Atmospheric Blocking
Sometimes the weather stops moving. The same forecast repeats for a week, and the reason is that the atmosphere has settled into a second state it can hold just as steadily as the ordinary one.
Most of the time the middle latitudes run west to east: waves ride along the jet and the weather changes every few days. Then, occasionally, a ridge parks itself over one part of the world and refuses to move. Storms divert around it. A heatwave or a cold spell sits in place. In 1979 Jule Charney and John DeVore asked whether this is a disturbance to the normal flow or a different flow altogether, and showed that a westerly current running over a mountain range can have more than one steady state at exactly the same forcing. Blocking, on that account, is not weather going wrong. It is the atmosphere sitting in its other chair.
Two States, One Atmosphere
What Charney and DeVore Did
Take a westerly current in a channel on a rotating planet, put a mountain in it, and keep the current going against friction by forcing it toward some target strength. Write the flow as a streamfunction, keep only the first few waves, and what is left is six ordinary differential equations in six numbers:
x2' = −(α₁ x1 − β₁) x3 − C x2 − δ₁ x4 x6
x3' = (α₁ x1 − β₁) x2 − γ₁ x1 − C x3 + δ₁ x4 x5
x4' = γ̄₂ x6 − C(x4 − x4*) + ε(x2 x6 − x3 x5)
x5' = −(α₂ x1 − β₂) x6 − C x5 − δ₂ x3 x4
x6' = (α₂ x1 − β₂) x5 − γ₂ x4 − C x6 + δ₂ x2 x4
The first number, x₁, is the strength of the westerly jet. Meteorologists call it the zonal index and have done since the 1940s. x₂ and x₃ are the two halves of the longest wave: how much of it sits over the mountain and how much sits a quarter wavelength downstream. The map at the top of the panel is not an artist’s impression of those numbers. It is the streamfunction they define, drawn directly.
x₁* is the only control. It is how hard the atmosphere is being pushed toward a strong jet, which in the real world is the pole-to-equator temperature difference. Every other coefficient in those equations follows from three fixed numbers: the shape of the channel, the beta effect, and how tall the mountain is.
The Result That Made This Famous
Set the forcing anywhere between 1.34 and 1.77 and press Start blocked, then Start zonal. Nothing about the equations changed between those two presses. Only where you started.
| blocked state | zonal state | |
|---|---|---|
| zonal index x₁ | 0.70 | 1.40 to 1.66 |
| standing wave | 0.49 | small |
| the wave | stands still over the mountain | travels |
| in time | a fixed point; perfectly steady | a cycle of 10.2 time units |
Measured at x₁* = 1.55 and 1.60. The blocked state is a fixed point: the ridge locks onto the mountain and simply stops, which is the mathematical form of the word quasi-stationary that every review of blocking uses. The zonal state is a limit cycle: the wave keeps going around, which is ordinary travelling weather. One system, one forcing, two completely different things to be.
The Basin, Which You Can Measure Yourself
Set the forcing to 1.55, press Start blocked, set the kick to +0.30 and press Kick the jet. The jet speeds up, wobbles, and comes back. Now set the kick to +0.50 and press it again. The block does not come back. It never comes back, because the flow has fallen out of one basin and into the other, and there is nothing in these equations to carry it home.
By bisection, the smallest kick to x₁ that ends the block:
| forcing x₁* | 1.35 | 1.45 | 1.55 | 1.65 | 1.75 |
|---|---|---|---|---|---|
| smallest kick that works | +0.679 | +0.574 | +0.457 | +0.319 | +0.119 |
The basin narrows steadily as the forcing rises, and at x₁* = 1.771 it closes entirely: the blocked state collides with the unstable flow that was dividing the two basins, and both vanish. That is the gold line in the branch diagram doing its one job. Its entire significance is that it is the edge, and an edge is only visible because there are two things for it to be between.
The Atmosphere Remembers Which Chair It Is In
Because both states exist over a range of forcing, the answer depends on history. Raise x₁* very slowly from 1.10 and the flow stays blocked the whole way up, until at 1.77 its basin closes and it jumps to the travelling state. Now lower the forcing again: it does not jump back at 1.77. It stays zonal all the way down to 1.34, where the travelling state is the one that disappears.
A gap of 0.43 in the forcing where the state of the atmosphere is not determined by the forcing at all, but by what it was doing last week.
Where the Block Comes From, If You Turn the Forcing Down
Slide the forcing below about 1.06 and the blocked state stops being perfectly steady. It does not collapse; it begins to breathe, swinging in x₁ between roughly 0.46 and 0.72 while the ridge stays locked over the mountain. The panel measures this as a complex pair of eigenvalues crossing zero at x₁* = 1.0590, which is a Hopf bifurcation, and it is why the blocked branch in the diagram goes from solid to dashed at that point while the flow visibly keeps blocking. A state can stop being a fixed point without ceasing to be a regime.
Why the Reading Is Open
The taxonomy asks one question of a weather system: does its own activity release the energy that drives it, or does it only carry energy delivered to it? A thunderstorm answers yes. A convection roll answers no. A block answers both.
On the supplied side, everything here is delivered. The jet is forced from outside; take x₁* away and the whole thing spins down against friction in ten time units with nothing to show for it. The mountain does not move. Nothing in the block releases any energy at all.
On the other side, the block is not a shape stamped on the flow by the forcing. The same forcing also permits a completely different flow, and which one you get is decided by the system’s own state. The mechanism is a genuine feedback: the standing wave over the mountain pushes back on the jet, the weakened jet lets the wave grow, and the stronger wave holds the jet down. The block is doing something to keep itself. It just is not making the energy with which it does it.
So this card is tagged reading open, in the same spirit as Global Circulation. The test as written does not resolve it, and the honest thing is to say which way each half points rather than to round the answer off.
What This Model Is Not
The multiple equilibria are contested, and have been since 1979. The objection is exactly the thing this panel is showing off: a six-mode truncation may manufacture steady states that a well-resolved atmosphere does not have. Higher-resolution models have had a much harder time finding clean multiple equilibria, and much of the modern literature explains blocking through wave breaking and transient eddies instead, mechanisms with no representation here at all. This page does not take a side. It shows what the low-order model does and says that the idea is disputed.
This model gets the lifetime of a block wrong, and not by a little. Real blocks last about a week to ten days, and the longest run two to three weeks. Here, in the range where both states exist, the blocked state is a stable fixed point: undisturbed, it lasts forever. Adding random gusts to the forcing does not fix this. Gusts small enough to leave the regime structure intact never end the block at all across two hundred thousand time units; gusts large enough to end it produce episodes lasting hundreds of time units, and larger ones still destroy the distinction between the two states rather than making transitions between them. The finite lifetime of a real block comes from things this model does not contain, namely the transient eddies that build the block and eventually flush it, and no tuning of what is here will produce it.
Time here is not calibrated. The equations are non-dimensional, and the panel reports non-dimensional time units. The customary reading, which this page offers as a convention and not as a measurement, is that the damping time 1/C = 10 units is the ten-day spin-down of friction in the boundary layer, making one unit about a day. On that convention the travelling state’s 10.2-unit cycle is a ten-day wave, which is the right order for a synoptic wave and is the only check available.
Barotropic. One layer, no vertical structure, no temperature, no moisture, no seasons. The mountain is a single smooth wave rather than the Rockies and the Himalayas.
Charney, J. G. & DeVore, J. G. (1979). Multiple flow equilibria in the atmosphere
and blocking. Journal of the Atmospheric Sciences, 36(7), 1205–1216.
Crommelin, D. T., Opsteegh, J. D. & Verhulst, F. (2004). A mechanism for atmospheric regime
behavior. Journal of the Atmospheric Sciences, 61(12), 1406–1419. Source of the
six-mode form and the coefficients used here.
Kautz, L.-A., Martius, O., Pfahl, S., Pinto, J. G., Ramos, A. M., Sousa, P. M. & Woollings, T.
(2022). Atmospheric blocking and weather extremes over the Euro-Atlantic sector, a review.
Weather and Climate Dynamics, 3(1), 305–336. Source of the description of blocks as
long-lasting, quasi-stationary and self-sustaining, and of the seven-to-ten-day typical
duration.