Correction and downscaling of NWP wind speed forecasts

METEOROLOGICAL APPLICATIONSMeteorol. Appl. 14: 105–116 (2007)Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/met.12

Correction and downscaling of NWP wind speed forecasts

Tom Howard* and Peter ClarkTom Howard, Met Office, FitzRoy Road, Exeter, EX1 3PB, UK

ABSTRACT: NWP models typically parametrize the effects of unresolved orography, often through use of an effective(orographic) roughness. Whilst this parametrization realistically models the orographic drag on the synoptic-scale flow, itcreates two problems for the assimilation of wind observations from high ground. First, the artificially increased surfacestress causes a reduction in the predicted wind speed at the standard wind observing height of 10 m, and second, thespeed-up over the unresolved summits is not modelled.

A method is described for reconciling observed and modelled wind speeds. The method is based on the linear theoryof neutral boundary-layer flow over hills and includes a resolution of both the problems described above. The method isapplied to both the assimilation of observations and the creation of an improved 10 m wind analysis. The method has beenon trial in the Met Office’s nowcasting system; significant improvements are demonstrated, particularly during strong windevents.

The simplified model presented here is not claimed to represent the full complexities of the boundary layer, butnevertheless produces computationally cheap, low-level wind forecasts, which are a significant improvement on the existingoutput from the Unified Model. Crown Copyright 2007. Reproduced with the permission of the Controller of HMSO.Published by John Wiley & Sons, Ltd

KEY WORDS downscaling; orographic roughness; orographic drag; nowcasting; windspeed; speedup; mesoscale; linear theory

Received 31 January 2006; Revised 26 February 2007; Accepted 26 February 2007

1. Introduction

Numerical weather prediction (NWP) systems of neces-sity run with a spatial resolution which cannot representthe impact of all orography explicitly. Figure 1 illustratestypical NWP model orography in contrast to a high-resolution orographic dataset. The impact of orographywhich is not resolved must be represented by some formof parametrization. This parametrization must, at least,represent the additional drag that is produced by surfacefriction and pressure forces. This additional drag occurs,at least in part, in a region close to the mean surfacewithin the boundary layer. The result is that care must betaken in interpreting model output in the region where thedrag occurs. Unfortunately, this is the region where windinformation is especially required from NWP systems.

Many applications, such as damage forecasts, sailingand pollutant dispersion, require a wind speed forecastclose to the surface, where the model wind represents,at best, an area average and, at worst, a quantity whichis strongly dependent on the parametrized orography.So a method is needed to estimate the wind profilenear the surface. This method would preferably take intoaccount local orography. Any correction technique must

* Correspondence to: Tom Howard, Met Office Joint Centre forMesoscale Meteorology, Meteorology Building, University of Reading,PO Box 243, Earley Gate, Reading, Berkshire, RG6 6BB, UK.E-mail: [email protected]

be much cheaper to run than the host NWP model, oth-erwise running at higher resolution would be a viablealternative. Such a technique can be considered a com-bination of a correction technique and a technique toadd smaller-scale information to the model, i.e. down-scaling.

This paper describes a simple and computationallycheap technique based on linear flow theory, whichis applied to the Met Office’s operational UnifiedModel. This model uses a parametrization of additionalboundary-layer drag due to orography which is imple-mented as an effective (‘orographic’) roughness (Grantand Mason, 1990). The additional drag is estimated frominformation about unresolved orography. The additionaldrag is applied at the model surface and added to thesurface stress arising from the land surface. The com-bined surface stress is then used to define an effectiveroughness length.

The orographic roughness (OR) concept is usedbecause observations over moderate terrain show that,above a certain height, the wind profile is indistinguish-able from that which would arise from a flat surface witha roughness substantially higher than the actual rough-ness due to the vegetation etc. on the surface (Grant andMason, 1990). Models are generally already formulatedto treat the roughness of flat surfaces, so implementationis straightforward. However, the concept has only beenextensively verified for flows close to neutral, though itis assumed to apply at a wider range of stabilities.

Crown Copyright 2007. Reproduced with the permission of the Controller of HMSO.Published by John Wiley & Sons, Ltd.

106 T. HOWARD AND P. CLARK

200 300 400 500

Easting (km)

0

500

1000

1500

2000

Hei

ght (

m)

Figure 1. A transect through Northing 804.025 km illustrating the MetOffice’s mesoscale model orography (the smoother line) in contrast to a1-km resolution orographic dataset. The location of WMO observation

station 03 065 (Cairngorm) is shown by the filled circle.

The difficulty of interpreting wind profiles is a conse-quence of the need for parametrization, not the particularmethod. Alternative approaches which use a verticallydistributed stress divergence (e.g. Wood et al., 2001),may yield more correct mean profiles over the depth theyare applied, but they still need interpretation to derivelocal winds.

The aim of the technique described in this paper isthus to provide a simple and cheap method of recover-ing a realistic wind profile in the lower boundary layerfrom NWP using only a small amount of additionalinformation and, furthermore, providing an approximatecorrection for local topography. The technique assumesthat flow is close to neutral and makes use of resultsfrom linear theory. The method is therefore not expectedto perform well where flow separation dominates, orin strongly stable or unstable conditions. The simplifiedmodel presented here is not claimed to represent the fullcomplexities of the boundary layer but nevertheless pro-duces computationally cheap, low-level wind forecasts,which are a significant improvement on the existing out-put from the Unified Model.

The method described here has been implemented inthe Met Office’s NIMROD nowcasting system. This com-bines NWP model output with observations to provide amore spatially detailed analysis and forecast system. Theinverse of the correction for local topography is used tocorrect the input NWP fields using recent surface obser-vations in a manner which is less dependent on localtopography. The correction is thus more characteristic ofthe underlying flow and can be advected with it.

2. The NIMROD system

NIMROD is a nowcasting system built and operatedby the Met Office (Golding, 1998) using an optimalcombination of very-short-range forecasting techniquesand mesoscale NWP. Though originally conceived as aprecipitation system, it has been extended to include avariety of parameters including low-level wind.

The Nimrod wind nowcasting scheme uses current syn-optic observations to adjust the mesoscale model windforecast. A difference field is formed by spreading the(model-observation) differences in both speed and direc-tion using a recursive filter. The difference field is addedto the model forecast to produce the analysis. Ideally,

the analysis correction would be advected with the gra-dient wind and applied to subsequent mesoscale modelforecasts, on the assumption that the error was char-acteristic of the flow. However, in practice, the modelerror over high ground was dominated by unrealisticallylow wind speeds, partly due to the unresolved orogra-phy parametrization, and partly because some observingstations see locally enhanced wind speeds due to localorography. Advecting the analysis correction leads tounrealistically high wind speeds downstream, often overthe sea, so the advection of the correction could not beused. This further motivated the need for a correctionto the model winds, which effectively compensates forthe orographically induced component of observed windspeed.

3. Correction and downscaling method

An extensive literature exists describing the flow overhills of small slope using linearized forms of the equa-tions of motion. Jackson and Hunt (1975) introduced alinear theory for neutral boundary layer flow over an iso-lated hill. Mason and Sykes (1979) essentially extendedthis solution to three dimensions. Mason and King (1985)took a slightly different approach, but both (and otherrelated papers) are based upon the concept of matching asolution in an ‘outer layer’, where the perturbation stressdivergence is small compared with the perturbation pres-sure gradient, to a solution in the ‘inner layer’, where theperturbation stress divergence dominates and the verticalvariation of perturbation pressure can be neglected, theperturbation pressure being taken from the outer layersolution. Mason and King (1985) outline a number ofalternative solutions, labelled A–D.

The different solutions depend largely on the choice ofwind and length scale to define the inner layer depthand the choice of advecting wind in the inner layer.While Jackson and Hunt (1975) use a single characteristiclength scale determined by the width of their isolated hill,later derivations choose a separate length scale for eachwavenumber in a Fourier decomposition of the orographyand hence different choices of advecting wind. An outlinederivation of models A–D is given in the Appendixto motivate the tuning performed below. In order toderive as simple a system as possible, an approach whichrequired a Fourier transform of the unresolved orographywas avoided (in part, because it is not obvious howto combine this with a spatially varying wind field).Thus, the unresolved orography is represented using acharacteristic length scale. This approach thus has muchin common with model B and Jackson and Hunt (1975)but has included the refinements C and D to show thatthese refinements, for the most part, lead to changesin coefficients rather than the fundamental mathematicalform, and that these coefficients can, in some sense,be regarded as tuneable, based on the spectrum ofwavenumbers.

Let us suppose we have a numerical model of a certainresolution. This model already contains a parametrization

Crown Copyright 2007. Reproduced with the permission of the Controller of HMSO. Meteorol. Appl. 14: 105–116 (2007)Published by John Wiley & Sons, Ltd. DOI: 10.1002/met

CORRECTION OF NWP WIND SPEEDS 107

of model OR, and it is desired to correct near-surfacewinds for local orography which is not resolved in themodel. Assume the following are given:

1. Winds on model levels, or fixed heights above themodel orography, denoted uM.

2. A local vegetative roughness length, z0. At present,the roughness length used in neutral conditions in themodel is used, though a truly local roughness lengthcould be used.

3. OR parameters H, the average peak/trough amplitudeand A/S, the silhouette area per unit horizontal area.

The OR parametrization represents the effects ofunresolved orography on the boundary layer flow asan enhanced surface roughness, and hence surface drag.The wind profiles which result are not realistic in aregion close to the surface. The linear analysis providesan estimate of the depth scale over which the localorographic perturbations apply. In practice, this depthscale is very comparable with that over which observedprofiles over complex terrain asymptote to the logarithmicprofile (Grant and Mason, 1990). It should be noted thatthe linear analysis is not strictly consistent with the ORparametrization as the latter includes the effects of flowseparation. However, the OR parametrization providesthe impact of the unresolved orography as a whole,

while our analysis provides an estimate of the local windprofile. The inconsistency is only relevant to the extentthat the local flow is separated, which occurs wherethere is steep orography (which, of course, represents alimitation, at least in principle, on the technique).

We reason as follows: the unresolved orography overan area determines the wind speed well away from theorography. This wind speed then determines the localwind profile given the vegetative roughness length in theabsence of modification by local orography. This localflow is then further modified by local orography, so thecorrection technique has two stages. The first reconstructsthe local wind profile in equilibrium with the vegetativeroughness and is termed the roughness adjustment (RA).The second then modifies this for local orography, andis termed the height correction (HC). The method isillustrated in Figure 2.

3.1. Outer layer reference height

The roughness adjustment corrects the wind profile onlyin that part close to the model surface where the ORparametrization has produced an unrealistic profile. Thisis a natural step, but it is important that the heightbelow which the adjustment is made is carefully chosen.Over gentle terrain it is expected that the impact of theOR is to be much shallower than over more hilly or

'Real'orography

Heightcorrection

λ

Roughnessadjustment

Smoothedmodelorography

u(z)

u(z)

u(z)

href

UHC(z)

uM(href)

uM(z)

Figure 2. Schema showing the roughness adjustment and height correction and illustrating nomenclature. The roughness adjustment (right) showsthe input model profile (solid) and the profile adjusted to local vegetative roughness (dotted). The height correction (left) shows this profile

(dotted) and a typical perturbed profile over a crest (solid).



mountainous regions. The height is thus variable withterrain. If chosen too low, the profile will not be fullycorrected, so winds will remain unrealistically low. Ifchosen too high, assumptions about local equilibrium willbe invalid.

For a sinusoidal ridge across the flow, the silhou-ette area (per unit length of ridge) of one wavelengthis the peak-to-trough height, H, so, for real orography,the quotient H/(A/S) represents a length scale, whichmay be thought of as a characteristic wavelength. Inour implementation, these parameters are derived froma dataset with approximately 100 m resolution over theUK and this characteristic wavelength is found to vary,over the UK, from about 800 m over lowland regionsto about 3 km over highland regions. These are reason-able length scales in the absence of other information.A more sophisticated approach might integrate the oro-graphic stress over the spectrum of wavelengths (see, forexample, Wood et al.), leading to numerically differentresults. The exact wavelength may thus be regarded astuneable, so the reference (outer layer) height is taken tobe:

href = ak−1 (1)

where the local wavenumber, k, is given by

k = πA/S

H/2(2)

and a is a tuneable parameter of order 1. The factor a

is included for completeness and in practice, results arenot sensitive to small variations in a. No direct attempthas been made to optimize a, and in the subsequent textit takes the value 1. Any deviation from this manifestsitself in the optimization of the overall scheme describedlater.

Table I shows typical values of subgrid orogra-phy parameters for the Met Office UK mesoscalemodel. These values all give characteristic wavelengthsH/(A/S) in the region of 2–5 km.

We take the outer layer wind speed to be the NWPmodel wind speed at the reference height:

UA = uref = uM (href ) (3)

With a = 1, href varies from about 300 m over south-ern England to about 800 m over the Alps. This reflectsthe expected variation with terrain described above but

Table I. Typical values of subgrid orography parameters forthe Met Office UK mesoscale model.

Location A/S H/2 Nom(m)

Southern England 0.02 20Highest Pennines 0.05 100W. Highlands of Scotland 0.12 200Alps 0.20 500

one might, perhaps, suspect that the southern Englandvalue is a little high, which may reflect an under-estimateof A/S resulting from too coarse resolution source data.These heights would be expected to be within the neutralboundary layer given the OR but are likely to be higherthan the depth of the logarithmic layer due to local rough-ness. However, the results are not very sensitive to theprecise height chosen so long as it is high enough tobe within the genuine (i.e. observable) logarithmic pro-file due to the OR. In practice, Grant and Mason (1990)found the profile to be logarithmic down to heights fairlyclose to the peak orography. This might suggest that afuture enhancement could be investigated.

3.2. Roughness adjustment

We assume a neutral logarithmic wind profile below thereference height:

u(z1) = u∗κ

ln(z1/z0) = uM (href )ln(z1/z0)

ln(href /z0)(4)

where z1 is the height above local orography at whichthe wind speed is to be calculated (always 10 m in thiswork), z0 is the vegetative roughness length and κ is VonKarman’s constant, taken to be 0.4 here.

3.3. Height correction

Let us assume that the sub-grid orography is representedby a sinusoidal trough/ridge system with amplitude h0

and wavenumber k. At present, the OR scheme isisotropic, so it is assumed that this is oriented acrossthe wind. Since a single characteristic length scale isassumed (the inverse wavenumber), analysis starts withModel B (see Appendix). However, it is also recognizedthat the refinement of model D, to use the unperturbedwind at height z as the inner layer wind in the definitionof C, should bring some of the advantages of model D aswell as being easier to compute (as no inner layer depthscale is needed). The solution for this case is given byEquation (5):

uHC = e−kzUA kh|z=0

1 −

K0

(+√

Cz)

K0

(+√

Cz0

) (5)

in the nomenclature given in the Appendix.We may write

h|z=0(x) = hL − hM (6)

where hL is the height of the true orography at thelocation required and hM is the effective orography at thelocation in the NWP model. This is obtained by simplebilinear interpolation from the NWP gridded orography.Note that uHC given by Equation (5) is complex – aphase shift is introduced into the flow in the innerlayer and so there is asymmetry between two points at



the same height on the upwind and downwind slopesof the hill. This is responsible for the pressure drag.Since directional dependence is beyond the scope of thiswork, the imaginary part of the Bessel function ratiois ignored, which is equivalent to taking an average ofthe upwind and downwind perturbations. The techniquecould, however, be extended by noting the direction ofthe local slope at the reference wavenumber. We estimatethe velocity perturbation, then, from (5) and (6):

uHC = e−kzUA Re

1 −

K0

(+√

Cz)

K0

(+√

Cz0

)

× k(hL − hM ) (7)

The derivation in the Appendix is for a sinusoidalridge/trough system with wavevector aligned with theflow. However, the result applies for any wave compo-nent aligned with the flow, and random, homogeneoustopography with the same characteristic wavenumber inall directions is assumed.

3.4. Overall adjustment

The initial trial method may be expressed as an estimatedwind speed, u, (representative for the smooth modelorography) multiplied by a correction factor (based onthe height difference between the model and ‘real’orography) which amplifies the wind speed to a figurerepresentative of the ‘real’ orography as follows:

u = u

(1 + βe−kz UA

uk(hL − hM )

)(8)

where

β = Re

1 −

K0

(+√

Cz)

K0

(+√

Cz0

)

(9)

This can be rewritten:

u = u

(1 + β

αe−kzk(hL − hM )

)(10)

whereα = u

UA(11)

If the magnitudes of terms in (10) are examined, thefollowing may be concluded:

1. β represents the modified inner layer wind profile.In practice, it has little systematic impact and mightperhaps be ignored (i.e. set to 1).

2. α is the ratio of the flat surface wind at measurementheight above local orography to that at the referenceheight, so must be (substantially) less than 1. Itdepends more on the height ratio than on the largervegetative roughness length. This is largest overforests and urban areas – an extreme might be 1 m,

so with href = 1000 m (fairly substantial orography)and measurements at 10 m, this ratio would be 1/3.

3. The exponential decay factor is close to 1 near thesurface except for the smallest hills.

4. If A/S is less than 1/π (which is steep!) then k(hL −hM ) ≤ 1

5. Thus, the term α causes most of the amplificationand the maximum wind speed is likely to be theroughness-corrected wind multiplied by a factor ofabout (1 + 1/α) (or (1 + 1/α2) under Model D). Inpractice, a limiter is necessary for the speedup (or theheight difference).

In the final version of the scheme, tuning parameterswere introduced giving:

u = k1u

(1 + k2

β

αe−kzk(hL − hM )

)(12)

The introduction of k2 can be seen as a move towardsModel D, with k2 as a representative value of 1/α forthe domain. The full model D was not implemented asthis makes the model more sensitive to the precise localvalue of the wavenumber k and the local roughness, butthis could be investigated in future.

Note that this analysis has assumed neutral stabilitythroughout. This is regarded as reasonable as it isexpected to have most impact in strong winds; as willbe shown below it has significant success. However, themethod could be extended to include stability given anestimate of surface heat flux, using an Obukhov lengthbased on the heat flux and local friction velocity fromthe analogue of (4). The form of the inner-layer profileβ is more difficult, but is probably not very importantcompared with the major terms, which are the flat-surfaceroughness correction factor α and the height scale forexponential decay. The method is likely to fail completelyin cases of strong stability with blocked flow, and is likelyto be less accurate in cases with strong surface heatingthough it should be remembered that the OR scheme isstill applied in these conditions so some correction maybe better than none.

3.5. Observation correction

If, for clarity and brevity, the tuning parameters areincorporated into the definitions of u and F, Equation (10)can be written:

u(z1) = u(1 + F) (13)

whereF = F(z, href , z0, hL − hM )

The term F (given neutral stability) depends onlyon terms, which are constant at each location, and themeasurement height. Thus F can be pre-calculated.

In the above downscaling, the roughness-correctedwind to fine scale orography has been discussed. Withinthe NIMROD system an objective analysis of surface



fields including wind speed is performed starting fromthe NWP model background. When analysing errors inthe NWP wind field it makes more sense to correct theobservation to obtain an ‘unperturbed’ or ‘NWP-model-orography’ wind speed observation as follows:

uob = uob

1 + F(14)

where uob is the ‘raw’ observed wind speed (i.e. at theobservation station height). This is then compared withthe roughness-corrected NWP wind to generate an error,which is spread through the recursive filter. The resultinganalysis field is then downscaled to the orography.

4. Results and tuning

This section describes results of applying the downscal-ing technique to input NWP fields. Initial implementa-tion of the method showed significant benefits, but thereremained a suggestion of overall bias and some depen-dence of the error on actual to model height difference.In the initial trial, covering a 4-week period (1–28 July2002), application of the new method changed the overallwind speed bias from −1.43 m s−1 to 0.36 m s−1, sug-gesting an over-correction, but, though much more sym-metrical, the error distribution still showed more grossunder- than overestimates (this is essentially method 2abelow). The tuning parameters introduced above weretherefore investigated. While both could, in principle,be optimized together, there was concern that a bias inheight difference could manifest itself in the value of k1.Separate optimizations were therefore performed: first,stations close to the NWP model orography were used tofix k1, then the remaining stations to determine the valueof k2. Finally, it was noted that errors appear asymmetric,in that, stations below the NWP orography show differ-ent (generally smaller magnitude) errors from those of anequal distance above, so separate optimizations were per-formed for the two sets, producing k2lo and k2hi . Thesevarious methods are summarized in Table II.

Data were used from four cases with particularlystrong winds: 1500 UTC 30 September 1998, 0000UTC 27 December 1998, 0900 UTC 26 November 1999and 0400 UTC 15 August 2002. The sites used are

Table II. Summary of tuning methods.

Method Summary

0 No adjustment1 Roughness adjustment only2a Height correction applied, based on model

windspeed2b Height correction applied, based on observed

windspeed3a As 2b, but with tuning: k1 and one value of k2

3b As 2b, but with tuning: k1, k2hi and k2lo

shown in Figure 3. In these cases, the most extremeerrors approach under-forecasts of 30 ms−1. The tuningparameters obtained from this data by minimization ofthe RMS anomaly as described above were:

Method 3a: k1 = 0.74, k2 = 0.6

Method 3b: k1 = 0.68, k2hi = 1.4, k2lo = 0.0

In method 3b, the correction factor for sites belowNWP orography, k2lo , was given a lower bound of zero,which proved the optimal fit. The value k2hi greater than1, is consistent with the earlier remark that k2 can bethought of as a value of 1/α representative of the entiredomain.

Comparison of methods requires care. Method 2aappears better for sites below the NWP orographyand method 2b appears better for sites above. This isbecause using method 2a corrects the anomaly to thestation height: ‘hill’ anomalies are increased, ‘valley’anomalies are reduced. Using method 2b, all anomaliesare represented consistently.

Overall summary statistics are shown in Table III forthe four cases.

The parameters derived above were applied to ninecases from a separate period with less strong winds –every 12 UTC hours from 3 to 11 September 2002.Results are summarized in Table IV. The results are lessdramatic, mainly because the overall wind speed is less.The maximum observed wind speed in this dataset was∼17 m s−1 as opposed to ∼38 m s−1 for the four strongwind cases. Overall, 3b is confirmed as the most effective.As an aside, applying the same optimization to these ninecases yields:

Method 3a: k1 = 0.87, k2 = 0.6

Method 3b: k1 = 0.78, k2hi = 1.8, k2lo = 0.0

These are similar to those above but allow for a greaterinfluence from the height correction. The first set has beenadopted as a more conservative solution.

Histograms for the anomalies are shown in Figure 4(strong wind cases) and Figure 5 (lighter wind cases).A non-linear scale has been used for the frequencies toemphasize the occasional serious errors. The scale usedis log2 (frequency + 1); this is a convenient logarithmicscale as it maps zero to zero and one to one. It is clearfrom these figures that, while the impact of the RA aloneon the RMS error is small, the bias is generally better and,perhaps more significantly, the error distribution is muchmore symmetric, the number of extreme under-forecastsbeing greatly reduced.

5. Summary and conclusions

A technique has been developed which both adjustsNWP-derived low-level wind speed for the artificialimpact of the NWP model’s unresolved orographic drag



Figure 3. Locations of Met. Stations used in testing the scheme.

Table III. Summary statistics for the six methods applied to thefour strong wind cases. Mean: mean anomaly (m/s). RMS: RMSanomaly (m/s). Extremes: Number of anomalies with absolute

value greater than 10m/s out of a total of 585 observations.

Method Mean RMS Extreme

0 –2.05 4.88 301 1.87 4.53 182a 1.84 4.56 252b 1.91 3.95 133a 0.05 3.01 63b 0.68 2.66 2

scheme and downscales to the unresolved orographyitself. While it is likely that a purely statistical schemecould have been derived given the same general form, theproposed scheme has several advantages. It is stronglyphysically based, and can be applied in the absence ofobservations. Correction factors are based on physicalparameters characteristic of sites of interest; though somestatistical tuning has yielded benefits, the untuned schemeperforms well and tuning parameters are close to 1.The method also provides corrections at an arbitrary

Table IV. Summary statistics for the six methods applied tothe nine less-strong wind cases. Mean: mean anomaly (m/s).RMS: RMS anomaly (m/s). Extremes: Number of anomalieswith absolute value greater than 10 m/s out of a total of 1991

observations.

Method Mean RMS Extreme

0 −1.50 2.63 131 0.05 2.34 42a −0.22 2.39 22b −0.34 2.78 163a −1.09 2.66 163b −0.47 1.82 1

height above the ground. Furthermore, in principle, themethod can be used with NWP models of differentresolution – only the parameters describing unresolvedorography need be changed.

The correction for the orographic drag scheme (‘rough-ness correction’) uses local roughness, but is novel inthat the height used as a reference wind depends uponthe amplitude and slope of unresolved orography.



10

−30 −20 −10 0 10 20

anomaly [m/s]

−30 −20 −10 0 10 20

anomaly [m/s]

−30 −20 −10 0 10 20

anomaly [m/s]

method 0

method 2a

method 1

log2

(fre

q+1) 8

6420

10

log2

(fre

q+1) 8

6420

−30 −20 −10 0 10 20

anomaly [m/s]

−30 −20 −10 0 10 20

anomaly [m/s]

method 3a method 3b10

log2

(fre

q+1) 8

6420

10

log2

(fre

q+1) 8

6420

−30 −20 −10 0 10 20

anomaly [m/s]

method 2b10

log2

(fre

q+1) 8

6420

10

log2

(fre

q+1) 8

6420

Figure 4. Histograms showing frequency of anomalies for each of the six different methods applied to four cases in which strong winds wereobserved.

The downscaled component (‘height correction’) is lin-ear in the orographic height difference between the siteof interest and the NWP model orography, the propor-tionality constant being based upon linear theory andso taking into account local factors such as representa-tive orographic wavelength and vegetative roughness. Anoptimization has shown that it is best to use the heightcorrection only for sites above the NWP model orogra-phy, where speedup occurs.

The scheme has been tuned and separately validated;significant reductions in RMS errors are possible, but,more crucially, errors in high wind cases are particularlyimproved.

The scheme is based on neutral linear theory. It couldbe extended to include stability if surface heat flux wereavailable from the model, though it is not clear how muchbenefit would be gained: it is likely that stability has itsbiggest impact at hilltop sites when low-level flow isblocked and so highly non-linear.

The roughness correction makes full use of localparameters (orographic wavelength and local roughness).The significance of the variation in A/S from site tosite requires investigation (for example, perhaps a fixedvalue could be used, so that the reference height onlydepends on the amplitude of unresolved orography).It is not clear how much the site-to site-variation inparameters other than the height difference contributesto the height correction, but examination of values ofthe overall correction suggest not very much, so it ispossible a simpler calculation could be adopted basedupon representative values.

Appendix: the neutral linear flow model.

Here we detail a simplified linear theory and summarizesome extensions to this theory. For a formulation of theproblem see Mason and King (1985) or Jackson and Hunt(1975).

Nomenclature.

The main variables are summarized in Figure 2. UA isthe constant linear advection speed which applies in bothinner and outer layer in the model identified by Masonand King (1985) as Model A.

Z is the height measured from the unperturbed surface(an independent variable).

z is the height measured from the perturbed orographicsurface. Since dz

dZ= 1, partial derivatives with respect

to z and Z are identical (i.e. ∂∂z

≡ ∂∂Z

) and, therefore,interchangeable in the following.

h = h(x, z) is the vertical displacement of streamlines.Since h is a perturbation order term, for any smooth

field φ the difference

∂φ

∂x− ∂φ

∂x

∣∣∣∣z constant

is one order higher than either of its terms and can bedisregarded for our purpose here. Thus the two terms,though not identical, can be used interchangeably in thefollowing.

τ is the unperturbed (effectively constant – see below)shear stress associated with the log-law approach flow inthe inner layer.



10

−30 −20 −10 0 10 20

anomaly [m/s]

−30 −20 −10 0 10 20

anomaly [m/s]

−30 −20 −10 0 10 20

anomaly [m/s]

method 0

method 2a

method 1

log2

(fre

q+1) 8

6420

10

log2

(fre

q+1) 8

6420

−30 −20 −10 0 10 20

anomaly [m/s]

−30 −20 −10 0 10 20

anomaly [m/s]

method 3a method 3b10

log2

(fre

q+1) 8

6420

10

log2

(fre

q+1) 8

6420

−30 −20 −10 0 10 20

anomaly [m/s]

method 2b10

log2

(fre

q+1) 8

6420

10

log2

(fre

q+1) 8

6420

Figure 5. Histograms showing frequency of anomalies for each of the six different methods applied to nine cases from September 2002.

τHC = τHC (x, z) is the shear stress perturbation.τ = τ(x, z) is the perturbed shear stress, i.e.

τ = τ + τHC (A1)

u = u(z) is the unperturbed log-law approach flowspeed.

uHC = uHC (x, z) is the inner-layer speed perturba-tion.

u = u(x, z) is the perturbed inner layer speed, i.e.

u = u + uHC (A2)

p is the perturbation kinematic pressure (i.e. perturba-tion dynamic pressure divided by the density, which isassumed constant).

uO is the outer-layer speed perturbation.w is the perturbation vertical velocity.

Outer Layer (inviscid flow, no boundary layer).

The outer layer linearized x-momentum equation is

UA∂uO

∂x= −∂p

∂x(A3)

We consider a single Fourier mode with wavevectoralong the mean flow so the form of the orographic surfaceis:

h|z=0 = h0eikx (A4)

Assume all perturbation variables have sinusoidal vari-ation in x with wavenumber k (though they may, ofcourse, have different phases).

Since UA is real and neither uO nor p have a meanor linear trend with x, Equation (A3) becomes

UA uO = −p (A5)

The outer layer linearized z-momentum equation is

UA∂w

∂x= −∂p

∂z(A6)

Continuity is described by

∂uO

∂x+ ∂w

∂z= 0 (A7)

Taking the divergence of the momentum equations andusing continuity we obtain

∇2p = 0 (A8)

Let p = p0ei(kx+mz), then

∇2p = 0 = −(k2 + m2)p ⇒ m2 = −k2 ⇒ m = ± ik

(A9)

so, since k is positive and real, for bounded p as z → ∞p = p|z=0e

−kz (A10)

The surface boundary condition is a feature whichdiffers between authors; w at the surface is determinedby a representative horizontal velocity scale, US, thus

w|z=0 = US∂h

∂x

∣∣∣∣z=0

= USikh



We shall postpone definition of US until later. Onsubstituting this into the z momentum equation and usingEquation (A10) we eventually obtain

p|z=0 = −UAUSkh (A11)

Substituting this into Equation (A5) we obtain

uO |z=0 = USkh (A12)

Stress term.

In the inner layer, we include the stress perturbation butassume that the pressure perturbation is as derived fromthe outer layer. The approach flow stress is taken to beeffectively constant or, more precisely, the height scalefor variations in the approach flow stress is assumed to bemuch greater than that for variations in the perturbationstress. This is equivalent to assuming that the approachflow can be well described by the log-law for thegreater part of the boundary layer (c/f Jackson and Hunt).Following the usual eddy viscosity hypothesis and mixinglength arguments, we have:

τ = κ2z2(

du

dz

)2

where κ is Von Karman’s constant (taken to be 0.4).For the perturbed inner layer, similarly:

τ = κ2z2(

du

dz

)2

Expanding u and dropping the second-order termleaves:

τ = τ + 2κzu∗(

∂uHC

∂z

)(A13)

where u∗ is the friction velocity of the approach flow,defined by τ = u2∗.

Differentiating (A13) w.r.t. z:

∂τ

∂z= ∂τ

∂z+ 2κu∗

∂

∂z

(z∂uHC

∂z

)= 2κu∗

∂

∂z

(z∂uHC

∂z

)

(since ∂τ∂z

= 0).It is implicit in the solution technique that the outer

layer depth scale is much larger than the inner layerdepth, which means, if u′′ (as defined below in EquationA17) is at least similar in magnitude to u0, that

∂uHC

∂z� ∂uO

∂z⇒ ∂uHC

∂z≈ ∂u′′

∂z

so∂τ

∂z≈ 2κu∗

∂

∂z

(z∂u′′

∂z

)(A14)

Inner layer.

For our simple model, we take, as a starting point, aconstant advection speed UI . The pressure field from the

outer layer is assumed to apply throughout the depth ofthe inner layer (which, incidentally, is not bounded). Sothe inner layer horizontal momentum equation becomes:

UI∂u

∂x= −∂p

∂x+ ∂τ

∂z(A15)

Again we assume sinusoidal variation of all perturba-tions with x. Thus:

UI ikuHC = −ikp + ∂τ

∂z(A16)

It is also convenient to define:

u′′ = uHC − UA

UIuO (A17)

which might be described as the perturbation attenuationcaused by the surface stress. Using Equation (A5) forp, our definition of u′′, Equations (A17) and (A14) weobtain:

UI iku′′ = ∂τ

∂z= ∂

∂z

{2κzu∗

∂u′′

∂z

}(A18)

or

u′′ − 4

C

∂

∂z

{z∂u′′

∂z

}= 0 (A19)

where

C = 2UI ik

κu∗

It is convenient to introduce the following substitution:

ξ 2 = Cz

Equation (A19) is then revealed as a modified Besselequation which, given the requirements of boundedness,has solution

u′′ = AK0

(+√

Cz)

(A20)

where K0 is a modified Bessel function of the secondkind.

Bottom boundary condition.

The bottom boundary condition is

uHC = 0 on z = z0

so

u′′|z=z0= − UA

UIuO

∣∣∣∣z=z0

≈ − UA

UIuO

∣∣∣∣z=0

which is a very good approximation assuming z0 is muchless than the scale height for decay of the outer layersolution. Using this in our expression for u′′ (Equation



(A20)) to derive A, writing in terms of uHC usingEquation (A17) gives

uHC = UA

UIuO

∣∣∣∣z=0

1 −

K0

(+√

Cz)

K0

(+√

Cz0

) (A21)

Substituting for uO |z=0 from Equation (A16):

uHC = UA

UIUS kh|z=0

1 −

K0

(+√

Cz)

K0

(+√

Cz0

) (A22)

withh|z=0 = kh0e

ikx

Overall solution.

The overall solution is a combination of the inner layersolution in the inner layer, which decays rapidly awayfrom the surface due to the Bessel function dependence,and the outer layer solution, which decays slowly (com-pared with the inner layer depth). The two may be con-veniently combined by including the exponential decay

uHC = UA

UIUS e−kzkh|z=0

1 −

K0

(+√

Cz)

K0

(+√

Cz0

)

Our overall solution is the inner layer solution givenby

u = u + uHC

However, it is convenient to think of the height cor-rection in terms of a multiplicative factor u = u(1 + F),so that

F = UAUS

u(z)UIe−kzkh|z=0

1 −

K0

(+√

Cz)

K0

(+√

Cz0

)

Velocity scales.

The solution, then, contains three characteristic windspeeds, all given by the input, unperturbed wind, atvarious heights in the profile. The first arises from theouter layer advection, the second arises from advec-tion within the inner layer, and enters the definitionof the height scale(s) in the inner layer, and the thirdis the surface speed in the outer layer solution. Thechoice of these scales has not been rigorously defined,or, at least, has not been agreed between authors. Tosome extent, choices have been made so as to opti-mize the agreement with numerical or physical experi-ments.

Mason and King (1985) performed a linearisation ofthe dimensional equations and presented various choicesof scales, identified as models A–D. Though introducedas refinements to the simpler solution, it is convenient

to discuss them as simplifications to the most complexsolution.

Model D.

The choice made by Mason and King is accepted assensible but arbitrary, and justified by comparison witha full non-linear numerical solution. The outer layersolution decays exponentially with a scale height href =k−1. A separate outer layer velocity scale is thereforeused for each wavenumber, given by the velocity atthis height. This ensures that winds at heights muchabove this have no influence on the solution at thiswavenumber, and, with a typically sheared environment,high wavenumber perturbations have reduced amplitude.This argument appears quite naturally if the analysis isperformed separately for each wavenumber of the flow.In this case, the inner layer depth is determined by thewavenumber, not a ‘characteristic length’ of the hill. Thesame velocity scale is used to determine the outer layersurface boundary condition.

The choice of inner layer velocity scale is difficult.Mason and King anticipate that the model will be appliedat a height above the region where stress divergence issignificant, and so suggest a velocity scale in the innerlayer equal to the velocity at the height at which theperturbation is to be calculated. This is obviously difficultto justify, in that it implies that the inner layer lengthscale varies with height, but it appears to give the best(and most extreme!) results. Thus, for model D

UA = US = u(href)

UI = u(z)

Model C.

The inner layer velocity scale is taken to be the unper-turbed speed at the inner-layer depth scale appropriatefor the wavelength. This depth scale is the height, zI, atwhich the stress divergence and perturbation advectionhave similar magnitudes. This means that

∣∣∣∣U(zI)∂uHC

∂x

∣∣∣∣ ≈∣∣∣∣2κu∗

∂

∂z

(z∂u′′

∂z

)∣∣∣∣ ≈∣∣∣∣2κu∗

zIuHC

∣∣∣∣Thus, assuming the neutral logarithmic wind profile

and a single wavenumber k, we obtain an implicitrelationship:

ln(zI/z0)kzI ≈ 2κ2

Thus, for model C

UA = US = u(href)

UI = u(zI)

Model B.

The same velocity scales are used for all wavenumbers,based on a characteristic length scale L of the hill, so thatk is replaced with L−1. In addition, the wind at the inner



layer height is used as the surface boundary condition forthe outer layer solution. Thus,

UA = u(href)

UI = US = u()

Model A.

This is essentially the same as Model B, except the outerlayer and inner layer solutions are not combined (sothe exponential decay with height is lost from the innerlayer). This is essentially the Jackson and Hunt (1975)solution.

List of symbols:

A/S The silhouette area of unresolved orographyper unit horizontal area – roughly equivalentto the typical slope.

a Tuneable parameter of order 1 relating href tok−1

α The ratio of the flat surface wind at measure-ment height above local orography to that atthe reference height, u(z)/UA

C2UI ik

κu∗

F Correction factor relating height correction toroughness-corrected wind speed.

H Average peak-to-trough amplitude of orogra-phy.

h(x, z) Vertical displacement streamlines by pertur-bation orography.

h0 Surface perturbation orography amplitude.hL Height of true orography.hM Height of model orography.href Reference (outer layer) height.

K0( ) Modified Bessel function of the second kind.k Characteristic wavenumber of unresolved

orographyk1, k2 Tuning parameters.

κ Von Karman’s constant (0.4)m Vertical wavenumber.p Perturbation kinematic pressure.

UA Outer layer (advection) wind speed.

UI Inner layer (advection) wind speed.US Outer layer surface wind speed.

u(x, z) Perturbed wind speed.u(z) Wind profile over model orography adjusted

for local roughness. Unperturbed flow overperturbation orography.

u∗ Surface friction velocity in unperturbed flow.u0 Outer layer wind speed perturbation.

uHC Correction to windspeed based on differ-ence between local and model orography, i.e.Height Correction. Inner layer speed pertur-bation.

uM Input winds from NWP model, on modellevels or fixed heights above orography.

uob Observed wind speed at a given location.uref Wind speed at reference height.

τ(x, z) Perturbed shear stress.τ Unperturbed shear stress.

τHC (x, z) Shear stress perturbation.w Perturbation vertical velocity.Z Height measured from the unperturbed sur-

face.z Height measured from the perturbed oro-

graphic surface.z0 Local vegetative roughness length.z1 Height above surface at which wind speed is

output (10 m in practice).

References

Golding BW. 1998. Nimrod: A system for generating automatic very-short-range forecasts. Meteorological Applications 5: 1–16.

Grant ALM, Mason PJ. 1990. Observations of boundary layer structureover complex terrain. Quarterly Journal of the Royal MeteorologicalSociety 116: 159–186.

Jackson PS, Hunt JCR. 1975. Turbulent wind flow over a lowhill. Quarterly Journal of the Royal Meteorological Society 101:929–955.

Mason PJ, King JC. 1985. Measurements and predictions of flow andturbulence over an isolated hill of moderate slope. Quarterly Journalof the Royal Meteorological Society 111: 617–640.

Mason PJ, Sykes RI. 1979. Flow over an isolated hill of moderateslope. Quarterly Journal of the Royal Meteorological Society 105:383–395.

Wood N, Brown AR, Hewer FE. 2001. Parametrizing the effectsof orography on the boundary layer: an alternative to effectiveroughness lengths. Quarterly Journal of the Royal MeteorologicalSociety 127: 759–777.


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Correction and downscaling of NWP wind speed forecasts