A Land Surface Model Is a Zip File of Today’s Climate

Anthropic’s newest model, Fable 5.2, had just come out, so I asked it to question me about land surface models the Socratic way, one question at a time, and twenty-two questions later I had talked myself out of most of what I thought “physically based” meant.

Author: Koutian Wu; GitHub: ktwu01

A land surface model, LSM for short, is the land half of a climate model. It decides how much rain soaks in, how much runs off, how much evaporates, and how much carbon the plants take up. People call these models physically based because they are full of equations with famous names on them. Here is the interrogation, as it happened.

Name one formula in an LSM that you are sure holds everywhere, at every scale.

Conservation of energy. Conservation of mass. That was easy.

Take water. Conservation says the change in soil water equals what comes in minus what goes out. Does that tell you how much goes out?

No. One equation cannot do that. You need something else to say how fast water moves, and that something else is where the trouble starts.

LSMs use Darcy’s law for how fast water moves down through soil. How did Darcy get it?

I did not know, so I was told. In 1856 Henry Darcy, a French engineer, was designing sand filters for the public fountains of Dijon. He filled a vertical iron pipe with river sand, about 2.5 meters tall and 0.35 meters across, ran water through it, and measured the flow. The flow went up in proportion to the pressure difference between the ends and down in proportion to the length of the sand. The constant of proportionality is what we now call hydraulic conductivity.

That is all of it. One pipe, uniform sand, fully wet, steady flow. He derived nothing. He measured.

A grid cell in an LSM is about 100 kilometers wide. How is the soil in it different from the sand in that pipe?

Heterogeneity. The cell holds clay, sand, rock, root channels, and cracks, and conductivity among them differs by orders of magnitude. The model gives the whole cell one number.

Can you measure that number in the field?

You can measure conductivity, yes. On a soil core about 10 centimeters across. The grid cell is 100 kilometers. That is six orders of magnitude between what the instrument touches and what the model needs.

So where does the grid cell’s number come from?

I did not know this either. Two steps. First, lookup: a global soil map gives the sand and clay fraction of each cell, and a regression turns texture into conductivity. A standard one is Cosby and colleagues, 1984, fitted to 1448 soil samples from the United States, each one core-sized. Second, tuning: when the model’s runoff or soil moisture misses the observations, someone changes the number until it stops missing.

So the pedigree reads: measured on small cores, regressed on American dirt, pasted onto the planet, then adjusted to make the output behave.

Is that number the same thing Darcy measured in his pipe?

No. It kept the name and lost the meaning.

Then what is left, at 100 kilometers, of what Darcy actually verified?

The form. Flow is proportional to the pressure gradient.

Darcy checked that proportionality in uniform wet sand. Suppose most of the water in a grid cell really drains through a few cracks and root holes. Is the cell’s total flow still proportional to its average gradient?

No.

So the parameter is a different object, and nobody has verified the form at this scale. And still, after tuning, the model’s runoff matches the river gauge.

What does the match prove?

That the parameter was bent to please the observations. It might be plain overfitting.

How do you tell a model that understood from a model that was fitted?

Split the data. Train on one part, validate on the other. If it holds on the part it never saw, the training meant something.

That works when both parts come from the same conditions. What is an LSM mainly used to predict?

The future. A hotter world with more CO2 and rain falling in different places. None of that is in the training set, and none of it is in the validation set. The standard test says nothing about the one job the model was built for.

Darcy’s law came from wet sand in a pipe. The formulas for exchange between ground and air were fitted to a 1968 field campaign on a flat square mile of wheat stubble in Kansas. The formula for how leaf pores open came from single leaves clamped in a lab chamber. Put a few dozen of these in one model. Under what conditions is every one of them used inside the range where it was tested?

Under useless conditions. The place would have to be a saturated sand column, a flat stubble field, and a leaf in a chamber at once. No such place exists. The modules were each validated somewhere, and nobody checked that the somewheres overlap. They do not.

Yet the models do not produce nonsense. Where evaporation is high and where rivers run large comes out roughly right. Every formula is out of bounds. What holds the model together?

I had no answer. There are two things. The first is conservation, my own opening answer. Over the long run a place cannot evaporate more water than falls on it, and cannot spend more energy on evaporation than the sun delivers. Climatologists have known since Budyko that these two ceilings fence the answer in, and a wrong formula cannot climb the fence. The second is tuning. One module runs high, another gets tuned low, and the errors cancel.

So who gets the credit for what the model gets right, the dozens of formulas, or conservation and tuning?

Conservation and tuning.

Conservation survives into a future climate. The errors were made to cancel under today’s climate. Will they still cancel?

Obviously not.

And this is what the models actually do. They agree with each other about the present, because the present is what they were tuned to. Ask them about 2100 and they scatter. In one comparison of eleven Earth system models, the land’s running total by 2100 came out anywhere from 165 billion tons of carbon released to 758 billion absorbed. The authors wrote that the models “do not even agree on the sign.”

You make the rule. What must a formula satisfy before it is allowed into a model that predicts the future?

Its assumptions must still hold in the future.

The future has not happened. How would you know?

My answer, verbatim, was “don’t fucking know.” Then: look back into history.

What makes a stretch of history a real test?

Lots of change. It has to leave today’s conditions behind. The last ice age qualifies. Twenty thousand years ago the planet was several degrees colder, CO2 was under 200 parts per million against more than 420 now, and pollen and old lake shorelines recorded where the forests and the water were.

Two formulas go back to the ice age, with no retuning allowed. One has a parameter fitted to today’s observations. The other has no free parameter and follows from conservation plus a physical limit. Which do you bet on?

The second.

Why?

I could not say, and the answer turned out to be the point of the whole exercise. Tuning presses today’s observations into a number. A tuned parameter is a small recording of the present climate. Change the climate and the recording is wrong. The parameter-free formula has no place to store such a recording. It leans on conservation and limits, and those do not care what century it is.

Then say it yourself. What are the physics formulas in an LSM?

A zip file. Today’s climate, compressed into a package small enough for humans to read and reason about.

That got one last push. Darcy’s law has held in laboratories for 170 years across every sand anyone has tried. It remembers nothing about Dijon in 1856. So where does the memory sit? In the parameters, and in the step that carries a number from a 10-centimeter core to a 100-kilometer cell. The equation is the container and the tuned number is the zip file. You can read the equations in an LSM all day and they will look like physics. What the model knows about the world is in the numbers, and the numbers know only today.

A zip file gives back what you put in. We put in the twentieth century, and we keep unzipping it to ask about the twenty-second.