MODEL DESCRIPTION

This is a very simple model of soil temperatures on the Moon’s Equator, intended to be used for teaching and learning about weather and climate. It’s just one-dimensional heat transfer through the soil forced by incoming solar radiation that heats the soil and outgoing thermal radiation that cools it at the surface.

The Moon is “tidally locked” to the Earth so that the time required to rotate once on its axis is exactly the same as the period of its orbit around the Earth (that is, the length of the Moon’s “day” is the same as the length of a Lunar “month:” 29 Earth days, 12 hours, 44 minutes).

The Moon has no atmosphere, and the “tilt” of its axis relative to the orbital plane of the Earth-Moon system around the Sun is negligible. Therefore the calculation of insolation at the Lunar surface is much simpler than the one you did for the Earth in problem 1 above. Assuming that the tilt of the Moon’s axis is zero, declination = 0. So at the Moon’s equator the cosine of the solar zenith angle is just equal to the cosine of the hour angle. Much simpler than the Earth!

On Earth the surface energy budget is given by Bonan’s equation 13.1:

\[ \begin{aligned} \rho c \frac{\Delta T}{\Delta t}\Delta z = (S\downarrow - S\uparrow + L\downarrow - L\uparrow) - H - \lambda E = G \end{aligned} \]

On the left-hand side, \(\rho\) is the density of the soil (kg m^-3), \(c\) is the heat capacity of the soil (J m^-3 K^-1), \(\Delta T\) is the change in soil temperature (Kelvin) over a time step (\(\Delta t\), seconds) for each soil layer whose thickness is \(\Delta z\) (meters). On the right-hand side the four terms inside the parentheses are the fluxes of shortwave radiation down and up, and longwave radiation down and up. \(H\) is the turbulent flux of sensible heat, \(\lambda\) is the latent heat of evaporation of water, \(E\) is the rate of evaporation of water from the surface, and \(G\) is the change in ground heat storage. All terms in this equation have the units of W m^-2.

The Moon’s surface energy budget is much simpler! There’s no water to evaporate and no turbulent air to carry sensible heat. There’s a little bit of longwave radiation emitted by the Earth that heats the Moon’s surface, but on the far side (which we never see) this is zero so we can neglect this. All of the upward flux of shortwave radiation is just the incoming solar radiation times the Moon’s albedo. To calculate upward longwave emission we can assume the surface emits as a blackbody. So the Moon’s surface energy budget is

\[ \begin{aligned} \rho c \frac{\Delta T}{\Delta t}\Delta z = S\downarrow (1-albedo) - \sigma T_S^4 \end{aligned} \]

In visible wavelengths, the Moon is surprisingly dark, with an albedo of just 7% (like asphalt!). Across the whole Solar spectrum it’s a little brighter, so please use the broadband albedo of 11%.

To calculate the temperature, divide the Lunar “soil” into 15 layers. Make the top layer thinnest (0.02 m = 2 cm), and then make each layer underneath 1.5 times as thick as the one above. This will get you 15 lunar soil temperatures down to about 17.5 meters depth.

The downward flux of heat at the top of each layer is proportional to the difference in temperature between adjacent layers according to Bonan’s equation 9.1:

\[ \begin{aligned} \rho c \frac{\Delta T}{\Delta t}\Delta z = -k \frac{\Delta T}{\Delta z} \end{aligned} \]

The proportionality constant \(k\) is called the “thermal conductivity.” The negative sign is there because the layers are counted down from the surface but the heat flux is defined as positive upward, away from the surface.

Changes in soil temperature are determined by the difference in the heat flux into and out of each layer. This is called the “heat flux divergence,” given by Bonan’s equation 9.2:

\[ \begin{aligned} F \uparrow = -k \frac{\Delta T}{\Delta z} = -(F_Z - F_{z+\Delta z}) \end{aligned} \]

There’s a nice worked example of this on Bonan page 133. The book also gives typical values of heat capacity and thermal conductivity for different kinds of soil.

The Lunar “soil” is actually made up of mineral grains blasted to smithereens by billions of years of impact bombardment. Unlike Earth soils, the pore space is not filled with water or air. It’s just vacuum, so the heat flow through the lunar soil is much slower than on Earth, depending on conductance and thermal radiation between adjacent grains!

According to Colozza (1991), the volumetric heat capacity of lunar regolith is about 4.4 million J m^-3 K^-1. Use this value for \(ρc\) in the equation above. The thermal conductivity of lunar “soil” is only about 0.025 W m^-1 K^-1. This is about 100 times less conductive than soils on Earth!

To calculate the changes in temperature with depth and time you’ll need to add solar radiation to the surface, subtract upward longwave, and step the heat flux equations forward in time. You’ll find that if you use a time step that’s too long, the numerical solution will become unstable and you’ll get crazy values (for example negative temperatures Kelvin). I find I can get away with a time step of a few hours.

The program that does the calculation is very simple, and the code that controls this website is surprisingly simple too! It is all written in the programming language R. You can read all about it on the “Website Code” tab to the right.