The implications of linearly varying eddy viscosity for wind-driven current profiles

Continental Shelf Research, Vol. 13, No. 1, pp. 1-24, 1993. 0278~343/93 $5.00 + 0.00 Printed in Great Britain. © 1992 Pergamon Press Ltd

T h e impl i ca t ions o f l inearly v a r y i n g eddy v iscos i ty for w ind-

dr iven current profi les

PETER D . CRAIG,* JOHN R . HUNTER* a n d BARBARA L. JOHNSTON*

(Received 21 March 1991; in revised form 4 September 1991; accepted 16 December 1991)

Abstract--A major problem in numerical circulation modelling is the specification of vertical turbulence closure. After the simplicity of a constant eddy viscosity, the next step towards reality appears to be an eddy viscosity described by

"13 = K U , Z

where K is Yon Karman's constant, u. is a friction velocity, and z is the distance away from the sea surface or bottom. Applied to the "Ekman problem", this eddy viscosity results in current profiles that have logarithmic boundary layers. The Ekman depth is replaced by a new depth of influence, Ku./f (where f i s the Coriolis frequency). If this depth is less than the water depth, the wind-forced current profile does not reach the bottom, and is unaffected by the bottom conditions. For water depths less than the depth of surface influence, currents through the whole water column are sensitive to the value of the bottom roughness. By contrast, changes to the surface roughness only affect currents in a thin layer near the surface.

Numerically, resolution of the log-layers at the sea surface and bottom requires the use of special techniques. The log-layers may be described analytically, or the vertical coordinate may be transformed to expand the near-boundary regions. For modelling the mid-water column, away from the log-layers, a constant eddy viscosity with a linear bottom slip condition represents the currents with reasonable accuracy. Appropriate values of the constant eddy viscosity and bottom friction parameter are determined empirically from the shear stress and bottom roughness by formulae that are consistent over a broad parameter range.

1. I N T R O D U C T I O N

EKMAN'S (1905) solution, for the depth-structure of currents in an infinite ocean forced by a steady surface wind, is the classic demonstration of the oceanographic consequence of a rotating frame of reference. His surface currents are angled (at 45 ° ) to the wind direction, and his subsurface currents spiral clockwise (in the northern hemisphere) as they diminish with increasing depth. Since 1905, ocean measurements have confirmed this spiralling tendency (e.g. RICHMAN et al., 1987), although details of the current structure do not exactly agree with Ekman's theory.

Ekman parameterized turbulence with a constant vertical eddy viscosity. Discrep- ancies between his solution and measurement are widely credited to misrepresentation of the eddy viscosity, and his solution has, over the years, been recalculated with a range of eddy viscosities (e.g. JORDAN and BAKER, 1980). There is no universally accepted

* CSIRO Division of Oceanography, GPO Box 1538, Hobart, Tasmania 7001, Australia.

1

2 P.D. CRAIG et al.

specification for the eddy viscosity, and there perhaps will never be, given that it is a gross simplification of highly complicated turbulent processes. A depth-independent eddy viscosity is still the most commonly used, both for its analytic simplicity, and for lack of realistic alternatives.

Amongst coastal-ocean numerical modellers, an approach that is finding increasing favour (e.g. DAVIES and JONES, 1990, 1991) is the adoption of so-called "higher-order" turbulence closure schemes. These schemes appear, for the most part, to be based on the "Rotta-Kolmogorov" model of turbulence proposed by MELLOR and YAMADA (1974, 1982). At its most complicated ("level 4", MELLOR and YAMADA, 1974), this model requires (for stratified flow) three empirical constants and seven mixing length assump- tions or, ultimately, 10 constants and one mixing length. At "level 2 TM, the turbulence closure is specified in terms of an eddy viscosity, which is determined from a simplified set of Reynolds-stress equations requiring five constants and a mixing length. The empirical constants appear reasonably robust, and the results of the level 2½ scheme encouraging (MELLOR and YAMADA, 1982).

In the level 2½ scheme, the eddy viscosity is constrained near horizontal surfaces to take the classical form

v = K u , z (1)

where K = 0.4 is von Karman's constant, u, is a friction velocity (defined as the square root of stress divided by density), and z is the (vertical) distance from a surface. The specification (1) is justified on the basis of mixing length arguments, and leads to logarithmic boundary layers similar to those observed (e.g. P~NDTL, 1952). The specification appears to have first been used in the context of atmospheric boundary layers by ELLISON (1956). It was discussed in the context of estuarine boundaries by BOWDEN et al.

(1959) and BOWDEN and HAMILTON (1975), and applied to the oceanic bottom boundary by THOMAS (1975) and the ocean surface layer by MADSEN (1977). JENTER and MADSEN (1988) review a range of uses of and observational justifications for an eddy viscosity of the form described by (1).

While the validity of (1) near the seabed seems to be well established, its accuracy near the sea surface is more questionable. A major problem is the difficulty of obtaining accurate measurements immediately below the sea surface. However, CHEUNG and STREET (1988) discuss a variety of field and tank observations, including their own, that support the concept of a surface log-layer in the ocean. Further, acoustic observations of near- surface bubbles by THORPE (1984), and also GRIFFITHS (1989), tend to favour a linear eddy- viscosity variation away from the surface.

Surface waves are understood to enhance the influence of turbulence on the mean flow. Observations by THORPE (1984) and KITAIGORODSKI et al. (1983) suggest that the region of intensified turbulence under breaking waves extends to a depth approximately 10 times the amplitude of the waves. In one of a series of papers on wave-current interactions, JENKINS (1987) modelled this surface region as a constant eddy viscosity layer, overlying deeper water in which (1) was assumed to be valid. He concluded that currents in the deeper water, at least, were insensitive to the near-surface eddy viscosities.

CHEONG and STREET'S (1988) observations, while confirming the depth-dependence of (1), suggest that the "constant" K may in fact be an increasing function of wind-speed. This possibility also appears to be supported by THORPE'S (1984) bubble analysis.

In general, the weight of evidence appears to support the conceptual correctness of a

Linearly varying eddy viscosity and wind-driven current profiles 3

linearly varying eddy viscosity near both the top and bottom ocean surfaces. An advantage that stems from a simple formulation like (1) is that it can lead to simplified dynamical situations that are amenable to analytic solution. In EKMAN'S (1905) constant eddy viscosity formulation, currents decay exponentially with distance from the surface. With a linear eddy viscosity as in (1), currents vary as Kelvin functions away from the boundary (SouLSBY, 1983). The linear eddy viscosity thus produces behaviour that is more difficult to grasp intuitively, since Kelvin functions do not have the familiarity of the exponential function. Further, without due care, the logarithmic properties of Kelvin's functions close to boundaries can lead to problems of resolution in a numerical representation.

In the present paper, we revisit the "Ekman problem", but this time with a linear eddy viscosity. The aim of the paper is to examine the implications of the linear eddy viscosity for numerical modelling. The problem is formulated and non-dimensionalized in Section 2, and the dependence of the current profiles on the governing non-dimensional parameters is discussed in Section 3. In Section 4 we examine numerical techniques that are capable of resolving the logarithmic boundary layers. These techniques are used in Section 5 to investigate the sensitivity of the current profiles to the eddy viscosity values in the mid- water column. Finally, in Section 6, we discuss the applicability of constant eddy viscosity approximations to the linear representation (1).

2. FORMULATION

The problem is posed in terms of the complex velocity W = U + i V , where U and V are horizontal velocity components. (Upper case notation is used to represent dimensional variables.) The equation for steady motion in an ocean of effectively infinite horizontal extent (with no horizontal pressure gradients) is

where Z is the vertical coordinate (positive upwards), v is the eddy viscosity and f is the (constant) Coriolis parameter (and i 2 = -1) . Motion is forced by a steady surface wind stress r assumed, for convenience, to be directed in the X(U) direction. The surface condition is then

dW r v = - , at Z = 0 (3)

dZ p

where p is the water density. At the seabed, Z = - H , a no-slip condition applies

W = 0, a t Z = - H .

As foreshadowed in the introduction, the eddy viscosity is assumed to vary linearly away from both the surface and the bottom. The simplest formulation satisfying this require- ment is a "bilinear" eddy viscosity, given by

/¢t t ,s(Z -- Z ) , 0 ~ Z ~ Z m v = (4)

KM,b(Z 2 ÷ H + Z), Zm - Z -> - H

where Z 1 and Z 2 are roughness lengths, and Zm is defined by

4 P . D . CP.~a~ et al.

u,s(Z1 - Zm) = U,b(Z2 + H + Zm). (5)

In (4) and (5), constants u,s and U,b are surface and bottom friction velocities. In the upper level, the friction velocity is defined explicitly from the wind stress by

2 77 U,s = -. (6)

P

In the lower level, Z < Zm, it is defined implicitly from the bottom stress by

2 dW U,b= V ~ a t Z = - H . (7)

JENTER and MADSEN (1988) used the definition (7) literally and determined the bottom friction velocity interactively. In the present study, however, we assume that the bottom stress will be dominated by some high frequency motion such as that due to tides or surface waves. In this case, the Win (7) will be that describing the high frequency dynamics, so that U,b can be regarded as an externally specified parameter for the wind-forced motion. This is an approach exactly analogous to the linearization of bottom friction in wind-driven models (e.g. HUNTER, 1975; PINGREE and GmFFITHS, 1987). The philosophy of a two-layer stress specification in (4) is basically the same as that adopted by DAVIES (1985). He postulated a surface-layer in which the stress was determined by the (instantaneous) wind, with a linearly varying (in depth) eddy viscosity, and a deeper layer, occupying most of the water column, in which the eddy viscosity was constant in both space and time and determined by the average (in time) tidal currents.

The roughness lengths Z1 and Z2 in (4) and (5) essentially define the minimum scale of the turbulence, determined by wave conditions at the surface, and morphology at the seabed. The matching depth Zm, in (5) can occur anywhere in the water column, depending on the relative magnitudes of u,s and U,b. In particular, the definition (5) expands the layer of weaker stress (i.e. smaller u,) relative to the other, as noted by JENTER and MADSEN (1988). They overcame this problem by specifying a discontinuous v. We will, for the present, ignore this problem, and retain the definitions (4) and (5), allowing continuity of both W and dW/dZ at Z = Zm.

It is convenient to non-dimensionalize Wby the scale u,JK, and all depths (including Z 1 and Z2) by the water depth H. With the introduction of lower case w and z to represent non-dimensional quantities, the governing equation then becomes

d (z(z) dW) d---zz ~ = iw, (8)

where

with z m now given by

V = ILl(Z1 - z), A(z) = f - ~ [L2(z 2 + 1 + z),

0 ~--- Z ~ Zm, (9) Zm>--Z > -- --1

Ll(zl - Zm) = L2(Z2 + 1 + Zm).

The constants L 1 and L 2 are defined by


(10)

and represent non-dimensional depth-scales. These scales are analogues of the Ekman depth which indicate, in some sense, the depth of influence of the boundaries. We shall examine in detail the actual meaning of L1 and L2 in the following sections.

The non-dimensional boundary conditions for (8) are

dw 1 dz=F, atz=O

1 (II)

and

w = 0, at2 = -1. (12)

The solution of (S), with boundary conditions (11) and (12) is relatively straightforward (see SOULSBY, 1983; JENTER and MADSEN, 1988). If we define new upper and lower level variables as

and

then

p*= L(zz+l+z) i

l/2

L2 ) Z,r.ZZ-1 (13b)

and

w = q(ber p1 + ibei pJ + bl(ker p1 + ikei pl), OLZZ.2, Pa)

w = a2(ber p2 + ibei p2) + b2(ker p2 + ikei p2), Z,ZZZ-1 (I4b)

in which ber, bei, ker and kei are zero-order Kelvin functions (e.g. ABRAMOWITZ and STEGUN, 1965), and al, bl, u2, b2 are constants determined from conditions (11) and (12) and the continuity of w and dwldz at z = z,.

The behaviour of the Kelvin functions for small arguments is tabulated in Table 1. As the argument tends to zero, each of ber, bei and kei approach as a constant value, while ker behaves logarithmically.

Table 1. The behaviour of zero-order Kelvin functions for small argument (adapted from ABRAMOWITZ and STEGUN, 1965)

x ber x bei x kei x ker x ker x + In (0.8905x)

0 1.000 0.000 -0.785 0.1 1.000 0.002 -0.777 2.:20

0.000 0.002

0.2 1.000 0.010 -0.758 1.733 0.008 0.3 1.000 0.022 -0.733 1.337 0.017 0.4 1.000 0.040 -0.704 1.063 0.030 0.5 0.999 0.062 -0.672 0.856 0.047

6 P.D. CRAIG et al.

3. CURRENT PROFILES

In this section, we will examine the structure of the current profiles described by (14), looking in particular at their sensitivity to the four parameters L1, L2, zl and z 2. JENTER and MADSEN (1988) present useful tables of the anticipated ranges of L 1 and z2. Basically, L1, ranges from a value of approximately 100 for storm winds in 2 m of water, to 0.02 for light winds in 200 m. The roughness length z2 meanwhile covers six orders of magnitude, from 2 X 10 - 3 for cobbles (or sand ripples) in 2 m, to 2 x 10 -9 for silt in 200 m. While L 2 may be expected to cover a similar range of values as L1, zl is rather more problematic. The specification of Zl is equivalent, by (11), to the specification of the eddy viscosity at the surface. There are empirical formulae for specifying either the surface roughness or eddy viscosity (e.g. DAVIES, 1985). If there is a wave-influenced surface layer, as discussed by JENKINS (1987) (see Section 1), then z = 0 in (11) becomes the base of the layer, and Zl is chosen to ensure continuity of the eddy viscosity. The implication of this is that Zl may be able to take considerably larger values than z 2. However, as noted in Section 1, the dynamics of the surface layer are not at all well established. Fortunately, as suggested by JENKINS (1987), and as we shall see, the subsurface currents are not sensitive to the choice of zl.

For presentation purposes, it is obviously necessary for us to work with a limited selection of parameter values. We will look only at cases with L1 = L2 so that, in most instances, we drop the subscripts and refer unambiguously to L. Three representative values of L will be examined: L = 0.1, when the surface stress has virtually no effect at the bottom, L = 1, when the frictional depth scale, or depth of influence of the surface, Ku,/f, is equal to the water depth, and L = 10, when the frictional depth-scale dominates the water depth. Further, we will use two values, 10 -3 and 10 6, for za and z2. As we shall see, this range of parameters, though limited, is sufficiently wide to allow the structure of the currents to be broadly characterized.

Figure l a - c shows profiles and hodographs of currents for L equal to 0.1, 1 and 10, respectively. In Fig. la and b, profiles for both zl = 10 -3 and zl = 10 .6 are shown. In Fig. lc, different curves are presented for z2 = 10 -3 and 10 -6. For Fig. la and b, changing z2 through this range made little difference to the solutions.

Probably the most noticeable feature in Fig. 1 is the surface logarithmic layers, the completely straight surface segments of the hodographs. In each case (including that for L = 10 which is not shown in Fig. 1), decreasing the surface roughness from 10 -3 to 10 -6 markedly increases the current strength. The reason for this is clear from (8). Near the surface, where zl - z is very small, the left-hand side of the equation scales as 1/(Zl - z) so that the right-hand side can be ignored to give

d ( (zl - z) dwl = ] (15)

This equation has the solution, using boundary condition (11), that

w = - l n ( z l - z) + cl (16)

for some constant Cl. Thus, the surface current is given by 6.9 + c 1 for Zx = 10 -3 and 13.8 + c1 for 10 -6. The difference of 6.9 between the two solutions is evident in the hodographs in Fig. 1. Conceptually, changing the surface roughness is the same as moving the surface from z = 10 -3 to z = 10 -6 on the In(z) curve.


(a) u - 1

I

V

- 1 0

I 10

I ~ 1 1 i 1 1 1 1

Y X

z l = 10 .3 z l : 10 "l;

u

- 1 0 13 - 7 1 I I _1_ I x l I I i I ~ i v i i ~ I I f -

Z L = 1 0 . 3

z t = 10 .5

"v"

0 I I I I i ~ z l I

z l = z ! = 1

£h

- 0 . 5 ~

- 1 . 0

Fig. 1. Current profiles and hodographs calculated for a bilinear eddy viscosity, for (a) L = 0.1, (b) L = i and (c) L = 10. Surface points are marked with an "x" and hodograph depths are marked with a " + " at intervals of 0.1. Curves are distinguished at their surface points by labelling according

to roughness length. Note that some curves are indistinguishable except at the surface.

It is obvious from Fig. 1 that, at least for cases where the roughness length is less than the log-layer thickness (that is, when the surface point falls on the straight part of the hodograph), the value of the surface roughness does not affect the solution below the log layer. In the central part of the water column, the magnitude of the currents changes much more slowly with depth (relative to the log-layer) and, as is to be expected from EKMAN'S (1905) original study, the currents spiral clockwise with increasing depth. As is also to be expected, the amount of spiralling, or the total change in direction over the depth,

8 P . D . CRAIO et al.

(b) u - 1

- 1

10

X X

f Zl = 10 "3

z t = 10 .6

u v

- 1 0 13 - 7

l= 10"3 = 10. 6 zt = I0 "3 zt

Zl = 10 "6

Fig. l(b).

- 0 . 5 '~

- 1 . 0

increases with decreasing L. The size of L indicates the ratio of frictional to rotational effects.

In Fig. la, with L = 0.1, there is no bottom boundary layer, because the currents have effectively decayed to zero before reaching the boundary. In each of Fig. lb and c, however, there is a distinct log-layer at the bottom. As has already been noted, the profile for L = 1 (Fig. lb) is virtually unaffected by the value of the bottom roughness, z2. For L = 10 (Fig. lc), however, the profile is clearly dependent on z 2.

In Fig. lc, decreasing the bottom roughness length increases the current magnitude at the top of the bottom log-layer, that is, it increases the "length" of the log-layer as observed in the hodograph. This effect is similar to that caused by variations in the surface roughness. The important difference is that, while changing the surface roughness affects

L i n e a r l y v a r y i n g e d d y v i s c o s i t y a n d w i n d - d r i v e n c u r r e n t p r o f i l e s 9

(c) l+1

- - W l r i i i i i , i

- - 1 0 1

10 I t I

Zl = 10"3 z2 = 10"3

X

X

f- z t = 10-3 z2 = 10 "6

- - 1 0 I I I I I

U

//:,2-'1°07, / / z2 10 6

13 - 7 I I I

z l = 10 -3 Z2 = 10 "6

V

0 I I

z 1 = 10-3 z2 = 10 -3

Fig . l ( c ) .

\ \

e~

- 0 . 5 '~

- 1 . 0

only the surface currents, changing the bottom roughness alters the whole profile. The decrease in bottom roughness reduces the eddy viscosity at the bottom boundary by three orders of magnitude, allowing stronger shears and, thus, higher velocities to develop in the bottom boundary layer. By contrast, the shear at the surface is fixed by the surface boundary condition. Decreasing the bottom roughness length not only increases current magnitudes throughout the profile, but also increases the current rotation with depth. As a point of interest, the near equality of the u-components in the two curves on Fig. lc is not a universal property for all values of L.

The case L = 1 (Fig. lb), when the friction depth equals the water depth, represents a transition. For this value of L, the current profile is rather insensitive to details of the bottom boundary condition. In fact, eliminating the log-layer by setting the bottom

10 P. D, CRAIG et al.

condition to dw/dz = 0 at z = - 1 makes little visual difference to the solution above z = -0 .9 . Thus, it appears to be a good rule-of-thumb that, if details of the bottom boundary layer are not required, and L < 1, then the profile can be adequately modelled by assuming a free-slip condition at the bottom (thus avoiding the need to resolve the bottom log-layer). For L > 1, however, the bottom boundary condition must be properly modelled.

On the basis of Fig. 1, we can speculate on an appropriate definition of the "log-layer thickness". In the surface log-layer, the dynamics are governed by (15) with solution (16). According to (16), the logarithmic behaviour, being real, occurs strictly in the direction of the wind. The base of the log-layer will be the depth at which the profile "deviates significantly" from (16). If (zl - z) scales as, say, a small parameter e, then w in (16) may be regarded as the zero-order solution in e. With this now denoted as w 0, the first-order solution then satisfies

d ( L ( z l - z) dwl] dz ] = iw° (17)

together with a zero-stress condition at the surface. Solving (17) yields

.Z 1 - - Z W 1 = l T (W0 "}- 2) + d (18)

where d is a constant that may be neglected in the present discussion. The factor i in (18) indicates that variations away from logarithimic behaviour will occur at right angles to the wind direction, demonstrating the influence of rotation. According to (18), the cross-wind velocity will be approximately 1% of the down-wind velocity at a depth z I - z = 0.01 L. On the hodographs, this is roughly the depth at which the curve appears, to the eye, to begin turning. Again, as another rule-of-thumb, the log-layer may be said to have a depth approximately one-hundredth of L.

The linear eddy viscosity is justified primarily on the grounds that it gives a reasonable representation of the logarithmic boundary layers. The suggestion from Fig. 1 is that, in the mid-water column where the currents do not vary rapidly with depth, the velocities are unlikely to be sensitive to the exact eddy viscosity specification. JENTER and MADSEN (1988) quote evidence that the linear eddy viscosity should only extend to approximately 0.1 L from the boundary. We examine the mid-water sensitivity in Section 5. However, we first develop techniques for treating more general eddy viscosity profiles.

4. NUMERICAL SOLUTION TECHNIQUES

The governing equation (8) can obviously be solved numerically with standard finite- difference techniques. The water depth is discretized into equal intervals, with an additional point above the surface to accommodate boundary condition (9), the second derivative is represented by a second-order accurate central difference, and determination of the solution then simply requires the inversion of a tridiagonal matrix. The disadvantage of this approach is that satisfactory resolution of the log-layers requires extremely fine discretization of the vertical domain. This is not so much of a problem in the present situation where only a single matrix inversion is required, but at the next level of


complexi ty , in a fo rmula t ion that involves t ime and space dependence , the computa t ion can become prohibi t ive.

A n al ternat ive approach is to t ransform the vertical coord ina te to

Op(z) = I z dz (19)

so that (8) becomes

d2w dqb2 = i B ( ~ ) w

in which B(qb) = A ( z ) . The surface bounda ry condi t ion is now

dw _ B((Ps), at dp = dps dq> zl

with

(20)

I 0 dz (21) ¢Ps = A(z )"

-1

The t ransformat ion (19) has the effect of expanding regions of small A relative to those of large A (since dz = AdOP).

Equa t ion (20) can now be solved using the same numerical technique as that described above for (8). To assess the approach , we define an er ror or difference 6 by

( } = - - 171] i W i ) , ( 2 2 )

= .=

where N is an arbi t rary n u m b e r of equally spaced discrete points in the vertical, wi is the analytic solut ion evaluated at point i, and ii,; is the numerical approximat ion. For all evaluat ions in the present paper , N was set to 401. Table 2 lists the surface value of w and

Table 2. Comparative performance of the three numerical schemes for the bilinear eddy viscosity solution, with parameter values L as listed, zl = z2 = 0.001 and discretization into n vertical points

Surface values of w Difference measure 6

L n Analytic Standard Transformed Combined Standard Transformed Combined

0.05 5 (3.6, -1.4) (0.2, -0.7) (3.5, -1.4) 0.68 0.99 11 (3.6, -1.4) (0.8, -1.1) (3.6, -1.4) 0.46 0.15 21 (3.6, -1.4) (1.4, -1.4) (3.6, -1.4) 0.31 0.04

101 (3.6, -1.4) (2.8, -1.4) (3.6, -1.4) 0.10 0.02

1 5 (5.4, -1.7) (2.0, -1.8) (5.9, -1.2) (4.1, -1.7) 0.32 0.64 0.17 11 (5.4, -1.7) (2.8, -1.8) (5.5, -1.6) (4.8, -1.6) 0.19 0.13 0.08 21 (5.4, -1.7) (3.4, -1.8) (5.4, -1.7) (5.2, -1.6) 0.13 0.03 0.05

101 (5.4, -1.7) (4.6, -1.7) (5.4, -1.7) (5.5, -1.6) 0.04 0.001 0.04

10 5 (10.7,-3.1) (5.1,-0.9) (9.1, -3.3) (11.1, -2.5) 0.55 0.26 0.10 11 (10.7, -3.1) (6.7,-1.3) (10.4,-3.2) (11.2,-2.6) 0.41 0.06 0.10 21 (10.7, -3.1) (7.7, -1.7) (10.6,-3.1) (11.2,-2.6) 0.30 0.02 0.10

101 (10.7,-3.1) (9.7, -2.6) (10.7, -3.1) (11.2, -2.6) 0.11 0.0006 0.10

12 p.D. CRAm et al.

the parameter 6 for numerical solutions of (8) and (20), described as the "standard" model and "transformed" model, respectively. The transformed model gives marked improvement in representation of the surface velocity, with agreement to within about 2% of the analytic solution for a discretization into only 11 points. The improvement over the full water column is marked for large values of L, but only marginal at L = 0.1. As an example of the appearance of the solutions, Fig. 2 shows profiles of the analytic solution (plotted at 401 points) compared with numerical solutions for L = 10, with numerical discretization into both 5 and 11 points.

For the bilinear A(z) of (9), (19) gives a logarithmic transformation in each half of the water column. Near the seabed, it is in this case the same as the log-transformation used by DAVIES and JONES (1990, 1991) to improve resolution in their tidal model. Their model specifies a level 2½ turbulence closure scheme in which, as discussed in Section 1, the eddy viscosity is constrained to behave linearly (with the vertical coordinate) near the boundary. In their case, then, the transformation will have an effect similar to that shown in Table 2 and Fig. 2.

The transformed coordinate do in (19) can be calculated analytically for A(z) defined by (9) or, indeed, for any piece-wise linear specification of A. For other forms of A, the coordinate dO may need to be generated numerically. This involves the calculation of dOt in (21) using a numerical integration scheme, followed by an iterative scheme, with repeated integrals, to calculate the inverse function z = z(dO) at discrete points. Combining Gauss- Legendre integration, with the z-interval ( -1 ,0) discretized into 100 points, together with Newton iteration, we were able to achieve a high degree of accuracy (6 < 0.0001). This technique is, however, computer intensive, and not well-suited to repeated use in, say, a time-dependent problem.

In many applications, however, A(z) will be specified to be piece-wise linear, so that dO can be calculated analytically. In this case, numerical solution of an equation having the form (20) will tend to be more efficient than that for the original equation (8).

A third solution technique for (8) is to treat the log-layers analytically, but the mid-water profile numerically. In the surface log-layer, the velocity takes the form (16), so that it is known to within an additive constant [Cl in (16)]. If we assume that the bottom of the log-layer is at -zt, then the surface condition (11) can be replaced by

dw 1 - - = - - , at z = - z v ( 2 3 ) dz za + zt

In the bottom log-layer, using the boundary condition (12) and reasoning similar to that leading to (16), the velocity is known to within a multiplicative constant c2

w = c21n (z2 + 1 + z) " z 2 (24)

Assuming, for convenience, the same log-layer depth zt, boundary condition (12), applied at the bottom, can now be replaced by the condition, derived from (24), that

dw w - -dz = / ," at z = - 1 + z t ( 2 5 )

(z2 + zl) In [ z2+z~} \ z2 /

applied at the top of the log-layer. In (25), as in (23), the unknown constant (c2) has been eliminated, so that the boundary condition is fully specified.


(a) U 7 10 i i i i I P J I I I i

~ standard combined

i - " '

V analytic transformed

-1C

U v

- 1 0 13 --7 0 1 [ [ I I I [ L, I I I ~ I Z ~ I I / I I P , ~ i k, I I

l/g . UY - 1 . 0

Fig. 2. Comparison of profiles and hodographs for analytic and numerical solutions for L = 10, zl = z2 = 0.001. Surface points are marked with an "x" and hodograph depths are marked at

intervals of 0.1. Vertical discretization is (a) 5 points and (b) 11 points.

With boundary conditions (23) and (25), the standard numerical technique, described in the first paragraph of this section, can be applied to the domain between the log-layers, without the need to resolve the layers themselves. Once w is determined (numerically) at z = zt and z = - 1 + zz, constants c 1 and c 2 can be calculated, and the solution in each of the boundary layers is given analytically by (16) and (24).

The behaviour of solutions calculated in this way is also indicated in Table 2, under the heading "combined" model , and on Fig. 2. For these solutions, the log-layer depth was set, after considerations described in Section 3, to zl -- 0.01 L. We also exper imented with zl set to 0.005 L and 0.015 L. For both L = 1 and 10, the solution accuracy was not significantly

14 P . D . CRAIG et al.

(b) - 1

I . .

U

I I I I I I I I I I

,standard

combined

, T

analytic transformed

1.l v

-i1 0 13 -r7 0 1

[.0

Fig. 2(b).

affected over the range of zl. There is a suggestion in Table 2 that the solution accuracy is compromised when zt is the same size as the discretization interval. Table 2 indicates that, with 11 points or more in the vertical, the "combined" model gives results of similar accuracy to the t ransformed model. The accuracy of the former does not, however, increase so rapidly with increasing discretization.

The "combined" model was not used for L = 0.1. In this case, zl is the same as zl, so that no improvement over the standard model can be expected. Low values of L (of order 0.1 and less) present a difficulty for these two models, in the sense that most of the solution variability is restricted to within approximately 2L of the surface (Fig. la) . Adequate discretization of this surface region can only be achieved with a discretization into a large number of points over the full water depth.


Thus, the combined model, incorporating analytic solutions near the boundaries with numerical solution in the interior, is considerably simpler and approaches the accuracy of the transformed model, solving (18). All of the numerical models encounter difficulties at low values of L, a problem that will be further discussed in Section 6.

5. MID-WATER EDDY VISCOSITY VALUES

To this stage, we have chosen to represent the mid-water-column eddy viscosity simply by extrapolating from the boundary layer values. This is a reasonably common approach, adopted for analytic simplicity (e.g. SOULSBY, 1983; JENa'ER and MADSEN, 1988). The justification is usually that the currents are unlikely to be sensitive to eddy viscosity values away from the boundary layers. SOULSBV (1983) suggested that, in reality, the eddy viscosity probably decays to the molecular viscosity outside the boundary layers.

Empirical studies of tidal currents, however, have tended to suggest that the eddy viscosity should take a larger constant value in the main part of the water column. BOWDEN et al. (1959), for example, proposed a continuous eddy viscosity, linear near the bottom, but constant above a depth of approximately 0.1 H. In another example, MAAS and VAN HAREN (1987) accurately modelled tidal currents in the North Sea using a constant eddy viscosity and a slip-condition at the bottom. In the Maas and van Haren study, L2 appears to be of order i (actually, around 0.24 and 1.4 for the anticlockwise and clockwise rotary components, respectively) and their lowest current meters were deployed at approximately 0.1 H, well above the log-layer. Their best-fit eddy viscosity was 1.4 × 10 -3 m e s -1 which, for their parameter values, would be achieved by an eddy viscosity of the form (4) at 0.01 H above the seabed.

Interestingly, DAVtES and JONES (1990) plotted eddy viscosity values, calculated with their level 2½ closure scheme, for the Me tide at a location in the mid-North Sea close to where MAAS and VAN HAREN'S (1987) measurements were made. The eddy viscosity does not vary greatly over the tidal cycle and has the approximate form of a linear increase from zero with height from the seabed to around 0.2 H, with a constant value thereafter. The mid-water eddy viscosity is about 0.02 m e s -1, an order-of-magnitude higher than MAAS and VAN HAREN'S (1987) estimation.

There are, of course, additional complications that arise with tidal, rather than steady, dynamics. In the case of non-steady forcing with angular frequency co, the inertial frequency f i n the formulation of the previous sections is replaced by co + f a n d co - f. In place of the depth-scale L in (10), there are now two depth-scales, and the velocity profile consists of two superimposed, counter-rotating components, with frequencies of co + f a n d co - f (e.g. SOULSBY, 1983). In the present context, however, the point is that tidal observations tend to confirm the concept of a constant mid-water eddy viscosity. The "cutoff depth", at which the eddy viscosity changes from linear to constant, will obviously determine its midwater value. JENTER and MADSEN (1988) quote support for a (non- dimensional) cutoff depth between 0.1 L and 0.15 L. These cutoffs will only affect the eddy viscosity profile for L values less than 5 and 3.3, respectively.

To demonstrate the sensitivity of the current profiles to the cutoff depth, Fig. 3 compares profiles for the strict bilinear eddy viscosity against eddy viscosities cut off at 0.1 L and 0.15 L. Figure 3a is for an L of 0.1, and Fig. 3b for L equal to 1. For both values of L, the behaviour is similar, although it is easier to distinguish in Fig. 3b. In each case, the surface currents in all three hodographs have similar magnitude, with a tendency to point further to the right of the wind as the cutoff depth decreases. As the cutoff depth

16 P . D . CRAIG et al.

(a) U

-I 1 ..~- i I I I Ineal

~ : - , . , o . ~ o,L

V

10

--1C

- - 1 0

I ~ b i l i n I ~1 f - I e.r

O.1L O.15L

U V

13 - 7 I I f I I I [ I I I I I I I v~, v l

O.15L

0.1L

Fig . 3.

1

~- bilinear

e~

--0.5 ~

- 1 . 0

Prof i l e s a n d h o d o g r a p h s s h o w i n g s e n s i t i v i ty to a m i d - w a t e r cu t -o f f o f the e d d y v i s c o s i t y , at b o t h 0.1 L and 0.15 L. S u r f a c e p o i n t s are a g a in m a r k e d w i th an "x" and h o d o g r a p h d e p t h s are

m a r k e d at in t e r v a l s o f 0.1. (a) L = 0.1 and (b) L = 1.

decreases, so too does the (constant) eddy viscosity in the central water column. The result of this is that, for smaller cutoff depths, the surface currents are stronger, and there is more shear and rotation of the currents with depth in the central part of the water column. The difference 6 of (22) has the values 0.61 and 0.49 for the two "cutoff" curves in Fig. 3a, and 0.39 and 0.22 in Fig. 3b.

If, as SOULSaY (1983) suggests, the eddy viscosity actually decreases away from the boundary layers, then the tendency towards stronger shear and rotation, observed in Fig. 3, would increase.

For cutoff depths set to a fixed fraction of L 1 and Le, a linearly varying mid-water eddy


(b) - I

I

V

- I (

U

I I t I I I

~ _ _ ~ j . ~ bilixear 'i

o.?sL ~o.,L

10 I I I I

-I0 I I I I I I

bilinear~.~O.l 5 L

U

lwl

V

13 -7 0 i I I I I I I I

- . 5 ~

1.0

Fig. 3(b).

viscosity joining the boundary layers will only be constant if L 1 = L2 (and Zl = z2). If L1 and L 2 are not equal, the options include a linearly varying mid-water eddy viscosity, or a constant value fixed, for example, at the surface cutoff depth.

For the specific case of L 1 = L2 (and Zl = z2), another reasonably common approach (e.g. BOWDEN and HAMILTON, 1975) is to specify an eddy viscosity quadratic over the water column

A = - L ( z 2 + z - - Z1) , - -1 -< z -< 0. (26)

In experiments over the three values of L (0.1, 1 and 10), we found the bilinear and quadratic solutions to be in close agreement, with 3 taking a value of 0.10 or less.

It is apparent from Fig. 3 that the current structure is sensitive to the eddy viscosity over

18 P .D. CRAm et al.

the full water column. There is an obvious need for further field investigation of mid-water eddy viscosity behaviour under wind-forced conditions, particularly in situations where L is of order 1 or less.

6. C O N S T A N T E D D Y V I S C O S I T Y A P P R O X I M A T I O N S

As we have already noted, an eddy viscosity constant with depth is still the most widely used in numerical modelling. It is perhaps a logical extension of the previous section to ask how well a constant eddy viscosity model can approximate the bilinear solution (14), which we may regard as one step closer to reality. In non-dimensional terms, the constant eddy viscosity problem, equivalent to (8), is expressed as

ed2w dz 2 = iw (27)

where E is the standard Ekman number defined by

E ~ u° fH 2 (28)

for constant eddy viscosity v0. The surface stress condition is

dw L dz E' a t z = 0 (29)

while at the bottom a linear slip condition is specified as

o d w = dz w, a tz = - 1 (30)

where

0 1 U0 rH

and r is the dimensional linear friction coefficient (e.g. PINGREE and GRIFFITHS, 1987). The application of a slip condition is an acknowledgement, in a constant eddy viscosity model, that the model cannot hope to simulate the bottom boundary layer in which the velocity is reduced to zero. The condition is often said to apply at the top of the seabed log-layer. However, because of the approximation involved in a constant eddy viscosity model, and because the log-layer depth is usually small compared with the total water depth, it is valid to apply (30) right at the seabed.

The aim of the present exercise is to determine the values of E and 0 that provide the best approximation to the bilinear solution. For a given set of parameters L, zl and z2 (e.g. Fig. 1), the "best" constant eddy viscosity solution is defined as the one that minimizes the error 6. Table 3 lists values of E and 0, determined using a numerical minimization routine, that give the best-fit profiles over a range of values of L, Zl and z2. Figure 4 shows the corresponding solutions for L = 0.1, 1 and 10.

Table 3 suggests that, over a wide range of L, at least from 0.5 to 10, the best-fit constant eddy viscosity is given approximately by


Table 3. Constant eddy viscosity parameters E and 0 which provide the best approximation (as estimated by 6) to profiles calculated with a bilinear eddy viscosity with

the listed values of L, zl and z2

L 21 2 2 E 0 6

0.1 0.001 0.001 0.0035 - - 0.43 0.2 0.001 0.001 0.014 - - 0.44 0.4 0.001 0.001 0.060 - - 0.43 0.6 0.001 0.001 0.11 - - 0.37 0.8 0.001 0.001 0.16 - - 0.33 1.0 0.001 0.001 0.20 0.8 0.27 2.0 0.001 0.001 0.41 0.8 0.17 4.0 0.001 0.001 0.87 0.9 0.12 6.0 0.001 0.001 1.3 0.9 0.10 8.0 0.001 0.001 1.8 0.9 0.10

10.0 0.001 0.001 2.2 0.9 0.09 10.0 0.001 10 -6 2.1 2.3 0.07

E = 0.2L (31)

or, in dimensional form,

v0 = 0.2Ku,sH.

One widely used approach in numerical modelling is to define the (depth-constant) eddy viscosity as

Vo = CU,sH (32)

where c is a constant. This is supposed to represent some depth-averaged form of the eddy viscosity. Thus, if v0 is the strict depth-average of the bilinear form (9), c would take the value 0.1 (ignoring zl and z2). A depth-average of the parabolic eddy viscosity (26) gives c = 0.067. CSANADY (1976) proposed a c value of 0.05. The relationship (31) is equivalent to a c in (32) of 0.08.

According to Fig. 4, for values of L of order 1 and larger, the constant eddy viscosity solution is a good approximation to the bilinear solution over most of the water column. It fails most dramatically in the strongly logarithmic layer near the bottom and, for the downwind component , near the surface. As L decreases (Fig. 4), velocities in the surface layer become more significant, relative to those deeper in the water column. For small values of L (L < 0.6, Fig. 3a), the constant eddy viscosity solution appears to begin to fit to the log- layer, rather than the mid-water, shape of the velocity profile. This is, apparently, the reason why the relationship (31) becomes less valid for small L.

As we have already noted, the velocity profile is insensitive to the bottom boundary condition for L < 1. In Table 3, constant eddy viscosity solutions were calculated with 0 = ~ when L < 1, so that only E was varied to give the best-fit solution. For L > 1, and for z2 = 0.001, the best-fit 0 takes a value very consistently around 0.9. Not surprisingly, however, the value of 0 is sensitive to z2. For L = 10, and z 2 = 10 -6, the best 0 is equal to 2.3.

20 P . D . CgaIG et al.

(a) 13. V

- 1 0 13 - 7 [ I~ I i ~ I I [ I I p I P I J i I I I I I ~ i

constant f - - " "

~ bilinear bilinear

0 1 I

c o n s t a n t

e~ - - 0 . 5

l:r

- 1 . 0

(b)

Fig. 4.

u v

- i 0 13 -7 E p r l ~ ~ T

constant

I ' constant

hillnear J

Im - 0 . 5 ~

- 1 . 0

C o m p a r i s o n of prof i les for b i l inea r and best-fi t cons tan t e d d y viscosi t ies . (a) L = 0 . 1 E = 0.0035, 0 = 0% (b) L = 1, E = 0.20, 0 = 0.8; (c) L = 10, E = 2.2, 0 = 0.9.

(c) U -1 o . 13 -7

constant

Fig. 4(c).

V

t I

bilinear


-o.5.~

-1.0

Equation (25) suggests that 0 may be estimated as

0 = zl ln(Zl/Z2) (33)

in which zl is the bottom log-layer depth, and z2 has been neglected by comparison with zt. Strictly, (33) should be applied at the top of the log-layer, z = - 1 + zt, rather than at the very bottom. For zl = 0.18, however, (33) gives 0 = 0.9 for z2 = 10 -3, and 0 = 2.2 for z 2 = 10 -6. This suggests that (33) may provide a reasonable estimation of 0, with zt independent of L and taking the (relatively large) value of 0.18.

It would be relatively simple to similarly exclude the surface log-layer from the fitting procedure by using (23) as the surface boundary condition. The procedure could then incorporate determination of the surface log-layer thickness, and the surface and bottom log-layers could be fitted to the profile analytically, as was done in Section 4.

7. CONCLUSION

For an ocean modelling formulation in which the eddy viscosity varies linearly with distance from the surface and seabed, the two parameters defined as L1 and L2 in (10) carry much of the information about the current profiles. Each parameter is the ratio of a depth of influence (or a decay depth) to the depth of the water. The parameters have real meaning in the sense that, for an L1 of 1 or less, a strictly wind-forced current profile, with no pressure gradients, will be virtually unaffected by the bottom boundary conditions. L~ and Le also represent the ratio of friction to rotation. As L1 increases, the profile becomes more unidirectional.

For L1 greater than 1, the current profile is sensitive to the value of the bottom

22 P.D. C~16 et al.

roughness, with current magnitudes increasing throughout the water column as the bottom roughness decreases. By contrast (for all values of LI), reducing the surface roughness increases the surface currents in a skin layer of depth equal to the roughness length, but does not affect the profile below this layer.

The linear eddy viscosity gives logarithmic layers at both the surface and bottom. These log-layers are clear in current hodographs as regions with no curvature, that is, they are completely dominated by frictional rather than rotational effects. At the surface, the logarithmic structure occurs only in the downwind velocity component and the depth of the log-layer, defined visually from the hodographs, is of the order of 0.01 L1. At this depth, rotation has caused a cross-wind velocity component of around 1% of the downwind component.

It is possible to calculate the currents inside the log-layers analytically, to within an additive constant at the surface and a multiplicative constant at the bottom. These analytic representations can be used as boundary conditions in a numerical model of the vertical structure, to overcome difficulties of resolution in the log-layers. The numerical solution provides the unknown constants to the boundary layer representations, and describes the mid-water current structure. For higher numerical accuracy at the same resolution, the vertical coordinate can be transformed functionally according to (19), to increase relative resolution in regions of small eddy viscosity.

The linear eddy viscosity should probably apply only close to the top and bottom surfaces and we have looked briefly at the consequencs of "cutting off" the eddy viscosity at a specified fraction (0.1 or 0.15) of L away from both the top and bottom boundaries. The current profiles are sensitive to the mid-water values of the eddy viscosity. At L = 1, for example, the range of variability introduced by varying the cutoff depths is such that it may be preferable, for convenience, to specify a constant eddy viscosity between the boundary log-layers.

In experiments with a constant eddy viscosity, we looked for the eddy viscosity and linear bottom friction parameter that gave the best-fit to the "bilinear" solution. For L of order 1 and greater, the best-fit eddy viscosity takes a dimensional value of about 0.08 u . s H [see (31)], which is consistent with values used in the literature. The linear bottom friction parameter is dependent on z2, but appears to have little dependence on L. It can be calculated using (33), based on the analytic structure of the bottom log-layer.

Small values of L (<0.5) create difficulties for numerical techniques. For small L (e.g. Fig. la), the current profile is restricted to an upper fraction of the water depth. For accurate numerical solutions in this case, it is the (non-dimensional) depth-scale L, rather than the total water depth, that must be appropriately discretized. In a realistic coastal domain, the water depth may easily vary over an order of magnitude, implying similar variation for L. If each water depth is discretized into the same number of points, as happens with the so-called sigma-transformation (e.g. DAVIES, 1985), then the numerical resolution in deeper water may not be satisfactory.

The discussion in this study has been centred mostly on L1 rather than L2, principally because the wind-stress is applied at the surface. Pressure gradients set up by the wind in a coastal region are felt right to the seabed, and establish bottom boundary layers, as in tidal dynamics (e.g. SOULSBY, 1983). For the bottom boundary, L 2 is obviously the relevant parameter. If L 1 and L 2 are both less than 1, the surface and bottom layers will be effectively independent of one another. For larger values of L1 and L2, the relative importance of the directly stress-induced currents (the "drift currents") and the pressure-


g r a d i e n t - i n d u c e d c u r r e n t s ( t h e " g r a d i e n t c u r r e n t s " ) will d e p e n d o n t h e spec i f i c g e o m e t r y

o f t h e r e g i o n b e i n g c o n s i d e r e d .

Acknowledgements We wish to thank Trevor McDougall and Chris Fandry for their comments on this manuscript, and Cathy Flanagan and Nikki Pullen for their typing. The project was partially funded by a grant from the Australian Research Council.

R E F E R E N C E S

ABRAMOWITZ M. and I. A. STE~UN (1965) Handbook of mathematical functions, Dover Publications, New York, 1046 pp.

BOWDEN K. F. and P. HAMILTON (1975) Some experiments with a numerical model of circulation and mixing in a tidal estuary. Estuarine and Coastal Marine Science, 3,281-301.

BOWDEN K. F., L. A. FAIRBAIRN and P. HUGHES (1959) The distribution of shearing stresses in a tidal current. Geophysical Journal of the Royal Astronomical Society, 2,288-305.

CHEUNG T. K. and R. L. STREET (1988) The turbulent layer in water at an air-water interface. Journal of Fluid Mechanics, 194, 133-151.

CSANADY G. T. (1976) Mean circulation in shallow seas. Journal of Geophysical Research, 71, 5389-5399. DAVIES A. M. (1985) Application of a sigma coordinate sea model to the calculation of wind-induced currents.

Continental Shelf Research, 4,389-423. DAVIES A. M. and J. E. JONES (1990) Application of a three-dimensional turbulence energy model to the

determination of tidal currents on the northwest European shelf. Journal of Geophysical Research, 95, 18143-18162.

DAVIES A. M. and J. E, JONES (1991) On the numerical solution of the turbulence energy equations for wave and tidal flows. International Journal for Numerical Methods in Fluids, 12, 17-41.

EKMAN V. W. (1905) On the influence of the earth's rotation on ocean currents. Arkivfoer Matematik, Astronomi och Fysik, 2, No. 11.

ELLISON T. H. (1956) Atmospheric turbulence. In: Surveys in mechanics, G. K. BATCHELOR and R. M. DAVIES, editors, Cambridge University Press, Cambridge, pp. 400-430.

GRIFFITHS G. (1989) Estimates of eddy diffusion in the upper ocean based on the backscatter strength of measurements of an acoustic doppler profiler. In: Advances in water modelling and measurement, M. H. PALMER, editor, BHRA (Information Services) Bedford, U.K., pp. 259-270.

HUNTER J. R. (1975) A note on quadratic friction in the presence of tides. Estuarine and Coastal Marine Science, 3,473-475.

JENKINS A. D. (1987) Wind and wave induced currents in a rotating sea with depth-varying eddy viscosity. Journal of Physical Oceanography, 7, 938-951.

JENTER H. L. and O. S. MADSEN (1988) Bottom stress in wind-driven depth-averaged coastal flows. Journal of Physical Oceanography, 19, 962-974.

JORDON T. F. and J. R. BAKER (1980) Vertical structure of time-dependent flow dominated by friction in a well- mixed fluid. Journal of Physical Oceanography, 10, 1091-1103.

K1TA1GORODSK1 S. A., M. A. DONELAN, J. L. LUMLEY and E. A. TERRAY (1983) Wave turbulence interactions in the upper ocean. Part II: Statistical characteristics of wave and turbulent components of the random velocity field in the marine surface layer. Journal of Physical Oceanography, 13, 1988-1999.

MAAS L. R. M. and J. J. M. VAN HAREN (1987) Observations on the vertical structure of tidal and inertial currents in the central North Sea. Journal of Marine Research, 45, 293-318.

MADSEN O. S. (1977) A realistic model of the wind-induced Ekman boundary layer. Journal of Physical Oceanography, 7,248-255.

MELLOR G. L. and T. YAMADA (1974) A hierarchy of turbulence closure models for planetary boundary layers. Journal of the Atmospheric Sciences, 31, 1791-1806.

MELLOR G. L. and T. YAMADA (1982) Development of a turbulence closure model for geophysical fluid problems. Reviews of Geophysics and Space Physics, 20,851-875.

PINGREE R. D. and D. K. GRIFFITHS (1987) Tidal friction for semidiurnal tides. Continental Shelf Research, 7, 1181-1209.

PRANDTL L. (1952) Essentials offluid dynamics, Blackie and Son, London, 452 pp.

24 P .D. CRAIG et al.

RICHMAN J. G., R. A. DE SZOEKE and R. E. DAVIS (1987) Measurements of near-surface shear in the ocean. Journal of Geophysical Research, 92,2851-2858.

SOULSBY R. L. (1983) The bottom boundary layer of shelf seas. In: Physical oceanography of coastal and shelf seas, B. JoHns, editor, Elsevier, Amsterdam, pp. 189-266.

THOMAS J. H. (1976) A theory of steady wind-driven currents in shallow water with variable eddy viscosity. Journal of Physical Oceanography, 5, 136-142.

THORPE S. A. (1984) On the determination of Kv in the near-surface ocean from acoustic measurements of bubbles. Journal of Physical Oceanography, 14, 855-863.

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