Thermal physics of the atmosphere

Professor Maarten Ambaum, University of Reading, UK


Maarten Ambaum is Professor of Atmospheric Physics and Dynamics at the University of Reading and former Head of Department at the Department of Meteorology in Reading. He has a degree in theoretical physics from the University of Utrecht, The Netherlands. His research interests span a wide range of topics in the physics and mathematics of the atmosphere and oceans. His more recent work dealt with the use of electrical processes in rainfall enhancement, statistical methods in climate science, thermodynamics of the climate system, and variability of the jet stream and the storm track. He is author of a textbook on thermal physics of the atmosphere. Some of his work was also part of an art-science collaboration between the Universities of Brighton, Reading, Exeter and Sussex.


Weather forecasting requires the description, understanding, and prediction of the fluid dynamics, the thermodynamics, and their interaction in the atmosphere. In this talk Professor Ambaum discussed some of the unique challenges around the thermodynamics of the atmosphere. The underlying physics is in principle mostly known (He pointed to some parts that are yet unknown), but the operation of that physics over a vast range of scales presents truly unique and exciting challenges.

My notes from the talk (if they don’t make sense then it is entirely my fault)

There is a straightforward link between classical thermodynamics, as taught in university physics departments, and thermal physics, as presented in atmospheric science,

The atmosphere is a fluid envelope and functions as a machine/engine. It receives energy, processes it and radiates it out.


Arnold Sommerfeld on Thermodynamics

“Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two points. The third time you go through it, you know you don’t understand it, but by that time you are so used to the subject, it doesn’t bother you anymore..”


Arnold Johannes Wilhelm Sommerfeld, ForMemRS (December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretical physics.

Albert Einstein on Thermodynamics

“A law is more impressive the greater the simplicity of its premises, the more different are the kinds of things it relates, and the more extended its range of applicability. (..) It is the only physical theory of universal content, which I am convinced, that within the framework of applicability of its basic concepts will never be overthrown.”

“The law that entropy always increases — the second law of thermodynamics — holds I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much worse for Maxwell equations. If it is found to be contradicted by observation – well these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of Thermodynamics, I can give you no hope; there is nothing for it but to collapse in deepest humiliation.”

“A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore, the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.”


Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics). His work is also known for its influence on the philosophy of science.

Homer Simpson on thermodynamics

In this house we obey the laws of thermodynamics


Homer Jay Simpson is a fictional character and the protagonist of the American animated sitcom The Simpsons.

Global energy budget

Energy in to the Earth (120PW) = the energy out of the Earth (120PW) although in reality 170PW is received from the Sun, 120PW is absorbed and 50PW is reflected.

120PW corresponds to 240W/m2 which is equivalent to 225 times the total UK energy demand,


Global circulation, atmospheric heat engine. The 10PW is responsible for the weather.

Predictive thermodynamics indicates maximum entropy production.


Results for the two-box model of the Earth system for varying poleward heat flux J. Solid lines: radiation temperature of equatorial region, T1, and polar region, T2. Dashed line: irreversible entropy production due to the poleward meridional heat flux. The entropy production is expressed per unit area on Earth.

Thermal Physics of the Atmosphere. Maarten H. P. Ambaum, Department of Meteorology, University of Reading. © 2010 by John Wiley & Sons, Ltd

It shows a parabolic shaped curve with a peak at 10PW.

The Earth’s atmosphere is a fluid envelope. Its thickness is equivalent to a postage stamp on a football.

Two-dimensional turbulence can be predicted.

Hydrostatic balance

In a fluid at rest the net force on any fluid parcel has to vanish. So, in a gravitational field the gravitational force has to be balanced by an internal force due to a decrease of pressure with height. This is called hydrostatic balance.

We can use thermodynamics to calculate the hydrostatic pressure decrease with height.

In fluid mechanics, a fluid is said to be in hydrostatic equilibrium or hydrostatic balance when it is at rest, or when the flow velocity for each parcel of fluid is constant over time. This occurs when external forces such as gravity are balanced by a pressure-gradient force. For instance, the pressure-gradient force prevents gravity from collapsing Earth’s atmosphere into a thin, dense shell, whereas gravity prevents the pressure gradient force from diffusing the atmosphere into space.



Although pressure and density decrease with altitude in the atmosphere, temperature remains relatively constant or even increases with altitude at certain regions, so it may well be used to distinguish between atmospheric layers. Based on temperature, the atmosphere is divided into four layers: the troposphere, stratosphere, mesosphere and thermosphere, as illustrated in diagram below. Based on composition and mixing, the atmosphere can be divided into two major layers, the homosphere and heterosphere, defined by whether the atmospheric gases are well mixed. The homosphere, starting from surface, includes the troposphere, stratosphere, mesosphere and the lowest part of the thermosphere. From there on lies the heterosphere where the chemical composition varies with altitude, and gases are stratified due to their molecular weight.


Structure of the atmosphere and its layers based on temperature distribution during summer (orange line) and winter (blue line). (Adapted from http://www.

There are some weird temperatures.

Integrating the hydrostatic equation between two height levels, we find that the mass between the two levels is proportional to the pressure difference. The mass M per unit area between two height levels is ≈ (p0 − p1)/g where we have assumed that g is constant between the two levels. In other words, equal pressure differences correspond to equal masses.

Global energy budget and the greenhouse effect

Mean surface temperature – Earth’s radiation temperature = greenhouse effect = 288 – 255 = 33K

The Sun’s radius is about RS = 6.96 × 108m and its surface temperature is about TS = 5780K. The Sun can be approximately treated as a black body. From Stefan’s law the total radiative energy output of the Sun can be found. When this radiation has reached the Earth’s orbit (average orbital radius rE = 149.5 × 109m) this energy is used to irradiate a much larger area, 4pr2E instead of 4pR2S. So the radiative energy flux per unit area S0 at the distance of the Earth can be found from 4pr2E S0 = 4pR2S sT4S .

The flux S0 is called the solar constant, or the total solar irradiance, and it has a value of about S0 = 1366 W m−2.

A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. (It does not only absorb radiation, but can also emit radiation. The name “black body” is given because it absorbs radiation in all frequencies, not because it only absorbs.)

The thermal energy radiated by a blackbody radiator per second per unit area is proportional to the fourth power of the absolute temperature.


Paths of radiative fluxes in a system made of a single, cloud-free slab atmosphere at temperature TA, with unit emissivity in the long-wave and zero emissivity in the short-wave part of the spectrum. The insolation is S0/4. The planetary short-wave albedo s is here provided by the Earth’s surface alone.

Greenhouse effect isn’t really the correct term. A better name is the blanket effect.

The effect of greenhouse gases alone cannot account for any temperature difference. The greenhouse effect is where molecules in the atmosphere absorb infrared radiation and radiate it in all directions. This means that that about one half is radiated downward toward Earth’s surface. The term cloud blanket effect is used to denote phenomenon in which the underside of a cloud reflects back down the infrared radiation that the Earth’s surface is radiating upward.

Planck Law

Planck’s law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T, when there is no net flow of matter or energy between the body and its environment.

This law is the foundation of modern thermodynamics

Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering processes. The equation of radiative transfer describes these interactions mathematically. Equations of radiative transfer have application in a wide variety of subjects including optics, astrophysics, atmospheric science, and remote sensing. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more realistic media, with complex multiple scattering effects, numerical methods are required.

Earth’s global energy budget by Kevin E. Trenberth, John T. Fasullo, and Jeffrey Kiehl

An update of the Earth’s global annual mean energy budget is given in the light of new observations and analyses. Changes over time and contributions from the land and ocean domains are also detailed.

The energy budget isn’t quite in balance

The image below shows the global annual mean Earth’s energy budget for the Mar 2000 to May 2004 period (W m−2). The broad arrows indicate the schematic flow of energy in proportion to their importance.


Global hydrological cycle is constrained by “basic” thermodynamics

Constraints on future changes in climate and the hydrologic cycle

Myles R. Allen & William J. Ingram

Nature volume 419, pages228–232(2002)

6.5%/7% increase in water vapour in atmosphere for each degree increase in temperature. However, rain is not limited by the amount of water in the atmosphere. It is limited by the energy budget of the atmosphere.

Below is a graph from Allen & Ingram (2002) showing the model response of rainfall under temperature changes from GHG increases. The dashed line marked “C-C” is the famous (in climate physics) Clausius–Clapeyron relation which, at current temperatures, shows a 6.5%/7% change in water vapour per ºC of warming. The red triangles are the precipitation changes from model simulations showing about half of that.

The Clausius-Clapeyron relation is an equation for a single-component system consisting of two phases in thermodynamic equilibrium at constant absolute temperature T and constant pressure P. A curve in a two-dimensional thermodynamical diagram that separates two phases in equilibrium is known as a coexistence curve. The Clausius–Clapeyron relation gives the slope of the coexistence curve in the P-T diagram:



Global-mean temperature and precipitation changes in AOGCM simulations (scatter plots), and probability distributions obtained by requiring consistency with recent observations (curves). Red triangles show global-mean temperature and precipitation changes in a wide-range of equilibrium CO2 – doubling experiments with simple thermodynamic (‘slab’) oceans, with the red line showing the best-fit line (least squares) linear relationship. Green diamonds show the same, at the time of CO2 doubling, for these CMIP-2 models for which data is available. Blue crosses are the green diamonds adjusted for disequilibrium in the CMIP-2 runs. This removes the bias with the best-fit line through the ‘slab’ experiments. All these points would lie on the dashed line labelled C-C if precipitation were to follow the Clausius-Clapeyron relation. The green dashed curve is the observationally constrained estimate of the distribution of global-mean temperature change at the time of CO2 doubling. The blue curve is the same, but adjusted for disequilibrium like the blue crosses. The red curve shows the distribution of global-mean precipitation changes implied by the blue curve, assuming the same straight-line relationship observed in the ‘slab’ experiments, with the same amount of scatter (assumed Gaussian).

AOGCM is Atmosphere-Ocean General Circulation Model WCRP Coupled Model Intercomparison Project

A slab ocean model configuration allows the user to run a full atmosphere model on top a much-simplified ocean model. The simplified ocean model is essentially a 0-dimension model running at every ocean point on the globe and meant to be an approximation of the well-mixed ocean mixed layer. The thermodynamic calculation should have a mixed-layer depth specified, and the temperature of the slab is calculated based on the depth and the surface energy fluxes. This configuration is useful for understanding the climate sensitivity of the whole coupled simulation, where, on the timescale of decades, the ocean mixed layer depth is the dominant player. It also useful for a simple analysis of the coupled system where only simple interactions with just the mixed layer ocean are of interest e.g. the Madden Julian Oscillation (i.e., situations in which the role of ocean dynamics is minimal).

So, adding carbon dioxide causes less energy to be radiated away. The atmosphere is warmer and can hold on to more water vapour meaning less rain.

Entropy for dry and moist air

Entropy is the measure of a system’s thermal energy per unit temperature that is unavailable for doing useful work. Because work is obtained from ordered molecular motion, the amount of entropy is also a measure of the molecular disorder, or randomness, of a system. The concept of entropy provides deep insight into the direction of spontaneous change for many everyday phenomena.

The atmosphere has been likened to a giant thermodynamic engine in which disorganized energy is transformed into the organized kinetic energy of the winds while the general circulation of the atmosphere can be regarded as simply being driven by temperature differences between the polar and equatorial regions.

The notions of entropy and its production in equilibrium and nonequilibrium processes form the basis of modern thermodynamics and statistical physics.

Entropy is a constraint on the structure of tropical convection. Entropy becomes more complex with increase presence of moisture in the atmosphere.

The temperature at the tropics is generally higher so you would expect the hotter air to go up. This not exactly what happens. Tropic convection can be a very rapid, violent motion.


These are illustrations of convective storms over cooler and warmer surface waters in the tropics.

The arrows going up and down inside the storm indicate the transport of ice and rain within the clouds. Ice is transported up into the anvil portion of the cloud where it can reflect sunlight. Rainwater gets removed from the storm, as depicted by the downward arrows and doesn’t have a chance to reflect sunlight. Over warmer waters, the height, thickness and cloud area of the anvil increase; the updrafts are stronger, and precipitation increases. Credit: NASA Graphics

Clouds are created in little blocks. It is colder at the top of a cloud than the top.

A tephigram is one of four thermodynamic diagrams commonly used in weather analysis and forecasting. The name evolved from the original name “T-f -gram” to describe the axes of temperature (T) and entropy (f) used to create the plot.

Reading a Tephigram


Cloud formation

An embryonic cloud droplet (molecular cluster) can be formed by collision of water vapor molecules. Once it exists, it may grow or decay depending on ambient water vapour pressure.


The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapour interface, such as the surface of a droplet. The vapour pressure at a convex curved surface is higher than that at a flat surface. The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials.

The Kelvin effect is surface tension on a drop of pure water. In small droplets water molecules have less connections to their surrounding molecules and easier to evaporate. Smaller drops have greater curvature and require greater vapour pressure to keep water molecules from evaporating away. As the drop size increases, the required saturation ratio for equilibrium decreases. If the required saturation ratio is exceeded, the drop will grow. But due to the finite size of the water molecule, the limit of surface tension occurs at 400% of relative humidity/saturation ratio.

Homogeneous nucleation of drops needs 400% relative humidity to start the formation of droplets by condensation. This never occurs in the atmosphere. Heterogeneous nucleation involves water molecules bonding to non-water particles (impurities) called condensation nuclei.

The Raoult effect occurs when impurities are added to the water and water drop formation occurs more easily. The impurities replace water molecules.

A Kohler curve combines both the curvature and solute effects i.e. the Kelvin and Raoult effect.

The Kohler curve becomes a maximum at the critical radius denoted by r.

The value of equilibrium saturation ratio at the critical radius is called the critical saturation ratio, and is denoted as S.

As the radius gets larger the equilibrium saturation ratio goes asymptotically to unity (relative humidity of 100%).

Adding more impurity to a droplet decreases S, and increases r.

You can replace the Raoult impurities with electric charges. These stabilise the drop and actually helps drop formation. The charges can be produced by cosmic rays and cloud seeding and the process is used in the Wilson cloud chamber.

Cosmic rays are high-energy protons and atomic nuclei which move through space at nearly the speed of light. They originate from the sun, from outside of the solar system, and from distant galaxies. Upon impact with the Earth’s atmosphere, cosmic rays can produce showers of secondary particles that sometimes reach the surface.

Cloud seeding is a type of weather modification that aims to change the amount or type of precipitation that falls from clouds by dispersing substances into the air that serve as cloud condensation or ice nuclei, which alter the microphysical processes within the cloud. The usual intent is to increase precipitation (rain or snow).

A cloud chamber, also known as a Wilson cloud chamber, is a particle detector used for visualizing the passage of ionizing radiation. It consists of a sealed environment containing a supersaturated vapour of water or alcohol. An energetic charged particle (for example, an alpha or beta particle) interacts with the gaseous mixture by knocking electrons off gas molecules via electrostatic forces during collisions, resulting in a trail of ionized gas particles. The resulting ions act as condensation centres around which a mist-like trail of small droplets form if the gas mixture is at the point of condensation. These droplets are visible as a “cloud” track that persist for several seconds while the droplets fall through the vapor. These tracks have characteristic shapes. For example, an alpha particle track is thick and straight, while an electron track is wispy and shows more evidence of deflections by collisions.

Charles Thomson Rees Wilson (1869–1959), a Scottish physicist, is credited with inventing the cloud chamber. Inspired by sightings of the Brocken spectre while working on the summit of Ben Nevis in 1894, he began to develop expansion chambers for studying cloud formation and optical phenomena in moist air. Very rapidly he discovered that ions could act as centres for water droplet formation in such chambers. He pursued the application of this discovery and perfected the first cloud chamber in 1911.


Clouds are formed when air contains as much water vapour (gas) as it can hold. This is called the saturation point, and it can be reached in two ways. First, moisture accumulates until it reaches the maximum amount the volume of air can hold. The other method reduces the temperature of the moisture filled air, which in turn lowers the amount of moisture it can contain. Saturation, therefore, is reached through evaporation and condensation, respectively. When saturation occurs, moisture becomes visible water droplets in the form of fog and clouds.

Condensation by itself does not cause precipitation (rain, snow, sleet, hail). The moisture in clouds must become heavy enough to succumb to gravity and return to earth’s surface. This occurs through two processes. In cold clouds ice crystals and water droplets exist side by side. Due to an imbalance of water vapor pressure, the water droplets transfer to the ice crystals. The crystals eventually grow heavy enough to fall to earth. In the second process, water droplets in warm clouds collide and change their electric charge. Droplets of unlike charge attract one another and merge, thereby growing until they have sufficient weight to fall.

Two basic ingredients are needed for precipitation water and impurities. Impurities are needed for condensation nuclei, sites on which water vapour may condense or deposit as a liquid or solid. Certain types and shapes of dust and salt particles, such as sea salts and clay, make the best condensation nuclei.

The precipitation ladder:

11. precipitation

10. droplet growth

9. buoyancy/cloudiness

8. condensation

7. saturation

6. humidification

5. cooling

4. expansion

3. ascent

2. dirty air

1. water vapor


NASA Apollo 11 image of the Earth above the Moon taken on 20 July 1969 at around 05h UTC (left) and the corresponding pseudo-image generated from a 29-hour 28-km resolution ECMWF forecast initialised from ERA40 data (right). Both images are centred over the western Pacific Ocean, with the equator running almost vertically and Australia visible on the left in brownish colour.

To celebrate the 50th anniversary of that first Moon landing, Philippe Lopez decided to investigate whether a numerical weather forecast obtained with the latest operational ECMWF Integrated Forecasting System (IFS) could reproduce the iconic image.

The comparison displayed above shows that the combination of initial conditions from the reanalysis and a recent version of the ECMWF IFS can simulate cloud patterns (in white) that agree reasonably well with those seen on the NASA image. This is remarkable given the limited number of observations that were used to produce ERA40 reanalyses for 1969 (no satellite data). It is also likely that the use of a more complex visible image simulator would provide an even more realistic image from the model.

Then he turned his attention to the famous “Blue Marble” image taken by the Apollo 17 mission on the way to the Moon on 7 December 1972. This image was compared with an IFS forecast initialised from ERA5, the latest reanalysis produced by ECMWF for the Copernicus Climate Change Service. The forecast was run for 11 hours at ECMWF’s current deterministic forecast resolution of 9 km. Here, this higher forecast resolution was necessary to account for the higher level of detail present in the NASA image compared to the Apollo 11 case.


NASA Apollo 17 image of the Earth taken on 7 December 1972 at 10h39 UTC (left) and the corresponding pseudo-image generated from an 11-hour 9-km resolution ECMWF forecast initialised from ERA5 reanalysis (right).

Answer to a question

The heat produced within the Earth due to radioactive decay has very limited effect on the weather/climate.

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