## Atmospheric Applications of the Monte Carlo Method

The Monte Carlo method can be applied to many problems of radiative transfer in planetary atmospheres. In the following sections we give a sample of such applications. We begin with irradiances in plane-parallel media, which are amenable to treatment by other methods, then show photon path length distributions, reflection and transmission by finite clouds, and finally irradiance, flux divergence, and heating rate profiles in a clear and in a horizontally heterogeneous cloudy atmosphere. For all these Monte Carlo calculations, the Henyey-Greenstein phase function (Sec. 6.3.2) was used. 20 30

### Optical Thickness

Figure 6.8: Transmissivity of a plane-parallel medium above a nonreflecting surface. The dotted curve is exponential attenuation, the thick solid curve with no circles is from the two-stream theory, and the solid circles are Monte Carlo calculations, all for an overhead sun. Open circles are Monte Carlo calculations for a 60° solar zenith angle. The asymmetry parameter is 0.85.

20 30

Optical Thickness

Figure 6.8: Transmissivity of a plane-parallel medium above a nonreflecting surface. The dotted curve is exponential attenuation, the thick solid curve with no circles is from the two-stream theory, and the solid circles are Monte Carlo calculations, all for an overhead sun. Open circles are Monte Carlo calculations for a 60° solar zenith angle. The asymmetry parameter is 0.85.

### 6.4.1 Irradiances in Plane-Parallel Media

Perhaps the simplest Monte Carlo calculations are of irradiances reflected by, transmitted by, and within a plane-parallel medium. An incident photon that emerges at any direction from the illuminated (top) boundary of such a medium is counted as contributing to the reflectivity, and we are spared the bother of keeping track of this direction. And similarly for photons emerging from the bottom boundary.

Figure 6.8 shows the transmissivity of a negligibly absorbing medium with properties similar to those of clouds at visible wavelengths calculated by the Monte Carlo method and compared with that calculated by the two-stream theory of Section 5.2. The two methods yield almost identical results, which gives us some confidence in the simple two-stream theory. Of course, this theory is limited to normal incidence, whereas the Monte Carlo method is indifferent to the direction of the incident photons. Figure 6.8 also shows transmissivity for a 60° solar zenith angle, which is never greater than that for overhead illumination. Can you give a simple physical explanation for this?

Monte Carlo calculations of the diffuse irradiance transmitted by a cloud are no more difficult than calculations of the total transmitted irradiance: incident photons that make a direct transit through the cloud are not counted as contributing to the diffuse irradiance. Figure 6.9 shows the diffuse downward irradiance below a plane-parallel medium (cloud) calculated by the Monte Carlo method and compared with that calculated by the two-stream theory. The agreement is quite good. Again, the two-stream theory is limited to normal incidence whereas the Monte Carlo method is not. This figure also shows the diffuse downward irradiance for a 60° solar zenith angle, which is greater than that for 0° illumination except at the smallest optical thicknesses. Optical Thickness

Figure 6.9: Diffuse downward irradiance normalized by the incident irradiance from the two-stream theory (thick solid line with no circles) and Monte Carlo calculations (solid circles) for an overhead sun. Open circles are Monte Carlo calculations for a 60° solar zenith angle. The asymmetry parameter is 0.85 and the medium is above a non-reflecting surface.

### Optical Thickness

Figure 6.9: Diffuse downward irradiance normalized by the incident irradiance from the two-stream theory (thick solid line with no circles) and Monte Carlo calculations (solid circles) for an overhead sun. Open circles are Monte Carlo calculations for a 60° solar zenith angle. The asymmetry parameter is 0.85 and the medium is above a non-reflecting surface.

Figure 6.8 shows total transmissivity and Fig. 6.9 diffuse transmissivity. For small optical thickness («0) the two are quite different: «1 for the former, «0 for the latter. With increasing optical thickness the two approach each other, as expected. For sufficiently large optical thickness essentially all the transmitted radiation is diffuse because of exponential attenuation of the direct radiation.

According to the two-stream theory, both upward and downward irradiances within a negligibly absorbing medium decrease linearly with increasing optical depth. The Monte Carlo calculations in Fig. 6.10, however, show that both irradiance profiles display some curvature, especially near the upper boundary. Moreover, for an optically thick medium, the curvature can be so extreme that the downward irradiance within the cloud can exceed the incident irradiance. This does not violate conservation of energy, which requires only that the difference between the two irradiances be constant with optical depth and the downward irradiance never be less than the upward, conditions the irradiances in Fig. 6.10 do indeed satisfy. But as Fig. 6.11 shows, this increase of the downward irradiance above the incident irradiance near the upper boundary occurs only for illumination near normal incidence.

Real clouds - horizontally and vertically inhomogeneous and without sharp boundaries -are vastly more complicated than mathematical clouds, and calculating irradiances within and outside such clouds is much easier than measuring these quantities for real clouds. How well do calculated irradiances compare with measured irradiances? Figure 6.12 shows irradiances in the middle of the visible spectrum for a stratocumulus cloud about 700 m thick and with an estimated optical thickness of 10-20. Unlike the mathematical clouds for which previous calculations were done, this real cloud is not vertically homogeneous. Nevertheless, the upward and downward measured irradiances are not jarringly different from the calculated ones. Both upward and downward irradiances decrease approximately linearly with increasing op Figure 6.10: Downward (J.) and upward (|) normalized irradiances within negligibly absorbing clouds of optical thickness 2 (top), 6.5 (center), and 48 (bottom) from two-stream theory and the Monte Carlo method. The asymmetry parameter is 0.85, the sun is overhead, and the surface below the clouds is nonreflecting. Horizontal irradiances are denoted by

Figure 6.10: Downward (J.) and upward (|) normalized irradiances within negligibly absorbing clouds of optical thickness 2 (top), 6.5 (center), and 48 (bottom) from two-stream theory and the Monte Carlo method. The asymmetry parameter is 0.85, the sun is overhead, and the surface below the clouds is nonreflecting. Horizontal irradiances are denoted by tical depth, the downward irradiance is greater than the upward, and the difference between them is approximately constant. And the relative difference between them is in rough accord with what we would expect from Fig. 6.10, which shows a steadily decreasing relative differ- Figure 6.11: Downward normalized irradiance within a negligibly absorbing cloud with total optical thickness 48 and asymmetry parameter 0.85 calculated by the Monte Carlo method. Open squares are for the sun overhead, solid squares for 60° solar zenith angle. The surface below cloud is nonreflecting.

Liquid Water Content (g m 3) 0.00 0.05 0.10 0.15 0.20 500 nm Irradiance (W m-2 nm-1)

Figure 6.12: Measured downward (right solid curve) and upward (left solid curve) irradiances at 500 nm within a stratocumulus cloud between about 950 m and 1700 m (dashed lines). The solar zenith angle was approximately 10°. The dotted curve is liquid water content. These curves were obtained from as-yet unpublished measurements provided by Peter Pilewskie. Photon Path Length (km)

Figure 6.13: Probability density for path lengths of photons that contribute to the downward radiance below cloud (J.) and to the upward radiance above cloud (|). The incident radiation is normal to a plane-parallel cloud of physical thickness 1 km, scattering mean free path 1/16 km, and asymmetry parameter 0.75. The cloud overlies nonreflecting ground.

Photon Path Length (km)

Figure 6.13: Probability density for path lengths of photons that contribute to the downward radiance below cloud (J.) and to the upward radiance above cloud (|). The incident radiation is normal to a plane-parallel cloud of physical thickness 1 km, scattering mean free path 1/16 km, and asymmetry parameter 0.75. The cloud overlies nonreflecting ground.

ence with increasing optical thickness, about 80% for an optical thickness of 2, about 60% for an optical thickness 6.5, and about 20% for an optical thickness of 48, whereas the relative difference is about 40% for the real cloud with an estimated optical thickness of 10-20.

### 6.4.2 Photon Path Lengths

Everything takes time. Nothing occurs in an instant. And yet up to this point time has been absent from our analyses. We have implicitly assumed, for example, that reflected photons appear instantaneously at a detector when a cloud is illuminated. This is usually an acceptable fiction because relevant distances, say of order kilometers, divided by the speed of light correspond to times of order millionths of a second. But suppose that a cloud is illuminated by a short duration pulse, as from a laser beam. A pulse width in length units may be of order 100 m, which corresponds to a pulse width in time units of order 0.1 ps. We may think of an idealized pulse as having a square shape: the beam is turned on instantaneously, is constant for a fixed time (pulse width), then is turned off instantaneously. Suppose that such a pulse illuminates a cloud illuminated at normal incidence. What is the shape of the reflected pulse in the backward direction and the transmitted pulse in the forward direction? All photons that contribute to the reflected and transmitted radiances are born equal but do not suffer the same fates in the cloud. Because "time and chance happeneth to them all" pulses are broadened or stretched by an amount depending on the scattering properties of the cloud. Calculating this pulse broadening is a task to which the Monte Carlo method is well suited.

Figure 6.13 shows calculated probability distributions of path lengths for those normally incident photons that contribute to the upward radiance above a plane-parallel cloud and to the downward radiance below the cloud; the negligibly absorbing cloud is 1 km thick with a scattering mean free path 1/16 km and asymmetry parameter 0.75. The most probable path length of photons contributing to the downward radiance is close to 1 km. Most of this ra-

1000 100 10 1

1000 100 10 1 12970 13070

Wavenumber (cm-1)

Figure 6.14: Absorption optical thickness of oxygen for 100 km of Earth's atmosphere. From Qilong Min.

12870

12970 13070

Wavenumber (cm-1)

13170

Figure 6.14: Absorption optical thickness of oxygen for 100 km of Earth's atmosphere. From Qilong Min.

diance comes from photons that make a direct path through the cloud. The most probable path length of photons contributing to the upward radiance is about half the thickness of the cloud. For both radiances the pulse is stretched considerably, about a factor of 20 or so for a pulse width of 100 m. Pulse broadening sets an upper limit on the pulse repetition rate (time between pulses) but also contains information about the cloud.

Photon path length distributions also can help us understand atmospheric radiative transfer. As an example, we consider upward and downward radiances at two adjacent wavenumbers in the near-infrared (wavelengths near 770 nm) where absorption in Earth's clear atmosphere is mostly by molecular oxygen. The absorption optical thickness of oxygen up to an altitude of 100 km is shown in Fig. 6.14. At wavenumber 12970 cm-1 the absorption optical thickness is negligible whereas at the nearby wavenumber 12965 cm-1 the absorption optical thickness is about 0.11. Monte Carlo calculations of the relative radiance difference (La - Ln)/Ln, where La is the radiance at the absorbing wavenumber and Ln is the radiance at the negligibly absorbing wavenumber, for the upward radiance at 100 km and the downward radiance at 0 km are shown in Fig. 6.15 for clear sky, a cloudy sky with a single cloud of scattering optical thickness 16 between 1.0 km and 1.6 km, and a cloudy sky with same total optical thickness but distributed differently: a low cloud of scattering optical thickness 8 between 1.0 km and 1.6 km and a high cloud of optical thickness 8 between 8.6 km and 10 km. The cloud is negligibly absorbing and has an asymmetry parameter 0.75.

For the clear sky the relative radiance difference at the two adjacent wavenumbers is almost the same, about 10%, for the upward (100 km) and downward (0 km) radiances and is negative because radiances are less at the absorbing wavenumber. A single low cloud changes these results: both relative radiance differences increase in magnitude and by approximately the same amount. But the same total cloud, when distributed equally between high and low altitudes, decreases the magnitude of the relative upward radiance difference and markedly increases the magnitude of the relative downward radiance difference. Why? Clear

### 1 Cloud Layer 2 Cloud Layers

Figure 6.15: Relative difference of radiances at two adjacent wavenumbers in the near-infrared, one for which absorption is negligible and one for which the absorption optical thickness, a consequence of absorption by molecular oxygen, is 0.11. This difference for the upward radiance at 100 km is denoted by that for the downward radiance at 0 km is denoted by j. The cloud layers are negligibly absorbing and have the same total optical thickness (16) but are distributed differently: a single layer between 1.0 km and 1.6 km and two layers of equal optical thickness, one between 1.0 km and 1.6 km, one between 8.6 km and 10 km.

Clear

### 1 Cloud Layer 2 Cloud Layers

Figure 6.15: Relative difference of radiances at two adjacent wavenumbers in the near-infrared, one for which absorption is negligible and one for which the absorption optical thickness, a consequence of absorption by molecular oxygen, is 0.11. This difference for the upward radiance at 100 km is denoted by that for the downward radiance at 0 km is denoted by j. The cloud layers are negligibly absorbing and have the same total optical thickness (16) but are distributed differently: a single layer between 1.0 km and 1.6 km and two layers of equal optical thickness, one between 1.0 km and 1.6 km, one between 8.6 km and 10 km.

To answer this we appeal to photon path length distributions. Figure 6.16 shows distributions for photons that contribute to the downward radiance at the negligibly absorbing wavenumber for the clear and two cloudy skies and at the absorbing waveneumber for the cloudy sky with equal optical thickness low and high clouds. The source of illumination is at 100 km and is normal to the plane-parallel atmosphere.

For the clear sky almost all path lengths of photons that contribute to the downward radiance are 100 km and hence so is the average path length. A single cloud layer changes the distribution but not by much. Two widely separated cloud layers, however, markedly change the photon path length distribution because of multiple scattering within each cloud and, more important, multiple scattering between clouds. This is why the downward radiance at the absorbing wavelength decreases. Longer path lengths expose photons to more chances of being absorbed by molecular oxygen.

The magnitude of the upward relative radiance difference with two cloud layers is less than that with one cloud because scattering by the high altitude cloud shields some incident photons from the most absorbing part of the atmosphere. Also shown in Fig. 6.16 is the photon path length distribution for the downward radiance at the absorbing wavenumber, which is not appreciably different from that at the negligibly absorbing wavenumber. Thus although we show mostly photon path length distributions at the negligibly absorbing wavenumber, our conclusions are valid for both wavenumbers. Figure 6.16: Path length distributions for photons contributing to the downward radiance at the ground for clear sky and two cloudy skies, one with a negligibly absorbing cloud between 1.0 km and 1.6 km, and one with two clouds of the same total optical thickness (16) equally shared by a low cloud (1.0-1.6 km) and a high cloud (8.6-10 km). Radiation is normally incident on a plane-parallel atmosphere at 100 km. Except for the dashed curve in the bottom panel, the wavenumber is that for which absorption is negligible. For the dashed curve, the molecular oxygen optical thickness is 0.11.

100 110 120 130 140 150 160 170 180 Photon Path Length (km)

Figure 6.16: Path length distributions for photons contributing to the downward radiance at the ground for clear sky and two cloudy skies, one with a negligibly absorbing cloud between 1.0 km and 1.6 km, and one with two clouds of the same total optical thickness (16) equally shared by a low cloud (1.0-1.6 km) and a high cloud (8.6-10 km). Radiation is normally incident on a plane-parallel atmosphere at 100 km. Except for the dashed curve in the bottom panel, the wavenumber is that for which absorption is negligible. For the dashed curve, the molecular oxygen optical thickness is 0.11.

### 6.4.3 Three-dimensional Clouds

The qualifier "three-dimensional" is in a sense redundant give that all clouds are three-dimensional. Their properties vary in all three spatial directions. A cloud with properties varying in only one direction is a figment of the imagination of modelers. A one-dimensional cloud is an idealization never realized in nature. This should always be kept in mind when assessing the pronouncements of modelers, who sometimes confuse model clouds with real ones. Plane-parallel homogeneous clouds are so much easier to deal with. Even though they don't exist, they ought to. Nor do rectangular clouds exist, but they are a closer match to those that do. 0.01

### 0.1 1 10 Cloud Aspect Ratio

Figure 6.17: Reflectivity (solid curve) for normal incidence of a negligibly absorbing columnar cloud with square cross section, vertical optical thickness 16, overlying a nonreflecting surface. The aspect ratio is the vertical geometrical thickness of the cloud relative to the length of its side. Reflectivity (dashed curve) of a cloud with the same vertical optical thickness but infinite in lateral extent. The asymmetry parameter is 0.75.

0.01

0.1 1 10 Cloud Aspect Ratio

Figure 6.17: Reflectivity (solid curve) for normal incidence of a negligibly absorbing columnar cloud with square cross section, vertical optical thickness 16, overlying a nonreflecting surface. The aspect ratio is the vertical geometrical thickness of the cloud relative to the length of its side. Reflectivity (dashed curve) of a cloud with the same vertical optical thickness but infinite in lateral extent. The asymmetry parameter is 0.75.

Figure 6.17 shows the reflectivity of a square columnar cloud of fixed vertical optical thickness but varying aspect ratio, the ratio of its geometrical thickness to the length of one of its sides. Absorption is negligible, the ground below cloud is non-reflecting, and incident radiation is vertically downward. For a sufficiently small aspect ratio, the reflectivity is close to that for a cloud infinite in lateral extent. For aspect ratios greater than about 1, the difference between a finite and an infinite cloud, otherwise identical, becomes appreciable. For large aspect ratios the reflectivity plunges because of leakage of photons out the sides of the cloud. You can observe the consequences of this while flying over a field of broken clouds, all with about the same vertical thickness but different lateral dimensions. The larger clouds are brighter than the smaller clouds, not because they mysteriously have become corrupted by some highly absorbing pollutant (an explanation we have encountered) but because leakage out the sides of a cloud is equivalent to absorption within it.

Figure 6.18 shows how leaked photons contribute to the irradiance distribution below cloud. For the highest aspect ratio (11), little radiation is transmitted into the geometrical shadow of the cloud, most incident radiation either reflected or transmitted through the sides. For the smallest aspect ratio (0.09), the irradiance in the shadow of the cloud is almost what would be transmitted if the cloud were infinite in lateral extent. For an appreciable distance beyond the geometrical shadow of this cloud the downward irradiance is 5-10% greater than the clear sky value. See Problem 4.15 for more about irradiances greater than clear sky values on days with broken clouds.

Now we turn to vertical radiances above and below cloud. Again, the physical thickness is 1 km, the optical thickness 16. Figure 6.19 shows radiances for clouds with four aspect ratios. For small aspect ratios « 1), the radiance hardly varies from cloud center to edge. And this is also true for large aspect ratios (> 1), although the magnitude of the radiance is Distance (km)  Figure 6.18: Downward irradiance distribution along a line formed by the intersection with the ground of a vertical plane through the center of columnar clouds 1 km thick with vertical optical thickness 16 and asymmetry parameter 0.75 but different aspect ratios: 11 (top), 1.4 (middle), 0.09 (bottom). The incident radiation is directly overhead and the irradiances are normalized by the incident irradiance. Outside the geometrical shadow region, shown by horizontal lines, the incident irradiance is subtracted from the total (open circles).

Distance (km)

Figure 6.18: Downward irradiance distribution along a line formed by the intersection with the ground of a vertical plane through the center of columnar clouds 1 km thick with vertical optical thickness 16 and asymmetry parameter 0.75 but different aspect ratios: 11 (top), 1.4 (middle), 0.09 (bottom). The incident radiation is directly overhead and the irradiances are normalized by the incident irradiance. Outside the geometrical shadow region, shown by horizontal lines, the incident irradiance is subtracted from the total (open circles).

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0.01

0.68

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0.01

0.68

0.00

Relative Distance from Cloud Center

1.00

Figure 6.19: Upward radiance above and downward radiance below square clouds 1 km thick with vertical optical thickness 16 and asymmetry parameter 0.75 and with the aspect ratios indicated. The horizontal axis is the position along a line formed by the intersection of a vertical plane through cloud center with the top and bottom boundaries. Cloud center is indicated by 0, cloud edge by 1. The downward radiances for the two largest aspect ratios are nearly identical. The normalization factor is the radiance of a diffuse reflector with reflectivity 1. Radiances calculated by Jonathan Petters for a plane-parallel cloud and a one-dimensional solution to the radiative transfer equation agree to at least three digits with Monte Carlo calculations for the 0.01 aspect ratio cloud.

much smaller. For an aspect ratio near 1, however, the gradient of radiance is appreciable, with higher radiances near the center than near the edge. This can be observed flying over broken clouds with comparable vertical and horizontal dimensions, and has nothing fundamental to do with a possibly different drop size distribution and liquid water content near cloud edge. For the almost plane-parallel cloud (smallest aspect ratio), we can estimate the vertical radiance by other than the Monte Carlo method. Radiances obtained both ways are in agreement, which gives us confidence that the scheme outlined in Section 6.3.6 for estimating radiances is sound. Figure 6.20: Monte Carlo calculations of downward and upward solar (top) and terrestrial (bottom) irradiance profiles for a typical tropical clear-sky atmosphere. Temperature and moisture profiles from Barker et al. (2003). The sun is overhead and the reflectivity of the diffusely reflecting ground is 0.2 over the solar spectrum. Terrestrial irradiances calculated using a standard method for plane-parallel media are indistinguishable from these Monte Carlo calculations. Terrestrial radiation results from Cole (2005).

Figure 6.20: Monte Carlo calculations of downward and upward solar (top) and terrestrial (bottom) irradiance profiles for a typical tropical clear-sky atmosphere. Temperature and moisture profiles from Barker et al. (2003). The sun is overhead and the reflectivity of the diffusely reflecting ground is 0.2 over the solar spectrum. Terrestrial irradiances calculated using a standard method for plane-parallel media are indistinguishable from these Monte Carlo calculations. Terrestrial radiation results from Cole (2005).

6.4.4 Solar and Terrestrial Irradiances, Flux Divergences, and Heating Rate Profiles

For Earth's thermodynamic internal energy to be constant over a sufficiently long time (e.g., a year) solar radiation absorbed by Earth must be equal to terrestrial radiation emitted by it. This was expressed by the radiative equilibrium equations in Section 1.6. These are what might be called zero-dimensional or global equations, specifying how much is absorbed and emitted but not where. Now we turn to "where" in the atmosphere but only in one dimension, the vertical.

Figure 6.20 shows Monte Carlo calculations of downward and upward solar and terrestrial irradiance profiles in a clear sky for a temperature and moisture profile typical of the tropics. Flux Divergence (W m 3)

Flux Divergence (W m 3) 0.006

0.009

Figure 6.21: Solar, terrestrial, and total negative flux divergences corresponding to the irra-diances in Fig. 6.20. Positive values correspond to net heating, negative to net cooling. Note the change in both horizontal and vertical scales between the two panels. Terrestrial radiation results provided by Jason Cole.

Separate profiles are shown for the solar irradiance between 0.2 pm and 0.7 pm, denoted for brevity as visible, and between 0.7 pm and 4.0 pm, denoted as infrared. The sharp decrease in the downward solar infrared irradiance below 10 km must be a consequence of absorption by something because the scattering optical thickness of the atmosphere is so small at these wavelengths. Figure 2.12 indicates that the likely culprit is water vapor. Between about 50 km and 15 km the visible downward irradiance decreases faster than the infrared irradiance because of absorption of ultraviolet (0.2-0.3 pm) radiation by ozone. The visible upward irradiance Figure 6.22: Depiction by shading of spatially varying integrated cloud liquid water content, from the surface to about 100 km, obtained from model calculations by Grabowski et al. (1998). The higher the brightness, the higher the liquid water content. The horizontal resolution is 2 km.

### 400 km

Figure 6.22: Depiction by shading of spatially varying integrated cloud liquid water content, from the surface to about 100 km, obtained from model calculations by Grabowski et al. (1998). The higher the brightness, the higher the liquid water content. The horizontal resolution is 2 km.

from 15 km to 50 km does not decrease with increasing altitude because most of the photons that ozone could absorb are removed on their downward trip through the atmosphere.

The irradiances in Fig. 6.20 as well as the flux divergences and heating rates in following paragraphs are spectrally integrated quantities. Easier said than done. Any of the absorption spectra in Chapter 2 or, closer to hand, Fig. 6.14, convey the unavoidable fact that Monte Carlo calculations - indeed, any calculations - of absorption at wavenumber intervals sufficiently small to resolve all absorption peaks and troughs over solar and terrestrial spectra would require computing time not measured in seconds, minutes, or even hours but days, weeks, months, years, possibly lifetimes. So we have to compromise and use absorption coefficients suitably averaged over a small number of bands: 32 solar bands, 12 terrestrial bands for the results in this section.

It is not irradiances per se that result in local atmospheric heating or cooling but rather their spatial rate of change, that is, solar and terrestrial flux divergences, which are shown in Fig. 6.21 for the irradiance profiles in Fig. 6.20. A negative flux divergence corresponds to heating (net transfer of radiant energy into a volume), a positive flux divergence corresponds to cooling (net transfer of radiant energy out of a volume). Almost all the action occurs in the lower 30 km of the atmosphere. Below about 15 km heating by absorption of solar radiation rapidly increases as does cooling by emission of terrestrial radiation. The two divergences are almost equal and opposite. The total flux divergence is shown on an expanded horizontal scale. Between about 8 km and 14 km the total flux divergence is positive, negative between about 8 km and the ground. The vertically integrated total flux divergence is negative because this profile is for a clear sky at solar noon.  Figure 6.23: Liquid water profile for the cloud field depicted in Fig. 6.22. The top panel shows, for each layer (altitude) of boxes, the fraction of the total number of boxes that contains liquid water. The bottom panel shows the average liquid water content calculated two ways: an average over all boxes at each altitude (open circles); an average over only those boxes at each altitude that contain liquid water (closed circles).

Figure 6.23: Liquid water profile for the cloud field depicted in Fig. 6.22. The top panel shows, for each layer (altitude) of boxes, the fraction of the total number of boxes that contains liquid water. The bottom panel shows the average liquid water content calculated two ways: an average over all boxes at each altitude (open circles); an average over only those boxes at each altitude that contain liquid water (closed circles).

Although the profiles in Figs. 6.20 and 6.21 were obtained by Monte Carlo calculations, they could have been obtained, and with much less time and effort, using any of the many methods for solving plane-parallel problems. Such methods, however, cannot be used to calculate profiles for a horizontally heterogeneous cloud field such as that shown in Fig. 6.22. This figure depicts by brightness differences (shading) horizontal variations in integrated liquid water content, obtained from model calculations, in a region 400 km on a side and 100 km deep Heating Rate (K day-1)

Figure 6.24: Horizontally averaged heating rates for solar (top) and terrestrial radiation (bottom) calculated by the Monte Carlo method for the cloud field depicted in Fig. 6.22. The solar zenith angle is 60° and the reflectivity of the diffusely reflecting ground is 0.2 at all solar wavelengths. The dots show results obtained from one-dimensional radiative transfer calculations in which each 2 km by 2 km vertical column with 44 levels is treated as infinite in lateral extent and an average calculated for the 40,000 vertical columns. Terrestrial radiation results from Cole (2005).

Heating Rate (K day-1)

Figure 6.24: Horizontally averaged heating rates for solar (top) and terrestrial radiation (bottom) calculated by the Monte Carlo method for the cloud field depicted in Fig. 6.22. The solar zenith angle is 60° and the reflectivity of the diffusely reflecting ground is 0.2 at all solar wavelengths. The dots show results obtained from one-dimensional radiative transfer calculations in which each 2 km by 2 km vertical column with 44 levels is treated as infinite in lateral extent and an average calculated for the 40,000 vertical columns. Terrestrial radiation results from Cole (2005).

composed of 40,000 columns with 44 levels. Thus the region is subdivided into 1,760,000 rectangular boxes each 2 km on a side but with variable thicknesses because equal vertical distances do not correspond to equal masses. Fewer than about 4% of the boxes contain liquid water, and as evidenced by Fig. 6.22, the horizontal distribution of liquid water is not uniform. Figure 6.23 shows the vertical distribution of liquid water, which peaks at a few kilometers and vanishes above about 10 km.

Vertical heating rates, expressed as a rate of temperature change (assuming no net evaporation or condensation), arithmetically averaged at each altitude over the entire region, are shown in Fig. 6.24 for both solar and terrestrial radiation. For each of the 40,000 vertical columns with 44 levels profiles also were calculated by the Monte Carlo method treating each  Figure 6.25: Fraction at each altitude of the total number of boxes with statistically significant differences between solar radiation heating rates computed by the Monte Carlo method applied to the full domain and applied to each vertical column as if it were infinite in lateral extent for the cloud field depicted in Fig. 6.22 (top). The arithmetic average of the significant heating rate differences for the boxes in each layer (bottom); the bin size is 0.5 K day-1. Solid circles are for boxes containing cloud liquid water, and open circles are for boxes with no liquid water.

Figure 6.25: Fraction at each altitude of the total number of boxes with statistically significant differences between solar radiation heating rates computed by the Monte Carlo method applied to the full domain and applied to each vertical column as if it were infinite in lateral extent for the cloud field depicted in Fig. 6.22 (top). The arithmetic average of the significant heating rate differences for the boxes in each layer (bottom); the bin size is 0.5 K day-1. Solid circles are for boxes containing cloud liquid water, and open circles are for boxes with no liquid water.

column as if it were infinite in lateral extent, and hence independent of all other columns, then all these profiles arithmetically averaged. Agreement between the two profiles is surprisingly good - on average.

The Monte Carlo method is a statistical sampling technique that gives only estimates of quantities (e.g., heating rates) that in principle could be obtained by solving deterministic  Figure 6.26: Fraction at each altitude of the total number of boxes with statistically significant differences between terrestrial radiation heating rates computed by the Monte Carlo method applied to the full domain and applied to each vertical column as if it were infinite in lateral extent for the cloud field depicted in Fig. 6.22 (top). The arithmetic average of the significant heating rate differences for the boxes in each layer (bottom); the bin size is 0.5 K day-1. Solid circles are for boxes containing cloud liquid water, and open circles are for boxes with no liquid water. Terrestrial radiation results provided by Jason Cole.

Figure 6.26: Fraction at each altitude of the total number of boxes with statistically significant differences between terrestrial radiation heating rates computed by the Monte Carlo method applied to the full domain and applied to each vertical column as if it were infinite in lateral extent for the cloud field depicted in Fig. 6.22 (top). The arithmetic average of the significant heating rate differences for the boxes in each layer (bottom); the bin size is 0.5 K day-1. Solid circles are for boxes containing cloud liquid water, and open circles are for boxes with no liquid water. Terrestrial radiation results provided by Jason Cole.

equations such as Eq. (6.15). Wherever there are estimates, uncertainties in their values are lurking in the background. A single Monte Carlo calculation gives an estimate, similar calculations give different estimates (see Fig. 6.3), and hence curves obtained from such calculations are plots of estimated mean values even if not explicitly noted. If one does a Monte

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101 ET Heating Rate Difference (K day 1)

Figure 6.27: Fraction of the total number of boxes with statistically significant differences between radiative heating rates computed by the Monte Carlo method applied to the full domain and applied to each vertical column as if it were infinite in lateral extent for the cloud field depicted in Fig. 6.22. Solid circles are for boxes containing cloud liquid water, open circles are for boxes with no liquid water. The bin size is 0.5 K day-1. Terrestrial radiation results provided by Jason Cole.

Heating Rate Difference (K day 1)

Figure 6.27: Fraction of the total number of boxes with statistically significant differences between radiative heating rates computed by the Monte Carlo method applied to the full domain and applied to each vertical column as if it were infinite in lateral extent for the cloud field depicted in Fig. 6.22. Solid circles are for boxes containing cloud liquid water, open circles are for boxes with no liquid water. The bin size is 0.5 K day-1. Terrestrial radiation results provided by Jason Cole.

Carlo calculation of, say, reflected irradiance by following the history of 108 photons, then does this again and again and again, each result will be slightly different. By doing several sets of such calculations, say 10 or more, one can calculate a mean and its standard deviation.

This was done to obtain the solar and terrestrial heating rate profiles (Fig. 6.24) for the heterogeneous cloud field depicted in Fig. 6.22. And the horizontally averaged heating rate profiles were surprisingly close - on average. But, how do the heating rates computed the two different ways compare from one box to the next? In each solar wavelength band 108 photons were injected at random points into the top (at 100 km) of the 400 km square cloud field; the solar zenith angle was 60°. A set of calculations was repeated 10 times and a heating rate mean and its standard deviation determined for each of the 1,760,000 boxes. Then we did 10

sets of Monte Carlo calculations for each of the 40,000 vertical columns taken to be infinite in lateral extent and with 108/4 x 104 = 2.5 x 103 incident photons in each solar band and again computed a heating rate mean and standard deviation. Because Monte Carlo calculations give only estimated means with their standard deviations we have to decide whether a calculated difference for a box between the two calculations is statistically significant.

Let Q be the mean heating rate and a its standard deviation for any of the 1,760,000 boxes calculated by assuming that each column is infinite in lateral extent. Let Qc be the mean heating rate and ac its standard deviation for that same box calculated by the Monte Carlo method for the three-dimensional heterogeneous cloud field. If Q + 3a < Qc + 3ac, we take the statistically significant difference to be Q - Qc (negative). If Q - 3a > Qc + 3ac, we take the statistically significant difference to be Q - Qc (positive). Solar heating rate differences are statistically significant for 56,690 boxes, 35,761 in cloudy sky, 20,292 in clear sky, all below the highest cloudy boxes. Terrestrial heating rate differences are statistically significant for 18,098 boxes, 15,885 in cloudy sky, 2,213 in clear sky.

Figure 6.25 shows the vertical distribution of the solar heating rate differences for both cloudy and clear sky and Fig. 6.26 shows the vertical distribution of the terrestrial heating rate differences. Figure 6.27 shows how the statistically significant solar and terrestrial heating rate differences are apportioned over all boxes, those with and without cloud water. The conclusions to be drawn from all these figures are that although treating each column in a heterogeneous cloud field as if it were infinite in lateral extent and then averaging over all columns agrees well with the average for the three-dimensional cloud field, box by box differences can be large, are mostly below the highest clouds, are greater for cloudy than for clear regions, and greater for solar radiation than for terrestrial radiation. All of these results are expected on physical grounds: clouds throw a monkey wrench into one-dimensional calculations, the more so at solar wavelengths because scattering by clouds at these wavelengths is not negligible. The average over columns does not, in general, represent any particular box well. The extent to which this affects climate and weather prediction by models is difficult to say, a variation on the perennial question, Do many wrongs make a right?