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

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1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 34: 3 25 (2008) Published online in Wiley InterScience ( Power-law velocity profile in a turbulent Ekman layer on 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 ɛ) for large Rossby numbers is proposed. The flow in the inner wall layer the outer wake layer are matched, using the Izakson Millikan Kolmogorov hypothesis; this leads to an open functional equation. Another open functional equation is obtained from the ratio of two successive derivatives of the basic functional equation; this admits two functional solutions, with power-law log-law velocity profiles. The envelope of the geostrophic-drag power law leads to the log law, determines the power-law index prefactor as a function of the surface Rossby number or the drag coefficient. The log laws power laws for velocity friction velocity, including the power-law constants, are universal, independent of the wall roughness. This universality is well supported by extensive experimental laboratory data. In traditional smooth-wall variables, there is no universality of scalings, different expressions are needed for different types of roughness. Approximate solutions of the power-law geostrophic drag cross-isobaric angle are also obtained. The power-law geostrophic-drag solution for each prescribed value of the power-law index is valid for a limited domain of Rossby numbers. Copyright 2008 Royal Meteorological Society KEY WORDS power-law log-law velocity profiles; Rossby-number similarity theory; neutral barotropic planetary boundary layer; Izakson Millikan Kolmogorov hypothesis Received 8 September 2007; Revised 6 May 2008; Accepted 5 June Introduction The pioneering paper of Prtl Tollmien (924), recognizing the inadequacy of the assumption of a constant eddy viscosity K to describe observed wind structure in a planetary boundary layer (PBL), allowed K to vary with height. Their model for the PBL, consisting of a single layer with K = cz m,where c = Ro 0.2 z m 0 G m = 0.843, is based on neutral flow in the constantstress boundary layer of a pipe. Here Ro = G/f z 0, where G is the geostrophic-wind velocity (for barotropic flow, the geostrophic wind is constant with height), f is the Coriolis parameter (positive north of the Equator), z 0 is the surface roughness length. The dimensionless number Ro is called the surface Rossby number. From Rossby-number similarity theory, the friction velocity, geostrophic resistance C g = /G cross-isobaric angle γ can be determined. Prtl Tollmien (924) were the first to propose an expression for the geostrophic-drag coefficient C g as a function of Ro,namelyC g = 0.48Ro 0.9. From dimensional analysis, Swimbank (974) proposed the power law, estimated the values of the constants, in geostrophic drag as * Correspondence to: Noor Afzal, Faculty of Engineering Technology, Aligarh Muslim University, Aligarh , India. noor.afzal@yahoo.com C g = 0.Ro 0.07 or C g = 0.3Ro /4 for the range 0 5 <Ro<0 9. Rossby Montgomery (935) provided an important insight into the nature of the PBL by dividing it into two layers. With this model, they obtained the resistance law expressing the geostrophic-drag coefficient the crossisobaric angle in terms of the surface Rossby number Ro (or Ro ). Kazanski Monin (96) derived precisely the same law from similarity theory. Further development of the theory proceeded as follows: Zilitinkevich et al. (967), Csanady (967), Gill (968), Blackadar Tennekes (968) Arya (975) extended the resistance law to stratified PBLs; Zilitinkevich Deardorff (974) Zilitinkevich (975, 989) extended the theory by accounting for variable PBL heights; Zilitinkevich Esau (2002, 2005) further extended the theory by accounting for the non-local effect of the free-flow stability. The various turbulent-flow models are presented in Hess Garratt (2002). Further, Garratt Hess (2002, p. 265, figure 2) present geostrophic-velocityprofile data (includes data from Izumi, Hay, Kerang, Western Weddell Sea, Caldwell in transitional roughwall variables (u/,z/z 0 ), Coleman (DNS) in fully smooth-wall variables (u/,z /ν), with smooth-torough conversion z 0 = 0.ν/ (z 0 is the aerodynamic roughness of the surface) in a semi-log plot based on log-law theory, but the failure of these data to fit a single universal curve in the inner layer region reflects Copyright 2008 Royal Meteorological Society

2 4 N. AFZAL the inherent experimental errors in the measured z 0 the presence of unsteadiness in the field over the l. The relation between the power-law log-law velocity profiles revives an issue raised by Prtl (935), who stated that the log law is the well-known limiting value obtained from the power law u = D y α + D 2 as α 0, D D 2, where u is the axial-velocity profile y is the normal distance from the wall. This proposition calls for accurate experimental data at high Reynolds number. Furthermore, because the pipe-flow velocity profile exhibits a very weak defect layer, a more severe test for the power-law velocity profile would be a high-reynolds-number turbulent boundary layer (where the overlap would not constitute the main body of flow ). The issue of the power-law velocity profile in a turbulent pipe, channel, boundary layer wall jet has been studied by various authors during the last decade. Relevant literature for a fully-smooth surface is presented in Afzal (2005a, 2005b). The analysis of the power-law turbulent velocity profile for transitional rough-pipe flow is described by Afzal, Seena Bushra (2006), for a transitional rough-surface boundary layer by Afzal (2007). Here we consider the simplest PBL in the steady state with horizontally-homogeneous neutral barotropic flow over a transitional rough surface. This basic state forms the cornerstone of our theoretical understing of the PBL. In the formula for the classical Rossby number, Ro = /f z 0, the coefficient /f is of the order of the outer depth δ of the turbulent Ekman layer. The Rossby number R = Ro adopted in this paper is itself based on the outer depth of a turbulent Ekman layer, δ = /f, where the constant of proportionality is of order unity (Rossby Montgomery, 935; Zilitinkevich, 989). In traditional log-law theory, the classical Rossby number Ro would suffice, because ln R = ln Ro + ln ln may be absorbed elsewhere (in the intercept). With the power law we do not have recourse to this device. We will show that, in the power-law theory of the turbulent Ekman layer, the depth-based Rossby number R plays a significant role in the overlap region. We will also compare the power-law theory with extensive direct-numerical-simulation (DNS), field laboratory data. We will analyse a two-layer asymptotic theory for open equations of a turbulent Ekman layer (without any closure model, such as eddy viscosity, mixing length or k ɛ), for large Rossby numbers. For a transitional rough surface, the inner length scale is z 0 the outerlayer scale is /f. We match the two layers using the Izakson Millikan Kolmogorov (IMK) hypothesis, to obtain an open functional equation. Another open functional equation, obtained from the ratio of two successive derivatives of the basic functional equation, admits two functional solutions, namely the power-law loglaw velocity profiles. The two functional solutions contain certain open constants that can be determined from matching, using either experimental data or a closure model. This is not surprising, as we are dealing with open equations of the mean turbulent motion. The analysis shows that both the log-law the power-law relations in the overlap region are universal, independent of the surface roughness. This universality in the inner overlap regions is supported by the velocity-profile DNS data of Coleman (999), Coleman et al. (2005) Shingai Kawamura (2004) on a fully-smooth surface, by the laboratory data of Caldwell et al. (972) on a transitional rough surface. For large values of z/z 0,the departure from universal behaviour is due to the influence of the outer layer. The envelope of the power-law friction factor relates the constants of the power law to those of the log law. The power-law constants are universal independent of the surface roughness. The geostrophic-drag predictions the cross-isobaric angle γ from power-law theory, as well as the log-law theory, are supported by extensive data from many sources, such as the laboratory data of Caldwell et al. (972) Howard Slawson (975), the DNS data of Shingai Kawamura (2004), Coleman (999), Coleman et al. (2005), Coleman et al. (990) Lin et al. (997), the turbulence model of Freedman Jacobson (2002): see Hess Garratt (2002, tables I V). 2. Equations of motion We will examine the flow geometry, in the adiabatic turbulent state, of an incompressible viscous fluid, over a transitional, rough, flat surface driven by a steady uniform pressure gradient subjected to steady system rotation. The flow above the boundary layer is geostrophic, i.e. the pressure gradient Coriolis force balance each other. The governing momentum Reynolds equations for the motion of the geostrophic wind are (Garrett, 994): ν 2 u z 2 + τ x ρ z = f(v V g); () ν 2 v z 2 + τ y ρ z = f(u U g). (2) Here the x direction is horizontally aligned along the surface stress, the y direction is also horizontal but perpendicular to the x direction, the z direction is vertically upwards. In the x y directions, the mean velocity components are u v, the Reynolds shear stresses are τ x = ρ u w τ y = ρ v w ; ν is the molecular kinematic viscosity, ρ is the fluid density, f is the Coriolis parameter (positive north of the Equator), U g V g are the geostrophic-wind components U g = ρf p y V g = ρf p x (3)

3 POWER-LAW VELOCITY PROFILE IN TURBULENT EKMAN LAYER 5 p is the static pressure. We analyse Equations () (2) for the semi-infinite Cartesian domain above a transitional flat surface in the horizontal x y directions: at z = 0, u = v = 0τ x = τ y = 0; as z, u U g, v V g τ x,τ y 0. At the upper boundary, the turbulence dies out. The geostrophic (horizontal) wind velocity G, the angle γ between the surface stress vector the surface geostrophic-wind vector (the cross-isobaric flow angle), the velocity components U g V g in the x y directions, are given by: U g = G cos γ V g = sgn G sin γ G = Ug 2 + V g 2. (4) tan γ = V g U g Here the sgn function takes the value + in the Northern Hemisphere in the Southern Hemisphere. This flow is statistically stationary, for sufficiently long times the turbulence forgets the initial conditions. The geostrophic-flow state is uniquely specified by G, f ν. The depth of the viscous Ekman layer is D = 2ν/f. The velocity scale G length scale D are used to define the Reynolds number Re as: Re = GD ν = G νf/2. (5) The Rossby number Ro roughness Rossby number Ro are: Ro = fz G 0 Ro = u, (6) fz 0 where the surface length scale z 0 is the aerodynamic roughness length adopted in the meteorological description of the atmospheric data = (τ S /ρ) /2 is the friction velocity, τ S being the surface shear stress. The outer depth δ of the turbulent Ekman layer is of the order of /f, without loss of generality it becomes δ = /f where is order unity (Rossby Montgomery, 935; Zilitinkevich, 989). The alternative Rossby number based on the outer depth of the turbulent Ekman layer is: 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 traditional log-law theory, the classical Rossby number Ro would suffice, because ln R = ln Ro + ln ln may be absorbed elsewhere (in the intercept). The parameter is a dimensionless form of the aerodynamic roughness scale z 0. The molecular viscous length scale near the wall ν/ is: = z 0 ν = z 0+. (8) The normal coordinate is z = y + d r, where the origin d r is located below the top of the roughness element, caused by irregular hydraulic roughness. This level lies between the protrusion bases heads, automatically satisfies the constraints 0 <d r <h d r = 0for a smooth surface. Clauser (954) proposed a method for determining the effective surface-roughness origin d r skin friction. The velocity scale is the friction velocity at the surface, the length scale is the outer turbulent depth δ, the inner length is z Outer layer In the outer layer, the velocity scale is the friction velocity the length scale is the turbulent depth δ. The outer variables are η = z/δ δ = /f. The horizontal velocity components u v are represented by velocity defects components: u U g = U(η) (9) v V g = V(η) the Reynolds-stress components: τ x = ρu 2 Ɣ x(η) τ y = ρu 2 Ɣ y(η). (0) The governing mean-momentum equations () (2) in the outer variables (9) (0) become: R 2 U η 2 + Ɣ x R 2 V η 2 + Ɣ y η = U The first-order equations, η = V. () Ɣ x η = V Ɣ y η = U, (2) with the outer boundary conditions U,V,Ɣ x,ɣ y 0as η, are non-viscous: they fail to satisfy the no-slip boundary condition on the surface, an inner layer near the wall region is needed Inner layer In the inner wall layer, the velocity scale is the friction velocity the length scale is the surface roughness scale z 0 used in the meteorological description of atmospheric data. The inner variable ζ is the velocity profiles u v are u = u + (ζ ) v = v + (ζ ) ζ = R η = z z 0, (3) (4)

4 6 N. AFZAL Reynolds shear stresses are τ x = ρu 2 τ x+(ζ ) τ y = ρu 2 τ. (5) y+(ζ ) The boundary-layer equations () (2), in terms of the inner variables (4) (5), become: 2 u τ x+ = R (v + V g+ ) 2 v τ y+ = R (u + U g+ ). (6) Integration gives: u + + τ x+ = + R ( ζv g+ ζ ( v + + τ y+ = R ζu g+ + ζ 0 u +dζ ) 0 v +dζ ). Here, R is just the inverse of the Rossby number Ro, which is a small parameter. Therefore the first-order inner equations are u + + τ x+ =, (7) v + + τ y+ = 0, (8) subject to the surface boundary conditions u + = v + = τ x+ = τ y+ = 0atζ = Matching In the turbulent motion, the equations are not closed unless a turbulence closure model is adopted. The approaches adopted by Izakson (937), Millikan (938) Kolmogorov (94) are model-free, appeal to an overlap hypothesis (Afzal, 976; Afzal Narasimha, 976). Thus, the IMK hypothesis may be stated as follows. In any turbulent flow, between the viscous the energetic scales there exists an overlap domain over which the solutions characterizing the flow in the two corresponding limits must match as the Reynolds number tends to infinity. The resemblance of the IMK hypothesis to conventional matching associated with a closed equation seems peculiar to turbulence theory. Matching the inner limit (ζ fixed, R ) of the outer expansions the outer limit (η fixed, R ) of the inner solution, using the IMK hypothesis, we arrive at the following functional equations connecting the unknown functions: for ζ η 0 in the overlap region, U g+ (R ) = U g = G cos γ V g+ (R ) = V g = sgn G sin γ. (2) For large values of the Rossby number (Ro ), we have U g+ (R ), matching dems that the left-h side must be unbounded for large ζ. Note that the matching would be impossible if u + were bounded as ζ. There are three variables ζ, η R, out of which two are independent as ζ = R η. We differentiate the functional equation (9) with respect to ζ, keeping η fixed, to yield u + = U g+ R R = U g+ R R ζ ζ u + = R U g+ R Further, differentiating the functional equation (9) with respect to η, keeping ζ fixed, to yield η U η = R U g+ R The above two matching conditions may be combined as ζ u + = R U g+ = η U R η. (22) 3. Log law from matching The integral of these relations gives the velocity profile as ku + = ln ζ (23) the geostrophic-resistance law as ku g+ = ln R A, (24) the outer velocity defect law becomes ku = ln η + A. (25) The stress at the surface has been assumed to have no y component, so that the wind in the surface layer also has no y component, v + (ζ ) = 0. (26) Matching relation (20) for the v component, we find: V g+ (R ) = V(0) = sgn B k (27) u + (ζ ) = U g+ (R ) + U(η), (9) v + (ζ ) = V g+ (R ) + V(η), (20) B 0 = B k. (28)

5 POWER-LAW VELOCITY PROFILE IN TURBULENT EKMAN LAYER 7 The relations (24) (27) (28), with A = A ln, become: k U g u = ln Ro A k V g u = sgn B. (29) The matching results (29) may also be expressed as: k u G = (ln Ro A) 2 + B 2 tan γ = sgn B (30) ln Ro A. The Reynolds shear stresses from velocity distributions in the overlap region are given by τ x+ = kζ R B k ζ τ y+ = R U (3) g+ζ + R k ζ(ln ζ ) Ɣ x = B k η R kη. Ɣ y = k η(ln η + A) The Reynolds shear τ x+ possesses a maximum at the point ζ x+,m = Ro B z x+,m G ν = Re, (32) B where the maximum value is τ x+,m = 2 k B Ro /2. (33) The composite solution for the velocity profiles u v from the inner outer solutions is then given by: k u u = ln ζ + BW u (η) k u v (34) = BW v (η) k u U g u = ln η + A + BW u (η) k v V g u = B ( W v (η) ). (35) Here W u is the wake function satisfying the boundary conditions W u (0) = 0W u ( ) =, W v is the wake function satisfying the conditions W v (0) = 0 W v ( ) =. At the edge of the boundary layer z = z δ, the geostrophic velocity is u = U g, the above relations become: A + B = ln η δ η δ = fz δ. (36) The outer-layer analysis with a closure model provides the two wake functions. If the outer-layer eddy viscosity is taken as constant, then the outer-layer analysis gives these wake functions, by a further extension of the work of Garratt Hess (2002), as: W u (η) = e βη( cos(βη) sin(βη) ), (37) W v (η) = e βη( cos(βη) + sin(βη) ), (38) where β = B/k. 4. Power law from matching In the classical Rossby number Ro = /f z 0, the outer depth δ of the turbulent Ekman layer is of the order of /f. The Rossby number R = Ro adopted in this paper is based on the outer depth δ = /f.in traditional log-law theory, the classical Rossby number Ro would suffice, because ln R = ln Ro + ln, ln may be absorbed elsewhere (in the intercept). For the power law there is no recourse to such a device. In the power-law theory of the turbulent Ekman layer presented here, the depth-based Rossby number R plays a significant role in the overlap region. The matching of the inner- outer-layer velocity profiles for large Reynolds numbers Ro is given by the relations (9) (22). For a fully-smooth surface, Izakson (937) considered a functional equation analogous to (9) for a smooth pipe, Millikan (938) considered its first derivative when proposing logarithmic laws. The approach of Izakson Millikan was extended to second order by Afzal (976) in fully-developed turbulent pipe or channel flow. Differentiating the functional equation (22) once more (or alternately differentiating the functional equation (9) twice), with respect to ζ,forη fixed, we get ( ζ u ) + = ( R R U g+ R ) R Based on the relation ζ = ηr, / R = η = ζ/r the above relation becomes ζ ( ) ( ) u+ U g+ = R R R R or ζ 2 2 u ζ u + = 2 U g+ U g+ R2 R 2 + R R In view of the matching relation (22), the first order derivative terms on both sides of above relation cancel each other, we get ζ 2 2 u + 2 = R 2 2 U g+, (39) R 2 whose solutions again give the log laws. The functional equation may be differentiated as many times as we please, but nothing new or inconsistent would be discovered. This differentiation is one of the keys for solving

6 8 N. AFZAL the functional equations, but other techniques may also be considered. We show here that these functional equations give an alternative solution of the power-law velocity distribution in the overlap region. The functional equation (39), when divided by the functional equation (22), gives the alternative functional equation: ζ 2 u + / 2 u + / = R 2 U g+ / R 2 U g+ / R, (40) the functional solution dems that ζ 2 u + / 2 u + / = α 2 U R g+ / R 2 U g+ / R = α, (4) where α is constant. Integrating these functional equations, we obtain: u + U g+ R = Jζα = JR α, (42) where J is a constant of integration. The two cases α = 0 α = 0 would arise during any further integration, giving respectively the log-law power-law velocity distributions friction factors. Integration of the relation (42) in the case α = 0gives: u + = Sζ α, (43) U g+ = SR α E, (44) S = J/α, (45) where E α are constants of integration. The matching relation (43) in the outer variable η yields: u + = S η α S = SR α. (46) Division of the matching relations (43) (44) yields an alternative velocity profile, from u ( Z ) α ( = E ), (47) U g δ S for E S this becomes the simple relation u/u g = (Z/δ) α. The power-law geostrophic resistance (44) becomes: U g = S( Ro ) α E. (48) Matching of the v component of velocity (20), for v (ζ ) = 0, requires: V g+ (Ro ) = sgn B 0. (49) The matching results (48) (49) may also be expressed as: G (S( Ro = ) u α E ) 2 + B 2 0 (50) tan γ = sgn B 0 S( Ro ) α E. (5) The formulae for the Reynolds shear stresses in the region of overlap between the inner outer layer become: τ x+ = Sα α R ζ B 0ζ τ y+ = R U g+ζ + R Ɣ x = B 0 η R Ɣ y = S + α η+α S + α ζ +α S α Y α (52). (53) The Reynolds shear stress τ x+ given by (52) has a maximum at ζ x+,m = ( S B 0 α( α)ro ) 2 α, (54) where the maximum value is τ x+,m = 2 α ( S ) α( α) α B 0 2 α Ro α 2 α. (55) The composite solution of the velocity profiles u v from the inner outer solutions yields: u u = Sζ α + B 0 u (η) v (56) u = B 0 v (η) u U g u = S (η α ) + B 0 ( u (η) ) v V g ( u = B 0 v (η) ). (57) Here u is the wake function satisfying the boundary conditions u (0) = 0 u ( ) =, v is the wake function satisfying the conditions v (0) = 0 v ( ) =. At the edge of the boundary layer z = z δ,the geostrophic velocity is u = U g, the above relations yield: B 0 = ηδ α η δ = fz. (58) δ 4.. Envelope of the friction-factor power law The friction-factor power law (48) with the assistance of relation (45) becomes: U g = J α exp(α ln R ) E, (59)

7 POWER-LAW VELOCITY PROFILE IN TURBULENT EKMAN LAYER 9 which forms a family of curves in the (U g+, ln R φ ) plane, parametrized by α. This family has an envelope that satisfies (59), the equation U g+ / α = 0 may be simplified as: α = ln( Ro ). (60) The skin-friction power law (59), after elimination of the power index α from relation (60) for the geostrophicresistance law, becomes the friction log law: provided k U g u = ln( Ro ) ke ke = A + ln, (6) J = (k exp()). (62) The power-law prefactor S from relation (45) may be expressed as: S = k exp()α = k exp() ln( Ro ). (63) In relations (60) (63), the term ln( Ro ) may be eliminated, using relation (6), to get alternative expressions for the power-law constants α S as α = k /U g (64) + E /U g In this case, ( Ug ) S = exp( ) + E. (65) C = C g cos γ = U g. (66) The relations (54) (55) for the maxima ζ x+,m τ x+,m may be simplified as ( exp( ) ζ x+,m = B ( α)ro ) 2 α τ x+,m = 2 α ( exp( ) ) α B ( α) 4.2. Approximate power-law model 2 α Ro α 2 α. The velocity distribution (47), in an approximate form where E S, becomes: u ( Z ) α. = (67) U g δ The geostrophic-resistance relation (50) may also provide a good approximation, as G = S( Ro ) α, (68) because the effect of the constant E is roughly counterbalanced by that of the constant B 0. The expressions (67) (68), based on the relations (60) (for the power index α) (63) (for the prefactor S), may be expressed in terms of Ro as: G = MRo n (69) tan γ = sgn B 0 MRo n, (70) where the power index n prefactor M are given by the relations: n = α (7) + α M = (S α ) +α = ( αk exp() α) +α. (72) The power-law geostrophic drag /G crossisobaric angle γ, for various values of the power-law index, are given in Table I. These may be compared with the power laws of Prtl Tollmien (924) (C g = 0.48Ro 0.9 ) Swimbank (974) (C g = 0. Ro 0.07, C g = 0.3Ro /4 ). 5. Results discussion We have analysed the geostrophic-velocity-profile DNS data of Coleman (999) Coleman et al. (2005) for Re = 2000, 000 on a fully-smooth surface, the DNS data of Shingai Kawamura (2004) for Re = 775, 600, 50, 400 on a fully-smooth surface, the laboratory data of Caldwell et al. (972) for Re = 753, 234, 59, 704. Shingai Kawamura (2004) Coleman (999) have presented data on a transitional rough surface in terms of fully-smooth wall variables (u/,z /ν). Caldwellet al. (972) presented the velocity-profile data (u/,z/z 0 ) with a shift of origin in z/z 0. Here again, the fit of the data to a single universal curve was not considered. Garratt Hess (2002, p. 265, figure 2) have also presented geostrophic-velocity-profile data, but these did not fall on a single universal curve because of inherent experimental errors. Figure (a) shows the geostrophic-velocity-profile data in semi-log plots in the transitional rough-wall variables (u/,z/z 0 ). A substantial log-law region exists for each Reynolds number, its domain increases as the Reynolds number increases. The data fall on a single Table I. The power-law geostrophic drag /G crossisobaric angle γ, for various values of the power-law index α. α /G tan γ / Ro /8.4896Ro /8 / Ro / 0.976Ro / / Ro / Ro /4

8 20 N. AFZAL universal curve in the inner region, consisting of a sublayer, the region between the sub-layer the log law, the overlap region. The departure from universal behaviour for large z/z 0 is due to the outer layer of the geostrophic flow. Figure (b) shows the geostrophic-velocity-profile data in log log plots in the wall variables (u/,z/z 0 ).A substantial power-law region exists for each Reynolds number, its domain increases as the Reynolds number increases. The data fall on a single universal curve in the inner region, consisting of a sub-layer, the region between the sub-layer the power law, the overlap region. The departure from universal behaviour for large z/z 0 is due to the outer layer of the geostrophic flow. Figure 2 shows the profile data for the geostrophic total velocity Q = u 2 + v 2 in the transitional roughwall variables (Q/,z/z 0 ), on a semi-log plot for the log-law representation on a semi log plot for the power-law representation. In each panel, the data fall on a universal curve in the inner region (consisting of a sub-layer region, the region between the sub-layer the overlap region, the overlap region). A substantial overlap region exists for each Reynolds number, whose domain increases as the Reynolds number increases. Figure 3 shows the profile data for the geostrophic (a) (b) (a) (b) Figure 2. As Figure, but for the mean geostrophic total velocity Q/, where Q = G = u 2 + v 2. cross velocity in the transitional rough-wall variables (v +,z/z 0 ). It shows a wake-like region, which depends on the Rossby number for large z/z 0 in the outer-layer geostrophic flow v +. The power-law velocity profiles in the inner outer regions of the overlap domain are governed by the relations: u + = Sζ α u + = S η α The velocity-defect power law is: u U g u = S (η α ) + E S = S( Ro ) α. (73). (74) Figure. Inner scaling of the mean-geostrophic streamwise velocity profile u/, from DNS data of Coleman (999) Coleman et al. (2005) Shingai Kawamura (2004) laboratory data of Caldwell et al. (972), for various values of the Reynolds number Re: (a) log-law representation in semi-log plot; (b) power-law representation in log log plot. Figure 3. As Figure (a), but for the mean geostrophic spanwise velocity profile v/.

9 POWER-LAW VELOCITY PROFILE IN TURBULENT EKMAN LAYER 2 The components of the geostrophic-drag power law satisfy: U g = S( Ro ) α E; (75) V g = sgn B 0. (76) cross- The power-law geostrophic drag G/ isobaric angle γ are given by G (S( Ro = ) u α E ) 2 + B 2 0 (77) tan γ = sgn B 0 S( Ro ) α E (78) respectively. The asymptotic expansion of the geostrophic drag (77) for large Reynolds number yields: Figure 4. Power-law prefactor S plotted against the reciprocal of the power-law index α, from DNS data from the smooth surface by Coleman et al., (2005) for Re = 2000, Coleman (999) for Re = 000, Shingai Kawamura (2004) for Re = 775, 600, 50, 400, the laboratory data of Caldwell et al., (972) for Re = 753, 234, 58, 704. The line is a result of the present work (63), a fit represented by (8). G = S( Ro ) α E + ( 2 B2 0 S( Ro ) α) + To lowest order, this becomes: G = S( Ro ) α E. (79) The DNS data of Coleman (999) Coleman et al. (2005) Shingai Kawamura (2004) on a fullysmooth surface the laboratory data of Caldwell et al. (972) on transitional rough surface, shown on log log plots in the transitional rough-wall variables (u/,z/z 0 ), reveal a substantial linear region representing the power-law domain of the overlap, as already shown in Figure (b). We have carefully estimated the power-law constants, consisting of the power-law index α the prefactor S, for each velocity profile, on a magnified scale (graphs not shown here), where the linear portion of each dataset is fitted with a straight line. The intercept of the straight line represents the prefactor S, its slope represents the power index α. Theseare estimated using the Microsoft Excel program. Figure 4 shows the prefactor S plotted against the reciprocal of the power-law index α, from the data of Coleman (999) Coleman et al. (2005), Shingai Kawamura (2004) Caldwell et al. (972). The least-squares fit to the data is represented by the relation: S = 0.9 α. (80) Figure 5. As Figure 4, but for the power-law index α as a function of the surface Rossby number Ro = /f z 0. The line is a result of the present work (60) with = 0.3. Figure 5 shows the power index α plotted against the Rossby number Ro, from relation (60), for the above data, with = 0.3 (Zilitinkevich, 989), using the relation: α = ln(0.3ro ). (8) Figure 6 shows the prefactor S plotted against Ro on a semi-log scale. The equation of best fit is: S = 0.9lnRo.5. (82) Figure 7 shows an alternative expression of the powerlaw index α in terms of the dimensionless friction velocity /U g, from the same data. In light of our prediction (64), this relation is expressed as: α = 2.5/U g.5 /U g. (83)

10 22 N. AFZAL Figure 6. As Figure 4, but for the power-law prefactor S as a function of the surface Rossby number Ro = /f z 0. The line is a result of the present work. Figure 8. As Figure 4, but for the power-law prefactor S as a function of the reciprocal of the geostrophic resistance ɛ = /U g. The line is a result of the present work (65). (a) Figure 7. As Figure 4, but for the power-law index α as a function of the geostrophic resistance ɛ = /U g. The line is a result of the present work. Figure 8 shows the power-law prefactor S plotted against the reciprocal of the dimensionless friction velocity, from the same data. In light of our prediction (65), this relation is expressed as: ( Ug ) S = exp( ) (84) The geostrophic-drag predictions have been analysed from extensive data from many sources: the laboratory data of Caldwell et al. (972) Howard Slawson (975); the DNS data of Shingai Kawamura (2004), Coleman (999), Coleman et al. (2005), Coleman et al. (990) Lin et al. (997); the turbulence model of Freedman Jacobson (2002): see Hess Garratt (2002, tables I, II, IV). Figure 9(a) shows the components of the geostrophic drag U g / plotted against the Rossby number Ro, from the above data, on a semi-log plot. The data are scattered it is not easy to draw an appropriate line. However, with Karman constant k = 0.4, the geostrophic-drag log law (6) obtained from the envelope of the geostrophicdrag power law fitted to the data becomes: k U g = ln Ro.2. (85) (b) Figure 9. Comparison of the power-law log-law geostrophic drag relations as a function of the roughness Rossby number Ro = /f z 0, from extensive field laboratory data from various sources. Our predictions are indicated by the lines (a) ku g / = 2.3lnRo.2 (b) kv g / = 2.7. The geostrophic-drag data at higher Rossby number are scattered about the log law (85), but for lower Rossby number (Ro 0 4 ) the systematic departure from the log law indicates the presence of higher-order effects. The power-law geostrophic drag U g / based on our prediction (75) has been estimated by using the relations (8) (82); results are shown in Figure 9(a) by the solid line. The results are in good agreement with the log-law relation (85), shown in the same figure as a dashed line.

11 POWER-LAW VELOCITY PROFILE IN TURBULENT EKMAN LAYER 23 Figure 9(b) shows the cross components of the geostrophic drag V g / plotted against the Rossby number Ro, from the same data. There is large scatter in the data, it is not possible to draw an appropriate asymptotic line for large values of the Rossby numbers Ro.The lines shown are based on the DNS data of Shingai Kawamura (2004), Coleman (999) Coleman et al. (2005). Figure 0 shows the geostrophic drag G/ plotted against the Rossby number Ro, from the same data. The data are scattered, it is not easy to draw an appropriate line. However, with Karman constant k = 0.4, the geostrophic-drag log law (obtained from the envelope of the geostrophic-drag power law) is fitted to the data as: G = 2.5lnRo 2, (86) where the higher-rossby-number data are scattered about the log law, for lower Rossby number (Ro 0 4 ) there is systematic departure from the log law (86), indicating higher-order effects. The power-law geostrophic drag G/ from prediction (79) subject to relations (8) (82) is shown by a solid line, which is in agreement with the log law (86) shown by a dashed line. The cross-isobaric angle γ from log-law theory is then given by: 2.7 tan γ = 2.3logRo. (87) Figure shows the variation of the cross-isobaric angle γ as a function of the roughness Rossby number Ro = /f z 0, from extensive field laboratory data from various sources. The data are widely scattered, but predictions from log-law theory (87) compare well with power-law theory (78). Our approximate geostrophic-drag relations (Table I) for power-law index α = /7, /0, /3, comparison with the data shown in Figure 2(a), are valid for limited domains of the Rossby number: α = /7 (n = /8) for 3.6 < log 0 Ro < 6, α = /0 (n = /) for 4.5 < log 0 Ro < 6.6, α = /3 (n = /4) for 6 < log 0 Ro < 8.4. Figure 2(a) also compares the Figure. Variation of tan γ as a function of the roughness Rossby number Ro = /f z 0, from power-law theory log-law theory, with extensive field laboratory data from various sources. (a) (b) Figure 2. The geostrophic-drag vector: (a) geostrophic drag G/ the power-law proposals of Prtl Tollmin (924) Swimbank (974), (b) cross-isobaric angle γ (expressed as tan γ ), as a function of the Rossby number Ro = G/f z 0, from the power law with index α = /7, /0, /3, log law general rational power law, with extensive field laboratory data from various sources. Figure 0. Comparison of the power-law log-law geostrophic drags present work G/ as a function of the Rossby number Ro = /f z 0, with extensive field laboratory data from various sources. geostrophic-drag power laws G/ = 0.48Ro 0.9 of Prtl Tollmien (924) G/ = 0.3Ro /4 of Swimbank (974) with the data; this shows the limitations of their validity. In Figure 2(b), the cross-isobaric angle γ as a function of the roughness Rossby number Ro = G/f z 0 for power-law index α = /7, /0, /3 is compared with extensive field laboratory data from various sources. These predictions are valid for the limited domains of the Rossby number described above.

12 24 N. AFZAL 6. Conclusions The classical Rossby number is Ro = /f z 0, the outer depth δ of the turbulent Ekman layer is of the order of /f. The Rossby number R = Ro adopted in this paper is based on the outer depth δ = /f. In traditional log-law theory, the classical Rossby number Ro would suffice, because ln R = ln Ro + ln ln may be absorbed elsewhere (in the intercept). For the power law we do not have recourse to such a device. In the power-law theory of the turbulent Ekman layer, the depth-based Rossby number R plays a significant role in the overlap region. We have compared the power-law theory with extensive DNS, field laboratory data. We have analysed the open equations of the mean turbulent geostrophic boundary layer in the two sublayers. Their matching in the overlap domain leads to a open functional equation, by the IMK hypothesis. An alternative functional equation, based on the ratio of two successive derivatives, leads to two functional solutions: the power-law log-law velocity profiles friction factors. The data for the geostrophic-velocity profile in transitional rough-wall variables (u/,z/z 0 ) lie on a single universal curve in the inner layer, which consists of a sub-layer, the region between the sub-layer the overlap region, the overlap region itself. The departure from the universal relation for large z/z 0 is the contribution of the outer layer away from the surface. Likewise, the predictions of geostrophic drag C g cross-isobaric angle γ as a function of Rossby number Ro are universal relations that are independent of the surface roughness. These facts from powerlaw theory log-law theory are well supported by extensive experimental data for all types of transitional surface roughness. The data on a semi-log plot show substantial universal logarithmic behaviour, u + = k ln(z/z 0 ),aspredicted by relation (23), in the overlap region. The domain of the log region increases as the Reynolds number increases. Here the Karman constant k is a universal number (for example, k = 0.4). The data on a log log plot show substantial universal power-law behaviour, u + = S(z/z 0 ) α, as predicted by relation (43) in the same overlap region. The domain of the power-law region increases as the Reynolds number increases. Here the power index α prefactor S are universal functions, independent of the transitional surface roughness. The power-law constants α, S E, the log-law constant k the surface roughness z 0 have to be determined by experiments, as the matching procedure will not determine them. The envelope of the geostrophicdrag power law shows that the power-law constants α S are functions of Ro or ɛ = /U g, as appropriate, but are independent of the surface roughness z 0. Hence, the power-law constants α S of a fullysmooth surface, with appropriate representation, would suffice. The power-law constants α S may be estimated from the relations (64) (65), provided the local geostrophic drag C = /U g is known. The surface roughness scale z 0 enters the picture only if we intend to estimate the power-law constants from the alternative expression (48) Table I containing Ro = /f z 0 Ro = G/f z 0 respectively. The approximate power-law velocity profile (67), the expressions (69) (70) of the geostrophic drag G/, the cross-isobaric angle γ as a function of the Rossby number Ro = G/f z 0, are valid for bulk-of-flow approximation. In particular, the approximate power-law solution for a given geostrophic-drag power-law index (for example, n = /8, /, /4), for the power law of Prtl Tollmien (924) for n = 0.9, for the power laws of Swimbank (974) for β = 0.07, /4, have been compared with the traditional log law, the general rational power law, extensive field laboratory data from various sources. The approximate power-law geostrophic drag cross-isobaric angle have limited domains of validity in terms of the Rossby number: α = /7 (n = /8) for 3.6 < log 0 Ro < 6, α = /0 (n = /) for 4.5 < log 0 Ro < 6.6, α = /3 (n = /4) for 6 < log 0 Ro < 8.4. Acknowledgements The author is grateful for the support of the All India Council of Technical Education, New Delhi. A Appendix: Notation A additive constant in geostrophic drag law (29) A additive term in velocity-defect law (25) geostrophic drag law (24) B constant in the skin-friction relation B 0 a constant in Equation (49) C g geostrophic-drag coefficient /G C constant in Equation (66) d r hydraulic roughness height of the surface D depth of viscous Ekman layer, (2ν/f ) /2 E additive term in skin-friction power law (44), B 0 f Coriolis parameter G geostrophic wind speed J constant of integration in power laws (42) k Von Kármán constant K outer-layer eddy viscosity n power index for geostrophic drag, α/(α + ) p(x,y) static geostrophic-pressure distribution Re Reynolds number, GD/ν Ro Rossby number, G/f z 0 Ro frictional Rossby number, /f z 0 R Rossby number based on outer depth, / fz 0 S prefactor in inner power law velocity profile (43) power law geostrophic drag (44)

13 POWER-LAW VELOCITY PROFILE IN TURBULENT EKMAN LAYER 25 S prefactor in outer power law velocity profile (46) u(z) geostrophic wind speed in x direction u velocity fluctuations in x direction friction velocity, (τ S /ρ) /2 U g geostrophic wind speed in x direction U(η) outer-velocity defect in x direction v(z) cross-geostrophic wind speed in y direction v velocity fluctuations in y direction V(η) outer-velocity defect in y direction V g geostrophic wind speed in y direction w velocity fluctuations in z direction W u (z) u component of wake function in log-law theory W v (z) v component of wake function in log-law theory x coordinate in direction of surface stress y coordinate perpendicular to surface stress z vertical coordinate z 0 aerodynamic roughness of the surface z 0+ dimensionless aerodynamic roughness, z 0 /ν α power index in the power-law velocity profile β a constant, B/k γ cross-isobaric angle of geostrophic wind Ɣ x (η) outer dimensionless Reynolds shear stress in x direction Ɣ y (η) outer dimensionless Reynolds shear stress in y direction δ depth of the turbulent Ekman layer constant in the boundary layer thickness of the Ekman layer, δ = /f ζ inner variable, z/z 0 η outer variable, z/δ ν molecular kinematic viscosity ρ density of fluid τ x Reynolds-shear-stress x component, ρ u w τ y Reynolds-shear-stress y component, ρ v w τ S shear stress on the surface dimensionless aerodynamic roughness, z 0+ u (z) u component of wake function in power law v (z) v component of wake function in power law References Afzal N Millikan argument at moderately large Reynolds numbers. 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