AN EMPIRICAL EXPRESSION FOR AERODYNAMIC

RESISTANCE IN THE UNSTABLE BOUNDARY LAYER

NEIL R. VINEY

Mathematics Department, University College, University of New South Wales, Canberra, A.C.T. 2600 Australia

(Received in final form 7 March, 1991)

Abstract. In unstable conditions, the set of equations defining the aerodynamic resistance to sensible heat transfer, rar cannot be solved analytically. An iterative technique must be used to obtain r, exactly, but this is cumbersome and time consuming. In this paper, a new, empirical equation is presented relating the ratio, Q, of the aerodynamic resistances in neutral and unstable conditions, to the bulk Richardson number, Ris. The equation takes the form Q = a + b(-RiB)C, where a, b and c are empirical functions of (z - d)/z,,. This model is shown to predict r, with a mean absolute error of 0.60 s m-r over the ranges -15 < Rin < 0 and 10 < (z - d)lz,, < 2300. Statistical comparison with other equations that have been proposed for Y, in unstable conditions indicates the superior precision of the model presented here.

a b C

CP d

g h H k L

Q r Ris T Ll

M* u V W x Z

zo

; Y

i %

G

Symbols

Regression coefficient (Equation 19), Regression coefficient (Equation 19), Regression coefficient (Equation 19)) Specific heat of air at constant pressure, J kg-’ K-i Zero-plane displacement, m Gravitational acceleration, m SC* Height of surface element, m Sensible heat flux, W mm2 Von Karman constant, Obukhov stability length, m Ratio of neutral to unstable ro, Resistance, s m-r Bulk Richardson number, Temperature, K Wind speed, m s-i Friction velocity, m s-i Stability function (Equation 18), Stability function (Equation 18), Stability function (Equation 18), Stability function (Equations 7 and 8), Reference height for air measurements, m Roughness length, m Regression coefficient (Equation 20), Regression coefficient (Equation 20), Regression coefficient (Equation 20), Regression coefficient (Equation 20), Monin-Obukhov stability parameter, Stability function (Equation 18), Air density, kg me3 Logarithmic profile function,

Boundary-Layer Meteorology 56: 381-393, 1991. 0 1991 Kluwer Academic Publishers. Printed in the Netherlands.

382 NEIL R. VlNEY

q Integral stability function,

Subscripts

f Air or aerodynamic Sensible heat

M Momentum s Surface

1. Introduction

The flux of sensible heat, H, between the surface and the air is

where p is the air density, cP is the specific heat of air at constant pressure, T, and T, are the temperatures of the surface and at some reference height in the air, respectively, and r, is the aerodynamic resistance to sensible heat. An analogous expression may be written for latent heat transfer, in which it is generally assumed that the aerodynamic resistance to latent heat is equal to that for sensible heat (Garratt, 1978).

In neutral conditions, this aerodynamic resistance is given by

r:, = Y,Y,l(k2u) ) (2)

where

k is the von Karman constant, z is the reference height for wind speed and air temperature measurement, d is the zero-plane displacement, and zom and z& are the roughness lengths for momentum and sensible heat, respectively.

In non-neutral conditions, the effects of buoyancy on the transfer of momentum and heat must be considered. Accordingly, the stability-corrected aerodynamic resistance, r,, is given by

?-a = (yh - *h)(Ym - ‘RJ(k*4 , (3)

where q,&) and Th(g are integral stability functions of J = (z - d)lL for momen- tum and sensible heat, respectively, and L is the Obukhov stability length defined by

AERODYNAMIC RESISTANCE IN THE UNSTABLE BOUNDARY LAYER 383

L = -u’*WY/, - q/z) k’g(Ts - To) ’

where g is the gravitational acceleration and

U * = kul(Y, - qm) ) (5)

is the friction velocity. In Equation (4), we have ignored the slight dependence of L on the difference between T, and the virtual temperature.

In stable conditions (L 3 0), the stability functions are (Dyer, 1974)

*&J) = *&) = maxi-5, -551, (6)

while in unstable conditions (L < 0), we have (Dyer and Hicks, 1970; Paulson, 1970)

Th(a = 21n((l +x*)/2), (7)

and

q&J) = ln((1 + x)*(1 + x*)/S) - 2 arctan + 7~/2 ,

with x = (1 - 16<)“4. For stable conditions, Equations (4-6) may be solved analytically (Choudhury

et al., 1986) to yield

*\Ir, = qh = (Y, - 10RiBY, -

- (Yz - 20RiBY,(Y, - Y,))“‘)l(2 - lORiB) ,

where RiB, the bulk Richardson number, is defined by

(9)

RiB = (z - d)g(T, - T,)I(u*T,) . (10)

Following from Equation (6), we must add the condition that T\Ir, and Th have minimum values of -5 in Equation (9). The stability-corrected aerodynamic resis- tance for stable conditions is then given directly by Equation (3).

Unfortunately, the non-linear forms of Equations (7) and (8) do not permit an analytical solution to Equations (4) and (5) in unstable conditions. Instead one must solve this set of equations by an iterative technique (Busch et al., 1976; Itier, 1980) to obtain an exact solution. This technique involves making an initial assumption of neutral stability (q,,, = *h = 0) in calculating a first estimate of r, (Equation 2) and L (Equation 4), whereupon the values of Vm and qh can be updated via Equations (7) and (8). New estimates of ra (Equation 3) and L can now be established. The iteration procedure can be repeated until the difference between successive estimates of r, falls below some arbitrary, pre-determined threshold value.

In addition to this exact solution, several approximation formulae, involving various simplifying assumptions and based on varying degrees of empiricism, have

384 NEIL R. VINEY

been proposed. Many of these formulae have been reviewed by Kalma (1989). In most of the following models, r, may be expressed as a stability-dependent pro- portion of r:. This stability dependence may be expressed in terms of the bulk Richardson number (Equation (lo)), although not all of the models were originally formulated explicitly in terms of Rt ‘B. Some of the models below have also been algebraically modified from the forms in which they were originally proposed.

Monteith (1973) presented the expression

r, = rL(l - 5RiB)-2. (11)

The derivation of Equation (11) involves the assumption that Equation (6) is valid for unstable conditions, and also invokes the Reynolds analogy, which assumes that the eddy diffusivities for sensible heat and momentum (and hence the respective resistances and roughness lengths) are equal. Neither assumption can be supported for l< -0.03 (Webb, 1970). Thus Equation (11) is strictly appropriate only in stable and near-neutral conditions. A similar expression was derived by Itier (1980), but explicitly for a stable atmosphere.

Hatfield (1983) adapted an expression from Monteith (1973) to give the non- neutral aerodynamic resistance to heat transfer as

r, = rL(l + 5Ris). (12)

This expression was first proposed to describe the aerodynamic resistance to momentum transfer (Monteith, 1973). As is the case with Equation (ll), its derivation relies on the Reynolds analogy and assumes that Equation (6) is valid for unstable conditions. As noted by Kalma (1989), Equation (12) also relies on the assumption that the friction velocity in non-neutral conditions (Equation (5)) may be approximated by its neutral value. This assumption severely compromises the validity of Equation (12), because r, is directly dependent on the stability- corrected friction velocity.

It can be shown that for -0.2 < Ri, < 0.2, Equation (12) represents the first two terms of the Maclaurin expansion of Equation (11). One would therefore expect Equations (11) and (12) to behave similarly in that stability range. However, like Equation (ll), Hatfield’s expression is best suited to stable and near-neutral unstable conditions. Finally, it will be seen that when RiB < -0.2, Equation (12) predicts a negative r,, which is physically meaningless.

Choudhury et al. (1986) found “with trials and errors” that in the unstable case, r, may be calculated by

r, = rL(l - 5RiB)-3’4. (13)

Choudhury et al. used k = 0.4 in obtaining Equation (13) and assumed that zOh = z,,l7.

Louis (1979) used data obtained from the iterative procedure to derive an empirical expression relating the term k2/((Y, - ~,)(YJ, - qh)) to RiB, from which one can obtain

AERODYNAMIC RESISTANCE 1N THE UNSTABLE BOUNDARY LAYER 385

9.4RiB -1

1 + 49.8k2(-Ri,(z - d)lz,,)“*/Y~ ’ (14)

Louis’ derivation uses the assumption that zom = zoh (and hence Y, = Yh). It also uses the flux-profile relationships proposed by Businger et al. (1971) for ‘Pm and qh, which are slightly different to those of Dyer and Hicks (1970), upon which Equations (7) and (8) are based. The value of k adopted by Louis (1979) was not stated, but given that the Businger et al. formulation was used, it is reasonable to assume k = 0.35 in Equation (14). The same value of k should be used in calculating r-L by Equation (2) for this model.

A slightly modified version of Equation (14) (using k = 0.4) was given by Mahrt and Ek (1984) in which

1.5RiB >

-1

1 + 75k*(--&(z/z,, + l))“*/(h(z/zo~ + 1))’ . (15)

This expression also assumes similarity of the roughness lengths for momentum and sensible heat.

Itier (1980) also used data from the iterative solution to derive an empirical equation. He related what was essentially the ratio between r, and rL to a modified parameterization of the Obukhov length. From this relationship, one can obtain the followlng expression for r,,

r, = rL(l + 6.5(-(zz,,,Jn2Y,Ri,/(z - d))3’4)-1 . (16)

Like Equation (14), this model was derived under the assumption that zom = z&, but no value of k is given.

Riou (1982) used theoretical arguments to derive a similar parameterization, from which one may obtain

ra = rL(l - 16(zz,,)“*Y,Ri,/(z - d))-3’4. (17)

Equation (17) is derived from the Dyer and Hicks (1970) flux-profile relationships. It is expected to give good estimates of r, for small values of z/z,, and small values of IRiB (Riou, 1982).

Byun (1990) solved Equations (4), (5), (7) and (8) analytically for 6 in terms of the gradient Richardson number. An approximation that was found to apply in the stable case was then used for unstable conditions to obtain IJ’ in terms of the bulk Richardson number. Byun assumed similarity of z& and zom, and used the Businger et al. (1971) flux-profile relationships with k = 0.35. However the form of the equations proposed by Byun is also amenable to expression in terms of the Dyer and Hicks (1970) flux-profile relationships with k = 0.4. Then 5 is given by

c= (zl(z - z,,)> ln(z/zom>(& - W - VW, if U2 > V3 (z/(z - z,,)) ln(z/z&(& - 2V”* cos(O/3)), ifU2SV3’ (18)

386 NEIL R. VINEY

TABLE I

Ranges of input data used in developing the diagnostic and comparative data sets

Input variable

Z

zlh T, Ts - T, IA

Range

2-10 m 2-3000

278-310 K 0.1-15.0 K 0.5-4.0 m s-l

where

U = (RiB - &)/48,

V = (Ris + &)/3 ,

w = (U’ - vy + ( uy3

and

8 = arccos(WP3”) .

In Equation (18), the inequality lJ2 > V3 corresponds approximately to the range in bulk Richardson number of -0.2097 < RiB < -0.0250. Values of RiB outside this range satisfy U2 c V3. However, to ensure computational stability, one should use the inequalities given in Equation (18) when evaluating 4’ by the Byun (1990) model. Substitution of l from Equation (18) into Equations (7) and (8) yields ‘P\lr, and qh, whereupon the aerodynamic resistance may be evaluated by Equation (3). The approximation analogy used by Byun (1990) cannot be applied to the case where zoh f z,,.

2. Model Development

Neutral values of aerodynamic resistance, r& were obtained from Equation (2), while the reference values of stability-corrected aerodynamic resistance, r,, were obtained by the iterative technique, over a range of unstable weather and surface conditions (Table I). The values for k = 0.4 and d = (2/3)h are taken from Brut- saert (1982), while the values of zom = 0.13h and z& = z&7 are taken from Monteith (1973) and Garratt (1978) respectively, and are considered appropriate for a variety of land surfaces. Over water, these parameterizations of zom and zoh

are unlikely to hold (Hicks et al., 1977), but that case will not be examined here.

AERODYNAMIC RESISTANCE IN THE UNSTABLE BOUNDARY LAYER 387

. co- . .

z/h=20 . . . .

. .

. . .

CD- . . . .

a z/h=50' . .

. . . . . . .

. . . . . -. . .

z/h=206 . . . .*. . “..

. . .

b- . . .

. . . . .

. . . z/h=2000 . . .

.. . . .

-15 -10 Ri, -5

0

Fig. 1. The relationship between the ratio Q = (Jr. and the bulk Richardson number, RiBI for selected values of z/h.

The quotient, Q = rL/r,, was then compared to the value of RiB, computed from Equation (10). Figure 1 shows that the general form of the relationship between Q and RiB is invariant for variations in T,, T, and U. That is, the data points lie on the same curve in the RiB - Q plane. The displacement of the curve, however, shows a slight dependence on z and h, but is conservative with constant z/h.

For each z/h class, regression equations of the form

Q = a + b(-Rig)C, (19)

were developed, where a, b and c are dimensionless regression coefficients. All the resulting equations had coefficients of determination, r2, of better than 0.999 and standard errors of the residuals, a,, of less than 0.04. Regression summaries are given in Table II.

The regression coefficients, a, b and c, in Table II were all found to be related to Y,, via equations of the form

y=a+Pln(y+(6-Y,)2), (20)

where CY, /3, y and S dimensionless empirical constants, and y represents a, b and c. These regressions are summarized in Table III.

With the full diagnostic data set (n = 2464) and with a, b and c given by Equation (20) and the data in Table III, Q can be predicted by Equation (19) (Figure 2)

388 NEIL R. VINEY

TABLE II

Statistical summary of regression analysis used in developing Equation (19). The number of data point in each class is n, the coefficient of determination is r’, and the standard error of the residuals is oc

(dimensionless)

zlh n a b c r2 u‘=

2 176 0.96783 1.76633 0.6834 0.99997 0.0104 3 176 1.00213 1.71517 0.6848 0.99985 0.0247 5 176 1.02502 1.61287 0.6697 0.99969 0.0345

10 176 1.02404 1.48931 0.6300 0.99960 0.0347 20 176 1.00359 1.39150 0.5822 0.99963 0.0282 30 176 0.98817 1.34274 0.5543 0.99968 0.0239 50 176 0.96812 1.28735 0.5210 0.99975 0.0188

100 176 0.94268 1.21975 0.4804 0.99983 0.0132 200 176 0.91994 1.15980 0.4450 0.99989 0.0092 300 176 0.90821 1.12755 0.4266 0.99992 0.0075 500 176 0.89478 1.08984 0.4055 0.99994 0.0058

1000 176 0.87919 1.04283 0.3803 0.99996 0.0044 2000 176 0.86585 1.00058 0.3583 0.99996 0.0037 3000 176 0.85895 0.97782 0.3467 0.99996 0.0035

TABLE III

Statistical summary of regression analysis used in evaluating the coefficients a, b and c (Equation (20))

Coefficient n a P Y s r* 0,

5 14 14 1.0591 1.9117 -0.05515 -0.22366 1.72 1.86 4.02 2.12 0.9978 0.9999 0.0029 0.0021 C 14 0.8437 -0.12431 3.49 2.79 0.9996 0.0026

with a mean absolute error of 0.014. The maximum absolute error is 0.197 (an underprediction), and occurs for large /RiBI.

3. Comparison with Other Expressions

The prediction precision of Equation (19) (with respect to the solution of the iterative model) was compared to those of Equations (11)-(M), by creating a new, completely randomized dataset of the variables in Table I. The random data were rejected for cases where RiB < -15. To enable a more detailed comparison at near-neutral unstable conditions, a second randomized data set was created with RiB limited to the range -0.2 < Ris < 0. Both these comparisons were per- formed with z& = z,,/7. To enable fairer comparisons for Equations (ll), (12) and (14)-(N), the predictions for both data sets were repeated using zOh = zom. The value of the von I&man constant was assumed to be 0.4 for all models, except Equation (14), for which k = 0.35 was adopted.

The results of all four comparison tests are given in Table IV. The negative values of the mean bias error indicate a tendency towards underprediction of r,.

AERODYNAMIC RESISTANCE IN THE UNSTABLE BOUNDARY LAYER 389

0 T-

2 4 6 IO Cl (predicted by Equation81 9)

Fig. 2. Predicted Q from the regression model (Equations (19) and (20)) versus the reference values of Q derived by the iteration technique for the diagnostic data set (not all points are shown). The

mean absolute error in the predictions is 0.014 (n = 2464).

In particular, the cases where MBE = - MAE are indicative of universal underpre- diction.

In Table 1V.a (zoh = 2,,/7), Equation (19) provides estimates of r, that are extremely close to the iterated solution for both stability ranges. The errors are an order of magnitude less than those of the other models. The predictions also show relatively little bias. When one uses zoh = zom (Table IV.b), the prediction precision of Equation (19) deteriorates markedly and it overpredicts ra in all cases and for both stability ranges.

None of Equations (11)-(H) provides good estimates over the wider stability range (Ri, > - 15) in Table IV.a(i), but Equations (14) and (18) give good results in Table IV.b(i). Most models (including Equation (13) for the wider stability range) are improved by assuming equality of the roughness lengths. All models predict better over the narrower stability range (Ri, > -0.2), but none approaches the precision achieved by Equation (19) in Table 1V.a. All of Equations (ll)- (18) underpredict in Table IV.a(i), and only Equation (17) overpredicts in Table IV.a(ii). The iterated values of r, are lower in Table 1V.b and consequently, only Equations (11) and (12), the two poorest models, and Equation (18), the best, underpredict. For the narrower stability range, the mean absolute errors of Equa- tion (18) in Table 1V.b are identical to those of Equation (19) in Table IV.a, but

390 NEIL R. WNEY

TABLE IV

The prediction accuracy of Equation (19) in comparison with Equations (11)-(18). MAE is the mean absolute error and MBE is the mean bias error (sm-‘). The means and ranges of the iterative solutions of r, (s m-i) for each data set are: Table IV.a(i), mean: 154.9, range: 15-598; Table IV.a(ii), mean: 118.1, range: 15-763; Table IV.b(i), mean: 100.7, range: 4-433; and Table IV.b(ii), mean: 83.2, range: 5-566. In each case the total

number of predictions is 100

Equation (i) RiB > -15.0 (ii) Ris > -0.2 MAE MBE MAE MBE

(a) zoh = zod

11 12 13 14 15 16 17 18 19

0’) Zoh = tom

11 12 13 14 15 16 17 18 19

130.61 -130.61

69.03 -68.42 32.61 -32.59 20.44 -19.76 39.59 -37.25 42.99 -34.23 21.32 -21.32 0.60 -0.05

80.35 -80.35

35.42 -28.69 6.34 3.57

14.71 14.71 11.76 0.82 17.09 3.68 3.48 -3.27

30.84 30.84

44.11 -44.11 44.66 -44.48 5.26 -2.84 7.43 -7.42 5.18 -4.02 8.83 -1.02

13.51 7.97 6.10 -6.10 0.55 -0.16

25.52 -25.51 27.46 -25.76 7.03 6.74 3.50 3.40 6.12 6.12 8.96 8.82

15.89 15.89 0.55 -0.55 8.91 8.91

are larger as a fraction of the mean aerodynamic resistance. However, Equation (19) predicts better over the wider stability range.

Some other general observations concerning Equations (11)-(B) emerge from Table IV. Equation (15) is a modification of Equation (14). Equation (15) predicts a higher value of r, than Equation (14) for all data points. Equation (15) predicts with better precision when .zoh = z&7; Equation 14 is better for &h = zom.

Equations (11) and (13) differ only in exponent. Equation (13) appears to be the better predictor of r,. Finally, the anticipated similarities between the predictions of Equations (11) and (12) are confirmed in Table IV.

4. Discussion

A wholly empirical expression like the one presented in this paper can only be as accurate as the data and information upon which it is based. In this case, much of that foundation is, itself, empirical. For instance, there is some conflict in the meteorological literature over the presently acceptable values of k, d/h, z,,lh and

AERODYNAMlC RESlSTANCE IN THE UNSTABLE BOUNDARY LAYER 391

zohlh. However, the small uncertainties in these quantities are insignificant when compared to the uncertainties associated with the flux-profile relationships them- selves. Indeed, the graphs presented by Dyer (1967) and Dyer and Hicks (1970) from which Equations (7) and (8) were deduced, exhibit considerable scatter, and have been subject to quite divergent independent analysis (e.g., Hicks et al., 1977). Furthermore, although the expressions for r, were tested for conditions ranging up to extreme instability (representing values of f as low as -3O), Equation (7) was derived from field data limited to the range, -2 < {< 0 and Equation (8) from the range -9 < f < 0.

Thus we can only test the precision of the model, subject to the assumptions that the underpinning information is valid and that it can be extrapolated beyond the stability range in which it was originally derived. The results presented in Table IV indicate that the model for r, described by Equations (19) and (20) has exceptional precision over a wide range of stability conditions. In particular, it is extremely precise for conditions ranging from neutral to moderately unstable: the conditions for which Equations (7) and (8) are considered valid. Furthermore, it compares extremely favourably with other expressions that have appeared in the literature. Its accuracy, however, has not been conclusively established, because we do not know the accuracy of the aerodynamic resistances calculated by the iteration procedure.

Figure 1 clearly indicates that, in addition to the stability parameter, RiB, the aerodynamic resistance also depends on the quotient, z/h, which represents the surface and measurement configuration. This z/h dependence can be deduced theoretically if one considers that from Equations (2) and (3), one can express r,lrL as (Y, - q,J(Yh - qVh)I(Y,Yh). Clearly this expression depends on z/h (a common factor in Y, and Y,), as well as atmospheric stability. Apart from the model proposed here, five of the other eight equations tested also include a factor that depends on z/h. These are Equations (14)-(18). In Tables IV.a(i) and IV.b(i) it appears that these equations perform slightly better than those depending only on stability. However over the narrower stability range (Tables IV.a(ii) and IV.b(ii)), Equation (13) is at least as good as Equations (14)-(17). The predictions of the other two equations (Equations (11) and (12)) remain poor, but these models are strictly more appropriate for stable rather than unstable conditions.

The only model that compares well with the Table 1V.a predictions of Equation (19) over both stability ranges is Equation (18) in Table IV.b(ii). However, as can be seen in Table IV.b(i), this model does not extend so well to more extreme instabilities. Furthermore, the analytical techniques used in deriving Equation (18) rely on equality of zom and zoh, and are not applicable t0 the case Of i&h = z,,/7. The model presented here has the added advantages over that of Byun (1990) in that it can be expressed as a single equation and does not require the evaluation of preliminary variables and inequalities.

The relatively poorer performance of Equation (19) with z& = zom (Table 1V.b)

392 NEIL R. VINEY

is unimportant. It is reasonable to assume that had Equation (19) been developed with the two roughness lengths equal, its resulting predictions would have a precision approaching those in Table 1V.a.

The tendency of all models to under-predict aerodynamic resistance in Table 1V.a has important implications for the consequent predictions of the energy fluxes. Clearly, by Equation (l), such an underprediction will lead to an overpre- diction of the sensible heat flux by the same proportion. If the latent heat flux is then derived as the residual in an energy balance approach, it will, in turn, be underpredicted. The relative error of this latent heat flux prediction could be very significant over a dry surface where the Bowen ratio is large. On the other hand, if the sensible and latent heat fluxes are calculated independently, by assuming that r, applies to both, both will be over-predicted. Independent measurements of the fluxes of net radiation and soil heat flux will then lead to non-closure of the surface energy balance equation.

By way of contrast, when similarity of z& and zom is assumed (Table IV.b), the consistent over-prediction of r, by most models at moderate to near neutral stabilities will result in slight under-predictions of H by Equation (1). However, it must be emphasised that use of the Reynolds analogy raises some doubts concem- ing the accuracy of the equations used in the iteration technique (Garratt, 1978; Brutsaert, 1982). In particular, the heights of the apparent sinks of momentum and heat near the surface are different. While the transport of both quantities is governed by molecular diffusion processes, momentum transfer is additionally affected by drag-induced local pressure gradients. Of the models tested, only Equations (13) and (19) explicitly account for this difference.

5. Conclusions

It has been shown that the relationship between the quotient t-L/r, and the bulk Richardson number, RiB, is sensitive to variations in z/h. Accordingly, the follow- ing expression for the aerodynamic resistance to sensible heat in unstable con- ditions has been developed:

where

r, = rL(a + b( -Rig)‘)-i

and

u = 1.0591 - 0.05521n(1.72 -t (4.03 - Y,)2) ,

b = 1.9117 - 0.2237 ln(1.86 + (2.12 - Y,)2) ,

c = 0.8437 - 0.1243 ln(3.49 + (2.79 - Y,)*) ,

Y, = ln((z - d)lz,,) .

This equation has been tested over a large range of RiB values and shown to

predict r,, with a mean absolute error of 0.60 s m-‘. Its predictions over this large range, as well as over a narrower range of moderately unstable Rill values, are more precise than those of the other models tested. In conjunction with the above expression, the use of the relationship, z,,/, = z ,,,,, 17, and a von K&man constant of 0.4, is recommended.

Acknowledgement

The author is indebted to Jetse Kalma for his comments on the manuscript.

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