Beljaars ECMWF PBL Parameterization

Meteorological Training Course Lecture Series

ECMWF, 2002 1

The parametrization of the planetary boundarylayerMay 1992

By Anton Beljaars

European Centre for Medium-Range Weather Forecasts

Table of contents

1 . Introduction

1.1 The planetary boundary layer

1.2 Importance of the PBL in large-scale models

1.3 Recommended literature

1.4 General characteristics of the planetary boundary layer

1.5 Conserved quantities and static stability

1.6 Basic equations

1.7 Ekman equation

2 . Similarity theory and surface fluxes

2.1 Surface-layer similarity

2.2 The outer layer

2.3 Matching of surface and outer layer (drag laws)

2.4 Surface boundary conditions and surface fluxes

3 . PBL schemes for atmospheric models

3.1 Geostrophic transfer laws

3.2 Integral models (slab, bulk models)

3.3 Grid point models with�

-closure

3.4�

-profile closures

3.5 TKE-closure

4 . List of symbols

REFERENCES

The parametrization of the planetary boundary layer

2 Meteorological Training Course Lecture Series

ECMWF, 2002

1. INTRODUCTION

1.1 The planetary boundary layer

Theplanetaryboundarylayer (PBL) is the region of theatmospherenearthesurfacewherethe influenceof the

surfaceis felt throughturbulent exchangeof momentum,heatandmoisture.The equationswhich describethe

large-scaleevolutionof theatmospheredonottakeinto accounttheinteractionwith thesurface.Theturbulentmo-

tion responsible for this interaction is small-scale and totally sub-grid and therefore needs to be parametrized.

Thetransitionregionbetweenthesurfaceandthefreeatmosphere,whereverticaldiffusiondueto turbulentmotion

takesplace,variesin depth.ThePBL canbeasshallow as100m duringnightoverlandandgoupto afew thousand

metreswhentheatmosphereis heatedfrom thesurface.ThePBL parametrizationdeterminestogetherwith thesur-

faceparametrizationthesurfacefluxesandredistributesthesurfacefluxesovertheboundarylayerdepth.Boundary

layerprocessesarerelatively quick; thePBL respondsto its forcing within a few hourswhich is fastcomparedto

thetimescaleof thelarge-scaleevolutionof theatmosphere,in otherwords:thePBL isalwaysin quasi-equilibrium

with the large-scale forcing.

1.2 Importance of the PBL in large-scale models

A number of reasons exists to have a realistic representation of the boundary layer in a large scale model:

• The large-scalebudgetsof momentumheatandmoistureareconsiderablyaffectedby the surface

fluxes on time scales of a few days.

• Model variables in the boundary layer are important model products.

• The boundary layer interacts with other processes e.g. clouds and convection.

Theimportanceof thesurfacefluxescanbeillustratedby estimatingtherecycle timeof thedifferentquantitieson

thebasisof typicalvaluesof thesurfacefluxes.Thenumbersin Table1 havebeenderivedfrom atypical runwith

the ECMWF model or are simply order of magnitudes estimations.

Alreadyfrom theseverycrudeestimatesit canbeseenthatthesurfacefluxesplayanimportantrole in forecasting

for themediumrange.It is obviousthatthesurfacefluxesarecrucial for the“climate” of themodel.With regard

to themomentumbudgetit shouldbenotedthat theEkmanspin-down time hasto beconsideredastherelevant

time scalebecauseit is anefficient mechanismfor spinningdown vorticity in theentireatmosphereby frictional

stress in the PBL only.

Thesecondimportantreasonto haveaPBL schemein alarge-scalemodelis thatforecastproductsareneedednear

thesurface.Thetemperatureandwind atstandardobservationheight(2 m and10m for temperatureandwind re-

spectively) areobviousproducts.It is importantto realizethatthePBL schemetogetherwith thelandsurfaceand

radiationschemeintroducesthediurnalpatternin thesurfacefields.Also theanalysedandforecastedfieldsof sur-

facefluxes(momentum,heatandmoisture)arebecomingmoreandmoreimportantasinput andverificationfor

wave models, air pollution models and climate models.

TABLE 1. GLOBAL BUDGETS(ORDEROF MAGNITUDE ESTIMATES)

Budget TotalSurface

fluxRecycle time

Water J m–2 80 W m–2 10 days7 107×



ECMWF, 2002 3

Finally it hasto berealizedthatotherprocessescannot beparametrizedproperlywithout having a PBL scheme.

Boundarylayercloudsareanobviousexample,but alsoconvectionschemesoftenusesurfacefluxesof moisture

in their closure.

1.3 Recommended literature

General textbook and introductory review on most aspects of the PBL:

Stull, R.B. (1988):An introduction to boundary layer meteorology. Kluwer publishers.

Introduction to turbulence:

Tennekes, H. and Lumley, J.L. (1972):A first course in turbulence. MIT press.

Atmospheric turbulence:

Nieuwstadt,F.T.M. and Van Dop, H. (eds. 1982): Atmosphericturbulence and air pollution

modelling. Reidel publishers.

Haugen, D.A. (ed. 1973):Workshop on micro meteorology. Am. Meteor. Soc.

Monin, A.S. and Yaglom, A.M. (1971):Statistical fluid dynamics. Vol I, MIT press.

Panofsky, H.A. and Dutton, J.A. (1984): Atmosphericturbulence: Models and methodsfor

engineering applications. John Wiley and sons.

Surface fluxes:

Oke, T.R. (1978):Boundary layer climates. Halsted press.

Brutsaert, W. (1982):Evaporation into the atmosphere. Reidel publishers.

1.4 General characteristics of the planetary boundary layer

Turbulence

Thediffusiveprocessesin theatmosphericboundaryaredominatedby turbulence.Themoleculardiffusioncanin

generalbeneglectedexceptin a shallow layer (a few mm only) nearthesurface.Thetime scaleof the turbulent

motionrangesfrom a few secondsfor thesmalleddiesto abouthalf anhour for thebiggesteddies(seeFig. 1 ).

Thelengthscalesvary from millimetresfor thedissipative fluctuationsto a few hundredmetresfor theeddiesin

thebulk of theboundarylayer. Thelatteronesdominatethediffusivepropertiesof theturbulentboundarylayer. It

is clear that these scales cannot be resolved by large-scale models and need to be parametrized.

Internal +potentialenergy

J m–2

(0.5%available)30 W m–2 8 days

Kineticenergy

J m–2 2 W m–2 10 days

Momentum kg ms–1

0.1 Nm–2 25 days(Eckman spin-down time: 4 days

TABLE 1. GLOBAL BUDGETS(ORDEROF MAGNITUDE ESTIMATES)

Budget TotalSurface

fluxRecycle time

4 109×

2 106×

2 105×



ECMWF, 2002

Figure 1. Spectrum of the horizontal wind velocity afterVan Der Hoven (1957). Some experimental points are

shown.

Stability

Thestructureof theatmosphericboundarylayeris influencedby theunderlyingsurfaceandby thestabilityof the

PBL (seeFig. 2).

Thesurfaceroughnessdeterminesto a certainextent theamountof turbulenceproduction,thesurfacestressand

theshapeof thewind profile. Stability influencesthestructureof turbulence.In anunstablystratifiedPBL (e.g.

duringday-timeover landwith anupwardheatflux from thesurface)theturbulenceproductionis enhancedand

theexchangeis intensifiedresultingin a moreuniform distribution of momentum,potentialtemperatureandspe-

cific humidity. In a stablystratifiedboundarylayer(e.g.duringnight-timeover land)theturbulenceproducedby

shearis suppressedby thestratificationresultingin a weakexchangeanda weakcouplingwith thesurface.The

qualitative impact of these aspect is illustrated inFig. 2

Diurnal pattern

Thetypical evolution of thePBL over landis illustratedin Fig. 3 for a 24 hourinterval. During daytime,with an

upwardheatflux from thesurface,theturbulentmixing is verystrong,resultingin approximatelyuniformprofiles

of potentialtemperatureandwind over thebulk of theboundarylayer. Theunstableboundarylayer is therefore

oftencalled“mixedlayer”. Nearthesurfaceweseeasuperadiabaticlayerandastrongwind gradient.Thetopof

themixedlayeris cappedby aninversionwhichinhibits theturbulentmotion(e.g.therisingthermals)to penetrate

aloft. Theinversionheightrisesquickly early in themorninganreachesa heightof a few kilometresduringday-

time. Whentheheatflux from thesurfacechangessignat night the turbulencein themixed layerdiesout anda

shallow stablelayernearthesurfacedevelops.Thenocturnalboundarylayerhasaheightof typically 50 to 200m

dependent on wind speed and stability.



ECMWF, 2002 5

Figure 2. The wind speed profile near the ground including (a) the effect of terrain roughness, and (b) to (e) the

effect of stability on profile shape end eddy structure. In (e) the profiles of (b) to (d) are re-plotted with a

logarithmic height scale. (Fig. 2.10 fromOke, 1978).



ECMWF, 2002

Figure 3. (a) Diurnal variation of the boundary layer on an “ideal” day. (b) Idealized mean profiles of potential

temperature( ), wind speed( � ) anddensity( ) for thedaytimeconvectiveboundarylayer. (c) Sameas(b) for

nocturnal stable layer. The arrows indicate sunrise and sunset.

θ ρv



ECMWF, 2002 7

Surface fluxes

Figure 4. Diurnal energy balance of (a) a Scots and Corsican pine forest at Thetford (England) on 7 July 1971,

and (b) a Douglas fir forest at Haney (B.C. Canada) on 10 July 1970, including (c) the atmospheric vapour

pressure deficit. In these figures Q* represents net radiation, QH the sensible heat flux, QE the latent heat flux and

∆Qs the soil heat flux. (Fig. 4.24 fromOke 1978).

Thesurfacefluxesof momentum,heatanmoistureareof crucialimportanceto themodelasthey affect theclimate



ECMWF, 2002

of themodel,themodelperformancein themediumrangeandplay an integral role in a numberof parametriza-

tions.Thediurnalpatternof thePBL over land is mainly forcedby theenergy budgetat thesurfacethroughthe

diurnalevolutionof thenetradiationat thesurfaceasillustratedin Fig. 4 . Duringdaytimethenetradiationis par-

titionedbetweentheheatflux into theground,thesensibleheatflux into theatmosphereandthelatentheatof evap-

oration.Thegroundheatflux is generallysmallerthan10%of thenetradiationduringdaytime.Thepartitioning

of availablenetradiationbetweensensibleandlatentheatis partof thelandsurfaceparametrizationscheme.The

fluxesaremuchsmallerduring night-timeandaremuchlessimportantfor the atmosphericbudgetsbut equally

relevant for the prediction of boundary layer parameters.

Theboundarylayerover seadoesnot have a distinctdiurnalpatternbut canbestableandunstabledependentof

theair typethatis advectedrelativeto theseasurfacetemperature.Strongcoldair advectionoverarelatively warm

seacanleadto extremelyhigh fluxesof sensibleandlatentheatinto theatmosphere.Warmair advectionover a

cold sea leads to a stable PBL but does occur less frequently.

It is importantto realizethatthethermodynamicsurfaceboundaryconditionsoverseaareverydifferentfrom those

over land.Overseathetemperatureandspecifichumidityarespecifiedandkeptconstantduringtheforecast.Over

landthesurfaceboundaryconditionfor temperatureandmoisturearenearlyflux boundaryconditions,becauseof

theconstraintimposedby thesurfaceenergybudget.In thelattercase,thefluxesaredeterminedby thenetradiation

at thesurface.This meansthat the total energy input into theatmosphere(sensible+ latentheat)is not somuch

determinedby theflow or by thePBL parametrizationbut by thenetradiationat thesurface.Theseahowever, is,

with its fixedSSTboundarycondition,aninfinite sourceof energy to themodel.Thespecificationof thePBL ex-

changewith theseasurfaceis thereforeextremelycritical. BeljaarsandMiller (1991)give anexampleof model

sensitivity to the parametrization of surface fluxes over tropical oceans.

PBL clouds

Cumulusclouds,stratocumuluscloudsandfog arevery muchpart of the boundarylayer dynamicsandinteract

strongly with radiation.

1.5 Conserved quantities and static stability

To describeverticaldiffusionby turbulentmotionwehaveto selectvariablesthatareconservedfor adiabaticproc-

esses.In thehydrostaticapproximationbothpotentialtemperatureθ anddry staticenergy areconserved for dry

adiabatic ascent or descent. They are defined as

(1)

(2)

Whenmoistprocessesareconsideredaswell it is necessaryto useliquid waterpotentialtemperatureθl or theliquid

water static energy � l and total water content� t:

(3)

(4)

(5)

θ �� 0 �⁄( )��

⁄ �� 0 �⁄( )0.286≈=

� �� +=

θl θ�

vapθ�� -------------

� l–=

� l �� vap � l–+=

� t � � l+=



ECMWF, 2002 9

Thestaticstability is determinedby thedensityof afluid parcelmovedadiabaticallyto a referenceheightin com-

parisonwith thedensityof thesurroundingfluid. Thevirtual potentialtemperatureθv andthevirtual dry staticen-

ergy are often used for this purpose:

(6)

(7)

1.6 Basic equations

To illustratetheclosureproblemwederivetheReynoldsequationsfor momentum,startingfrom thethreemomen-

tumequationsfor incompressibleflow in arectangularcoordinatesystemwith z perpendicularto theearthsurface:

(8)

(9)

Decomposition of each variable into a mean part and a fluctuating (turbulent) part:

(10)

After averaging,applicationof theBoussinesqapproximation(retaindensityfluctuationsin thebuoyancy terms

only) and applying the hydrostatic approximation:

(11)

(12)

Furthersimplificationsareobtainedby neglectingviscouseffectsfor largeReynoldsnumbers( � � /ν >> 1, where�is a lengthscale)andby assumingthatthe�� -scalesaremuchlargerthanthe� -scales(boundarylayerapprox-

imation):

θv θ 1

�v�------- 1–

� � l–+ θ 1 0.61� � l–+{ }= =

� v �� 1 0.61� � l–+{ } ��+=

∂ �∂ �------ � ∂ �

∂�------ � ∂ �∂�------ � ∂ �

∂�------ � �–+ + + =1ρ---∂�

∂�------– ν∇2 �+

∂ �∂ �------ � ∂ �

∂�------ � ∂ �∂�------ � ∂ �

∂�------ � �+ + + + =1ρ---∂�

∂�------– ν∇2 �+

∂ �∂ �------- � ∂ �

∂�------- � ∂ �∂�------- � ∂ �

∂�-------+ + + =1ρ---∂�

∂�------– ν∇ � �–+

∂ �∂�------ ∂ �

∂�------ ∂ �∂�-------+ + 0=

� � � ′ , ρ+ ρ0 ρ′ ,+= =

� ! � ′ , �+ " � ′ ,+= =

� # � ′+=

∂ �∂ �-------- � ∂ �

∂�-------- ! ∂ �∂�-------- # ∂ �

∂�-------- � !–+ + + =1ρ0-----∂"

∂�-------– ν∇2 � ∂∂�------ � ′ � ′ ∂

∂�------ � ′ � ′–∂

∂�------ � ′ � ′––+

∂ !∂ �------- � ∂ !

∂�------- ! ∂ !∂�------- # ∂ !

∂�------- � �+ + + + =1ρ0-----∂"

∂�-------– ν∇2 ! ∂∂�------ � ′ � ′ ∂

∂�------ � ′ � ′–∂

∂�------ � ′ � ′––+

� =1ρ0-----∂"

∂�-------–

∂ �∂�-------- ∂ !

∂�------- ∂ #∂�---------+ + 0=



ECMWF, 2002

(13)

Theseequationsdescribethehorizontalmomentumbudgetof the resolvedflow andhave extra termsdueto the

Reynoldsdecompositionandaveragingprocedure.Theterms and areknown astheReynoldsstresses

andrepresentverticaltransportof horizontalmomentumby unresolvedturbulentmotion.Thesearethetermsthat

need to be parametrized. Similar equations exist for potential temperature and specific humidity.

An equationthatis importantin someparametrizationschemesandgivesinsightin thedynamicsof turbulenceis

theequationfor turbulentkineticenergy. To derive it, theequationsfor thefluctuatingquantitiesaretakenby sub-

tractingtheequationsfor meanmomentumfrom theequationsfor totalmomentum.Thekineticenergy of thefluc-

tuationsis obtainedby multiplying theequationsfor thedifferentcomponentsby thevelocityfluctuationitself and

by adding the three equations for the three components. The result is:

(14)

(15)

Theleft handsideof thisequationrepresentstimedependenceandadvection.Theright handsidehassource,sink

andtransportterms.TermsI representmechanicalproductionof turbulenceby wind shearandconvert energy of

themeanflow into turbulence.Termtwo is theproductionof turbulenceby buoyancy andconvertspotentialenergy

of theatmosphereto turbulenceor vice versa.TermsIII andIV aretransportor turbulentdiffusiontermsbecause

they equalzerowhenvertically integratedover thedomain.They representtheverticaltransportof turbulenceen-

ergyby theturbulentfluctuationsandthepressurefluctuationsrespectively. Thelastterm(V) is thedissipationterm

which converts turbulence kinetic energy into heat by molecular friction at very small scales.

1.7 Ekman equation

To illustratethemechanismof Ekmanpumpingweconsiderthefollowing equationsfor thesteadystateboundary

layer:

(16)

(17)

∂ �∂ �-------- � ∂ �

∂�-------- ! ∂ �∂�-------- # ∂ �

∂�-------- � !–+ + +1ρ0-----∂"

∂�-------– ∂ � ′ � ′∂�---------------–=

∂ !∂ �------- � ∂ !

∂�------- ! ∂ !∂�------- # ∂ !

∂�------- � �+ + + +1ρ0-----∂"

∂�------- ∂ � ′ � ′∂�---------------––=

� 1ρ0-----∂"

∂�-------–=

� ′ � ′ � ′ � ′

$′ � ′

$ 12--- � ′2 � ′2 � ′2+ +( )=

∂$

∂ �------- � ∂$

∂�------- ! ∂$

∂�------- # ∂$

∂�-------+ + + = � ′ � ′∂ �∂�--------– � ′ � ′∂ !∂�-------–�ρ0----- � ′ρ′–

I I II

+∂

∂�------$

′ � ′ +

� ′ � ′ρ

------------ ε–

III IV V

�%! ! G–( )–∂ � ′ � ′

∂�--------------- where �&! G,–1ρ0-----∂"

∂�-------= =

� � � G–( )–∂ � ′ � ′

∂�--------------- where � � G,– 1ρ0-----–

∂"∂�-------= =



ECMWF, 2002 11

For a simpleclosurewith , andconstant' we rewrite theequationsin

complex notation:

(18)

Figure 5. (a) Hodograph of Ekman spiral; the vectors indicate the increase of wind speed and its veering with

height,(b) Illustrationof theageostrophicwind in thePBL of acyclone,and(c) verticalvelocity in cyclonedueto

Ekman pumping (Holton, 1979).

The solution with�%( ! ( 0 at the surface and�%()� ,! ( ! far away from the surface, reads:

� ′ � ′ ' ∂ � ∂�⁄–= � ′ � ′ ' ∂ ! ∂�⁄–=

i � � i !+( ) * � � G i ! G–( )– ' ∂2

∂� 2-------- � i !+( )=



ECMWF, 2002

(19)

Theverticalvelocity #,+ at thetopof thePBL is derivedfrom thecontinuityequationby integrationfrom thesur-

face to boundary layer depth- , where- is large.

(20)

Weseethattheverticalvelocityat thetopof thePBL is proportionalto thecurl of thesurfacestressandto thecurl

of thegeostrophicwind. This qualitative resultis in fact independentof how ' is exactly specified.Thevertical

velocity canbeeseenasa boundaryconditionfor theinviscid flow above theboundarylayer. Theinflow of air in

acyclonedueto friction at thesurfacecausesanascendingmotionat thetopof thePBL andspinsdown thevortex

by vortex compression(conservationof angularmomentum).This mechanismis very importantbecauseit trans-

fers the effect of boundary layer friction to the entire troposphere. Ekman pumping is illustrated inFig. 5.

2. SIMILARITY THEORY AND SURFACE FLUXES

Many physicalprocessesin theatmospherearetoocomplicatedto enablethederivationof parametrizationsonthe

basisof first principles.Turbulenceis suchanexample:althoughturbulentfluctuationsaredescribedby theequa-

tionsof motionfor acontinuumfluid, it is verydifficult to generatesolutionsof theseequationsandmoreover the

solutionsshow chaoticbehaviour on very shorttime scales.Most of thetime we areonly interestedin statistical

averages.It is thereforecommonpracticeto rely onempiricalrelationshipsbasedonobservations.Similarity the-

ory provides the framework to organize and group the experimental data.

Similarity theorystartswith theidentificationof therelevantphysicalparameters,thendimensionlessgroupsare

formedfrom thetheseparameters,andfinally experimentaldatais usedto find functionalrelationsbetweendimen-

sionlessgroups.Whenthefunctionalrelationsareknown, they canbeusedaspartof a parametrizationscheme.

Becauseexperimentaldatais alwaysnoisyandshowsalot of scatterit is importantto limit asmuchaspossiblethe

numberof dimensionlessgroups.It mayfor instancebedifficult to find anempiricalrepresentationof noisydata

whenmorethanoneor two independentparametersareinvolved.It is thereforeadvantageousto considerdifferent

partsof theboundarylayer(surfacelayer, outerlayer)anddifferentlimiting cases(neutral,veryunstableetc.)sep-

arately, to simplify theproblemandto limit thenumberof dimensionlessgroupsthatarerelevantat thesametime.

Theproceduresarestraightforward in principle(theBuckinghamPi dimensionalanalysismethod;seee.g.Stull,

1988),but requireexperienceandintuition in practice.Thejudgmentof whichparametersareimportantor unim-

portant is a crucial aspect of the analysis.

Similarity theoryis usedin virtually all PBL schemesthathavebeendevelopedfor atmosphericmodels.Different

aspects of the boundary layer similarity are discussed in the following sections.

2.1 Surface-layer similarity

Thesurfacelayer is theshallow fractionof thePBL nearthesurfacewherethefluxesareapproximatelyequalto

thesurfacevalues.Thesurfacelayeris thereforeoftencalledtheconstantflux layer. Althoughthedefinitionimplies

that�/. - is muchsmallerthan1 in thesurfacelayer, in practicetheupperlimit is oftendefinedastheheightwhere

� i !+( ) � G i ! G–( ) � G i ! G–( ) * �'-----

1 2⁄ �exp–=

#0 = ∂ �∂�--------– ∂ !

∂�-------– d�

0

0∫ =

1�---

∂ � ′ � ′∂�--------------- ∂ � ′ � ′

∂�---------------+–

='2�------

1 2⁄ ∂ � G

∂�-----------–∂ ! G

∂�-----------–



ECMWF, 2002 13

�1. -2( 0.1.

Therelevantparametersin thesurfacelayer for thewind profilesarethemomentumflux, thebuoyancy flux and

height above the surface

, (21)

, (22)

z (turbulence length scale).

Theheightabovethesurfaceis thelengthscalefor turbulencebecausethesizeof thetransportingeddiesisconfined

by theirdistancefrom thesurface.If thepotentialtemperatureandspecifichumidityareconsidered,wealsoneed

the kinematic fluxes of heat and moisture: and . The following scales are defined:

(23)

(24)

(25)

(26)

Sincethenon-neutralsurfacelayerhastwo relevantlengthscales(� and�

) all dimensionlessquantitiesarewritten

as function of�1. � . Examples of dimensionless gradients and turbulence intensities are:

(27)

(28)

(29)

(30)

(31)

1ρ--- τ0 � ′ � ′( )0

2 � ′ � ′( )02

+{ }1 2⁄

=

�ρ--- � ′ρ′( )0–

�θv----- � ′θ′v( )0=

� ′θ′( )0 � ′ � ′( )0

� *τ0

ρ----

1 2⁄(friction velocity)=

� � *3–

3 �θv----- � ′θv′( )

----------------------------

0

(Obukhov length)=

θ*

� ′θ′( )0–

� *---------------------- (turbulent temperature scale)=

� *

� ′ � ′( )0–

� *---------------------- (turbulent humidity scale)=

3 � � *

� ′ � ′–( )0

------------------------∂ �∂�-------- φM

��---- =

3 � � *

� ′ � ′–( )0

------------------------∂ !∂�------- φM

��---- =

3 �θ*------∂θ

∂�------ φH��----

=

3 �� *------∂ �

∂�------ φQ��----

=

σ 4� *------- f 5 ��----

=



ECMWF, 2002

(32)

note that (33)

where .

Thedimensionlessgradientfunctions(27)–(30)havebeenstudiedextensively with helpof field experimentsover

homogeneousterrain.Fig. 6 illustratesthis for φΜ with datafrom theKansasexperiment.Many empiricalforms

havebeenproposedfor thedimensionlessfunctions.Theexpressionsfor theunstablesurfacelayer(oftenreferred

to as the Dyer and Hicks profiles; seePaulson 1970,Dyer 1974 andHogstrom 1988 for a review) are:

(34)

(35)

(36)

(37)

and the linear relationship

(38)

asoriginally proposedby Webb(1970)is fairly consistentwith mostdatafor 0 6 �/. � 6 0.5(cf. Hogstrom,1988).

Analysisby Hicks (1976)of stablewind profilesabove theflat homogeneousterrainof the“Wangara”siteandby

Holtslag andDeBruin(1988)of Cabauwdataup to resultedin thefollowing expression(seealsoBel-

jaars and Holtslag, 1991)

(39)

(40)

where78( 1, 9:( 0.667,�;( 5 and<=( 0.35.Thisexpressionbehaveslike(38)for small�/. � valuesandapproaches

φM ~ 7 �1. � for large�1. � . The3/2 power in φH hasbeenintroducedto makeφH /φM proportionalto for

large values of�1. � (see discussion on stability functions in section3.3).

σθ

θ*------ f θ

��---- =

' M3 � � *------------ 1

φM-------=

' M

τ >d � d�⁄------------------

τ?d! d�⁄------------------= =

φM 1 16��----–

14---–

=

φH Q, 1 16��----–

12---–

for��---- 0<=

30.4=

φM φH Q, 1 5��---- for

��---- 0>+= =

� �⁄ 10≈

φM 1 7 ��---- 9 1 �@< ��----+ + < ��----–

exp+ +=

φH 1 123--- 7 ��----+

32---

9 1 � < ��----+ + < ��----–

exp+ +=

� �⁄( )1 2⁄



ECMWF, 2002 15

Figure 6.Comparisonof dimensionlesswind shearobservationswith interpolationformulas(afterBusingeretal.

1971;ζ ( �1. � ).



ECMWF, 2002

Figure 7. Wind profiles extrapolated from observed 10 m wind according to (2.1.21-22) with (2.1.25) and

(2.1.28) in comparison with observations up to 200 m (Cabauw). The upper and lower panel represents unstable

andstablestratificationrespectively. Lm indicatesthestabilityclassasanaverageoveranumberof observations.

The dots and error bars represent averages and standard deviations of observations (fromHoltslag 1984).

The gradient functions can be integrated to profiles for wind, potential temperature and specific humidity



ECMWF, 2002 17

(41)

(42)

(43)

(44)

with

(45)

(46)

where (47)

and

(48)

(49)

This canbederivedfrom thegradientfunctionswith helpof therelationshipφ = 1 – (η) dΨ . dη, whereη=�/. � .

Examplesof wind profilesaccordingto theempiricalformsof (45) and(48) areshown in Fig. 7 in comparison

with data.

It is clearthatthesefunctionsplayacrucialrole in parametrizationschemes.Thegradientfunctionsprovidedirect

expressionsfor exchangecoefficientsasthey relatefluxesto gradients.Theintegralprofile functionsaregenerally

usedin modelsthathave a modellevel in thesurfacelayerto specifytheprofilesbetweenthelowestmodellevel

andthesurface.Theexpressions(41) to (44)canin factbeseenasfinite differenceformsfor thelayerbetweenthe

surface and the lowest model level.

Limiting cases in surface layer similarity

In determininganalyticalformsfor thesimilarity functionsit is oftenusefulto considertheirasymptoticbehaviour.

Theneutrallimit (�/. � = 0) andthefreeconvectionlimit (A �/. � >>1)areof particularinterestbecausethey provide

information to limit the degrees of freedom in choosing analytical expressions.

In theneutrallimit with �/. � = 0, theObukhov lengthdropsoutasarelevantscaleandthereforethedimensionless

� τ >ρ3 � *

-------------�

� 0M---------

ΨM��----

–ln=

! τ?ρ3 � *

-------------�

� 0M---------

ΨM��----

–ln=

θ θs–� ′θ′–( )03 � *

----------------------�

� 0H--------

ΨM��----

–ln=

� � s–� ′ � ′–( )03 � *

----------------------�

� 0Q--------

ΨM��----

–ln=

ΨM 2 1 �+( ) 2⁄{ } 1 � 2+( ) 2⁄{ } �( ) π 2⁄+atan–ln+ln=

ΨH Q, 2 1 � 2+( ) 2⁄{ } for��---- 0<ln=

� 1 16��----–

14---

=

ΨM 7 ��----– 9 ��---- �<---–

< ��----– 9&�

<------–exp–=

ΨH 1 123--- 7 ��----+

32---

– 9 ��---- �<---–

< ��----– 9/�

<------– for��---- 0>exp–=



ECMWF, 2002

windgradienthastobeconstant.ThedimensionalwindgradientφM isbydefinitionequalto1becausetheempirical

constantis absorbedin thedefinitionandis calledtheVon Karmanconstant.Theexactvalueof theVon Karman

constanthasbeensubjectof considerabledebate.Valuesbetween0.35and0.42havebeenproposed,but 0.4is the

most widely accepted value now (e.g.Hogstrom, 1988).

Theotherlimiting casewith – �/. � >> 1 is calledthefreeconvectionlimit. A largevaluefor – �/. � implies that

theproductionof turbulenceby buoyancy is muchlarger thantheproductionby wind shearandthat the friction

velocity � * is irrelevantasscalingquantity. Theonly way to obtainindependenceof � * in thedimensionlesspo-

tentialtemperaturegradientexpressionis by choosinga1/3power law for – �/. � >> 1, as�

is proportionalto �by definition:

(50)

Alternatively a convective velocity scale can be defined based on the on the buoyancy flux

(51)

resulting in

(52)

The freeconvectionvelocity representsphysically thevelocity that largeeddies(e.g.thermals)typically have in

theconvective boundarylayer. It shouldbenotedthatequation(35) doesnot satisfytheconstraintasimposedby

thefreeconvectionlimit. In practicehowever (35) fits dataasaccuratelyasexpressionsthat tendto

for large –�/. � .

Anotherexampleof thefreeconvectionlimit is thescalingrelationfor thevariancesof velocity andtemperature

fluctuations (see alsoFig. 8):

(53)

(54)

With thedefinitionsof�

andθ* it canbeverifiedthat � * dropsoutof theserelations.Alternatively thesameforms

canbeexpressedin termsof thefreeconvectionvelocitywithoutusing � *. Theadvantageof theexpressionin z/L

is thatit providesa limiting form for large�/. � . Theseexamplesclearlyillustratethestrengthof similarity theory.

For theneutralandfreeconvectionlimit thesimilarity argumentsevenprovide theshapeof thefunctions,experi-

ments are only needed to determine the value of the constants (e.g. the Von Karman constant).

� *3

3 � � *

� ′θ′( )0

-------------------∂θ∂�------ φH

��---- ��----–

13---

∼=

� *s�θv----- � ′θ′v( )0�

13---

=

3 � � *s

� ′θ′( )0

-------------------∂θ∂�------ constant=

� �⁄–( ) 1– 3⁄

� ′2( )

12---

� *---------------- ��4 ��----–

13---

=

θ′2( )

12---

θ*-------------- � θ

��----–

13---–

=



ECMWF, 2002 19

Figure 8. Examples of free convection scaling for large –�/. � . Standard deviations of vertical velocity and

temperature fluctuations.



ECMWF, 2002

2.2 The outer layer

Theouterlayerof thePBL is theremainingpartof theboundarylayerabove thesurfacelayer. Thefluxesarenot

constantasin thesurfacelayerbut decreasemonotonicallyfrom their surfacevaluesto smallvaluesat thetop of

thePBL (seeFig. 9 for theconvectiveboundarylayer).Thesurfacelayerparametersarealsorelevantin theouter

layer. Additionally, theboundarylayerheight - (lengthscale)andin theextratropicstheCoriolis parameter� (in-

verse time scale) enter the problem.

Neutral PBL

If theboundarylayerheightis not determinedexternally (e.g.by a cappinginversion)but by theboundarylayer

processesthemselvesthentheboundarylayerheightmustbeproportionalto theonly lengthscalein theproblem

namely��B . � . Therelevantparametersare:� *, � , and� ; thevelocityprofilesareusuallywrittenasdifferenceswith

the geostrophic wind as so-called velocity defect laws:

(55)

(56)

Figure 9. Heightdependenceof thekinematicheatflux in theconvectiveboundarylayer. Thesolid line refersto

a model with entrainment; the dashed line refers to a model without entrainment.

� � G–

� *------------------- f 5 � � *

�--------- =

! ! G–

� *------------------ f C � � *

�--------- =



ECMWF, 2002 21

Figure 10. Example of mixed layer scaling for the horizontal and vertical turbulent velocity fluctuations.

Convective boundary layer (mixed layer scaling)

In theconvective boundarylayer thePBL heightis determinedby a cappinginversionat height - which canbe

interpretedasanindependentparameter. Formallyalsothefriction velocity � * andtheCoriolis parameter� playa

role.They areoftenomittedbecausethewind turningis smallin thewell mixedlayerand– �/. � (which is by def-

inition largerthan– 0.1 - . � ) is oftenlarge.Theremainingrelevantparametersare , - , and� � from which

we define:

(57)

(58)

Thedimensionlessgradientsof wind temperatureandspecifichumidityarenotveryinterestingin thiscasebecause

thegradientsareverysmall.Theturbulentexchangeis sostrongin theconvectiveboundarylayerthattheprofiles

arenearlyuniform (well mixed).Mixed layer scaling,however, appliesto velocity andtemperaturefluctuations

e.g.σ D /w*, σ E /w*, σ F /w* andσθ /θ* are functions of�/. - (seeFig. 10 for σ DHGIEJGKF ).

Stable boundary layer (local scaling)

Thesameprinciplesasusedfor theconvectiveboundarylayercouldbeappliedto thestableboundarylayer. How-

ever, the friction velocity can not be omitted as friction is the only source of turbulence production.

� ′θ′v( )0

velocity scale � *�θv----- � ′θ′v( )0 -

13---

=

temperature scale θ*� ′θ′( )0

� *-------------------–=



ECMWF, 2002

To limit thenumberof independentparametersNieuwstadt(1984)introducedtheideaof local scalingwhich can

bemadeplausiblein thefollowing way. In thestableboundarylayerturbulenceis generatedby shearproduction

anddestroyedby buoyancy andviscousdissipation.Sincebuoyancy is actingin theverticalcomponentit tendsto

limit theverticalextentoverwhichthemixing takesplaceandthereforelimits theturbulencelengthscale.Thede-

couplingof turbulencefrom thesurfacesuggestthat the local fluxesmight bemoresuitableasdirectscalingpa-

rameters.Thestrengthof this idealies in thefactthattheresultingfunctionsarenecessarilythesameastheones

for thesurfacelayersincethelocalfluxesareequalto thesurfacefluxeswhenthesurfaceis approached.Thismeans

thatexperimentaldatafrom thesurfacelayerandtheouterlayercanbecombined.For modelswith local closure

(asin theECMWFmodel)it alsomeansthattheclosureschemefor thesurfacelayerandtheouterlayercanbethe

same.

In theframework of local scalingtherelevantparametersare:the local fluxes and ,thebuoyancy pa-

rameter� /θv and height� . From these we define:

(59)

Localscalingimpliesthatdimensionlessquantitiesarefunctionsof �/. Λ insteadof �/. � . Examplesof localscaling

are given inFig. 12; the height dependence of momentum and heat fluxes is illustrated inFig. 11.

For large �/. Λ theturbulencelengthscaleis Λ andthepresenceof thesurfaceshouldhave no influence.This im-

plies that � shoulddropout of thescalingrelations.Theconsequencefor thedimensionlessgradientfunctionsis

thatthey becomelinearin �/. Λ for large�/. Λ becausethey have � asalengthscalein theirdefinition.Theprinciple

thatdimensionlessfunctionsbecomeindependentof � is called� -lessscaling.Theconsequencein Fig. 12 is that

the empirical functions tend to a constant for large�1. Λ.

Thedifferentscalingregimesof theboundarylayeraresummarizedin Fig.13 for theunstableandstablePBL.The

dimensionlessheight�/. - andthedimensionlessstabilityparameter- . � areusedto delineatethedifferentscaling

regimes.

2.3 Matching of surface and outer layer (drag laws)

Theprofile functionsfor theneutralouterlayerwritten in velocity defectform mustbelogarithmicfor

andmustmatchthesurfacelayerprofiles.With thecoordinatesystemalignedwith thesurfacewind

and neutral stability:

surface layer limit for :

(60)

outer layer limit for :

(61)

(62)

τ ρ⁄ � ′θ′v

Λ τ ρ⁄ 3 2⁄–3 �

θv----- � ′θ′v

----------------------- (local Obukhov length) .=

� -⁄ 0→� ′ � ′( )0 0=

�L� 0M⁄ ∞→

�� *-----

13--- �� 0M---------

ln=

� -⁄ 0→

� � G–

� *-------------------

13--- � �� *------

const+ln=

! 0=



ECMWF, 2002 23

Figure 11. Height dependence of the kinematic heat flux and the kinematic momentum flux in the stable

boundary layer (fromNieuwstadt 1984)

Matching the surface and outer layer forms leads to (Tennekes, 1973):

(63)�NM� *--------

13--- � *

� � 0M------------

O 3----–ln=



ECMWF, 2002

(64)

TheconstantsO andP canbemeasuredandexpressin dimensionlessform thedragandtheageostrophicangleof

thesurfacewind with respectto thegeostrophicforcing.Such“drag-laws” canbeapplieddirectly to modelsthat

donot resolvethePBL, but arealsoanimportantdiagnostictool. In modelsthatdoresolvetheboundarylayer, the

draglaw canbecomputed(e.g.in onecolumntests)andcomparedwith datafor thesteadystatePBL (seeFig.14).

For non-neutralflow it is oftenassumedthatO andP arefunctionsof thestabilityparameter . Such

an assumption is quite arbitrary but works fairly well in practice.

Figure 12.Two examplesof local scaling(a)exchangecoefficient for momentumand(b) thestandarddeviation

of vertical velocity in the stable boundary layer (fromNieuwstadt 1984).

! M� *--------

P 3----–=

µ � * � �( )⁄=



ECMWF, 2002 25

Figure 13. Scaling regions in the unstable (upper panel) and the stable (lower panel) PBL (fromHoltslag and

Nieuwstadt 1986).



ECMWF, 2002

Figure 14.ConstantsO andP in thegeostrophicdraglaw accordingto Wangaradata(ClarkeandHess1974)in

comparison with the ECMWF (CY34) model formulation after 9 hours of integration (as a function of stability

parameter� �RQ . � ).

2.4 Surface boundary conditions and surface fluxes

Theparametrizationof thePBL introducesat thesametime theneedfor boundaryconditionsat thesurface.The

boundaryconditionsaresimply vanishingvelocity andspecifiedtemperaturesandspecifichumiditiesat thesur-

face.Overseathetemperatureis specifiedandthespecifichumidity is thesaturationvalueat theseasurfacetem-

perature.Specificationof temperatureandspecifichumidity over land is partof the landsurfaceparametrization

scheme.In somemodels,however, thesurfaceenergy balanceequationis usedover landto eliminatethesurface

temperatureandsurfacespecifichumidity. This is oftenreferredto asthePenmanMonteithconcept(seeMonteith

1981), which is not discussed here.



ECMWF, 2002 27

Lookingmorecarefullyat theboundaryconditions,onerealizesthatanadditionalparametrizationproblemarises.

Theboundaryconditionsareconnectedto thesurfaceroughnesslengthswhichareextremelyimportantto thesur-

facefluxes.To illustratethis we assumethatwe have a modellevel nearthesurfacein theconstantflux layerat a

specifiedheight� 1 (in theECMWF model m). Thesurfacelayerprofilesareusedto specifytheboundary

condition:

(65)

(66)

(67)

(68)

Thelogarithmicprofilesarewritten in away that � , ! , θ and � have theirsurfacevaluesfor � = 0 (adisplacement

heightequalto the roughnesslengthhasbeenaddedto thevertical coordinate).The integrationconstantsin the

logarithmicprofilesarecalledroughnesslengthsbecausethey arerelatedto thesmallscalesurfaceinhomogenei-

tiesthatdeterminethesurfacedrag.In generaltheviscousforcesplaynoroleatall in thesurfacedrag(exceptover

averysmoothsea);thepressureforcesontheroughnesselementstransferthewind stressto thesurface(form drag)

anddosomoreefficiently thanviscousforcesbecausetheheightof roughnesselementstendsto belargerthanthe

thicknessof theviscouslayer. For heatandmoisturethesituationis slightly different.Formdraghasnoequivalent

for scalarquantities.The air-surfacetransferis throughan inter-facial sub-layerwheremoleculartransferdom-

nates. This implies that the roughness lengths for momentum, heat and moisture are generally different.

TABLE 2. TYPICAL ROUGHNESSLENGTHS S B (Wieringa1986)

Ground cover Roughness length (m)

Water or ice (smooth)(over waterT 0M increaseswith wind speed

10-4

Mown grass 10-2

Long grass, rocky ground 0,05

Pasture land 0.20

Suburban housing 0.6

Forest, cities 1–5

32≈

� τ > 0

ρ3 � *

-------------� 1

� 0M--------- 1+

ΨM

� 1�----- –ln=

! τ? 0

ρ3 � *

-------------� 1

� 0M--------- 1+

ΨM

� 1�----- –ln=

θ θs–� ′θ′–( )03 � *

----------------------� 1

� 0H-------- 1+

ΨM

� 1�----- –ln=

� � s–� ′ � ′–( )03 � *

----------------------� 1

� 0Q-------- 1+

ΨM

� 1�----- –ln=



ECMWF, 2002

Figure 15.Variationof theeffectiveroughnesslength asafunctionof , whereλ is thewavelengthof

2-dimensional sinusoidal orography with amplitude7 . Different models are compared (Taylor et al. 1989).

It is importantto realizethatthespecificationof thesurfaceroughnesslengthsis animportantaspectof thepara-

metrizationsincethethey determineto a largeextentthesurfacefluxes.Theroughnesslengthis asurfacecharac-

teristicandis relatedto its geometry, but theonly wayto defineit is by meansof thelogarithmicprofile.Measuring

thelogarithmicprofile well away from thesurfaceis theonly way of determining� 0M for a uniform terrain.For a

largescalemodelwith grid spacingsof theorderof 100km many terraininhomogeneitiesexist thataffect thesur-

facedrag.Theeffectsof sub-gridterraineffectsandsub-gridorography areaccountedfor by definingand“effec-

tive roughnesslength”, which is definedas the roughnesslength that gives the correct surface flux in the

parametrization.Theroughnesslengthsareoftenderivedfrom land-usemapswith anextra contribution from the

varianceof thesub-gridorography (seeTable2). A databasefor landuseis provide by Wilson andHenderson-

Sellers(1985).A review of sub-gridaveragingis providedby Mason(1988)andTayloretal. (1989).Theissueof

sub-gridaveragingof � 0H and � 0Q hasonly very recentlyreceived attention(seeWood andMason1991;Mason

1991;Beljaars and Holtslag, 1991).

Fig.15 illustratesempiricaldataontherelationbetweentheamplitudeof nottoosteepsub-gridorography andthe

aerodynamicroughnesslength.Steeporography is moredifficult to parametrize(seeMason,1988for a review).

Thedifferencebetweentheroughnesslengthfor heatandmomentumcanbequitelargeasillustratedin Fig. 16 .

Recentstudiessuggestthatterraininhomogeneitieshavea largeimpacton theratio � 0H and� 0M (seeMason1991).

Specificationof thesurfaceroughnesslengthsoverseais evenmoreimportantthanover landparticularlyfor heat

andmoisturebecausewith thefixedboundaryconditionsfor temperatureandmoisturetheseais in principlean

infinite sourceof energy to the model.Currently the ECMWF modelhasequalvaluesfor momentumheatand

� 0Meff λ � 0M⁄



ECMWF, 2002 29

moisture all specified with help of the Charnock relation

(69)

(70)

Figure 16. Temperature difference over the inter-facial sub-layer for short grass. By definition:

. Thisfigureillustratesthepossibledifferencebetweentheroughnesslengthfor

momentum and for heat particularly when an “effective value” is used. (Cabauw data;Beljaars and Holtslag,

1991).

This parametrizationdoesnot satisfythesmoothsurfacelimit at low wind speedwhereviscousforcesdominate

(with scalingon ν . � B ). FurthermoreSmith (1988)arguesin a recentreview that the increaseof the roughness

lengthwith increasingwind speeddoesnotapplyto heatandmoisture.An alternativewhichcombinesthesmooth

surface limit with the conclusions of Smith’s review is:

(71)

(72)

(73)

� 0M max 0.018� *

2

�----- 1.15 10 5–× m, =

� 0H � 0Q � 0M= =

θ0 θs– θ*

3⁄( ) � 0M � 0H⁄( )ln=

� 0M 0.018� *

2

�----- 0.11ν� *-----+=

� 0 U 1.4 10 5–× 0.40ν� *-----+=

� 0Q 1.3 10 4–× 0.62ν� *-----+=



ECMWF, 2002

The two formulations are compared with data byLarge and Pond (1982) in theFigs. 17– 19 .

Figure 17. Neutral transfer coefficients for momentum between the sea surface and the 10 m level. The

observations were published byLarge and Pond (1982); the thick solid and dashed line representEqs. (69) and

(71) respectively. V MN is related to� 0M by means of:V MN

3 � 1 � 0M⁄( )ln⁄{ }2=



ECMWF, 2002 31

Figure 18. Neutral transfer coefficients for heat between the sea surface and the 10 m level. The observations

were published byLarge and Pond (1982); the thick solid and dashed line representEqs. (69) – (70) and(71) –

(72), respectively. V HN is relatedto � 0M and� 0H by meansof: , where� 1 is the 10 m level (normally the height of the lowest model level).

Figure 19.Eqs. (69), (70), (71) and(73), respectively. V QN is related to� 0M and z0Q by means of:

, where� 1 is the10m level (normallytheheightof thelowestmodel

level.

3. PBL SCHEMES FOR ATMOSPHERIC MODELS

DifferentPBL schemesaredescribedin thischapterrangingfrom verysimpledraglaws to formulationsbasedon

theturbulencekinetic energy equation.Sincethegeostrophicdraglaws arebecominglessandlesspopular, they

will only bedescribedbriefly. Mostattentionwill bepaidto (vertical)grid pointparametrizationswith simpleclo-

sure for the exchange coefficient as they are the most popular ones.

V HN

3 � 1 � 0M⁄( )ln⁄{ }3 � 1 � 0H⁄( )ln⁄{ }=

V QN

3 � 1 � 0M⁄( )ln⁄{ }3 � 1 � 0Q⁄( )ln⁄{ }=



ECMWF, 2002

3.1 Geostrophic transfer laws

Geostrophictransferlaws canbeusedto introducesomeeffectsof surfacefluxesin modelsthathave hardlyany

or no levelsin theboundarylayer. They arebasedonboundarylayersimilarity for thesteadystateboundarylayer.

A relationis assumedbetweenthemodelvariablesat the lowestmodellevel andthefluxesat thesurface,where

thelowestmodellevel is assumedto beat thetop of theboundarylayeror above theboundarylayer. Suchanap-

proachmakesonly sensein modelswith low resolutionnearthesurface.Thesimplestform of a bulk transferlaw

expressesthesurfacefluxesof momentum,heatandmoistureinto themodelvariables� G, ! G, θG, � G above the

boundary layer and surface valuesθs and� s in the following way:

(74)

(75)

(76)

(77)

The index G hasbeenusedhereto indicatevariablesabove theboundarylayer; for wind it canbe thoughtof as

geostrophicwind. Theconstants , and arechosenfrom field dataor adjustedon thebasisof model

performance.Differentvaluescanbechosenfor landandsea.Advantagesof thisschemeare:(i) theschemeis very

simpleand(ii) it is afirst orderapproximationfor theinfluenceof surfacefluxes.Disadvantagesare:(i) thescheme

doesnothaveadiurnalcycle,(ii) therearenoeffectsof stability, (iii) theempiricalcoefficientsareveryuncertain,

(iv) there are no forecast products in the PBL and (v) the physics of the PBL is poorly represented.

A second“bulk” scheme,which is very similar to thefirst one,introducesstability effectsin thetransfercoeffi-

cients , and in accordance with the similarity theory discussed inSection 2.3.

(78)

where , or , (79)

and , . (80)

TheconstantO dependsonstabilityandis thesameoneasin Eq.(63). ExperimentaldataonA is shown in Fig. 14

(ClarkeandHess1974).Thisformulationis slightly morerealisticthanthefirst onebut is verymuchlimited to the

simplesituations,for whichthetransferlawsareknown.However, with anappropriatelandsurfaceschemefor the

surfacetemperature and surfacemoisture , this schemeallows for diurnal variation of the surfacedrag

throughthe stability dependenceof parameterO . The morecomplicatedcaseof the unsteadyPBL, caseswith

strongadvection,baroclinicor inversioncappedPBL’sarenotexpectedto bewell capturedby bulk transferlaws.

� ′ � ′( )0 V M UG � G–=

� ′ � ′( )0 V M UG ! G–=

� ′θ′( )0 V H UG θG θs–( )–=

� ′ � ′( )0 V Q UG � G � s–( )–=

V M V H V Q

V M V H V Q

V M V H V Q13--- � *

� � 0M------------

O 3----–ln2–

= = =

O O � *

� �------ = O O -�----

=

� * � ′ � ′( )02 � ′ � ′( )0

2+{ }

12---

=� � *

3–3 �

θv-----

� ′θv′( )0

------------------------------------=

θs � s



ECMWF, 2002 33

3.2 Integral models (slab, bulk models)

In integral modelstheentireboundarylayer is characterizedby a limited numberof parameters(e.g.PBL depth,

meanvelocity, meantemperatureetc.).Prognosticequationsarederivedfor thesePBL characteristicsratherthan

for thebasicvariables.Themainadvantageis thatthePBL doesnothaveto beresolvedby theverticallayerstruc-

ture of the model.

Thebasicassumptionfor integralmodelsis thatasimilarity form existsfor themodelvariablese.g.for potential

temperature:

, (81)

where is a temperaturescaleand - theboundary-layerheight.Whentheform of f is known from field experi-

ments,Eq.(81)canbesubstitutedin theequationsof motionandintegratedfrom thesurfaceto � (W- . Theresulting

differential equations for and- describe the entire PBL without resolving it explicitly in the vertical.

Figure 20. Schematic profiles ofand in a mixed layer model.

Thisapproachis illustratedwith helpof aslabmodelfor theconvectiveboundarylayer(mixedlayer).In themixed

layertheturbulentexchangeis sufficiently strongto make theprofilesof wind, potentialtemperatureandspecific

θ θG–

θ*--------------- f

�----

=

θ*

θ*

θ θ′ � ′



ECMWF, 2002

humidityapproximatelyuniformin thebulk of theboundarylayer(thesimplestpossibleexpressionfor f). Thenear

surfacepartshows gradients,but thesurfacelayer is relatively shallow anddoesthereforenot contributesignifi-

cantlyto thebudgetsof thetotalboundarylayer. Herewe limit to thebudgetof potentialtemperaturebecauseit is

themainquantitythatfeedsbackto thedynamicsof themixedlayer. Quantity representsthepotentialtemper-

aturein thebulk of themixedlayerandits timeevolution is determinedby heatfluxesat thesurfaceandthetopof

themixedlayer(Fig.20 illustratesthetemperaturestructureof thewell mixedlayer;seeDriedonksandTennekes,

(1984)):

, (82)

, (83)

, (84)

. (85)

In theseequations representsthepotentialtemperaturejumpat thetopof theboundarylayer, theentrain-

mentvelocity, thelargescaleresolvedverticalvelocityatheight- , andγ thestablelapserateabovethemixed

layer. Equations(82)–(85) describetheenergy budget,theheightandthetemperaturestepat thetop of themixed

layerandhaveasimplephysicalinterpretation.Therateof changeof theenergy of themixedlayeris proportional

to thedifferenceof heatflux at thesurfaceandthetop(Eq.(82)); theheightchangeof themixedlayerwith respect

to thelargescaleascentor descentis by definitiontheentrainmentvelocity (Eq.(83)); thetemperaturestepat � (- changesdueto temperaturechangesof the mixed layer andduea shift of the mixed layer with respectto the

stablelapserate(Eq.(84)); andtheentrainmentheatflux is theamountof air thatis entrainedtimesthetemperature

jump (Eq. (84)). This setof equationsis not closed;it hasto besupplementedby equationsfor thesurfacefluxes

andby anentrainmentassumption.Thesurfacefluxescanbeobtainedby usingsurfacelayerformulationsasdis-

cussed inSection 2.4. For the heat flux,Eq. (67) can be used with� somewhere in the mixed layer and .

To parametrize the entrainment we consider the equation for turbulent kinetic energy:

(86)

This equationdescribesthebalancebetweenshearproduction(I), buoyancy productionor destruction(II), trans-

portby turbulentvelocityandpressurefluctuations(III) anddissipation(IV). Theorderof magnitudeof thediffer-

ent terms typical for the entire mixed layer depth can be expressedin terms of velocity and length scales

characteristic for the mixed layer:

I:

II:

θm

- dθm

d�--------- � ′θ′( )0 � ′θ′( )0–=

d-d�------- � e � L+=

d∆θm

d�-------------- γ � e

dθm

d�---------–=

� ′θ′( )0 � e∆θ–=

∆θm � e

� L

θ θm=

∂$

∂ �------- = � ′ � ′∂ �∂�--------– � ′ � ′∂ !∂�-------–�ρ0----- � ′θv′+

I( ) I I( )

∂∂�------

$′ � ′ � ′ � ′

ρ------------+

+ ε .–

I I I( ) IV( )

� *3 -⁄� θv⁄( ) � ′θv′( )0 � *

3 -⁄+



ECMWF, 2002 35

III:

IV:

Theseestimateshavebeenmadeby assumingasinglelengthscale- asturbulencelengthscaleandfor theestima-

tion of thederivativesandby assumingthattheturbulentfluctuationshave magnitude (see(57) for its defini-

tion). It is importantto realizeherewhat themechanismof entrainmentis. Theenergy is producedby shearand

buoyancy in thebulk of themixedlayer. Most of theenergy is transferredto smallerandsmallerscalesandulti-

matelyconvertedinto heatby viscousdissipation.Someof it is diffusedupwardsby theturbulentmotionitself (e.g.

thermals)andpenetratesinto theinversionwhereit is destroyedby thenegative buoyancy (Fig. 21 illustratesthe

energy budgetof themixedlayer).Sotheentrainmentat thetop of themixedlayer is fed by upwarddiffusionof

turbulent kinetic energy produced in the bulk of the mixed layer.

Sincethedominanttermsareof theorderof , it is to beexpectedthat thebuoyancy flux at � (X- is also

proportional to this quantity. An obvious closure therefore is:

(87)

wherethe“entrainment”constant is determinedfrom experimentaldata.Thenumericalvalueof is about

0.2 (Driedonks and Tennekes 1984).

Theadvantagesof integralmodelsareobvious:(i) they arerelatively simple,(ii) they donot rely onverticalreso-

lution and(iii) they arephysically realisticfor simplecasesasthemixedlayer. Disadvantagesare:(i) this typeof

modelsis difficult to implementasthey involve a moving boundarycondition(thetop of theboundarylayer) for

theatmosphereabove thePBL, (ii) thesimilarity profilesarelessobvious for e.g.thestableboundarylayerand

virtually unknown for morecomplicatedboundarylayersase.g.thebaroclinicor cloudyboundarylayerand(iii)

thetransitionbetweendifferentscalingregimes(e.g.fromunstabletostable)aredifficult tohandle.Formixedlayer

modelsin particularit hasto beaddedthatthey performverywell for awell mixedquantityaspotentialtempera-

ture, but quite poorly for less well mixed quantities as momentum and specific humidity.

Becauseof thetechnicalcomplicationsof having a moving lower boundaryconditionin a forecastmodel,Dear-

dorff (1972)proposedto combinethe integral boundarylayerapproachwith a fixedverticalgrid structure.This

schemewasspeciallydesignedfor GCM’swith poorverticalresolution.Thediagnosedboundarylayerheightcan

bebelow andabovethelowestmodellevel (Fig.22). Theboundarylayerheightcanbeseenasanadditionalmodel

level at theheightwherethestructureof theatmospherechangesdrastically. After every time step,with thePBL

scheme the grid points inside the boundary layer are adjusted to the mixed-layer profile.

� *3 � *

3+( ) -⁄� *

3 � *3+( ) -⁄

� *

� *3 -⁄

� ′θv′( )0 V e � ′θv′( )0=

V e V e



ECMWF, 2002

Figure 21.Normalizedtermsof theturbulencekineticenergy equation.Thenormalizationis on . (Fig. 5.4

from Stull 1988)

Figure 22. Grid point structure inDeardorff’s (1972) PBL model.

3.3 Grid point models with Y -closure

General description

A fairly simpleclosurefor modelsthatresolvetheboundarylayerexplicitly, is the' -typeclosure.In analogywith

molecular diffusion, it is assumed that the fluxes are proportional to the gradients (Louis, 1979):

� *3 -⁄



ECMWF, 2002 37

, (88)

. (89)

Theexchangecoefficient ' is notapropertyof thefluid asin moleculardiffusion,but dependsontheflow. Surface

layer similarity provides the formulation for' near the surface (i.e. in the lowest model layer):

, (90)

whereZ[( 3 � . Useof thesameexpressionin theouterlayer (above the lowestmodellevel), would beconsistent

with theideasof local scaling(Section2.2) for thestableboundarylayer, providedthat , and arepro-

portionalto � for large �/. � . In this way thelengthscale� dropsout for large � . Two problemsarisefrom sucha

simpleextensionof surfacelayersimilarity bothrelatedto thelackof anupperboundto Z : (i) thediffusioncoeffi-

cientbecomesverylargein neutralsituationsand(ii) with differentfrom in stablesituations(necessaryfor

reasonsto beexplainedlater),localscalingbreaksdown and Z remainsanessentialpartof theparametrization.To

limit the length scaleBlackadar (1962) proposed the following expression:

(91)

This formulationobeys surfacelayersimilarity andis consistentwith local scalingfor thestableboundarylayer.

Theonly additionalparameterthathasbeenintroducedis theasymptoticlengthscaleλ, which is oftenthesame

for momentumandheat.The ideais that the turbulencelengthscaleis limited by the boundarylayer height.In

someparametrizationsλ ~ �RQ . � or λ ~ - , where- is adiagnosedboundarylayerheight.Sincetheresultsarenot

very sensitive to the exact value ofZ , this parameter is often chosen constant.

Expressions(90) containφ which dependson �/. � in thesurfacelayerandthereforeon thesurfacefluxes.Above

thesurfacelayer theφ-functionsdependon Z . 3 � insteadof �/. � to beconsistentwith the ideasof local scaling.

An inconvenientaspectof Eq. (90) is that it containsfluxeswhich makestheclosurean implicit one.To find an

explicit closure we substitute the closure assumptions in the definition of the Obukhov length and obtain:

(92)

This is a relationbetweenthegradientRichardsonnumberand Z . � which canbesolvednumericallywhentheφ-

functionsarespecified.Insteadof closureassumption(90) it is muchmoreefficient to specifystability functions

that depend on the Richardson number because then we have the fluxes explicitly in terms of model variables:

(93)

where and , and by definition: .

� ′ � ′ ' M∂ �∂�-------- ,–= � ′ � ′ ' M

∂ !∂�-------–=

� ′θ′ ' H∂θ∂�------ ,–= � ′ � ′ ' H

∂ �∂�------=

' MZ 2

φM2

------- dUd�------- , ' M

Z 2

φMφH------------- dU

d�-------= =

φM φH φQ

φM φH

1Z---

13 �------ 1λ---+=

� * �θv-----

dθv

d�--------

dUd�-------

2------------- Z3 �-------

φH

φM2

-------= =

' M Z 2\M

dUd�------- , ' H Z 2\

HdUd�------- ,==

\M

\M

� *( )= \H

\H

� *( )= \M φM

2– , \H φMφH( ) 1–= =



ECMWF, 2002

Equation(93) is theclosureasit is actuallyusedin many models.Thefunctions and arespecifiedasan-

alytical functionsof� * or aretabulated.In bothcasesthey arederivedfrom empiricaldataon and which

arefunctionsof �/. � (or Z . 3 � ), sotheindependentvariablehasbeentransformedfrom �/. � (or Z . 3 � ) to� * with

Eq. (92).

Figure 23. The vertical grid structure near the surface in the ECMWF model (19-level version and 31-level

version). Solid lines represent model levels; dashed lines are flux levels.

Thegradientsasneededin theclosureassumptionarecomputedfrom thegrid point valuesby meansof a finite

differenceapproximation(cf. Fig. 23 ). A staggeredgrid is usedwith flux levelsbetweenthefull modellevels.A

simplefinite differenceapproximation(assuminglinearly interpolatedprofilesbetweenmodellevels)is notaccu-

ratenearthesurface,wheretheprofilesof wind temperatureandmoistureshow pronouncednon-linearbehaviour

particularlynearthesurface.To deriveafinite differenceform for thesurfacelayerweassumethattheflux is con-

stantwith heightandintegratetheclosureassumptionsfrom thesurfaceto � ( � 1 where� 1 is theheightof thelowest

model level. This results in the usual surface layer profiles:

(94)

(95)

(96)

(97)

\M

\H

φM φH

� 1

τ > 0

ρ3 � *

-------------� 1

� 0M--------- 1+

ΨM

� 1�----- –ln=

! 1

τ? 0

ρ3 � *

-------------� 1

� 0M--------- 1+

ΨM

� 1�----- –ln=

θ1 θs–� ′θ′–( )03 � *

----------------------� 1

� 0H-------- 1+

ΨH

� 1�----- –ln=

� 1 � s–� ′ � ′–( )03 � *

----------------------� 1

� 0Q-------- 1+

ΨH

� 1�----- –ln=



ECMWF, 2002 39

Theseexpressionscanbeinterpretedasanexactfinite differencerepresentationof thelowestmodellayer. Unfor-

tunatelyit is animplicit formulationasit contains . Again we canrelatethe“flux-stability parameter”

to the bulk Richardson number :

(98)

(99)

Thisequationrelates� * b to � 1 /

�, � 1/� 0H and� 1/� 0M. For computationalefficiency thestability functionsareusually

representedin termsof thebulk Richardsonnumber� * b. Thesurfacefluxesof momentum( and ), heat(]W^

and water vapour_a`�b are expressed in model variables at the lowest model level in the following way:

(100)

(101)

(102)

(103)

where

(104)

(105)

(106)

With the neutral transfer coefficients defined as:

(107)

� �⁄ � �

⁄� * b

� * b�θv-----

� 1 θv1 θv0–( )

U12

--------------------------------=

� 1�-----� 1

� 0H-------- 1+

ΨH

� 1�----- –ln

� 1

� 0M--------- 1+

ΨM

� 1�----- –ln

2----------------------------------------------------------------=

τ > τ?

τ >ρ----- V M U1 � 1=

τ?ρ----- V M U1 ! 1=

]ρ �� ---------– V H U1 θ1 θs–( )=

`ρ----– V Q U1 � 1 � s–( )=

V M V MN\

M

� * b

� 1

� 0M---------

� 1

� 0H--------, ,

=

V H V QN\

Q� * b

� 1

� 0M---------

� 1

� 0H--------, ,

=

V Q V MN\

M

� * b

� 1

� 0M---------

� 1

� 0H--------, ,

=

V MN

3� 1

� 0M---------

ln

--------------------

2

=



ECMWF, 2002

(108)

(109)

The closureassumptionsarehiddennow in the empirical functionsof the Richardsonnumber, which arenever

measureddirectly. The reasonis that they dependon the surfaceconditions(surfaceroughnesslengths)which

makesit difficult to determinethemdirectlyfrom data.Experiments,however, doprovidedataonthe or func-

tions.Fromtherewe canderive numericallythe \ functionsfor differentsurfaceroughnesslengthsandmake an

empiricalfit to beusedin theforecastmodel.However the\ functions(104)–(106)dependon threedifferentpa-

rametersandit canbedifficult to obtainsimpleempiricalexpressionsthatareaccuratefor all possiblevalesof the

parameters.In many modelsthe threeroughnesslengthsarethesamewhich reducesthenumberof independent

parameters to 2, but the empirical functions can still be fairly inaccurate as will illustrated subsequently.

The stability functions

Themaindifficulty with ' -closureis thespecificationof thestability functions.Louis et al. (1982)give anover-

view of how thesefunctionhaveevolvedin theoperationalECMWFmodelastheresultof adjustmentsonthebasis

physicalargumentsandmodelperformance.ThefunctionsthathavebeenusedsinceDecember1981are(Louiset

al. 1982):

(110)

(111)

(112)

(113)

where9W( 5, �c( 5, and<2( 5, and

V HN

3� 1

� 0H--------

ln

--------------------

2

=

V QN

3� 1

� 0Q--------

ln

--------------------

2

=

φ Ψ

\M 1

29 � * 0

1 V 0 � * 0–( )1 2⁄+----------------------------------------- for

� * 0 0<–=

\H 1

39 � * 0

1 V 0 � * 0–( )1 2⁄+----------------------------------------- for

� * 0 0<–=

\M

1

1 29 � * 0 1 < � * 0+( ) 1 2⁄–+-------------------------------------------------------------- for

� * 0 0>=

\H

1

1 39 � * 0 1 < � * 0+( )1 2⁄+------------------------------------------------------------ for

� * 0 0>=

V 0 39/�3 2 1

� 1

� 0M---------+

1 2⁄

1� 1

� 0M---------+

ln2

------------------------------------- in the surface layer, and=

V 0 Z 2

3� 2-------- above the surface layer.=



ECMWF, 2002 41

Extensivediagnosticsof operationalforecastshasshown thatthediffusionwastoostrongin themiddleandupper

troposphere,wherethejet streamwaspartially smearedout by theactionof theverticaldiffusionscheme.It was

thereforedecidedto make a roughestimateof theboundarylayerheightandto switchoff theschemeabove the

boundarylayerin stablesituations.Thischangewasintroducedoperationallyin January1988.Thepoorperform-

anceof theschemeabovetheboundarylayercanbeattributedto theshapeof thestability functionin theRichard-

son number range of 0.1 to 1.

An alternative form of thestability functions(the\ _ � *d^ functions)canbederivedfrom theφ functions(34)–(35)

in unstablesituationsand(39)–(40) in stableflow. We will call themtheM.O. functions.Thecomparisonof the

stability functionsis shown in Fig. 24 . We seethat for positive Richardsonnumbersthe functionsderived from

and dependon thesurfaceroughnesslength.Themostpronounceddifferenceabove thesurfacelayer is

thattheM.O. functionsgivemuchsmallervaluesabove� * = 0.1in comparisonto theLouisetal. (1982)functions.

It shouldberealizedthattheshapeof thestability functionsis subjectto anongoingdebate(Hogstrom1988).The

difficultiesoriginatefrom limitations in theconceptof local diffusionandfrom uncertaintiesin theexperimental

data.An exampleof thefirst difficulty is theunstableor convectiveboundarylayerwherethediffusionis not local.

Thereasonthatalocalschemestill performsreasonablywell is thatthecoefficientsbecomelargeandthereforethe

profilesvirtually uniform irrespective of theexactmagnitudeof thediffusioncoefficients.For quantitiesthatare

not well mixed in theconvective boundarylayerase.g.moistureor wind in a baroclinicPBL, a local schemeis

expectedto belessappropriate.A secondexampleof thesameproblemis diffusionin stablystratifiedflow in the

intermittentregime.Localscalingworksfairly well in thefully turbulentregimeupto Richardsonnumbersof 0.2.

Beyond0.2,turbulencebecomesintermittentandthediffusivecharacteristicsarenotwell understood.Also wave-

turbulenceinteractionis a relevantprocessin this range.The reportedmagnitudeof thediffusioncoefficientsin

clear-air turbulence(Richardsonnumbersbetween0.2and1) variesby andorderof magnitudefrom onestudyto

the other(KennedyandShapiro1980;Uedaet al. 1981;Kondo et al. 1978;Nieuwstadt1984;Kim andMahrt

1991).Theshapeof functions and asexpressedby Eqs.(39)–(40) is reasonablywell documented(seeBel-

jaarsandHoltslag1991)for weakandmoderatestability. Theratio , however, is inspiredby theideathat

this ratio shouldbecomeproportional� * for largeRichardsonnumbers(seeEq. (92)). This particularasymptotic

behaviour is muchlessdocumented,but is quit relevantto thetransformationfrom �/. � to� * in Eq.(92). Another

parameterthatdeterminesto alargeextentthediffusionin theclear-air turbulencedomainis theasymptoticlength

scaleλ. A simpleparametrizationstrategy is to selectstability functionswhich arereasonablywell documented

from� * = 0 to

� * = 0.2,applyphysicalconstraintsfor largeRichardsonnumbersasin Eqs.(39)–(40) andadjust

λ to obtainreasonablediffusioncoefficientsin theclearair turbulencedomaine.g.onthebasisof databy Kennedy

and Shapiro (1980).

One column simulations

To illustratethecharacteristicsof ' -closure,asinglecolumnversionof theverticaldiffusionschemehasbeenin-

tegratedover9 hourswith constantgeostrophicforcingandconstantforcingfrom thesurface.In thisway, anearly

steadystateis obtained.Theselectedforcing is aconstantgeostrophicwind of 10 m/sanda constantsurfaceheat

flux. Thesurfaceroughnesslengthis 0.1m. Fig. 25 shows theO andP parametersin thedraglaw of Eqs.(63) –

(64)astheresultof theparametrizationwith Louisetal. (1982)stabilityfunctionsandwith theM.O. stabilityfunc-

tions.Thecomparisonwith empiricaldatashowslittle difference;themaindifferenceis thattheM.O schemecov-

ersa muchlargerdimensionlessstability rangethantheLouis et al. schemefor thesamerangeof surfacefluxes.

Thedifferencebetweenthetwo schemesis moreobviousin theFigs.26 and27 for a downwardheatflux of 20

W/m.Firstof all weseathatthesurfacestressis muchsmallerwith theM.O. schemeandthatthePBL is lessdeep.

Boundarylayerheightswith theM.O. schemearemuchmorerealisticandcomparemuchbetterwith dataasfor

instanceexpressedby theempiricalequilibriumheightsproposedby Zilitink evich (1972)or observationsby Nieu-

wstadt (1981).

φM φH

φM φH

φM φH⁄



ECMWF, 2002

With arelatively low numberof grid pointsin thePBL, significantdiscretizationerrorscanbeexpected.Theshal-

low stableboundarylayercouldbeparticularlyaffectedasit is only resolvedby two or threemodellevels.The

effect of resolutionon thestructureof thestableboundarylayeris shown in Fig. 28 , wherea simulationwith the

19-level modelresolutionis comparedto a simulationwith threetimesasmany levels.We seethattheresultsare

very similar andthatthelow resolutionsimulationis fairly accurate.Theunexpectedrobustnessof thenumerical

schemeis relatedto thetreatmentof thesurfacelayerwhereaconsiderablepartof theshearandtemperaturegra-

dientsareconcentrated.In thesurfacelayertheschemeusestheintegral profile functions(94)–(97) asa bulk for-

mulationratherthanafinite differenceform. It is known from otherapplicationsthattheintegralprofile functions

are quite accurate also when extrapolated outside the surface layer (VanWijk et al. 1990).



ECMWF, 2002 43

Figure 24. Stability functions according toLouis et al. (1982) and derived from the universalφ-functions (M.O.

scheme) above the surface layer and for different surface roughness lengths in the surface layer.



ECMWF, 2002

Figure 25.O andP parametersin thegeostrophicdraglaw assimulatedwith theLouisetal. (1982)functionsand

the M.O. stability functions.



ECMWF, 2002 45

Figure 26. Flux profiles in the stable boundary layer withLouis et al. (1982) functions (solid) and iteratively

obtained functions from the M.O.φ-functions (dashed).



ECMWF, 2002

Figure 27. AsFig. 25, but for wind and temperature profiles



ECMWF, 2002 47

Figure 28. Illustration of resolution dependence of a 9-hour integration for the stable boundary layer in one

column mode.

Concluding remarks

Theadvantageof ' -closurewith localexpressionsfor thediffusioncoefficientsis thatit is fully consistentwith the

grid point formulationof themodelandthatthesameschemecanbeusedfor stable,unstable,baroclinic,land,sea



ECMWF, 2002

etc.A disadvantageis thata numberof modellevelsareneededto resolve theboundarylayer. On theotherhand,

it is quite remarkablethat theschemeperformsfairly well with 2 levels in a shallow stableboundarylayer. The

reasonis thatthesurfacelayerapproximationis exactin thesensethatthesurfacelayerprofilebetweenthesurface

andthelowestmodellevel is representedexactly. Theresolutionis only neededto obtainareasonableshearstress

profile,but flux profilesaregenerallyfairly linearasfunctionof heightandcanberepresentedwith a few points

only. Properresolutionis neededhoweverto resolveinversions.Sincethepositionof theinversionis variable,good

resolutionis neededtroughout thelower troposphere(Beljaars1991).This is obviouslyadisadvantagecompared

to mixedlayermodelswherea dedicatedmodelvariableexiststo keeptrackof theinversionheight.Anotherdis-

advantageis thattheclosureis localanddoesnotaccountfor the“integral” aspectsof e.g.theconvectiveboundary

layer. Althoughthephysicsof theconvective boundarylayeris not well describedwith local closure,this is not a

realproblemin practicebecausea well mixedlayer is alwaysproducedwhentheexchangecoefficientsaresuffi-

ciently large.Local closurehasits limitationsof courseasit doesnot describeaspectsascounter-gradientfluxes,

not well-mixedparametersetc.Themaindisadvantageof local closureis that it doesnot describeentrainmentat

thetopof themixedlayer, becausethenon-localmechanismof entrainmentis not representedatall by a localclo-

surescheme.With localclosuretheentrainmentrateis generallyzeroasthelocalclosureproducesverysmallex-

change coefficients in stable stratification.

In summarywecansaythe' -typeclosureis very realisticfor thestablePBL if theresolutionis sufficient;alsoin

the unstable boundary layer, well-mixed quantities are well represented, but entrainment is not modelled at all.

3.4 Y -profile closures

An interestingclosurewhichhasbeenproposedfor low resolutionmodelsis the' -profileclosure(seeTroenand

Mahrt1986andHoltslagetal. 1990).Theexchangecoefficientsarenotexpressedin localgradientsbut asintegral

profilesfor theentirePBL. Theboundarylayerheightandthesurfacefluxesareusedasscalingparameters.Also

countergradienttermsareincludedin theformulation.Theexchangecoefficientsareprescribedasfollowsfor the

stable boundary layer and the unstable surface layer (�/. - < 0.1):

(114)

(115)

(116)

(117)

(118)

(119)

In theunstableouterlayerthesameexpressionsareusedexceptthat and"fe areevaluatedat � = 0.1 - andthat

a counter gradient term is introduced in the heat and moisture equations:

� ′ � ′ ' M∂ �∂�-------- , � ′ � ′ ' M

∂ !∂�------- ,= =

� ′θ′ ' H∂θ∂�------ , � ′ � ′ ' H

∂ �∂�------ ,= =

' M

3 � s� 1�----–

2=

' H

3 � s� 1�----–

2"8e 1–=

� s

� *

φM-------=

"8e φH

φM-------=

� s



ECMWF, 2002 49

(120)

(121)

where is evaluatedat the top of thesurfacelayer (�1. - = 0.1). If thestability functionsobey freeconvection

scaling, this closure will automatically provide the usual free convection velocity scale:

(122)

(123)

Thisclosureisagainanimplicit closurein thesensethattheexpressionsfor thefluxescontainthefluxes.In practice

this is solvedby treatingthesurfacelayerexactly asis donein thelocal closurescheme.After thesurfacefluxes

have been computed on the basis of the profiles at the previous time step, the exchange coefficients are specified.

Theboundarylayerheightis anessentialpartof theparametrizationin this schemeandhasto bediagnosed.The

following expression is used

(124)

where� * c is a critical Richardsonnumberwith a numericalvalueof about0.5,θvs is a virtual temperaturesome-

wherein thesurfacelayer(e.g.10m) andθv (h) is thevirtual potentialtemperatureat thetopof theboundarylayer.

For theconvectiveboundarylayerthesurfacetemperatureis augmentedwith thetemperatureexcessthatthether-

mals have with respect to their surroundings:

(125)

Oneof themainadvantagesof this schemeis that it is extremelyrobustwith respectto numericalstability. The

explicit calculationof thediffusioncoefficientsin thelocal ' -closurecanintroducestability problemsevenif the

solutionmethodof thediffusionequationis fully implicit (KalnayandKanamitsu1988;GirardandDelage1990;

Beljaars1991).The' -profilemethodis muchmorestableasit doesnotallow for oscillationsin theprofileof ex-

change coefficients.

A secondadvantageis that the non-localaspectsof this closureallow for the inclusion of entrainmentin the

scheme.However, in theformulationasproposedby TroenandMahrt, theentrainmentis highly dependenton the

details of the numerical formulation and on the vertical resolution.

A disadvantageof theschemeis thatit maybedifficult to developgeneralizationsfor cloudyandbaroclinicbound-

ary layers.

� ′θ′ ' H∂θ∂�------ γ θ–

, � ′ � ′ ' H∂ �∂�------ γ g–

,= =

γ θ V � ′θ′( )0

� s-------------------- , γ g V � ′ � ′( )0

� s-------------------- , V 8≈= =

� s

� s � *3 � 1 � *

3+( )1 3⁄

, V 8 ,= =

� *�θv----- � ′θ′v( )0 -

1 3⁄=

-� * c �h-( )2 ! -( )2+{ }

�θv-----

θv -( ) θvs–{ }------------------------------------------------------=

θvs θv V � ′θ′v( )0

� s---------------------+=



ECMWF, 2002

3.5 TKE-closure

Thesimplest“higher order” closureis thespecificationof thediffusioncoefficientswith a prognosticvariablefor

theturbulencekineticenergy (TKE). Many differentformulationshavebeenproposed;mostof themare“truncat-

ed” versionsof full 2ndorderclosureschemes.A goodoverview of thedifferentapproximationsis givenby Mellor

andMellor andYamada(1974).Implementationsin thecontext of forecastmodelsarediscussedby Delage(1974),

Miyakodaand Sirutis(1977),Manton(1983),Thierry andLacarrerre(1983),Smith(1984)andDtmenil (1987).

An example of such a closure is the one proposed byThierry and Lacarrerre (1983):

(126)

(127)

(128)

(129)

(130)

(131)

d$

d�------- � ′ � ′∂ �∂ �--------– � ′ � ′∂ !∂ �-------–�ρ0----- � ′ρ′ ∂

∂�------$

′ � ′ � ′ � ′ρ

------------+ – ε ,––=

' M Vfi[Zji $ 1 2⁄ , KH αH'lk , Km α m�' M ,= = =

ε V ε

$ 3 2⁄

Z ε-----------=

$′ � ′ � ′ � ′

ρ------------+ 'lm d

$d�-------–=

1Z ε---- 13 �------

V LE1

------------- 13 �------V LE2

-------------+ n

1n

2–V LE5

Z s------------+ +=

n1 = 1 1 V LE3

-3 �------+ ⁄

n2 = 1 1 Vfopm 4

�-----–

⁄ for�

0<

= 0 for�

0>

1Z s--- = 0 locally unstable

=�θ---

∂θ∂�------

1 2⁄ $ 1 2⁄– locally stable

V ε 0.125 , Vfi 0.5 , αH 1 , α m 1 ,= = = =

V LE1 15 , V LE2 5 , V LE3 0.005 , V LE4 1 , V LE5 1.5 ,= = = = =

1Zji-----

13 �------V LK1

------------- 13 �------V LK2

-------------+ n ′1 n ′2–

V LK5

Z s------------+ +=



ECMWF, 2002 51

Thepurposeof usingtheTKE equationis to have a prognosticequationfor theturbulentvelocity scale.A diag-

nosticequationis still neededfor thelengthscales.Moreadvancedmodelsincludeanequationfor thelengthscale

(Mellor andYamada1982)or for thedissipationrate(DeteringandEtling 1985,Beljaarset al. 1987,Duynkerke

and Driedonks 1987). Here we limit to diagnostic length scales.

Thebackgroundof theclosureis relatively simple:wereplacetheunknown termsby dimensionallycorrectexpres-

sionsin known modelvariables.Thenatureof theexpressionsis suchthat they representthephysicsof thecon-

sideredterms.Finally, adjustableconstantsare usedto obtain a good fit with data.In the exampleabove, the

diffusioncoefficientsarechosenproportionalto a turbulentvelocityscaleandaturbulentlengthscale.Bothscales

aresupposedto representthelargetransportingeddies,andthevelocityscaleis derivedfrom theturbulentkinetic

energy. Thelengthscaleis3 � nearthesurface,proportionalto - (boundarylayerheight)in neutralandunstable

PBL’s andlimited by theObukhov�

in very stableflow. Thediffusionof TKE is describedin thesameway as

diffusionof momentumandheati.e.down-gradient.Dissipation(transformationinto heatby viscousforces)takes

placeat the smallestturbulent scaleswhich arenot modelledat all. The conceptof energy cascadeis usedhere

which saysthattheenergy lossat largescalesis equalto thedissipationat smallscalesbecauseno energy canbe

lost in thedecayof turbulencefrom largescalesto smallscales.We thereforecanmodelthedissipationin terms

of largescalevelocity andlengthscales.Theamountof energy lost perunit of time on thecascadeby the large

eddiesis proportionalto$

divide by theeddyturn-over-time . This explainstheparametrizationof ε in

terms of large eddy characteristics.

TheTKE equationincludesadvectionterms,but theturbulencetermshaverelativelyshorttimescaleswhichmakes

thattheequationis in quasi-equilibriummostof thetime.To simplify theequation,theadvectiontermscanbene-

glected,whereasthetimedependencecanberetainedsimplyasa techniqueto “iterate” thesolutiontowardsequi-

librium. The main advantageof TKE closureis that it is the simplestclosurewith somenon-localturbulence

aspects.By meansof thediffusionof turbulencefrom theproductionareain themixedlayertowardsthecapping

inversion,thisclosureis capableof describingentrainment(althoughnotnecessarilyquantitatively correct).Apart

from thisadvantageit alsoenablestheintroductionof moreadvancedcloudparametrizationschemesSmith1984;

DuynkerkeandDriedonks1987).It is disappointing,however, thatexperiencesofaratECMWFhasindicatedthat

theschemeis moredemandingonverticalresolution.Fromtwo differentstudies(Dtmenil 1987;Manton1983)it

wasconcludedthattheTKE closurehashardlyadvantagesover thesimplerLouis (1979)closurewith theresolu-

tion thatis availablein thePBL with the19 level ECMWFmodel.A morefundamentalproblemwith higherorder

closureschemesis thattheseschemesdonotnecessarilyreproducethesurfacelayersimilarity laws.Thesesurface

layerlawsarealwaysneededto specifythesurfaceboundaryconditionat thelowestmodellevel. Thismayresult

in an inconsistency between the lowest model layer and the layers aloft.

4. LIST OF SYMBOLS

Transfer coefficients for momentum, heat and moisture

n ′1 = 1 1 V LK3-3 �------+

⁄

n ′2 = 1 1 V LK4

�-----–

⁄ for�

0<

= 0 for�

0>

V LK1 15 , V LK2 11 , V LK3 0.0025 , V LK4 1 , V LK5 3 .= = = = =

Z $ 1 2⁄⁄

V M V H V Q, ,



ECMWF, 2002

Neutral transfer coefficients for momentum, heat and moisture

Specific heat at constant pressure

Turbulence kinetic energy

Evaporation

Coriolis parameter

Functional dependence in general

Acceleration of gravity

Boundary layer height

Stability functions of Richardson number

Heat flux

Von Karman constant

Turbulent diffusion coefficients for momentum and heat

Turbulence length scale

Obukhov length

Latent heat of vaporization

Pressure

Surface pressure

Specific humidity of water vapour

Liquid water specific humidity

Total water specific humidity

Gas constant for dry air

Gas constant for water vapour

Richardson number

Bulk Richardson number

Dry static energy

Liquid water static energy

Virtual dry static energy

Temperature

Resolved or mean��q� and� velocities

Horizontal vector _r�q� ! ^Geostrophic wind components

Friction velocity

Total velocities in��s� and� directions

V MN V HN V QN, ,

�� $

`�f

�-\

M\

H,

]3

' M ' H,

Z��

vap

�� 0

�� l

� t��

vap� *� * b

�� l

� v

�� !t#, ,

U

� G ! G,

� *

� �u�, ,



ECMWF, 2002 53

Turbulent parts of the��q� and� velocities

Entrainment rate

Resolved vertical velocity

Convective velocity scale in the surface layer

Convective velocity scale in the outer layer

Horizontal coordinate (usually west–east)

Horizontal coordinate (usually north–south)

Vertical coordinate (height above the surface)

Dissipation of turbulence kinetic energy

Dimensionless wind and temperature gradients

Potential temperature gradient above mixed layer

Dimensionless stability corrections in wind and temp. profiles

Potential temperature

Liquid water potential temperature

Virtual potential temperature

Mixed layer temperature

Temperature jump at the top of the mixed layer

Turbulent temperature scale

Asymptotic turbulence length scale

Local Obukhov length

Standard deviation

Turbulent stress

Surface stress

Horizontal components of stress

Horizontal components of surface stress

Kinematic viscosity of air

Density

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� ′ � ′ � ′,,

� e

� L

� *s

� *

��

ε

ϕM φH,

γ

ΨM ΨH,

θ

θl

θv

θm

∆θm

θ*

λ

Λ

σ

τ

τ0

τ > τ?,

τ > 0 τ? 0,

ν

ρ



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Beljaars ECMWF PBL Parameterization