When our son was ten years old, he came home from a town fair in Battleground, Indiana, with an unwanted pet—a goldfish in a plastic bag. The family rushed out to buy a fish bowl and food and plopped the golden-red animal into it. In three days, it was dead!
It turns out that you can’t just put a gold fish in a fish bowl. When it metabolizes its food and expels its waste, it builds up toxic levels of ammonia unless you add filters or plants or treat the water with chemicals. In the end, the goldfish died because it was asphyxiated by its own pee.
It’s a basic rule—don’t pee in your own fish bowl.
The same can be said for humans living on the surface of our planet. Polluting the atmosphere with our wastes cannot be a good idea. In the end it will kill us. The atmosphere may look vast—the fish bowl was a big one—but it is shocking how thin it is.
Turn on your Apple TV, click on the screen saver, and you are skimming over our planet on the dark side of the Earth. Then you see a thin blue line extending over the limb of the dark disc. Hold! That thin blue line! That is our atmosphere! Is it really so thin?
When you look upwards on a clear sunny day, the atmosphere seems like it goes on forever. It doesn’t. It is a thin veneer on the surface of the Earth barely one percent of the Earth’s radius. The Earth’s atmosphere is frighteningly thin.
Consider Mars. It’s half the size of Earth, yet it cannot hold on to an atmosphere even 1/100th the thickness of ours. When Mars first formed, it had an atmosphere not unlike our own, but through the eons its atmosphere has wafted away irretrievably into space.
An atmosphere is a precious fragile thing for a planet. It gives life and it gives protection. It separates us from the deathly cold of space, holding heat like a blanket. That heat has served us well over the eons, allowing water to stay liquid and allowing life to arise on Earth. But too much of a good thing is not a good thing.
Common Sense
If the fluid you are bathed in gives you life, then don’t mess with it. Don’t run your car in the garage while you are working in it. Don’t use a charcoal stove in an enclosed space. Don’t dump carbon dioxide into the atmosphere because it also is an enclosed space.
At the end of winter, as the warm spring days get warmer, you take the winter blanket off your bed because blankets hold in heat. The thicker the blanket, the more heat it holds in. Common sense tells you to reduce the thickness of the blanket if you don’t want to get too warm. Carbon dioxide in the atmosphere acts like a blanket. If we don’t want the Earth to get too warm, then we need to limit the thickness of the blanket.
Without getting into the details of any climate change model, common sense already tells us what we should do. Keep the atmosphere clean and stable (Don’t’ pee in our fishbowl) and limit the amount of carbon dioxide we put into it (Don’t let the blanket get too thick).
Some Atmospheric Facts
Here are some facts about the atmosphere, about the effect humans have on it, and about the climate:
Fact 1. Humans have increased the amount of carbon dioxide in the atmosphere by 45% since 1850 (the beginning of the industrial age) and by 30% since just 1960.
Fact 2. Carbon dioxide in the atmosphere prevents some of the heat absorbed from the Sun to re-radiate out to space. More carbon dioxide stores more heat.
Fact 3. Heat added to the Earth’s atmosphere increases its temperature. This is a law of physics.
Fact 4. The Earth’s average temperature has risen by 1.2 degrees Celsius since 1850 and 0.8 degrees of that has been just since 1960, so the effect is accelerating.
These facts are indisputable. They hold true regardless of whether there is a Republican or a Democrat in the White House or in control of Congress.
There is another interesting observation which is not so direct, but may hold a harbinger for the distant future: The last time the Earth was 3 degrees Celsius warmer than it is today was during the Pliocene when the sea level was tens of meters higher. If that sea level were to occur today, all of Delaware, most of Florida, half of Louisiana and the entire east coast of the US would be under water, including Houston, Miami, New Orleans, Philadelphia and New York City. There are many reasons why this may not be an immediate worry. The distribution of water and ice now is different than in the Pliocene, and the effect of warming on the ice sheets and water levels could take centuries. Within this century, the amount of sea level rise is likely to be only about 1 meter, but accelerating after that.
Balance and Feedback
It is relatively easy to create a “rule-of-thumb” model for the Earth’s climate (see Ref. [2]). This model is not accurate, but it qualitatively captures the basic effects of climate change and is a good way to get an intuitive feeling for how the Earth responds to changes, like changes in CO2 or to the amount of ice cover. It can also provide semi-quantitative results, so that relative importance of various processes or perturbations can be understood.
The model is a simple energy balance statement: In equilibrium, as much energy flows into the Earth system as out.
This statement is both simple and immediately understandable. But then the work starts as we need to pin down how much energy is flowing in and how much is flowing out. The energy flowing in comes from the sun, and the energy flowing out comes from thermal radiation into space.
We also need to separate the Earth system into two components: the surface and the atmosphere. These are two very different things that have two different average temperatures. In addition, the atmosphere transmits sunlight to the surface, unless clouds reflect it back into space. And the Earth radiates thermally into space, unless clouds or carbon dioxide layers reflect it back to the surface.
The energy fluxes are shown in the diagram in Fig. 3 for the 4-component system: Sun, Surface, Atmosphere, and Space. The light from the sun, mostly in the visible range of the spectrum, is partially absorbed by the atmosphere and partially transmitted and reflected. The transmitted portion is partially absorbed and partially reflected by the surface. The heat of the Earth is radiated at long wavelengths to the atmosphere, where it is partially transmitted out into space, but also partially reflected by the fraction a’a which is the blanket effect. In addition, the atmosphere itself radiates in equal parts to the surface and into outer space. On top of all of these radiative processes, there is also non-radiative convective interaction between the atmosphere and the surface.
These processes are captured by two energy flux equations, one for the atmosphere and one for the surface, in Fig. 4. The individual contributions from Fig. 3 are annotated in each case. In equilibrium, each flux equals zero, which can then be used to solve for the two unknowns: Ts0 and Ta0: the surface and atmosphere temperatures.
After the equilibrium temperatures Ts0 and Ta0 are found, they go into a set of dynamic response equations that governs how deviations in the temperatures relax back to the equilibrium values. These relaxation equations are
where ks and ka are the relaxation rates for the surface and atmosphere. These can be quite slow, in the range of a century. For illustration, we can take ks = 1/75 years and ka = 1/25 years. The equilibrium temperatures for the surface and atmosphere differ by about 50 degrees Celsius, with Ts = 289 K and Ta = 248 K. These are rough averages over the entire planet. The solar constant is S = 1.36×103 W/m2, the Stefan-Boltzman constant is σ = 5.67×10-8 W/m2/K4, and the convective interaction constant is c = 2.5 W m-2 K-1. Other parameters are given in Table I.
| Short Wavelength | Long Wavelength |
| as = 0.11 | |
| ts = 0.53 | t’a = 0.06 |
| aa = 0.30 | a’a = 0.31 |
The relaxation equations are in the standard form of a mathematical “flow” (see Ref. [1]) and the solutions are plotted as a phase-space portrait in Fig. 5 as a video of the flow as the parameters in Table I shift because of the addition of greenhouse gases to the atmosphere. The video runs from the year 1850 (the dawn of the industrial age) through to the year 2060 about 40 years from now.
The scariest part of the video is how fast it accelerates. From 1850 to 1950 there is almost no change, but then it accelerates, faster and faster, reflecting the time-lag in temperature rise in response to increased greenhouse gases.
What if the Models are Wrong? Russian Roulette
Now come the caveats.
This model is just for teaching purposes, not for any realistic modeling of climate change. It captures the basic physics, and it provides a semi-quantitative set of parameters that leads to roughly accurate current temperatures. But of course, the biggest elephant in the room is that it averages over the entire planet, which is a very crude approximation.
It does get the basic facts correct, though, showing an alarming trend in the rise in average temperatures with the temperature rising by 3 degrees by 2060.
The professionals in this business have computer models that are orders of magnitude more more accurate than this one. To understand the details of the real climate models, one needs to go to appropriate resources, like this NOAA link, this NASA link, this national climate assessment link, and this government portal link, among many others.
One of the frequent questions that is asked is: What if these models are wrong? What if global warming isn’t as bad as these models say? The answer is simple: If they are wrong, then the worst case is that life goes on. If they are right, then in the worst case life on this planet may end.
It’s like playing Russian Roulette. If just one of the cylinders on the revolver has a live bullet, do you want to pull the trigger?
Matlab Code
function flowatmos.m
mov_flag = 1;
if mov_flag == 1
moviename = 'atmostmp';
aviobj = VideoWriter(moviename,'MPEG-4');
aviobj.FrameRate = 12;
open(aviobj);
end
Solar = 1.36e3; % Solar constant outside atmosphere [J/sec/m2]
sig = 5.67e-8; % Stefan-Boltzman constant [W/m2/K4]
% 1st-order model of Earth + Atmosphere
ta = 0.53; % (0.53)transmissivity of air
tpa0 = 0.06; % (0.06)primes are for thermal radiation
as0 = 0.11; % (0.11)
aa0 = 0.30; % (0.30)
apa0 = 0.31; % (0.31)
c = 2.5; % W/m2/K
xrange = [287 293];
yrange = [247 251];
rngx = xrange(2) - xrange(1);
rngy = yrange(2) - yrange(1);
[X,Y] = meshgrid(xrange(1):0.05:xrange(2), yrange(1):0.05:yrange(2));
smallarrow = 1;
Delta0 = 0.0000009;
for tloop =1:80
Delta = Delta0*(exp((tloop-1)/8)-1); % This Delta is exponential, but should become more linear over time
date = floor(1850 + (tloop-1)*(2060-1850)/79);
[x,y] = f5(X,Y);
clf
hold off
eps = 0.002;
for xloop = 1:11
xs = xrange(1) +(xloop-1)*rngx/10 + eps;
for yloop = 1:11
ys = yrange(1) +(yloop-1)*rngy/10 + eps;
streamline(X,Y,x,y,xs,ys)
end
end
hold on
[XQ,YQ] = meshgrid(xrange(1):1:xrange(2),yrange(1):1:yrange(2));
smallarrow = 1;
[xq,yq] = f5(XQ,YQ);
quiver(XQ,YQ,xq,yq,.2,'r','filled')
hold off
axis([xrange(1) xrange(2) yrange(1) yrange(2)])
set(gcf,'Color','White')
fun = @root2d;
x0 = [0 -40];
x = fsolve(fun,x0);
Ts = x(1) + 288
Ta = x(2) + 288
hold on
rectangle('Position',[Ts-0.05 Ta-0.05 0.1 0.1],'Curvature',[1 1],'FaceColor',[1 0 0],'EdgeColor','k','LineWidth',2)
posTs(tloop) = Ts;
posTa(tloop) = Ta;
plot(posTs,posTa,'k','LineWidth',2);
hold off
text(287.5,250.5,strcat('Date = ',num2str(date)),'FontSize',24)
box on
xlabel('Surface Temperature (oC)','FontSize',24)
ylabel('Atmosphere Temperature (oC)','FontSize',24)
hh = figure(1);
pause(0.01)
if mov_flag == 1
frame = getframe(hh);
writeVideo(aviobj,frame);
end
end % end tloop
if mov_flag == 1
close(aviobj);
end
function F = root2d(xp) % Energy fluxes
x = xp + 288;
feedfac = 0.001; % feedback parameter
apa = apa0 + feedfac*(x(2)-248) + Delta; % Changes in the atmospheric blanket
tpa = tpa0 - feedfac*(x(2)-248) - Delta;
as = as0 - feedfac*(x(1)-289);
F(1) = c*(x(1)-x(2)) + sig*(1-apa)*x(1).^4 - sig*x(2).^4 - ta*(1-as)*Solar/4;
F(2) = c*(x(1)-x(2)) + sig*(1-tpa - apa)*x(1).^4 - 2*sig*x(2).^4 + (1-aa0-ta+as*ta)*Solar/4;
end
function [x,y] = f5(X,Y) % Dynamical flow equations
k1 = 1/75; % 75 year time constant for the Earth
k2 = 1/25; % 25 year time constant for the Atmosphere
fun = @root2d;
x0 = [0 0];
x = fsolve(fun,x0); % Solve for the temperatures that set the energy fluxes to zero
Ts0 = x(1) + 288; % Surface temperature in Kelvin
Ta0 = x(2) + 288; % Atmosphere temperature in Kelvin
xtmp = -k1*(X - Ts0); % Dynamical equations
ytmp = -k2*(Y - Ta0);
nrm = sqrt(xtmp.^2 + ytmp.^2);
if smallarrow == 1
x = xtmp./nrm;
y = ytmp./nrm;
else
x = xtmp;
y = ytmp;
end
end % end f5
end % end flowatmos
This model has a lot of parameters that can be tweaked. In addition to the parameters in the Table, the time dependence on the blanket properties of the atmosphere are governed by Delta0 and by feedfac for feedback of temperature on the atmosphere, such as increasing cloud cover and decrease ice cover. As an exercise, and using only small changes in the given parameters, find the following cases: 1) An increasing surface temperature is moderated by a falling atmosphere temperature; 2) The Earth goes into thermal run-away and ends like Venus; 3) The Earth initially warms then plummets into an ice age.
By David D. Nolte Oct. 16, 2022
References
[1] D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd Ed. (Oxford University Press, 2019)
[2] E. Boeker and R. van Grondelle, Environmental Physics (Wiley, 1995)
[3] Recent lecture at the National Academy of Engineering by John Holdren.
Galileo Unbound 