Thermal Structure of the Troposphere

Thermal Structure of the Troposphere

The preliminary Excel model of the thermal structure of the troposphere is based on a global energy budget. The model calculates the flow of energy between the troposphere and the surface of the Earth based on incident sunlight and atmospheric and surface optical and thermal properties specified by the analyst. The model also estimates the thermal structure of the troposphere which is defined as the temperature as a function of altitude including the surface of the planet and ranging to 10 km. The textbook standard temperature profile is described by the normal lapse rate of 6.5K per km. The global average surface temperature is reported to be 288K and temperature decreases 6.5K/km typically in the troposphere before rising and falling and rising again in the thin upper atmosphere where extreme ultraviolet is absorbed by ozone and other gases.

The model fills the niche between simple global energy budgets that treat the atmosphere as a single layer and advanced research models that are extremely complex. Most of the mass of the atmosphere is in the troposphere and the troposphere model has many simplifying assumptions especially concerning the upper atmosphere. The model is preliminary because there are many effects that can be incorporated if time allows and because many assumptions and approximations need to be investigated further. Specifically, the model ignores the relationship between composition and optical and thermal properties, distribution of water vapor with altitude, the upper atmosphere, wavelength-dependent effects other than the broad bands of short vs. long wavelength, latitude dependent effects, and variations due to clouds and due to surface features like oceans vs. land.

The model was used to investigate the Earth’s atmosphere with and without the benefit of latent heat transport, the effect of variations in greenhouse gas long wavelength absorption, the effect of a faint young Sun (four billion years ago), the effect of a highly reflecting surface (Snowball Earth nearly one billion years ago), and the effects of higher solar flux and extreme greenhouses as found on Venus.

Figure 1 The spectrum of sunlight includes ultraviolet, visible, and near infrared wavelengths much of which is absorbed or scattered by the atmosphere even without clouds present

Figure 1 shows the spectrum of sunlight incident on the Earth’s atmosphere when the Sun is directly overhead. On a cloudless day, the direct beam that reaches the Earth’s surface is reduced by scattering losses shown in blue and by absorption shown in black. The atmosphere of the planet is continuously intercepting a solar flux of 1372 watts per square meter if the slight ellipticity of the Earth’s orbit is ignored. Locally this flux varies from zero on the side of the rotating planet that is facing away from the Sun to 1372 w/m2 where the Sun is directly overhead, such as the Equator at local noon on an Equinox.

The nearly spherical planet intercepts a disk of sunlight equal to πR2, where R is the average radius of planet, which bulges at the Equator due to the rapid rotational velocity. Although this solar power is non-uniformly distributed over a sphere of surface area 4πR2, the average global flux is 343 w/m2 at all times, 24 hours a day all year long. The local daily average at the Equator on an Equinox is 1372w/m2/2 or 686 w/m2. In the winter, the daily average at a pole is zero. The global average flux is therefore one fourth of the peak or 343 w/m2.

Some of the flux incident on the top of the atmosphere is absorbed or scattered before reaching the surface of the planet. Absorption and scattering varies with path length through the atmosphere which varies with altitude, latitude, seasons, and time of day. Absorption and scattering by clouds and by clear skies varies with wavelength and with composition, especially moisture content. Absorption by the Earth’s surface varies from deep blue oceans to bare land to vegetation.

Figure 2 All of the solar energy absorbed by the Earth’s surface and atmosphere is radiated as long wavelength infrared “blackbody” radiation

Figure 2 shows that the energy radiated in the long wavelength infrared has the same area as the energy absorbed by the Earth even though their spectral distributions are very different. All of the flux intercepted by the Earth is returned to space in order to maintain thermodynamic equilibrium. Since the Earth’s albedo is about 30%, the remaining 70% of the incident flux is absorbed by the atmosphere or surface which radiates it away as blackbody radiation into space. The atmosphere and surface of the Earth transfer energy by radiation, convection, conduction, and evaporation/condensation of moisture, but only radiation can transport energy into space. Figure 3 below shows the absorption spectrum of several major triatomic greenhouse gases that trap outgoing long wavelength radiation and help elevate the average surface temperature of the Earth to 288K well above the effective atmospheric temperature of 255K.

Figure 3 The Earth’s average surface temperature is 288K because of long wavelength absorption by greenhouse gases

Figure 4 Simplified Average Global Energy Flow Budget annotated with model parameters

Figure 4 is an average global energy budget diagram (simplified from Salby page 45) that quantifies the flow of energy on Earth. Unlike Salby, this diagram combines the properties of the clear sky and the clouds to provide an unambiguous interpretation of the budget factors. It is possible to estimate the surface temperature of the Earth from the energy flux radiated by the surface. Figure 5 shows the thermal structure of the atmosphere including the normal lapse rate of the troposphere which is the focus of attention in the preliminary model.

Figure 5 Thermal structure of the atmosphere showing the 6.5K per km normal lapse rate in the troposphere

Figure 6 Density vs. altitude in the troposphere

The preliminary model of the thermal structure of the troposphere includes inputs for the incident solar flux and the absorption and scattering properties of the atmosphere and the surface of the Earth. It also has a factor to account for ocean evaporation which transports latent heat to the atmosphere and a preliminary formula to represent non-radiative transport within the atmosphere due to convection. The troposphere includes about the 80% of the mass of the atmosphere that is below 10 km altitude and it is a fully convective region unlike the upper atmosphere.The density chart in Figure 6 shows that most of the atmosphere is in the troposphere and is the justification for the current emphasis on this region. The twenty points shown in the density chart are used to weight absorption and scattering fractions in the Excel model.

In our model, the troposphere is divided into 20 layers represented by 20 rows in the Excel spreadsheet. The model could have been divided into 10 layers or 40 layers or any other number. Energy is transported by convection as well as electromagnetic radiation throughout the troposphere, so the concept of layers is a numerical modeling term, not a description of physical features. The model calculates the energy flow in each layer and estimates the temperature of the Earth’s surface and each layer of the troposphere by using a modified version of the Stefan Boltzmann blackbody radiation law that accounts for the partial transparency of the atmosphere. The outputs of the model include all of the fluxes in the energy budget and a temperature profile in the troposphere which can be compared to the standard lapse rate of 6.5K per km.

The model inputs can be varied to analyze the effect of variations of in the properties of the atmosphere, the surface of the planet, and the output of the Sun which is known to change over billions of years. The model relates properties of the planet to energy flow but it does not derive the properties from fundamental characteristics like the density and composition of the atmosphere, oceans, and land. A more advanced model could include these relationships but would probably need to include wavelength dependent effects. A more advanced model could include local variations from the global averages and could include horizontal energy flow on the Earth’s surface.

Altitude dependent properties can be modeled in different rows of the current model, but the cases analyzed in this study had properties that only varied with relative atmospheric density. The main goal of the current effort was to develop a model and to exercise it to investigate the potential for using Excel to model global energy flow.

Figure 7 Schematic diagram of incoming and outgoing SW and LW energy flow, latent heat, and convection

Figure 7 shows a diagram of energy flow summarizing the processes that were modeled. The diagram shows how quickly a multi-layer model becomes complex. The particles in the atmosphere scatter short wavelength (ultraviolet, visible, and near infrared) sunlight in all directions, some of it generally upward and some of it generally downward. The model assumes that each layer scatters SW equal amounts up and down and that this energy is simply included in the general flow of incoming or outgoing SW radiation and subject to additional scattering and absorption.

The incoming SW flux is the flux exiting the bottom of a layer and equals the incoming flux from the adjacent layer above it reduced by the total absorption and half of the scattered incoming of the layer itself. The outgoing SW flux is the flux exiting the top of a layer and equals the outgoing flux from the adjacent layer below it reduced by the total absorption and half of the scattered outgoing SW flux of the layer itself. The incoming SW flux for the top layer is the incident SW flux from the Sun. The outgoing SW flux for the lowest layer is the SW scattered by the surface.

Layers of the atmosphere absorb SW flux, long wavelength (LW) flux, sensible heat from conduction, and latent heat (LH) from evaporation/condensation. Regardless of the source, energy absorbed by a layer of the atmosphere is re-radiated in all directions as LW infrared, some of it generally upwelling and some of it generally downwelling. In the model, radiant LW flux either goes up or down and it is simply included in the general flow of LW radiation that is either incoming or outgoing. In the model, LW is either transmitted or absorbed and re-radiated; it is not scattered by aerosols or particulates or by the surface of the Earth.

The incoming LW flux is the flux exiting the bottom of a layer and equals the incoming flux transmitted and radiated downward by the adjacent layer above it reduced by the total absorption of the layer itself. The outgoing LW flux is the flux exiting the top of a layer and equals the outgoing flux transmitted and radiated by the adjacent layer below it reduced by the total absorption of the layer itself. The incoming LW flux for the top layer is zero because no LW flux is incident from space. The outgoing LW flux for the lowest layer is the LW radiated by the surface.

The model includes an input for radiative transport of LW flux from the surface. In the absence of sensible and latent heat, this factor would be unity. For the baseline global energy budget model, this factor is approximately 0.79 and the remaining 0.21 is transported by latent heat (0.18) and sensible heat (0.03). For a planet with a solid surface, latent heat is zero and the factor is 0.97. As in all of the preliminary analyses, these factors are taken as fixed properties of a system and are not related to the composition or other fundamental characteristics of the system.

Since the model only has a moderate number of layers and they are only partially transmitting, there is a temperature gradient across the layer which influences the fraction of absorbed energy radiated up vs. down. There is no simple way to determine the fraction that is upwelling, so reasonable estimates are based on the assumption that the effective temperature of downwelling radiation is lower than the effective temperature of upwelling radiation of the layer directly below it.

Figure 8 Formulas in preliminary Excel model spreadsheet

Figure 8 shows the preliminary model formulas in their Excel spreadsheet. Each layer of the atmosphere is represented by a different row, starting at the top of the troposphere and the planetary surface is in the row below the atmospheric layers. Some of the columns of the spreadsheet have color coded cells to highlight locations of SW and LW flux formulas.

The inputs are shown in dark red text in pink shaded cells and the key outputs are shown in red text. In addition to the flux calculations, the last three columns show temperature calculations based on the equation for blackbody radiation in a partially absorbing medium like the atmosphere. The inputs and outputs are scattered all over the spreadsheet. It is possible to add altitude dependent effects by altering the rows of some of the columns instead of equating the rows. In this model, the only altitude dependent effect used is density.

The most important feature of the model is that the cells have many circular references that account for the feedback loops as LW flux is radiated back and forth between layers of the atmosphere and the surface of the Earth. Excel has a special option that allows the user to automatically override the circular reference error message and to iterate the spreadsheet calculation until it meets a convergence criteria or performs a specific number of iterations.

Figure 9 Model inputs and outputs for baseline case T1 and goals and for cases T2, G1, and G2

In order to make it easy to run the model and to interpret the results, the spreadsheet inputs are linked to the inputs in the first column of a second spreadsheet shown in Figure 9. The cells below the inputs are linked to the outputs of the spreadsheet. The second column lists ten goals for the output fluxes of the baseline case: these values correspond to the global energy budget values in Figure 4.