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5.3 planetary boundary layer momentum equations 123

−f!u − ug

"− ∂v′w′

∂z= 0 (5.19)

where (2.23) is used to express the pressure gradient force in terms of geostrophicvelocity.

5.3.1 Well-Mixed Boundary Layer

If a convective boundary layer is topped by a stable layer, turbulent mixing canlead to formation of a well-mixed layer. Such boundary layers occur commonlyover land during the day when surface heating is strong and over oceans when theair near the sea surface is colder than the surface water temperature. The tropicaloceans typically have boundary layers of this type.

In a well-mixed boundary layer, the wind speed and potential temperature arenearly independent of height, as shown schematically in Fig. 5.2, and to a firstapproximation it is possible to treat the layer as a slab in which the velocity andpotential temperature profiles are constant with height and turbulent fluxes varylinearly with height. For simplicity, we assume that the turbulence vanishes at thetop of the boundary layer. Observations indicate that the surface momentum fluxcan be represented by a bulk aerodynamic formula3

#u′w′

$

s= −Cd

%%V%% u, and

#v′w′

$

s= −Cd

%%V%% v

Fig. 5.2 Mean potential temperature, θ0, and mean zonal wind, U , profiles in a well-mixed boundarylayer. Adapted from Stull (1988).

3 The turbulent momentum flux is often represented in terms of an “eddy stress” by defining, forexample, τex = ρou′w′. We prefer to avoid this terminology to eliminate possible confusion withmolecular friction.

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第五章：量纲分析

聂绩

1 相似原理

下标 t表示实物，下标 m表示模型。

δl =ltlm

δv =vtvm

δt =tttm

δt =δlδv

δF = FtFm

δF = FtFm

= ρtτtatρmτmam

= δρδ3lδlδ2t

= δρδ2l δ2v

2 无量纲化

∂v∂t + v ·∇v = −1

ρ∇p+ gk + µρ∇

2v + F

LV T

∂v′∂t′ + v′ ·∇v′ = − P

ΠV 21ρ′∇p′ + L

V 2gk + µΠV L

1ρ∇

2v′ +!F

ΠV 2L2F ′

1

第五章：量纲分析

聂绩

1 相似原理

下标 t表示实物，下标 m表示模型。

δl =ltlm

δv =vtvm

δt =tttm

δt =δlδv

δF = FtFm

δF = FtFm

= ρtτtatρmτmam

= δρδ3lδlδ2t

= δρδ2l δ2v

2 无量纲化

∂v∂t + v ·∇v = −1

ρ∇p+ gk + µρ∇

2v + F

LV T

∂v′∂t′ + v′ ·∇v′ = − P

ΠV 21ρ′∇p′ + L

V 2gk + µΠV L

1ρ∇

2v′ +!F

ΠV 2L2F ′

1

2 ⽆量纲化 2

Sr = LV T

Eu = PΠV 2

Fr =!

V 2

gL

Re = ΠV Lµ

Ne ="F

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=( LV T )t

( LV T )m

= δlδvδt

= 1

EutEum

= δpδρδ2v

= 1

FrtFrm

= δ2vδlδg

= 1

RetRem

= δµδρδvδl

= 1

NetNem

= δFδρδ2vδ

2l= 1

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5.1 atmospheric turbulence 119

Noting that the mean velocity fields satisfy the continuity equation (5.5), we canrewrite (5.7), as

Du

Dt= Du

Dt+ ∂

∂x

!u′u′

"+ ∂

∂y

!u′v′

"+ ∂

∂z

!u′w′

"(5.8)

whereD

Dt= ∂

∂t+ u

∂

∂x+ v

∂

∂y+ w

∂

∂z

is the rate of change following the mean motion.The mean equations thus have the form

Du

Dt= − 1

ρ0

∂p

∂x+ f v −

#∂u′u′

∂x+ ∂u′v′

∂y+ ∂u′w′

∂z

$

+ Frx (5.9)

Dv

Dt= − 1

ρ0

∂p

∂y− f u −

#∂u′v′

∂x+ ∂v′v′

∂y+ ∂v′w′

∂z

$

+ Fry (5.10)

Dw

Dt= − 1

ρ0

∂p

∂z+ g

θ

θ0−

#∂u′w′

∂x+ ∂v′w′

∂y+ ∂w′w′

∂z

$

+ Frz (5.11)

Dθ

Dt= − w

dθ0

dz−

#∂u′θ ′

∂x+ ∂v′θ ′

∂y+ ∂w′θ ′

∂z

$

(5.12)

∂u

∂x+ ∂ v

∂y+ ∂w

∂z= 0 (5.13)

The various covariance terms in square brackets in (5.9)–(5.12) represent turbu-lent fluxes. For example, w′θ ′ is a vertical turbulent heat flux in kinematic form.Similarly w′u′ = u′w′ is a vertical turbulent flux of zonal momentum. For manyboundary layers the magnitudes of the turbulent flux divergence terms are of thesame order as the other terms in (5.9)–(5.12). In such cases, it is not possible toneglect the turbulent flux terms even when only the mean flow is of direct interest.Outside the boundary layer the turbulent fluxes are often sufficiently weak so thatthe terms in square brackets in (5.9)–(5.12) can be neglected in the analysis oflarge-scale flows. This assumption was implicitly made in Chapters 3 and 4.

The complete equations for the mean flow (5.9)–(5.13), unlike the equations forthe total flow (5.1)–(5.5), and the approximate equations of Chapters 3 and 4, arenot a closed set, as in addition to the five unknown mean variables u, v, w, θ , p,

there are unknown turbulent fluxes. To solve these equations, closure assump-tions must be made to approximate the unknown fluxes in terms of the fiveknown mean state variables. Away from regions with horizontal inhomogeneities


2.7 thermodynamics of the dry atmosphere 51

Taking the logarithm of (2.44) and differentiating, we find that

cpD ln θ

Dt= cp

D ln T

Dt− R

D ln p

Dt(2.45)

Comparing (2.43) and (2.45), we obtain

cpD ln θ

Dt= J

T= Ds

Dt(2.46)

Thus, for reversible processes, fractional potential temperature changes are indeedproportional to entropy changes. A parcel that conserves entropy following themotion must move along an isentropic (constant θ ) surface.

2.7.2 The Adiabatic Lapse Rate

A relationship between the lapse rate of temperature (i.e., the rate of decrease oftemperature with respect to height) and the rate of change of potential tempera-ture with respect to height can be obtained by taking the logarithm of (2.44) anddifferentiating with respect to height. Using the hydrostatic equation and the idealgas law to simplify the result gives

T

θ

∂θ

∂z= ∂T

∂z+ g

cp(2.47)

For an atmosphere in which the potential temperature is constant with respect toheight, the lapse rate is thus

−dT

dz= g

cp≡ #d (2.48)

Hence, the dry adiabatic lapse rate is approximately constant throughout the loweratmosphere.

2.7.3 Static Stability

If potential temperature is a function of height, the atmospheric lapse rate, # ≡−∂T/∂z, will differ from the adiabatic lapse rate and

T

θ

∂θ

∂z= #d − # (2.49)

If # < #d so that θ increases with height, an air parcel that undergoes an adiabaticdisplacement from its equilibrium level will be positively buoyant when displaced


5.1 atmospheric turbulence 119

Noting that the mean velocity fields satisfy the continuity equation (5.5), we canrewrite (5.7), as

Du

Dt= Du

Dt+ ∂

∂x

!u′u′

"+ ∂

∂y

!u′v′

"+ ∂

∂z

!u′w′

"(5.8)

whereD

Dt= ∂

∂t+ u

∂

∂x+ v

∂

∂y+ w

∂

∂z

is the rate of change following the mean motion.The mean equations thus have the form

Du

Dt= − 1

ρ0

∂p

∂x+ f v −

#∂u′u′

∂x+ ∂u′v′

∂y+ ∂u′w′

∂z

$

+ Frx (5.9)

Dv

Dt= − 1

ρ0

∂p

∂y− f u −

#∂u′v′

∂x+ ∂v′v′

∂y+ ∂v′w′

∂z

$

+ Fry (5.10)

Dw

Dt= − 1

ρ0

∂p

∂z+ g

θ

θ0−

#∂u′w′

∂x+ ∂v′w′

∂y+ ∂w′w′

∂z

$

+ Frz (5.11)

Dθ

Dt= − w

dθ0

dz−

#∂u′θ ′

∂x+ ∂v′θ ′

∂y+ ∂w′θ ′

∂z

$

(5.12)

∂u

∂x+ ∂ v

∂y+ ∂w

∂z= 0 (5.13)

The various covariance terms in square brackets in (5.9)–(5.12) represent turbu-lent fluxes. For example, w′θ ′ is a vertical turbulent heat flux in kinematic form.Similarly w′u′ = u′w′ is a vertical turbulent flux of zonal momentum. For manyboundary layers the magnitudes of the turbulent flux divergence terms are of thesame order as the other terms in (5.9)–(5.12). In such cases, it is not possible toneglect the turbulent flux terms even when only the mean flow is of direct interest.Outside the boundary layer the turbulent fluxes are often sufficiently weak so thatthe terms in square brackets in (5.9)–(5.12) can be neglected in the analysis oflarge-scale flows. This assumption was implicitly made in Chapters 3 and 4.

The complete equations for the mean flow (5.9)–(5.13), unlike the equations forthe total flow (5.1)–(5.5), and the approximate equations of Chapters 3 and 4, arenot a closed set, as in addition to the five unknown mean variables u, v, w, θ , p,

there are unknown turbulent fluxes. To solve these equations, closure assump-tions must be made to approximate the unknown fluxes in terms of the fiveknown mean state variables. Away from regions with horizontal inhomogeneities

��

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x¶

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-



Fig. 5.3 Balance of forces in the well-mixed planetary boundary layer: P designates the pressuregradient force, Co the Coriolis force, and FT the turbulent drag.

means is needed to determine the vertical dependence of the turbulent momentumflux divergence in terms of mean variables in order to obtain closed equations forthe boundary layer variables. The traditional approach to this closure problem isto assume that turbulent eddies act in a manner analogous to molecular diffusionso that the flux of a given field is proportional to the local gradient of the mean. Inthis case the turbulent flux terms in (5.18) and (5.19) are written as

u′w′ = − Km

!∂u

∂z

"; v′w′ = − Km

!∂ v

∂z

"

and the potential temperature flux can be written as

θ ′w′ = − Kh

!∂θ

∂z

"

where Km(m2s− 1) is the eddy viscosity coefficient and Khis the eddy diffusivityof heat. This closure scheme is often referred to as K theory.

The K theory has many limitations. Unlike the molecular viscosity coefficient,eddy viscosities depend on the flow rather than the physical properties of thefluid and must be determined empirically for each situation. The simplest modelshave assumed that the eddy exchange coefficient is constant throughout the flow.This approximation may be adequate for estimating the small-scale diffusion ofpassive tracers in the free atmosphere. However, it is a very poor approximationin the boundary layer where the scales and intensities of typical turbulent eddiesare strongly dependent on the distance to the surface as well as the static stability.Furthermore, in many cases the most energetic eddies have dimensions comparableto the boundary layer depth, and neither the momentum flux nor the heat flux isproportional to the local gradient of the mean. For example, in much of the mixedlayer, heat fluxes are positive even though the mean stratification may be very closeto neutral.





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∂z

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!∂ v

∂z

"


θ ′w′ = − Kh

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128 5 the planetary boundary layer

It can be shown that√

i = (1 + i) /√

2. Using this relationship and applying theboundary conditions of (5.28), we find that for the Northern Hemisphere (f > 0),A = 0 and B = −ug. Thus

u + iv = −ug exp!−γ (1 + i) z

"+ ug

where γ = (f/2Km)1/2.Applying the Euler formula exp(−iθ) = cos θ − isin θ and separating the real

from the imaginary part, we obtain for the Northern Hemisphere

u = ug

#1 − e−γ z cos γ z

$, v = uge−γ z sin γ z (5.31)

This solution is the famous Ekman spiral named for the Swedish oceanographerV. W. Ekman, who first derived an analogous solution for the surface wind driftcurrent in the ocean. The structure of the solution (5.31) is best illustrated by ahodograph as shown in Fig. 5.4, where the zonal and meridional components ofthe wind are plotted as functions of height. The heavy solid curve traced out onFig. 5.4 connects all the points corresponding to values of u and v in (5.31) forvalues of γ z increasing as one moves away from the origin along the spiral.Arrowsshow the velocities for various values of γ z marked at the arrow points. Whenz = π/γ , the wind is parallel to and nearly equal to the geostrophic value. It isconventional to designate this level as the top of the Ekman layer and to define thelayer depth as De ≡ π/γ .

Observations indicate that the wind approaches its geostrophic value at about1 km above the surface. Letting De = 1 km and f = 10−4 s−1, the definitionof γ implies that Km ≈ 5 m2 s−1. Referring back to (5.25) we see that for acharacteristic boundary layer velocity shear of |δV/δz| ∼ 5×10−3 s−1, this valueof Km implies a mixing length of about 30 m, which is small compared to the depthof the boundary layer, as it should be if the mixing length concept is to be useful.

Qualitatively the most striking feature of the Ekman layer solution is that, likethe mixed layer solution of Section 5.3.1, it has a boundary layer wind component

Fig. 5.4 Hodograph of wind components in the Ekman spiral solution. Arrows show velocity vectorsfor several levels in the Ekman layer, whereas the spiral curve traces out the velocity variationas a function of height. Points labeled on the spiral show the values of γ z, which is anondimensional measure of height.

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ji nie's group · january 27, 2004 9:4 elsevier/aid aid 5.3 planetary boundary layer momentum...

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