How far downwind a surface is remembered

A second length at the wall is this collection’s account of roughness: a rough surface changes the wall law by a single length, the roughness height, and the resulting profile is logarithmic with a shifted origin.

That is a statement about a surface in equilibrium with the flow above it. This essay is about what happens when the surface changes, which it does at every field boundary, coastline, hedge and building in the world — and about how long the flow takes to notice.

The internal layer #

Wind blowing over one surface arrives at a change — grass to trees, land to sea, smooth to rough — and the flow immediately above the new surface begins to adjust. The adjustment spreads upwards as an internal boundary layer, growing from the change downwind.

Above that layer nothing has happened yet. The flow there is still in equilibrium with the surface upstream of the change, and it will be until the layer reaches it.

The layer’s height grows as the fetch to the four-fifths power — a well-established empirical relation with the new surface’s roughness setting the coefficient, 0.75 here on a roughness length of 0.03 metres. That exponent is close enough to one that people sometimes quote a ten-to-one rule for the layer’s growth, and the layer is not the interesting quantity.

The layer is not the equilibrium layer #

The interesting quantity is smaller, and this is where the factor of ten in the rule of thumb comes from. The internal layer itself reaches 95.8 metres by the time a ten-metre measurement is in equilibrium with the new surface over crops; the equilibrium sublayer inside it is a tenth of that.

The layer itself climbs quickly: 2.35 metres at ten metres of fetch, 4.09 at twenty, 8.50 at fifty, and 12.4 at eighty. Inside it the flow has been disturbed by the new surface; only in its lowest tenth — 0.235 metres at ten metres of fetch, 1.24 at eighty — has it reached a new equilibrium — a logarithmic profile with the new roughness in it. In between is a transition region belonging to neither surface.

So a measurement in equilibrium with the new surface has to be made inside a layer that is a tenth of one that is itself a fraction of the fetch — the 0.1 is the coefficient that turns a ten-to-one growth rule into the hundred-to-one fetch rule, and it is doing all the work. Putting the two together gives the requirement everybody in micrometeorology quotes:

About a hundred metres of uniform fetch for every metre of height. A ten-metre mast over crops needs 958 metres; a fifty-metre one needs seven kilometres.

The rule is a middle rather than a law #

Because the layer grows as a power of the fetch rather than linearly, the ratio is not a constant. Over the heights and roughnesses anybody actually works at it runs from 36 to 214 — a factor of six.

Three surfaces, six heights, one relation:

Written as a ratio of fetch to height the table runs from 36 to 214 — a factor of six around the hundred everybody quotes. It is largest where measurements are highest and surfaces smoothest, which is exactly the combination used for wind resource assessment and for boundary-layer meteorology: a 50-metre mast over short grass needs 10.7 kilometres of uniform upwind surface, not five. The hundred-to-one rule is optimistic in the cases where it is relied on most.

What a measurement in the wrong place is measuring #

The consequence is not a small error; it is a measurement of a different surface. A profile taken a kilometre downwind of a change from short grass to scattered trees has an internal layer about 150 metres deep with an equilibrium sublayer of 15, so everything above fifteen metres on that mast belongs to the grass or to neither, and only the bottom of it belongs to the trees.

Fitting a single logarithmic profile through the whole thing returns a roughness length and a friction velocity that describe neither surface, and both are used downstream: the roughness goes into a model and the friction velocity goes into a flux.

What “in equilibrium” is asking for #

It is worth being precise about the condition, because it is stronger than it first appears and that is why it is so expensive.

Equilibrium means the turbulence at the measurement height has adjusted to the new surface — not merely that the mean velocity has. The mean adjusts first and the stresses adjust after it, because the stresses are second moments and depend on the whole history of the eddies rather than on the local gradient. That is what a mean profile cannot tell anybody applied to a surface change: matching the mean is easy and matching what carries the momentum is not.

So the fetch requirement depends on what is being measured. A mean wind speed needs less fetch than a momentum flux, which needs less than a scalar flux, which needs less than a spectrum — and the fetches quoted here are for the middle of that range.

That ordering also says which measurements are safest to make on a short fetch. A wind speed used only as a wind speed is fairly robust; the same measurement used to infer a roughness length is not, because a roughness length is read off the profile’s shape and the shape is exactly what has not adjusted.

What the solver computed, and how it was checked #

The internal layer’s growth is the standard empirical relation and the equilibrium sublayer is a tenth of it, which is the conventional figure. Neither is derived here, and the essay is arithmetic on two accepted relations rather than a computation of the adjustment — which is a genuinely hard problem involving the whole downstream evolution of the turbulence.

Two checks. That the growth exponent is exactly four fifths, over four decades of fetch — it reads 0.800000000 — which tests the arithmetic rather than the physics. And that the fetch-to-height ratio at ten metres over an ordinary surface is near a hundred, within a factor of one and a half; it reads 95.8 over crops — because the rule of thumb is what the essay is testing, and a construction that failed to reproduce it would be a construction with an error in it.

Why the memory is so long #

A layer that grows with no length in the problem at all is the layer with no length in it, and the contrast is the point: here the fetch is the length, which is why the answer is a distance rather than a shape.

The physical reason is worth stating because it explains why the requirement is so demanding.

Information about the surface travels upwards by turbulent mixing, at a rate set by the vertical velocity fluctuations — which are of order the friction velocity, a few per cent of the wind speed. It travels downwind at the wind speed. The ratio of the two is the fetch-to-height ratio for the layer itself, which is about ten; the extra factor of ten to reach the equilibrium sublayer is what turns it into the 36 to 214 the table reads.

So the ratio of the two is the ratio of the wind speed to the friction velocity, which is twenty or thirty in the atmosphere. Every metre the layer climbs costs twenty or thirty metres of fetch, and the factor of ten between the layer and its equilibrium part multiplies that to a few hundred. The computed ratios bear that out: 36 over the smoothest surface, 214 over the roughest, with the familiar hundred sitting in the middle of a factor of six rather than standing as a law.

That is a transport argument rather than a diffusion one — turbulence carries the information rather than molecular viscosity — and it is why the growth is nearly linear in the fetch rather than a square root. The measured exponent is 0.800000000, and the fifth it falls short of one is the small departure from linearity that the layer’s own thickening produces: a diffusive growth would give a half, and the difference between 0.5 and 0.8 is the difference between an eddy carrying the information and a molecule doing it.

The same statement, in the streamwise direction #

This is the atmospheric form of a result this collection has already computed twice.

A layer that is an integral of everything upstream finds a laminar boundary layer whose thickness is a weighted integral of the external velocity over the whole surface upstream. The mechanism is the same and the numbers are much smaller, because molecular diffusion is slower than turbulent mixing relative to the convection — which is the wrong way round from what the phrase suggests and is exactly why a turbulent layer is more forgiving.

A duct that forgets everything but one number finds that a duct’s memory of its inlet is a single mode. The internal layer is the same structure with the wall condition changed instead of the inlet, and the same conclusion holds: what survives is one shape whose amplitude is decaying.

The footprint, which is the honest version of the question #

Asking whether the fetch is long enough is a yes-or-no version of a question whose real answer is a distribution, and the modern treatment says so.

An instrument at a height measures a flux that arrived from somewhere upwind. Not from one place: from a footprint, a weighted area whose weight peaks a few hundred heights upwind and whose tail extends much further. The instrument reports one number and that number is an integral of the surface over that weighting.

Read that way the fetch question becomes: how much of the footprint is the new surface? And the answer is never all of it, because the footprint’s tail has no end — the same heavy-tailed structure the wall kernel in the wall the fluid is listening to has, for the same reason, with turbulent transport replacing molecular.

That is the right way to think about it and the wrong way to design a campaign, which is why the rule of thumb survives. A footprint calculation tells an analyst what was measured; a fetch rule tells a field team where to put the mast.

Where this changes an answer #

Wind resource assessment. A mast is put where the developer can get access, which is rarely in the middle of a kilometre of uniform ground. The measured wind speed is then extrapolated to hub height with a logarithmic law fitted to the profile — a profile that has two surfaces in it — and the error propagates into an energy yield as roughly the cube.

Flux measurements. An eddy-covariance tower measures the exchange of carbon dioxide or water between a surface and the atmosphere. The flux it measures comes from a footprint upwind, whose extent is a few hundred times the measurement height, and if the surface changes inside that footprint the flux belongs to a mixture. Footprint modelling exists entirely because of this essay’s subject.

Airport wind measurements. The wind reported to an aircraft comes from a sensor at ten metres in a place chosen for other reasons. Its relation to the wind at fifty metres over the runway depends on what is between them.

And model validation. A measurement used to validate a surface-layer scheme has to be in equilibrium with the surface the scheme is being given, or the comparison is between a model of one surface and a measurement of two.

What can be done when the fetch is not available #

Since most sites do not have a kilometre of uniform anything, it is worth saying what is done instead, because the options are unequal.

Measure lower. The requirement scales with the height, so a two-metre mast needs a fifth of the fetch a ten-metre one does. That works for surface fluxes and not for wind resource, where the measurement is wanted high precisely because the extrapolation from low is what is being avoided.

Select by wind direction. Almost every site has some directions with long uniform fetch and some without. Filtering the record by direction gives a clean subset at the price of most of the data and of a bias, since the wind direction is correlated with the weather.

Model the adjustment. Two-surface models exist and are used, and they need the upstream roughness, which is the quantity that is hardest to know — so the correction has an uncertainty comparable with the error it is correcting.

Or accept the mixture and say so. The honest option, and it is compatible with using the data, provided the reported quantity is described as what it is: a measurement over a landscape rather than over a surface.

The general shape of that list is familiar from every measurement in this collection that reports an accumulated quantity — the instrument in the answer is where it is stated in general.

What the picture cannot show #

The two-surface profile is drawn as two logarithmic segments joined at a height, and a real one has a smooth transition region between them that is neither. Fitting through that region is what produces the error the essay is about, and the figure’s sharp join makes the problem look easier to spot than it is.

Nothing here shows the turbulence doing the transporting. The internal layer grows because eddies carry the surface’s influence upwards, and the eddies are the thing this collection’s solver cannot compute.

The same arithmetic, at three other scales #

The structure — an adjustment that spreads upwards while the flow carries it downwind — repeats at scales from a hedge to a continent, and the ratio is roughly the same each time because it is set by the ratio of the wind speed to the friction velocity.

A hedge or a building. The disturbance is felt for tens of heights downwind and is fully adjusted only after hundreds, which is why a wind measurement near a building is a measurement of the building.

A coastline. Air moving from sea to land meets a roughness change of two orders of magnitude, and the internal layer takes tens of kilometres to reach the height of a wind farm. Offshore wind resource assessment lives entirely inside this problem.

And a continental land-use change. The layer that grows from a change of surface at the scale of a region is the whole atmospheric boundary layer, and its adjustment time is a day — which is one of the reasons regional climate models are sensitive to their land-surface description in a way that looks disproportionate.

In every case the number to compute first is the same one: the height wanted, times a few hundred.

The same question asked of a wake rather than a surface is the last essay in this field, on the same machinery.

What this makes of a single measurement #

The practical consequence is that a wind measurement is a statement about an area, not about a point, and the area has to be worked out before the measurement means anything.

The height sets the depth of the memory and the fetch sets its extent. An instrument at ten metres is reading a layer that has grown over the last kilometre or so of surface; one at two metres is reading a much shorter fetch and therefore a much smaller patch.

So two instruments on one mast are not measuring the same site. The upper one is reporting a kilometre of upwind terrain and the lower one a hundred metres of it, and if the terrain changed between them the profile between the two instruments is not a profile of anything.

And a site classification is a statement about the wrong thing if it describes the ground under the mast rather than the ground upwind of it, which is why modern siting standards are written in terms of distance to the nearest roughness change rather than in terms of what the mast is standing on.

Who found it, and when #

The internal boundary layer was identified in the 1950s and the four-fifths growth law is Elliott’s, from 1958, refined by Wood and others since. The tenth-depth equilibrium sublayer is a later and softer result — it is quoted variously as a tenth or a fifth, which is itself part of the factor of six.

Footprint modelling, which is the modern form of the question, dates from the 1990s and asks it properly: not “is the fetch long enough” but “where did the flux this instrument is measuring actually come from”, which is a distribution rather than a distance.

Limits recorded rather than smoothed over #

The equilibrium condition is about the turbulence, not the mean. Stated above and worth repeating as a limit: the fetches quoted are for a momentum flux. A mean wind speed needs less, a scalar flux needs more, and a spectrum needs more again — because each is a higher moment of a field that adjusts from the bottom up, and the higher moments are the last to arrive. That ordering is the same one what a mean profile cannot tell anybody sets out, with distance downwind standing in for time.

Neutral stratification. Everything above assumes a neutral atmosphere. In unstable conditions the layer grows much faster and in stable ones much more slowly, by factors of several — so the fetch requirement is a strong function of the time of day, and the numbers here are the middle of it.

One change, and a step change. The relation is for a single abrupt change from one uniform surface to another. Real landscapes are a sequence of changes at every scale, and the flow is never in equilibrium with anything.

The tenth is a convention. The equilibrium sublayer’s depth as a fraction of the internal layer is quoted between a tenth and a fifth depending on the author and the quantity being measured. That factor of two is inside the essay’s factor of six and is the least defensible number in it.

The instrument is assumed ideal. Everything above is about the flow. A real anemometer has its own response time, its own flow distortion and its own mounting, and three buffer layers, one friction is a reminder that what a model says about a wall and what an instrument reports near one are separate questions.

And nothing here is computed from the equations. Both relations are empirical, the arithmetic is arithmetic, and the essay’s contribution is to put the two together and say what the product means for an instrument.