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2021-01-18
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Next week's colloquium by Prof. Jennifer Kay (University of Colorado, Boulder, USA) is entitled How do clouds affect global warming?
.
Gases and liquids are collectively known as fluids. In a case of good timing, you are already learning basic aspects of the physics of fluids in PHY152. The sections of Knight's textbook that we make use of here are 14.1 Fluids, 14.2 Pressure, and 14.3 Buoyancy.
Knight also describes key concepts of thermal physics in Chapters 18 and 19, and you may have already heard of some of them.
When reading these parts of Knight, don't worry about the details of calculational examples, just focus on the main ideas. The total reading here from the bits of Chapters 18 and 19 is about 10 pages.
The material below is provided in case you want it, but you can skip it if you run out of time. I hope it helps explain some aspects of climate physics for you in an accessible way. It should be useful not only for the Week 2 colloquium, but also for the Week 3 and Week 10 colloquia. This stuff is roughly five pages long.
The simplest type of atmospheric model assumes that the only dependence of interesting variables is on altitude, and ignores heat transfer. Intuitively, we expect that the air pressure should be greater lower down in the atmosphere, because the weight of all the air further up is pushing down on it. We can be more quantitative if we consider an atmospheric region close enough to Earth's surface that the acceleration due to gravity $g$ and the temperature $T$ are both roughly constant over a range of altitudes $h$ above the ground.
Let the pressure be $p$ and the air density be $\rho$. As we know from Knight chapter 14, the magnitude of a pressure difference in a fluid between two heights is $\rho\, g\, \Delta h$, where $\Delta h$ is the magnitude of the height difference. For us, air pressure at height $(h+dh)$ above Earth's surface must be lower than at height $h$ because there is less weight of air pushing down on it, so
$$
dp = - \rho\,g\,dh \,.
$$
To be able to find the pressure as a function of height $p(h)$, we need to know how the density of the air $\rho$ is related to the air pressure $p$ and other variables. Such an equation for a fluid is known as its equation of state. If we assume that air is the simplest possible type of gas, called an ideal gas
(a concept discussed in Knight 18.6), its equation of state is
$$
p \,V = N\, k_B \,T \,,
$$
where $V$ is the volume, $N$ is the number of molecules, $k_B$ is Boltzmann's constant, and $T$ is the temperature in Kelvins. We do not need to know where this comes from in order to use it. Rearranging, we can find the density of the gas: it is the number of molecules $N$ times the mass $m$ of a molecule divided by the volume $V$,
$$
\rho = {\frac{N\,m}{V}} = {\frac{p \, m}{k_B\, T}}\,.
$$
Putting the above ingredients together, we have
$$
dp = -{\frac{p\,m\,g}{k_B\, T}} dh = -{\frac{p}{H}} dh \,, \quad {\rm where}
\quad H = {\frac{k_BT}{mg}} \,.
$$
$H$ is known as the scale height. $H$ is constant in our simple model, because $g$ and $T$ are assumed to be approximately constant, while $m$ and $k_B$ are always constant. So we have $dp/p = - dh/H$. We know how to integrate an equation like this: it gives
$$
p(h) = p_0 \exp\left( -{\frac{h}{H}}\right)\,,
$$
where $p_0$ is the pressure at $h=0$. Finally, since the equation of state in our simple model has linear proportionality between $p$ and $\rho$ at constant $T$, this implies that the air density $\rho$ also falls off exponentially with the same height dependence. So the air is indeed thinner higher up - exponentially thinner.
Fun fact: the reduced atmospheric pressure higher up reduces the boiling point of water, e.g. at 10,000 ft / 3,048 m it is only 89.8 degrees Celsius. This makes cooking while camping at high altitude a new kind of skill!
Thinning of the air at high altitude is why commercial airliners pressurize the air inside the cabin, so that pilots and crew and passengers don't pass out or die from lack of oxygen. The fuselage of the plane has to be strong enough to withstand the pressure differential. Airliners also heat the air using waste heat from the engines, because the outside temperature up at 35,000 ft is too cold even for a prairie boy.
Generally, heat can be transferred by conduction, convection, or radiation. Radiation emitted by the Sun peaks in the visible part of the electromagnetic spectrum (at green), and also has noticeable power in the near-infrared and a bit in the ultraviolet (UV). The surface of the Earth is colder than the surface of the Sun, so it emits EM radiation at longer wavelengths, in the near infrared (IR). If Earth had no atmosphere, radiation would be the only relevant mechanism of heat exchange between the Earth and outer space, and the surface of the earth would be below freezing. Air temperature would also exponentially fall with altitude. We have mostly liquid oceans because Earth has an atmosphere that acts kinda like a blanket.
The atmosphere of Earth contains mostly nitrogen and oxygen gases, small variable amounts of water vapour, and trace amounts of other gases like argon and carbon dioxide. Their molecules or atoms can absorb radiation from certain parts of the electromagnetic spectrum. These can then re-emit energy, in any direction. Which kinds of species absorb EM radiation at what wavelength depends on the kinds of transitions involved: atomic transitions of electrons (generally in the UV), vibrational modes of molecules (near-IR), or rotational modes of molecules (far-IR). Visible light tends to fall in the gap between common electronic and vibrational transitions, and so atmospheric gases do not absorb very much sunlight. Gases that absorb radiation in the longer wavelength near-IR range emitted by earth are called greenhouse gases. Water vapour, carbon dioxide, methane, and ozone are the four main culprits that help trap heat this way, and water vapour is the most important among them.
Air is a pretty poor thermal conductor, so conduction is only important to atmospheric dynamics very close to Earth's surface. Most of the rest of the energy transfer relevant to atmospheric physics occurs via convection. Convection is more important at lower altitudes; radiation is more important high up in the atmosphere. Convection cells happen when hot air rises in one part of the system and cold air sinks elsewhere, making a moving train of air. An example is in an enclosed room in winter when you have a heater going: heated air rises to the ceiling, then moves whereever it can go (away from the heater), sinks down the colder opposite wall, and then returns back across along the floor to the heater.
Clouds happen when the temperature gets low enough that water vapour condenses into liquid droplets or ice crystals. These tend to nucleate around tiny particles of dust, sea salt, etc. that got kicked up by winds and atmospheric convection cells and stayed aloft because they were so lightweight. Precipitation falls when growing droplets/crystals get too big to stay up. Clouds are relevant to radiative heat transfer partly because they have an albedo (nonzero reflectance) and reflect some incoming sunlight back out to space (more than atmospheric gases do). They also reflect some radiation emitted up by earth back down, and contribute partly to the greenhouse effect. Clouds are involved in convective heat transfer as well.
How do physicists model Earth's climate? They need to know about three key components -- the atmosphere, oceans, and land masses -- and the interactions between them, such as the transfer of heat and molecules. Earth's climate is a complicated physical system because it is governed by gnarly nonlinear differential equations which couple all three of these ingredients together. Analytical methods are not sufficient to solve them; numerical computation is needed.
What does nonlinearity mean? You have already met examples of linear physical systems. These are relatively simple to analyze because they give proportionate responses. If you change the initial conditions for a linear system a wee bit, and dynamically evolve forward in time, the result later on will be a proportionate change from the original. A related fact is that solutions of linear differential equations obey the superposition principle. However, linearity is not generic for systems in Nature, and in particular it does not apply to fluid dynamics. Earth's climate is an inherently nonlinear system.
Nonlinear systems behave in a qualitatively different fashion than linear ones: they do not respond to inputs proportionately. As a consequence, they can exhibit much more complicated behaviours, such as chaos and the so-called butterfly effect. Quantitatively speaking, the butterfly effect is about the sensitivity to initial conditions of a deterministic nonlinear system: small changes early on can result in exponentially large differences in later behaviour. More colloquially: a butterfly flapping its wings in Brazil now could affect the weather in Texas later! This is why climate modelling is inherently hard: your model has to have pretty good spatial resolution and your computer has to be extremely powerful to be able to produce reliable predictions for the future for various parts of the world with differing climate. One reason why it took so long for climate scientists to make predictions with some degree of confidence is that the computing power needed to get more precise only became available in relatively recent years.
When thinking about levels of complexity for modelling our planet's atmosphere, it can help to think about dimensionality. To see this, suppose that we work with a spherical coordinate system with the origin at the centre of the Earth: radius $r$, latitude $\theta$, and longitude $\varphi$. A one-dimensional model of the atmosphere might only consider $r$ dependence as a first approximation. The next level of complexity would be to remember that temperature depends on latitude, because sunlight hits Earth's surface more obliquely closer to the poles. So our model should include both $r$ and $\theta$ dependence. We should also add in the effects of longitude, because land masses and oceans and atmospheric weather are obviously not the same in all timezones even if we stick to the same latitude. So for completeness we should insist on tracking the dependence of all variables of interest on $r, \theta$, and $\varphi$. Of course, everything generically depends on time $t$ as well, so modelling is really a 4D problem. Finally, since planet Earth does not just sit motionless in space, but instead rotates about its (tilted) axis while getting continuous heat input from the Sun and radiating heat out into space, this makes everything even more interesting!
Because hot air rises, warmer temperatures near the equator produce upwelling of air (until a height called the tropopause where it runs out of extra buoyancy), while colder temperatures near the poles make the air sink. This gives rise to atmospheric convection cells that transport heat and moisture. If the Earth did not rotate, large scale atmospheric motions would circulate between the poles and the equator in a single simple back-and-forth pattern. But the earth also rotates on its axis, towards the East, and this changes the picture.
Because of planetary rotation, the linear velocity of a piece of land at the equator is much faster than that of a piece of land near the North or South pole. So if you tried to throw a ball from Toronto to a more northerly friend in Canada, it would land to your right of them, because their horizontal speed is lower and they haven't caught up to it yet. If on the other hand your northerly friend tried to throw you a ball, it would get deflected to their right, because you are moving faster than them and you moved ahead of the ball. This is why early people firing cannonballs over any significant distance tended to miss their targets! (and Flat Earthers have no way of explaining this fact, haha.)
So in the Northern hemisphere, air currents get deflected towards the right by planetary rotation. In the Southern hemisphere, they get deflected to the left. This phenomenon is named after Coriolis, and it is the reason why low pressure weather systems rotate clockwise in the southern hemisphere and counterclockwise in the northern hemisphere. The Coriolis effect is strongest near the poles and absent at the equator, which is why hurricanes rarely form in equatorial regions and never cross the equator. It also gives rise to three large-scale convection cells between the pole and equator in each hemisphere, with the first sinking of air happening at around 30 degrees latitude, demarcating the edge of the Hadley cell. This might sound kinda boring, but it gives rise to the phenomena of tradewinds, tropical rain belts and hurricanes, subtropical deserts, and the jet streams! The next two convection cells going up in latitude are known as Ferrel and polar cells.
Atmospheric winds at the surface of the earth affect the motions of the oceans, because of friction. Oceanic currents also feel the Coriolis effect, because water is also a fluid free to move about. Surface ocean currents driven in circles are known as gyres. Heat transfer from warmer low latitudes to colder high latitudes occurs via Western boundary currents of any given ocean (because planetary rotation is Easterly), like the Gulf Stream of the North Atlantic.
How about heat transfer for oceans? Consider the role of the Sun's radiation. Because Earth's surface is about 71% oceans, solar input is important. The top of the ocean definitely feels sunlight, but this does not have much of an effect deeper down than a few hundred metres because hotter water does not sink, it rises. This barrier to oceanic mixing at different depths is known as the thermocline and it is more important at low and mid latitudes. Closer to the poles, the story is more seasonal, because of sea ice formation.
When sea ice forms, driven by prolonged cooling, it is freshwater ice without salt in it, so the remaining water gets saltier. Saltier water is more dense, so it sinks, and this process is important to oceanic convection. This salinity based pumping of surface water down into the depths drives the water to move horizontally, until it can come up again and then move across to complete a convection loop. This is known as thermohaline circulation. The energy needed to pump the dense cold water up again is provided by planetary heat gradients. Antarctica has a special role to play in the story of the deep motions of the oceans, because the region around 60 degrees South is the only latitude on Earth where seawater can flow all the way around in longitude without bumping into land.
In a complicated system, certain phenomena can get magnified or damped down by what are known as feedback loops. An example is spreading of the deadly coronavirus. If we do nothing to try to prevent the spread (like wearing masks, avoiding indoor gatherings, mandating paid sick leave for all workers, and materially supporting parents so kids can stay home from schools), the number of people infected spreads exponentially -- because one person infects more than one other, and each of those goes on to infect more people, et cetera. This is known as a positive feedback loop. (Note: this scientific word choice convention was unfortunate: spreading the coronavirus is definitely not a positive thing!). A negative feedback loop would be when we do everything in our power to dampen down the spread of the coronavirus, so that it does not grow exponentially fast in the population, instead it dies away exponentially, getting us back to a stable state of things. Negative feedbacks regulate change, positive feedbacks amplify it.
Uncontrolled positive feedbacks can give rise to runaway processes -- an example is the runaway greenhouse effect on Venus. Could this happen on Earth? Yes it could, if humans continue being horribly cavalier about emitting greenhouse gases and warming up the atmosphere. Why could it happen? Consider the role of water vapour. The warmer the air, the more water vapour it can hold. (Cold air is drier, which is why you need more moisturizer in winter.) The more water vapour in the atmosphere, the more it helps to trap planetary heat. This makes the air even warmer, which can hold even more water vapour. And so forth. A saving grace is that more water vapour also gives rise to more clouds, and clouds act as a moderate negative feedback, because they have an albedo and reflect more sunlight, cooling things down.
So if humans would phase out -- and even better, stop -- emitting greenhouse gases from burning fossil fuels, we have a hope of stabilizing global warming at reasonable levels, so that people in the very hottest parts of the planet don't die of heat or be forced to migrate to other places which will meet them with xenophobia and racism. As a physicist with a social conscience, I think we should be doing everything we possibly can to move away from extracting and burning fossil fuels, in Canada especially. In particular, I think the Alberta Tar Sands should stay in the ground. Government should be financially encouraging fossil fuel workers to retrain for renewable energy projects. Renewable energy would actually be enough to power a greener Canadian economy, if we could generate the political will to switch. While doing that, it will be crucial not to concentrate the disadvantages created by renewable energy projects on already disadvantaged groups, like Indigenous communities.
2021-01-18