between H and 6.4. Neutral conditions
50 51 52 53 54 55 55 56 57 59 59 59 60 60 61 61 61 62 64 64 65 66 66 66 67 67 67 67 68 68 69 70 72 72 74 74 75 76 77 78 80 82 83 84 84 84 85 85 86 87
4 6.5 Monin-Obukhov similarity theory (MOST) 6.5.1 MOST flux-profile relationship 6.5.2 Integrated MOST functions 6.5.3 MOST relationships between fluxes and standard deviations 6.5.4 Note on MOST and the neutral and free convection limit* 6.5.5 Fluxes of water vapor 6.6 T-q-qx similarity and fluxes of other gasses 6.7 Aerodynamic resistances 6.8 Exercises Chapter 7 How to measure or model vertical fluxes? 7.1 Introduction 7.2 The eddy-correlation method 7.3. Flux-profile relationships 7.3.1 The free-convection approximation 7.4 The Bowen ratio method 7.5 The Penman-Monteith equation 7.5.1 Concept 7.5.2 The governing equations 7.5.3 Numerical and analytic solutions 7.5.4 Differences between grass, forest and open water a. Grass b. Forest c. Open water 7.5.5 Operational application of the Penman-Monteith equation Step 1 Step2 7.5.6 Extrapolation towards the surface* 7.5.7 Sensitivity of LvE and surface temperature to wind speed 7.5.7.1 Introduction 7.5.7.2 Equilibrium state 7.5.7.3 Wind sensitivity LvE 7.5.7.4 Wind sensitivity Surface temperature 7.6 Advection: a case study 7.6.1 Introduction 7.6.2 Experimental 7.6.3 Some results. 7.6.3.1 Correlation coefficients 7.7 The standard deviation or variance method 7.8 Scintillometer method 7.9 Soil-vegetation-atmosphere transfer models 7.9.1 The Jarvis-Stewart model 7.9.2 A-gs model 7.10 Exercises 7.10 MATHCAD and EXCEL modules 7.10.1 Flux-profile relationships: derive yourself 7.10.2 Businger-Dyer and free convection 7.10.3 Excel modules using real data 7.10.4 Wind sensitivity energy balance and Penman-Monteith Chapter 8 Water temperature of a well mixed lake Theory
88 89 90 91 92 93 94 94 97 98 98 98 99 101 102 103 103 104 105 106 106 106 106 106 106 106 107 107 107 108 108 110 111 111 112 113 117 119 119 120 121 121 122 123 123 123 123 123 125 125
5 Chapter 9 Well-mixed convective boundary layer model: Interaction with a vegetated land surface 128 9.1. General aspects 128 9.2. PBL growth 128 9.3. "Thermodynamics" and conservation laws 130 9.3.1 Governing equations 130 9.3.2 Closure assumption 131 9.3.3 Coupling to the surface layer; description Excel module 131 9.3.4 Some exercises 132 9.3.4.1 Reference case 132 9.3.4.2 Sensitivity studies 133 9.4 Approximate analytic solutions for the convective case 134 9.5 Excel and MATHCAD modules 135 9.5.1 A MATHCAD module on the CBL 135 9.5.2 A coupled mixed layer Penman-Monteith model 135 Chapter 10 Miscellaneous Issues 136 10.1 Introduction 136 10.2 Dew 136 10.2.1 Introduction 136 10.2.2 Dewrise 136 10.2.3 When does dewfall occur? 136 10.3 Frost 137 10.3.1 Night-frost 138 10.3.2 Crop protection against night-frost 138 10.3.2.1 Agricultural practice 138 10.3.2.2 Night-frost protection 139 10.3.4 Frost penetration depth into the soil 139 10.4 Fog 140 10.4.1 Fog control 141 10.5 Propagation of sound in the atmosphere 141 10.5.1 Velocity of sound 141 Chapter 11 Literature 144 Introduction to meteorology 144 Turbulence 144 Atmospheric surface and boundary layer 144
6
Chapter 1 An Introductory Survey These notes are dealing with micro-meteorological phenomena occurring in the air layer immediately above land surfaces. Since we all live in this air layer most people are familiar with these phenomena. However, most people are unacquainted with the details and the physical background. Simple questions such as: "Why do the air temperature, wind speed and water vapor pressure at, say, 2 m show a diurnal variation?" can be answered only after one has studied the physical details. It is the aim of this text to provide the background of the various land surface processes with emphasis on the evaporation process. As introduction we examine data collected on a calm sunny day in June at Cabauw, the Netherlands, located at about 52 0N and 5 0E.
1.2 A calm sunny summer day at Cabauw In order to introduce the reader to the subjects we will deal with we will show some data that were gathered at and near a 200 m tower on a calm sunny summer day in June. This tower is located at Cabauw, the Netherlands, at about 52 N about 40 km from the Dutch coast and is operated by the Royal Netherlands Meteorological Institute (KNMI). We will examine half-hourly values of the air temperature, the specific humidity, the wind speed as well as the various components of the surface energy budget. Furthermore, we will pay attention to soil heat flux and soil temperatures. In first instance we just describe that what is observed without trying to give an explanation. Thereafter we will try to explain the various observed features.
1.2.1 Air temperature In Figure 1.1 the diurnal variation of the air temperature at 0.6, 2 and 200 m in the atmosphere are depicted. The time is expressed in Greenwich time (UT), so that the local noon is about at 11.40 (UT) and 12.40 (MET; day-light saving time). The sunrise and sunset on the 30th of June are at 3:23 and 20:04 UT respectively. Some features are: The temperature at 0.6 m reaches its maximum first. At 200 m the maximum is reached about 1 hr later; All maximum values are reached well after local noon; In the early morning the minimum temperature at 0.6 m is reached around sunrise, but the data show scatter around that time. At 200 m the minimum temperature is reached several hours later; The temperature curves are not symmetrical with respect to noon and to the time the maximums occur neither. (They are not a sin-function); At daytime the vertical temperature differences are smaller than that at night..
7
30
Celcius
25
20
15
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0
3
6
9
12 time (UT)
15
18
21
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Figure 1.1 Air temperature at 0.6 (full), 2 (dot) and 200 m (dash-dot). From a atmospheric thermodynamic point of view the potential temperature g z , with g the is more relevant. Using the simple expression ( z ) T ( z ) cp acceleration of gravity (9.8 m s-2) and cp =1005 J kg-1 K-1 the data plotted in Figure 1.1 were converted into potential temperatures (see Figure 1.2). 30
Celsius
25
20
15
10
Figure 1.2
0
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4
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12 UT
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at 0.6 (full), 2 (dot), 80 and 140 m (dash-dot and dash).
8
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height m
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0
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20 potential temperature Cels ius
25
30
Figure 1.3 Vertical profiles of potential temperature at about 2 and 14 UT. Figure 1.2 is very similar Figure 1.1, except that it is seen that during daytime the -curves for 80 and 140 m almost coincide. The vertical - profiles at about 2 and 14 UT are shown in Figure 1.3. It is seen that is constant with height for level above about 30 m at 14 UT. During nighttime the temperature increases with height, but the rate-of-change varies with z. In both curves the vertical gradients of are largest near the surface.
1.2.2 Specific Humidity Since specific humidity q is conserved for vertical motion in the absence of phase changes, we use q as water vapor variable. In Figure 1.4 the diurnal variation of q is depicted at the same levels as in Figure 1.1. Compared to the temperature, q is behaving quite differently. In the night q tends to become smaller and to increases after sunrise. However, just after 8 UT q drops sharply, whereas it rises again around 16 UT. Note that the wind speed was low, by which horizontal effect are expected to be small and that we are dealing with clear cloudless day without fog. So, what explains this irregular behavior? We will come back on this question later. Vertical profiles of q at about 2 and 14 UT are depicted in Figure 1.5. It is seen that the nighttime curve show a maximum at 40 m, whereas q at 14 UT decreases continuously with height, but above 80 m the vertical differences are small. In both examples the air has the lowest q at 200 m. Near the ground the vertical gradient of q is large at 12 UT. At 2 UT q appears to increase with height in the first 40 m of the atmosphere.
9
12
specific humidity g/kg
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0
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12 UT
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Figure 1.4 Specific humidity at 0.6 (full), 2 (dot), 80 and 140 m (dash-dot and dash).
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height m
150
100
50
0
6
6.5
7 7.5 8 Specific humidity g/kg
8.5
9
Figure 1.5 Vertical profile of specific humidity at about 2 (full) and 14 UT (dot).
10
1.2.3 Wind speed The diurnal variation of the horizontal wind speed at 10, 20, 140 and 200 m respectively are shown in Figure 1.6. Since a calm day was selected the wind speed at 10 m is low. There is no clear diurnal variation at that level.
10
wind speed m/s
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12 UT
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Figure 1.6 Vertical profile of wind speed at 10 (full), 20 (dot), 140 (dash) and 200 (dash-dot) m
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height
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0
1
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3 4 wind speed m/s
5
6
Figure 1.7 Vertical wind profile at 2 (full) and 14 UT (dot).
7
11 During daytime the vertical differences are small, see Figure 1.7, in which the wind profiles at 2 and 14 UT are shown. Remarkably, at 2 UT the wind profile has a maximum at 140 m. This is known as a low level yet. At 14 UT the wind speed is fairly constant above 40 m.
1.2.4 The energy balance of the earth's surface An important 'driving force' of the 'physics' of the air and soil layers near the landatmosphere interface is the incoming solar radiation, denoted also as incoming shortwave radiation. In addition, the surface receives infrared radiation emitted by the atmosphere. This will be called outgoing longwave radiation. The latter is the result of what is called the Greenhouse effect. A portion of the incoming shortwave radiation is reflected upwards by the surface. This is the outgoing shortwave radiation. Finally, the surface itself emits outgoing longwave radiation. The net radiation at the surface is defined through: Q*
K
K
L
L
(1.1)
in which K means shortwave, L is longwave, the arrow-indices denote incoming or outgoing and Q* is the net radiation. The units are W m-2.. At the surface net radiation is partitioned into sensible heat flux H flowing into the atmosphere, energy used to evaporate liquid water LvE and energy used to heat the soil G: Q*
H
Lv E G
(1.2)
Details on Equations (1.1) and (1.2) will be given later. 1.2.4.1 Radiation The diurnal cycle of K , K , L L and Q* is shown for our sunny day in June in Figure 1.8. It is seen that the incoming and outgoing shortwave radiation curves are close to a sine function between sunrise and sunset (3:23 and 20:04 UT respectively). The longwave radiation components show a diurnal variation also, but the, first, the amplitudes are much smaller than those of the shortwave components, whereas, during nighttime their values are non-zero. The net radiation curve is sine-like also during daytime and almost constant at - 55 W m-2 during most nighttime hours. For comparison we calculated the daily (24 hr) mean values of the terms of the radiation budget and we obtained the following values:
K = 354. K215 = 359, K = 79, L = 336, L = 435 and Q* = 157 all in W m-2. K215 is the measured incoming solar radiation at the top of the tower, i.e. at 215 m. It is seen that on this sunny summer day the mean incoming longwave radiation is only slightly less than the incoming shortwave (solar) radiation at the surface. The ratio of K and K is 22%. This ratio is denoted as the surface albedo.
12
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W/m2
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0 0
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Figure 1.8 Diurnal cycle of K (full), K (dot), L (x's), L (+'s) and Q* (dash).
1200
1000
W/m2
800
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0 0
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UT
Figure 1.9 Diurnal cycle of K (full), K215 (dot), Kex (dash).
13
The incoming solar radiation that would occur without the atmosphere is denoted as the extraterrestrial radiation Kex. A measure for the turbidity of the entire atmosphere is the difference or the ratio of K and Kex. The turbidity of the lowest 215 m is characterized by the difference between K and K215. In Figure 1.9 K , K215 and Kex are compared. Around noon the ratio of Kex and K is 78%. This ratio drops to about 65% just after sunrise and just before sunset. In the morning this ratio is slightly greater than in the afternoon. To illustrate the significance of the incoming longwave radiation in Figure 1.10 the measured incoming short and longwave radiation (no measured at Lindenberg, Germany (where the German Weather Service maintain similar observation facilities as at Cabauw) are plotted. The mean daily values are now: K = 59 and L = 275 W m-2. So, in winter the daily L >> K . It is seen that for this day both K and L show irregular behavior. This is due to the effects of clouds. It is seen that around 10 MET there clouds cause a dip in K , but a rise of L . Apparently there were clouds also during nighttime. 400
W/m2
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MET
Figure 1.10 Incoming short- and longwave radiation at Lindenberg, 31 January 2000
1.2.4.2 Sensible, latent and soil heat fluxes We now consider the right-hand-side terms of Equation (1.2), i.e. - using a more precise nomenclature - the sensible heat flux density H, the latent heat flux density LvE and the soil heat flux density G all expressed in W m-2. Note that E is the vertical
14 water vapor flux density (kg s-1 m-2) and Lv the latent heat of vaporization (J kg-1). The diurnal variation of H, LvE, G and Q* is depicted in Figure 1.11 500
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W m-2
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Figure 1.11 Diurnal cycle of Q* (full), H (dot), LvE (o's) and G (+'s). It is seen that on this day the energy used for evaporation (LvE) exceeds H. Around 17 UT H becomes negative. The maximum values of H, LvE and G are about 190, 260 and 55 Wm-2 on this selected day, respectively. Note that in this example in the measured values of LvE are < 0 between 0 and 4 UT, whereas LvE remain positive after sunset. G is slightly negative at night. 1.2.4.3 Soil temperature and heat flux Soil temperature data are available at depth 0, that means at the interface of the grass cover and the soil layer, and at a depth of 2 cm, denoted as Ts0 and T2 respectively. In additional, we evaluated an effective surface temperature T0 from the outgoing longwave radiation L by assuming that the emissivity of the surface (for an explanation see later) is s = 0.98. In Figure 1.12 the diurnal cycle of the surface and soil temperatures are plotted. Note that there is a time shift in the moment the maximum values are reached, i.e. the radiation derived surface temperature T0 reaches its maximum first (about 45 min after local noon), next the temperature at the grasssoil interface and, next, the soil temperature at 2 cm Ts2 almost 2 hours later. Note that T0 reaches a maximum of 37.9 0C. Apparently, close to the surface the temperature gradients are large. To illustrate that further, in Figure 1.13 the air temperatures profiles at 0.6, 2 and 200 m are plotted as well the radiation derived surface temperature T0. The maximum difference between the T0 and the air temperature at 0.6 m is greater than that between 2 an 200 m.
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Figure 1.12 The diurnal cycle of T0 (full), Ts0 (x's) and Ts2 (o's).
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Figure 1.13 Diurnal cycle of T0 (full), and T at 0.6 (dot), 2 (o's) and 200 (+'s) m.
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Figure 1.14 Diurnal cycle of the soil heat flux density at 0 (full), 5 (dot) and 10 (x's) cm.
Two soil heat flux plates inserted at a depth of 5 and 10 cm below the surface provided data of the soil heat flux density at these depth, G5 and G10 respectively. In Figure 1.14 G5 and G10 and the extrapolated soil heat flux at the grass-soil interface determined with a procedure that will be explained later are plotted. Note that the amplitude the soil heat flux decreases with depth and that there is a phase shift between the two curves. Moreover, G is maximal just after local noon. It is stressed that here we are dealing with a grass covered surface. It will be seen later that bare soil behaves differently.
1.3 Summary The air temperatures reach their maximum first at the ground after local noon. Close to the ground the air temperature is minimal around sunset and at 200 m a couple of hours later. Diurnal temperature curves are not 'sine-like'. At daytime the vertical temperature gradients are smaller than that at night. Close to the ground the vertical gradients are the largest. The temperature differences at 0.6 m and the surface are often greater than those between 2 and 200 m around noon.. During most of the daylight the potential temperature is constant with height > 20 m. The specific humidity is fairly well mixed during daytime also, but shows a much irregular behavior. In some periods the first 200 m of the atmosphere moistens (q is increasing)¸ but in others it dries up. The latter occurs even during daytime when evaporation is positive. The wind speed at 10 m show hardly varied during the day. The vertical differences are significant during night and are small during day. The vertical wind speed gradients are largest near the surface. During nighttime a so-called low-level jet can occur in the first 200 m of the atmosphere.
17 The daily (24 hr) mean of the incoming shortwave radiation is only slightly greater than that of the incoming longwave radiation. The ratio of incoming solar and the extraterrestrial radiation dependents on solar angle. At noon this ratio is maximal (78%) and just after/before sunrise/sunset is about 65%. In the morning this ratio is slightly greater than in the afternoon. The difference between the incoming solar radiation at 2 and 125 m is small. The daily averages differ 5 Wm-2, which is about 1%. Soil temperatures are retarded with respect to the surface temperature. The phase shift increases with depth. At the same time the amplitude of a temperature curve decreases with depth also. The soil heat flux behaves similarly, except that it become maximal earlier than the temperature, notably just after local noon.
1.4. First interpretation and course outline 1.4.1 Simple picture It is the objective of this course to provide the reader a theoretical framework which with the experimental data as presented above can be explained and how the different phenomenon can be described in mathematical form suitable for application in numerical models. Our approach will be that we will formulate some questions based on a further analysis of the data shown above. This has been done to show the reader that, for instance, the vertical transfer of heat in the lowest layer of the atmosphere is governed by equations that differ from those describing e.g. heat transfer in a solid. For this purpose, we adapt a simple picture of the physical processes occurring near the atmosphere-land interface. First of all, we assume that the surface as well the environmental conditions are horizontally homogeneous. That means that horizontal gradients of temperature, humidity, wind speed etc. are so small that these do not affect the time evolution of these quantities. Furthermore, in this picture, the solar radiation is considered the 'main driving force' of land surface processes. The sun is supplying the energy required to maintain these processes. In first instance, it supplies energy to the earth's surface. Evidence given above on incoming solar radiation reveals that the surface receives about 70% of the incoming solar radiation at the top of the atmosphere, but that the lowest layer 200 m of the atmosphere is almost transparent for solar radiation, since the difference between the measured values at 214 m and the ground is only 1%. So, in first instance we adopt the simple picture that the air layer near the surface is transparent for solar radiation, at least in absence of clouds, mist of fog. The surface reflects 22% of the incoming solar radiation back into space. Next, we consider the longwave radiation transfer. Now, it cannot be assumed that the atmosphere does not interact with infrared radiation. Details will be given later, but the main issue is that air has strong absorption bands in the infrared region. As a result, the atmosphere absorbs and emits radiation in the infrared. This feature explains that the surface receives day and night (incoming) longwave radiation.. On the other hand, the earth's surface emits longwave radiation also. This is the outgoing longwave radiation component depicted above. Note that on a daily bases this term is the largest of the radiation balance. In order to explain main features, we assume, in first instance, that infrared (=longwave) radiation does not affect the 'thermodynamics' of the lowest part of the atmosphere directly, i.e. it is assumed that the heating and cooling effects of the vertical divergence (change with height) of infrared radiation can be ignored. In general, this assumption is incorrect, in particular in the first mm or so above the
18 surface. Furthermore, under certain conditions, it neither holds for thicker air layers near the ground. Nevertheless, in most of this course we will neglect effects of vertical longwave radiation divergences. Fortunately, in most cases this assumption is correct to understand 'first-order' features. In this way we arrive at the following simple picture of what happens at the atmosphere-land interface. Radiation is the 'driving force' of the time evolution of the surface energy budget, in particular net amount of radiation received by the earth's surface. According to equation (1.1) Q* is the sum of incoming short and longwave radiation minus the sum of outgoing short and longwave components. A partitioning of Q* over other energy forms takes place at the surface. According to equation (1.2) a part of Q* is transformed into sensible heat that is flowing into the atmosphere. This term H causes a temperature rise of the atmosphere. This increase of the temperature starts at the surface and, next, propagates further into the atmosphere. Mechanisms for this vertical transport of sensible heat into the atmosphere are fairly complicated and will be a important topic of this course. Similarly, a portion of Q* is used to heat the soil, but the heat transfer mechanisms in the soil differ from that in the atmosphere. Also here the soil temperature rise starts at the surface and, next, penetrates into deeper soil layers. Finally, a part of Q* is used to evaporate the available liquid water at the surface. This process brings water vapor into the atmosphere, which will be transferred from the surface to higher atmospheric levels by transport mechanism that appear to be similar to that of sensible heat. After sunset the shortwave radiation is absent and the surface receives less longwave radiation from the atmosphere than it emits. As a result, the surface starts to cool.
1.5 Vertical heat transfer into the atmosphere In this section we will analyze the temperature data presented above in order to illustrate the particular properties of the vertical transfer of heat in the lowest air layers just above the surface. Fir this purpose will consider the governing equations describing the vertical heat flows under horizontally homogenous conditions. These read as:
QH
cpK
T z
(1.3)
cp
T t
(1.4)
and
QH z
In equations (1.3) and (1.4) QH is the vertical flow of sensible heat per unit time (t) and area across a horizontal surface at level z. Furthermore, is the density of medium and cp is the specific heat (at constant pressure). In the context of this course QH can be either H or G. However, when equation (1.3) is applied to the atmosphere, T has to be replaced with the potential temperature . The point we want to make here is that the so-called temperature diffusion coefficient K behaves quite differently in the atmosphere than in the soil. Using the data presented in the previous section we will make an estimate of K at different levels.
19
To show this, we first apply equation (1.3) to calculate the vertical temperature gradient in the atmosphere from the measured heat flux H using a value for K listed in handbooks for 'still air', i.e. at 20 0C the thermal diffusion coefficient amounts about 2 10-5 m2 s-1. As example, we use the maximum value of H plotted in Figure 1.11, i.e. H = 189 W m-2. With = 1.2 kg m-3 and cp = 1005 J kg-1 K-1 we 189 obtain form equation (1.3) that close to the ground 7836 z 1.2 *1005 * 2 *10 5 K m-1. The measured value is of the order of 1 K m-1 so apparently, the value K = 2 -5 2 -1 10 m s that refers to 'still air' does not apply to the atmosphere near the ground. The difference between the calculated and the measured values is more than 3 orders of magnitude! Secondly, we estimate from the measured curves shown in Figure 1.2 and Figure 1.13 typical for the change of H with height in the first 10 m of the atmosphere. This appears to be less than 7 % of H at the ground. Consequently, we can ignore, in first instance, the height variation of H in the first 20 m. Next, we apply (1.3) in finite difference form to evaluate typical values for K, using the measured values of depicted in Figure 1.2 and the measured values of H shown in Figure 1.11 i.e. we will use:
K i (t )
H (t ) cp
(t ) zi
(1.5)
i
where index i refers to the level and of the air i is the mean change in time of layer between zi+1 and zi. at time t. In this way we find that at 12.25 UT K increases with height from 7.4 10-3 m2s-1 at about 30 cm to 14.1 m2s-1 at 15 m!. Furthermore, similar calculations for the entire diurnal variation of K reveal that at a given level K is smaller at night compared to daytime values. For instance at 2.25 UT K increases with height from 3.2 10-3 m2s-1 at about 30 cm to only 0.4 m2s-1at 15 m. If one determines the daily (24-hourly) means of the vertical temperature gradient and the mean K and, next, with (1.3) a 'mean heat flux density is calculated, one do not get the measured mean heat flux which is in our example 43 W m-2. In general, the use of daily averages of vertical gradients is not related in a straightforward way to the vertical fluxes. This is due to the fact that the processes are highly non-linear, in particular, the difference between night and day create this nonlinearity. In spring and autumn, it can be possible that the daily mean vertical gradient of temperature suggest a downward mean heat flux, whereas the actual heat flux is upwards. See exercise 1. So, this simple analysis of the measured - and H data reveals that the thermal diffusion coefficient is strongly height and time dependent. See Figure 1.15 and Figure 1.16 ad well exercise 1.1. They increase with height and show a diurnal variation with a minimum at night and a maximum during around noon. Because in the lowest atmosphere heat and water vapor (or other gasses) are transferred vertically by the same mechanisms, the corresponding atmospheric diffusion coefficients for
20 these gasses are increasing with height also. An important objective of this course is to find a suitable explanation for this striking feature. It appears that, in general, this can occur in turbulent flows. Turbulence will be the 'key-word' in chapters of these notes dealing with vertical transfer of sensible heat and water vapor (etc.) in the lowest layers of the atmosphere. Flows in the soil are not turbulent so the heat transfer processes are in the soil differ from that in the atmosphere. They are more similar to heat transfer in homogeneous solids, but it will be shown in Chapter 3 that also the soil medium has its own peculiar behavior. The experimental data collected at Cabauw on our sunny day in June, does not allow a detailed analyses to show these features. 0.4
15
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height m
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15
height m
Figure 1.15 Calculated thermal diffusion coefficient K (m2s-1) as function of height at 2.25 UT (left panel) and at 12.25 UT right panel.
3
K
2
1
0
0
10
20
30
UT Figure 1.16 Diurnal variation of calculated thermal diffusion coefficient K (m2s-1) at 6 m.
21
1.6 Some other examples Of course, the environmental conditions of 1 day in June at 1 location cannot be representative for all situations. For that reason we present here some other examples. In Figure 1.17 the four terms of the energy balance equation are depicted for Cabauw also, but now for a very dry year that occur in 1976. It is seen that after 8 MET the evaporation does not increase, but 'levels off'. This indicates that the plants close their stomata. It also illustrates the fact that in the Dutch climate droughts are never very severe. It is seen also that around noon H > LvE.
Figure 1.17 Energy balance at Cabauw in the very dry year 1976 in the Netherlands To illustrate that water bodies behave differently in Figure 1.18 and Figure 1.19 the annual cycle of the 4 energy balance terms are given for two adjacent lakes in the Netherlands. The first has a depth of 5 m and the second is 15 m deep. Because water can be in a turbulent state also the effective depth that is affected by radiation etc. is much larger for water than for soil. As a result the effective heat capacity of water bodies is large. This makes that the system gets a 'memory'. For example, the water temperature at a certain moment t is not only determined by the incoming radiation etc., but also by that what happened in the past. As a result lake evaporation is not in phase with net radiation. For the 5 m lake the phase shift is, say, 10 days, whereas for the 15 m deep lake this shift is already 1 month. Furthermore, the behavior of the term G, which is now the time change of the heat stored in the water body (per unit area), is a large compared to Q*. For the 15 m deep lake their amplitude is almost the same. Note that in the spring G is positive and in the autumn negative. This implies that in the spring most of Q* is used to heat the water rather than to evaporate water, whereas in the autumn the water temperature decreases and the energy that is released in this way is used for evaporation. In general, H is small for lakes.
22
Figure 1.18 Annual cycle energy budget term of a lake (5 m)
Figure 1.19 As Figure 1.18, except for a depth of 15 m
1.7 Course outline The objectives of this course is to discuss in detail the physical background of the four components of the surface energy balance equation (1.2), i.e. the net radiation, soil heat flux and the two fluxes into the atmosphere H and LvE. In Chapter 2 we will introduce the properties of air and water vapor and the physical background of the relevant quantities. Furthermore, the vertical structure of the atmosphere is discussed and a schematic division of the atmosphere into different layers is presented as well. In Chapter 3 radiation is dealt with. The physical laws governing radiation are introduced and the physical relevance of the four components of the surface radiation budget is discussed. Furthermore, it is shown how these terms might be estimated from simple weather data collected at standard meteorological stations. In Chapter 4 soil temperatures and soil heat fluxes are discussed and models, inclusive numerical ones, are introduced. These describe soil temperatures and soil
23 heat flux in relation to the surface temperature or 'driving forces' at the surface. The impact of soil moisture on various soil thermal properties will be dealt with also. The Chapter 5 an introduction to turbulence is presented and some definitions of variables describing turbulent flows are introduced. In Chapter 6 a detailed description is given how the turbulent fluxes for sensible heat H, momentum and mass (water vapor, carbon dioxide or other gases) can be described as function of (measurable) vertical gradients or variances using dimensional analyses. ` In Chapter 7 methods of observation and models are treated that are based on the results obtained in the previous chapters. A simple model for the temperature and energy balance of a well-mixed lake is presented in Chapter 8. This chapter is meant for students with a hydrological background. In Chapter 9 a model describing the atmosphere-land interaction for daytime conditions is dealt with. Some miscellaneous subjects such as dew formation, fog, night-frost and propagation of sound are discussed in Chapter 10.
Exercise In the Excel-file "April 1_2002_example.xls" 10-min values of temperature at different heights are given, as well as the four components of the radiation balance and the sensible heat flux density H for 1 April 2002 at Cabauw, (52 N 5 E). a. Determine the mean temperatures at 2 and 10 m and calculate the mean value of H also. Discuss the result. Note that if H is (per definition) positive it is upwards by definition. b. Plot the diurnal variation of the radiation components. Determine their averages also. Compare the incoming short and longwave radiation. Between 0 and 5 UT the incoming longwave radiation show variation. Explain this. c. Estimate the change of H with height using the temperature data of 2 and 10 m.
MATHCAD module In the file CABAUW DATA._30_June.mcd the data of the calm sunny day used above to illustrate several features are analyzed. At the end some exercises are given. In the locked areas some hints can be found. This module is meant also to introduce the inexperienced reader in working with Mathcad.
24
Chapter 2 Properties of air and water vapor 2.1 Composition of air Under conditions prevailing in the lowest terrestrial atmosphere air can be considered a mixture of perfect gases. The most important gaseous constituents of so-called dry air are listed in Table 2.1., where dry air is air form which (hypothetically) water vapor has been excluded. Table 2.1 Composition of 'dry' air Gas
N2 O2 Ar CO2 dry air
Molecular mass kg kmol-1
Molar (volume) fraction
Mass fraction
28.013 31.999 39.848 44.010 28.976
0.7809 0.2095 0.0093 0.0003 -
0.7552 0.2315 0.0128 0.0005 -
Secific gas constant J kg-1K-1 296.80 259.83 208.13 188.92 287
miRi/m J kg-1K-1 224.15 60.15 2.66 0.09 -
Except CO2 the relative fractions of the main components listed in Table 2.1 are remarkably constant up to about 80 km. It is well known that the concentration of carbon dioxide has increased significantly in the last 50 years, but due to its relatively low contribution to dry air compared to N2 and O2 this CO2 enhancement can be ignored here. A mixture of perfect gases behaves as a perfect gas also.
2.2 Properties of perfect gas According to Charles and Boyle-Gay Lussac the so-called equation of state of a perfect gas reads as:
p
R* T m
RT
(2.1)
where p is the pressure (N m-2), is the density of air (kg m-3), T is the absolute temperature (K), m is the molecular mass (weight) and R* is universal gas constant. R* The latter is 8314 J kmol-1 K-1. It is common to denote R as the specific gas m constant of the gas mixture. For the main atmospheric constituents R is listed in Table 2.1. According to Dalton the pressure of a mixture of gases is the sum of the partial pressures of the individual gas constituents, i.e. the pressure that would occur if this individual gas occupies the same volume of the mixture at the same temperature. Moreover, according to Dalton, each component of the mixture does occupy the same volume V as this mixture. Let Mi be the mass in volume V be the volume of the gas M i Then the mixture of constituent i and fi = Mi/M its mass fraction, where M
25
Mi . Since the density of the mixture equals V the density of the ist component reads also as density of component i is
i
i
M , V
(2.2)
fi
Applying (2.1) to the mixture with joint specific gas constant Rmix and (weighed harmonic) mean molecular mass m , we get
p
R* T m
Rmix T
(2.3)
after some algebra that
m
1 fi mi
(2.4)
and
Rmix
R* m
(2.5)
In this way we find for dry air that the mean molecular mass, here denoted as md , and the joint specific gas constant, here denoted as md and Rd are md
28.976 kg kmol-1
and Rd = 287 J kg-1 K-1. Air can be considered as a mixture of dry air and water vapor. In the next we will denote the joint specific gas constant of air (so inclusive water vapor) simply with R and its mean molecular mass with m. Because the water vapor concentration is variable R and m are dependent of the water vapor content. For many practical applications these variations can be ignored. In this course we often will take R and m constant, i.e. R = 290 J kg-1 K-1 and m = 29 kg kmol-1. See also the exercise at the end of this chapter. For water (vapor) the specific gas constant is 462 J kg-1K-1, because its molecular mass is about 18 kg kmol-1.
2.3 Hydrostatic equilibrium If the atmosphere is at rest (no wind) the air pressure at a level z is determine by the weight of the air that is present above this level. Inevitably, the air pressure decreases with height. The decrease of the pressure, -dp, when z is increasing with dz is given by
dp
g ( z )dz
(2.6)
26
in which g = 9.81 m s-2 is the acceleration of the gravity, and (z) the air density at level z. With equation (2.1) this leads to
dp p
g dz R T
d ln( p)
(2.7)
For an isothermal atmosphere (in which T is constant with height)¸equation (2.7) yields: p( z )
p(0)e
gz RT
(2.8)
where p(0) is the pressure at level z = 0. Equations (2.7) and (2.8) are valid for non-moving air, however, it appears that under most real-life conditions they are very good approximations. Only if horizontal accelerations are of the same order as g equations (2.7) and (2.8) are violated. This can occurs in thunderstorms of when air is forced to move against steep slopes or buildings.
2.4 Atmospheric thermodynamics The specific heat of air at constant volume cv and the specific heat at constant pressure cp are related as follows: cp - cv = R
(2.9)
Strictly speaking cv and cp depend on water vapor concentration. In most practical applications we can use the 'dry-air' values which are: cv = 718 J kg-1K-1 and cp = 1005 J kg-1K-1. The first law of thermodynamics, which is the conservation of energy, applied to an air parcel to which an amount Q is added yields: Q
cv dT
pd
in which = 1/ is the specific volume. The reader is invited to show that this with equation (2.9) this can be written as
Q
c p dT
dp
Processes for which Q = 0 are denoted as adiabatic. It appears that motions of air parcels are approximately adiabatic, as long as water vapor does not change phase. This is called a dry- adiabatic process. Then, according to equation then
c p dT
dp
27 Combining equation
dT T
R dp cp p
with the equation of state (2.13)
Because in general dx/x = dln(x) and a dln(x) = dln(xa), equation (2.13) can be integrated from reference level (denoted with index o) to level z. Next, we denote the temperature at reference level by which quantity is denoted as potential temperature. This definition of together with integration of equation (2.13) gives
T
p0 p
R cp
.
(2.14)
Combination of equation and the hydrostatic equation (2.7) yields that the change of air temperature with height in an atmosphere in adiabatic equilibrium equals: dT dz
g cp
(2.15)
So, if an air parcel is lifted vertically in an adiabatic motion its temperature decrease 9.8 K per km. Note that the 'energy content' of this parcel is unchanged. So, the 'normal' temperature T is not a proper 'indicator' for this energy content. On the other hand, does not change if for the vertical adiabatic motion. Consequently, we will use rather than T if we consider the thermodynamics of the atmosphere. For instance, vertical gradient of will cause a vertical heat flux, whereas according to (2.15) if the vertical temperature gradient equals -g/cp there is no heat flux, because the atmosphere is assumed to be in adiabatic equilibrium. dT In meteorology it is common to denote as the lapse rate. When dz < 0 it is said that there is an inversion. Using the mean lapse rate as indicator the atmosphere can be divided in different layers. This is shown in Figure 2.1 where the mean mid-latitude temperature is depicted as function of height. In this figure the names of the different layers are given also. It is seen that in the troposphere is positive, i.e. the mean temperature decrease with height. In the stratosphere < 0, so there is an inversion. It will be shown later that in such a layer vertical motions are suppressed.. The lapse rate in the mesosphere is positive again, whereas it is negative in the thermosphere. The tropopause separates the troposphere and stratosphere. Its height depends of latitude and season. Above the tropics it can reach a height of almost 20 km, whereas in at the poles in winter it is about at 7 km. Above the mid-latitude the north-south change of height of the tropopause is relatively fast, in particular in the winter hemisphere. Most features determining weather and climate occur in the troposphere. The tropopause can be considered a 'lid" on the atmosphere as far these features are concerned, however, it must be stressed that interaction between the
28 troposphere and stratosphere across the tropopause are important for several atmospheric phenomena. This is outside the scope of this course. It appears to make sense to divide the troposphere in sub-layers also (see Figure 2.2). This is due to the fact that there is a layer above the surface that directly 'feels' the presence of the earth's surface. This layer is called atmospheric boundary layer (ABL) or - in some textbooks - the planetary boundary layer (PBL). A main property of the PBL is that PBL flows, usually, are turbulent. Another characteristics is the PBL temperature show a clear diurnal variation. It has been shown above that close to the ground the vertical gradients of temperature, humidity and wind speed are the largest in, say, the first 10 to 20 m above the surface. This layer is often denoted as atmospheric surface layer or more briefly surface layer. A more detailed inspection of physical processes near and just above vegetated surfaces justifies the introduction of the so-called friction sub-layer. See Figure 2.3.
Figure 2.1 Vertical structure of the entire atmosphere
29
Figure 2.2 Schematic layers within the toposphere
Figure 2.3 Layers close to the ground
30
2.5 Water vapor Atmospheric water can appear in three possible phases: the solid, the liquid and the gaseous phase. This is due the fact that the triple point of water is within the temperature range occurring in the atmosphere as a result of the distance earth-sun and the natural greenhouse effect in which water substances play an important role. Note that for instance at Mars the greenhouse effect is so small that the temperatures there are well below freezing point, by which the liquid phase is either absent or very rare. The gaseous form of water is called water vapor or humidity.
2.5.1 Thermal effects of phase changes Phase changes are accompanied with changes in latent heat. This reflects on the thermal regime in the atmosphere. This is illustrated by the following example. It is assumed that in a sample of moist air (a mixture of dry air and water vapor) of 1 kg, 2 gram of water vapor condensates. Since the latent heat of vaporization Lv is about 2.5 106 J kg-1, an amount of heat of 0.002. 2.5 106 = 5000 J is released. It is assumed next that all this heat is used to increase the temperature of the air sample at constant pressure. Because the specific heat at constant pressure of air is approximately cp = 1000 J kg-1 K-1 and the mass of the air sample still very close to 1 kg, the temperature rise is about 5 K.
2.5.2 Water vapor variables It appears that even in the 21st century it is difficult to measure the water vapor concentration. There exist a number of different water vapor variables which are a measure for the water vapor concentration. It appears that different measuring techniques require different water vapor variables. Moreover, a number of water vapor variables are introduced for theoretical reasons. a. Absolute humidity (symbol v units kg m -3). The most straightforward way to quote the amount of water vapor is the water vapor density that is denoted as absolute humidity. Optical absorption methods allows the direct measure directly v, e.g. with an infrared or with an ultraviolet hygrometer. b. Water vapor pressure (symbol ea units N m-2, hPA or mbar) This is the partial pressure of water vapor. In most textbooks it is denoted as e, but because we will use computer algebra where e is reserved for the base of the natural logarithm, we will use here ea. Applying the equation of state to the water vapor component of air, we obtain. ea
v
RvT
(2.16)
in which Rv is the specific gas constant of water vapor. Since the molecular mass of water (H2O) is about 18, Rv = 462 J kg-1 K-1, see above. Although ea cannot be measured directly is is often used in theories for evaporation. c. Mixing ratio and specific humidity (symbol r and q, dimensionless) If an air parcel is lifted towards a higher level, inevitably its volume increase due to expansion. Moreover, its pressure drops. This implies that for vertical motions neither the absolute humidity nor the water vapor pressure remain constant. On the other hand the mass fraction, e.g. the ratio of the mass of the water vapor over the total mass of
31 the air parcel will not change. Because the various components of air occupy the same volume, a mass ratio is a ratio of densities also. Based on these thoughts, two water variables that are almost identical can be defined, notably The mixing ratio defined by v
r
(2.17)
d
where
d
is the density of the dry air component.
The specific humidity is defined by q
v
(2.18)
So, the difference between r and q is that r is the mass fraction of water vapor with respect to the mass dry air and q with respect to the total mass of the air sample. Note that v
(2.19)
d
This leads to
q
r
(2.20)
r 1
Because both r and q often are much less than 1, this means that for many practical applications q r . According to the law of Dalton, the partial pressure of dry air is the total pressure minus the water vapor pressure. Applying the equation of state (2.1) to dry air and to water vapor as well yields after some algebra:
r
Rd e a Rv p ea
(2.21)
The ratio of the specific gas constant for dry air and water vapor is an important number in meteorology. It equals 0.621 and is close to 5/8. Using equation (2.20) and Rd ea usually is 2 2.5 3 3.5 4 4.5 5 40 26 11 44 30 16 3 47 34 21 9 50 38 26 14 2 53 41 30 18 7 55 44 33 23 12 2 58 47 37 27 17 7 60 50 40 31 21 12 2 62 52 43 34 25 16 7 63 55 46 37 29 20 12 65 57 48 40 32 24 16 ew 0.5 1 1.5 2 2.5 3 3.5 4 5 6 7 8 6.11 91 82 74 65 57 48 40 32 16 6.57 92 83 75 67 59 51 43 35 20 5 7.06 92 84 76 68 60 53 45 38 23 9 7.58 92 85 77 70 62 55 48 41 27 13 8.13 93 85 78 71 64 57 50 43 30 17 4 8.72 93 86 79 72 65 59 52 45 33 20 8 9.35 93 86 80 73 67 60 54 48 35 24 12 1 10.02 93 87 80 74 68 62 56 50 38 27 15 5 10.73 94 87 81 75 69 63 57 51 40 29 19 8 11.48 94 88 82 76 70 64 59 53 42 32 22 12 12.28 94 88 82 77 71 66 60 55 44 34 24 15 13.12 94 89 83 77 72 67 62 56 46 37 27 18 14.02 94 89 84 78 73 68 63 58 48 39 30 21 14.97 95 89 84 79 74 69 64 59 50 41 32 23 15.98 95 90 85 80 75 70 65 60 51 42 34 26 17.05 95 90 85 80 75 71 66 62 53 44 36 28 ew 1 2 3 4 5 6 7 8 10 12 15 20 17.05 90 80 71 62 53 44 36 28 13 18.18 90 81 72 63 54 46 38 30 15 1 19.37 90 81 72 64 55 47 40 32 18 4 20.64 91 82 73 65 57 49 41 34 20 7 21.97 91 82 74 66 58 50 43 36 22 10 23.38 91 83 74 66 59 51 44 37 24 12 24.86 91 83 75 67 60 53 46 39 26 14 26.43 92 83 76 68 61 54 47 40 28 16 0 28.09 92 84 76 69 62 55 48 42 30 18 3 29.83 92 84 77 69 63 56 49 43 31 20 5 31.67 92 85 77 70 63 57 50 44 33 22 7 33.61 92 85 78 71 64 58 52 46 34 24 9 35.65 92 85 78 71 65 59 53 47 36 25 11 37.79 93 85 79 72 66 59 53 48 37 27 13 40.05 93 86 79 72 66 60 54 49 38 28 15 42.42 93 86 79 73 67 61 55 50 39 30 17 44.92 93 86 80 73 67 62 56 51 41 31 18 47.54 93 86 80 74 68 62 57 52 42 32 20 1 50.29 93 87 80 74 69 63 58 52 43 33 21 3 53.18 93 87 81 75 69 64 58 53 44 35 22 5 56.21 93 87 81 75 70 64 59 54 44 36 24 7 59.4 94 87 81 76 70 65 60 55 45 37 25 8 62.73 94 87 82 76 70 65 60 55 46 38 26 9 66.23 94 88 82 76 71 66 61 56 47 39 27 11 69.9 94 88 82 77 71 66 61 57 48 39 28 12 73.74 94 88 82 77 72 67 62 57 48 40 29 13 es 2.86 3.09 3.34 3.61 3.90 4.21 4.54 4.90 5.27 5.68 6.11
es-ei 0.26 0.26 0.25 0.24 0.22 0.20 0.17 0.14 0.10 0.05 0
> d This case refers to solar radiation and air molecules. According to Rayleigh's law the scattering into a particular direction, e.g. perpendicular to the solar rays, is proportional to 1/ . In the visible, blue is scattered much more than red solar radiation. Among other features, this explains why clean, clear atmospheres are blue and the reddish color of the sun when it sets. Law of Mie; ≈ d The law of Mie determines scattering properties when the diameter of the aerosols or cloud particles is about equal to . On of the features of Mie's law is that scattering is almost independent of . As a consequence, the scattered light is white. An example of the effect of particle size on the color of scattered light is the smoke of cigarettes. The particles produces directly by the burning process are very small and scattered light is blue. However, after inhalation the particles grow due to the high humidity in the human lungs. There they grow to 'Mie-sizes' and exhaled smoke scatters visible light white. Similarly, hygroscopic aerosols with 'Rayleigh-size' can grow to 'Miesize' when the humidity of the atmosphere increases. Their appearance becomes whiter. Laws of geometric optics: 0 dz and w'T ' < 0), the flux Richardson number is positive, i.e. Rf > 0. In that case u g is the only TKE production term, because now u ' w' w'T ' tends to destroy z T TKE. u g Furthermore, it is said that if R f > w'T ' z T turbulence is primarily generated mechanically, the conditions are (near) neutral. This is all a matter of definition. Even nowadays the flux Richardson number is difficult to measure and in numerical weather forecast models there is a need to express physical quantities in term of state variables such as and u. For those reasons the gradient Richardson number has been introduce defined by
Ri
g T
z u z
2
.
(5.31)
It is noted that according to the K-theory given by equations(5.12) and (5.13) Rf
Kh Ri . Km
Moreover, experiments has revealed that under unstable conditions
(5.32)
82
z L
(L < 0)
Ri
(5.33)
and that for some stable conditions
z L with
Ri or Ri 1 5 Ri
z and 0 L
1 5
(0
0.5) expression (5.34) does not hold.
5.8.1 The Obukhov length Obukhov considered the unstable surface layer and remarked about the flux Richardson number Rf the following. Since within the surface layer the covariances u decreases rapidly with w'T ' and u ' w' are approximately constant with height, but z height, R f is increasing with height also. That means that buoyancy production
u g w'T ' increases with respect to the mechanical term u ' w' . This is an important z T feature, further away from the surface buoyancy becomes more important as TKE u production term, whereas the significance of u ' w' declines when z increases. z g u u ' w' , This implies also that it inevitable that there will be a level where w' T ' z T i.e. where the two production terms are equal. Then Rf = -1. Obukhov argued that this level, that will vary with environmental conditions, because it u depends on w'T ' , u ' w' and is a proper length scale to characterize the turbulent z state of the surface layer. This is the physical background of the Obukhov length introduced earlier through expression (5.27), i.e. L
T u*2 g *
(5.27)
Unfortunately, according to (5.27), L is negative under unstable conditions, whereas at level -L the flux Richardson number is close to, not exactly, -1. In one of the exercises we will come back to this feature. The point we want to make here is that, although the Obukhov length has been introduced on physical rationale, its final definition through equation (5.27), is from a pedagogical point of view not very fortunate. It will
83 be seen later that we can consider L also as a length scale characterizing the turbulence in the surface layer that has to be compared to the other length scale z. In that picture L is a rather arbitrary combination of physical quantities that play a role in vertical transfer processes in the surface layer.
5.9 Exercises 1. Show that R f z 2. Show that L
Kh Ri . Kf U 3f u*3
.
z and Tf and *. L 4. We consider an air parcel density p and temperature Tp in surrounding air with density and absolute temperature T. Show that the net upward force per unit of mass of surrounding air acting on this parcel due to a) gravity and b) to ' 'Archimedes' is given by g with ' p. 3. Derive a similar relation between
5. The parcel in 4 has a vertical velocity of w'. Show that the net upward force (per unit air mass) performs specific (per unit mass) power is approximately w' T ' . (Power is performed work per unit time). g T
84
Chapter 6 Fluxes in the surface layer and similarity theory 6.1. Introduction The section deals with relationships between 'measurable state variables' -in first instance, the vertical gradient of the time averaged (potential) air temperature and horizontal wind speed- and vertical fluxes, in first instance, the vertical fluxes of sensible heat (density) and momentum. Since, these relationships cannot be derived from first physical principles, these relationships are based on dimensional-analysis. We first summarize the different steps to be made in dimensional-analysis: 1. 2. 3. 4.
Select which variables are relevant for the problem; Form dimensionless groups of these variables; Perform field experiments under a wide range of environmental conditions; Fit empirical relationships between the measured values of dimensionless groups. 5. If one can form only one dimensionless group then this group must be constant. The theory does not give the mathematical form of these empirical relationships. That means that there is a certain freedom in choosing the mathematical functions.
6.2. Surface layer or constant-flux layer We confine ourselves to flat and horizontally homogeneous terrain and stationary conditions. The latter means that the statistical properties of the turbulent flow do not vary in time. Moreover, it is assumed that surface heating and wind shear cause the vertical flux of heat and momentum. 'External non-local effects" are negligible. This means that we confine ourselves to z m z0, where m is a constant that is about 10. So, finally, we will consider the flow in a relatively thin layer denoted as the surface layer that is defined by: 10 z0 < z < 0.1 h.
(6.1)
Experiments (see the evidence shown in Chapter 1) show that this surface layer is so thin that the change with height of the vertical fluxes across it can be neglected compared to the magnitude of the fluxes itself. Therefore, the surface layer is often called also the constant flux layer.
85 Finally, it is noted that within the surface layer vertical turbulent transfer of heat and momentum is expected to be due primarily to the 'exchange' of eddies with a typical size of the order z. This means that in the surface layer z is a significant physical parameter in the dimensional-analysis that will be applied in the next sections. Therefore z is one of the 'relevant' variables. This is a crucial feature of turbulent flows near the surface.
6.3. Free convection We start with the simple case that conditions are very unstable when the wind speed is very low and surface heating is very large. In that limit we can ignore wind shear and the flux of momentum, and that there is surface heating only. This case is denoted as 'free-convection'. Free-convection conditions occur at low wind speeds and high solar radiation, so these are confined to daytime. The surface heating will create temperature differences and thus density differences in the gravity field of the earth, resulting in "Archimedes forces". A force per unit of mass is (according to Newton's law) an acceleration. An air parcel that has a temperature that differs T' (K) from the surrounding air with absolute temperature T, T' experiences a vertical acceleration (= force per unit mass) of g (the "Archimedes" T acceleration). Note that we have used the approximation (5.2). Let the vertical velocity of the parcel be w', then the work done by the Archimedes force per unit of mass and unit of time is (vertical path per second multiplied by the w' T ' acceleration) g . It is seen that we have presented here a sort of derivation for the T g buoyancy production term w'T ' in the TKE budget equation (5.29). The term T g w'T ' can be regarded as the only driving force of the turbulent ‘free-convective' T g surface layer. In this way we arrive at the conclusion that , w'T ' and z (as argued T above) are 'relevant' variables for free convection.
6.3.1 Free convection flux-profile relationship For free-convection case we are seeking for a relation between the vertical gradient of and w'T ' , which is a direct measure for the z sensible heat flux density. For this problem we can form one (independent) dimensionless group only: the mean potential temperature
w' T ' z2
g T
3 2
.
z
Since only one dimensionless group ca be formed , inevitably, this group is constant (see step 5 in the introduction).
86
So, we obtain:
C f z2
w' T '
3 2
g T
,
z
(6.2)
with Cf is a constant. In literature a value of e.g. Cf = 1.15 can be found. This results leads to the following simple expression for the sensible heat flux density:
c pC f z 2
H
3 2
g T
,
z
(6.3)
With the constant flux assumption, i.e. H is constant with z, we can integrated this expression and we arrives at: c pC f H 3 z1
g T 1 3
1
z2
2
1 3
3 2
(6.4)
3 2
6.3.2 Free-convection relationship between H and In a similar way one can derive a relationship between the standard deviation of temperature, T, and w'T ' . Turbulent motion in the free-convective surface layer will lead to temperature fluctuations. It is to be expected that T increases when the surface heating, thus w'T ' , will increase. Now the only dimensionless group that can be formed is: w' T ' gz T
T
3 2
,
which leads to the simple expression:
H
c p C fT
g z T
3 2 T
Similarly, we find for the vertical wind speed
H
c p C fw
2 w
g z T
(6.5) w:
(6.6)
87 where CfT and Cfw are empirical constants.
6.4. Neutral conditions Now we consider the case where turbulence is generated mainly by wind shear and surface heating can be ignored. The result of the wind shear is the momentum flux , which is the friction force per unit area exerted by the flow on the surface. It is recalled that in micrometeorology it is common to define the friction by: u*
, so
u* is a measure of . We want to related u* to the gradient of the mean horizontal wind u u velocity . The relevant variables for this case are now u*, z and . Again one z z dimensionless group can be formed that has to be constant and we find:
u*
u z
z
(6.7)
where is a constant denoted as the von Karman constant. Its numerical value is still under debate, but here we will apply the (rounded) value of = 0.4. Note that (6.7) with equation (5.13) leads to Km
u* z .
(6.8)
Next, we allow some surface heating, but in such a way that wind shear is still the main generator of turbulence (buoyancy production is assumed to be much smaller). Moreover, we expect that under neutral conditions the same ‘eddies’ transport heat and momentum. Then, it is reasonable to assume that the exchange coefficients for heat and momentum are the same, i.e. Km
Kh
u* z .
(6.9)
This leads to an estimate for the sensible heat flux density under near-neutral conditions:
H
c p u* z
H
cp z
2
z
, which with (6.7) reads as
u . z z
(6.10)
In integrated form the expressions found above can be written as:
u*
u1 u 2 z ln 1 z2
(6.11)
88 and
H
cp
2
u1 u 2
z z ln 1 ln 1 z2 z2
1
2
,
(6.12)
where indices 1 and 2 refer to levels z1 and z2 respectively, so, for instance u1 is the wind speed at z1. Equation (6.11) is applied often with the lowest level taken at the so-called surface roughness length z0 for momentum, i.e. the level at which the wind speed extrapolated towards the surface is (apparently) zero. Then (6.11) reads as: u*
u z ln z0
(6.13)
where u is measured at level z. The roughness length can be regarded as a surface property (although is also a property of the flow). The value of z0 varies between 0.0002 m for very smooth surfaces with a fetch of at least 5 km (open sea), to 0.5 mm for snow and small water bodies, 3 cm for grass and short vegetation, 10 cm for low vegetation (higher than grass), 0.5 m for parkland, and about 1 m for a forest. Over tall vegetation, it i difficult to fulfill the requiring z >> 10 z0 and another semiempirical length scale is introduced, the so-called zero-plane displacement, d through:
u*
u z d ln z0
.
(6.14)
The idea is that eddies determining effectively the vertical fluxes have not a size that scales with z but with (z-d). For crops a rule of thumb is that z0 is about 1/7 and d is about 2/3 of the of the vegetation height. Note that the actual values of these quantities depend very much on the 'architecture' and 'structure' of the vegetation. So, they have to be applied with caution, especially for sparse vegetation.
6.5 Monin-Obukhov similarity theory (MOST) Above we discussed two ‘limit’ cases, notably the free-convection case, where buoyancy production is dominant and the neutral case, where wind shear production is much greater than buoyancy. Now we consider the general case that both production terms are playing a role and we follow the similarity arguments used by Monin and Obukhov.
89
g , w'T ' , u* and z, from which T we now can two independent length scales can be formed, notably the Obukhov length L and z, where L is introduced in equation (5.27). We recall that In this general case, the list of 'relevant' parameters is
T u*2 g *
L
(6.15)
z can be formed which is a L dimensionless height. According to similarity theory all second dimensionless groups z that can be formed must be a function of . This is the essential of the MoninL Obukhov similarity theory (MOST). From these two length scales a dimensionless group
6.5.1 MOST flux-profile relationship According to the rationale of MOST the dimensionless temperature and wind speed gradients can be written as z z
h *
z L
(6.16)
and the dimensionless wind speed gradients as u z z u*
in which
m
h
z L
z and L
(6.17)
m
z L
are universal functions of
introduced because experiments reveal that then at
z . Note that the constant L
is
z = 0 these universal MOST L
functions are 1, so h 0 = m 0 = 1. Several field experiments have been carried out to determine the universal z z functions h and m . Moreover, Businger (1980) presented a 'heuristic' L L derivation of the MOST functions. This appears to be consistent with experimental data, however, there is still no unanimity on the final mathematical form of the functions. In this course we will adopt the so-called Businger-Dyer expressions that reads: for unstable conditions ( L < 0)
90
h
z L
m
z L
2
z 1 16 L
1 2
(6.18)
and for stable situations, i.e. L > 0
h
z = L
m
z z = 1 5 L L
Note that (6.19) is confined to the stability region 0
0)
Ri 1 5 Ri
z L
Ri
1 5
(6.23)
z . L
, with
(6.24)
6.5.2 Integrated MOST functions Because it is rather difficult to measure vertical gradients accurately, it is convenient to integrate the -functions between two levels, by which only observations of and u at these levels are required. It is common to introduce for this purpose
x
z L
z L 0
1
x
'
'
d ' , with index x = h or m.
(6.25)
This leads (after some algebra) to the following general expressions for H and u*:
91
cp
H z ln u1 zu 2
m
zu1 L
2
u1 u2
m
zu 2 L
m
zu 2 L
1
2
z ln 1 z2
(6.26) h
z1 L
h
z2 L
and
u1 u 2
u*
z ln u1 zu 2
m
z u1 L
(6.27)
In these expressions is measured at levels z 1 and z 2, which might be different from the levels at which u is observed (zu1 and zu2). The levels with index 1 are the lowest. It can be shown that (6.18)and (6.19) lead to the following:
a. Unstable case ( L < 0 ):
h
z L
1 x2 2 ln , with x 2
m
z L
2 ln
1 x 2
ln
1 x2 2
z 1 L
1
4
2 arctan( x)
(6.28)
(6.29)
2
b. Stable case ( L > 0 ): z L
h
m
z L
5
z L
Note that (6.30) is confined to the stability region 0
0) it is found that T
2
(6.35)
1.5
(6.36)
*
and w
u*
6.5.4 Note on MOST and the neutral and free convection limit* The results obtained for the semi-empirical MOST functions found in the previous sections obey the conditions that they transform into the expressions found for neutral z situations. So, for 0 the MOST functions are equal to the functions found for L neutral conditions. For still unknown reasons, in the free convective situation this does not apply to the MOST flux-profile functions. Confining ourselves to sensible heat it can be shown that h fulfils the free-convection results only if in the limit the expression
1 2 h
3 3
, with ,
z / L becomes constant (show*), so for 1
z / L 3 . This is large values one expects that h becomes proportional to not what has been found experimentally. Expression (6.18) yields for large 1
z / L 2 . This might be due to the fact that the values h proportional to experimental data refer to a limited range. Recent findings appear to substantiate this. Note that the relationships found for the standard deviations do obey the freeconvection limit requirements. That means that for large values (6.33) yields formula (6.5)
93
6.5.5 Fluxes of water vapor In the above we discussed temperature and wind speed only. This has been done because gradients of horizontal wind speed and potential temperature can be considered as the ‘driving forces’ of turbulence. In this section we will first deal with water vapor. The vertical flux density of water vapor, E, is called evaporation also. (its units are kg m-2 s-1). This naming is due to the fact that in the surface layer E is ‘driven’ by evaporation of liquid water at the surface. This can be evaporation of open water, of rainwater on plants, soil moisture or water present in plants. In the latter case the actual evaporation process takes place in the cavities of the stomata in the plant leaves. Note that it is common to denote the energy flux density related to the flux density of water vapor, i.e. the quantity LvE as the flux density of latent heat. Its units are W m -2. This is due to the fact that water vapor in the atmosphere is a latent heat source: when water vapor condensates heat is released. This feature is very important for many meteorological processes. The release of heat due to condensation of water vapor is one of the most important energy sources for e.g. thunderstorms, cyclones and tornados. Note that in the surface layer the actual evaporation process take place at the surface, so Lv , which is slightly temperature dependent, has to be taken at surface temperature. For practical reason we will take Lv at air temperature. The error we will make is negligibly small in most cases. Strictly speaking, the similarity theories we presented above are valid only for dry air. For moist air one has to account for the density effects of water vapor, in particular there where buoyancy was discussed. In meteorology it is common to account for water vapor buoyancy effects by replacing the ‘normal’ or potential temperature with the virtual temperature, Tv, or potential virtual temperature, v. Moreover, we have to replace w' ' T with w' Tv . Because Tv T (1 0.61q) , w' Tv depends on both w' T and w' q ' . It can be shown* that approximately:
w' Tv '
w' T ' 0.61T w' q' .
(6.37)
As a consequence the adapted Obukhov length reads as: L
T g
u*2 *
0.61 T q*
.
(6.38)
It will be clear that in cases where the sensible heat flux is small compared to the latent heat flux, the correction cannot be ignored. Approximately, the water vapor flux correction is L
Ld 1
0.07
,
in which Ld is the Obukhov length without the correction, so Ld Bowen ratio, i.e. the ratio of the sensible over the latent heat flux.:
(6.39) T u*2 , and g *
the
94 c p w' T ' Lv w' q'
H Lv E
(6.40)
Note that under unstable conditions L is negative and that its absolute value decreases when increases. For simplicity, we will ignore the water vapor effects on L in most cases.
6.6 T-q-qx similarity and fluxes of other gasses Experience has shown that under 'MOST' conditions temperature, specific humidity and specific concentrations of other gasses behave 'similar'. That means that all MOST relationships found above for T can be applied in a similar way to It is said that T, q and qx (the 'specific' density of gas x). This T-q-qx similarity leads to equality of the turbulent exchange coefficients for temperature, water vapor, CO2 and other gasses: Kh
Ke
(6.41)
Kx
In the same way; h
e
(6.42)
x
This implies that
h
z L
h
z = L
e
z L
x
z L
z 1 16 L
e
z = L
x
z z = 1 5 L L
1 2
(L < 0)
(L >0. 0
(6.43)
z L
0.5 )
(6.44)
Of course, then the integrated -functions (6.28)-(6.30) found above for temperature and sensible heat applies to q and qx also:
h
z L
e
z L
x
z for all L. L
(6.45)
It should be noted that the T-q and the T-qx similarity is not generally applicable. It is still a subject for research.
6.7 Aerodynamic resistances It is common in micrometeorology to write the integrated flux-profile relationships in terms of a resistance defined in section Turbulence see e.g. (5.20). The above results yield MOST expressions for these resistances.
95 For momentum we find in this way (with z1 > z2):
rm
z ln 1 z2
m
z1 L
m
z ln 1 z2
z2 L
m 2
u*
z1 L u1
m
2
z2 L
(6.46)
u2
and for the fluxes of sensible heat , water vapor, carbon dioxide etc.:
ln rh
z1 z2
m
z1 L
m
z2 L 2
ln
z1 z2
h
z1 L
h
z2 L
(6.47)
u1 u 2
It is common to call rh the aerodynamic resistance for sensible heat flux. Because of T-q-qx similarity rh = re = rx. In the following we not distinguish any longer between rh , re or rx and we will use simply one resistance, denoted as ra that will be called the aerodynamic resistance. So:
ln ra
z1 z2
m
z1 L
m
z2 L 2
ln u2
u1
z1 z2
h
z1 L
h
z2 L
.
(6.48)
96
Table 6.1 MOST functions z/L(0) 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
m,
m
m
0.0040 0.0079 0.0118 0.0157 0.0195 0.0233 0.0271 0.0308 0.0345 0.0381 0.0731 0.1054 0.1354 0.1636 0.1902 0.2153 0.2392 0.2619 0.2836 0.4613 0.5945 0.7023 0.7934 0.8725 0.9427 1.0059 1.0634 1.1162 1.4947 1.7391
m
m
1.0050 1.0100 1.0150 1.0200 1.0250 1.0300 1.0350 1.0400 1.0450 1.0500 1.1000 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000 5.0000 5.5000 6.0000
-0.0050 -0.0100 -0.0150 -0.0200 -0.0250 -0.0300 -0.0350 -0.0400 -0.0450 -0.0500 -0.1000 -0.1500 -0.2000 -0.2500 -0.3000 -0.3500 -0.4000 -0.4500 -0.5000 -1.0000 -1.5000 -2.0000 -2.5000 -3.0000 -3.5000 -4.0000 -4.5000 -5.0000
,
m
and
m
as function of z/L. h
h 0.99209 0.98437 0.97683 0.96946 0.96225 0.95520 0.94830 0.94155 0.93495 0.92848 0.87039 0.82199 0.78087 0.74536 0.71429 0.68680 0.66227 0.64018 0.62017 0.48795 0.41523 0.36761 0.33333 0.30715 0.28630 0.26919 0.25482 0.24254 0.17408 0.14286 h 1.0050 1.0100 1.0150 1.0200 1.0250 1.0300 1.0350 1.0400 1.0450 1.0500 1.1000 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000 5.0000 5.5000 6.0000
0.0080 0.0158 0.0236 0.0313 0.0389 0.0464 0.0538 0.0611 0.0684 0.0756 0.1436 0.2056 0.2626 0.3154 0.3646 0.4108 0.4542 0.4953 0.5343 0.8436 1.0661 1.2413 1.3863 1.5103 1.6187 1.7151 1.8021 1.8812 2.4312 2.7726 h
-0.0050 -0.0100 -0.0150 -0.0200 -0.0250 -0.0300 -0.0350 -0.0400 -0.0450 -0.0500 -0.1000 -0.1500 -0.2000 -0.2500 -0.3000 -0.3500 -0.4000 -0.4500 -0.5000 -1.0000 -1.5000 -2.0000 -2.5000 -3.0000 -3.5000 -4.0000 -4.5000 -5.0000
97
6.8 Exercises 1. Obukhov introduced the length scale L, now called Obukhov length, as the height where the buoyancy production rate of TKE is equal to the mechanically generated TKE. He estimated that this height is -L (per definition L is negative under unstable conditions for which the two TKE production terms are positive). Show that Obukhov's estimate is not exact. Hint:
z L
First,
show 1 4
z 1 16 L
m
that
z L
m
Rf .
Next,
numerically
solve
equation
for
-z/L.
2. In literature the bulk-Richardson number is introduced, defined by: g z g z 0 0 RiB z z z , in which (z) and 0 are the 2 T uz T uz 2 potential temperature at heights z and z0. Show that 2
z L
RiB
z ln z0 ln
m
z z0
h
z L
for unstable conditions. Note that
z L
h ,m
z0 L
can be ignored in most cases. RiB z z ln And for stable cases: . L 1 5 RiB z0 3. Calculate Km and Kh as function of height for different values for L at u* = 0.25. For example for L = -100000, -10000, -1000, -100, -10, -1, 1, 10, 100, 1000, 10000, 100000 at levels 0.1, 0.2, 0.3, 0.5, 0.7, 1, 2, 3, 5, 7, 10, 20 and 30 m. 4. How does Ri vary with height for the L values used in 3. 5. Derive Kh from the free convection expression (6.3). 6. Adopt the constant flux layer assumption. How does vary convection case given by expression (6.3)?
7. Show that H
cp
2
z2
3 2
z
g z T L
1 2 h
3 2
.
z
with z in the free
98
Chapter 7 How to measure or model vertical fluxes? 7.1 Introduction Using the results of chapter 5 and 6 we now can discuss methods to measure the vertical flux densities of momentum, sensible heat, water vapor, CO2 and other gasses. In the next we will discuss the different methods.
7.2 The eddy-correlation method We simply recall the results obtained in Chapter 5, in particular equations (5.5)-(5.8) that reads: H
E
Fx
c p w'T '
(7.1)
w'u '
(7.2) (7.3)
w'q'
w' q x '
in which q x
(7.4) x
.
Moreover, we repeat the remarks we made in Chapter 5 on the eddy correlation method:
The eddy-correlation method requires fast response sensors since every variation in the quantity to be measured needs to be tracked (even within a second the temperature can vary by one degree!). Since quantities in turbulent flows are so variable in time, one needs a large amount of samples to accurately determine mean values, variances and covariances (i.e. fluxes). Therefore, a commonly used sampling rate is 20 Hz (20 samples per second). The averaging time over which means, variances and covariances are determined is limited by the (in)stationarity of the conditions (e.g. the air temperature rises quickly in the morning, which makes the mean temperature over a long period of time a badly determined quantity. A commonly used averaging period is 10 min. The above requirements imply that we have to collect 12000 samples for each turbulent quantity. So for the four fluxes mentioned above we need fast response sensors for the 3 wind components u, v and w, temperature, specific humidity and CO2, with which at least 6 times 12000 = 72000 sample has to be collected each 10 min time intervals.
99 There are a large number of detailed remarks that can be made on the eddy-correlation approach, but this is outside the scope of this course.
7.3. Flux-profile relationships In general, the vertical fluxes can be obtained by measuring the potential temperature and horizontal wind speed to determine the flux density of sensible heat H and momentum . To measure evaporation specific humidity has to be observed at two levels also and, if one wants to measure the flux density of CO2 or other gasses the specific density qx also at two levels. In all cases the potential temperature and horizontal wind speed at two levels are needed because they are related to 'mechanically generated turbulence (wind shear) and 'buoyancy' (vertical gradients of ) that 'transfers' water vapor, CO2 etc. Consequently, H and (or u*) has to be determined anyway to evaluated L. So, we start with the set of governing equations from which H and u* and next L can be determined. . These are (6.26) and (6.27) which are repeated and renumbered here:
H cp
u*
2
z ln u1 zu 2
m
z u1 L
m
u1 zu 2 L
u2
1
z ln 1 z2
2
(7.5) h
z1 L
u1 u 2 z ln u1 zu 2
m
z u1 L
(7.6)
zu 2 L
m
h
z 2 L
and (5.27) defining L L
T u*2 g *
Furthermore
(7.7)
*
is defined by
*
w' T ' and H u*
w' T ' . cp
We recall that = 0.4, g = 9.8 m s-2, cp = 1005 J kg-1 K-1. Furthermore is air density, T mean absolute air temperature of the surface layer, zu1 and zu2 are the levels t which the horizontal wind speed is measured, similarly, z and z 2 the levels where potential temperature is observed. Furthermore, the functions h and m are given by equations (6.28) and (6.29) if at the lowest level z is greater than at z 2, i.e. under unstable conditions, and by (6.30) for stable conditions (if at the lowest level is less than that at the highest). We will not repeat these expressions here. The set of governing equations to determine H, u* and L cannot for unstable conditions be solved analytically due to the complex character of (6.28) and (6.29),
100 but, fortunately an iterative numerical method provides accurate approximate solutions. In this procedure the following steps are followed: 1. 2. 3. 4.
Start with a value of L for instance 106 m; Evaluated with (7.5) and (7.6) H (and thus *) and u*; Determine a new value of L with (7.7); Repeat this process until the values of H, u* and L do not change anymore using a certain criterion that determines the desired accuracy..
When one chose zu1 = z and zu2 = z 2 (next denotes as z1 and z2 respectively) and when z2/z1 is less about 6, it can be shown that the following analytic expressions yield fairly accurate approximate solutions for unstable conditions ( at the lowest level z is greater than at z 2): : 2
H cp
u*
L
ln
u 2
z2 z1
u
1 16 Rib
1 16 Rib
z ln 2 z1
3 4
(7.8)
1 4
(7.9)
z1 z 2
(7.10)
Rib
in which 2 measuring level.
1
(< 0 in this case) and
u
u2
u1 , indices 1 and 2 refer to the
and Rib is an 'effective' bulk-Richardson number, in this case given by:
Rib
z1 z 2 ln
z2 z1
g T
u
2
.
(7.11)
If zu1 = z and zu2 = z 2 there is an exact analytic solution for stable condition in virtue of the simple mathematical linear form of expressions (6.30). These are:
H cp
2
z ln 2 z1
u 2
1 5Ribs
2
(7.12)
101
u*
u
1 5Ribs
(7.13)
1 5Ribs Ribs
(7.14)
g T
(7.15)
z ln 2 z1
and
L
z2
z1
z ln 2 z1 where now
Rib
z2
z1
u
2
It should be noted that these results for the stable case are derived from equation (6.30) that is valid for a limited stability range. One has to check whether Ribs does not exceeds 0.2, because then (6.30) is not valid. If Ribs > 0.2 H and u* has to be taken equal to zero. This is due to the fact that when Ribs > 0.2 turbulence vanishes, by which there is no turbulent flux. After H, u* and next L have been determined the evaporation E can be calculated using 2
E z ln u1 zu 2
m
z u1 L
u1 zu 2 L
m
u 2 q1 q 2 z q1 ln z2
z q1 h
(7.16)
zq2 h
L
L
where zq1 and zq2 are the heights at which specific humidity q is measured. Similarly, the flux density of CO2 is obtained from 2
FCO2 ln
z u1 zu 2
m
z u1 L
u1
m
u 2 qCO2 1 zu 2 L
ln
qCO2 2
z q1 z2
z q1 h
L
zq2 h
L
where now zq1 and zq2 are the heights at which specific density qCO2
CO2
is
measured.
7.3.1 The free-convection approximation According to equation (6.4) that is repeated here, H can be found from only:
at 2 levels
102 g T
c pC f H 3 z1
1 3
1
z2
2
3 2
3 2
1 3
(7.17)
Experience shows that during daytime equation (7.17) is a good approximation if the MOST is considered to be the "truth", provided levels z1 and z2 are not too far from the surface. For instance if one uses z1 = 1 m and z2 = 2m.
7.4 The Bowen ratio method Above we have already introduced the Bowen ratio that is defined as the ratio of sensible and latent heat flux densities: c p w' T ' . Lv w' q'
H Lv E
(7.18)
Now we make use of the fact that for T and q the aerodynamic resistances are the same, so that we can write: cp Lv
(7.19)
q
that -using the relations between water vapor pressure and the definition of the psychometric constant can be rewritten as (7.20)
ea
in which
, q or ea) at levels z1 Rv c p p and z2, and the psychometric constant given by equation (2.30), i.e. Rd Lv and ea the water vapor pressure. x
x2
x1 , x1 and x 2 the mean value of x (either
A suitable method to measure ea is using a psychrometer consisting of a wetand dry-bulb thermometer at two levels, yielding T and Tw (the wet-bulb temperature). It can be shown that approximately
sw
Tw T
1
1
in which sw is the derivative of the saturation water vapor pressure at Tw. Next we apply the energy balance equation (1.2) that is repeated here
(7.21)
103 Q*
H
(7.22)
Lv E G
We recall that Q* is the net radiation and G is the soil heat flux density. Next, simple algebra then gives:
H
Q* G 1
(7.23)
Q* G 1
(7.24)
and
Lv E
It is seen that the Bowen ratio method requires observation of wet- and dry-bulb temperatures at two levels and measurements of Q* and G. The advantage is that nu wind speed observations are needed, but, the prize for this is that the method does not yield u*. The Bowen method is not applicable when the Bowen ratio is close to -1. Inevitably, then Q* - G is close to zero and expressions (7.23) and (7.24) are very inaccurately. It is noted that over irrigated crops surrounded by dry fields can be about -1. So, the Bowen ratio method is not recommended for irrigated crops in dry environmental conditions. If one wants to measure fluxes of other gasses in a 'Bowen-like' way, one can determine in additional 'mass Bowen ratio’ defined by
x
E Fx
w' q' w' q x'
q qx
(7.25)
In this way the flux density Fx can be obtained from E determined with (7.24).
7.5 The Penman-Monteith equation 7.5.1 Concept In order to give the reader an impression of the behavior of the various terms of the surface energy balance, in this module a simple Soil-Atmosphere-Vegetation Transfer (SVAT) model will be considered, which is the so-called big-leaf model based on the evaporation formula by Penman and Monteith. It applies to vegetation completely covering the ground. The essential simplification made in this approach is that the vegetation layer is replaced with a single big-leaf, in which stomatal cavities (stomata) are thought. Within these stomata the actual evaporation process takes place. The big-leaf has a temperature T0 and it is assumed and that there the air is saturated at T0. The big leaf is able to close the stomata in cases of water stress in the root zone. Closing the stomata will cause an increase of the surface resistance rs. If the stomata are closed entirely, rs is infinitely large. If there is no water stress and, furthermore, the vegetation is in optimal condition (no diseases etc.), rs has its
104 minimum value. For most arable crops the minimum value of rs is about 40 s m-1. For trees this amounts about 150 s m-1. In dry periods rs can become 500 s m-1 or higher. If the vegetation is wet rs = 0.
Figure 7.1 Conceptual picture of big-leaf model
7.5.2 The governing equations Close to the ground the difference between T and is small, and provided that one defines the potential temperature using the surface pressure as reference, then q is linked to the water vapor pressure ea. Furthermore, we apply the expression found for the aerodynamic resistance for the layer between 'standard' level z and 'the ground' so that we obtain the following equations: Q* Q*
H Lv E
H
Lv E G
(1
)K
cp
T0
(7.26) s
(L
T ra
c p es T0 ea ra rs
4 0
T )
(7.27) (7.28) (7.29)
105 In equation (7.29) is the psychrometer constant defined by equation (2.30) and is rs is the so-called surface resistance (or stomatal resistance). It has the same units as ra (i.e. s m-1) and it describes the effect that plants are able to close or open their stomata, the very small, cavities in the leaves where the actual transpiration process takes place. It was Monteith who introduced this approach in the equation that derived earlier by Penman for open water (for which rs = 0). For that reason this approach is called the Penman-Monteith method.
7.5.3 Numerical and analytic solutions In principle, this set of equations (7.26) - (7.29) can be solved numerically, also accounting for the fact that the aerodynamic resistance is stability dependent according to expression (6.48). Note that in this application in (6.48) the lowest wind speed level has to be chosen at the roughness length z0, that there the wind speed is zero, and the lowest level the so-called roughness length for heat z0h at which the actual temperature equals the surface temperature T0 used in (7.28) and (7.29). In the MATHCAD module a full numerical solution of the governing equations is presented. A comparison is made with the approximate analytic solution that is obtained below. An approximate analytic solution is obtained by assuming that the aerodynamic resistance is a function of wind speed only, and by applying the 'Taylor-expansion' approximation: ew (T0 )
ew (T ) s T0
(7.30)
T
Then, using (7.26) - (7.29) and some algebra, it can be shown that s Q* G Lv E s
cp ra
es T
r 1 s ra
ea
(7.31)
This is the well-known Penman-Monteith equation. We recall that Q* is the net radiation, E the evaporation, Lv is the latent heat of vaporization (which is about 2.5 106 J kg-1), ew T the saturation water vapor pressure at air temperature T at reference level, e is the water vapor pressure at reference level, s is the derivative of ew at T, the psychometric constant (at sea level and 20 0C about 0.66 hPa K-1), ra the aerodynamic resistance, rs the surface resistance, is the air density, and cp the -1 -1 specific heat of air at constant pressure (1005 J kg K ). Note that in equation (7.31) E is the transpiration, because it is assumed that the vegetation covers the ground completely, by which the bare soil evaporation can be ignored. In addition, the vegetation is assumed to be dry, so that there is no intercepted rainwater or dew.
106
7.5.4 Differences between grass, forest and open water The Penman-Monteith equation is very suitable to point out the different behavior of different surface types. As an example we will discuss here a typical grass field, a dense forest and open water. We will confine ourselves to typical summer and daytime conditions. a. Grass At optimal conditions (no water shortage, no diseases etc.) grass has a minimum values of rs of about 30 s m-1. Its albedo is typically 25%, and the soil heat flux is small (about 10% of net radiation). It is a rather 'smooth' surface, by which ra is relative large. b. Forest Under the same conditions as grass, forest has a relatively high rs of about 150 s m-1, provided the canopy is dry. Its albedo is smaller: about 15%, it is much rougher than grass, so its ra is smaller. Some authors found that ra = 10 s m -1 is a suitable value for most forests and environmental conditions. Its soil heat flux is small also, but the heat storage term in the canopy cannot be neglected if conditions change rapidly. If canopies are wet, it appears that wet forests lose water through evaporation much faster than grass. This is a very important feature in wintertime. Then E can be 30% of precipitation for a forest, whereas then grass has a very low E. c. Open water Due to turbulent motions in the water body, effectively a much thicker layer of water is involved in energy transfer processes near the surface than a vegetated land surface is. This means that the G term can be very large for open water. We will deal with this feature in a separate section. Moreover, water has a low albedo (about 6%), is aerodynamic 'smooth', so its ra is large (ignoring effects of waves) and rs = 0.
7.5.5 Operational application of the Penman-Monteith equation In operational practice in hydrology and irrigation the Penman-Monteith equations are used to determine the so-called crop water requirements of various agricultural crops in semi-arid regions where irrigation is needed. This is an empirical approach and it is outside the scope of this course to discuss it in details. Recently, the FAO (Food and Agricultural Organization of the UN) developed a method that briefly is as follows: Step 1 Determine the hypothetical quantity ETR, which is the outcome of the PenmanMonteith equation using weather data collected at a 'nearby' station for a 'standard' grass field optimally supplied with water. This standard grass has a prescribed ra, albedo and rs. Soil heat flux is estimate also in the latest version of the FAO method. Step2 The water requirements of a particular crop is obtained by multiplying ETR with a crop constant kc that depends on several external factors: ETcrop = kc ETR.
107 Further details will be presented by dr. Benoit Duchemin. In the past, in the Netherlands a slightly different approach was used by first calculating the 'open water evaporation', i.e. the outcome of the Penman equation using prescribed values for ra,, and albedo for water, and G =0 and rs =0). Nowadays, KNMI uses for ETR the much more simple Makkink formula:
ETR
0.65
s s
K ,
(7.32)
which appears to be good approximation of evaporation of a 'standard' grass field in the Netherlands in summertime.
7.5.6 Extrapolation towards the surface* As mentioned above in the Penman-Monteith approach one applies the MOST for the whole layer between the surface and level z. Tacitly, the "MOST profiles" of temperature, specific humidity and wind speed that are linked to the fluxes of momentum, heat and water vapor through MOST, have been extrapolated towards the surface. Because these profiles become logarithmic close to the ground this is not trivial. It is common to assign to the surface a "roughness length" for momentum, sensible heat, water vapor and other gasses. These are defined as that level zx where apparently the extrapolated MOST profile give a mean value of , u, q or qx that is equal to actual value at the surface value. First, this definition is rather vague because the surface values are not easy to define uniquely. Second, in practice the surface values are difficult to measure, except for wind speed for which we can assume that u = 0 at z = z0, the roughness length for momentum, discussed above. There is still a lot of uncertainties about the roughness lengths for heat. In many z meteorological models one uses a fixed ratio for ln 0 where z0h is the roughness z oh length for heat, i.e. the apparent height at which the extrapolated 'MOST T-profile' equals the surface temperature. In remote sensing applications, where the surface temperature is obtained e.g. from satellites, the value of z0h appears to be a very 'sensitive' quantity.
7.5.7 Sensitivity of LvE and surface temperature to wind speed 7.5.7.1 Introduction It is important to note that, in principle, the Penman-Monteith formula is a descriptive equation. This is due to the fat that it is to be expected that the input data op the PMformula are not independent of the evapotranspiration itself. In any case, this applies to the air temperature and water vapor pressure at reference height. As example we consider an oasis surrounded by a desert. The air over a desert is dry and hot, whereas over the oasis it is cool and humid. The evapotranspiration of the desert is zero and will be positive for the oasis. So, the second term of the PM-equation and LvE cannot be considered to be independent. Since the net radiation and soil heat flux both depend on the surface temperature, which in its turn will be related to LvE, Q* and G are expected to be not independent of LvE. As a consequence, strictly speaking, the
108 PM-equation is not suitable for sensitivity studies, because it does not account for feedback mechanism between LvE and the input variables. Nevertheless, the PMequation is suitable to reveal some feature on the relationship between wind speed and LvE and wind speed and surface temperature. It appears that the so-called equilibrium evapotranspiration define by
s Q* G s
Lv Eeq
(7.33)
plays a role in this context. 7.5.7.2 Equilibrium state It can be shown that for wet surfaces and the case that the air is close to saturation, LvE is given by equation (7.33). Then the Bowen ratio is given by
eq
(7.34)
s
It can be shown that LvE can be equal to its equilibrium value also when the surface is not wet, by which rs > 0. That occurs when
rseq
1
eq
c p e s T ea Q* G
(7.35)
Defining the Priestley-Taylor parameter by Lv E (7.36) Lv E eq it follows from above that in the equilibrium state aPT = 1. Obviously, in that state LvE is independent of ra and thus of wind speed. PT
7.5.7.3 Wind sensitivity LvE Using the results obtained in the previous section, it can be argued that if the surface resistance rs > rseq that then LvE deceases with increasing wind speed, whereas, when rs < rseq LvE increases with increasing wind speed. Because, according to equation (7.35) rseq contains terms that are expected to have a diurnal variation, the question how 24-hourly averaged values of LvE change with wind speed cannot be answered. As an example, we calculated with a model that will be described later (see 7.6.3) LvE for the sunny calm day in June at Cabauw, described in Chapter1. It appeared that during daytime a suitable value of rs = 270 s m-1,i.e. then the modeled H agrees with the measured one. Next we changed the wind speed only. In Figure 7.2 the calculated LvE are plotted for three wind speeds, 1, 4 and 7 m s-1 respectively. In addition in Figure 7.3 the Priestley-Taylor parameter PT is depicted. It is seen that if PT < 1 LvE appears to decrease with increasing wind speed indeed and when PT > 1 LvE increases with increasing wind speed. Around the transition hour at 15 UT this is not exactly true, but in main lines these results agree with the analyses derived from the Penman-Monteith equation..
109 U
0
1
U
3
4
U
7
6
300
250
LvE W/m2
200
150
100
50
0
50
0
5
10
15
20
25
UT
Figure 7.2 Calculated LvE using input data at Cabauw discussed in Chapter 1 for wind speed u = 1 (solid), 4 (dot) and 7 m s-1 (dash).
2
alpha PT
1.5
1 1
0.5
0
5
10
15 UT
Figure 7.3 As Figure 7.2 but for
PT
110
7.5.7.4 Wind sensitivity Surface temperature Since we solve the entire set of equations that governs the Penman-Monteith approach, the surface temperature T0 is obtained also with the numerical method we applied. This allows us to demonstrate the sensitivity of T0 to wind speed. For the selected cases shown in Figure 7.2 we calculated the corresponding surface temperatures and the results are depicted in Figure 7.4. 50
Surface temperatures
40
30
20
10
0
0
5
10
15
20
25
UT
Figure 7.4 As Figure 7.2, except for T0
It is seen that the surface temperature is quite sensitive to wind speed. Inspection of equation (7.29) for LvE suggests that through es(T0) which is a quite strong function of T0 , LvE is expected to be also very sensitive to T0 variations due to wind speed changes. However, there is a feedback mechanism that tends to decrease the wind speed effects on LvE. This is related to the fact that the aerodynamic resistance will change with wind speed also. It should be noted that the evidence shown in the section on wind sensitivity are highly dependent on environmental conditions. Here we took as example a surface with a relative high surface resistance and a low value for z0h. It must concluded that, as a results, the T0 sensitivity to wind speed is highly dependent on these factors also. This conclusion implies that the use of remotely sensed T0 in order to estimate LvE using the equations from which Penman-Monteith has been derived, should be applied with caution.
111
7.6 Advection: a case study 7.6.1 Introduction
L y s im e te r E T / N e t R a d ia tio n
LvE is generally some fraction of net radiation (Q*) for climates that have sufficient rainfall to support the evapotranspiration (ET). In areas where the air mass is strongly modified by dry, desert conditions, however, the ratio of LvE to Q* can exceed 2. Fig. XX illustrates this for Kimberly, Idaho, where lysimeter measurements of 24-hour ET of full cover alfalfa divided by the corresponding Rn are presented. The measured ET, i.e. LvE expressed in energy units, where is the latent heat of vaporization, exceed the available energy (that is close to Q* for daily values, because the daily soil heat flux density, G, is usually small) by 50% for most of August and September. During this period, dry air is advected from the large desert areas upwind of Kimberly.
Kimberly, Idaho 1969-71 Indication of Advection
3
2.5 2 1.5 1 0.5 0 50
100
150 200 Day of Year
250
300
Figure 7.5 Fraction of ET (in energy units) of net radiation in Kimberly (Idaho, USA); daily lysimeter values. (Data from James Wright (USDA-ARS, Kimberly). Considering the surface energy balance see (1.2), Q* G
Lv E
H
(7.37)
one can see that under the conditions mentioned above, i.e. LvE > Q* and G is negligible, the sensible heat flux density, H, must be negative. The required additional energy, needed to maintain the high evaporation rate, must be supplied by extracting sensible heat from the lower atmosphere.
112 If we consider large horizontally homogeneous fields where the atmospheric flow is in equilibrium with the underlying surface, the air temperature and humidity in the atmospheric surface layer are well adapted to the irrigated field and NO longer have the properties of the dry upwind terrain. Crucial for our considerations is that a negative H implies that the atmosphere just above the surface is stably stratified and the negative buoyancy effects suppress turbulent motions. The turbulence, needed for vertical transfer of water vapor, therefore, can only be generated in a mechanical way. This means that LvE can exceed Q* only if there is enough wind to offset the damping effects of stability. Under calm conditions it is to be expected that daily ET cannot exceed Q*. For small fields with finite size, on the other hand, things become more complicated. We define a small field as a field where under certain conditions the wind fetch is too small to adapt the advected, dry air mass to the irrigated surface. In that case, at any distance from the edge separating the irrigated field and the dry upwind desert, the influence of the upwind dry terrain will increase with increasing wind speed. For that reason ET is expected to increase with increasing wind speed due to the fact that dry desert air is forced to flow over the wet irrigated surface. From this common sense reasoning, we deduce that Q* of irrigated fields in dry deserts is enhanced by wind in two ways: 1. For large fields, i.e. large in the sense that the flow has adapted to the irrigated surface, wind enhances the mechanically generated turbulence needed to maintain vertical transfer under stable conditions. This is denoted as regional advection; 2. For small fields, i.e. small in the sense that the flow has not been fully adapted to the irrigated field, wind forces dry, 'non-adapted' desert air over the irrigated field, which enhances evaporation. This case is often called local advection. Considering vertical exchange of eddies or air parcels under conditions that H < 0 and LvE > 0, it is expected that upward moving eddies contain relatively cool and wet air, whereas downward moving parcels will be warm and dry. Consequently, the correlation coefficient of turbulent temperature and humidity measurements, RTq, is expected to be negative. Conversely, under 'normal' conditions, i.e. both H and LvE are > 0, RTq is expected to be positive. So, RTq seems an appropriate indicator for advection conditions. Also correlation coefficients between horizontal wind speed and temperature and humidity might be good indicators.
7.6.2 Experimental The RAPID experiment was carried out between 25 August and 19 September 1999 in an agricultural area of 70 x 25 km, located in Idaho, USA, 20 KM SOUTHEAST of Twin Falls. Staff members of the University of Idaho, Wageningen University, Campbell Scientific, Inc., Utah State University and USDA-ARS, Kimberly, Idaho, participated in RAPID. Four eddy-correlation systems were deployed, all consisting of a CSAT3 3D sonic anemometer and a Krypton hygrometer. These systems were mounted at 3 and 10 m above the ground. In addition, two Bowen ratio systems and sensors to measure net radiation and its components and soil heat flux were installed. In this note we will use mainly data of the eddy-correlation system operated at 3m by Wageningen University. The micrometeorological equipment was installed between two centre-pivot irrigated alfalfa fields of approximately 1 mile by 1 mile. Towards
113 the west, the dominant wind direction, beyond the field adjacent to the equipment, two more irrigated fields, of respectively alfalfa and wheat were grown.
7.6.3 Some results. In Figure 7.6 the various components of the energy balance and the wind speed at 3 m are depicted for a day with low wind speed. Note that H and LvE have been corrected for non-closure of the surface energy balance. It is seen that during daytime H is positive and LvE is not exceeding Q*. In Figure 7.7 the same quantities are plotted but now for two successive days with wind speeds at 3 m exceeding 3 m s-1. Now H is negative both during day- and night time and LvE > Rn. At night LvE is positive. Under nighttime conditions H and G are the only available energy sources to feed LvE as Rn is negative. This means that H must be negative, i.e. directed towards the surface, and, consequently, the stratification is stable. H is then expected to be related directly to wind speed, since wind is the only turbulence generator. In Figure 7.8 this feature is illustrated by plotting for the same data set the night time values of H is against u. It is seen that H is related almost linearly to u for those selected data.. 800
8
600
400 4
200
2
0
-200
0
0
600
1200
1800
Local Time Rn
LvE_EC corr
H_EC corr
U
Figure 7.6 Energy balance terms on a calm day.
U [m s-1]
Energy (W m-2)
6
114
800
8
600
4
200
2
0
-200
0
0
600
1200
1800
0
600
1200
1800
Local Time Rn
LvE_EC corr
H_EC corr
U
Figure 7.7 As Figure 7.6, except for toe days with high wind speed.
0
-50
-100
-150
-200 0
1
2
3
4
5
6
-1
U [m s ]
Figure 7.8 H shown in Figure 7.8 as plotted against wind speed at 3 m.
U [m s-1]
400
HEC corrected [Wm -2]
Energy (W m-2)
6
115 We solved the set of equations that governs the Penman-Monteith equation, i.e. we solved the set of equations formed by (7.26) - (7.29). Moreover, we accounted for the fact that net radiation is a function of the surface temperature: Q*
(1
)K
s
(L
Ts4 )
(7.38)
Finally, we used for ra equation (6.48). By try-and-error we 'tuned' model parameters and we found that rs = 20 s m-1 when Rn - G > 0 and 100 s m -1 otherwise, zom = 0.005 m, z0h = z0m /10 and
s
= 0.98.
In Figure 7.9 the modelled and measured H and LvE are compared for the two days as depicted inFigure 7.7. In Figure 7.10 and Figure 7.11 we compared the modelled and measured values of H and LvE separately. Furthermore, in Figure 7.12 the calculated and observed surface temperatures are compared.. Although with some scatter, especially for H, the overall picture is that the tuned model describes the selected high-wind situation well. 800
Energy (W m-2)
600
400
200
0
-200 0
600
1200 LvE_Model
1800
0 Local Time
LvE_EC corr
600 H_Model
1200
1800
H_EC corr
Figure 7.9 Modeled and measured H and LvE
116
50
y = 0.98x r 2 = 0.64
HEC corrected [Wm 2]
0
-50
-100
-150
-200 -200
-150
-100
-50
0
50
H Model [Wm2]
Figure 7.10 Comparison between modeled and measured H
600
y = 0.99x r 2 = 0.97
LvEEC corrected [Wm 2]
500
400
300
200
100
0 0
100
200
300
400
500
L v E Model [Wm2]
Figure 7.11 As Figure 7.10, except for LvE
600
117
25
y = 0.98x r 2 = 0.97
Ts measured [Wm 2]
20
15
10
5
0 0
5
10
15
20
25
T s model [Wm2]
Figure 7.12 Comparison between measured and modeled surface temperature
The numerical scheme consists of two iteration loops. The first concerns the evaluation of the aerodynamic resistance ra and the second the iterative solutions of the governing equations for the Penman-Monteith equation allowing the non-linear term in the net radiation, i.e. the term containing T04, and the non-linear term in equation (7.29), i.e. the term containing es(T0). As input, the measured incoming short and longwave radiation are used as well as the measured soil heat flux, and the wind speed, air temperature and specific heat at 3 m. Our results show that large irrigated fields (of alfalfa) can be affected by regional advection indeed, provided the wind speed is high. Then the evaporation can be significant also at night. Our findings agree with our picture of turbulent flows over irrigated fields in dry conditions. It suggests that in future, crops such as alfalfa that apparently do not close their stomata when transpiration is high, might be not used in arid regions where wind speeds can be high. It is likely, that other crops do close their stomata at high evaporation rates. More research is needed.
7.6.3.1 Correlation coefficients A correlation coefficient between two turbulence variable x and y is defined by R xy
x' y ' x
y
(7.39)
118 Under advective conditions, the sensible heat flux is downwards and evaporation is upwards. Therefore, it is to be expected that the correlation coefficient between temperature and specific humidity is negative. In Figure 7.13 this is confirmed. Here RTq is depicted for a selection of the data to which Figure 7.7 refers also. So, the sign of RTq is a measure for the occurrence of advection. Unfortunately, this is a quantity that is not easy to measure on a routine weather station. 1
0
-0.5
-1 0
600
1200
1800
0
600
1200
1800
Time (UTC) RuT
Ruq
RTq
Figure 7.13 Correlation coefficients RuT, Ruq and RTq
1
0.5
RTq
Correlation Coeficients
0.5
0
-0.5
-1 -1
-0.5
0
0.5
R uT
Figure 7.14 RuT versus RTq
1
119
In Figure 7.13 the correlation coefficients between the horizontal wind speed component u and T, RuT , and between u and q, Ruq, are plotted also. This result suggest that there is a relationship between RuT and RTq. This is confirmed by Figure 7.14 in which RuT is plotted versus RTq for all available RAPID data for which the wind speed exceeded 4 m s-1. This results can be used in future on a standard weather data, because there is a tendency that cup anemometers and wind vanes are replaced with a 2-D sonic anemometer that provides both horizontal wind speed and wind direction. But this instrument allows high frequency sampling and provides the air temperature also. So, RuT can be determined with a 2-D sonic anemometer. Adopting the results shown in Figure 7.14, in its turn, this yields RTq that appeared to be a proper indicator for advection. Next, for advective situations the numerical model (that is in fact an extension of the Penman-Monteith equation) can be applied to calculate the actual evaporation.
7.7 The standard deviation or variance method Using the results obtained in Chapter 6 for standard deviations and fiction velocity, notably equations (6.33), (6.35), (6.27) using (6.30) it is possible to determine H from measurements of the standard deviation of temperature T and the wind speed at a single level, provided that the roughness length for momentum z0 is known. The governing equations are for unstable conditions:
z 2.9 1 28.4 L
T *
u*
z ln z0
m
u z L
1 3
(7.40)
m
(7.41)
z0 L
and the definition of the Obukhov length given by (5.27) T u*2 g *
L
`(7.42)
This set can be solved iteratively making use of (6.29):
m
z L
1 x 2 ln 2
1 x2 ln 2
2 arctan( x)
2
with x
z 1 L
1
4
.
7.8 Scintillometer method Recently, scientific research has revealed that an optical method to determine the area averages of H and LvE, called scintillometry, is very promising. Copies of recent papers in scintillometry are available on request. It is outside the scope of this course to discuss all theoretical aspects of this novel approach. Here we will confine
120 ourselves to the theory behind the so-called large aperture scintillometer. This instrument yields the so-called temperature structure parameter CT2 which is a sort of spatially averaged standard deviation of temperature. So, the main advance is that a C2z scintillometer yield spatially averaged information. It can be shown that T2 is a *
dimensionless parameter and, consequently, according to MOST
2 T 2 *
C z
is a function of
z/L. It has been found that for unstable conditions:
CT2 z
2 3
z 4.9 1 9 L
2 *
(7.43)
next we recall that:
u*
z ln z0
m
u z L
m
(7.44)
z0 L
T u*2 g *
L
`(7.45)
and
m
z L
1 x2 ln 2
1 x 2 ln 2
2 arctan( x)
2
with x
z 1 L
1
4
.
Also, this set can be solved iteratively. Over tall vegetation in the above expressions z ahs to be replaced with z - d , see equation (6.14). It appears that if z is large and wind speed is small the so-called free-convection approximation holds, i.e. a fair approximation is obtained by H fc
0.576 c p z C
2 T
4 3
(7.46)
Moreover, it is noted that under humid conditions CT2 has to be corrected for effects of humidity on the refraction index of air. Details are outside the scope of these introductory notes on the subject.
7.9 Soil-vegetation-atmosphere transfer models In the last decades several so-called soil-vegetation-atmosphere transfer models (SVAT's) has been developed. Several of these are based on the governing equations for the Penman-Monteith equation. It is outside the scope of this introductory course
121 to deal with the details of the different SVAT's that are published in open literature in the last decade. We will give here a very brief description of some SVAT's.
7.9.1 The Jarvis-Stewart model In the Jarvis-Stewart approach the surface resistance rs is described in terms of known environmental parameters as follows: gs
g 0 g K K gT T g D D g
(7.47)
in which g s 1 / rs is the so-called surface conductance, g 0 the maximum conductance under unlimited conditions, and the functions g K , g T , g D and g describe the influence on gs by shortwave radiation, temperature, water vapor deficit at standard level (D = es(T) - ea, see Chapter 2) and water stress, characterized by is the leaf water potential , respectively. The g-functions have been determined from experimental data. This implies that the Jarvis-Stewart model is mainly empirical, but the approach based also on plant physiological knowledge.
7.9.2 A-gs model This approach is based on the idea that the complex photosynthesis process can be summarized to a simple chemical reaction: CO2+H2O + light
CH2O +O2
The plants take up the required CO2 from the atmosphere through the stomata in the leaves. Thus, similarly, to the way plants transpire. If the stomatal openings are large the CO2 uptake is large as well as the transpiration. In this way it photosynthesis and transpiration are interrelated. In the A - gs approach gs is described through:
gs
1.6
An Cs
Ci
(7.48)
where An is the net photosynthetic rate, Ci is the intercellular CO2 concentration and Cs the CO2 concentration at the leaf.. The factor 1.6 describes the difference between molecular diffusion of CO2 and H2O into air. An is a function of photosynthetic active radiation (PAR), temperature, water deficit and water stress. In this way the JarvisStewart and the A-gs models are similar. An advantage of the A-gs is that transpiration is described in connection to photosynthesis. In current climate models the A-gs approach has been implemented recently.
122
7.10 Exercises 1. The wind speed at height of 10 m above a surface with roughness length z0m = 2 cm is 5 m s-1. The conditions appear to be neutral. Determine: a. The wind speed at 2 m; b. The friction velocity u*; c. The aerodynamic resistance rm for the layer between 2 m and zom. Note that = 0.4. 2. The roughness length for momentum z0m can be determined experimentally by measuring the wind speed at at least 2 levels. Derive a general expression for z0m. How can z0m be determined graphically? 3. Above a well-mixed lake the following is observed (30-min values): Incoming shortwave radiation Incoming longwave radiation Air temperature at 1 m Dew point at 1 m Air temperature at 2 m Dew point at 2 m
350 Wm-2 300 Wm-2 15.0 0C 13.00C 14.5 0C 13.00C
The water temperature is 15.5 0C and it appears that this temperature has increased in the latest hour 0.02 degree. The water depth is 5m. The heat flux at the bottom of the lake can be ignored. The albedo of water is 0.06. Determine the net radiation, the heat storage term G and the flux densities of sensible and latent heat H and LvE. 4.
Give a brief derivation of the Penman-Monteith equation. Give the assumptions and discuss the various ways the Penman-Monteith equitation is applied in practice by the FAO.
5. Determine the sensible heat flux density and evaporation E in mm/hr from the following data (30 min-values): Net radiation Soil heat flux density Air temperature at 2 m Water vapor pressure at 2 m Aerodynamic resistance Surface resistance Air pressure
250 W m-2 30 W m-2 15 0C 14 hPa 60 s/m 40 s/m 1007 hPa
123 6. The surface temperature appears to be equal to the air temperature at 2 m above a grass field. The net radiation is 400 W m-2 and the soil heat flux density 30 W m-2. Determine LvE. 7. A forest has been replaced with grass. Discuss the main micrometeorological effects. 8. The following data sets are given: set 1 Height u q 0 (m) (m/s) ( C) (g/gk) 1 2
10.1 11.5
10.0 10.0
10.3 10.3
set 2 u (m/s)
( C)
q (g/gk)
2.3 2.6
20.0 19.8
17.9 17.5
0
a. What is the stability (stable, neutral or unstable) of set I and set 2? b. Calculate for sets 1 and 2 the sensible and latent heat flux density and the friction velocity u*. c. Determine the evaporation E in mm/hr. Assume for simplicity that the air density is 1.2 kg m-3 and that the psychrometer constant is = 0.66 hPA K-1.
7.10 MATHCAD and EXCEL modules The following MATHCAD modules are available:
7.10.1 Flux-profile relationships: derive yourself In this module the reader is trained to drive flux-profile relationships from experimental data.
7.10.2 Businger-Dyer and free convection This module contains procedures to determine the fluxes using the Businger-Dyer relationships numerically. In addition the approximate analytic solutions are given and - for the unstable case- the free convection solution. The reader can use this module to compare the differences.
7.10.3 Excel modules using real data In the Excel file Veldprakflux.xls data collected at the experimental field of the Wageningen University are presented from which the fluxes using e.g. flux profile relationships can be derived.
7.10.4 Wind sensitivity energy balance and Penman-Monteith In the MATHCAD module entitled Wind sensitivity EB Cabauw.mcd contains many features. Firstly, it contains a fully iterative procedure to solve the non-linear governing equations from which the Penman-Monteith formula has been derived. It is applied to the input date collected at Cabauw on 30 June 1995, i.e. the same data set presented in Chapter 1. By changing the wind speed, while taking all other environmental quantities constant, the sensitivity of the energy balance terms can be
124 evaluated. In first instance a surface resistance is chosen that is derived from the data set itself. The wind sensitivity appears to be related with this surface resistance. The module deals also with the aerodynamic resistance that, on one hand is solved fully iteratively and accounting for stability effects and on the other hand is approximated with simple wind functions. This allows a comparison between the actual and the estimated values. Furthermore, a comparison can be made between actual value of LvE, i.e. the solution obtained with the full numerical procedure, and the analytic solution given by the Penman-Monteith equation. The module allows also investigating the behavior of the surface temperature and its sensitivity to wind speed or the roughness length for heat z0h.
125
Chapter 8 Water temperature of a well mixed lake This chapter is dealing with the annual cycle of the water temperature and the terms of the energy balance of a lake. We confine ourselves to lakes which have a constant depth are well mixed. That means that the water temperature is constant with depth and this implies also that the surface temperature T0 equals the mean water h 1 Twater dz . A simple model will be discussed that is based on the temperature: Twater h0 governing equations from which the Penman-Monteith formula has been derived (see section 7.5). The main difference will be the description of the term G, whic is now the change of the heat stored in the water body (per unit time and area). The model enables as to describe the time evolution of the temperature and the terms of the energy balance of well-mixed lakes under various meteorological conditions.
Theory Measurements have revealed that for most lakes the heat flow at the bottom of the lake from the water towards the underlying ground is very small and can be ignored. With the condition that the water is well mixed, the heat storage term, G, in the energy balance equation can be written as: T0 (8.1) G w cw h t Where w cw is the volumetric heat capacity of water that equal 4.2 106 J m-3 K-1, h is the water depth, t is the time. In addition, the governing equations as used for the derivation of the Penman equation apply, notably: Q*
H Lv E
H
Lv E G
cp
T0
(8.2)
T
(8.3)
ra
c p ew T ra
e
(8.4)
We recall that Q* is the net radiation, H the sensible heat flux density, E the evaporation, Lv is the latent heat of vaporization (which is about 2.5 106 J kg-1), ew T0 is the saturation water vapor pressure at water temperature T0. Moreover T and e are the air temperature and water vapor pressure at reference level, the psychometric constant (at sea level and 20 0C about 66 Pa K-1), ra the aerodynamic resistance, is the air density, and cp the specific heat of air at constant pressure (1005 J kg-1 K-1). Assuming that water is black in the infrared we can write Q*
1 r K
L
T04
(8.5)
126 Next we apply a Taylor expansion approximation for ew(T0) and T04 towards the wetbulb temperature Tw at reference level, i.e.: ew (T0 )
ew (Tw ) s w T0
(8.6)
Tw
with sw the derivative of ew at wet-bulb temperature and T04
Tw4
4Tw3 T0
(8.7)
Tw
This leads (after some algebra) to the following simple differential equation for T0:
T0 t
Te
T0
(8.8)
'
in which the Te is the so-called equilibrium temperature and ’ is a time constant, which are given by:
Te
Tw
'
w
Qw* X
(8.9)
cw h X
(8.10)
with 4 Tw3
X
cp s ra
(8.11)
and Qw*
(1 rw ) * K
L
Tw4 .
(8.12)
where Qw* is the net radiation if the water temperature would be Tw. This hypothetical quantity depends solely on meteorological input data. In order to allow analytic solutions, we use a simple 'wind function' formulation for ra (often used in the last century). In particular we will use a relation proposed by Sweers (1976): cp ra
Lv f u2
A0 c A u1
cu 2 u2 (W m-2 hPa-1,)
(8.13)
in which A0 = 5 km2 and A is the effective area of the lake. The latter depends on the wind direction and the shape of the lake. Furthermore, u2 is the wind speed at 2 m measured over the lake. The empirical constants cu1 and cu2 are taken as 3.6 and 2.7 respectively. If is assumed that “weather” is constant during 1 time step t the equilibrium
127 temperature, Te , which can be considered the “driving force” in equation (8.8) varies in steps. In that case the solution of equation (8.8) is given by
T0 t n
T0 t n
1
Ten T0 t n
1
1 e
t '
(8.14)
in which t is the time step in s, T0 t n the water temperature at tn, which is the time at the end of nth decade, Ten is the equilibrium temperature during time step t . In the first time step an initial value of T0 is required. After having evaluated the time evolution of T0 with (8.8), the terms of the energy balance can be solved using equations (8.2)-(8.4).
128
Chapter 9 Well-mixed convective boundary layer model: Interaction with a vegetated land surface 9.1. General aspects The model refers to the situation that turbulence is generated in the atmospheric or planetary boundary layer (PBL), primarily by surface heating. Moreover, this turbulence is so vigorous that the bulk of the PBL is well-mixed. This means that all conserved quantities, such as potential temperature ( ), specific humidity (q) are constant with height within the PBL. Above the PBL there is the non-turbulent 'free' atmosphere, where the potential temperature increases linearly with height and the specific humidity decreases linearly. It is said that the PBL is capped by an 'inversion', which has a 'strength' given by the slope of for z > h, where h is the PBL height At the top of the PBL, i.e. at level z = h, there is a sudden jump in and q. The magnitudes of these jumps play a role in the 'entrainment' process to be discussed later. The lowest 10% of the PBL is the surface layer (or constant flux layer). Here the and q vary with height and it is assumed that the K-theory, relating the fluxes of sensible heat and water vapor to the vertical gradients of and q respectively. In the Driedonk-Tennekes model it is assumed that the thickness of the surface layer and the entrainment layer are small, so that their heat capacities can be ignored. This schematic this picture leads to and q profiles as depicted in Fig. 1. It is important to note that in slab-models, such as that by Driedonks and Tennekes, the profiles are prescribed. This implies that a detailed description of turbulence itself is not needed. This makes that this type of models is relatively simple compared to other PBL-models, such as LES, or higher order closure models. Note that in the wellmixed layer first-order closure (K-theory) is violated. Note that for the time being we will ignore effects of water vapor on the density of air. This means that we will use instead of the virtual potential temperature.
9.2. PBL growth Surface heating creates "thermals" that will rise due to buoyancy. Some of these thermals will get vertical speeds that are high enough to penetrate into the stable free atmosphere aloft the PBL. This causes the PBL to grow, i.e. h increases in time as long as there is surface heating. The growth rate (dh/dt, with t the time) depends on several factors such as H and the 'strength' of the capping inversion. Due to the growth of the PBL, air just above the PBL will be brought into the PBL itself. This is called entrainment. Since this air is warmer and dryer than the PBL air, the entrainment process creates at z = h a downward flux of sensible heat and a "negative" downward flux of water vapor. So the entrainment process tends to heat the PBL and tend to make it dryer.
129
z
q
dq dz
z
z h
q
d dz
h
h
q(t0 + ++dt t
t0
t0 + t
t0
q
t0 + t
z h
130 Figure 10.1 Schematic profiles of potential temperature and specific humidity at two time steps. Note that the specific humidity can either become dryer or wetter in the second time step.
9.3. "Thermodynamics" and conservation laws Besides the fact that the surface sensible heat flux (H) is the driving (generating) force for turbulence and that it causes h to increase, it affects the temperature of the PBL as well, of course. This process has to obey the laws of thermodynamics, in particular "the conservation of energy" law. To account for this law, one must consider the heat input at the surface, but also the heat input at z = h due to the entrainment process. For water vapor the laws of conservation of mass must apply. For simplicity phase changes will be ignored.
9.3.1 Governing equations The laws of conservation of heat leads to d m dt
w' T ' 0
dqm dt
w' q' 0
w' T ' h
(9.1)
h
w' q' h
(9.2)
h
Due to the fact that the form of the profiles as shown in Fig. 1 are assumed to be conserved, "mathematics" (see Garratt, pag. 155) leads to equations for the 'entrainment fluxes' at the top of the PBL: dh dt
wh
(9.3)
dh dt
wh .
(9.4)
w' T ' h
w' q' h
q
Also, it can be shown, using straightforward mathematics, that: d dt d q dt
q
dh dt
wh
d m dt
(9.5)
dh dt
wh
dqm . dt
(9.6)
In equations (9.3)-(9.6) the effect of subsidence, i.e. vertical motions due large-scale meteorological processes, is described by subsidence speed wh.
131
9.3.2 Closure assumption Assuming that the surface fluxes w'T0 and w' q' 0 are know as well as
,
q
and wh,
the set of equations (9.1)- (9.6) are not closed, i.e. we have 7 unknowns (h, m, qm, , q, w' Th and w' q' h ) and only 6 equations. We need to make one closure assumption. For conditions where only surface heating generates turbulence, i.e. where mechanically generated turbulence is absent, it appears that w' Th is proportional to w'T0 : w' Th =-cF w'T0 ,
(9.7)
Several authors proposed cF = 0.2. Driedonks showed that in many early morning cases this leads to unrealistic results, because then wind affects the growth of the (still) shallow PBL also. A more complete closure formula is given by Tennekes. w' T ' h
c F w' T ' 0
A
u*3T0 gh
(9.8)
where A is a second empirical constant. Note that if one takes A = 0, one gets back equation (9.7).
9.3.3 Coupling to the surface layer; description Excel module The simple ‘slab’-model for the convective well-mixed boundary layer is coupled to the ‘big-leaf’ model describing the flux-input, i.e. we will use the governing equations on which the Penman-Monteith formula is based: Q*
H Lv E '
H
Lv E G
c p w' T'0 Lv w' q' 0
(9.9)
cp
T0
Lv
m
ra q sat T , p q m ra rs
(9.10) (9.11)
where Lv is the latent heat of vaporization of water at surface temperature, the air density at the surface, cp the specific heat of air at constant pressure, ra the aerodynamic resistance, rs the surface resistance. In the Excel module rs is prescribed, whereas ra is evaluated from the prescribed wind speed at the top of the surface layer, the calculated H, using simplified flux-profile relationships that will not be described in detail here. In this procedure the roughness lengths for momentum (z0) and for heat (z0h) are playing a part. These have to be given. Effects of surface temperature on net radiation are accounted for:
132 Q* (1 r ) K
L
s
T04
(9.12)
in which r is the surface albedo, K the incoming shortwave radiation, L the incoming longwave radiation, s the emissivity of the surface, the StephanBoltzmann constant and where T0 , the surface temperature is in K. The soil heat flux density G is taken as a simple fraction of K : G = Gfac Q *
(9.13)
For L we either will use the measured value or the simple parameterization 4
L
atm
abs
60C
(9.14)
where atm is the apparent emissivity of the (clear) atmosphere and C the cloud cover. For simplicity we take simply
=
m
in K, and
(9.15)
0.8
atm
abs
This simplification is used in order to make the model not too complex. Since it is meant here for education purposes only, this approximation is acceptable. However, the reader is cautioned not to use the model for research.
The set of equations referring to the well-mixed boundary layer and the surface layer fluxes are solved using a simple forward finite difference scheme. This yields the surface fluxes H and LvE, the aerodynamic resistance ra, the surface temperature T0, and the boundary layer parameters, h, m, qm, , q, w' Th and w' q' h .
9.3.4 Some exercises 9.3.4.1 Reference case Use the following initial values as 'typical' for a Dutch clear sky summer day The surface parameters are typical for grassland. Table 9.1 Input parameters referring to 'typical grass in the Netherlands' Quantity Amplitude global rad. albedo zom zoh Gfac pressure rs
value
units 700 0.23 0.05 0.005 0.1 100000 40
W/m2 m m Pa s/m
Initial values thetam
290 K
133 Dtheta h qm Delta q gam theta gam q U
3 100 7 -0.3
K m g/kg g/kg
0.0075 K/m -.0015 -.0015 g/kg per m 2 m/s
Run the model and discuss the results; inspect the charts for h, fluxes, temperatures and qm. Use this run as a reference case. 9.3.4.2 Sensitivity studies The program allows a large number of sensitivity studies. This is due to the large number of variables that are playing a role in the problem. The student is free to carry out his/her own sensitivity study. In the next we suggest a number of these studies.
Sensitivity study on 'external' forgings The model describes the interaction between the surface (layer) and the PBL. The time evolution of the different PBL parameters are determined by "internal or local" and 'external or non-local" variables. The surface parameters can be considered "internal" parameters. The solar radiation and the wind speed are "external" parameters. Also, the "strengths" of the inversion aloft the PBL can be regarded to be "external" factors, which are related to large-scale meteorological processes. The subsidence is of course an external parameter also. Note that the default value of wh (the vertical velocity due to subsidence) is taken zero. It can be changed in cell V20. The most straightforward sensitivity study in this category is that in which the incoming short wave radiation is varied. This can be done by changing the amplitude in cell V2. In addition, the wind speed can be varied. The strength of the inversion, described by is another important external parameter. For instance, what is the effect on dh/dt if is taken very large? Sensitivity study on initial conditions There are several issues that are related to the initial conditions. One example is the initial value of m. Run the model in the 'reference' mode, but for m = 273, 283, 293 and 303 K and consider the surface fluxes and the Bowen ratio. One can study also the effects of changes of the initial values of h and . Sensitivity study on wind and surface roughness The program allows a study of the sensitivity of h, the mixed layer and surface potential temperatures and the Bowen ratio (see note Micrometeorology) on changes in wind speed and the roughness lengths for momentum (z0m). Note that we use the wind speed at z = h/10. Then the effective roughness length is greater than the z0m one founds from wind profile data gathered up to, say, 5m. This is
134 due to the fact that obstacles such as isolated trees, cows, ditches etc. are seen at larger heights, and these tend to increase the roughness of the surface. Typical values for a flat sand desert grass, tall arable crops (maize) and forest are 1, 5, 10 and 75 cm. The surface temperatures can be monitored remotely with nowadays satellites. In this application the roughness length for heat, z0h, plays an important role. For its definition also the lecture notes on Micrometeorology. In literature one can find a wide range of values for this quantity. As a rule of thumb one uses z0h is about z0m/10, but many investigators reported much smaller values. In any case, the student can study the sensitivity of the surface temperature and its difference with the air temperature on z0h or the ratio z0m/z0h
Sensitivity study on water (vapor) variables One category of sensitivity studies is related to the 'water availability" at the surface, which in this model is described via the surface resistance, rs. An arable crop 'optimally' supplied with water has and rs of about 40 s/m. The FAO uses a value of about 70 m/s for normal water supply conditions. Dry will corresponds to 100 and very dry to 500 m/s. Note that for forests rs is much higher (see lecture note on Micrometeorology). A wet canopy has a rs = 0. In this category of sensitivity studies one can pay attention to all water vapor related variables, such as qm, q and q as well as the entrainment flux of water vapor at z=h. An interesting issue is the time evolution of qm. Discuss the factors that tend to increase qm and the factors that tend to decrease qm. Which variables play a role in this process? Closure assumption and the encroachment model u*3T0 were gh obtained empirically for a number of sunny summer days in the Netherlands. In literature other values have been found. The student can study the sensitivity of e.g. h on changes of these constants.
The constants cF and A in the closure assumption w' T ' h
c F w' T ' 0
A
9.4 Approximate analytic solutions for the convective case It appears that for the convective case, i.e. the case where mechanically generated turbulence can be neglected and closure assumption (9.7) applies the set of equations listed above for m, and h has approximate analytic solutions. These read: h(t )
m
(t )
2
1 2c e
00
I (t )
0
1 ce h(t ) 1 2ce
(9.16)
(9.17)
135
ce h(t ) 1 2ce
(t )
(9.18)
t
w' T ' 0 dt ' , i.e. the integrated value of the surface kinetic heat flux w'T ' 0
With I (t ) t0
from
time
t0
until
time
t,
and
parameter
0
h0
0
1 2 m,
h02
and
and h h0 that are depending on the initial values of denoted as m0, and h0. It appears that soon after t0, I(t) >> 0, so that that in the convective case the PBL height increases in time proportional to I (t ) . In a MATHCAD module the analytic solutions are compared with the full numerical ones. 00
m0
0
9.5 Excel and MATHCAD modules 9.5.1 A MATHCAD module on the CBL The MATHCAD "well mixed CBL Introduction. mcd" is dealing with the basics of the well mixed convective boundary layer (CBL) theory described in 9.4. A simple forward difference numerical scheme is used to solve the set of governing differential equations. The approximate analytic solution is given as well.
9.5.2 A coupled mixed layer Penman-Monteith model In the Excel file "Mixed_Layer_PM.xls" contains the module the coupled model described in section 9.3.3.
136
Chapter 10 Miscellaneous Issues 10.1 Introduction In this chapter some environmental issues will be discussed that are important for everyday life. We will confine ourselves to mainly qualitative descriptions.
10.2 Dew 10.2.1 Introduction Formation of dew on plants is appears to play an important role in the energy and water budgets of vegetation and, last but not least for the development of plant diseases. Dew on a given surface is the process that water vapor near this surface condensates, i.e. it refers to the phase change of water vapor into liquid water. This will occur when the temperature of this surface is lowered such that the dew point (see chapter 2) at the given the given environmental conditions is reached. So, a 'cooling engine' is needed for the dew process. Two types of dew can be distinguished: Dewfall Then the water vapor needed for dew comes from the atmosphere. Dewrise In that case the water vapor required for dew originates from the soil.
10.2.2 Dewrise Dewrise is caused mainly by molecular diffusion in the pores between the soil particles. A detailed description of this process is outside the scope of this course. We confine ourselves here to mentioning that over a wide range of soil moisture conditions air in these pores is close to saturation. That implies that vertical water vapor gradients are driven by vertical temperature gradients, i.e. the water vapor flow in the soil is from 'warm-to-cold'. This implies that dewrise needs a cool surface warmer deeper soil layers. This will happens at night. Another feature is salt. This tends to lower the saturation water vapor pressure. So salt at the surface tends to enhance the formation of dew. This applies also to dewfall.
10.2.3 When does dewfall occur? We recall that dewfall requires that the surface temperature drops to dew point. Furthermore, dewfall is downwards water vapor flow, thus it is in fact a negative evaporation. For that reason, we can apply the theory on evaporation described in previous chapters. First, we present a qualitative description. At the moment dewfall starts, the surface will be covered with a water layer and, as a consequence, the air just above the surface will be saturated. A negative evaporation implies that the water vapor pressure at standard level is greater than the surface water vapor pressure that equals the saturation value at surface temperature. So, dewfall will be maximal when the air is close to saturation. Because the surface temperature must be smaller than the air temperature at standard level, the sensible heat flux will be negative also, i.e. towards the surface. Similarly, the soil heat flux is negative also, so towards the surface also. The sole energy balance term that can provide the necessary cooling is the net radiation. This term will be negative only
137 during nighttime when the outgoing longwave radiation is greater than the incoming longwave radiation. Consequently, dew can occur only during nighttime. In addition, it is remarked that at the phase change of water vapor to water heat is released. The formation of 1 mm of dew per hour (= 1/3600 kg per m-2 s-1) is accompanied by released heat of about 690 W m-2. At clear nights net radiation is seldom less than 100 W m-2, so the expected maximum dew is of the order of 0.1 mm/hr (because not all net radiation is used for dew). Dewfall can be significant only if the atmosphere is turbulent. Since the situation is stable and it is seen in the previous chapters that at low wind speeds buoyancy can suppress turbulence entirely, this feature implies that dewfall cannot occur at very low wind speeds. On the other hand, at high wind speeds the relative significance of the sensible heat flux will increase by which the surface temperature tends to increase above dew point. Experiences show that dewfall is confined to a rather limited range of wind speeds at reference level. The MATHCAD module Wind sensitivity EB Cabauw.mcd allows studies on dewfall by adapting the input data to dewfall conditions. Based on the above considerations, dewfall is maximal when the air is (almost) saturated, clouds are absent and soil heat flux is zero. Because the surface is wet, the surface resistance is then zero also. As a result, according to the Penmans s 4 L L T04 . Monteith the maximum dewfall is given by: atm T s s It is assumed that the surface is 'black' (T is the air temperature and T0 is the surface temperature and atm is the apparent emissivity of the clear atmosphere). Note that under dewfall conditions the evaporation equals the equilibrium value given by equation (7.33). Then the Bowen ratio is given by equation (7.34). So the partitioning of the 'available' net radiation over sensible and latent heat flux depends on temperature, because s is T dependent. Note that 0.5 when T = 20 0C (at sea level).
s
is about 1.5 at T = 0 0C and about
In summary conditions favorably for dewfall are: Clear nights (incoming longwave radiation small); Fully vegetated surface (soil heat flux small); High relative humidity (water vapor pressure high); Wind speeds between about 0.5 and 4 m/s;. Dewfall in summer is greater then in winter due to fact that
s
is smaller at
higher temperatures. It is noted that if the surface temperature drops below freezing point (0 0C) water vapor sublimates directly into ice. In that case hoarfrost is formed. The latent heat of sublimation (i.e. the amount of heat released when 1 kg of water vapor sublimates into ice) is at 0 0C 2.834 *106 J kg-1.
10.3 Frost For several reasons frost is important. Examples are penetration of frost into the soil, which causes freezing or damaging of water and gas pipes and/or communication
138 cables. Soil properties will change after frost, which is of importance for agriculture. The freezing of crops due to night frost is of interest for agriculture also for obvious reasons. In very cold climates ice lenses can be formed in the soil that are able to lift houses.
10.3.1 Night-frost Night-frost will occur due to radiative cooling at the surface. Now, it is not required that the air is turbulent. So night-frost will be most 'intense' when G, H and T04 (T is the LvE all are zero and the energy balance equation is reduced to atm T t air temperature and T0 is the surface temperature and atm is the apparent emissivity of the clear atmosphere). So the surface temperature then is given by T0 4 atm T , where T0 and T are expressed in K. Example: if atm = 0.8 and T = 278, T0 = 262 K. We recall that this refers to the most 'intense' night-frost conditions, which occur when: Clear nights (no clouds); Dry air ( atm small); Low wind speeds (no turbulence); Fully vegetated surface (small soil heat flux) On peat soils night-frost occurs earlier than on clay. On dry soils night-frost occurs earlier than on wet soils. In the Netherlands most crops are most sensitive to frost damage in the month of May. In this month the probability that night-frost occurs is lower than e.g. in April, but the crop development makes plants more vulnerable. This explains the existence of Ice Saints. In the Netherlands the period between the 11th - 13 th of May are called the 3 Ice Saints. In Germany there are 5 Ice Saints, indicating that crop damage caused by frost is an important issue in West Europe. Night-frost can even occur in July. Severe night-frost events in NO Netherlands occurred on the 15th and 16th of July 1921 and on 19th of July 1971. Most plants and trees 'harden' against frost damage in cold winters. The last decade is characterized by very warm winters in West Europe. Due to these warm winters crops and plants became more vulnerable for frost damage.
10.3.2 Crop protection against night-frost 10.3.2.1 Agricultural practice The effect of vegetation cover on soil heat flux implies that for crops that do not cover the ground entirely, weeding is important. Then soil heat flux is enhanced. Because cold air is flowing to the lowest parts in the terrain, night-frost sensitive crops must not be grown in these depressions in the landscape. For the same reason, solid walls have to be avoided on slopes. Fruit trees are extremely frost sensitive in spring when they are blossoming. For operational reasons these trees are small. Because under stable conditions the air temperature increases with height, taller trees as less frost sensitive.
139 10.3.2.2 Night-frost protection In the past several methods have been investigated. Examples are: heating with stoves in the field, creation of wind with ventilators (by which warmer upper air is mixed with cold air at the surface), two methods appear to be economically feasible, notably: 1. Covering crops with material acting as radiation shield (suitable for low crops only); 2. Spraying the crop with water. The latter method is based on the fact that when water freezes heat is released and the temperature is kept on 0 0C. The disadvantage is that the ice coverage can be so large that the plants will be damaged. It is important to give enough water; otherwise the ice will drop below freezing point. 10.3.4 Frost penetration depth into the soil In Chapter 4 the theory on soil temperature is presented. This allows for instance the calculation of the soil temperature after a sudden temperature drop at the surface with which one can determine after how many days freezing starts at a particular depth. The sudden temperature change at the surface then mimics the start of a period with frost. Such calculations reveal that the results depend very much on soil properties. Unfortunately, this approach is too complicated for practical applications. So there is a need of more simple rules of thumb for the penetration of frost into the soil. Such a simple rule is the following: frost penetration depth z = a
J
(10.1)
where a is a parameter depending on soil type and J is minus the sum of daily mean temperatures, where the summation starts at the first day when T < 0 and stops at the moment J > 0. N
J
Tk k 1
For example, if for 5 successive days the mean daily air temperatures measured at reference level are: 2, -3, 1, -6, -3, then J = -(-3 +1-6-3) = 11. In Table 10.1 the value of parameter a is given for different soil types. Note that the use of these values gives the penetration depth in cm. Table 10.1 Soil type Light sandy soils Dry grassland on clay Wet grassland on clay Grassland on peat
a cm(K-1/2 day-1/2) 6 4 3 2
Example: for the J found above we get for light sand soils a frost penetration depth of almost 20 cm. For grassland on peat this is only 7 cm. So far we ignored the effects of snow. Snow is a very poor heat conductor, so it prevents the soil from freezing. It is a well-known 'fact of life' that the penetrations
140 depth of frost is not very large in regions where snow occurs, even when the temperatures are low. A rule of thumb is to adapt a to snow as follows:
as
0.1 a 0.1 d
(10.2)
where a is given in Table 10.1. Note that when thaw sets in, first, the top layer will 'melt''. As a result there exists a frozen layer in the soil that at the top and the bottom is at freezing point of water. So there will be no heat flux across this layer. Therefore, the unfrozen top layer is thermally isolated from deeper soil layers by which the temperature can rise significantly in this layer. This triggers plant activities, however, this frozen layer does not allow vertical transfer of water neither. As a result plants can dry out after the set in of a warm period after frost.
10.4 Fog The occurrence of fog is an important environmental issue for road, water and air traffic. Fog is a cloud at the ground. At temperatures above freezing point of water, fog consists of small water droplets. When the temperature is below zero Celsius, fog is a mixture of ice particles and super cooled water droplets. Assuming that condensation of water vapor in the atmosphere occurs when the relative humidity exceeds 100%, then mist will be formed when the air close to the ground is brought to saturation. In this way various mechanism for fog formation can be obtained: Adiabatic cooling This occurs when air is forced to rise against a hill or mountain. Then adiabatic cooling can lead to saturation. Adding of water vapor In this way fog can be formed above lakes and swamps. Isobaric cooling without mixing Due to radiatve cooling that causes night-frost and dew also, the air close to the ground can reach dew point. This is called radiative fog. Isobaric mixing Because the saturation water vapor is a non-linear function of temperature it is possible that two air masses that both are not saturated become saturated after mixing. This can lead to fog near frontal systems. Isobaric cooling in combination with adding of water vapor often occurs over grassland near ditches. This leads to ditch fog. Arctic sea smoke that occurs when cold air formed over ice flows over nearby water is another example as well as. fog is formed over cooling ponds in wintertime. In coastal areas fog can be formed when warm air flows from land over cooler sea. Then air is cooled isobaric and moisture is added. Examples are fog near San Francisco in the USA.
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Fog areas can be moved horizontally by wind. This is denoted as advection fog. So far is has been assumed that condensation occurs when the relative humidity is 100 %. In general, this is not true. In entirely clean air condensation will take place at relative humidity of 400 % or more! The condensation process requires so-called condensation nuclei. In 'natural' conditions there is a sufficient number of condensation nuclei, by which, usually, condensation takes place when relative humidity is close to 100 % indeed. This is rather fortuitous! In polluted air condensation can occur at relative humidity's less than 100 %. In cities where coal is used for house heating fog can be formed in unsaturated air. In the decades fog was formed on New Year's Eve due to condensation nuclei brought in the atmosphere by fireworks.
10.4.1 Fog control Because fog causes a lot of economical damage for instance for airports, several attempts has been undertaken to control fog. Some of these methods, which have not been very successfully, are: Mechanical mixing When air that contains fog is mixed with drier air aloft the fog layer, the resulting mixture can be unsaturated. This idea has been used to try to control fog. Air has been mixed for instance with a helicopter. For thick fog layers this method is not suitable. Spraying with giant condensation nuclei By spraying with giant condensation nuclei the process of precipitation might be triggered. The giant nuclei form water drops that are larger than the 'naturally formed' fog droplets. The larger drops fall faster than the smaller ones by which they can grow to the so-called coalescence process. This increases their falling velocity more, by which they can collide with smaller fog droplets. Finally, the drops are so large that they fall as raindrops: fog disappears due to rain. Super-cooled fog layers can be sprayed with freezing nuclei. Then the formation of ice particles is triggered. In its turn, the so-called Bergeron-Findeisen effect is started. The water vapor pressure around the ice particles is less than that around water droplets. When an ice particle is close to a water droplet a water vapor flow is triggered from the water (high pressure) towards the ice (low pressure) particle. As a result the ice particle grows and can reach such a size that gets a high falling velocity. The fog disappears due to snow. In nature fog can disappear due to the Bergeron-Findeisen effect when fog cools (e.g. by radiative effects) below -7 Celsius.
10.5 Propagation of sound in the atmosphere In modern society sound pollution is an important environmental issue. For that reason we will deal briefly with how sound propagates in the air layers close to the ground. We will confine ourselves to 'main lines'.
10.5.1 Velocity of sound In still air the velocity of sound is given by
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cp
(10.3) RT cv where cp and cv are the specific heat at constant pressure and volume respectively, R is the specific gas constant of air and T the absolute temperature. Not that cp, cv and R are dependent on specific humidity. See chapter 2. In moving air one has to add the wind speed component in the propagation direction of the sound wave. This additional contribution by wind to the velocity of sound can be positive and negative. According to the law of Snell a sound wave that travels in an air layer with velocity c1 into an air layer with different environmental conditions where the velocity of sin i sin r sound is c2 is refracted according to: where the angles i and r are c1 c2 defined in the following:
c
Layer 1 with c1
i
Layer 2 with c2 > c1 r
When the temperature changes with height the sound velocity changes with height also. This implies that in unstable air (T decreases with height) sound wave tend to reflect upward, whereas in stable air the sound waves are refracted downward and may partially focused. The result is that during still nights when a ground inversion is formed sound can be heard remarkably loud. During daytime the audibility of the same sound (e.g. rail roads) source is poor. A nice example is the audibility of thunder that is affected by the temperature profile under thunderstorms. At a certain distance from the thunderstorm thunder is inaudible. Wind shear effects enhance this LvE.
143 The influence of wind affects the velocity of sound near the surface, because there the wind speed vector increases with height. As a result, sound waves traveling in the direction of the wind are refracted downward, whereas sound wave that move against the wind are bended upwards. The audibility of sound sources increases if they are in the direction of the wind and decreases if they are in the opposite direction. Usually, the wind and temperature effects are acting both. A joint temperature-wind shear effect on the audibility of thunder has lead to the countryman saying: "the calm before the storm".
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Chapter 11 Literature Introduction to meteorology McIlveen, R., 1991: Fundamentals of weather and climate, Chapman & Hall, UK
Turbulence Tennekes, H. and Lumley, J.L., 1972: A first course in turbulence, MIT Press, Cambridge, USA
Atmospheric surface and boundary layer Haugen D.A. (ed.), 1973: Workshop on micrometeorology, AMS, Boston. Garratt, J.R., 1992: The atmospheric boundary layer, Cambridge Univ. Press, UK Nieuwstadt, F.T.M. and Van Dop, H. (Eds), 1982: Atmospheric Turbulence and air pollution, Reidel, Dordrecht. Stull, R.B., 1988: An introduction to boundary layer meteorology, Kluwer, Dordercht
