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Inertially coupled Ekman layers
Dynamics of Atmospheres and Oceans 35 (2002) 27–39
Inertially coupled Ekman layers John A.T. Bye∗ School of Earth Sciences, The University of Melbourne, Victoria 3010, Australia Accepted 9 July 2001
Abstract The inertial coupling model of the surface shear stress at the sea surface (Bye, 1995) which takes account of the surface wavefield, has been applied to couple the Ekman layers of the ocean and atmosphere. We determine the surface shear stress and geostrophic drag coefficient, under barotropic conditions. The results are expressed in terms of the shear between the inertially weighted (i.e. velocity × square root of the density) relative geostrophic velocities in the two fluids, in which the reference velocity need not be specified, a priori. We find, in particular, that the deflection of the relative surface geostrophic wind to the surface shear stress in naturally occurring seastates, is about 9◦ . In the application of the analysis to general circulation models, it is argued that, since the inertially weighted relative geostrophic velocities in air and water are of similar magnitude, this implies that the surface shear stress can be significantly reduced by the current component of the inertially weighted geostrophic shear, with a corresponding reduction in importance of the Ekman transport. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Ekman layer; Inertial coupling; Geostrophic velocities
1. Introduction One of the central concepts in dynamical oceanography and meteorology is the Ekman layer. As originally introduced by Ekman (1905), it describes the frictional adjustment near the surface of a rotating fluid. In order to obtain the solution of the problem for the coupled air–sea system, it is necessary to formulate the viscous stresses in each fluid separately, and also to match the flows at the interface between the two fluids. In this paper, we consider two fluids, each of constant density and viscosity, as treated by Ekman (1905), but apply a different matching condition at the interface, which takes account of the inertia of the system. ∗ Fax: +61-38344-7761. E-mail address: [email protected] (J.A.T. Bye).
0377-0265/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 6 5 ( 0 1 ) 0 0 0 8 3 - 5
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2. Inertial coupling The inertial coupling hypothesis, in contrast to the viscous coupling hypothesis which is traditionally assumed is that at the sea surface, the shear stress is represented by the two aerodynamic bulk relationships (Bye, 1995) which take account of the wavefield, in air τ s = ρ1 K u(z) − (u0 + uL ) × (u(z) − (u0 + uL )) (1a) and in water
τ s = ρ2 K (u0 + εuL ) − u(−z) × ((u0 + εuL ) − u(−z))
(1b)
in which z = 0 denotes the mean interfacial level, and u(z) and u(−z) are, respectively, the air velocity at height (z) and the water velocity at depth (−z), u0 is a reference velocity, since it is common to both fluids, uL is the spectrally averaged phase velocity of the wave spectrum, and εuL √ is the spectrally averaged surface Stokes velocity (the surface Stokes drift), where ε = ρ1 /ρ2 ,K(±z) is a drag coefficient, ρ 1 and ρ 2 are, respectively, the densities of air and water, and τ s is the interfacial shear stress (Fig. 1). On equating (1a) and (1b) we obtain the inertial coupling relation (Bye, 1995) 1/2
1/2
τ s = 41 K|ρ1 (u(z) − u0 ) − ρ2 (u(−z) − u0 )| 1/2
1/2
× (ρ1 (u(z) − u0 ) − ρ2 (u(−z) − u0 ))
(2)
and the auxiliary equation u0 =
εu(z) + u(−z) − 2εuL 1+ε
Fig. 1. Cartoon of the inertially coupled planetary boundary layers.
(3)
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The expressions (1)–(3) are assumed to apply within a wave boundary layer which extends upwards and downwards from the interface. At the edges (z = ±zB )of the wave boundary layer the fluid velocity tends to the values u1 = u(zB ) where u1 is the surface wind, and u2 = u(−zB ) where u2 is the surface current. In the following discussion, the relations will be applied at these levels, except where mentioned otherwise, such that 1/2
1/2
1/2
1/2
τ s = K1 |ρ1 (u1 − u0 ) − ρ2 (u2 − u0 )|(ρ1 (u1 − u0 ) − ρ2 (u2 − u0 ))
(4)
where KI = (1/4)K, and also εuL = 21 (ε(u1 − u0 ) + (u2 − u0 ))
(5)
The inertial coupling relation follows directly from assuming that the velocity shears in the wave boundary layers of the two fluids are of similarity form. The conditions under which (4) and (5) yield the familiar forms of the classical drag law description of the near surface momentum transfer are discussed in Section 3.1, after the Ekman analysis has been presented. The similarity relation between the two wave velocities (uL and εuL ) can be checked directly using wave spectral models (Bye, 1988, Bye et al., 2001), and an extended discussion on the derivation of the inertial coupling relation is given in Bye and Wolff (2001a).
3. Inertially coupled Ekman layers In the coupled Ekman layers the equations are, ∂τ ∂p1 ρ1 f × u1 = z − ∂z ∂x ρ2 f × u2 =
∂τ z ∂p2 − ∂z ∂x
(6) (7)
in which τ z is the horizontal shearing stress, p is pressure, and f = 2ω sin φ is the Coriolis parameter in which ω is the angular velocity of rotation of the Earth, and φ is latitude. We obtain below the solution for the planetary boundary layers of the two fluids coupled by (4), assuming that in each fluid the viscosities are constant outside the wave boundary layer. This model differs from the classical Ekman solution due only to the use of inertial coupling in place of viscous coupling. The results of the analysis are derived for the northern hemisphere (f > 0). For the constant viscosity model, the components of shearing stress are related to the frictional velocities by the expressions τs1x = Q1 (−uˆ 1 + vˆ1 ),
τs1y = Q1 (−uˆ 1 − vˆ1 )
(8)
in the air, where Q1 = ρ1 (fv1 /2)1/2 and τs2x = Q2 (uˆ 2 − vˆ2 ),
τs2y = Q2 (uˆ 2 + vˆ2 )
(9)
in the water, where Q2 = ρ2 (fv2 /2)1/2 , and u1 = ug1 + uˆ 1 , and u2 = ug2 + uˆ 2 in which the geostrophic velocities satisfy the equations, f × ug1 = −
1 ∂p1 ρ1 ∂x
and
f × ug2 = −
1 ∂p2 ρ2 ∂x
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For fully turbulent flow in each fluid, the fluid viscosities (v1 ) and (v2 ) are represented by the similarity expressions, v1 =
Aku2∗ , f
v2 =
Akw∗2 f
(f > 0)
in which k is von Karman’s constant, u∗ = (|τ s1 |/ρ1 )1/2 and w∗ = (|τ s2 |/ρ2 )1/2 and A is a constant of 0(10−1 ) (Garratt, 1992), from which we obtain, 1/2 1/2 Ak Ak 1/2 1/2 Q1 = ρ1 |τ s1 |1/2 , Q2 = ρ2 |τ s2 |1/2 2 2 The expressions (8) and (9) for the shearing stress in each fluid are now equated with the inertial coupling relation (4) thus τ s1 = τ s = τ s2
(10)
and on rearranging (4), to yield, 1/2
1/2
1/2
τ s = KI |τ s |1/2 (ρ1 (u1 − u0 ) − ρ2 (u2 − u0 ))
(11)
and defining the inertially weighted surface velocities, 1/2
U 1 = ρ 1 u1 ,
1/2
U 2 = ρ2 u2
and substituting (8), (9) and (11) in (10), we obtain, Fsx = α(−Uˆ 1 + Vˆ1 ) = α(Uˆ 2 − Vˆ2 ) = Uˆ 1 − Uˆ 2 + Gx
(12)
Fsy = α(−Uˆ 1 − Vˆ1 ) = α(Uˆ 2 + Vˆ2 ) = Vˆ1 − Vˆ2 + Gy
(13)
and
where α = (Ak/2KI )1/2 , and G = U g1 − U g2 , in which U g1 = U g1 − ρ1 u0 and 1/2
U g2 = U g2 − ρ2 u0 are the inertially weighted relative geostrophic velocities, u0 being 1/2 1/2 1/2 the reference velocity, where U g1 = ρ ug1 , U g2 = ρ ug2 , and Uˆ 1 = ρ uˆ 1 , Uˆ 2 = 1/2
1
1/2
2
1
1/2
ρ2 uˆ 2 , and F s = τ s /(KI |τ s |1/2 ). Eqs. (12) and (13) are linear equations for the frictional velocities. The solutions are, Uˆ 1 = −Uˆ 2
(14)
and Uˆ 2 =
1 ((2 + α)Gx + αGy , (2 + α)Gy − αGx ) ((2 + α)2 + α 2 )
from which we obtain, α ((α + 1)Gx − Gy , Gx + (α + 1)Gy ) Fs = ((1 + α)2 + 1)
(15)
(16)
Note that throughout, the inertially weighted velocities are denoted by upper case symbols.
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In polar form, these expressions yield, G = (G cos φ, G sin φ)
(17)
where φ is the angle of G to the x-axis, and Uˆ 2 =
G (2 + α)2 + α 2
(cos(φ − δ), sin(φ − δ))
(18)
where δ = tan−1 (α/(2 + α)) and Fs =
αG (α + 1)2 + 1
(cos(φ + γ ), sin(φ + γ ))
(19)
in which γ = tan−1 (1/(α + 1)) and γ + δ = π/4. Uˆ 2 lies at an angle (δ) to the right hand side of G (and Uˆ 1 lies at an angle (π −δ) to the left-hand side), and Fs (and τ s ) lie at an angle (γ ) to the left-hand side of G (Fig. 2), and the geostrophic drag coefficient, K0 = |τ s |/G2 , is given by the expression, K0 =
α2 KI 1 + (1 + α)2
The sum of the inertially weighted surface velocities, U 1 + U 2 = U g1 + U g2 or alternatively, U 1 + U 2 = U g1 + U g2
(20)
where U 1 = U 1 − ρ1 u0 and U 2 = U 2 − ρ2 u0 . From (5), we also have, 1/2
ρ1 uL = 21 (U 1 + U 2 ) 1/2
1/2
(21)
Fig. 2. Orientation of the surface shearing stress (τ s )and the inertially weighted velocities ( Uˆ 1 , Uˆ 2 and G) in the coupled boundary layers.
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and, on substituting (20) in (21), we obtain, u0 =
εug1 + ug2 − 2εuL
(22)
1+ε
and hence (3) is valid also for the geostrophic velocities, and yields, 1/2
(23a)
1/2
(23b)
G = 2ρ1 (ug1 − (uL + u0 )) G = 2ρ2 ((εuL + u0 ) − ug2 )
which show that the inertially weighted geostrophic shears in each fluid are equal, and also that G is the inertially weighted shear across the planetary boundary layer, or, in short, the planetary shear. 3.1. Representation in the surface shear stress reference frame In the frame of reference of the surface shear stress, which is assumed to lie along 0x, i.e. τsy = 0, we obtain, + (1/2)U ((1/2) + α)Ug1 g2 1 U1 = (24) , (Vg1 + Vg2 ) (1 + α) 2 and
U 2
=
+ ((1/2) + α)U (1/2)Ug1 g2 1 , (Vg1 + Vg2 ) (1 + α) 2
from which U 1 − U 2 =
α Gx , 0 1+α
(25)
(26)
and hence, on using (26) and (21), we obtain the velocity shear in the water, defined as the difference between the vector sum of the surface Stokes drift and the reference velocity, and the surface current, εuL + u0 − u2 =
(1/2)α [Gx , 0] (1 + α) ρ 1/2
(27)
2
and similarly, the velocity shear in the air, defined as the difference between the surface wind, and the vector sum of the spectrally averaged phase velocity and the reference velocity, u1 − uL − u0 =
(1/2)α [Gx , 0] (1 + α) ρ 1/2
(28)
1
In accordance with the original hypothesis (Eq. (1)), both these vectors lie along the direction of the surface stress (τ s ) with their magnitudes in the ratio (ε), i.e. the velocity shear in the
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wave boundary layer is tangential to the surface shear stress. Normal to the surface shear stress, we have, v(z) = v0 + vL
(29)
v(−z) = v0 + εvL
(30)
and
such that the normal velocities ( in the wave boundary layer) are constant in each fluid, and from (21) or (24) for the air, we obtain, + U ) (Ug1 1 1 g2 (31) v(z) − v0 = − 1/2 2 (1 + α) ρ1 and from (21) or (25) for the water, we have, v(−z) − v0 = ε(v(z) − v0 )
(32)
Using (29) and (30), the surface shear stress along 0x is, τsx = ρ1 KI |(u1 − u0 ) − (u2 − u0 )/ε| ((u1 − u0 ) − (u2 − u0 )/ε)
(33)
Next, on introducing the relation between the Stokes shear and the Ekman shear, εuL = r(u0 − u2 ) where r is a constant, (5) yields, r(u1 − u2 ) uL = (1 + ε + 2r)
(34)
and on substituting (34) in (33), we obtain, τsx = ρ1 KI |F |(F )|u1 − u2 |(u1 − u2 )
(35)
where F = 2(r + 1)/(1 + ε + 2r). In a frame of reference in each fluid, moving at the respective normal velocities (29) and (30), (34) and (35) are a formally complete description of the momentum exchange process. Furthermore, in horizontally homogeneous conditions in which the surface shear stress is a constant vector, they would be also valid at a fixed measurement station. Thus, the classical drag law description of the near surface momentum transfer is the limiting form of the more general inertial coupling relation, applicable in horizontally homogeneous conditions. The familiar forms ( (34) and (35)), however, obscure the fundamental structure of the inertial coupling relation, from which they have been derived.
4. Interpretation of the inertially coupled dynamics The structure of the solution of Section 3 depends only on the parameter α, which has the following significance. From (1b), we have directly that, (1/4)w∗2 KI = εu − u + u 2 L 2 0
(36)
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and on substituting (36) in the definition of α, and eliminating Ak in favour of the viscosity (v2 ) in the Ekman layer in the water, we obtain, |εuL − u2 + u0 | α= (37) wE where wE = w∗2 /(2v2 f )1/2 is the Ekman scale velocity for the ocean. Thus, ␣ is the ratio of the velocity shear in the wave boundary layer to the velocity shear in the Ekman layer (note that an identical result for ␣ can also be obtained for the atmospheric boundary layer). There are two limiting conditions (α → 0) and (α → ∞). 4.1. The Ekman limit (α → 0) For, α → 0, δ → 0, γ → π/4, and K0 /K1 → 0, Uˆ 2 → 1/2G and U 1 = U 2 . This is the Ekman limit in which the surface shearing stress lies at an angle 45◦ to the left-hand side of the inertially weighted geostrophic shear (f > 0). In this situation, from (5), uL + u0 = u1
and
εuL + u0 = u2
and hence the velocity shear in the wave boundary layer in each fluid is negligible, and by reference to the aerodynamic relations (1), it is apparent that the thickness of the wave boundary layer, zB → 0. 4.2. The Stokes limit (α → ∞) For, α → ∞, δ → π/4, γ → 0 and, K0 /K1 → 1, Uˆ 1 → 0, Uˆ 2 → 0 and U 1 = U g1 , U 2 = U g2 , and thus the wave boundary layer attains its maximum extent. In the Stokes limit, the shear stress lies in the direction of the planetary shear, and the frictional circulation in the constant viscosity layers is negligible. The above two limits define the range of possible states for the coupled system, in which the outer frictional layers are fully turbulent and the presence of the wavefield is taken account of by the inertial coupling relationship in which the surface Stokes drift and the spectrally averaged phase velocity are in the ratio (ε). We now seek to evaluate ␣ from observations in order to determine which limit is the best model for naturally occurring seastates. 4.3. Naturally occurring seastates In order to evaluate α, estimates of the parameters, A, k and KI , are required. We assume that k ∼ 0.4, and A ∼ 0.2 (Garratt, 1992), and estimate KI from (36). Observations (Bye, 1988) indicate that εuL + u0 − u2 /w∗ ∼ 12, and hence K I ∼ 1.7 × 10−3 . This estimate is similar to observations of the 10 m drag coefficient over the ocean (K10 ). The drag coefficient, KI , however, refers to the height (zB ) equal to the thickness of the wave boundary layer, which is not a fixed height. 1 From these estimates we obtain α ∼ 5, which 1 K10 can be obtained from K by assuming that a logarithmic velocity profile occurs between the edge of the I wave boundary layer (zB ) and 10 m, and using an estimate of zB , see Bye (1996).
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yields δ = 36◦ , γ = 9◦ and K0 = 0.7 KI . This state is clearly much closer to the Stokes limit than the Ekman limit.
5. Velocity structure of the coupled boundary layers Using (24) and (25), (27) and (28), (21), (31) and (32) the ratios of the magnitudes of the velocities for a non-zero relative geostrophic wind in the wave boundary layer, are, in the air, |u1 − u0 | : |u1 − u0 − uL | : |uL | : |v(z) − v0 | 1/2
[(1 + 2α)2 + 1]
1/2
: α : [(1 + α)2 + 1]
:1
relative surface wind : velocity shear in the wave boundary layer : spectrally averaged phase velocity : relative normal velocity and in the water, |u2 − u0 | : |εuL +u0 u2 | :ε|uL | :ε|v(z) − v0 | √ 2 : α : [(1+α)2 +1]1/2 : 1 relative surface current : velocity shear in the wave boundary layer : surface Stokes drift : relative normal velocity where from (31), (v(z) − v0 ) = −(1/2)(1/(1 + α)) (ug1 − u0 ), and from (21), using (20), uL = (1/2)(ug1 − u0 ). Using the velocity ratios, we also find that, r = −(1 + α). Hence, the Ekman shear and the Stokes shear in the wave boundary layer are of opposite sign. In the Ekman limit (α → 0) the velocity shears become negligible in comparison with the relative normal velocity, such that the outer frictional (Ekman spiral) circulations dominate. and the wave and surface velocities would coincide in each fluid. In the Stokes limit (α → ∞), on the other hand, the velocity shear in the wave boundary layer is dominated by the Stokes drift, and also the Ekman spiral circulations are of negligible importance. In this situation the wave velocities (uL and εuL ) in the inertial coupling relation (1), which are derived from irrotational wave theory, and therefore would be expected to take no direct role in the energy transfer process, also define the essential turbulent processes which maintain the wave boundary layer (through the velocity shear,) i.e. the critical layer mechanism in the air, and the energy dissipation in the water.For the naturally occurring seastate (α = 5) the respective velocity ratios are: air
11.0 : 5.0 : 6.1 : 1
water
1.4 : 5.0 : 6.1 : 1
in which the velocity shears in the wave boundary layers of each fluid attain about 80% of their respective wave velocities (the spectrally averaged phase velocity in air, and the surface Stokes drift in water). The relative normal velocity in air is 8% of the relative geostrophic velocity, and r = −6, such that the ratio of Eulerian shear to the Stokes shear in the wave
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boundary layer is –1:6. The similarity of the spectrally averaged Stokes drift profile and the logarithmic velocity profile which occurs in turbulent boundary layers is well known, e.g. Bye (1988). We note that the turbulent properties of the wave boundary layer are not specified (except to estimate the constant eddy viscosities in the outer frictional layers). In this respect, our analysis differs from McWilliams et al. (1997), who also take account of both Stokes and Ekman velocities. In their work, the constant viscosity layers are continued to the sea surface where they are matched with the surface stress and surface Stokes drift. In our analysis, the matching occurs at the edges of the constant stress wave boundary layers. Fig. 3(a) shows the velocity structure of the coupled boundary layers due to a non-zero geostrophic wind (ug1 ) relative to the reference velocity (u0 ), for a naturally occurring sea state ( in which α = 5) in the Northern Hemisphere. The surface wind (u1 ) lies to the left-hand side of the wavefield, and the surface current lies to the right hand side, and the Ekman spirals in the two fluids rotate clockwise. The velocity hodograph indicates a smooth transition between the Ekman spirals and the wave boundary layer in which the component of velocity normal to the surface shear stress is constant within each fluid. In the air, the wind direction in the Ekman spiral and in the wave boundary layer rotates anticlockwise,
Fig. 3. Velocity hodographs for a naturally occurring seastate (α = 5), driven by (a) a non-zero geostrophic wind (ug1 ); and (b) a non-zero geostrophic current (ug2 ). The air velocities are normalised by the factor (ε) with respect to the water velocities, and the velocities in each fluid are shown to scale. The dashed line shows the velocity hodograph (directed from the non-zero geostrophic velocity), on which the dots in the Ekman spirals indicate the position of the current vector at intervals of (1/4)Di , where Di = π(2υi /|f |)1/2 is the Ekman depth in air (i = 1) or in water (i = 2). The directions of rotation in the Ekman spirals are shown by the circulatory arrows.
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whereas in the water, the current direction in the Ekman spiral and the wave boundary layer rotates clockwise. The planetary shear in the water is due solely to the Stokes drift, which lies at an angle of 9◦ to the right hand side of the surface shear stress (for α = 5), and this angle increases to 45◦ at the base of the wave boundary layer. The ratios of the magnitudes of the velocities in the wave boundary layer for a non-zero geostrophic current can be found in a similar way, and Fig. 3(b) shows the corresponding velocity structure, relative to u0 . Note that the surface shear stress (τ s ) is oppositely directed for the non-zero geostrophic current. In the general situation, the velocity fields due to the geostrophic circulations are additive. 5.1. The significance of the relative velocity frame Throughout the analysis, the results have been expressed in a relative velocity frame, ie. in terms of the velocities, u1 − u0 , u2 − u0 , and ug1 − u0 , ug2 − u0 . Thus: 1. The u0 is a reference velocity, in a similar manner to the reference velocity which occurs in the classical double Ekman spiral solution, e.g. Kraus and Businger (1994). 2. The inertial weighting of the relative velocities, however, indicates that the importance of the relative oceanic geostrophic velocity in the frictional circulation and the surface shear stress, is augmented by the factor, 1/ε ≈ 30, in comparison with the relative atmospheric geostrophic velocity. This is the distinctive feature of the inertially coupled Ekman layers. 3. The results of the analysis are independent of u0 , such that the reference frame in which the reference velocity is measured need not be specified, a priori. In order to apply the results for proscribed geostrophic velocities in the two fluids, however, u0 must also be known. In particular, the surface shear stress is predicted by (16), which is expressed in terms of the planetary shear, which involves the relative geostrophic velocities of the two fluids, which depend on the geostrophic velocity and the reference velocity. Locally, in space and time, u0 is the Eulerian velocity common to both fluids. 4. The magnitude of the atmospheric geostrophic velocity is normally much greater than the Eulerian velocity, and hence the frictional circulation driven by the atmosphere is almost independent of the ocean dynamics (which is a comforting result for the meteorologist, as a detailed knowledge of the oceanic general circulation appears not to be of critical importance). For the frictional circulation driven by the ocean, however, the Eulerian velocity and the oceanic geostrophic velocity are of similar magnitude. Hence, the reference velocity, here, is of critical importance. 5. The conclusion from (4) is that the surface shear stress in the local reference frame cannot be accurately predicted by an analysis of the coupled Ekman layers alone. A general circulation model, or an observational programme, which gives the reference velocity field (u0 (x, y, t)), is also required. Eq. (22) indicates that this can be achieved from the determination of the geostrophic velocities, ug1 and ug2 and also the wave velocity, uL , obtained from a wave model. 6. All the wave models have the common property that they are formulated in the Earth reference frame, in which u0 = 0. Bye and Wolff (1999) have incorporated some simple wave models in the specification of the surface shear stress in a fine resolution quasi-geostrophic oceanic model, and, in particular, it was found that significant relative oceanic geostrophic velocities (ug2 − u0 ) occur in the planetary boundary layer.
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7. The implications are that, in general, the current component, (ug2 − u0 )/ε, of the planetary shear (G) is not insignificant in comparison with the wind component, ug1 − u0 , in the determination of the surface shear stress τ s . This result has important consequences for the general circulation of the ocean (Bye and Wolff, 1999). 8. In summary, the Ekman dynamics at any location, as observed and modelled theoretically in this paper, are independent of u0 . The prediction of the frictional circulations of the two fluids, however, requires that u0 be known from a dynamical model (or from observation). Numerical experiments with simple wave models for u0 , indicate that in determining the surface shear stress, the frictional circulation driven by the ocean is of similar importance to that driven by the atmosphere.
6. Ekman transports The Ekman transport in the atmosphere is 1 T E1 = (−τsy , τsx ) ρ1 f and, in the ocean, is 1 T E2 = (τsy , −τsx ) ρ2 f
(38)
(39)
On substituting for τ s and using (19) we obtain for the atmospheric Ekman transport, 1 α2 T E1 = (40) KI G2 (−sin (φ + γ ), cos (φ + γ )) ρ1 f (α + 1)2 + 1 For the Stokes limit (40) reduces to the expression KI T E1 = (−Gy , Gx )G ρ1 f
(41)
and for the naturally occurring sea state (α = 5) we have T E1 =
0.7KI G2 (sin(φ + γ ), cos(φ + γ )) ρ1 f
(42)
where γ = 9◦ . Thus, in the Stokes limit, the Ekman transport lies 90◦ to the left-hand side (in the Northern Hemisphere) of the geostrophic motion. In naturally occurring seastates, the angle of deflection is 99◦ to the left-hand side, and for the same drag coefficient (KI ) the magnitude is reduced by the factor (0.7). Similar results can be obtained for the oceanic Ekman transport which lies in the opposite direction to the atmospheric Ekman transport and has the same mass transport.
7. Discussion The application of the inertial coupling surface stress relation to the coupled planetary boundary layers yields sets of relations in terms of the planetary shear, see Section 3.
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The important property of the planetary shear is that the effect of the surface ocean geostrophic velocity is not insignificant. The resulting frictional relations depend on the parameter (␣). For α → 0, the Ekman limit is attained in which the effect of the wavefield would become negligible, and the structure of the planetary boundary layer would be identical with the classical double Ekman spiral (Kraus and Businger, 1994; Bye, 1986), since the wave boundary layer, shrinks to a negligible thickness. For α → ∞, on the other hand, in the Stokes limit the frictional circulation occurs only within the wave boundary layer, and the Ekman transport is normal to the planetary shear. Observations indicate that the Stokes limit is closely approached in the ocean, in which the geostrophic drag coefficient (K0 ) is about 70% of the drag coefficient (KI ), which is evaluated at the edge of the wave boundary layer, and the rotation of the surface stress from the geostrophic flow is about 9◦ (to the left-hand side in the Northern Hemisphere, and to the right hand side in the Southern Hemisphere), such that the Ekman transport is almost normal to the planetary shear. In the Southern Ocean, in which the surface geostrophic wind and current velocities are almost zonal, a recent series of simulations using a two layer fine resolution quasi-geostrophic model (Bye and Wolff, 1999, Bye and Wolff, 2001b) indicates that the effect of the current on the planetary shear (G) reduces the surface shear stress on average by about 50%. Thus, the Ekman transport is also about 50% of the classical estimate based on the wind field alone.
Acknowledgements Helpful discussions with Professor B.A. Kagan during a visit to Adelaide supported by the Department of Industry, Science and Tourism under the Bilateral Science and Technology Collaboration Program are gratefully acknowledged. Helpful comments by the referees are also acknowledged. The manuscript was typed by Mrs Carol Walding. References Bye, J.A.T., 1986. Momentum exchange at the sea surface by wind stress and understress. Quart. J.R. Met. Soc. 112, 501–510. Bye, J.A.T., 1988. The coupling of wave drift and wind velocity profiles. J. Mar. Res. 46, 457–462. Bye, J.A.T., 1995. Inertial coupling of fluids with large density contrast. Phys. Lett. A 202, 222–224. Bye, J.A.T., 1996. Coupling ocean-atmosphere models. Earth–Sci. Rev. 40, 149–162. Bye, J.A.T., Wolff, J.-O., 1999. Atmosphere–ocean momentum exchange in general circulation models. J. Phys. Oceanogr. 27, 671–692. Bye, J.A.T., J.-O. Wolff, 2001a. Momentum transfer at the ocean-atmosphere interface: the wave basis for the inertial coupling approach. Ocean dynam (in press). Bye, J.A.T., Wolff, J.-O., 2001b. Quasi-geostrophic modelling of the coupled ocean atmosphere. Math. Comput. Model. 33, 609–617. Bye, J.A.T., Makin, V.K., Jenkins, A.D., Huang, N.E., 2001. In: Jones, I.S.F., Toba, Y. (Eds.), Coupling Mechanisms in Wind stress over the ocean. Cambridge University Press, Cambridge (in press) Ekman, V.W., 1905. On the influence of the Earth’s rotation on ocean-currents. Ark. Math. Astron. Fys. 2, 1–53. Garratt, J.R., 1992. The atmospheric boundary layer. Cambridge University Press, Cambridge, 316 pp. Kraus, E.B., Businger, J.A., 1994. Atmosphere–Ocean Interaction, 2nd Edition. Oxford University Press, Oxford. McWilliams, J.C., Sullivan, P.P., Moeng, C.-H., 1997. Langmuir turbulence in the ocean. J. Fluid Mech. 334, 1–30.
Inertially coupled Ekman layers John A.T. Bye∗ School of Earth Sciences, The University of Melbourne, Victoria 3010, Australia Accepted 9 July 2001
Abstract The inertial coupling model of the surface shear stress at the sea surface (Bye, 1995) which takes account of the surface wavefield, has been applied to couple the Ekman layers of the ocean and atmosphere. We determine the surface shear stress and geostrophic drag coefficient, under barotropic conditions. The results are expressed in terms of the shear between the inertially weighted (i.e. velocity × square root of the density) relative geostrophic velocities in the two fluids, in which the reference velocity need not be specified, a priori. We find, in particular, that the deflection of the relative surface geostrophic wind to the surface shear stress in naturally occurring seastates, is about 9◦ . In the application of the analysis to general circulation models, it is argued that, since the inertially weighted relative geostrophic velocities in air and water are of similar magnitude, this implies that the surface shear stress can be significantly reduced by the current component of the inertially weighted geostrophic shear, with a corresponding reduction in importance of the Ekman transport. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Ekman layer; Inertial coupling; Geostrophic velocities
1. Introduction One of the central concepts in dynamical oceanography and meteorology is the Ekman layer. As originally introduced by Ekman (1905), it describes the frictional adjustment near the surface of a rotating fluid. In order to obtain the solution of the problem for the coupled air–sea system, it is necessary to formulate the viscous stresses in each fluid separately, and also to match the flows at the interface between the two fluids. In this paper, we consider two fluids, each of constant density and viscosity, as treated by Ekman (1905), but apply a different matching condition at the interface, which takes account of the inertia of the system. ∗ Fax: +61-38344-7761. E-mail address: [email protected] (J.A.T. Bye).
0377-0265/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 6 5 ( 0 1 ) 0 0 0 8 3 - 5
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2. Inertial coupling The inertial coupling hypothesis, in contrast to the viscous coupling hypothesis which is traditionally assumed is that at the sea surface, the shear stress is represented by the two aerodynamic bulk relationships (Bye, 1995) which take account of the wavefield, in air τ s = ρ1 K u(z) − (u0 + uL ) × (u(z) − (u0 + uL )) (1a) and in water
τ s = ρ2 K (u0 + εuL ) − u(−z) × ((u0 + εuL ) − u(−z))
(1b)
in which z = 0 denotes the mean interfacial level, and u(z) and u(−z) are, respectively, the air velocity at height (z) and the water velocity at depth (−z), u0 is a reference velocity, since it is common to both fluids, uL is the spectrally averaged phase velocity of the wave spectrum, and εuL √ is the spectrally averaged surface Stokes velocity (the surface Stokes drift), where ε = ρ1 /ρ2 ,K(±z) is a drag coefficient, ρ 1 and ρ 2 are, respectively, the densities of air and water, and τ s is the interfacial shear stress (Fig. 1). On equating (1a) and (1b) we obtain the inertial coupling relation (Bye, 1995) 1/2
1/2
τ s = 41 K|ρ1 (u(z) − u0 ) − ρ2 (u(−z) − u0 )| 1/2
1/2
× (ρ1 (u(z) − u0 ) − ρ2 (u(−z) − u0 ))
(2)
and the auxiliary equation u0 =
εu(z) + u(−z) − 2εuL 1+ε
Fig. 1. Cartoon of the inertially coupled planetary boundary layers.
(3)
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29
The expressions (1)–(3) are assumed to apply within a wave boundary layer which extends upwards and downwards from the interface. At the edges (z = ±zB )of the wave boundary layer the fluid velocity tends to the values u1 = u(zB ) where u1 is the surface wind, and u2 = u(−zB ) where u2 is the surface current. In the following discussion, the relations will be applied at these levels, except where mentioned otherwise, such that 1/2
1/2
1/2
1/2
τ s = K1 |ρ1 (u1 − u0 ) − ρ2 (u2 − u0 )|(ρ1 (u1 − u0 ) − ρ2 (u2 − u0 ))
(4)
where KI = (1/4)K, and also εuL = 21 (ε(u1 − u0 ) + (u2 − u0 ))
(5)
The inertial coupling relation follows directly from assuming that the velocity shears in the wave boundary layers of the two fluids are of similarity form. The conditions under which (4) and (5) yield the familiar forms of the classical drag law description of the near surface momentum transfer are discussed in Section 3.1, after the Ekman analysis has been presented. The similarity relation between the two wave velocities (uL and εuL ) can be checked directly using wave spectral models (Bye, 1988, Bye et al., 2001), and an extended discussion on the derivation of the inertial coupling relation is given in Bye and Wolff (2001a).
3. Inertially coupled Ekman layers In the coupled Ekman layers the equations are, ∂τ ∂p1 ρ1 f × u1 = z − ∂z ∂x ρ2 f × u2 =
∂τ z ∂p2 − ∂z ∂x
(6) (7)
in which τ z is the horizontal shearing stress, p is pressure, and f = 2ω sin φ is the Coriolis parameter in which ω is the angular velocity of rotation of the Earth, and φ is latitude. We obtain below the solution for the planetary boundary layers of the two fluids coupled by (4), assuming that in each fluid the viscosities are constant outside the wave boundary layer. This model differs from the classical Ekman solution due only to the use of inertial coupling in place of viscous coupling. The results of the analysis are derived for the northern hemisphere (f > 0). For the constant viscosity model, the components of shearing stress are related to the frictional velocities by the expressions τs1x = Q1 (−uˆ 1 + vˆ1 ),
τs1y = Q1 (−uˆ 1 − vˆ1 )
(8)
in the air, where Q1 = ρ1 (fv1 /2)1/2 and τs2x = Q2 (uˆ 2 − vˆ2 ),
τs2y = Q2 (uˆ 2 + vˆ2 )
(9)
in the water, where Q2 = ρ2 (fv2 /2)1/2 , and u1 = ug1 + uˆ 1 , and u2 = ug2 + uˆ 2 in which the geostrophic velocities satisfy the equations, f × ug1 = −
1 ∂p1 ρ1 ∂x
and
f × ug2 = −
1 ∂p2 ρ2 ∂x
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For fully turbulent flow in each fluid, the fluid viscosities (v1 ) and (v2 ) are represented by the similarity expressions, v1 =
Aku2∗ , f
v2 =
Akw∗2 f
(f > 0)
in which k is von Karman’s constant, u∗ = (|τ s1 |/ρ1 )1/2 and w∗ = (|τ s2 |/ρ2 )1/2 and A is a constant of 0(10−1 ) (Garratt, 1992), from which we obtain, 1/2 1/2 Ak Ak 1/2 1/2 Q1 = ρ1 |τ s1 |1/2 , Q2 = ρ2 |τ s2 |1/2 2 2 The expressions (8) and (9) for the shearing stress in each fluid are now equated with the inertial coupling relation (4) thus τ s1 = τ s = τ s2
(10)
and on rearranging (4), to yield, 1/2
1/2
1/2
τ s = KI |τ s |1/2 (ρ1 (u1 − u0 ) − ρ2 (u2 − u0 ))
(11)
and defining the inertially weighted surface velocities, 1/2
U 1 = ρ 1 u1 ,
1/2
U 2 = ρ2 u2
and substituting (8), (9) and (11) in (10), we obtain, Fsx = α(−Uˆ 1 + Vˆ1 ) = α(Uˆ 2 − Vˆ2 ) = Uˆ 1 − Uˆ 2 + Gx
(12)
Fsy = α(−Uˆ 1 − Vˆ1 ) = α(Uˆ 2 + Vˆ2 ) = Vˆ1 − Vˆ2 + Gy
(13)
and
where α = (Ak/2KI )1/2 , and G = U g1 − U g2 , in which U g1 = U g1 − ρ1 u0 and 1/2
U g2 = U g2 − ρ2 u0 are the inertially weighted relative geostrophic velocities, u0 being 1/2 1/2 1/2 the reference velocity, where U g1 = ρ ug1 , U g2 = ρ ug2 , and Uˆ 1 = ρ uˆ 1 , Uˆ 2 = 1/2
1
1/2
2
1
1/2
ρ2 uˆ 2 , and F s = τ s /(KI |τ s |1/2 ). Eqs. (12) and (13) are linear equations for the frictional velocities. The solutions are, Uˆ 1 = −Uˆ 2
(14)
and Uˆ 2 =
1 ((2 + α)Gx + αGy , (2 + α)Gy − αGx ) ((2 + α)2 + α 2 )
from which we obtain, α ((α + 1)Gx − Gy , Gx + (α + 1)Gy ) Fs = ((1 + α)2 + 1)
(15)
(16)
Note that throughout, the inertially weighted velocities are denoted by upper case symbols.
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In polar form, these expressions yield, G = (G cos φ, G sin φ)
(17)
where φ is the angle of G to the x-axis, and Uˆ 2 =
G (2 + α)2 + α 2
(cos(φ − δ), sin(φ − δ))
(18)
where δ = tan−1 (α/(2 + α)) and Fs =
αG (α + 1)2 + 1
(cos(φ + γ ), sin(φ + γ ))
(19)
in which γ = tan−1 (1/(α + 1)) and γ + δ = π/4. Uˆ 2 lies at an angle (δ) to the right hand side of G (and Uˆ 1 lies at an angle (π −δ) to the left-hand side), and Fs (and τ s ) lie at an angle (γ ) to the left-hand side of G (Fig. 2), and the geostrophic drag coefficient, K0 = |τ s |/G2 , is given by the expression, K0 =
α2 KI 1 + (1 + α)2
The sum of the inertially weighted surface velocities, U 1 + U 2 = U g1 + U g2 or alternatively, U 1 + U 2 = U g1 + U g2
(20)
where U 1 = U 1 − ρ1 u0 and U 2 = U 2 − ρ2 u0 . From (5), we also have, 1/2
ρ1 uL = 21 (U 1 + U 2 ) 1/2
1/2
(21)
Fig. 2. Orientation of the surface shearing stress (τ s )and the inertially weighted velocities ( Uˆ 1 , Uˆ 2 and G) in the coupled boundary layers.
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and, on substituting (20) in (21), we obtain, u0 =
εug1 + ug2 − 2εuL
(22)
1+ε
and hence (3) is valid also for the geostrophic velocities, and yields, 1/2
(23a)
1/2
(23b)
G = 2ρ1 (ug1 − (uL + u0 )) G = 2ρ2 ((εuL + u0 ) − ug2 )
which show that the inertially weighted geostrophic shears in each fluid are equal, and also that G is the inertially weighted shear across the planetary boundary layer, or, in short, the planetary shear. 3.1. Representation in the surface shear stress reference frame In the frame of reference of the surface shear stress, which is assumed to lie along 0x, i.e. τsy = 0, we obtain, + (1/2)U ((1/2) + α)Ug1 g2 1 U1 = (24) , (Vg1 + Vg2 ) (1 + α) 2 and
U 2
=
+ ((1/2) + α)U (1/2)Ug1 g2 1 , (Vg1 + Vg2 ) (1 + α) 2
from which U 1 − U 2 =
α Gx , 0 1+α
(25)
(26)
and hence, on using (26) and (21), we obtain the velocity shear in the water, defined as the difference between the vector sum of the surface Stokes drift and the reference velocity, and the surface current, εuL + u0 − u2 =
(1/2)α [Gx , 0] (1 + α) ρ 1/2
(27)
2
and similarly, the velocity shear in the air, defined as the difference between the surface wind, and the vector sum of the spectrally averaged phase velocity and the reference velocity, u1 − uL − u0 =
(1/2)α [Gx , 0] (1 + α) ρ 1/2
(28)
1
In accordance with the original hypothesis (Eq. (1)), both these vectors lie along the direction of the surface stress (τ s ) with their magnitudes in the ratio (ε), i.e. the velocity shear in the
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33
wave boundary layer is tangential to the surface shear stress. Normal to the surface shear stress, we have, v(z) = v0 + vL
(29)
v(−z) = v0 + εvL
(30)
and
such that the normal velocities ( in the wave boundary layer) are constant in each fluid, and from (21) or (24) for the air, we obtain, + U ) (Ug1 1 1 g2 (31) v(z) − v0 = − 1/2 2 (1 + α) ρ1 and from (21) or (25) for the water, we have, v(−z) − v0 = ε(v(z) − v0 )
(32)
Using (29) and (30), the surface shear stress along 0x is, τsx = ρ1 KI |(u1 − u0 ) − (u2 − u0 )/ε| ((u1 − u0 ) − (u2 − u0 )/ε)
(33)
Next, on introducing the relation between the Stokes shear and the Ekman shear, εuL = r(u0 − u2 ) where r is a constant, (5) yields, r(u1 − u2 ) uL = (1 + ε + 2r)
(34)
and on substituting (34) in (33), we obtain, τsx = ρ1 KI |F |(F )|u1 − u2 |(u1 − u2 )
(35)
where F = 2(r + 1)/(1 + ε + 2r). In a frame of reference in each fluid, moving at the respective normal velocities (29) and (30), (34) and (35) are a formally complete description of the momentum exchange process. Furthermore, in horizontally homogeneous conditions in which the surface shear stress is a constant vector, they would be also valid at a fixed measurement station. Thus, the classical drag law description of the near surface momentum transfer is the limiting form of the more general inertial coupling relation, applicable in horizontally homogeneous conditions. The familiar forms ( (34) and (35)), however, obscure the fundamental structure of the inertial coupling relation, from which they have been derived.
4. Interpretation of the inertially coupled dynamics The structure of the solution of Section 3 depends only on the parameter α, which has the following significance. From (1b), we have directly that, (1/4)w∗2 KI = εu − u + u 2 L 2 0
(36)
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and on substituting (36) in the definition of α, and eliminating Ak in favour of the viscosity (v2 ) in the Ekman layer in the water, we obtain, |εuL − u2 + u0 | α= (37) wE where wE = w∗2 /(2v2 f )1/2 is the Ekman scale velocity for the ocean. Thus, ␣ is the ratio of the velocity shear in the wave boundary layer to the velocity shear in the Ekman layer (note that an identical result for ␣ can also be obtained for the atmospheric boundary layer). There are two limiting conditions (α → 0) and (α → ∞). 4.1. The Ekman limit (α → 0) For, α → 0, δ → 0, γ → π/4, and K0 /K1 → 0, Uˆ 2 → 1/2G and U 1 = U 2 . This is the Ekman limit in which the surface shearing stress lies at an angle 45◦ to the left-hand side of the inertially weighted geostrophic shear (f > 0). In this situation, from (5), uL + u0 = u1
and
εuL + u0 = u2
and hence the velocity shear in the wave boundary layer in each fluid is negligible, and by reference to the aerodynamic relations (1), it is apparent that the thickness of the wave boundary layer, zB → 0. 4.2. The Stokes limit (α → ∞) For, α → ∞, δ → π/4, γ → 0 and, K0 /K1 → 1, Uˆ 1 → 0, Uˆ 2 → 0 and U 1 = U g1 , U 2 = U g2 , and thus the wave boundary layer attains its maximum extent. In the Stokes limit, the shear stress lies in the direction of the planetary shear, and the frictional circulation in the constant viscosity layers is negligible. The above two limits define the range of possible states for the coupled system, in which the outer frictional layers are fully turbulent and the presence of the wavefield is taken account of by the inertial coupling relationship in which the surface Stokes drift and the spectrally averaged phase velocity are in the ratio (ε). We now seek to evaluate ␣ from observations in order to determine which limit is the best model for naturally occurring seastates. 4.3. Naturally occurring seastates In order to evaluate α, estimates of the parameters, A, k and KI , are required. We assume that k ∼ 0.4, and A ∼ 0.2 (Garratt, 1992), and estimate KI from (36). Observations (Bye, 1988) indicate that εuL + u0 − u2 /w∗ ∼ 12, and hence K I ∼ 1.7 × 10−3 . This estimate is similar to observations of the 10 m drag coefficient over the ocean (K10 ). The drag coefficient, KI , however, refers to the height (zB ) equal to the thickness of the wave boundary layer, which is not a fixed height. 1 From these estimates we obtain α ∼ 5, which 1 K10 can be obtained from K by assuming that a logarithmic velocity profile occurs between the edge of the I wave boundary layer (zB ) and 10 m, and using an estimate of zB , see Bye (1996).
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yields δ = 36◦ , γ = 9◦ and K0 = 0.7 KI . This state is clearly much closer to the Stokes limit than the Ekman limit.
5. Velocity structure of the coupled boundary layers Using (24) and (25), (27) and (28), (21), (31) and (32) the ratios of the magnitudes of the velocities for a non-zero relative geostrophic wind in the wave boundary layer, are, in the air, |u1 − u0 | : |u1 − u0 − uL | : |uL | : |v(z) − v0 | 1/2
[(1 + 2α)2 + 1]
1/2
: α : [(1 + α)2 + 1]
:1
relative surface wind : velocity shear in the wave boundary layer : spectrally averaged phase velocity : relative normal velocity and in the water, |u2 − u0 | : |εuL +u0 u2 | :ε|uL | :ε|v(z) − v0 | √ 2 : α : [(1+α)2 +1]1/2 : 1 relative surface current : velocity shear in the wave boundary layer : surface Stokes drift : relative normal velocity where from (31), (v(z) − v0 ) = −(1/2)(1/(1 + α)) (ug1 − u0 ), and from (21), using (20), uL = (1/2)(ug1 − u0 ). Using the velocity ratios, we also find that, r = −(1 + α). Hence, the Ekman shear and the Stokes shear in the wave boundary layer are of opposite sign. In the Ekman limit (α → 0) the velocity shears become negligible in comparison with the relative normal velocity, such that the outer frictional (Ekman spiral) circulations dominate. and the wave and surface velocities would coincide in each fluid. In the Stokes limit (α → ∞), on the other hand, the velocity shear in the wave boundary layer is dominated by the Stokes drift, and also the Ekman spiral circulations are of negligible importance. In this situation the wave velocities (uL and εuL ) in the inertial coupling relation (1), which are derived from irrotational wave theory, and therefore would be expected to take no direct role in the energy transfer process, also define the essential turbulent processes which maintain the wave boundary layer (through the velocity shear,) i.e. the critical layer mechanism in the air, and the energy dissipation in the water.For the naturally occurring seastate (α = 5) the respective velocity ratios are: air
11.0 : 5.0 : 6.1 : 1
water
1.4 : 5.0 : 6.1 : 1
in which the velocity shears in the wave boundary layers of each fluid attain about 80% of their respective wave velocities (the spectrally averaged phase velocity in air, and the surface Stokes drift in water). The relative normal velocity in air is 8% of the relative geostrophic velocity, and r = −6, such that the ratio of Eulerian shear to the Stokes shear in the wave
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boundary layer is –1:6. The similarity of the spectrally averaged Stokes drift profile and the logarithmic velocity profile which occurs in turbulent boundary layers is well known, e.g. Bye (1988). We note that the turbulent properties of the wave boundary layer are not specified (except to estimate the constant eddy viscosities in the outer frictional layers). In this respect, our analysis differs from McWilliams et al. (1997), who also take account of both Stokes and Ekman velocities. In their work, the constant viscosity layers are continued to the sea surface where they are matched with the surface stress and surface Stokes drift. In our analysis, the matching occurs at the edges of the constant stress wave boundary layers. Fig. 3(a) shows the velocity structure of the coupled boundary layers due to a non-zero geostrophic wind (ug1 ) relative to the reference velocity (u0 ), for a naturally occurring sea state ( in which α = 5) in the Northern Hemisphere. The surface wind (u1 ) lies to the left-hand side of the wavefield, and the surface current lies to the right hand side, and the Ekman spirals in the two fluids rotate clockwise. The velocity hodograph indicates a smooth transition between the Ekman spirals and the wave boundary layer in which the component of velocity normal to the surface shear stress is constant within each fluid. In the air, the wind direction in the Ekman spiral and in the wave boundary layer rotates anticlockwise,
Fig. 3. Velocity hodographs for a naturally occurring seastate (α = 5), driven by (a) a non-zero geostrophic wind (ug1 ); and (b) a non-zero geostrophic current (ug2 ). The air velocities are normalised by the factor (ε) with respect to the water velocities, and the velocities in each fluid are shown to scale. The dashed line shows the velocity hodograph (directed from the non-zero geostrophic velocity), on which the dots in the Ekman spirals indicate the position of the current vector at intervals of (1/4)Di , where Di = π(2υi /|f |)1/2 is the Ekman depth in air (i = 1) or in water (i = 2). The directions of rotation in the Ekman spirals are shown by the circulatory arrows.
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37
whereas in the water, the current direction in the Ekman spiral and the wave boundary layer rotates clockwise. The planetary shear in the water is due solely to the Stokes drift, which lies at an angle of 9◦ to the right hand side of the surface shear stress (for α = 5), and this angle increases to 45◦ at the base of the wave boundary layer. The ratios of the magnitudes of the velocities in the wave boundary layer for a non-zero geostrophic current can be found in a similar way, and Fig. 3(b) shows the corresponding velocity structure, relative to u0 . Note that the surface shear stress (τ s ) is oppositely directed for the non-zero geostrophic current. In the general situation, the velocity fields due to the geostrophic circulations are additive. 5.1. The significance of the relative velocity frame Throughout the analysis, the results have been expressed in a relative velocity frame, ie. in terms of the velocities, u1 − u0 , u2 − u0 , and ug1 − u0 , ug2 − u0 . Thus: 1. The u0 is a reference velocity, in a similar manner to the reference velocity which occurs in the classical double Ekman spiral solution, e.g. Kraus and Businger (1994). 2. The inertial weighting of the relative velocities, however, indicates that the importance of the relative oceanic geostrophic velocity in the frictional circulation and the surface shear stress, is augmented by the factor, 1/ε ≈ 30, in comparison with the relative atmospheric geostrophic velocity. This is the distinctive feature of the inertially coupled Ekman layers. 3. The results of the analysis are independent of u0 , such that the reference frame in which the reference velocity is measured need not be specified, a priori. In order to apply the results for proscribed geostrophic velocities in the two fluids, however, u0 must also be known. In particular, the surface shear stress is predicted by (16), which is expressed in terms of the planetary shear, which involves the relative geostrophic velocities of the two fluids, which depend on the geostrophic velocity and the reference velocity. Locally, in space and time, u0 is the Eulerian velocity common to both fluids. 4. The magnitude of the atmospheric geostrophic velocity is normally much greater than the Eulerian velocity, and hence the frictional circulation driven by the atmosphere is almost independent of the ocean dynamics (which is a comforting result for the meteorologist, as a detailed knowledge of the oceanic general circulation appears not to be of critical importance). For the frictional circulation driven by the ocean, however, the Eulerian velocity and the oceanic geostrophic velocity are of similar magnitude. Hence, the reference velocity, here, is of critical importance. 5. The conclusion from (4) is that the surface shear stress in the local reference frame cannot be accurately predicted by an analysis of the coupled Ekman layers alone. A general circulation model, or an observational programme, which gives the reference velocity field (u0 (x, y, t)), is also required. Eq. (22) indicates that this can be achieved from the determination of the geostrophic velocities, ug1 and ug2 and also the wave velocity, uL , obtained from a wave model. 6. All the wave models have the common property that they are formulated in the Earth reference frame, in which u0 = 0. Bye and Wolff (1999) have incorporated some simple wave models in the specification of the surface shear stress in a fine resolution quasi-geostrophic oceanic model, and, in particular, it was found that significant relative oceanic geostrophic velocities (ug2 − u0 ) occur in the planetary boundary layer.
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7. The implications are that, in general, the current component, (ug2 − u0 )/ε, of the planetary shear (G) is not insignificant in comparison with the wind component, ug1 − u0 , in the determination of the surface shear stress τ s . This result has important consequences for the general circulation of the ocean (Bye and Wolff, 1999). 8. In summary, the Ekman dynamics at any location, as observed and modelled theoretically in this paper, are independent of u0 . The prediction of the frictional circulations of the two fluids, however, requires that u0 be known from a dynamical model (or from observation). Numerical experiments with simple wave models for u0 , indicate that in determining the surface shear stress, the frictional circulation driven by the ocean is of similar importance to that driven by the atmosphere.
6. Ekman transports The Ekman transport in the atmosphere is 1 T E1 = (−τsy , τsx ) ρ1 f and, in the ocean, is 1 T E2 = (τsy , −τsx ) ρ2 f
(38)
(39)
On substituting for τ s and using (19) we obtain for the atmospheric Ekman transport, 1 α2 T E1 = (40) KI G2 (−sin (φ + γ ), cos (φ + γ )) ρ1 f (α + 1)2 + 1 For the Stokes limit (40) reduces to the expression KI T E1 = (−Gy , Gx )G ρ1 f
(41)
and for the naturally occurring sea state (α = 5) we have T E1 =
0.7KI G2 (sin(φ + γ ), cos(φ + γ )) ρ1 f
(42)
where γ = 9◦ . Thus, in the Stokes limit, the Ekman transport lies 90◦ to the left-hand side (in the Northern Hemisphere) of the geostrophic motion. In naturally occurring seastates, the angle of deflection is 99◦ to the left-hand side, and for the same drag coefficient (KI ) the magnitude is reduced by the factor (0.7). Similar results can be obtained for the oceanic Ekman transport which lies in the opposite direction to the atmospheric Ekman transport and has the same mass transport.
7. Discussion The application of the inertial coupling surface stress relation to the coupled planetary boundary layers yields sets of relations in terms of the planetary shear, see Section 3.
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The important property of the planetary shear is that the effect of the surface ocean geostrophic velocity is not insignificant. The resulting frictional relations depend on the parameter (␣). For α → 0, the Ekman limit is attained in which the effect of the wavefield would become negligible, and the structure of the planetary boundary layer would be identical with the classical double Ekman spiral (Kraus and Businger, 1994; Bye, 1986), since the wave boundary layer, shrinks to a negligible thickness. For α → ∞, on the other hand, in the Stokes limit the frictional circulation occurs only within the wave boundary layer, and the Ekman transport is normal to the planetary shear. Observations indicate that the Stokes limit is closely approached in the ocean, in which the geostrophic drag coefficient (K0 ) is about 70% of the drag coefficient (KI ), which is evaluated at the edge of the wave boundary layer, and the rotation of the surface stress from the geostrophic flow is about 9◦ (to the left-hand side in the Northern Hemisphere, and to the right hand side in the Southern Hemisphere), such that the Ekman transport is almost normal to the planetary shear. In the Southern Ocean, in which the surface geostrophic wind and current velocities are almost zonal, a recent series of simulations using a two layer fine resolution quasi-geostrophic model (Bye and Wolff, 1999, Bye and Wolff, 2001b) indicates that the effect of the current on the planetary shear (G) reduces the surface shear stress on average by about 50%. Thus, the Ekman transport is also about 50% of the classical estimate based on the wind field alone.
Acknowledgements Helpful discussions with Professor B.A. Kagan during a visit to Adelaide supported by the Department of Industry, Science and Tourism under the Bilateral Science and Technology Collaboration Program are gratefully acknowledged. Helpful comments by the referees are also acknowledged. The manuscript was typed by Mrs Carol Walding. References Bye, J.A.T., 1986. Momentum exchange at the sea surface by wind stress and understress. Quart. J.R. Met. Soc. 112, 501–510. Bye, J.A.T., 1988. The coupling of wave drift and wind velocity profiles. J. Mar. Res. 46, 457–462. Bye, J.A.T., 1995. Inertial coupling of fluids with large density contrast. Phys. Lett. A 202, 222–224. Bye, J.A.T., 1996. Coupling ocean-atmosphere models. Earth–Sci. Rev. 40, 149–162. Bye, J.A.T., Wolff, J.-O., 1999. Atmosphere–ocean momentum exchange in general circulation models. J. Phys. Oceanogr. 27, 671–692. Bye, J.A.T., J.-O. Wolff, 2001a. Momentum transfer at the ocean-atmosphere interface: the wave basis for the inertial coupling approach. Ocean dynam (in press). Bye, J.A.T., Wolff, J.-O., 2001b. Quasi-geostrophic modelling of the coupled ocean atmosphere. Math. Comput. Model. 33, 609–617. Bye, J.A.T., Makin, V.K., Jenkins, A.D., Huang, N.E., 2001. In: Jones, I.S.F., Toba, Y. (Eds.), Coupling Mechanisms in Wind stress over the ocean. Cambridge University Press, Cambridge (in press) Ekman, V.W., 1905. On the influence of the Earth’s rotation on ocean-currents. Ark. Math. Astron. Fys. 2, 1–53. Garratt, J.R., 1992. The atmospheric boundary layer. Cambridge University Press, Cambridge, 316 pp. Kraus, E.B., Businger, J.A., 1994. Atmosphere–Ocean Interaction, 2nd Edition. Oxford University Press, Oxford. McWilliams, J.C., Sullivan, P.P., Moeng, C.-H., 1997. Langmuir turbulence in the ocean. J. Fluid Mech. 334, 1–30.