power-law velocity profile in a turbulent ekman layer on a transitional rough surface

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETYQ. J. R. Meteorol. Soc. 134: 1113–1125 (2008)Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/qj.285

Power-law velocity profile in a turbulent Ekman layeron a transitional rough surface

Noor Afzal*Aligarh Muslim University, India

ABSTRACT: A two-layer asymptotic theory of mean momentum in a turbulent Ekman layer without any closure model(such as eddy viscosity, mixing length, or k − ε) for large Rossby numbers is proposed. The flow in the inner walllayer and the outer wake layer are matched, using the Izakson–Millikan–Kolmogorov hypothesis; this leads to an openfunctional equation. Another open functional equation is obtained from the ratio of two successive derivatives of the basicfunctional equation; this admits two functional solutions, with power-law and log-law velocity profiles. The envelope ofthe geostrophic-drag power law leads to the log law, and determines the power-law index and prefactor as a function of thesurface Rossby number or the drag coefficient. The log laws and power laws for velocity and friction velocity, includingthe power-law constants, are universal, and independent of the wall roughness. This universality is well supported byextensive experimental and laboratory data. In traditional smooth-wall variables, there is no universality of scalings, anddifferent expressions are needed for different types of roughness. Approximate solutions of the power-law geostrophicdrag and cross-isobaric angle are also obtained. The power-law geostrophic-drag solution for each prescribed value of thepower-law index is valid for a limited domain of Rossby numbers. Copyright 2008 Royal Meteorological Society

KEY WORDS power-law and log-law velocity profiles; Rossby-number similarity theory; neutral barotropic planetaryboundary layer; Izakson–Millikan–Kolmogorov hypothesis

Received 8 September 2007; Revised 6 May 2008; Accepted 5 June 2008

1. Introduction

The pioneering paper of Prandtl and Tollmien (1924),recognizing the inadequacy of the assumption of aconstant eddy viscosity K to describe observed windstructure in a planetary boundary layer (PBL), allowed K

to vary with height. Their model for the PBL, consistingof a single layer with K = czm, where

c = 0.0895Ro−0.12z1−m0 G

and m = 0.843, is based on neutral flow in the constant-stress boundary layer of a pipe. Here Ro = G/f z0,where G is the geostrophic-wind velocity (for barotropicflow, the geostrophic wind is constant with height), f

is the Coriolis parameter (positive north of the Equa-tor), and z0 is the surface roughness length. The dimen-sionless number Ro is called the ‘surface Rossby num-ber’. From Rossby-number similarity theory, the fric-tion velocity u∗, geostrophic resistance Cg = u∗/G andcross-isobaric angle γ can be determined. Prandtl andTollmien (1924) were the first to propose an expressionfor the geostrophic-drag coefficient Cg as a function ofRo, namely Cg = 0.148Ro−0.119. From dimensional anal-ysis, Swimbank (1974) proposed the power law, and esti-mated the values of the constants, in geostrophic drag as

* Correspondence to: Noor Afzal, Faculty of Engineering and Technol-ogy, Aligarh Muslim University, Aligarh 202 02, India.E-mail: [email protected]

Cg = 0.111Ro−0.07 or Cg = 0.113Ro−1/14 for the range105 < Ro < 109.

Rossby and Montgomery (1935) provided an importantinsight into the nature of the PBL by dividing it into twolayers. With this model, they obtained the resistance lawexpressing the geostrophic-drag coefficient and the cross-isobaric angle in terms of the surface Rossby number Ro

(or Ro∗). Kazanski and Monin (1961) derived preciselythe same law from similarity theory. Further develop-ment of the theory proceeded as follows: Zilitinkevichet al. (1967), Csanady (1967), Gill (1968), Blackadar andTennekes (1968) and Arya (1975) extended the resis-tance law to stratified PBLs; Zilitinkevich and Deardorff(1974) and Zilitinkevich (1975, 1989) extended the the-ory by accounting for variable PBL heights; and Zilitinke-vich and Esau (2002, 2005) further extended the theoryby accounting for the non-local effect of the free-flowstability.

The various turbulent-flow models are presented inHess and Garratt (2002). Further, Garratt and Hess(2002, p. 265, figure 2) present geostrophic-velocity-profile data (includes data from Izumi, Hay, Kerang,Western Weddell Sea, Caldwell in transitional rough-wall variables (u/u∗, z/z0), and Coleman (DNS) in fullysmooth-wall variables (u/u∗, zu∗/ν), with smooth-to-rough conversion z0 = 0.11ν/u∗ (z0 is the aerodynamicroughness of the surface) in a semi-log plot based onlog-law theory, but the failure of these data to fit asingle universal curve in the inner layer region reflects

Copyright 2008 Royal Meteorological Society

1114 N. AFZAL

the inherent experimental errors in the measured u∗ andz0 and the presence of unsteadiness in the field over theland.

The relation between the power-law and log-law veloc-ity profiles revives an issue raised by Prandtl (1935), whostated that the log law is the well-known limiting valueobtained from the power law

u = D1yα + D2

as α → 0, D1 → ∞ and D2 → −∞, where u is theaxial-velocity profile and y is the normal distance fromthe wall. This proposition calls for accurate experimen-tal data at high Reynolds number. Furthermore, becausethe pipe-flow velocity profile exhibits a very weak defectlayer, a more severe test for the power-law velocity pro-file would be a high-Reynolds-number turbulent bound-ary layer (where the overlap would not constitute the‘main body of flow’). The issue of the power-law veloc-ity profile in a turbulent pipe, channel, boundary layerand wall jet has been studied by various authors duringthe last decade. Relevant literature for a fully-smooth sur-face is presented in Afzal (2005a, 2005b). The analysis ofthe power-law turbulent velocity profile for transitionalrough-pipe flow is described by Afzal, Seena and Bushra(2006), and for a transitional rough-surface boundarylayer by Afzal (2007).

Here we consider the simplest PBL in the steady statewith horizontally-homogeneous and neutral barotropicflow over a transitional rough surface. This basic stateforms the cornerstone of our theoretical understandingof the PBL. In the formula for the classical Rossbynumber, Ro∗ = u∗/f z0, the coefficient u∗/f is of theorder of the outer depth δ of the turbulent Ekman layer.The Rossby number R∗ = �Ro∗ adopted in this paperis itself based on the outer depth of a turbulent Ekmanlayer, δ = �u∗/f , where the constant of proportional-ity � is of order unity (Rossby and Montgomery, 1935;Zilitinkevich, 1989). In traditional log-law theory, theclassical Rossby number Ro∗ would suffice, becauseln R∗ = ln Ro∗ + ln � and ln � may be absorbed else-where (in the intercept). With the power law we donot have recourse to this device. We will show that, inthe power-law theory of the turbulent Ekman layer, thedepth-based Rossby number R∗ plays a significant role inthe overlap region. We will also compare the power-lawtheory with extensive direct-numerical-simulation (DNS),field and laboratory data.

We will analyse a two-layer asymptotic theory foropen equations of a turbulent Ekman layer (without anyclosure model, such as eddy viscosity, mixing lengthor k − ε), for large Rossby numbers. For a transitionalrough surface, the inner length scale is z0 and the outer-layer scale is u∗/f . We match the two layers usingthe Izakson–Millikan–Kolmogorov (IMK) hypothesis, toobtain an open functional equation. Another open func-tional equation, obtained from the ratio of two succes-sive derivatives of the basic functional equation, admits

two functional solutions, namely the power-law and log-law velocity profiles. The two functional solutions con-tain certain open constants that can be determined frommatching, using either experimental data or a closuremodel. This is not surprising, as we are dealing with openequations of the mean turbulent motion. The analysisshows that both the log-law and the power-law relationsin the overlap region are universal, and independent ofthe surface roughness. This universality in the inner andoverlap regions is supported by the velocity-profile DNSdata of Coleman (1999), Coleman et al. (2005) and Shin-gai and Kawamura (2004) on a fully-smooth surface, andby the laboratory data of Caldwell et al. (1972) on atransitional rough surface. For large values of z/z0, thedeparture from universal behaviour is due to the influ-ence of the outer layer. The envelope of the power-lawfriction factor relates the constants of the power law tothose of the log law. The power-law constants are uni-versal and independent of the surface roughness. Thegeostrophic-drag predictions and the cross-isobaric angleγ from power-law theory, as well as the log-law the-ory, are supported by extensive data from many sources,such as the laboratory data of Caldwell et al. (1972) andHoward and Slawson (1975), the DNS data of Shingaiand Kawamura (2004), Coleman (1999), Coleman et al.(2005), Coleman et al. (1990) and Lin et al. (1997), andthe turbulence model of Freedman and Jacobson (2002):see Hess and Garratt (2002, tables I–V).

2. Equations of motion

We will examine the flow and geometry, in the adiabaticturbulent state, of an incompressible viscous fluid, over atransitional, rough, flat surface driven by a steady uniformpressure gradient and subjected to steady system rotation.The flow above the boundary layer is geostrophic, i.e. thepressure gradient and Coriolis force balance each other.The governing momentum Reynolds equations for themotion of the geostrophic wind are (Garrett, 1994):

ν∂2u

∂z2 + 1

ρ

∂τx

∂z= −f (v − Vg); (1)

ν∂2v

∂z2 + 1

ρ

∂τy

∂z= f (u − Ug). (2)

Here the x direction is horizontally aligned along thesurface stress, the y direction is also horizontal butperpendicular to the x direction, and the z direction isvertically upwards. In the x and y directions, the meanvelocity components are u and v, and the Reynolds shearstresses are τx = −ρ〈u′w′〉 and τy = −ρ〈v′w′〉; ν is themolecular kinematic viscosity, ρ is the fluid density, f

is the Coriolis parameter (positive north of the Equator),Ug and Vg are the geostrophic-wind components

Ug = − 1ρf

∂p∂y

Vg = 1ρf

∂p∂x

(3)

Copyright 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 134: 1113–1125 (2008)DOI: 10.1002/qj

POWER-LAW VELOCITY PROFILE IN TURBULENT EKMAN LAYER 1115

and p is the static pressure. We analyse Equations (1)and (2) for the semi-infinite Cartesian domain abovea transitional flat surface in the horizontal x and y

directions: at z = 0, u = v = 0 and τx = τy = 0; and asz → ∞, u → Ug, v → Vg and τx, τy → 0. At the upperboundary, the turbulence dies out.

The geostrophic (horizontal) wind velocity G, theangle γ between the surface stress vector and the surfacegeostrophic-wind vector (the cross-isobaric flow angle),and the velocity components Ug and Vg in the x and y

directions, are given by:

Ug = G cos γ

Vg = −sgn G · sin γ

G =√

U 2g + V 2

g

tan γ = VgUg

. (4)

Here the sgn function takes the value +1 in the North-ern Hemisphere and −1 in the Southern Hemisphere. Thisflow is statistically stationary, and for sufficiently longtimes the turbulence ‘forgets’ the initial conditions. Thegeostrophic-flow state is uniquely specified by G, f andν. The depth of the viscous Ekman layer is D = √

2ν/f .The velocity scale G and length scale D are used todefine the Reynolds number Re as:

Re = GD

ν= G√

νf/2. (5)

The Rossby number Ro and roughness Rossby numberRo∗ are:

Ro = Gf z0

Ro∗ = u∗f z0

}, (6)

where the surface length scale z0 is the aerodynamicroughness length adopted in the meteorological descrip-tion of the atmospheric data and u∗ = (τS/ρ)1/2 is thefriction velocity, τS being the surface shear stress. Theouter depth δ of the turbulent Ekman layer is of the orderof u∗/f , and without loss of generality it becomes δ =�u∗/f where � is order unity (Rossby and Montgomery,1935; Zilitinkevich, 1989). The alternative Rossby num-ber based on the outer depth of the turbulent Ekman layeris:

R∗ = �Ro∗. (7)

In the power-law theory of the turbulent Ekman layer,considered here, the depth-based Rossby number R∗plays a significant role in the overlap region. In traditionallog-law theory, the classical Rossby number Ro∗ wouldsuffice, because ln R∗ = ln Ro∗ + ln � and ln � may beabsorbed elsewhere (in the intercept).

The parameter is a dimensionless form of theaerodynamic roughness scale z0. The molecular viscouslength scale near the wall ν/u∗ is:

= u∗z0

ν= z0+. (8)

The normal coordinate is z = y + dr, where the ‘ori-gin’ dr is located below the top of the roughness element,caused by irregular hydraulic roughness. This level liesbetween the protrusion bases and heads, and automati-cally satisfies the constraints 0 < dr < h and dr = 0 fora smooth surface. Clauser (1954) proposed a method fordetermining the effective surface-roughness origin dr andskin friction u∗. The velocity scale is the friction velocityat the surface u∗, the length scale is the outer turbulentdepth δ, and the inner length is z0.

2.1. Outer layer

In the outer layer, the velocity scale is the friction velocityu∗ and the length scale is the turbulent depth δ. The outervariables are η = z/δ and δ = �u∗/f . The horizontalvelocity components u and v are represented by velocitydefects components:

u − Ug = u∗U(η)

v − Vg = u∗V (η)

}(9)

and the Reynolds-stress components:

τx = ρu2∗�x(η)

τy = ρu2∗�y(η)

}. (10)

The governing mean-momentum equations (1)–(2) inthe outer variables (9)–(10) become:

1R∗

∂2U

∂η2 + ∂�x

∂η= −�V

1R∗

∂2V

∂η2 + ∂�y

∂η= �U

. (11)

The first-order equations,

1

∂�x

∂η= −�V

1

∂�y

∂η= �U1

, (12)

with the outer boundary conditions U, V, �x, �y → 0 asη → ∞, are non-viscous: they fail to satisfy the no-slipboundary condition on the surface, and an inner layernear the wall region is needed.

2.2. Inner layer

In the inner wall layer, the velocity scale is the frictionvelocity u∗ and the length scale is the surface rough-ness scale z0 used in the meteorological description ofatmospheric data. The inner variable ζ is

ζ = R∗η = z

z0, (13)

and the velocity profiles u and v are

u = u∗u+(ζ )

v = u∗v+(ζ )

}(14)


1116 N. AFZAL

and Reynolds shear stresses are

τx = ρu2∗τx+(ζ )

τy = ρu2∗τy+(ζ )

}. (15)

The boundary-layer equations (1)–(2), in terms of theinner variables (14)–(15), become:

1

∂2u+∂ζ 2 + ∂τx+

∂ζ= −R−1∗ �(v+ − Vg+)

1

∂2v+∂ζ 2 + ∂τy+

∂ζ= R−1∗ �(u+ − Ug+)

. (16)

Integration gives:

1

∂u+∂ζ

+ τx+ = 1 + R−1∗ �(ζVg+ − ∫ ζ

0 v+dζ)

1

∂v+∂ζ

+ τy+ = R−1∗ �(−ζUg+ + ∫ ζ

0 u+dζ)

.

Here, R−1∗ � is just the inverse of the Rossby numberRo∗, which is a small parameter. Therefore the first-orderinner equations are

1

∂u+∂ζ

+ τx+ = 1, (17)

1

∂v+∂ζ

+ τy+ = 0, (18)

subject to the surface boundary conditions u+ = v+ =τx+ = τy+ = 0 at ζ = 0.

2.3. Matching

In the turbulent motion, the equations are not closedunless a turbulence closure model is adopted. Theapproaches adopted by Izakson (1937), Millikan (1938)and Kolmogorov (1941) are model-free, and appeal to anoverlap hypothesis (Afzal, 1976; Afzal and Narasimha,1976). Thus, the IMK hypothesis may be stated as fol-lows.

In any turbulent flow, between the viscous and theenergetic scales there exists an overlap domain overwhich the solutions characterizing the flow in the twocorresponding limits must match as the Reynolds numbertends to infinity.

The resemblance of the IMK hypothesis to conven-tional matching associated with a closed equation seemspeculiar to turbulence theory. Matching the inner limit(ζ fixed, R∗ → ∞) of the outer expansions and the outerlimit (η fixed, R∗ → ∞) of the inner solution, using theIMK hypothesis, we arrive at the following functionalequations connecting the unknown functions:

u+(ζ ) = Ug+(R∗) + U(η), (19)

v+(ζ ) = Vg+(R∗) + V (η), (20)

for ζ → ∞ and η → 0 in the overlap region, and

Ug+(R∗) = Ugu∗ = G

u∗ cos γ

Vg+(R∗) = Vgu∗ = −sgn G

u∗ sin γ

. (21)

For large values of the Rossby number (Ro∗ → ∞),we have Ug+(R∗) → ∞, and matching demands that theleft-hand side must be unbounded for large ζ . Note thatthe matching would be impossible if u+ were boundedas ζ → ∞.

There are three variables ζ , η and R∗, out of whichtwo are independent as ζ = R∗η. We differentiate thefunctional equation (19) with respect to ζ , keeping η

fixed, to yield

∂u+∂ζ

= ∂Ug+∂R∗

∂R∗∂ζ

= ∂Ug+∂R∗

R∗ζ

ζ∂u+∂ζ

= R∗∂Ug+∂R∗

Further, differentiating the functional equation (19)with respect to η, keeping ζ fixed, to yield

η∂U

∂η= R∗

∂Ug+∂R∗

The above two matching conditions may be combinedas

ζ∂u+∂ζ

= R∗∂Ug+∂R∗

= η∂U

∂η. (22)

3. Log law from matching

The integral of these relations gives the velocity profileas

ku+ = ln ζ (23)

and the geostrophic-resistance law as

kUg+ = ln R∗ − A1, (24)

and the outer velocity defect law becomes

kU = ln η + A1. (25)

The stress at the surface has been assumed to have noy component, so that the wind in the surface layer alsohas no y component, and

v+(ζ ) = 0. (26)

Matching relation (20) for the v component, we find:

Vg+(R∗) = −V (0) = −sgnB

k(27)

andB0 = B

k. (28)



The relations (24) and (27)–(28), with A = A1 − ln �,become:

kUgu∗ = ln Ro∗ − A

kVgu∗ = −sgn B

. (29)

The matching results (29) may also be expressed as:

k Gu∗ = √

(ln Ro∗ − A)2 + B2

andtan γ = −sgn B

ln Ro∗ − A.

(30)

The Reynolds shear stresses from velocity distributionsin the overlap region are given by

τx+ = 1 − 1kζ

− R−1∗ �Bk

ζ

τy+ = −R−1∗ Ug+ζ + R−1∗ �k−1ζ(ln ζ − 1)

}(31)

and�x = 1 − �B

kη − R−1∗ � 1

kη

�y = k−1η(ln η − 1 + A)

}.

The Reynolds shear τx+ possesses a maximum at thepoint

ζx+,m =√

Ro∗B

zx+,mGν = Re 1√

�B

, (32)

where the maximum value is

τx+,m = 1 − 2

k

√B

Ro−1/2

∗ . (33)

The composite solution for the velocity profiles u andv from the inner and outer solutions is then given by:

k uu∗ = ln ζ + BWu(η)

k vu∗ = −BWv(η)

}(34)

andku − Ug

u∗ = ln η + A + BWu(η)

kv − Vg

u∗ = −B(1 − Wv(η)

) . (35)

Here Wu is the wake function satisfying the boundaryconditions Wu(0) = 0 and Wu(∞) = 1, and Wv is thewake function satisfying the conditions Wv(0) = 0 andWv(∞) = 1. At the edge of the boundary layer z =zδ , the geostrophic velocity is u = Ug, and the aboverelations become:

A + B = − ln ηδ

ηδ = f zδu∗

}. (36)

The outer-layer analysis with a closure model providesthe two wake functions. If the outer-layer eddy viscosityis taken as constant, then the outer-layer analysis gives

these wake functions, by a further extension of the workof Garratt and Hess (2002), as:

Wu(η) = 1 − e−βη(cos(βη) − sin(βη)

), (37)

Wv(η) = 1 − e−βη(cos(βη) + sin(βη)

), (38)

where β = B/k.

4. Power law from matching

In the classical Rossby number Ro∗ = u∗/f z0, the outerdepth δ of the turbulent Ekman layer is of the orderof u∗/f . The Rossby number R∗ = �Ro∗ adopted inthis paper is based on the outer depth δ = �u∗/f . Intraditional log-law theory, the classical Rossby numberRo∗ would suffice, because ln R∗ = ln Ro∗ + ln �, andln � may be absorbed elsewhere (in the intercept). Forthe power law there is no recourse to such a device.In the power-law theory of the turbulent Ekman layerpresented here, the depth-based Rossby number R∗ playsa significant role in the overlap region.

The matching of the inner- and outer-layer velocityprofiles for large Reynolds numbers Ro∗ → ∞ is givenby the relations (19) and (22). For a fully-smooth sur-face, Izakson (1937) considered a functional equationanalogous to (19) for a smooth pipe, and Millikan (1938)considered its first derivative when proposing logarithmiclaws. The approach of Izakson and Millikan was extendedto second order by Afzal (1976) in fully-developed turbu-lent pipe or channel flow. Differentiating the functionalequation (22) once more (or alternately differentiating thefunctional equation (19) twice), with respect to ζ , for η

fixed, we get

∂

∂ζ

(ζ

∂u+∂ζ

)= ∂

∂R∗

(R∗

∂Ug+∂R∗

)∂R∗∂ζ

Based on the relation ζ = ηR∗, ∂ζ/∂R∗ = η = ζ/R∗ theabove relation becomes

ζ∂

∂ζ

(∂u+∂ζ

)= R∗

∂

∂R∗

(R∗

∂Ug+∂R∗

)

or

ζ 2 ∂2u+∂ζ 2 + ζ

∂u+∂ζ

= R2∗∂2Ug+∂R2

∗+ R∗

∂Ug+∂R∗

In view of the matching relation (22), the first orderderivative terms on both sides of above relation canceleach other, and we get

ζ 2 ∂2u+∂ζ 2 = R2

∗∂2Ug+∂R2

∗, (39)

whose solutions again give the log laws. The functionalequation may be differentiated as many times as weplease, but nothing new or inconsistent would be discov-ered. This differentiation is one of the keys for solving


1118 N. AFZAL

the functional equations, but other techniques may alsobe considered.

We show here that these functional equations give analternative solution of the power-law velocity distributionin the overlap region. The functional equation (39),when divided by the functional equation (22), gives thealternative functional equation:

ζ∂2u+/∂ζ 2

∂u+/∂ζ= R∗

∂2Ug+/∂R2∗

∂Ug+/∂R∗, (40)

and the functional solution demands that

ζ∂2u+/∂ζ 2

∂u+/∂ζ= α − 1

R∗∂2Ug+/∂R2

∗∂Ug+/∂R∗ = α − 1

, (41)

where α is constant. Integrating these functional equa-tions, we obtain:

∂u+∂ζ

= Jζα−1

∂Ug+∂R∗ = JRα−1∗

, (42)

where J is a constant of integration. The two cases α = 0and α �= 0 would arise during any further integration,giving respectively the log-law and power-law velocitydistributions and friction factors.

Integration of the relation (42) in the case α �= 0 gives:

u+ = Sζα, (43)

Ug+ = SRα∗ − E, (44)

andS = J/α, (45)

where E and α are constants of integration. The matchingrelation (43) in the outer variable η yields:

u+ = S1ηα

S1 = SRα∗

}. (46)

Division of the matching relations (43) and (44) yieldsan alternative velocity profile, from

u

Ug=

(Z

δ

)α(1 − E

S1

)−1, (47)

and for E S1 this becomes the simple relation u/Ug =(Z/δ)α . The power-law geostrophic resistance (44)becomes:

Ug

u∗= S(�Ro∗)α − E. (48)

Matching of the v component of velocity (20), forv1(ζ ) = 0, requires:

Vg+(Ro∗) = −sgn B0. (49)

The matching results (48) and (49) may also beexpressed as:

G

u∗=

√(S(�Ro∗)α − E

)2 + B20 (50)

andtan γ = −sgn B0

S(�Ro∗)α − E. (51)

The formulae for the Reynolds shear stresses in theregion of overlap between the inner and outer layerbecome:

τx+ = 1 − Sα

ζ 1−α − R−1∗ �B0ζ

τy+ = −R−1∗ Ug+ζ + R−1∗S

1 + αζ 1+α

(52)

and�x = 1 − �B0η − R−1∗

S1α

Y 1−α

�y = S11 + α

η1+α

}. (53)

The Reynolds shear stress τx+ given by (52) has amaximum at

ζx+,m =( S

B0α(1 − α)Ro∗

) 12−α

, (54)

where the maximum value is

τx+,m = 1 − 2 − α

1 − α

( S

B0α(1 − α)

) 12−α

Ro− 1−α

2−α∗ . (55)

The composite solution of the velocity profiles u andv from the inner and outer solutions yields:

uu∗ = Sζα + B0�u(η)vu∗ = −B0�v(η)

}(56)

and

u − Ugu∗ = S1(η

α − 1) + B0(�u(η) − 1)

v − Vgu∗ = −B0

(1 − �v(η)

) . (57)

Here �u is the wake function satisfying the boundaryconditions �u(0) = 0 and �u(∞) = 1, and �v is thewake function satisfying the conditions �v(0) = 0 and�v(∞) = 1. At the edge of the boundary layer z = zδ , thegeostrophic velocity is u = Ug, and the above relationsyield:

B0 = −ηαδ

ηδ = f zδu∗

}. (58)

4.1. Envelope of the friction-factor power law

The friction-factor power law (48) with the assistance ofrelation (45) becomes:

Ug

u∗= J

αexp(α ln R∗) − E, (59)



which forms a family of curves in the (Ug+, ln Rφ) plane,parametrized by α. This family has an envelope thatsatisfies (59), and the equation ∂Ug+/∂α = 0 may besimplified as:

α = 1

ln(�Ro∗). (60)

The skin-friction power law (59), after elimination ofthe power index α from relation (60) for the geostrophic-resistance law, becomes the friction log law:

kUgu∗ = ln(�Ro∗) − kE

kE = A + ln �

}, (61)

providedJ = (k exp(1))−1. (62)

The power-law prefactor S from relation (45) may beexpressed as:

S = 1

k exp(1)α= 1

k exp(1)ln(�Ro∗). (63)

In relations (60) and (63), the term ln(�Ro∗) maybe eliminated, using relation (61), to get alternativeexpressions for the power-law constants α and S as

α = k−1 u∗/Ug

1 + Eu∗/Ug(64)

andS = exp(−1)

(Ug

u∗+ E

). (65)

In this case,

C1 = Cg cos γ = u∗Ug

. (66)

The relations (54) and (55) for the maxima ζx+,m andτx+,m may be simplified as

ζx+,m =(exp(−1)

B(1 − α)Ro∗

) 12−α

and

τx+,m = 1 − 2 − α

1 − α

(exp(−1)

B(1 − α)

) 12−α

Ro− 1−α

2−α∗ .

4.2. Approximate power-law model

The velocity distribution (47), in an approximate formwhere E S1, becomes:

u

Ug=

(Z

δ

)α

. (67)

The geostrophic-resistance relation (50) may also pro-vide a good approximation, as

G

u∗= S(�Ro∗)α, (68)

because the effect of the constant E is roughly counter-balanced by that of the constant B0. The expressions (67)and (68), based on the relations (60) (for the power indexα) and (63) (for the prefactor S), may be expressed interms of Ro as:

u∗G

= MRo−n (69)

andtan γ = −sgn B0MRo−n, (70)

where the power index n and prefactor M are given bythe relations:

n = α

1 + α(71)

and

M = (S�α)− 1

1+α = (αk exp(1)�−α

) 11+α . (72)

The power-law geostrophic drag u∗/G and cross-isobaric angle γ , for various values of the power-lawindex, are given in Table I. These may be compared withthe power laws of Prandtl and Tollmien (1924) (Cg =0.148Ro−0.119) and Swimbank (1974) (Cg = 0.111Ro−0.07, Cg = 0.113Ro−1/14).

5. Results and discussion

We have analysed the geostrophic-velocity-profile DNSdata of Coleman (1999) and Coleman et al. (2005)for Re = 2000, 1000 on a fully-smooth surface, theDNS data of Shingai and Kawamura (2004) for Re =775, 600, 510, 400 on a fully-smooth surface, and thelaboratory data of Caldwell et al. (1972) for Re =1753, 1234, 1159, 704. Shingai and Kawamura (2004)and Coleman (1999) have presented data on a transi-tional rough surface in terms of fully-smooth wall vari-ables (u/u∗, zu∗/ν). Caldwell et al. (1972) presented thevelocity-profile data (u/u∗, z/z0) with a shift of origin inz/z0. Here again, the fit of the data to a single universalcurve was not considered. Garratt and Hess (2002, p. 265,figure 2) have also presented geostrophic-velocity-profiledata, but these did not fall on a single universal curvebecause of inherent experimental errors.

Figure 1(a) shows the geostrophic-velocity-profile datain semi-log plots in the transitional rough-wall variables(u/u∗, z/z0). A substantial log-law region exists foreach Reynolds number, and its domain increases as theReynolds number increases. The data fall on a single

Table I. The power-law geostrophic drag u∗/G and cross-isobaric angle γ , for various values of the power-law index α.

α u∗/G tan γ

1/7 0.2198Ro−1/8 1.4896Ro−1/8

1/10 0.1446Ro−1/11 0.9761Ro−1/11

1/13 0.11066Ro−1/14 0.7470Ro−1/14


1120 N. AFZAL

universal curve in the inner region, consisting of a sub-layer, the region between the sub-layer and the log law,and the overlap region. The departure from universalbehaviour for large z/z0 is due to the outer layer of thegeostrophic flow.

Figure 1(b) shows the geostrophic-velocity-profile datain log–log plots in the wall variables (u/u∗, z/z0). Asubstantial power-law region exists for each Reynoldsnumber, and its domain increases as the Reynolds num-ber increases. The data fall on a single universal curvein the inner region, consisting of a sub-layer, the regionbetween the sub-layer and the power law, and the overlapregion. The departure from universal behaviour for largez/z0 is due to the outer layer of the geostrophic flow.

Figure 2 shows the profile data for the geostrophictotal velocity Q = √

u2 + v2 in the transitional rough-wall variables (Q/u∗, z/z0), on a semi-log plot for thelog-law representation and on a semi–log plot for thepower-law representation. In each panel, the data fallon a universal curve in the inner region (consisting ofa sub-layer region, the region between the sub-layer andthe overlap region, and the overlap region). A substantialoverlap region exists for each Reynolds number, whosedomain increases as the Reynolds number increases.

Figure 3 shows the profile data for the geostrophic

(a)

(b)

Figure 1. Inner scaling of the mean-geostrophic streamwise velocityprofile u/u∗, from DNS data of Coleman (1999) and Colemanet al. (2005) and Shingai and Kawamura (2004) and laboratorydata of Caldwell et al. (1972), for various values of the Reynoldsnumber Re: (a) log-law representation in semi-log plot; (b) power-law

representation in log–log plot.

(a)

(b)

Figure 2. As Figure 1, but for the mean geostrophic total velocity Q/u∗,where Q = G = √

u2 + v2.

cross velocity in the transitional rough-wall variables(v+, z/z0). It shows a wake-like region, which dependson the Rossby number for large z/z0 in the outer-layergeostrophic flow v+.

The power-law velocity profiles in the inner and outerregions of the overlap domain are governed by therelations:

u+ = Sζα

u+ = S1ηα

}. (73)

The velocity-defect power law is:u − Ug

u∗ = S1(ηα − 1) + E

S1 = S(�Ro∗)α

}. (74)

Figure 3. As Figure 1(a), but for the mean geostrophic spanwisevelocity profile v/u∗.



The components of the geostrophic-drag power lawsatisfy:

Ug

u∗= S(�Ro∗)α − E; (75)

Vg

u∗= −sgn B0. (76)

The power-law geostrophic drag G/u∗ and cross-isobaric angle γ are given by

G

u∗=

√(S(�Ro∗)α − E

)2 + B20 (77)

and

tan γ = −sgn B0

S(�Ro∗)α − E(78)

respectively. The asymptotic expansion of the geostrophicdrag (77) for large Reynolds number yields:

G

u∗= S(�Ro∗)α − E + 1

2B2

0

(S(�Ro∗)α

)−1 + · · ·

To lowest order, this becomes:

G

u∗= S(�Ro∗)α − E. (79)

The DNS data of Coleman (1999) and Coleman et al.(2005) and Shingai and Kawamura (2004) on a fully-smooth surface and the laboratory data of Caldwellet al. (1972) on transitional rough surface, shown onlog–log plots in the transitional rough-wall variables(u/u∗, z/z0), reveal a substantial linear region represent-ing the power-law domain of the overlap, as alreadyshown in Figure 1(b). We have carefully estimated thepower-law constants, consisting of the power-law indexα and the prefactor S, for each velocity profile, on amagnified scale (graphs not shown here), where the lin-ear portion of each dataset is fitted with a straight line.The intercept of the straight line represents the prefactorS, and its slope represents the power index α. These areestimated using the Microsoft Excel program.

Figure 4 shows the prefactor S plotted against thereciprocal of the power-law index α−1, from the dataof Coleman (1999) and Coleman et al. (2005), Shingaiand Kawamura (2004) and Caldwell et al. (1972). Theleast-squares fit to the data is represented by the relation:

S = 0.9

α. (80)

Figure 4. Power-law prefactor S plotted against the reciprocal ofthe power-law index α−1, from DNS data from the smooth sur-face by Coleman et al., (2005) for Re = 2000, Coleman (1999)for Re = 1000, Shingai and Kawamura (2004) for Re = 775, 600,510, 400, and the laboratory data of Caldwell et al., (1972) forRe = 1753, 1234, 1158, 704. The line is a result of the present work

(63), a fit represented by (81).

Figure 5. As Figure 4, but for the power-law index α as a function ofthe surface Rossby number Ro∗ = u∗/f z0. The line is a result of the

present work (60) with � = 0.3.

Figure 5 shows the power index α plotted againstthe Rossby number Ro∗, from relation (60), for theabove data, with � = 0.3 (Zilitinkevich, 1989), using therelation:

α = 1

ln(0.3Ro∗). (81)

Figure 6 shows the prefactor S plotted against Ro∗ ona semi-log scale. The equation of best fit is:

S = 0.9 ln Ro∗ − 1.5. (82)

Figure 7 shows an alternative expression of the power-law index α in terms of the dimensionless frictionvelocity u∗/Ug, from the same data. In light of ourprediction (64), this relation is expressed as:

α = 2.5u∗/Ug

1 − 1.5u∗/Ug. (83)


1122 N. AFZAL

Figure 6. As Figure 4, but for the power-law prefactor S as a functionof the surface Rossby number Ro∗ = u∗/f z0. The line is a result of

the present work.

Figure 7. As Figure 4, but for the power-law index α as a functionof the geostrophic resistance ε = u∗/Ug. The line is a result of the

present work.

Figure 8 shows the power-law prefactor S plottedagainst the reciprocal of the dimensionless friction veloc-ity, from the same data. In light of our prediction (65),this relation is expressed as:

S = exp(−1)(Ug

u∗+ 0.55

). (84)

The geostrophic-drag predictions have been analysedfrom extensive data from many sources: the laboratorydata of Caldwell et al. (1972) and Howard and Slawson(1975); the DNS data of Shingai and Kawamura (2004),Coleman (1999), Coleman et al. (2005), Coleman et al.(1990) and Lin et al. (1997); and the turbulence modelof Freedman and Jacobson (2002): see Hess and Garratt(2002, tables I, II, IV).

Figure 9(a) shows the components of the geostrophicdrag Ug/u∗ plotted against the Rossby number Ro∗, fromthe above data, on a semi-log plot. The data are scatteredand it is not easy to draw an appropriate line. However,with Karman constant k = 0.4, the geostrophic-drag loglaw (61) obtained from the envelope of the geostrophic-drag power law fitted to the data becomes:

kUg

u∗= ln Ro∗ − 1.2. (85)

Figure 8. As Figure 4, but for the power-law prefactor S as a functionof the reciprocal of the geostrophic resistance ε = u∗/Ug. The line is

a result of the present work (65).

(a)

(b)

Figure 9. Comparison of the power-law and log-law geostrophic dragrelations as a function of the roughness Rossby number Ro∗ = u∗/f z0,from extensive field and laboratory data from various sources. Ourpredictions are indicated by the lines (a) kUg/u∗ = 2.3 ln Ro∗ − 1.2

and (b) kVg/u∗ = −2.7.

The geostrophic-drag data at higher Rossby numberare scattered about the log law (85), but for lowerRossby number (Ro∗ ≤ 104) the systematic departurefrom the log law indicates the presence of higher-ordereffects. The power-law geostrophic drag Ug/u∗ basedon our prediction (75) has been estimated by using therelations (81) and (82); results are shown in Figure 9(a)by the solid line. The results are in good agreement withthe log-law relation (85), shown in the same figure as adashed line.



Figure 9(b) shows the cross components of the geo-strophic drag Vg/u∗ plotted against the Rossby numberRo∗, from the same data. There is large scatter in the data,and it is not possible to draw an appropriate asymptoticline for large values of the Rossby numbers Ro∗. Thelines shown are based on the DNS data of Shingai andKawamura (2004), Coleman (1999) and Coleman et al.(2005).

Figure 10 shows the geostrophic drag G/u∗ plottedagainst the Rossby number Ro∗, from the same data.The data are scattered, and it is not easy to draw anappropriate line. However, with Karman constant k =0.4, the geostrophic-drag log law (obtained from theenvelope of the geostrophic-drag power law) is fitted tothe data as:

G

u∗= 2.5 ln Ro∗ − 2, (86)

where the higher-Rossby-number data are scattered aboutthe log law, and for lower Rossby number (Ro∗ ≤ 104)there is systematic departure from the log law (86), indi-cating higher-order effects. The power-law geostrophicdrag G/u∗ from prediction (79) subject to relations (81)and (82) is shown by a solid line, which is in agree-ment with the log law (86) shown by a dashed line. Thecross-isobaric angle γ from log-law theory is then givenby:

tan γ = 2.7

2.3 log Ro∗ − 1. (87)

Figure 11 shows the variation of the cross-isobaricangle γ as a function of the roughness Rossby numberRo∗ = u∗/f z0, from extensive field and laboratory datafrom various sources. The data are widely scattered, butpredictions from log-law theory (87) compare well withpower-law theory (78).

Our approximate geostrophic-drag relations (Table I)for power-law index α = 1/7, 1/10, 1/13, and compar-ison with the data shown in Figure 12(a), are validfor limited domains of the Rossby number: α = 1/7(n = 1/8) for 3.6 < log10 Ro < 6, α = 1/10 (n = 1/11)for 4.5 < log10 Ro < 6.6, and α = 1/13 (n = 1/14)for 6 < log10 Ro < 8.4. Figure 12(a) also compares the

Figure 10. Comparison of the power-law and log-law geostrophic dragspresent work G/u∗ as a function of the Rossby number Ro∗ = u∗/f z0,

with extensive field and laboratory data from various sources.

Figure 11. Variation of tan γ as a function of the roughness Rossbynumber Ro∗ = u∗/f z0, from power-law theory and log-law theory,


(a)

(b)

Figure 12. The geostrophic-drag vector: (a) geostrophic drag G/u∗ andthe power-law proposals of Prandtl and Tollmin (1924) and Swimbank(1974), and (b) cross-isobaric angle γ (expressed as tan γ ), as afunction of the Rossby number Ro = G/f z0, from the power law withindex α = 1/7, 1/10, 1/13, log law and general rational power law,


geostrophic-drag power laws G/u∗ = 0.148Ro−0.119 ofPrandtl and Tollmien (1924) and G/u∗ = 0.113Ro−1/14

of Swimbank (1974) with the data; this shows the limita-tions of their validity. In Figure 12(b), the cross-isobaricangle γ as a function of the roughness Rossby numberRo = G/f z0 for power-law index α = 1/7, 1/10, 1/13 iscompared with extensive field and laboratory data fromvarious sources. These predictions are valid for the lim-ited domains of the Rossby number described above.


1124 N. AFZAL

6. Conclusions

• The classical Rossby number is Ro∗ = u∗/f z0, andthe outer depth δ of the turbulent Ekman layer is ofthe order of u∗/f . The Rossby number R∗ = �Ro∗adopted in this paper is based on the outer depthδ = �u∗/f . In traditional log-law theory, the classicalRossby number Ro∗ would suffice, because ln R∗ =ln Ro∗ + ln � and ln � may be absorbed elsewhere(in the intercept). For the power law we do not haverecourse to such a device. In the power-law theoryof the turbulent Ekman layer, the depth-based Rossbynumber R∗ plays a significant role in the overlapregion. We have compared the power-law theory withextensive DNS, field and laboratory data.

• We have analysed the open equations of the meanturbulent geostrophic boundary layer in the two sub-layers. Their matching in the overlap domain leads toa open functional equation, by the IMK hypothesis.An alternative functional equation, based on the ratioof two successive derivatives, leads to two functionalsolutions: the power-law and log-law velocity profilesand friction factors.

• The data for the geostrophic-velocity profile in transi-tional rough-wall variables (u/u∗, z/z0) lie on a sin-gle universal curve in the inner layer, which consistsof a sub-layer, the region between the sub-layer andthe overlap region, and the overlap region itself. Thedeparture from the universal relation for large z/z0 isthe contribution of the outer layer away from the sur-face. Likewise, the predictions of geostrophic drag Cg

and cross-isobaric angle γ as a function of Rossbynumber Ro are universal relations that are independentof the surface roughness. These facts from power-law theory and log-law theory are well supported byextensive experimental data for all types of transitionalsurface roughness.

• The data on a semi-log plot show substantial universallogarithmic behaviour, u+ = k−1 ln(z/z0), as predictedby relation (23), in the overlap region. The domainof the log region increases as the Reynolds numberincreases. Here the Karman constant k is a universalnumber (for example, k = 0.4).

• The data on a log–log plot show substantial universalpower-law behaviour, u+ = S(z/z0)

α , as predictedby relation (43) in the same overlap region. Thedomain of the power-law region increases as theReynolds number increases. Here the power index α

and prefactor S are universal functions, independent ofthe transitional surface roughness.

• The power-law constants α, S and E, the log-law con-stant k and the surface roughness z0 have to be deter-mined by experiments, as the matching procedure willnot determine them. The envelope of the geostrophic-drag power law shows that the power-law constants α

and S are functions of Ro∗ or ε = u∗/Ug, as appro-priate, but are independent of the surface roughnessz0. Hence, the power-law constants α and S of a fully-smooth surface, with appropriate representation, wouldsuffice.

• The power-law constants α and S may be esti-mated from the relations (64)–(65), provided the localgeostrophic drag C1 = u∗/Ug is known. The surfaceroughness scale z0 enters the picture only if we intendto estimate the power-law constants from the alter-native expression (48) and Table I containing Ro∗ =u∗/f z0 and Ro = G/f z0 respectively.

• The approximate power-law velocity profile (67), theexpressions (69) and (70) of the geostrophic dragG/u∗, and the cross-isobaric angle γ as a functionof the Rossby number Ro = G/f z0, are valid forbulk-of-flow approximation. In particular, the approx-imate power-law solution for a given geostrophic-dragpower-law index (for example, n = 1/8, 1/11, 1/14),for the power law of Prandtl and Tollmien (1924)for n = 0.119, and for the power laws of Swimbank(1974) for β = 0.07, 1/14, have been compared withthe traditional log law, the general rational power law,and extensive field and laboratory data from varioussources. The approximate power-law geostrophic dragand cross-isobaric angle have limited domains of valid-ity in terms of the Rossby number: α = 1/7 (n =1/8) for 3.6 < log10 Ro < 6, α = 1/10 (n = 1/11) for4.5 < log10 Ro < 6.6, and α = 1/13 (n = 1/14) for6 < log10 Ro < 8.4.

Acknowledgements

The author is grateful for the support of the All IndiaCouncil of Technical Education, New Delhi.

A Appendix: Notation

A additive constant in geostrophic drag law (29)A1 additive term in velocity-defect law (25) and

geostrophic drag law (24)B constant in the skin-friction relationB0 a constant in Equation (49)Cg geostrophic-drag coefficient u∗/G

C1 constant in Equation (66)dr hydraulic roughness height of the surfaceD depth of viscous Ekman layer, (2ν/f )1/2

E additive term in skin-friction power law (44),B0

f Coriolis parameterG geostrophic wind speedJ constant of integration in power laws (42)k Von Karman constantK outer-layer eddy viscosityn power index for geostrophic drag, α/(α + 1)

p(x, y) static geostrophic-pressure distributionRe Reynolds number, GD/ν

Ro Rossby number, G/f z0

Ro∗ frictional Rossby number, u∗/f z0

R∗ Rossby number based on outer depth, �u∗/f z0

S prefactor in inner power law velocity pro-file (43) and power law geostrophic drag (44)



S1 prefactor in outer power law velocity pro-file (46)

u(z) geostrophic wind speed in x directionu′ velocity fluctuations in x directionu∗ friction velocity, (τS/ρ)1/2

Ug geostrophic wind speed in x directionU(η) outer-velocity defect in x directionv(z) cross-geostrophic wind speed in y directionv′ velocity fluctuations in y directionV (η) outer-velocity defect in y directionVg geostrophic wind speed in y directionw′ velocity fluctuations in z directionWu(z) u component of wake function in log-law

theoryWv(z) v component of wake function in log-law

theoryx coordinate in direction of surface stressy coordinate perpendicular to surface stressz vertical coordinatez0 aerodynamic roughness of the surfacez0+ dimensionless aerodynamic roughness, z0u∗/να power index in the power-law velocity profileβ a constant, B/k

γ cross-isobaric angle of geostrophic wind�x(η) outer dimensionless Reynolds shear stress in

x direction�y(η) outer dimensionless Reynolds shear stress in

y directionδ depth of the turbulent Ekman layer� constant in the boundary layer thickness of the

Ekman layer, δ = �u∗/fζ inner variable, z/z0

η outer variable, z/δ

ν molecular kinematic viscosityρ density of fluidτx Reynolds-shear-stress x component, −ρ〈u′w′〉τy Reynolds-shear-stress y component, −ρ〈v′w′〉τS shear stress on the surface dimensionless aerodynamic roughness, z0+�u(z) u component of wake function in power law�v(z) v component of wake function in power law

ReferencesAfzal N. 1976. Millikan argument at moderately large Reynolds

numbers. Phys. Fluids 19: 600–602.Afzal N. 2005a. ‘Scaling of power law velocity profile in wall-bounded

turbulent shear flows’. AIAA-2005-0109, 43rd AIAA AerospaceSciences Meeting and Exhibit, 10–13 Jan 2005, Reno Nevada.

Afzal N. 2005b. Analysis of power law and log law velocity profiles inoverlap region of a turbulent wall jet. P. R. Soc. A 461: 1889–1910.

Afzal N. 2007. Power law velocity profile in the turbulent boundarylayer on transitional rough surfaces. J. Fluid Eng. 129: 1083–1100.

Afzal N, Narasimha R. 1976. Axisymmetric turbulent boundary layersalong a circular cylinder with constant pressure. J. Fluid Mech. 74:113–129.

Afzal N, Seena A, Bushra A. 2006. Power law turbulent velocity profilein transitional rough pipes. J. Fluid Eng. 128: 548–558.

Arya SPS. 1975. Geostrophic drag and heat transfer relations for theatmospheric boundary layer. Q. J. R. Meteorol. Soc. 101: 147–161.

Blackadar AK, Tennekes H. 1968. Asymptotic similarity in neutralbarotropic planetary boundary layers. J. Atmos. Sci. 25: 1015–1022.

Caldwell DR, van Atta CW, Heland KN. 1972. A laboratory studyof the turbulent Ekman layer. Geophysical Fluid Dynnamics 3:125–160.

Clauser FM. 1954. Turbulent boundary layers in adverse pressuregradients. J. Aeronautical Sciences 21: 91–108.

Coleman GN. 1999. Similarity statistics from a direct numericalsimulation of the neutrally stratified planetary boundary layer. J.Atmos. Sci. 56: 891–900.

Coleman GN, Ferziger JH, Spalart PR. 1990. A numerical study of theturbulent Ekman layer. J. Fluid Mech. 213: 313–348.

Coleman GN, Johnstone R, Ashworth M. 2005. ‘DNS of the turbulentEkman layer at Re = 2000’. In ERCOFTAC Workshop Direct andLarge-Eddy Simulation-6, Poitiers, France, 12–14 September 2005,1–8.

Csanady GT. 1967. On the resistance law of a turbulent Ekman layer.J. Atmos. Sci. 24: 467–471.

Freedman FR, Jacobson MZ. 2002. Transport-dissipation analyticalsolutions to the E–e turbulence model and their role in predictionsof the neutral ABL. Boundary-Layer Meteorol. 102: 117–138.

Garratt JR. 1994. The Atmospheric Boundary Layer. CambridgeUniversity Press.

Garratt JR, Hess GD. 2002. ‘Neutrally stratified boundary layer’.Pp 262–271 in Holton JR, Pyle J, Curry J (eds), Encyclopedia ofAtmospheric Sciences. Academic Press.

Gill AE. 1968. Similarity theory and geostrophic adjustments. Q. J. R.Meteorol. Soc. 94: 586–588.

Hess GD, Garratt JR. 2002. Evaluating models of the neutral, barotropicplanetary boundary layer using integral measures. Part I: Overview.Boundary-Layer Meteorol. 104: 333–358.

Howard GC, Slawson PR. 1975. The characteristics of a laboratoryproduced turbulent Ekman layer. Boundary-Layer Meteorol. 8(2):201–219.

Izakson AA. 1937. On formula for the velocity distribution near walls.Tech. Phys. USSR 4: 155–159.

Kazanski AB, Monin AS. 1961. On the dynamic interaction betweenthe atmosphere and earths surface. Izv. AN SSSR Geophys. 5:514–515.

Kolmogorov AN. 1941. The local structure of turbulence inincompressible viscous fluid for very large Reynolds numbers. Dolk.Akad. Nauk. SSSR 30: 9–13 (reprinted in P. R. Soc. A 434: 9–13,1991).

Lin CL, Moeng CH, Sullivan PP, McWilliams JC. 1997. The effect ofsurface roughness on flow structures in a neutrally stratified planetaryboundary layer flow. Phys. Fluids 9(11): 3235–3249.

Millikan CB. 1938. ‘A critical discussion of turbulent flow in channelsand circular tubes’. Pp 386–392 in Proc. 5th International Congressof Applied Mechanics.

Prandtl L. 1935. ‘The mechanics of viscous fluids’. Pp 34–208 inDurand WF (ed), Aerodynamics Theory, Vol.3. California Instituteof Technology: Pasadena, California.

Prandtl L, Tollmien W. 1924. Die Windverteilung ber dem Erdboden,errechnet aus den Gesetzen der Rohrstrmung. Z. Geophys. 1: 47–55.

Rossby CG, Montgomery RB. 1935. The layers of frictional influencein wind and ocean currents. Pap. Phys. Oceanogr. Meteorol. 3(3):101 pp.

Shingai K, Kawamura H. 2004. A study of turbulence structure andlarge-scale motion in the Ekman layer through direct numericalsimulations. J. Turbul. 5: 013 (http//jot.iop.org/).

Swimbank WC. 1974. The geostrophic drag coefficient. Boundary-Layer Meteorol. 7: 125–127.

Zilitinkevich SS. 1975. Resistance laws and prediction equations for thedepth of the planetary boundary layer. J. Atmos. Sci. 32: 741–752.

Zilitinkevich SS. 1989. Velocity profiles, the resistance law and thedissipation rate of mean flow kinetic energy in a neutrally and stablystratified planetary boundary layer. Boundary-Layer Meteorol. 46:367–387.

Zilitinkevich SS, Deardorff JW. 1974. Similarity theory for theplanetary boundary layer of time-dependent height. J. Atmos. Sci.31: 1449–1452.

Zilitinkevich SS, Esau IN. 2002. On integral measures of the neutral,barotropic planetary boundary layers. Boundary-Layer Meteorol.104: 371–379.

Zilitinkevich SS, Esau IN. 2005. Resistance and heat/mass transfer lawsfor neutral and stable planetary boundary layers old theory advancedand re-evaluated. Q. J. R. Meteorol. Soc. 131: 1863–1892.

Zilitinkevich SS, Laikhtman DL, Monin AS. 1967. Dynamics of theboundary layer in the atmosphere. Izv. AN SSSR Fiz. Atm. 3(3):297–333.


power-law velocity profile in a turbulent ekman layer on a transitional rough surface

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