Weather reading open

Convective Self-Aggregation

Give the atmosphere an ocean that is the same temperature everywhere and sunlight that falls the same everywhere, and the rain will not fall everywhere. It gathers into a few clumps and leaves the rest of the world dry.

Every other card in this set has something for the weather to organise around. The convection roll has a warm plate under it. The hurricane has a patch of warm ocean. The block has a mountain range. This one has nothing. The sea surface temperature is uniform, the forcing is uniform, the domain wraps around on itself so there is no edge and no centre, and the starting state is random noise with no structure at any size. A pattern appears anyway, and once it has appeared it holds. There is no difference for the convection to gather around, so it makes one.

Something Out of Nothing In Particular

a published model, solved · 10,000 km square, wrapped

gross moist stability M,

fastest-growing pattern, predicted,

pattern actually there,

unevenness of the moisture,

fraction of the world that is moist,

day,

starting

The curve on the right is not fitted to the run; it is computed from the same equations. The panel linearises its own model about the uniform state, works out the growth rate of every pattern size by finding the eigenvalues of a three-by-three matrix, and draws the result. The blue dot is the size the theory says should win. The red dot is the size actually present in the picture on the left. They are calculated by completely separate routes, and the only reason they agree is that the arithmetic is right. The dashed line on the lower plot is the predicted growth rate laid over the measured one, again with nothing fitted.

A Published Model, Not a Toy Written Here

shallow water coupled to moisture

The storm-cluster panel runs a budget that was written for that page and says so loudly. This one does not. It runs a set of equations out of the literature, unchanged:

u_t = −f k × u − g ∇h − αu h_t + H ∇·u = F_h(q) − λh q_t + Q ∇·u + ε∇·(qu) − κ∇²q = F_q(q)

with h standing in for the temperature of the middle troposphere and q for the moisture low down. The published constants are used as published: g = 10, H = 30 m, Q = 15 m, μ₁ = 1/36000 s⁻¹, κ = 10⁵ m²/s, and a domain ten thousand kilometres square.

The Loop, In One Paragraph

why a wet place gets wetter

A patch that happens to be a little moister than its surroundings condenses a little more water, which warms the column. A warm column pushes air out at the top. Air pushed out at the top has to be replaced from below, so air flows in near the ground, and the air near the ground is the moist air. So the moist patch pulls in more moisture, which makes it moister. Meanwhile the air that went out at the top has to come down somewhere, and where it comes down it dries the column, which makes that column push air out less, which makes it drier still.

Whether that runs away is decided by one number, the gross moist stability:

M = 1 − Q μ₂ / (H μ₁)

which compares how much moisture the circulation imports against how much the column exports. When M is negative the circulation brings in more than it takes out, and the uniform state cannot hold. The slider sets M directly. Push it above zero and nothing ever happens, however long you wait.

Not Every Size Grows

the curve on the right, and what it predicts

Linearise those three equations about the uniform state and the growth rate of a pattern of size L falls out as the largest eigenvalue of a small matrix. It is not positive everywhere. Patterns smaller than about 540 km are wiped out by the sideways spreading of moisture; patterns bigger than a few thousand kilometres are too big for the temperature to stay even across them and the damping kills them. In between there is a window, and the fastest growth sits at about 1,600 km with a doubling time of 18 hours.

quantity	predicted	measured in the run
growth rate of a single seeded mode	exact	within 0.2 to 1.5%
size of the pattern when it first appears	1,633 km	1,429 to 2,000 km
smallest world that can aggregate at all	538 km	between 560 and 580 km

That last row is the one worth pausing on. A small enough world cannot do this, because the only patterns that can grow do not fit inside it. Real simulations of this phenomenon have the same problem and it is a known and irritating feature of the subject: shrink the domain and the effect vanishes, which for years made people wonder whether it was real.

Press “Stir It Flat”

the pattern is maintained, not merely made

That button erases the moisture field completely and leaves everything else alone: the winds, the temperature, all of it. The world becomes uniformly damp everywhere, which is the state the whole page began in.

before stirring unevenness 1.311, 22% of the world moist one day later 0.380 three days later 1.243 five days later 1.370 twenty days later 1.238, 16% of the world moist

It comes straight back. Not the same pattern; a different one, in a different place, but the same kind of pattern with the same statistics. This is not a structure the initial conditions happened to contain. It is the state the system goes to. Press Start again a few times and watch five completely different worlds land on the same numbers:

seed	unevenness	moist fraction
7	1.303	18.8%
101	1.298	19.8%
2027	1.284	18.5%
55555	1.291	19.6%
918273	1.301	19.8%

A spread of 1.5% in the unevenness across five worlds that look nothing like each other. The same thing the dendritic growth panel says about frost: the arrangement is an accident, and what can be measured from it is not.

And “Turn the Feedback Off”

nothing is preserved

That button moves M from −0.5 to +0.2 on a pattern that is fully formed. The pattern does not sit there. It unwinds:

day 0 unevenness 1.303 day 2 0.329 day 6 0.028 day 10 0.003 day 14 0.000

Two weeks and the world is uniform again with no trace of what was there. The structure is not a thing that was built and now stands. It is a thing that is being made continuously, and it exists only for as long as the making continues.

The Basin, Which Is the Whole World

what the two buttons above actually demonstrate

Those two experiments are a basin measurement, and it is worth saying so in those words because the answer is unusual.

Start with what Stir it flat shows. The uniform state is not a rival to the aggregated one. It is not a place the system can sit. It is an unstable equilibrium sitting inside the aggregated state’s basin, which is exactly what M < 0 means and is the only thing it means. Wipe the moisture out completely and you have not left the basin; you have put the system on a knife edge inside it, and it falls off within a day.

So how big is the basin? It is everything. There is no initial condition of this model, at M = −0.5, that does not end up aggregated: not the uniform state, not noise of any amplitude, not a mature pattern that has been shredded. That is a strictly larger basin than anything else in this set has. The magnetic pendulum has three basins with a fractal boundary between them. The block has a basin with a measurable edge that closes at a fold. This has one basin and no edge.

What varies instead is which aggregated state you land in, and that is where the interesting scatter lives. Five different starting noises give five worlds that share almost no structure: different numbers of clumps, in different places, of different shapes. And they agree on the unevenness to 1.5% and on the moist fraction to about a point. The basin is in the statistics and not in the arrangement, which is the same thing the dendritic growth panel measures about frost and says in the same words.

There is one way to make the basin empty, and it is not a change to the state at all. Shrink the world below about 560 km and there is no aggregated state to have a basin: the only patterns that could grow do not fit. The basin does not shrink as the domain shrinks. It exists at full size, and then it is gone.

Why the Reading Is Open

the hardest case in the set, and the most interesting

The taxonomy asks whether the system’s own activity releases the energy that drives it. The answer here splits cleanly in two and neither half wins.

Everything energetic is delivered. The warmth comes from the ocean, the ocean is warmed by the sun, and the atmosphere cools to space. Take that away and there is no convection to organise. On the plain reading of the test this is a supplied system, and the Turn the feedback off experiment shows it has no reserves of any kind.

And yet the organisation is entirely its own. Every other weather card gets its shape from a shape in the world outside: a plate, a coast, a mountain, a temperature gradient. Here the boundary conditions contain no information about the pattern at all: not its size, not its position, not how many clumps there will be. Those come from the equations, and the equations are the system. The supply is uniform; the structure is not; and nothing external made the difference. That is a genuinely different situation from a convection roll, and the test as written does not have a place to put it.

So the card is tagged reading open, alongside Atmospheric Blocking and Global Circulation. The honest position is that this page is a good argument that the test needs a second question, not that the answer to the first one is yes.

What This Model Is Not

the limits

One layer, and no radiation. There are no clouds here, no water vapour absorption, no longwave cooling, no surface fluxes. In the cloud-resolving simulations where this phenomenon was found, the feedback that does most of the work is radiative: dry columns radiate heat to space more efficiently and so sink harder. That mechanism is not in these equations. What is here instead is the moisture-circulation feedback, packed into the single number M. The pattern that comes out looks like the real thing and arises for a related reason, but it is not the same mechanism and the page will not pretend otherwise.

Two things here are authored rather than published. The damping rates α and λ were not in the part of the paper that could be read, and are set equal at one over five days; the answers change by less than a few percent for any damping slower than about three days, and the sweep is in the scope note. And the numerical scheme is a choice: a staggered grid with forward-backward time stepping, checked against the analytic growth rates to about one percent.

No rotation is switched on. The model supports it and the paper uses it, and with rotation the clumping stops at a size set by the rotation rate rather than running to the size of the world. The panel leaves it at zero because the point being made is about having no imposed structure of any kind, and a rotation rate is an imposed structure.

The picture is the moisture anomaly, not a satellite image. Gold means drier than average and blue means moister; there is no cloud field being computed.

Weather and Climate Dynamics, 5, 1153 (2024), A simple model linking radiative–convective instability, convective aggregation and large-scale dynamics. Open access. Source of the three equations, of the piecewise-linear coupling functions, of every published constant used here, and of the gross-moist-stability instability criterion.
Bretherton, C. S., Blossey, P. N. & Khairoutdinov, M. (2005). An energy-balance analysis of deep convective self-aggregation above uniform SST. Journal of the Atmospheric Sciences, 62(12), 4273–4292.
Wing, A. A., Emanuel, K., Holloway, C. E. & Muller, C. (2017). Convective self-aggregation in numerical simulations: a review. Surveys in Geophysics, 38, 1173–1197. Source of the definition used on the card: convection spontaneously organising into isolated clusters despite spatially homogeneous boundary conditions and forcing.

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