Wind drift in a homogeneous equilibrium sea1

R. M. Samelson∗2

College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, OR, USA3

∗Corresponding author address: R. M. Samelson, 104 CEOAS Admin Bldg, College of Earth,

Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331-5503, USA.

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E-mail: rsamelson@coas.oregonstate.edu6

Generated using v4.3.2 of the AMS LATEX template 1

ABSTRACT

In a homogeneous equilibrium sea, the only mean divergence of momentum

flux is that from the vertical gradient of the vertical flux of horizontal momen-

tum. An exact result for the surface wind drift in this case is derived from

a universality and symmetry argument under the assumption that both fluids

are neutrally stratified and that the only length scale in each fluid is the ra-

tio of friction velocity to Coriolis parameter. The resulting surface wind drift

is approximately 3.5% of the vector difference of the geostrophic wind and

geostrophic ocean current outside the respective momentum boundary layers.

The surface-wave processes that drive Stokes’ drift, and therefore also the

Stokes’ drift itself, are in this setting indistinguishable from the rest of the

wave-turbulent shear flow dynamics that force the mean flow, and this wind

drift response therefore must include any Stokes’ drift that may occur. A

physical or effective roughness of the surface introduces an additional length

scale that breaks the symmetry and may alter the surface wind drift and the

near-surface velocity fields. This effect is explored with a semi-analytical

log-Ekman model of the coupled boundary layers that includes a simple em-

pirical wave-correction factor φ . Values φ ≈ 0.6 and z∗0 ≈ 2 cm of the wave-

correction factor and ocean surface roughness length, chosen to fit a classi-

cal empirical relation between wind drift and 10-m winds, are found to give

log-Ekman solutions that also fit observed near-surface ocean current profiles

better than solutions with no wave correction (φ = 1).

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1. Introduction28

When the wind blows over the sea surface, the surface of the ocean is set in motion. Challenges29

to observing this surface wind drift and the accompanying near-surface ocean currents include the30

motion of the interface, the small vertical extent of the shear layer, and the sometimes ill-defined31

nature of the interface in the presence of breaking waves. The problem of determining this drift32

and its dependence on the wind continues to be of interest, both for intrinsic scientific reasons33

and because of its practical importance for understanding and predicting the motion of oil spills34

and other floating materials or objects. Among other novel technologies, a new generation of35

remote sensing instruments that use microwave Doppler scatterometry to measure the motion of36

the uppermost centimeters of the water column (Ardhuin et al. 2017; Rodriguez et al. 2018, 2019)37

both promise new measurements relevant to this problem, and invite complementary progress that38

would improve understanding and aid interpretation of the new measurements.39

The case considered here is that of a homogeneous equilibrium sea. The most general meaning40

of homogeneous is implied, so the ocean is taken to have uniform density and neutral stability.41

The equilibrium-sea condition means also that the surface wave field is in equilibrium with the42

local wind, so that mean wave amplitudes are constant and there are no horizontal divergences of43

wave momentum fluxes. In this case, there are no temporal or horizontal gradients of any fields44

– except, of course, the large-scale pressure gradients associated with the geostrophic wind and45

current outside the momentum boundary layers – and the only mean divergence of momentum46

flux is that from the vertical gradient of the vertical flux of horizontal momentum. In this setting,47

the surface-wave processes that drive Stokes’ drift (e.g., Phillips 1977), and therefore also the48

Stokes’ drift itself, are in this sense indistinguishable from the rest of the wave-turbulent shear49

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flow dynamics that contribute to this flux divergence, and the equilibrium surface and near-surface50

velocity response must be characterized more generally as a wind drift.51

A general argument is first given that is based only on an assumption of perfect turbulent uni-52

versality (Reynolds’ number similarity) for the oceanic and atmospheric surface boundary layers,53

and is shown to yield an exact expression for the surface wind drift as a function of the geostrophic54

wind and current outside the boundary layer and the ratio of the atmospheric and oceanic fluid den-55

sities. Violations of the assumed perfect universality must however arise from asymmetry near the56

interface in the dimensionless framework. An analytical, depth-dependent, eddy-viscosity model57

of the turbulence is then used to explore the effects of this asymmetry. A new wave-correction58

factor is introduced, and model profiles with and without the wave-correction factor are compared59

to observations of surface wind drift and near-surface shear.60

2. General formulation61

a. Dimensional equations and boundary conditions62

Consider the case of a homogeneous, equilibrium sea under a homogeneous atmosphere, so that63

in the suitably averaged horizontal momentum equations, all temporal and horizontal gradients64

vanish except the large-scale pressure gradients associated with the geostrophic wind and current65

outside the momentum boundary layers. Let the total mean atmosphere and ocean horizontal66

velocities be given by V∗ = (V x∗ ,V

y∗ ) and U∗ = (Ux

∗ ,Uy∗ ) , respectively. These velocities then67

satisfy the corresponding mean momentum equations,68

ρa f k× V∗ = −Ga−dτ∗dz∗

, z∗ > 0

ρo f k× U∗ = −Go−dτ∗dz∗

, z∗ < 0 (1)

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Here the positive constants ρa, ρo and f are the ocean and atmosphere densities and the Coriolis69

parameter, respectively; Ga and Go are the constant large-scale pressure gradients; τ∗(z) is the ver-70

tical turbulent flux of horizontal momentum; (x,y,z) are Cartesian coordinates, positive eastward,71

northward, and upward, respectively; k is the upward unit vector; and the air-sea interface is at72

z∗ = 0, with the atmosphere above (z∗ > 0) and the ocean below (z∗ < 0). It is assumed that the73

horizontal velocities approach their geostrophic values far from the interface,74

V∗→ V∗G =1

ρa fk×Ga as z∗→ ∞

U∗→ U∗G =1

ρo fk×Go as z∗→−∞. (2)

The deviations of the velocities from the geostrophic values,75

V∗ = V∗− V∗G, U∗ = U∗− U∗G, (3)

then satisfy76

ρa f k×V∗ =dτ∗dz∗

, z∗ > 0

ρo f k×U∗ =dτ∗dz∗

, z∗ < 0, (4)

and77

V∗→ 0 as z∗→ ∞, U∗→ 0 as z∗→−∞. (5)

The standard conditions of continuity of velocity and stress at the interface of two viscous fluids78

(Batchelor 1967) are taken to hold at the air-sea interface z∗ = 0:79

V∗(z∗ = 0) = U∗(z∗ = 0), (6)

or, equivalently,80

V∗(z∗ = 0)−U∗(z∗ = 0) =−(V∗G− U∗G), (7)

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and81

τ∗(z∗→ 0+) = τ∗(z∗→ 0−) = τ∗0, (8)

where z∗ → 0+ and z∗ → 0− denote the limits as z∗ = 0 is approached from above and below,82

respectively.83

b. Dimensionless equations84

The fundamental velocity and time scales v∗ = |τ∗0/ρa|1/2, u∗ = |τ∗0/ρo|1/2 and f−1 may be85

used to express the equations (4) in dimensionless form:86

k×V =dτ

dz, z > 0

k×U =dτ

dz, z < 0, (9)

where the dimensionless height and depth are87

z =z∗Da

, Da =v∗f, z > 0; z =

z∗Do

, Do =u∗f, z < 0; (10)

and the dimensionless velocity deviations are V = V∗/v∗ and U = U∗/u∗. The corresponding88

dimensionless boundary conditions are:89

V→ 0 as z→ ∞, U→ 0 as z→−∞, (11)

V−αU =−(VG−αUG) and τ = 1 at z = 0, (12)

where90

α =u∗v∗

=

(ρa

ρo

)1/2

(13)

is the ratio of the two friction velocities, and (UG,VG) = (U∗G/u∗,V∗G/v∗) are the dimensionless91

geostrophic velocities.92

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3. An estimate of wind drift93

The dimensionless equations (9)-(13) for the velocity deviations V and U depend on the di-94

mensionless geostrophic velocities VG and UG only through their difference V∗G/v∗−U∗G/u∗ =95

(V∗G−αU∗G)/v∗ = VG−αUG as expressed in the interface condition (12). The velocity devia-96

tions V and U will therefore be unchanged if VG and UG are replaced by V′G and U′G, where97

V′G = VG +∆V, U′G = UG +1α

∆V, (14)

and ∆V is arbitrary. Now, choose ∆V so that V′G =−U′G, i.e., so that VG+∆V =−(UG+∆V/α),98

which is satisfied if99

∆V =− α

1+α(VG +UG). (15)

The dimensionless equivalent of equations (1) for the modified total velocities U = U+U′G and100

V = V+V′G, with the associated far-field boundary conditions, are then anti-symmetric about the101

interface z = 0:102

k× (V−V′G) =dτ

dz, z > 0

k× (U−U′G) =dτ

dz, z < 0,

V→ V′G as z→ ∞, U→−V′G as z→−∞. (16)

That is, with z→−z and U→−V, the equation and far-field boundary condition for z < 0 trans-103

form into the corresponding equation and boundary condition for z> 0. Given the antisymmetry of104

the field equations and the far-field boundary conditions (16), it is then consistent to assume that105

the dimensionless modified total velocity profile is itself antisymmetric, with U(−z) = −V(z).106

This in turn implies that U = V = 0 at z = 0, that is, that U(0) =−U′G =−V(0) = VG.107

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With the velocity deviations V = U = 0 at z = 0 known in terms of the modified geostrophic108

velocities V′G and U′G, the original total velocity at z = 0 can be computed:109

U(0) = U(0)+UG =−U′G +UG = V′G +UG = VG−α

1+α(VG +UG)+UG, (17)

and the dimensional wind drift is110

U∗(0) =α

1+α(V∗G−U∗G)≈ α V∗G, (18)

where the approximation assumes |V∗G|� |U∗G| and α� 1. For ρa = 1.25 kg m−3 and ρ0 = 1025111

kg m−3, α ≈ 0.035, and consequently, under these assumptions, the surface ocean wind-drift112

current – which by this argument includes all effects that might be associated with wave-driven113

Stokes’ drift in a homogeneous equilibrium sea – will be approximately 3.5% of the geostrophic114

vector wind.115

A hidden assumption of the universality and symmetry argument yielding (18) is that the dimen-116

sionless air-sea interface has identical characteristics when viewed from above and below. This117

assumption is almost certainly violated for the air-sea interface. The large difference in the scale118

depths Da = v∗/ f and Do = u∗/ f , which are related by Do = αDa with α ≈ 0.035, means that119

the dimensionless roughnesses of the interface will appear very different even if the dimensional120

roughnesses are nearly equal. This is true whether the roughness is directly physical, as from cap-121

illary waves, or effective and virtual, as from vertical transport of momentum by gravity waves and122

gravity-wave breaking. From a general point of view, the amplitude of vertical displacements from123

the surface wave field introduces additional dimensional scales, beyond Da and Do, on which the124

dimensionless fields may depend. This in turn means that the dimensionless turbulent fields are125

likely to have different, rather than equivalent universal, characteristics near the interface, despite126

the antisymmetry in the far field.127

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4. A log-Ekman model128

a. Formulation129

To examine the effects of interface asymmetry on the surface wind-drift and near-surface cur-130

rents, a specific model of the turbulent stress is required. A standard approach is to related the131

stress to the velocity profile through eddy viscosities that may depend on the distance |z∗| from the132

air-sea interface:133

τ∗ = ρa K+∗ (z∗)

dV∗dz∗

, z∗ > 0; τ∗ = ρ0 K−∗ (z∗)dU∗dz∗

, z∗ < 0. (19)

The eddy viscosities are taken to have the log-layer form near the interface and the constant134

Ekman-layer form far from the interface,135

K+∗ (z∗) =

κ v∗ (z+∗0 + z∗), |z∗|< |z+∗1|

κ v∗ (z+∗0 + z+∗1), |z∗| ≥ |z+∗1|,

K−∗ (z∗) =

φκ u∗ (z−∗0− z∗), |z∗|< |z−∗1|

φκ u∗ (z−∗0− z−∗1), |z∗| ≥ |z−∗1|,

(20)

where κ is the von Karman constant, φ is a dimensionless constant, u∗ and v∗ are friction velocities,136

and z±∗0 are roughness lengths for the atmosphere and ocean, respectively; z∗ is the distance from137

the interface at z∗ = 0. Thus K±∗ increases linearly with distance from the interface in each case,138

until its respective maximum values K±∗,max are reached at the distances |z±∗1|, after which K±∗139

remains constant and equal to K±∗,max at all further distances from the interface.140

This model, with the log-layer form of the eddy viscosities and with φ = 1, and with the interface141

and boundary conditions given above, is similar to the steady version of the coupled boundary-142

layer model of Lewis and Belcher (2004). The model differs from Lewis and Belcher (2004) in143

that the eddy viscosities have constant values far from the interface; for the solutions described144

here, the dimensionless depths z±1 at which the constant values are attained are chosen to give145

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maximum dimensionless eddy viscosities K±(|z±| ≥ |z±1 |) = 0.03, but the profiles are relatively146

insensitive to this choice.147

In the absence of rotation ( f = 0), the form (20) gives logarithmic solutions for the velocity148

profiles in the regions 0 < |z∗|< z±∗1. The dimensionless constant φ is analogous to the inverse of a149

Monin-Obukhov stability function, and allows a departure of the near-interface ocean current shear150

from the universal neutral profile. This departure may be attributed physically to the influence of151

surface waves on the ocean near-surface turbulence and φ may be considered an empirical wave-152

correction factor.153

The fundamental velocity and time scales v∗ = |τ∗0/ρa|1/2, u∗ = |τ∗0/ρo|1/2 and f−1 may be154

used to express the equations (4) with (19) in dimensionless form:155

ddz

[K+(z)

dVdz

]− iV = 0, z > 0;

ddz

[K−(−z)

dUdz

]− iU = 0, z < 0; (21)

where the dimensionless height and depth are156

z =z∗Da

, Da =v∗f, z > 0; z =

z∗Do

, Do =u∗f, z < 0; (22)

the dimensionless velocities V∗/v∗ and U∗/u∗ have been written in complex form,157

V =V x + iV y, U =Ux + iUy, (23)

and the eddy viscosities have been scaled by v∗Da = v2∗/ f and u∗Do = u2

∗/ f ,158

K+(z) =1

v∗DaK+∗ , K−(−z) =

1u∗Do

K−∗ . (24)

With κ+ = κ and κ− = φκ , the dimensionless eddy viscosities may be written in a unified form,159

K±(z) =

κ± (z±0 ± z), |z|< |z±1 |

κ± (z±0 ± z±1 ), |z| ≥ |z±1 |.

(25)

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The corresponding dimensionless boundary conditions are:160

V → 0 as z→ ∞, U → 0 as z→−∞, (26)

V −αU =−(VG−αUG) and K+dVdz

= K−dUdz

at z = 0, (27)

along with161 ∣∣∣∣K+dVdz

∣∣∣∣= ∣∣∣∣K−dUdz

∣∣∣∣= 1 at z = 0, (28)

which is an additional constraint on the solution that follows from the definitions of the unknown162

scaling velocities v∗ and u∗.163

b. Standard form164

For |z| ≥ |z±1 |, the dimensionless eddy viscosity K± is independent of z, and (21) are the classical165

Ekman (1905) equations, which have exponential solutions. For |z| ≤ |z±1 |, where the diffusivities166

K± depend linearly on z, Ellison (1959) has shown that the resulting equations may be solved in167

terms of Hankel or modified Bessel functions. In the latter case, both equations in (21) can be168

reduced to a single standard form by the substitution169

ζ± = (1+ i)

[2

κ±(z±0 ± z)

]1/2

, (29)

in terms of which (21) takes the form170

ζ2 d2W

dζ 2 +ζdWdζ−ζ

2W = 0. (30)

In (30), W and ζ may be taken to represent either V and ζ+ or U and ζ−, as the same equation171

results in both cases. The equation (30) is a modified Bessel equation of order zero (Abramowitz172

and Stegun, 1964, p. 374, eq. 9.6.1), with solutions K0(ζ ) and I0(ζ ), the modified Bessel functions173

of order zero.174

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Solutions to (21) may thus be obtained analytically in terms of elementary functions, using175

K0(ζ ) and I0(ζ ) in |z| ≥ |z±1 , the classical exponential Ekman solutions in |z| ≥ |z±1 |, the bound-176

ary conditions (26)-(28), and imposing appropriate matching conditions at z = z±1 . The required177

matching conditions may be derived by integrating (21) across an infinitesimal interval enclos-178

ing z = z±1 ; the result is that V and dV/dz, and U and dU/dz, must be continuous at z = z±1 ,179

respectively:180

W (z→ z±1 |−) =W (z→ z±1 |+),dWdz

(z→ z±1 |−) =dWdz

(z→ z±1 |+), (31)

where W again represents either V or U , and z→ z±1 |± denotes the limits approaching z±1 from181

above (|+) and below (|−). From (29), it follows also that182

dζ±

dz=± 2i

κ±ζ±. (32)

c. Dimensionless solutions183

Let the x-direction be chosen in the direction of the geostrophic wind VG, where, without loss184

of generality, VG is taken to represent the difference of geostrophic wind and current, and UG is185

taken to be zero. Following this approach, (21) with (25) and the boundary conditions (11), (12)186

and (31) give the solution187

V (z) =

A+K0[ζ

+(z)]+B+ I0[ζ+(z)], 0 < z≤ z+1 ,

A+1 exp[−(1+ i)(z− z+1 )/δ+], z > z+1 ,

U(z) =

A−K0[ζ

−(z)]+B− I0[ζ−(z)], z−1 ≤ z < 0,

A−1 exp[(1+ i)(z− z−1 )/δ−], z < z−1 ,

where188

A+ =a2(c3d4− c4d3)

DVG, B+ =−a1(c3d4− c4d3)

DVG,

A− =d4(a1b2−a2b1)

DVG, B− =−d3(a1b2−a2b1)

DVG, (33)

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and189

a1 = − 2iκζ

+1

K1(ζ+1 )+

1+ iδ+

K0(ζ+1 ), a2 =

2iκζ

+1

I1(ζ+1 )+

1+ iδ+

I0(ζ+1 ),

b1 = K0(ζ+0 ), b2 = I0(ζ

+0 ), b3 =−αK0(ζ

−0 ), b4 =−αI0(ζ

−0 ),

c1 = −(κz+0

)1/2 K1(ζ+0 ), c2 =

(κz+0

)1/2 I1(ζ+0 ),

c3 = −(φκz−0

)1/2 K1(ζ−0 ), c4 =

(φκz−0

)1/2 I1(ζ−0 ),

d3 =2i

φκζ−1

K1(ζ−1 )− 1+ i

δ−K0(ζ

−1 ), d4 =−

2iφκζ

−1

I1(ζ−1 )− 1+ i

δ−I0(ζ

−1 ),

δ+ = [2κ(z0 + z+1 )]

1/2, δ− = [2φκ(z0− z−1 )]

1/2,

D = (a1b2−a2b1)(c3d4− c4d3)+(a1c2−a2c1)(b3d4−b4d3). (34)

The condition (28) requires additionally that190

(κz+0

)1/2 ∣∣−A+K1(ζ+0 )+B+ I1(ζ

+0 )∣∣= 1, (35)

and the solutions for A+1 and A−1 are obtained from191

A+1 = A+K0(ζ

+1 )+B+ I0(ζ

+1 ), A−1 = A−K0(ζ

−1 )+B− I0(ζ

−1 ). (36)

The corresponding dimensionless total velocity profiles are then U(z)+UG for z≤ 0 and V (z)+VG192

for z≥ 0, which satisfy the original dimensionless boundary conditions and in addition have stress193

of unit magnitude at the interface.194

d. Conversion to dimensional form195

For a given fluid, the roughness length and the geostrophic velocity are typically related to each196

other and to the stress at the surface, and consequently these three parameters are generally not197

independent. For the atmosphere, the relation is encoded in the neutral drag coefficient CDN , which198

relates the 10-m neutral wind V∗10N to the stress,199

v2∗ =CDN V 2

∗10N , (37)

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with the downward stress direction generally taken parallel to V∗10N . Under the further assumption200

that the wind profile is logarithmic from the surface through 10 meters, i.e., that V∗(z∗)/v∗ =201

(1/κ) ln[(z+∗0 + z∗)/z+∗0], CDN may be expressed in terms of the roughness length,202

CDN =κ2

ln2(z∗10/z+∗0)=

κ2

ln2(z10/z+0 ), (38)

where z+∗0 is the dimensional roughness length, z10 = z∗10 f/v∗, and it has been assumed that z+∗0�203

z∗10. For the calculations here, it is assumed that the roughness length is given by the Charnock204

relation for the atmospheric roughness length over the sea surface,205

z+∗0 =αcv2∗

g, (39)

where αc ≈ 0.02 is a dimensionless constant and g = 9.81 m s−2 is the acceleration of gravity.206

The dimensionless roughness length z+0 is then a function of the dimensional friction velocity v∗,207

z+0 =αc f

gv∗. (40)

Consequently, the unknown friction velocity v∗ can be obtained from z+0 ,208

v∗ =g

αc fz+0 , (41)

and this value of v∗ may be used to relate the dimensionless solutions obtained for a given value209

of z+0 to dimensional solutions.210

5. Wind and current profiles211

a. Standard case212

Solutions of these coupled neutral boundary layer equations with the Charnock relation (39)213

show both the Ekman-layer turning of the velocity deviation and the constant-stress log-layer214

structure near the air-sea interface (Figs. 1,2). Under the assumption that the dimensional ocean215

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and atmospheric roughness lengths are equal (z−∗0 = z+∗0) and that φ = 1 (no wave correction), the216

stress and velocity deviation profiles are nearly symmetric and anti-symmetric, respectively, and217

the ocean surface velocity is directed nearly parallel to the geostrophic wind, despite the difference218

in dimensionless roughness lengths and consistent with the estimate (18) derived in Section 3 for219

the exactly symmetric and anti-symmetric case of equal dimensionless roughness lengths.220

Note that the standard 10-m height at which atmospheric winds are typically defined for use in221

bulk stress parameterizations is located near the top of the atmospheric log layer in these solutions222

(Figs. 1,2). In contrast, the ocean log layer thickness in these solutions is smaller by a factor223

of α and so is only approximately 30 cm, much smaller than the amplitudes of surface-wave224

deformations of the overlying interface. This illustrates both the difficulty of observing the ocean225

log layer – if indeed it may exist in the ocean – and the asymmetry of the interface geometry and226

dynamics in the dimensionless context.227

The wind profile also shows, unexpectedly, a rapid clockwise rotation with depth through ap-228

proximately 10◦ within a few roughness lengths of the interface (e.g., Fig. 1, middle left-center229

panel). Hodographs of the wind and current profiles (e.g., Fig. 2, middle panel) reveal the origin230

of this rotation to be kinematic, rather than dynamic. It arises because, in these solutions, the wind231

is measured in a stationary reference frame (or, more generally, relative to the geostrophic ocean232

current), rather than relative to the moving ocean surface. Consequently, because of the assumed233

continuity of the velocity field at the interface, the wind velocity does not approach zero as the234

interface is reached, but instead approaches the surface ocean velocity. Because the stress and235

vertical wind shear in the constant-stress layer are directed to the left of the surface ocean current,236

the wind measured in the stationary frame rotates as the interface is approached. If the wind were237

measured relative to a reference frame moving with the surface ocean current, there would be no238

near-surface rotation of the wind. In contrast, the vector stress is constant through the interface239

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and into the log-layer regimes, regardless of the reference frame, because it depends upon the wind240

shear rather than the absolute wind (e.g., Fig. 1, middle and lower right-center and right panels).241

b. Wind drift with wave correction: φ < 1242

A standard empirical rule-of-thumb for the surface wind drift, originating in studies of oil-spill243

spread, is that it will be directed 15◦ to the right (in the northern hemisphere) of the 10-m wind244

at 3% of the wind speed (e.g., Weber 1983). For the log-Ekman model with κ− = κ = 0.4 and245

no wave-correction (φ = 1), it is possible to choose the ocean roughness length z−∗0 so that the246

resulting surface ocean velocity matches either one of these two conditions, but not both. The 3%247

of wind speed criterion is matched for z−∗0 = 4.9×10−4 m, but the corresponding angular deflection248

is only 11◦ (Fig. 3). Quantitative uncertainty bounds for the empirical rule-of-thumb numbers are249

not available, but it seems reasonable to expect that this 25% difference in the deflection relative250

to the 10-m wind direction would be observationally distinguishable and significant. Similarly,251

the 15% deflection is matched for z−∗0 = 8× 10−2 m, but the corresponding drift speed is only252

about 2%, rather than 3%, of the 10-m wind speed, a 30% difference that again can be expected to253

be observationally distinguishable and significant. Consequently, it appears from this comparison254

that the standard log-Ekman model, with no wave correction, can be rejected as an adequate model255

of the surface wind drift.256

If the log-Ekman model is generalized to include the wave correction, so that both the roughness257

length z−∗0 and the wave-correction constant φ are free parameters, then it is possible to find solu-258

tions of the log-Ekman model for which the surface ocean velocity matches both of the empirical259

rule-of-thumb numbers simultaneously. For a fixed roughness length, decreasing φ so that κ− < κ260

systematically increases the ratio of surface current speed to 10-m wind speed, with values of this261

ratio reaching 10% for z−∗0 ≈ 10−3 m and φ ≈ 0.25 (Fig. 3). Alternatively, increasing z−∗0 with262

16

φ fixed systematically increases the deflection angle, with values of this angle reaching 35◦ for263

z−∗0 ≈ 1 m (Fig. 3). For a given atmospheric surface roughness length of z+∗0 = 0.12 m, correspond-264

ing to a 10-m wind speed of approximately 7 m s−1, the solution that matches both rule-of-thumb265

numbers for surface wind drift has z−∗0 ≈ 2.3 cm and φ ≈ 0.23/0.4 = 0.575 (Fig. 3).266

c. Near-surface shear: comparison with observations267

The near-surface ocean velocity profiles from the log-Ekman model may also be compared to268

near-surface ocean current observations. A convenient, though not comprehensive, set of obser-269

vations for this purpose are those used by Lewis and Belcher (2004) in their related comparison,270

originally published by Briscoe and Weller (1984); Price et al. (1987); Chereskin (1995); Wijffels271

et al. (1994) and previously summarized by Price and Sundermeyer (1999). These observations272

are compared here with three solutions of the log-Ekman model, to illustrate the influence of the273

wave-correction parameter φ on the fits to the observed subsurface profiles. The observations had274

mean 10-m wind speeds of 6, 7, and 8 m s−1, but for simplicity all are compared here with the log-275

Ekman solutions selected from described above, which have approximately 7 m s−1 10-m wind276

speeds. The observations are made dimensionless by the observed friction velocities and Coriolis277

parameters, and compared to dimensionless model profiles.278

The three log-Ekman solutions chosen for the comparison all match the 3% rule-of-thumb cri-279

terion: the first is the standard solution with no wave correction (φ = 1) and z−∗0 = 4.9×10−4 m;280

the second is the optimal solution with φ ≈ 0.58 and z−∗0 ≈ 2.3 cm, which also matches the 15◦281

criterion; and the third has φ = 0.25 and z−∗0 ≈ 50 cm, which has an angular deflection near 25◦282

(Fig. 3). The comparison is limited by the small data set and the absence of data very near the283

surface, but the second, optimal-wind-drift solution appears to match the observations better than284

the other two solutions (Fig. 4). Especially if attention is restricted to the data points in the upper285

17

half of the spanned dimensionless depth interval, for which the observed velocities are more than286

twice the friction velocity and are thus likely more reliable statistically than the weaker, deeper287

velocities, the standard, (φ = 1) profile has too rapid a decay with depth of the current magnitude288

and too slow a rotation of the current with depth. In contrast, the third solution, with the more289

extreme wave correction φ = 0.25, has too slow a decay with depth of the current magnitude, and290

has a current rotation with depth that fits only the data points that show most rapid rotation. The291

second, optimal-wind-drift solution, with φ ≈ 0.58, matches the near-surface velocity magnitude292

points well, and gives a better match of rotation with depth than does the standard profile. The293

second solution matches best the deflection-angle data points that show the slowest rotation with294

depth, and in that sense is complementary to the third solution, suggesting that a suitably-weighted295

optimal fit to both the surface-wind-drift rule and the subsurface profile observations would yield296

a solution with parameters intermediate to the second and third solutions.297

The wind and current profiles for the optimal-wind-drift solution with φ ≈ 0.58 and z−∗0 ≈ 2.3298

cm are similar to the standard solution, but show subtle yet systematic differences (Figs. 5,6).299

With the much larger roughness length, the log-layer regime for the optimal solution is truncated300

relative to that for the standard solution, existing for only one decade, z−∗0 < |z−∗ | < 10z−∗0 ≈ 20301

cm, a distance from the interface an order of magnitude smaller than the amplitude of the interface302

deformation by the surface wave field. A small difference in the hodograph of near-surface vector303

ocean stress is also apparent, with the component of stress perpendicular to the geostrophic wind304

(or wind-current difference) systematically enhanced by roughly 20% relative to that parallel to305

the geostrophic wind.306

18

6. Discussion307

The starting point of this work is the thesis that in a homogeneous, equilibrium sea, the only308

relevant divergences of mean momentum fluxes are the vertical gradients of vertical transport of309

horizontal momentum. One consequence of this assertion is that dynamical representations of the310

resulting mean current profiles should be able to reproduce observed profiles without the prescrip-311

tive introduction of a Stokes’ drift velocity component derived independently through kinematic312

analysis of Lagrangian trajectories in the fluctuating wave field. Rather, it should be possible313

to find an appropriate representation of the mean vertical fluxes of horizontal momentum, which314

will then support dynamically the total surface-trapped mean shear, including any component that315

might often be associated with the wave field and identified as Stokes’ drift.316

Although it is possible that more comprehensive and rigorous comparisons with data could yield317

fits that are sufficiently accurate to make the log-Ekman model a useful general representation of318

mean current profiles in the ocean surface boundary layer, the primary goal of these examples is to319

provide a concrete illustration of one method of incorporating wave effects on near-surface shear320

directly into a dynamical model. To the extent that this approach is successful, it would seem a321

plausible and potentially more dynamically consistent alternative to the mixed kinematic-dynamic322

approach, in which the results from a kinematic analysis of Lagrangian Stokes’ drift in a free wave323

field are inserted directly into the momentum balance for the mean current field.324

The generalized log-Ekman model formulated here differs from previous models primarily in325

allowing an empirical wave-correction, through the factor φ in (20), to the ocean eddy-viscosity326

profile near the interface. This correction is analogous to the Monin-Obukhov stability correc-327

tions for stratified boundary layers, which may similarly be interpreted as modifying the effective328

value of the von Karman coefficient κ . The improvement in the fit to observations for the result-329

19

ing optimal-wind-drift solution relative to the standard solution (Figs. 4) is not dramatic, because330

of the relatively small differences in all observed and modeled quantities, but it is similar to the331

improvement that was obtained by Lewis and Belcher (2004) in their related model through the332

more customary introduction of an explicit, independent Stokes’ drift component. From a physi-333

cal point of view, the correction φ < 1 suggest that the surface gravity wave motions suppress the334

spatial growth of turbulent eddies with distance from the interface, relative to the spatial growth335

that would occur adjacent to a rigid boundary. It seems plausible that the straining motions asso-336

ciated with wave-orbital velocities could systematically distort turbulent eddies and inhibit their337

development in this way.338

7. Summary339

The departure of the surface wind drift from exact proportionality to the vector difference of340

the geostrophic wind and currents outside the momentum boundary layers evidently arises from341

asymmetries in the dimensionless structure of the air-sea interface and associated motions near the342

interface, which penetrate much more deeply into the dimensionless ocean boundary layer than343

into the dimensionless atmospheric boundary layer. In the absence of this asymmetry, the gen-344

eral argument given here provides an analytical expression for the exact proportionality that relies345

only on an assumption of turbulent universality. Although the assumption of perfect universality346

is almost certainly violated, the resulting expression nonetheless appears to provide a useful ap-347

proximation to the empirical 3%-15◦ rule-of-thumb for the surface wind drift, with the observed348

rotation to the right of the 10-m wind evidently reflecting primarily the opposite rotation of the349

10-m wind relative to the geostrophic wind above.350

The effect of the interface asymmetry is evidently to modify the near-surface characteristics of351

the turbulence in the ocean boundary layer, relative to that in a similar stress boundary layer at a352

20

rigid boundary. The explicit model solutions described here show that some of these effects can353

be represented and explored by introducing a wave-correction factor into the dimensionless log-354

layer equations. This hypothesis could presumably be examined further and more directly through355

detailed analysis of near-surface ocean current and turbulence measurements, or through numerical356

simulations that incorporate a moving free-surface or perhaps a more idealized representation of357

wave-orbital velocities and their effect on the turbulent motion field.358

Acknowledgments. This research was supported by the National Aeronautics and Space Admin-359

istration (NASA) Ocean Vector Winds Science Team, NASA Grant NNX14AM66G. I am grateful360

to E. Dever and R. de Szoeke for helpful comments on an early presentation of some of these361

results.362

21

References363

Ardhuin, F., Y. Aksenov, A. Benetazzo, L. Bertino, P. Brandt, and E. Caubet, 2017: Measuring364

currents, ice drift, and waves from space: the Sea Surface KInematics Multiscale Monitoring365

(SKIM) concept. Ocean Sci., 14, 337–354.366

Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge University Press.367

Briscoe, M. G., and R. A. Weller, 1984: Preliminary results from the Long-Term Upper Ocean368

Study (LOTUS). Dynamics of Atmospheres and Oceans, 8, 243–265.369

Chereskin, T. K., 1995: Direct evidence for an ekman balance in the California Current. Journal370

of Geophysical Research, 100, 18 261–18 269.371

Ekman, V. W., 1905: On the influence of the Earth’s rotation on ocean currents. Arkiv Mat. As-372

tronom. Fyzik, 2, 1–53.373

Ellison, T. H., 1959: Atmospheric turbulence. Surveys in Mechanics, G. K. Batchelor, and R. M.374

Davies, Eds., Cambridge University Press, 400–430.375

Lewis, D. M., and S. E. Belcher, 2004: Time-dependent, coupled, Ekman boundary layer solutions376

incorporating Stokes drift. Dynamics of Atmospheres and Oceans, 37, 313–351, doi:10.1016/j.377

dynatmoce.2003.11.001.378

Phillips, O. M., 1977: The dynamics of the upper ocean. 2nd ed., Cambridge University Press.379

Price, J. F., and M. A. Sundermeyer, 1999: Stratified Ekman layers. Journal of Geophysical Re-380

search: Oceans, 104, 20 467–20 494.381

Price, J. F., R. A. Weller, and R. R. Schudlich, 1987: Wind-driven ocean currents and ekman382

transport. Science, 238, 1534–1538.383

22

Rodriguez, E., M. Bourassa, D. Chelton, J. T. Farrar, D. Long, D. Perkovic-Martin, and R. Samel-384

son, 2019: The Winds and Currents Mission Concept. Frontiers, submitted.385

Rodriguez, E., A. Wineteer, D. Perkovic-Martin, T. Gal, B. Stiles, N. Niamsuwan, and R. Monje,386

2018: Estimating ocean vector winds and currents using a Ka-band pencil-beam doppler scat-387

terometer. Remote Sensing, 10, 576.388

Weber, J. E., 1983: Steady wind- and wave-induced currents in the open ocean. Journal of Physical389

Oceanography, 13 (3), 524–530, doi:10.1175/1520-0485(1983)013〈0524:SWAWIC〉2.0.CO;2.390

Wijffels, S., E. Firing, and H. Bryden, 1994: Direct observations of the ekman balance at 10◦N in391

the Pacific. Journal of Physical Oceanography, 24, 1666–1679.392

23

LIST OF FIGURES393

Fig. 1. Dimensionless atmosphere (V = V +VG, green) and ocean (U , blue) velocity and stress394

profiles for (z+∗0,z−∗0) = (0.12,0.86)× 10−3 m and φ = 1 vs. (upper panels) dimensionless395

height and depth z, (middle) log10(z+0 + z) for atmosphere only, and (lower) log10(z

−0 − z)396

for ocean only. (left panels) Dimensionless atmosphere (green) and ocean (blue) velocity397

components in x (solid) and y (dashed) directions vs. height and depth. (left center) Angle398

of atmosphere and ocean velocities vs. height and depth. (right center) Dimensionless stress399

components in x (solid) and y (dashed) directions vs. height and depth. (right) Angle of400

stress vectors vs. height and depth. . . . . . . . . . . . . . . . . . 25401

Fig. 2. Hodographs of dimensionless atmosphere (green) and ocean (blue) velocity and stress pro-402

files for (z+∗0,z−∗0) = (0.12,0.86)× 10−3 m and φ = 1: (upper panel) V = V +VG and U ,403

(middle) V and αU , (lower) atmosphere (dashed green) and ocean (blue) downward stress404

τ = τ∗/τ∗0. The corresponding values at the heights z±1 (+), z+1 0 (×), z = 0 (· or ◦) are405

shown. In the upper panel, the points |V10N |V10/|V10| (red ◦) and |V10|τ0/|τ0| (green �) are406

also shown, where V10N is the 10-m equivalent neutral wind computed from the stress. . . . 26407

Fig. 3. Log-Ekman model solutions vs. ocean roughness length z−∗0 for φ =408

{1,0.75,0.575,0.5,0.375,0.25} ({blue,cyan,black,green,magenta,red}, respectively).409

(upper panel) Surface wind drift speed as a fraction (%) of geostrophic wind speed. (lower410

panel) Deflection angle of surface wind drift relative to 10-m wind or surface stress (◦,411

positive counter-clockwise from wind or stress direction). Solutions matching the 3% rule412

are indicated in both panels (dots). . . . . . . . . . . . . . . . . . 27413

Fig. 4. Observations (∆ - LOTUS3; � - TPHS; ∇ - EBC) taken from Table 2 of Lewis and Belcher414

(2004) and log-Ekman model profiles of ocean (upper and middle panel) dimensionless415

current speed and (lower panel) rotation angle relative to geostrophic wind direction vs. di-416

mensionless depth, for solutions in Fig. 3) matching the 3% rule with φ = {1,0.575,0.25}417

({blue,black,red}, respectively). The depths of z−1 at which the eddy viscosity reaches its418

maximum value are indicated (dots with corresponding colors) and arbitrarily normalized419

eddy viscosity profiles (thin lines, corresponding colors) are also shown in the lower panel.420

The observed rotation angles were reported relative to the observed 10-m wind or surface421

stress direction and have been adjusted accordingly, using the model 10-m wind or surface422

stress directions, which were effectively identical for the three solutions depicted. . . . . 28423

Fig. 5. As in Fig. 1 but for (z+∗0,z−∗0) = (0.12,22.8)×10−3 m and φ = 0.575. . . . . . . . . 29424

Fig. 6. As in Fig. 2 but for (z+∗0,z−∗0) = (0.12,22.8)×10−3 m and φ = 0.575. . . . . . . . . 30425

24

0 20 40(U,V)=(U*/u*,V*/v*)

-1

-0.5

0

0.5

1

(z,z

+)

= z

* f/(u

*,v*)

-200 0

arctan(V/U) (o)

-1

-0.5

0

0.5

1

(z,z

+)

= z

* f/(u

*,v*)

(v*,V

10,V

G* ,u

*,U

0;z

0+*,z

0* ) = (0.25,7.1,9.2,0.0086,0.2;0.12,0.86) (m s-1; mm)

-1 0 1downward stress

-1

-0.5

0

0.5

1

(z,z

+)

= z

* f/(u

*,v*)

-200 0

downward stress angle (o)

-1

-0.5

0

0.5

1

(z,z

+)

= z

* f/(u

*,v*)

0 20 40(U,V)=(U*,V*)/u*

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0+z)

= lo

g 10(z

0* +

z*)

f/v*

-20 0 20

arctan(V/U) (o)

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0+z)

= lo

g 10(z

0* +

z*)

f/v*

-1 0 1downward stress

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0+z)

= lo

g 10(z

0* +

z*)

f/v*

-200 0

downward stress angle (o)

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0+z)

= lo

g 10(z

0* +

z*)

f/v*

0 20 40(U,V)=(U*,V*)/u*

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0-z)

= lo

g 10(z

0* -

z*)

f/u*

-200 0

arctan(V/U) (o)

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0-z)

= lo

g 10(z

0* -

z*)

f/u*

-1 0 1downward stress

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0-z)

= lo

g 10(z

0* -

z*)

f/u*

-200 0

downward stress angle (o)

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0-z)

= lo

g 10(z

0* -

z*)

f/u*

FIG. 1. Dimensionless atmosphere (V = V +VG, green) and ocean (U , blue) velocity and stress profiles for

(z+∗0,z−∗0) = (0.12,0.86)× 10−3 m and φ = 1 vs. (upper panels) dimensionless height and depth z, (middle)

log10(z+0 + z) for atmosphere only, and (lower) log10(z

−0 − z) for ocean only. (left panels) Dimensionless atmo-

sphere (green) and ocean (blue) velocity components in x (solid) and y (dashed) directions vs. height and depth.

(left center) Angle of atmosphere and ocean velocities vs. height and depth. (right center) Dimensionless stress

components in x (solid) and y (dashed) directions vs. height and depth. (right) Angle of stress vectors vs. height

and depth.

426

427

428

429

430

431

432

25

0 10 20 30 40 50

(Ux,Vx)=(Ux* /u*,V

x* /v*)

-10

-5

0

5

10

(Uy ,V

y )=(U

y */u*,V

y */v*)

(v*,V10,VG* ,u*,U0;z0

+*,z0* ) = (0.25,7.1,9.2,0.0086,0.2;0.12,0.86) (m s- 1; mm)

0 1 2 3 4 5

( Ux,Vx)=(Ux* ,Vx

* )/v*

-1

-0.5

0

0.5

1

( U

y ,Vy )=

(Uy *,V

y *)/v *

0 0.2 0.4 0.6 0.8 1 1.2downward x-stress

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

dow

nwar

d y-

stre

ss

FIG. 2. Hodographs of dimensionless atmosphere (green) and ocean (blue) velocity and stress profiles for

(z+∗0,z−∗0) = (0.12,0.86)× 10−3 m and φ = 1: (upper panel) V = V +VG and U , (middle) V and αU , (lower)

atmosphere (dashed green) and ocean (blue) downward stress τ = τ∗/τ∗0. The corresponding values at the

heights z±1 (+), z+1 0 (×), z = 0 (· or ◦) are shown. In the upper panel, the points |V10N |V10/|V10| (red ◦) and

|V10|τ0/|τ0| (green �) are also shown, where V10N is the 10-m equivalent neutral wind computed from the stress.

433

434

435

436

437

26

10-3 10-2 10-1 100 101

z*0 (m)

0

1

2

3

4

5

6

7

8

9

10

|U*0

|/|V *1

0| 1

00

10-3 10-2 10-1 100 101

z*0 (m)

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

(U*0

)-(V

*10) (

o )

6.9 7 7.1 7.2 7.3 7.4 7.5 7.6V*10 (m s-1)

0

1

2

3

4

5

6

7

8

9

10

|U*0

|/|V *1

0| 1

00

6.9 7 7.1 7.2 7.3 7.4 7.5 7.6V*10 (m s-1)

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

(U*0

)-(V

*10) (

o )

FIG. 3. Log-Ekman model solutions vs. ocean roughness length z−∗0 for φ = {1,0.75,0.575,0.5,0.375,0.25}

({blue,cyan,black,green,magenta,red}, respectively). (upper panel) Surface wind drift speed as a fraction (%)

of geostrophic wind speed. (lower panel) Deflection angle of surface wind drift relative to 10-m wind or surface

stress (◦, positive counter-clockwise from wind or stress direction). Solutions matching the 3% rule are indicated

in both panels (dots).

438

439

440

441

442

27

0 5 10 15 20 25|U|=|U*|/u*

-0.4

-0.3

-0.2

-0.1

0

z=z *f/u

*

100 101

|U|=|U*|/u*

-0.4

-0.3

-0.2

-0.1

0

z=z *f/u

*

-150 -100 -50 0

arctan(Uy/Ux) (o)

-0.4

-0.3

-0.2

-0.1

0

z=z *f/u

*

FIG. 4. Observations (∆ - LOTUS3; � - TPHS; ∇ - EBC) taken from Table 2 of Lewis and Belcher (2004)

and log-Ekman model profiles of ocean (upper and middle panel) dimensionless current speed and (lower panel)

rotation angle relative to geostrophic wind direction vs. dimensionless depth, for solutions in Fig. 3) matching

the 3% rule with φ = {1,0.575,0.25} ({blue,black,red}, respectively). The depths of z−1 at which the eddy

viscosity reaches its maximum value are indicated (dots with corresponding colors) and arbitrarily normalized

eddy viscosity profiles (thin lines, corresponding colors) are also shown in the lower panel. The observed

rotation angles were reported relative to the observed 10-m wind or surface stress direction and have been

adjusted accordingly, using the model 10-m wind or surface stress directions, which were effectively identical

for the three solutions depicted.

443

444

445

446

447

448

449

450

451

28

0 20 40(U,V)=(U*/u*,V*/v*)

-1

-0.5

0

0.5

1

(z- ,z

+)

= z

* f/(u

*,v*)

-200 0

arctan(Vy/Vx,Uy/Ux) (o)

-1

-0.5

0

0.5

1

(z- ,z

+)

= z

* f/(u

*,v*)

(v*,V

10,V

G* ,u

*,U

0;z

0+*,z

0* ) = (0.25,7.1,9.2,0.0086,0.21;0.12,23) (m s-1; mm)

-1 0 1downward stress

-1

-0.5

0

0.5

1

(z- ,z

+)

= z

* f/(u

*,v*)

-200 0

downward stress angle (o)

-1

-0.5

0

0.5

1

(z- ,z

+)

= z

* f/(u

*,v*)

0 20 40V=V*/v*

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0+z)

= lo

g 10(z

0* +

z*)

f/v*

-20 0 20

arctan(Vy/Vx) (o)

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0+z)

= lo

g 10(z

0* +

z*)

f/v*

-1 0 1downward stress

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0+z)

= lo

g 10(z

0* +

z*)

f/v*

-200 0

downward stress angle (o)

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0+z)

= lo

g 10(z

0* +

z*)

f/v*

0 20 40U=U*/u*

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0-z)

= lo

g 10(z

0* -

z*)

f/u*

-200 0

arctan(Uy/Ux) (o)

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0-z)

= lo

g 10(z

0* -

z*)

f/u*

-1 0 1downward stress

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0-z)

= lo

g 10(z

0* -

z*)

f/u*

-200 0

downward stress angle (o)

-8

-7

-6

-5

-4

-3

-2

-1

0

log

10(z

0-z)

= lo

g 10(z

0* -

z*)

f/u*

FIG. 5. As in Fig. 1 but for (z+∗0,z−∗0) = (0.12,22.8)×10−3 m and φ = 0.575.

29

0 10 20 30 40 50

(Ux,Vx)=(Ux* /u*,V

x* /v*)

-10

-5

0

5

10

(Uy ,V

y )=(U

y */u*,V

y */v*)

(v*,V10,VG* ,u*,U0;z0

+*,z0* ) = (0.25,7.1,9.2,0.0086,0.21;0.12,23) (m s- 1; mm)

0 1 2 3 4 5

( Ux,Vx)=(Ux* ,Vx

* )/v*

-1

-0.5

0

0.5

1

( U

y ,Vy )=

(Uy *,V

y *)/v *

0 0.2 0.4 0.6 0.8 1 1.2downward x-stress

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

dow

nwar

d y-

stre

ss

FIG. 6. As in Fig. 2 but for (z+∗0,z−∗0) = (0.12,22.8)×10−3 m and φ = 0.575.

30