The balance that is its own error

Air moving across a rotating planet feels a sideways force proportional to its speed — a force that exists only because the frame is turning, and which a bathtub is far too small to notice. Set that force against the pressure gradient and the two balance: the wind blows along the isobars rather than across them, faster where they are closer together, and the whole of large-scale meteorology follows. That is geostrophic balance, it is the first thing anybody learns about the atmosphere, and it works.

It works because it neglects something, and what it neglects is the acceleration of the air as it goes round a curve. The licence for that is the Rossby number

$$\mathrm{Ro} = \frac{V}{fR},$$

with $f$ the Coriolis parameter and $R$ the radius of curvature of the flow, and the rule is the familiar one: geostrophic balance holds when the Rossby number is small.

The unusual thing about this case is how exactly that rule can be checked.

The quadratic, and the identity that falls out of it #

Keep the centripetal acceleration and the balance for circular flow becomes the gradient wind:

$$\frac{V^2}{R} + fV = fV_g \quad\text{(cyclonic)}, \qquad -\frac{V^2}{R} + fV = fV_g \quad\text{(anticyclonic)},$$

where $V_g$ is the geostrophic wind — the speed the pressure gradient would give if the curvature were ignored. Divide the first by $fV$:

$$\frac{V_g}{V} = 1 + \frac{V}{fR} = 1 + \mathrm{Ro}.$$

The geostrophic wind overestimates a cyclonic wind by exactly the Rossby number. Not to leading order; exactly, for circular flow. The anticyclonic branch gives $1 - \mathrm{Ro}$ with the same exactness, so the geostrophic wind underestimates there.

This is the only case in this collection where the accuracy of an approximation and the value of the group that licenses it are literally the same number. There is nothing to invert and no tolerance to choose: one per cent accuracy needs $\mathrm{Ro} = 0.01$, ten per cent needs 0.1, and $\mathrm{Ro} = 1$ is a hundred per cent error.

What that does to the folklore threshold #

The threshold “Ro ≪ 1” is therefore unusually easy to interpret and unusually badly used. Synoptic weather systems have $\mathrm{Ro}$ around 0.1: geostrophic balance is a ten per cent approximation for them, which is why forecasters use it for reasoning and never for numbers.

A tropical cyclone has $\mathrm{Ro}$ between 1 and 10 near the eyewall — the balance there is not approximate, it is absent, and the flow is in cyclostrophic balance instead, with the pressure gradient held up by the centripetal term alone and the Coriolis force a correction.

A tornado has $\mathrm{Ro} \sim 10^3$. The planet’s rotation contributes nothing whatever, which is the same conclusion the bathtub reaches by a different route and for the same reason.

The interesting band is the middle one. Nothing in the geostrophic approximation announces its own error, so a synoptic-scale calculation gives a wind ten per cent too fast and looks entirely reasonable, and the error is systematic rather than random: cyclonic winds are always over-predicted and anticyclonic ones always under.

The limit that is not an accuracy at all #

The anticyclonic quadratic is $V^2 - fRV + fRV_g = 0$, whose roots are real only while $fR \ge 4V_g$ — that is, while

$$\frac{V_g}{fR} \le \frac{1}{4}.$$

Past that there is no balanced anticyclonic flow at all. Not an inaccurate one; none. No wind whatever can balance that pressure gradient with the curvature going that way.

That is a threshold of an entirely different kind from everything above, and it is the same shape of result as a nozzle that will pass no more however hard it is pushed. It is a discriminant vanishing, so it is exact, sharp, at an algebraic number, and has no tolerance anywhere in it — the same shape of result as the critical Stokes number of an eighth and as the maximum a wedge can turn.

And it has a consequence anybody can check by looking at a weather chart. Anticyclones are broad and gentle; cyclones can be small and violent. A 500 km high at 45°N cannot support a geostrophic wind above 12.9 m/s; a 50 km one at 20°N cannot support 0.62 m/s. There is no such thing as a small tight high. Lows have no such limit, which is why hurricanes exist and anti-hurricanes do not.

Nothing in the geostrophic equation contains a hint of it #

Worth stating plainly, because it is the sharpest available example of what a dropped term takes with it.

Geostrophic balance is $fV = fV_g$, that is $V = V_g$. It is linear, it has exactly one root, and it has that root for every value of every parameter. There is no configuration of a geostrophic atmosphere in which no wind exists. The limit at a quarter is produced entirely by the term the approximation drops, and no amount of care in applying the approximation could reveal it.

That is the general hazard with an asymptotic simplification and it is more serious than losing accuracy. A linearised equation loses not only precision but whole classes of behaviour — roots that vanish, solutions that bifurcate, limits that exist — and the loss is silent, because the simplified equation goes on producing answers.

The numbers, on systems that exist #

Putting the identity to work on real systems makes the size of the error concrete, and it is larger than the reputation of the approximation suggests.

The third row is the one worth staring at. A perfectly ordinary depression, the kind that crosses western Europe twice a week, has a Rossby number of a third — so the geostrophic wind computed from its isobars is a third too fast, systematically, and the correction is not a refinement but the difference between a gale and a strong breeze.

Forecasters have always known this and have always used the gradient wind for numbers. What the identity adds is that the correction is not something to be estimated: it is $1/(1+\mathrm{Ro})$, and the Rossby number is computable from the chart.

Why the error has a sign #

The systematic sign is worth its own paragraph, because it is what makes the approximation dangerous rather than merely imprecise.

An unbiased error averages out over many systems; a signed one accumulates. Geostrophic balance over-predicts every cyclone and under-predicts every anticyclone, so a climatology built from geostrophic winds has too much cyclonic circulation and too little anticyclonic — which is a bias in the mean vorticity of the analysed atmosphere, and mean vorticity is not a quantity anybody wants a bias in.

The reason for the sign is the same geometry that produces the anticyclonic limit. Going round a cyclonic curve requires a net inward force, so part of the pressure gradient goes into turning the air and only the rest is left to be balanced by Coriolis — hence less wind than geostrophy asks for. Going round an anticyclonic curve requires a net outward force, so the Coriolis force must exceed the pressure gradient — hence more wind.

One term, dropped, and both the systematic bias and the hard limit come from it.

What the balance gives when it is differentiated, and the constant it cannot supply #

The balance is used far more often in its differentiated form than in the one this essay has been testing, and the differentiated form has a gap in it that belongs beside everything else here.

Take geostrophic balance at two heights and subtract. The pressure gradient at the upper level differs from the one below by the weight of the air between them, which depends on its temperature — so the vertical shear of the geostrophic wind is fixed by the horizontal temperature gradient:

$$\frac{\partial \mathbf{V}_g}{\partial z} \;\propto\; \hat{z}\times\nabla T .$$

That is the thermal wind relation, and it is the most-used diagnostic in the subject, because a temperature field is measurable everywhere by radiosonde and satellite while a wind field is not. It explains the jet stream in one line: the pole is colder than the equator, so the westerly wind must increase with height, and it does so until the temperature gradient reverses at the tropopause — which is exactly where the jet’s core sits. The jet is not driven at its own altitude; it is the accumulated shear of everything below it.

And it inherits this essay’s error unchanged. The relation is the vertical derivative of a balance that is wrong by the Rossby number, so the shear it gives is wrong by the same fraction, with the same sign, in the same systematic direction.

But the more interesting defect is that it gives a derivative and not a value. Integrating the shear upward from some level gives the wind at every other level relative to that one, and nothing in the thermal wind relation says what the wind at the starting level is. The constant of integration is not determined by the temperature field, however perfectly it is measured.

In the atmosphere that gap is easily closed, because the surface wind is measured. In the ocean it was not, and the consequence occupied physical oceanography for most of a century. A hydrographic section — a line of stations, each lowering instruments to measure temperature and salinity with depth — gives the density field, and therefore the geostrophic shear, and therefore the currents at every depth relative to one. Which one is unknown. The classical fix was to assume a level of no motion, usually at a depth of a kilometre or two, on the argument that the deep ocean is quiet; every published current from a hydrographic section for decades rested on that assumption, and the transport of the Gulf Stream is a number that moved as the assumption was revised.

It is the same missing number this collection keeps meeting, arriving in a new subject: the equations are exact, the data are exact, and the answer is determined only up to a constant that the physics inside the domain does not fix. And the resolution is the one that has resolved every other instance in these essays — the constant is supplied from a boundary. Satellite altimetry measures the sea surface’s height directly, the surface pressure gradient follows from it, and the geostrophic current at the top of the water column is then known rather than assumed. The whole depth then follows by integrating the shear downwards.

Which is why the phrase satellite oceanography names a change of subject rather than an improvement in instruments. What arrived was not a better measurement of the ocean’s interior. It was the one number the interior could never contain.

Where the number is small enough for real #

Two places in the atmosphere and ocean genuinely satisfy the balance to a per cent, and both are large.

The mid-latitude jet stream has speeds around 50 m/s, radii of curvature of a few thousand kilometres and $f \approx 10^{-4}$: $\mathrm{Ro} \approx 0.05$. Five per cent, which is why the thermal wind relation — the vertical derivative of geostrophic balance — is quantitatively useful for diagnosing the jet from a temperature field.

The ocean’s large-scale circulation has speeds of centimetres a second over hundreds of kilometres: $\mathrm{Ro} \sim 10^{-3}$. Geostrophy there is good to a part in a thousand, which is why sea-surface height measured by satellite altimetry can be turned directly into surface currents, and why that measurement works at all.

Below the balance sits the layer where it fails: within an Ekman depth of the boundary, friction enters, the wind crosses the isobars, and the entire vertical transport of the ocean and atmosphere depends on the small angle by which it does.

The same structure, one term further out #

The gradient wind is itself an approximation — it assumes circular streamlines and steady flow — and keeping the next term gives the cyclostrophic and inertial balances, each with its own regime. Writing all of them as a single momentum balance:

$$\underbrace{\frac{V^2}{R}}_{\text{centripetal}} + \underbrace{fV}_{\text{Coriolis}} = \underbrace{\frac{1}{\rho}\frac{\partial p}{\partial n}}_{\text{pressure}},$$

three terms, and every named balance in dynamical meteorology is one of them being dropped — the same kind of bookkeeping that decides which terms a boundary layer keeps. Which one may be dropped is decided by the Rossby number, and the accuracy of dropping it is, by the identity above, that same number — which makes this the tidiest instance in the collection of a group doing exactly the job it is advertised for, while the threshold attached to it does not.

The one thing the number does decide correctly #

It is worth ending the argument fairly. The Rossby number is not a bad group; it is an unusually good one, and this essay is the only place in the collection where the accuracy threshold and the balance threshold coincide in form even while differing in value.

What it decides correctly is which balance to use at all. Below about 0.1 the flow is geostrophic and the pressure gradient is held by Coriolis; between 0.1 and 10 the gradient wind is needed and both terms matter; above 10 the Coriolis term can be dropped instead and the flow is cyclostrophic. Those are genuine regime boundaries, they are what the number was invented to mark, and they are approximately at the values the folklore gives.

The mistake is a narrower one than “the number is wrong”. It is that the same number is asked to do two jobs — say which terms matter, and say how accurate an answer is — and it does the first well and the second only if the answer is read as a percentage rather than as a pass.

What the picture cannot show #

Circular flow, and only circular flow. The exact identity $V_g/V = 1 \pm \mathrm{Ro}$ holds for steady circular streamlines. Real weather systems are elliptical, unsteady and embedded in a larger flow, and the curvature of a streamline differs from the curvature of a trajectory whenever the system is moving — which for a synoptic system moving at 15 m/s is not a small difference.

Frictionless, and above the boundary layer. Everything here is inviscid. Within about a kilometre of the ground the wind crosses the isobars by twenty to thirty degrees and neither balance describes it.

The Coriolis parameter is taken as constant. Its variation with latitude is what gives Rossby waves, and Rossby waves are the reason weather moves; nothing in this essay has any of that in it.

And the anticyclonic limit is a limit on balanced flow. Air does not simply cease to exist past a quarter. What happens instead is that the flow becomes unsteady, or adjusts by radiating gravity waves until the gradient falls back within the limit — which is a real and observed process and is outside a steady balance entirely.

Who found it, and when #

Buys Ballot stated the empirical rule in 1857 — with the wind at one’s back in the northern hemisphere, low pressure lies to the left — and the dynamical explanation followed within a decade. The gradient wind is in Ferrel’s work of the 1850s and 1860s. The anticyclonic limit has been in textbooks since Brunt in the 1930s, usually as a two-line remark.

Rossby’s name attached to the number in the 1930s and 1940s, when the scale analysis that produced it became the foundation of numerical weather prediction: deciding which terms to keep was not a matter of elegance but of what the machines of the day could integrate, and the first successful forecast in 1950 used a filtered equation set chosen by exactly this argument.

The surprising connection is that the accuracy and the licence are the same number here and nowhere else in this collection. Every other group needs a solution before its threshold can be found; this one carries its own. And it is still quoted with an adjective rather than a percentage, which suggests the difficulty is not that the arithmetic is hard.

Where the ladder goes next #

Above this rung is the balance that includes the unsteadiness — the difference between streamline and trajectory curvature, which is what makes a deepening system’s wind field differ from a steady one’s — and beyond that potential vorticity, which is the conserved quantity that makes the whole subject tractable and which this collection has not built.

Beside it sits the bathtub, where the same number is enormous and the same balance is absent, and the Ekman layer, where friction breaks it. Below it is the general account of what a group’s own value is worth, in which this is the row whose two numbers are closest to being the same statement.