Linking

It seems my notions about links are wildly unusual (and thanks Jules for adding them Wednesday). My idea is that since we're discussing science here, if you disagree with some point I make, or have a point to add yourself, then it is a good idea to include a link or full reference to a good source. One of the things about doing science is that you don't take people's reporting as gospel. Scientists are people, and people make mistakes. So it's a good idea to make it easy for people to check out the full original source. When you've described something well, they can be thankful for your much better description. Or maybe they can learn more about the topic. Wins all around.

Maybe there's something about how the blog world works that makes this a bad idea. If so, let me know how and why. In the mean time, please make it easy for people to follow up the science you bring up.

Math in Climatology

I'm doing my best to keep the math to a minimum here, because one can indeed understand a lot of the fundamentals without it, also because I know many people suffer from math anxiety. Then a large contribution from the fact that I know if I start down the road towards using math, I'll use a lot of it.

Now, on the math anxiety side, I highly recommend Sheila Tobias' Overcoming Math Anxiety. Unlike your probable expectations, she is not some outsider who never had the problem herself. She had a big case of it, but decided that physics was interesting and she'd bite the bullet and try to learn the math needed. She did, along the way dealing with a number of anxieties and false impressions about mathematics. One of the major problems shared by many people is the notion that only certain 'special' people can do mathematics. If you're talking the seriously hard core stuff that wins Fields Medals, that's probably true. If you're thinking about the level that I'm using here, the answer is ... nonsense. Anybody without a serious learning disability can learn this, regardless of age, gender, race, religion, part of world you live in, etc.

Here's a sampling of courses or material that one would almost certainly encounter on the way to being able to study climatology professionally:

Algebra I
Algebra II
Trigonometry
Differential Calculus
Integral Calculus
Calculus of Several Variables
Probability and/or Statistics
Ordinary Differential Equations

That's just warming up. You'd also likely encounter several of:
Linear Algebra
Partial Differential Equations (probably multiple courses)
Complex Analysis
Numerical methods (probably multiple courses)
Statistics (in a multi-term sequence)


... and probably several more that aren't leaping to my mind right now.

Building the Simplest Climate Model

Where did the simplest model come from? Nice that I showed a model that gives realistic answers, but even better to know how it works!

The model is built on one of the great conservation laws of science -- conservation of energy. If you track all the ways that energy enters your 'system', and all the ways that it leaves, you can know quite a lot about it even without knowing very much about what's going on inside. That word 'system' may not mean what you're thinking. What I mean here is that we draw an imaginary box around, in this case, the earth, and see what passes through the box.

We have, in general, three ways to get energy through the box. We can carry some material from one side to the other (advection or convection). But the earth doesn't gain or lose material from space, so can't gain or lose energy this way. Another way is to conduct heat across the boundary. A frying pan does this in making the handle hot even though the flame is a good distance away. But with a vacuum on the other side of our box, there's nothing to conduct to. The third way is as radiation. Everything emits some radiation. The hotter is is the more it emits. A traditional thermos tries to minimize all three -- the inside is glass, a poor conductor, the glass is almost totally surrounded by a vacuum to cut down conduction even further, it is sealed, to prevent convection, and the inner wall of the thermos is reflective, to prevent the wall from gaining or losing heat by radiation to your drink.

In looking at what can cross the boundary of the earth system we have only two things -- radiation from the sun, and radiation from the earth. We all know that many things can happen inside the box: storms, building and melting ice sheets, forest growth and decay, and so on. But they're all inside. For our simplest model of the temperature of the earth we don't need to know about those, just income versus outgo. (Well, more to it than this, but hang on for now.)

The sun's radiation all comes in to the box, passing through a disk whose area is pi*r^2, where r is the radius of the earth and pi is the usual 3.14159... Same as the sun and moon look like disks to us, we look like a disk to the sun. Some of that solar energy gets bounced right back out. The fraction of the incoming solar energy that gets bounced out is called the albedo. For the earth, it is about 0.30 averaged over the whole planet, through the whole year. Particular surfaces can be much higher (snow can be 0.8) or much lower (oceans can be 0.06). But since we're only concerned with what passes through the box, we don't need to worry about those details yet.

The earth also emits radiation, as does everything that's not at absolute zero. For an ideal black body, that emission is proportional to s*T^4. This is the Stefan-Boltzmann law, and s is the Stefan Boltzmann constant, equal to 5.67e-8 Watts per square meter per K^4. Every square meter of the earth radiates like this. Since the earth is (very nearly) a sphere, the total area is 4 * pi * r^2. Again, r is the radius of the earth.





Conservation laws are written in terms of a balance of income and outgo. If the earth is in energy balance, then the following equation is exact. If it isn't, we'll have to look at how far out we are.
Income = pi * r^2 * S

Outgo =
pi * r^2 * S * a (albedo reflecting energy immediately away from the earth)
+ 4 * pi * r^2 * s * T^4

For the conservation law being Income = Outgo, we get
pi * r^2 * S = pi * r^2 * S * a + 4 * pi * r^2 * s * T^4

We notice two things rapidly. First, pi * r^2 is in every term. We can divide the equation by that and simplify. Second, the solar term is on both sides. We can subtract the albedo term from both sides to get all the solar-related terms on one side, and the earth on the other. So now we have:

S*(1-a) = 4*s*T^4

If we solve for T, we're back to the equation I gave in the original note

It's remarkable that the size of the earth doesn't show up here. Actually, it does -- that 4 is the remaining part. It's 4 because the entire area of the (spherical) earth emits energy to space, while the sun only contributes energy through a circular disk, and the area of the sphere is 4 times larger than the area of that disk.

This leads to a point that I see commonly missed. When solar observers talk about how much the solar output changes, they're referring to the S -- seen by the disk of the earth. During a solar cycle, it varies by about 1 Watt per square meter. In looking at climate, though, we think in terms of the spherical earth we're standing on, rather than the disk in space a satellite is using. When climate people talk about Watts per square meter, we're talking about 4 times as many square meters as solar observers. To make the two comparable, climatologists often divide the equation by 4, meaning that when a solar observer says 1 Watt per square meter, climate folks will treat it as 0.25. Not because we don't think the sun's important, but because the 0.25 gives a description that we can compare to earthly climate processes we observe all over the sphere.

On Monday, there'll be a third post in this set -- analyzing this model. In the mean time, a project for the more mathematically skilled out there: How is this model affected by using a proper oblate spheroid for the earth? Is it? And, for everyone, try out the spreadsheet version and see how it behaves as you change the parameters.

Ah, I should note that the S above isn't exactly the S in the original. Here it's the solar constant divided by mean earth-sun distance (in AU), while I left the two separate in the original. As tamino showed, the two are awfully close to each other since the mean earth-sun distance (over a year) doesn't change much even as the orbit varies on tens of thousand year time scales.

A different and more important matter if you're not used to this is that the temperatures will come out in Kelvin -- the scientific absolute temperature scale. Celsius is used, but when we are concerned about energy, it's best to use Kelvin (K). Fahrenheit is probably never used in doing science (even if someone did, it never gets published that way). Kelvin is Celsius plus 273, so water freezes at about 273 K, boils at about 373 K. (Or Celsius is Kelvin minus 273.) The earth's black body temperature is about 255 K, or -18 C.

Science on the way

No, not articles I'm going to be writing, but a sampling of sessions that are coming at the fall meeting of the American Geophysical Union:

Session names here, descriptions below.

Satellite Geodesy of the Polar Oceans
Progress in Quaternary Geochronology in Polar Regions
Cryospheric Climate Data Records
Rapid Arctic Environment Change
Environmental Impacts of a Shrinking Arctic Sea Ice Cover
Long-Term Trends in the General Circulation of the Atmosphere: Observations, Simulations, Mechanisms, and Impacts
Large Scale Cryosphere – Climate Connectivity
Ice Sheet Hydrology and Dynamics

This is just a small sampling of the sessions that will be going on. To see more, check out the AGU for the Fall meeting's program. The sessions turn on those good science questions: what are the data, how good are they, how can we get and retain better, what does it mean, how well do we understand what's already going on, what might happen in the future, and what would that mean, etc.

Now for the details, happy reading:

Satellite Geodesy of the Polar Oceans
New satellite geodetic techniques and observations are providingunprecedented views of the polar oceans, their ice cover as well as the topography and tectonic fabric of the seafloor below. Satellite altimeter measurements of ice and snow elevation from radar and laser
systems, such as Envisat and ICESat are providing wide area, continuous, information on ice thickness changes. Satellite measurements of sea surface topography and gravity for example from GRACE offer the potential for significant advances in our understanding of the Polar
Oceans, particularly when combined with in-situ observations. Marine gravity fields from satellite altimetry as well as satellite and surface gravimetry are also providing new maps of the tectonic structure and bathymetry of the poorly mapped polar oceans. The current satellite
capabilities will be significantly enhanced in the near future with the launch of new missions such as GOCE, CryoSat-2 and IceSat-2. We welcome contributions on all aspects of satellite geodesy of the polar oceans with emphasis on applications of these data to science problems in the
areas of oceanography, the marine cryosphere and tectonics.


Progress in Quaternary Geochronology in Polar Regions
The main purpose of this session is to provide an opportunity for discussion of recent progress and challenges in developing chronologies in Arctic and Antarctic regions. Session organizers hope to include, but are not limited to: research in the fields of ice-core chronology,
tephrochronology, dendrochronology and DNA analysis, as well as cosmogenic nuclide, radiocarbon and luminescence dating. Session conveners encourage a discussion of new developments in chronological methods with regard to different paleoclimate archives and the advantages and weaknesses of various methods.

Cryospheric Climate Data Records
The generation of cryospheric climate data records (CDRs) is a critical step in providing the necessary information for scientists, decision-makers, and stakeholders to make adaptive choices that could improve the nation's resiliency to environmental change and variability, maintain our economic vitality, and improve the safety and comfort of U.S. citizens. This session will bring together recent efforts in the development of CDRs over Arctic and Antarctic ocean, ice, and
terrestrial surfaces, including snow cover, sea and land ice, melt onset, surface temperature, sea surface temperature, etc. Contributions are encouraged covering the full spectrum of CDR development and use, including initial formulation of algorithms, validation of CDRs, and use
of CDRs for scientific inquiry.

Rapid Arctic Environment Change
The extent of Arctic perennial sea ice was reduced by another million square kilometers between the winters of 2007 and 2008 with seasonal ice occupying the North Pole region in mid-winter for the first time in the observational record. There were major warm temperature anomalies in the
central Arctic in fall 2007 and accelerated ice drift along the Transpolar Drift Stream in spring and summer 2007. Many impellent science issues remain: Is the summer ice extent more influenced by
initial conditions or by summer forcing? What is the relative role of dynamics, thermodynamics, and feedbacks in Arctic ice reduction? Will the current change continue, accelerate, or decelerate? Has Arctic change crossed into a new state or can it be reversed? What are the
regional and global impacts? Fortunately, the International Polar Year program has been ongoing with international research efforts providing new and crucial results on Arctic change and its impacts. In this regard, this session calls for presentations on 2007-2008 Arctic change, the historical context of recent change, predictions of change, and impacts of Arctic change on ocean, land, and atmosphere.

Environmental Impacts of a Shrinking Arctic Sea Ice Cover
Arctic sea ice extent at the end of the summer melt season has declined sharply over the period of satellite observations and is projected to disappear entirely as concentrations of atmospheric greenhouse gases continue to rise. The record low ice extent of September 2007 served as an exclamation point on the downward trend and further raised concern that the Arctic may be on the verge of rapid transition to a seasonal ice cover. While the factors forcing this trend have and will continue to be widely studied, less attention has been paid to the environmental impacts of current and future sea ice loss. Continued loss of the ice cover may result in strong rises in atmospheric temperature and water vapor content, not just at and near the surface, but extending through a considerable depth of the troposphere. Changes in humidity and the
boundary layer structure are likely to alter cloud conditions, a key determinant of the surface energy balance. Through atmospheric transports, warming will likely extend well beyond areas of ice loss, potentially influencing arctic land areas, glaciers, ice caps, and the Greenland ice sheet. Extensive open water areas will promote increased wave action and coastal erosion. Ice loss may in turn have impacts on patterns of atmospheric circulation and precipitation not just within
the Arctic, but potentially extending into middle latitudes. While evidence is growing that some of these effects are already occurring, they are likely grow in coming decades. This session will address emerging and projected environmental impacts of arctic sea ice loss through both observational and modeling studies.

Long-Term Trends in the General Circulation of the Atmosphere: Observations, Simulations, Mechanisms, and Impacts
An increasing body of evidence indicates that key-elements of the large-scale atmospheric general circulation have undergone significant change over the past several decades, which may be an important indicator of climate change. Examples include widening of the Hadley cell, changes in the position of the subtropical jets and extratropical storm tracks, lifting of the tropopause, and trends in the annular modes. However, there is still much to be learned about the observational evidence of these changes, their causes, and their societal and environmental impacts. Several causal mechanisms have been suggested, such as changes in tropospheric stability, extratropical eddy activity, and stratospheric dynamics. But it is still unclear which may
be most appropriate and whether there exists a single unifying theory. This session will bring renewed focus to this issue by examining further evidence for these processes, exploring their cause(s), and improving our theoretical understanding of their linkages. We invite studies on all aspects of long-term change in the general circulation, particularly papers that combine observational, modeling, and theoretical approaches.

Large Scale Cryosphere – Climate Connectivity
High-latitude regions are recognized as being critically sensitive to anthropogenic climate change, and are also subject to large scale climate phenomena such as the northern and southern annular modes. The cryosphere is a dominant but highly variable feature in these regions, and exists in numerous forms including snow cover, ice sheets, permafrost and sea ice. Therefore components of the cryosphere can be expected to interact with climate variability and change, either as a passive responder to climate, as an instigator of climate perturbations, or through feedback mechanisms that affect both facets of the polar environment. Moreover, this cryosphere – climate connectivity is not constrained to high latitudes, but can also occur in snow-covered mid-latitudes and hence potentially affect the global climate system. This session brings cryospheric, atmospheric, hydrologic and climate scientists together to share recent advances in our current understanding of this connectivity. Investigations are solicited involving all components of the cryosphere, and both climate variability and change. Studies that target large, regional – continental scales are particularly encouraged.

Ice Sheet Hydrology and Dynamics
The extent to which fluctuations in basal and surface hydrology affect ice sheets is a topic of widespread and recent concern; changes in the flow of the Greenland Ice Sheet correlate with changes in its surface hydrology. Melting at the Antarctic Peninsula has factored in the disintegration of ice shelf sections, and exchanges of water between lakes at the base of the Antarctic Ice Sheet could modulate mass losses through episodic drainage and lubrication. Although relationships between the hydrology and flow of mountain glaciers have been studied for much of the past century, attention has only recently turned to ice sheets, so a detailed understanding is lacking. This session will explore the effects hydrology has on ice sheet dynamics by collating the results of field experiments, satellite observations, and numerical modeling. Contributions from each of these areas are encouraged.

The Simplest Meaningful Climate Model

It's a surprise to me that it is possible to construct a climate model that is meaningful in the space of a blog post. At least you can if you're interested in atmospheric temperatures. This isn't the case for the oceans, which is interesting for different reasons.

Here it is: T = (S*(1-a)/r^2/4/s)^(1/4)

1 line, easily dumped into your spreadsheet program of choice or calculator. I'll give some sample spreadsheet code at the bottom.

T is the temperature the earth radiates to space with.
S is the solar constant, which is about 1367 Watts per square meter (on a plane face-on to the sun, which our spheroidal earth isn't -- that's why the division by 4)
a is the albedo of the earth -- how much of the sun's energy is bounced straight back to space (now an observable quantity, it's about 0.30)
r is the average earth-sun distance, 1.00 Astronomical Unit
s is the Stefan-Boltzman constant, 5.67*10^(-8)
and ^ means 'to the power of' (x^y key on calculators, this symbol in spreadsheets).

If we plug in this values, we see that T is 255 K (-18 C, 0 F). ... and this matches the observations fairly well. (I've rounded several of the numbers above, and this one -- research the figures yourself and plug them in, and then go looking on your own for reference figures on how much energy the earth radiates to space and its equivalent temperature. This is one of the 'projects' I mentioned)

What happened? Certainly where I live (Washington DC area) seldom gets that cold, and I did say this was a meaningful model. The thing is, the model is meaningful for two reasons, first, that it does match up with the observations fairly well (temperature as radiated to space -- we have satellites circling the earth to check this figure now). Second, the model does not match up to the figure we really want -- the temperatures at the surface.

This tells us that the model is missing something important that affects the surface temperature, but not the temperature the earth shows to space. Whatever it is, its average effect is about 33 K (60 F).

But first, play around with those numbers and see what happens to temperature. In the next part, I'll go in to how to derive this model. Third part will be some looking at analyzing it. There's more here than may meet the eye.

Spreadsheeting:
If you're comfortable with naming variables, just paste this formula in to a1 and name the variables S, r, a, s elsewhere. The figures in a1 will be the temperature.
= (S*(1-a)/r^2/4/s)^(1/4)

S: 1367
a: 0.30
r: 1.0000
s: 5.67e-8 (cannot change)

if you're less comfortable with your spreadsheet, use this formula in a1:
= (b1*(1-b2)/b3^2/4/b4)^(1/4)

b1: 1367
b2: 0.30
b3: 1.0000
b4: 5.67e-8 (cannot change)