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Chapter 3

Boundary Layer

Ichiro TAMAGAWARiver Basin Research Center, Gifu University1-1 Yanagido, Gifu, 501-1193, JAPANPhone/Fax: +81-58-293-2430E-mail: [email protected]: http://tama.cive.gifu-u.ac.jp/˜tama/

3.1 Introduction

‘Boundary Layer’ is discussed in this chapter. By the way, what is ‘boundary’?Webster’s II dictionary explains that boundary is ‘something that marks a limitor border.’ In our context, ‘boundary’ is border part of the atmosphere. It is thelower border part of the atmosphere, because the other borders are undefinableor unclear. Here the ‘Boundary Layer’ is the lowest part of the atmosphere.Through it, momentum, heat, water vapor, and other materials, are exchangedbetween the earth surface and atmosphere. We often call it ‘atmospheric bound-ary layer’ to distinguish from the one in fluid mechanics.

3.1.1 Boundary layer in fluid mechanics

Some readers know the word ‘boundary layer’ appearing in in hydraulics, fluiddynamics or fluid mechanics. It is the same idea as the one in meteorology buttheir introducing way are sometimes different.

Fluid mechanics mainly treated water movement in old time, especially waterin pipe, channel, tank or river for engineering purpose. Water can be treatedas incompressible fluid, in which the density is not changed by pressure. Themovement can be described by the Navier-Stokes equation:

∂u∂t

+ (u · ∇)u = −1ρ∇p + f + ν∇2u (3.1)

2 CHAPTER 3. BOUNDARY LAYER

Here the letters are follows:

u: velocity vector ∇ :(

∂∂x , ∂

∂y , ∂∂z

)

ρ : density p : pressuref : force to unit mass .

The last term of Eq.3.1 shows viscosity. In many cases, the viscosity term isnot important except the region very near boundary such as wall. The region isboundary layer. In the other part, the viscosity term can be neglected. Therethe powerful potential theory can be applied to solve the flow so often.

When the Reynolds number of the flow is large enough, that is, the speedof the flow is large and/or the spatial scale of the flow is large, the state ofthe flow becomes turbulence. The turbulent flow shows irregular and no steadymotion. The turbulent motion plays similar effect as viscosity to the mean flow.It transports momentum and smoothes the flow pattern. Even in such a case,the equation of mean flow can be expressed in similar form as

∂u∂t

+ (u · ∇)u = −1ρ∇p + f + νt∇2u, (3.2)

where νt is turbulent viscosity coefficient. The νt is much larger than ν and isnot constant but the function of the flow itself. But the form of Eq.3.2 is sosame as Eq.3.1 that the boundary layer is also formed along the wall.

In boundary layer, the following approximations are applied with good ac-curacy.

• The flow (velocity vector) is parallel to the surface.

• The difference of the velocity (gradient of the velocity) exist only thedirection normal to the surface.

• The flow is almost steady.

Those can be understand from the thinness of the boundary layer. The flowin the thin layer along the surface can not avoid to go along the surface. Itsvelocity varys from zero at surface into some value at the most outer position ofthe thin layer. Therefore the gradient normal to the surface is large enough toneglect those in other two directions. Thin layer quickly reaches the equivalentstate to the outer flow due to small mass in it.

3.1.2 Boundary layer in Meteorology

motion in the atmosphere

In meteorology, we consider ‘air’ as the fluid. The air is compressible fluid. Itsdensity varys both by pressure and temperature. And also air is mixed gasso that its density can achage by the amount of water vapor1 In meteorologicalcondition, the following three features about compressibility are enough usually.

1The mass of water molecule is lighter than nitrogen or oxygen.

3.1. INTRODUCTION 3

The one is the density change due to the vertical pressure change of hydro-static balance, as the readers know the the air density decreases as the heightbecome high. The second is the change due to the temperature change. Thehot air is lighter than cold air, as the readers know. And the last is due to watervapor content.

Other than the density change, the effect of the earth rotation, which iscalled Colioris effect, must be included in the equation of motion especially inlarge scale motion. The Reynolds number of the atmospheric boundary layerflow is very large. The equation of motion must include the effect of turbulenceas mentioned later. Turbulent motion of the air has important roll especially inboundary layer. That is almost same as the definition of boundary layer.

Thermal stratification

First let us consider the density change. Meteorologists prefer to use ‘virtualpotential temperature’ than density itself. It can treat the three kinds of densitychange together. Start the discussion from dry air. The potential temperatureis defined as the temperature that the air will have after moving to the referencepressure level (1000hPa) without heat exchanging between the moving air andits environment2. The potential temperature is conserved in kinetic process.The ‘no-heat exchange’ is good approximation to the motion in atmosphere.

The equation showing the potential temperature (θ) of the air with pressurep (hPa) and temperature T (K) is

θ = T

(p0

p

)R/Cp

, (3.3)

where p0 is the reference pressure (1000hPa), R is the gas constant of theair (287 N m kg−1K−1) and Cp is the specific heat at constant pressure (1005J kg−1K−1).

To add the effect of water vapor, the virtual temperature Tv is used insteadof T . Tv is defined as

Tv = T

{1 +

(1ε− 1

)q

}∼ T (1 + 0.61q), (3.4)

where ε is the ratio of the molecular mass of water vapor to that of dry air(0.622) and q is specific humidity(kg kg−1), which is the ratio of water vapormass to the total air mass including water vapor in same volume. Tv is thetemperature that the dry air of the same density as the air considered wouldbe have under same pressure. When Tv is used in Eq.3.3 instead of T for moistair,

θv = Tv

(p0

p

)R/Cp

= T (1 + 0.61q)(

p0

p

)R/Cp

(3.5)

is obtained.2It is called as ‘adiabatic condition’


θv is a very good indicator of the density of moving air. Using it, air densitiesat different altitude can be compared directly. If the θv of lower air is smallerthan that of upper air, the lower air would be heavier than the upper one aftermoving to same altitude. In such a case, the vertical motion of air is verysuppressed, because the rising air becomes heavier than the surrounding airand feels negative buoyancy (downward force). And descending air feels positivebuoyancy (upward force) in the same way. Therefore vertical motion shows waveform and can not grow so much. This layer with this thermal stratification iscalled as stable layer. And if θv of lower air is larger than the upper, the verticalmotion is enhanced. The rising air feels positive buoyancy and rises more anddescending air feels negative buoyancy and so on. The vertical motion becomesactive convection. That layer is called unstable layer. Table 3.1 shows the aboverelationship. Those stability condition is mentioned as thermal stratification.

stability motion∂θv

∂z > 0 stable wave, inactive turbulence∂θv

∂z = 0 neutral∂θv

∂z < 0 unstable convection, active turbulence

Table 3.1: thermal stratification and stability

The stability of the atmosphere is stable almost everywhere, because theconvection in the unstable layers transports heat and water vapor to reduceunstability to make unstable the layer into neutral or stable layer. The tem-perature of atmosphere usually decreases as altitude becomes large, but thepotential temperature increases. The boundary layer near land surface in day-time is a rare sample of unstable layer.

3.2 Structure

Atmospheric boundary layer changes its depth and characteristics as time goesby. The heat from surface has large effect to its character through the modifi-cation of the thermal stratification. The detail of heat exchange at the surfacewill be discussed later. The atmosphere is cooled or heated from the bottom,according to the surface cooling (in nighttime) or heating (in daytime). Theabove is typical for the boundary layer in fine day over land. The water vaporand low altitude clouds may play main roll over the ocean, because of the weakheating/cooling and strong water vapor supply from the surface. Strong cloudconvection like cumulonimbus causes strong wind and turbulence into the layernear surface. In such a case, boundary layer can not be distinguished with cloudsystem. Although above, here the boundary layer in fine day over land will bediscussed for simplicity. It will provide the basic physical idea for understandingactual boundary layer.

If the one dimensional boundary layer is considered, that is, the horizontalinhomogeneity can be neglected, the conceptual boundary layer structure over

3.2. STRUCTURE 5

land is following:

• the surface layer lays near the surface.

• nocturnal boundary layer lays over the surface layer in nighttime.

• mixed layer lay over surface layer in daytime.

• mixed layer become residual layer in nighttime.

The schematic picture is Fig.3.1.

Convective Mixed Layer

Residual Layer

Noctarnal Layer

surface layer

surface layer

interfacial layer

sun rise sunset

Free atmosphere

(neutral)

(unstable)(stable)

(stable)

(very stable)

about 1 km

Time of day

Height

Figure 3.1: Schematic picture of one dimensional boundary layer structure. Thisfigure is downgraded one of the similar figures in Stull 1988 and Garratt 1992.

surface layer

The air near surface, typically under several tens meter above the ground, thevertical motion of air is suppressed by the surface and the friction by the surfacecauses large decrease of wind speed near surface. Small scale turbulence, mainlycaused by wind speed vertical difference (called as wind shear) is dominant inthe flow. Thermal stratification modifys the turbulence by buoyancy effect. Thesurface layer will be appeared in the later section.

mixed layer

In daytime, the surface is heated by the solar radiation. The heat through thesurface layer increases the θv of the upper part of the boundary layer from itsbottom side. Then the stability of the surface layer becomes super adiabatic(absolutely unstable) and that of the upper boundary layer become neutral by


the active convection in the layer. The latter is called as a convective boundarylayer, which is characterized as well mixed layer.

Heated air at bottom moves upward but it will meets the warmer (or lighter)air at some altitude because upper atmosphere has higher θv. That limits con-vective motion under some height. The air goes even more higher altitude thanthe equilibrium level using its inertia and entrains the upper air of outer re-gion into the convective boundary layer. The entrained air will be mixed wellin the boundary layer. Then the sharp difference between the boundary layerand the atmosphere outside is formed at top of the convective boundary layer.This is called an interfacial layer, which makes upper edge of the convectiveboundary layer. The thermal stratification of it is very stable. It is sometimescalled capping inversion.3 It spreads out horizontally with undulating form likehummocks as updrafts under it.

The buoyancy energy supply from the bottom and also the upper entrain-ment contribute the growth of the convective boundary layer. In the convectiveboundary layer, the air inside is well mixed by the convection. The slab model,in which u, θv and other variables are constant, is often used in simple theoret-ical discussion. The well mixed feature is sometimes visualized by the behaviorof the smoke from a tall chimney.

nocturnal layer and residual layer

In nighttime in fine day, the air cooled from the bottom. θv decreases fromthe bottom, then the thermal stability goes to stable. The turbulence and itsmixing become to be suppressed. Wind in the layer becomes weak by decreasingthe momentum transfer from the upper air. Lower air becomes cooler and morehumid, because of deducing heat and water vapor transfer into upper layer.

Wave motion can be seen with intermittent turbulence. Pure wave motiondoes not transfer heat, momentum and so on. The vertical moved air returns toits height without mixing. Sporadic occurring unstability may contribute theirtransfer process. The transfer process remains unclear until now. And also,radiative heat transfer process increases its importance with weak turbulenttransfer.

The nocturnal layer decouples the surface below and the air above by itsweak mixing due to stable stratification. Sometime it contributes nocturnal jetor drainage flow on the slope.

The former mixed layer air decrease the turbulent energy after decreasing orstopping buoyancy supply from surface. The layer is called as a residual layer.It has the caps on top and bottom. It is characterized as a decaying turbulentmotion.

3.3. TURBULENT TRANSPORT 7

potential temp. 2002 Nov. 17 at Kinomoto

297 296 295 294 293 292 291 290 289 288 287 286 285 284 283 282 281 280 279

0 5 10 15 20

Hour (JST)

0

500

1000

1500

2000

2500

3000

3500

Figure 3.2: Potential temperature distribution measured at Kinomoto, Shigaprefecture, Japan in 17 November 2002 by HyARC, Nagoya university duringthe intensive observation 2002 of Lake Biwa project.

3.2.1 Observed Data

Fig. 3.2 is an observed potential temperature change at Kinomoto near LakeBiwa, Japan during intensive observation 2002 of Lake Biwa project. Ra-diosonde by HyARC of Nagoya university sounded it. The weather was fineat the observed day. The above mentioned features can be seen. The conceptskeep useful to understand it. Please compare Fig.3.2 with Fig.3.1. The pointsare:

• Nocturnal layer formed after 16 JST, which is under a few hundreds meter.

• Convective boundary layer formed after 10 JST, which reached about700m,

• while unstable stratified surface layer formed under about 200m.

3.3 Turbulent transport

Except very stable cases, the flow in the boundary layer is turbulent. Let thevariables such as u, w, T , q, divide into the mean and fluctuation (turbulence)as

x = x + x′

3‘inversion’ is used to mention about the state that temperature (not θv) increases asaltitude become large. The situation is inverse to usual case.


, where x represents u, w, and so on. In the discussion of surface layer, uis usually set into the mean wind direction for simplicity. The x shows somekind of average of x. The way of average should be defined appropriately, suchas ensemble averaging, time averaging, and space averaging, according to itscontext. x′ is zero in usual definitions.

The averaged form of the second term of Eq.3.1 become

(u · ∇)u = (u · ∇)u +

∇ · u′u′∇ · u′v′∇ · u′w′

(3.6)

The equation of continuity is used to deduce Eq.3.6 and u, v, w are the com-ponents of u. The term u′u′, u′v′, u′w′ and so on are called as Reynolds stressterms, which are the direct expressions of eddy viscosity or turbulent viscosity.They show the contribution of turbulence to mean flow by transporting themomentum. The convergence of them accelerates the mean velocity. And samekinds of turbulent transports exist for scaler variables such as heat, water vapor,CO2, and so on.

For example, the averaged equation of potential temperature can be deducedfrom the equation of conservation. The conservation equation is

∂θ

∂t+ (u · ∇) θ = S (3.7)

where S shows the effect of non adiabatic heating, such as molecular diffusion,the convergence of radiation flux, heat conduction from surface, and so on. Thenthe both side of equation is averaged into

∂θ

∂t+ (u · ∇) θ = −∇θ′u′ + Sθ (3.8)

Also that for specific humidity q is

∂q

∂t+ (u · ∇) q = −∇q′u′ + Sq (3.9)

where Sq shows source for water vapor such as the evaporating rate of the waterat the point.

The turbulent motion is characterized by its irregularity. Only the statisticalfeatures are required to be known for many purpose. They are described in thenext section in very simple surface layer.

3.3.1 surface layer turbulence

The discussion here is limited only for a very simple case. The followings areassumed.

• The surface is so homogenous to be able to ignore the horizontal variationof every variables in the surface layer.

3.3. TURBULENT TRANSPORT 9

• The surface layer reaches equilibrium to environment so quickly that ev-erytime it is in equilibrium state and its time change can be ignored.

• There are no convergence of radiation, no phase change of water or nofloating obstacle (rain drop, snow flake, sand, and so on) in the surfacelayer.

• Colioris force can be ignored after the comparison with other terms.

Fri Sep 4 14:00:00 1998

U (m/s)

0 500 1000 1500

1.5

3.5

5.5

7.5

1.5

3.5

5.5

7.5

V (m/s)

-2.6

-0.6

1.4

3.4

-2.6

-0.6

1.4

3.4

W (m/s)

-0.8

0.2

1.2

-0.8

0.2

1.2

T (deg)

4.8

5.8

6.8

7.8

4.8

5.8

6.8

7.8

q (kg/kg)

4.0e-035.0e-036.0e-037.0e-038.0e-03

4.0e-035.0e-036.0e-037.0e-038.0e-03

0 500 1000 1500

z/L (moist) -6343.62 z/L (dry) -6276.52

Wind Dir 34.8Means: U W T q RH 3.107 0.105 4.97 0.00457 49.0Tau(Pa) H(W/m2) LE(W/m2)-0.00006 78.4 145.6 sigma_w/u* sigma_T/T* sigma_q/q* 38.804 -0.054 -0.062

Figure 3.3: Sample of turbulence measurement and its results. Data weretaken over Tibetan plateau in China during intensive observation period ofGAME/Tibet 1998. The values appearing are with explanation in the figure.‘sigma’ shows the standard deviation.

The equations of mean state can be reduced into

∂u′w′

∂z= 0 (3.10)


∂w′θ′

∂z= 0 (3.11)

∂w′q′

∂z= 0 (3.12)

The above equations show that u′w′, w′θ′ and w′q′ are constant in the surfacelayer. The x′w′(x = u, θ or q) shows the amount of x transported by turbulenceper unit time and per unit area of horizontal plane. This kind of amount is calledflux. Eqs.3.10 – 3.12 show that the flux is constant in the surface layer. Theassumption used here might be thought to be oversimplification, but the resultswith constant flux assumption are good approximation for the fluxes over flatterrains. The flux from (or into) surface passes through surface layer with smallconsumption into (or from) upper air. The fluxes in Eqs.3.10 – 3.12 have therelationships to shearing stress(τ), heat flux(H) or surface evaporation(E) as

τ = −ρu′w′ (3.13)

H = ρCp

(p

p0

)R/Cp

w′θ′ (3.14)

E = ρw′q′ (3.15)

where p is pressure at the surface and turbulent fluctuations of p and ρ areignored. The reader should remember that the heat H causes H/(ρCp) temper-ature change at constant pressure condition to understand the factor observedin Eq.3.14. τ , H and E should be mentioned to be averaged value.

The turbulent fluctuations (u′, w′, θ′ and q′) can be measured with fastresponse sensors. Sonic anemometer-thermometer is a standard sensor used tomeasure wind velocity and temperature. The measurement is based on the factthat the traveling speed of sonic wave is a sum of that in static air and windvelocity. It was developed during late 1960s and now it is very stable and ac-curate sensors manufactured by several companies. For humidity fluctuation,the optical sensor using the absorption of water vapor in infrared or ultravioletband is used. It needed much care about their unstability, but recently somegood infrared sensors was started to use. The measured turbulent fluctuationsare used to evaluate surface fluxes with Eqs.3.13–3.15. This technique is calledas eddy covariance (or correlation) method. It is thought to be reliable becauseit is a direct measurement. Fot that purpose, it is usually done with about10Hz data sampling and about 30 minutes averaging. The duration of 30 min-utes is supported by the spectral gap around 1 hour, but not everytime andeverywhere. The goodness of the flux estimation depends on the applicabilityof the assumptions mentioned before into the measurement condition. Fig.3.3is a sample of turbulence measurement and the fluxes by eddy covariance tech-nique. The data were taken at Amdo over Tibetan plateau under GAME/Tibetproject.

Recently, the eddy covariance observation comes to be popular even to beapplied over forest or city buildings. In such a case, sometimes the correction isrequired due to its departure from ideal horizontal homogeneous and statistically

3.4. LOGARITHMIC LAW AND MONIN-OBUKHOV SIMILARITY 11

steady condition. The discussion based on the integrate equation of Eqs.3.8 and3.9 or so is necessary in such a case.

3.4 Logarithmic law and Monin-Obukhov simi-larity

There are many statistical theory about turbulence. A few of them are intro-duced below.

In the surface layer, the flux (u′w′, w′T ′ and so on) is important, because itis constant and mean states are balanced with it.

The neutral surface layer is simple due to no buoyancy. In this case, the ver-tical distribution of mean wind speed is known to be expressed with logarithmicfunction of the hight from the surface. This is explained as follows.

The magnitude of u′w′ is estimated, by thinking that the fluctuations arecaused by eddy mixing of the profile of mean wind as follows:

u′w′ ∼ |u′||w′| (3.16)

|u′|,|w′| ∼ `∂u

∂z(3.17)

where ` shows the representative size of eddy, that is representative length scaleof turbulent mixing. ` must be affected by the distance to the surface. So ` ∝ z.Then

|u′w′| = −u′w′ =(

kz∂u

∂z

)2

(3.18)

where k is a constant, which is called as ‘von Karman’ constant. The valueof k is about 0.4 by experiments. The scaling parameter of turbulent velocity

u∗ =√|u′w′| is often used, which is called as friction velocity. Using u∗, Eq.3.18

becomes

u∗ = kz∂u

∂z(3.19)

u =u∗k

log(

z

z0

)(3.20)

Here z0 is an integral constant called as roughness length. z0 showsthe height at which the logarithmic wind would become zero, if logarith-mic law could be applied. z0 is usually treated as a constant whichis determined by surface condition. Kondo 1994 shows z0 as follows:

z0(m) z0(m)bare soil 10−4 water (2ms−1 wind) 0.27×10−4grass land (0.1m∼1m) 0.01∼0.15 forest 0.3 ∼ 1rural town 0.2∼0.5 city 1 ∼ 5

For the thermally stratified surface layer, that is for stable or unstable layer,the correction to the buoyancy effect is necessary. The turbulent kinetic energy


equation is used to obtain adequate stability parameter for describing turbulentstatistics. The turbulent kinetic energy equation reduced by the surface layerassumption used here is as follows:

∂e

∂t= −u′w′

∂u

∂z+

g

θv

w′θ′v −∂w′e∂z

− 1ρ

∂p′w′

∂z− (viscous dissipation) (3.21)

Here e shows turbulent kinetic energy (e = u′2+v′2+w′22 ). The right side should

be zero due to the steadiness of the surface layer. The first term of the leftside of the equation shows turbulent production by wind shear. And the secondterm does production or destruction by buoyancy. The third term and fourthterm show turbulent transport of e and redistribution of e, respectively. Forsttwo terms are important to describe turbulent statistics. The second termvanishes in neutral layer, that is the logarithmic law theory considers the firstone only. The second term adds energy production in unstable condition andthe produced energy is consumed by viscous dissipation. In stable condition, thebuoyancy term also reduces turbulent kinetic energy produced by the first term.Therefore the ratio of the two term can be a good non-dimensional parameter.The following parameters are used to evaluate the ratio and used as a stabilityparameter.

Ri =g

θv

∂θv

∂z∂u∂z

(3.22)

z

L=

kz q

θvθv∗

u∗2 (3.23)

where θv∗ = −w′θ′v/u∗. Ri is called Richardson number, and z/L is calledMonin-Obukhov stability.

Ri can be evaluated only by mean variables. It is convenient to use tocompute turbulent statistics from mean values. But z/L includes turbulentparameters u∗ and θ∗. Nevertheless, z/L is widely used in describing turbulentcharacteristics with u∗, θ∗ and other scaling parameters. Both Ri and a/Lbecome negative value in unstable condition, zero in neutral and positive instable condition.

The logarithmic law is expanded by including stability parameter Ri orz/L, which is found by Monin and Obukhov during 1940s and 1950s. The re-lationship was experimentally confirmed during late 1960s and 1970s by severalintensive observations. To mentioning Monin-Obukhov similarity theory, thenon-dimensional shear functions are introduced.

Eq.3.19 can be arranged as

kz

u∗∂u

∂z= 1 (3.24)

Eq.3.24 can be read as follows: the shear of the mean flow can be non-dimensionalized by u∗ and z. k is defined as the value of the equation becomes


1. Then Eq.3.24 can be expanded as

kz

u∗∂u

∂z= φm

( z

L

)(3.25)

where φm is a non-dimensional function of stability, called as a non-dimensionalshear function of wind speed4.

In the same way, the shear of θ, q can be expressed as

kz

θ∗∂θ

∂z= φh

( z

L

)(3.26)

kz

q∗∂q

∂z= φe

( z

L

)(3.27)

where θ∗ and q∗ are the scaling parameters for the turbulent fluctuations ofpotential temperature and of specific humidity, defined θ∗ = w′θ′/u∗ and q∗ =w′q′/u∗. The subscripts m, h e are the initial letters of momentum, heat andevaporation respectively. The functional forms and parameters of φm, φh and φe

can not determined only by theory. They need to be determined by experiments.The form in Eqs.3.28 and 3.29 is so-called Businger-Dyer type expression.

The constants are slightly different among observations. Here the values aretaken from the summary of the textbook by Garratt (Garratt 1992).

φm =

{1 + 5 z

L

(0 ≤ z

L < 1)

(1− 16 z

L

)1/4 (−5 < zL < 0

) (3.28)

φh =

{1 + 5 z

L

(0 ≤ z

L < 1)

(1− 16 z

L

)1/2 (−5 < zL < 0

) (3.29)

for q and other scaler variables, the form of non-dimensional shear function issaid to be same, but the observational results are not so many. In many case,the assumptions (horizontal homogeneity and steadiness) used here are not wellapplicable. For very unstable condition ( z

l ¿ −1), energy of turbulence issupplied only from buoyancy force. Therefore u∗ is no more good parameter tonon-dimensional shear functions. In such a case, dimensional analysis leads thefollowing relationships: φm ∝ (

zL

) 13 and φh ∝

(zL

)− 13 . Kadar and Yaglom 1989

recommended as follows:

φm = 0.4( z

L

) 13

(3.30)

φh = 0.5( z

L

)− 13

(3.31)

and H is directly related to the temperature difference as

H = ρCpAh

(θ(z)− θ(surface)

)1+a(3.32)

4This kind of discussion is called as dimensional analysis. It is widely used in treatingcomplicated phenomena.


as the parameter in Eq.3.32, Kondo 1994 said (Ah, a) = (1.2 × 10−3, 1/3) forz = 1m, and Tamagawa 1996 said (2.7 × 10−3, 0.4) for z = 20m. Accordingto the experiment in laboratory on heat transfer, the following relationship isobtained

Nu = ARa1/3∼0.4 (3.33)

Nu and Ra are Nusselt number and Rayleigh number respectively. Detailsshould be referred to the textbook about heat transfer mechanics. In any way,both values above are consistent this relationship. This kind of care on freeconvection is sometimes necessary to estimate H under strong heating.

Here the relationships between mean values and turbulence with Monin-Obukhov similarity are described. Integrating Eqs.3.25–3.26 with constant fluxapproximation, the difference between two altitude can be expressed as

u(z2)− u(z1) =∫ z2

z1

φm

( z

L

)dz

=u∗kz

{log

(z2

z1

)+ ψm

(z2

L

)− ψm

(z1

L

)}(3.34)

θ(z2)− θ(z1) =∫ z2

z1

φh

( z

L

)dz

=θ∗kz

{log

(z2

z1

)+ ψh

(z2

L

)− ψh

(z1

L

)}(3.35)

q(z2)− q(z1) =∫ z2

z1

φe

( z

L

)dz

=q∗kz

{log

(z2

z1

)+ ψe

(z2

L

)− ψe

(z1

L

)}(3.36)

where ψm, ψh and ψe are defined as ψx =∫

φx−1zL

d(

zL

)(x shows m, h or e). Or

integrating from parametric heights showing surface like z0 in Eq.3.20 to heightz,

u(z) =∫ z

z0

φm

( z

L

)dz

=u∗kz

{log

(z

z0

)+ ψm

( z

L

)− ψm

(z0

L

)}(3.37)

θ(z)− θ(surface) =∫ z

zT

φh

( z

L

)dz

=θ∗kz

{log

(z

zT

)+ ψh

( z

L

)− ψh

(zT

L

)}(3.38)

where zT shows the height that θ would be the surface value with supposingnon-dimensional shear function could be applicable. But for water vapor, thesurface value can not be defined except on the water. The β showing water


availability or evaporation efficiency is introduced as

β(q(z)− qs0

)=

∫ z

zq

φe

( z

L

)dz

=q∗kz

{log

(z

zq

)+ ψe

( z

L

)− ψe

(zq

L

)}(3.39)

where qs0 shows the saturation specific humidity at surface mean temperature.In Eqs.3.34–3.39, ψm,h,e (z/z0,T,q) set to be zero and zq = zT usually. The β setto be 1 on the water surface and 0 on non-evaporating surface.

Using Eqs.3.34 – 3.36 or Eqs.3.37 – 3.39, u∗, θ∗ and q∗ can be estimated fromthe difference of mean wind speed, potential temperature and specific humidityat two observation level or from one observation and adequate parameters.

For example, if the observed mean values at z1 and z2 are ob-tained, the tentative values of u∗, θ∗, q∗ can be obtained by assumingz/L = 0 in Eq.3.34–3.36. And z/L can be calculated by using theobtained u∗ and so on using Eq.3.23. And then use Eq.3.34–3.36again but with obtained z/L to get refined u∗ etc. The parametersu∗, θ∗, q∗ become to show no change after several iteration.

u∗, θ∗ and q∗ can be easily converted into shearing stress τ , sensible heatflux H and water vapor flux E by

τ = ρu∗2 (3.40)

H = −ρCp

(p

p0

)R/Cp

u∗ θ∗ (3.41)

E = −ρu∗ q∗ (3.42)

Therefore the relationship between the profiles and fluxes are obtained. Theobservation of profiles of wind, temperature and humidity can be used to esti-mate heat balance at the surface mentioned below. But the required accuracy ofthe observation is very high. The small errors that is not a problem to measurethe variables themselves often cause serious errors to obtain the differences inEqs.3.34–3.36. The careful calibration of the sensors is required. The observerhad better to remind the fact that the direct measurement of Eqs.3.13–3.15sometimes becomes easier, especially for H, although the cost of turbulent mea-suring sensors are much higher than usual sensors for mean values.

3.4.1 surface heat budget

On the idealized plane surface, the energy comes into surface must go upwardor downward because the surface can not store energy due to zero depth. Insuch a case, heat budget equation:

(1− α)Qs + Ql ↓ −Ql ↑= H + λE + G (3.43)


is applied to the surface. Here, Qs is the short wave (visible) radiation fluxdownward, Ql ↓ and Ql ↑ are the infrared radiation fluxes coming into and goingup from the ground respectively, α is refractivity of the surface (albedo), λ is thelatent heat of vaporization of water and G is heat flux into ground. The left sideof equation is often called net radiation (Qn = (1 − α)Qs + Ql ↓ −Ql ↑). Thisequation gives boundary conditions to energy equations both for atmosphereand ground. For example, the stability of boundary layer is affected by surfacecondition (α, z0 and so on) by limiting available energy through Eq.3.43. Themeteorological simulation model includes the same kind of equation.

In unideal case such as on city or forest, the surface can hardly set to bethought and one more layer called as canopy layer is considered in modeling.The heat storage term of the canopy should be added to Eq.3.43 to expand.Recent numerical models treats the complicated surface condition as a layer orseveral layers called as canopy layer or surface model.

3.5 For more study

The very rough sketch of boundary layer and a little detail of surface bound-ary layer are described above. The textbooks listed below are recommendedtextbooks to study boundary layer more and also reference of this chapter.

R. S. Stull An introduction to boundary layer meteorology, Kluwer AcademicPublishers, 1988

J. R. Garratt The atmospheric boundary layer, Cambridge university press,1992

The aboves are comprehensive textbooks of atmospheric boundary layer. Manyobservational results and discussions are found in them.

R. G. Fleagle and J. A. Businger An introduction to atmospheric physics(2nd ed.), Academic press, 1980

This is a textbook for whole atmospheric physics but the discussion about sur-face layer and observational instruments are very educative.

J. C. Kaimal and J. J. Finnigan Atmospheric boundary layer flows, Ox-ford university press, 1994

This book has detail discussion surface layer flow and measurement techniquesand instruments. It is good for surface layer observational researcher.

Other references

J. Kondo 水環境の気象学 (Meteorology of water environment), Asakura corp.1994 (in Japanese).

3.5. FOR MORE STUDY 17

I. Tamagawa Turbulent Characteristics and Bulk Transfer Coefficients overthe Desert in the HEIFE Area, Boundary-Layer Meteorology, 77, pp.1–20, 1996.

B. A. Kadar and A. M. Yaglom Mean fields and fluctuation moments inunstable stratified turbulent boundary layers, Journal of fluid mechanics,212, pp.637–662, 1990.