A NOTE ON FINITE-DIFFERENCE SCHEMES FOR

THE SURFACE AND PLANETARY BOUNDARY LAYERS

P. A. TAYLOR*

Dept. of Mathematics, University of Toronto, Toronto 181, Ont., Canada

and

YVES DELAGE*

Dept. of Physics, University of Toronto, Toronto 181, Ont., Canada

(Received 7 April, 1971)

Abstract. The solution of the planetary boundary-layer equations by finite-difference methods hasrecently become very popular. Among recent papers using such methods, several use somewhatarbitrary finite-diffeience meshes and some do not make use of a constant flux or wall layer near theground. It is shown that the use of finite differences right down to the ground can be a veryinaccurate procedure when used in conjunction with an eddy viscosity or mixing lengthproportional to (z + zo) or z near the ground. Such an approach can lead to results that arehighly dependent on the finite-difference scheme used and virtually independent of the roughnesslength, zo. A scheme using an expanding grid, based on the form chosen for mixing length or eddyviscosity, is proposed which gives good results with or without a surface layer in the case of a neutrallystratified atmosphere.

1. Introduction

In this note, which is primarily expository in intent, we consider the time independentmean flow of a neutrally stratified, constant density, lower atmosphere above homo-geneous terrain with surface roughness z. The main aim of the paper is to considerdifferent vertical finite-difference schemes for the planetary boundary layer but it willbe convenient to consider finite differences for the surface layer (0 <z < 20 m, say,which can be approximated by a constant flux layer) to illustrate some aspects of theproblem. In either case the simple problems considered here give rise to ordinarydifferential equations in z with U or U=(U, V) as the dependent variables. We seekvelocity profiles U(z) or U (z) satisfying boundary conditions U =0 on z =0 and U = UHon z=zH. The two-point boundary-value problems are to be solved numerically.Several methods are available (see Keller, 1968) for solving problems of this type. Ifwe are solely interested in the steady-state, homogeneous terrain problem, then a'shooting' method is perhaps the most convenient. Blackadar (1962, 1965) and Taylor(1967, 1969b) have used this approach and have obtained accurate solutions to theirequations for velocity profiles in the planetary boundary layer. If, in the long run,we wish to consider problems with spatial and/or temporal variations in surfaceroughness and, in the more general problem allowing for non-neutral stratification,wish to include variations in the surface temperature or heat flux, then we mustdiscretize the z-axis at a relatively small number of fixed gridpoints. In the steady-state,

* Present affiliation: Dept. of Oceanography, University of Southampton, Southampton, England.

Boundary-Layer Meteorology 2 (1971) 108-121. All Rights ReservedCopyright © 1971 by D. Reidel Publishing Company, Dordrecht-Holland

A NOTE ON FINITE-DIFFERENCE SCHEMES

homogeneous terrain problem, this corresponds to using a 'finite-difference' methodto solve the boundary-value problem. We wish to consider two aspects of this approach:(a) what points should we choose as discretization points? and (b) should we treat thelayer close to the ground in a special way by introducing a 'wall layer' within whichwe specify the form of the velocity profile?

We consider only a relatively simple mixing-length () approach to the problem ofrelating the kinematic shear stress, r, to the mean velocity field although similarsituations could arise with methods based on the turbulent energy equation. Modelswhich use a constant eddy viscosity (e.g., all but § 8 of Delage and Taylor, 1970)are not subject to the difficulties discussed here.

2. Finite Differences Within the Surface Layer, no Wall Layer

This case is considered largely to illustrate the type of difficulties that can occur. Sup-pose that we have a constant flux layer (O<z (<ZH) in which

dz-=0. (1)

dz

If we use the standard mixing-length model we will have

dUT1 / 2 = k(z + Zo) d (2)

dz

The boundary-value problem is completed by demanding that

(a) U=0 on z=0

and

(b) U= UH on z = ZH

where, typically, ZH > z o .The analytic solution to this problem can be expressed as:

U* (Z + z~U = In ° (3)

k U= k Z /where

u, =kU/ln ZH + Z (4)

Equations (1) and (2) can be combined to give a second-order ordinary differentialequation d2 U dU

( + Zo) d2 + d = 0. (5)

We will try to solve the boundary-value problem (Equation (5) with U(zH)=UH;U(0) =0) by a finite-difference method.

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P. A. TAYLOR AND YVES DELAGE

Consider first a simple, equally spaced grid with spacing h and N+ 1 points z(I),I=0, 1,..., N, so that z(0) =0 and z(N) =zH. If we use central differences for dU/dzand d2 U/dz2 , we have N- 1 simultaneous equations for U(I); I= 1,..., (N- 1). Theseare

(U(I + 1)- 2U(1)+ U(I -1)) U(I + 1)-U(I-l)(z (I)+ ZO) + = 0h2 2h

(6)where U(0)=0 and U(N)=UH.

In many of the schemes we wish to discuss for the planetary boundary layer, thefinite-difference grid is so coarse that z(1)>z o . To illustrate the difficulty we neglectthe zo term entirely in the finite-difference Equations (6). This is a reasonable approxi-mation to make but the solutions we obtain will then clearly be independent of theroughness length. If we set zo =0 in the differential Equation (5), we introduce asingularity at z =0. In the limit as zO0, the solution of the boundary-value problem(with a discontinuity at z = 0) is:

U= UH for O< z <H

U=0 when z=0.

Figure 1 shows the finite-difference solutions to this problem. The resulting solutionsare clearly highly dependent on the grid spacing and not a very good representationof the exact solution !

Several authors have used a finite-difference grid near the ground that essentiallysets z(I+ 1)=2z(I). The solutions obtained with this grid depend upon the finitedifference representation of dU/dz and d2 U/dz2 . In § 8 of an earlier paper (Delageand Taylor, 1970), using such a finite difference grid close to the ground, the authors

Fig. 1. Finite-difference results for the constant flux layer with zo = 0 and a uniform grid.

110

" cI..*

i i

A NOTE ON FINITE-DIFFERENCE SCHEMES

committed a combination of errors that, in the present context, will give a plausiblelooking but completely erroneous profile. With zo ignored in the equation for mixinglength and a finite-difference scheme near the ground with z(I+ 1) =2z(1), we usedcentral differences based on z(I+ 1), z(l) and the point z(I)-(z(I+ 1)-z(I)). Thislatter point is always z=0. Solving the finite-difference equations yields the simple

result U(I) = U(I + 1)

and, if we work from U(N) down towards the ground, we obtain the profile shown inFigure 2. This is independent of N except for the portion of the profile in 0<z<z(l).Again the finite-difference solution is meaningless.

1.0

zZH

0.7'

0.5(

02

0U/UH

Fig. 2. Finite-difference results for the constant flux layer with zo = and z(I+ 1) = 2z(I).

The two cases used as illustration are perhaps extremes in that they ignore z. Thesecond scheme would always give poor results but the constant spacing scheme withzo included in the equations will give a reasonable approximation to the exact solutionif sufficient grid points are chosen. In order to achieve this, we would have to haveN of order ZH/Z,, which in most cases is impractical. Clearly it is quite possible tochoose other non-uniform grids and appropriate finite-difference representationsand achieve better results.

Perhaps the soundest approach to the problem of non-uniform grids is to applya coordinate transformation

= (Z); = F-(O) (7)and to then use equal spacings and central differences in the new variable . This is thebasic idea behind the 'expanding grid' used by Luther (1970) and is to be preferredover the somewhat arbitrary choice of grid points made by Estoque and Bhumralkar(1970), Pandolfo (1971) and others.

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P. A. TAYLOR AND YVES DELAGE

In the case of the boundary-value problem defined by Equation (5) plus the bounda-ry conditions, a suitable coordinate transformation is achieved by setting

= In + (8)zo

With this transformation Equation (5) becomes

d2 U2= 0. (9)

A finite-difference solution of Equation (9) using equally spaced discretizationpoints and central differences will, apart from possible round-off errors, always givethe exact solution.

3. Wall Layers

One frequently used way of alleviating the difficulties that arise because of the 'nearlysingular point' at z = 0 in Equation (5) is to consider a shallow 'wall layer', 0 z < zw,within which we attempt to solve the governing equations by some sort of approxima-tion. This usually takes the form of assuming that the layer 0<z zw is a constantflux layer with, in the neutral case, a logarithmic velocity profile. For our analysis ofthe surface layer this solution is of course exact and valid in the whole region. Itbecomes an approximate solution, valid only in the wall layer, if we consider timedependent or horizontally heterogeneous problems for the surface layer or if we con-sider the equations for the planetary boundary layer as in § 4.

For roughness change problems, Taylor (1969a, c) has used a wall layer in conjunc-tion with coordinate transformations similar to Equation (8) and obtained satis-factory results. Peterson (1969) in effect uses a wall layer in his approach to the rough-ness change problem. His governing equations are somewhat different but he appearsto obtain satisfactory results using a wall layer in conjunction with discretizationpoints that are equally spaced in z.

The degree of accuracy obtained with a scheme of this type will depend on the depthof the wall layer. As an illustration we consider a simple case for our basic problem(Equation (5)) with ZH/Zo=100, N=10 and a wall layer 0<z<z(2), where z(2)=20zo. We use the assumed logarithmic profile in the wall layer to give values for U at

TABLE I

Velocity profiles, U/UH, calculated with equally spaced discretization points.ZH/ZO = 100, N= 10. 'Exact' value of u*/UH = 0.0866.

z/ZH 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Exact profile 0.520 0.660 0.744 0.805 0.852 0.891 0.924 0.952 0.977 1.00With wall layerzwzH=0.2 0.515 0.653 0.739 0.800 0.849 0.888 0.922 0.951 0.977 1.00No wall layer 0.433 0.595 0.695 0.767 0.823 0.869 0.909 0.943 0.973 1.00

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A NOTE ON FINITE-DIFFERENCE SCHEMES

z=z(l) where this is required. The finite difference results, together with the exactsolution and the solution with no wall layer, (but with the z0 term retained) are givenin Table I. The results without a wall-layer are poor. On the other hand in cases whereh is small compared to z (e.g., h/zo=0.1), we find good agreement with the exactsolution whether we use a wall layer or not.

4. The Planetary Boundary Layer

For the planetary boundary layer the equations for U and V, the components of themean horizontal velocity, may be written as

drx.d. _ fV (10)dz

dry- = f (- VU) (11)dz

where Tr and ry are the horizontal components of the shear stress, f is the coriolisparameter and (Ug, 0) is the geostrophic wind which we assume to be independent ofheight. We have taken the x-axis parallel to the geostrophic wind. In terms of eddyviscosity (K) and mixing-length () theories it is now customary to write

dU= K -- (12)

dz

and toset K = /21 (13)

where = lr. This is equivalent to

=dz = 1dz) + . (14)

These equations have been used by Blackadar (1962), Taylor (1969b), Pandolfo(1971) and others. In addition, we use Blackadar's (1962) form for the mixing length

k (z + Zo)I (15)

1 + kz/(

For the test cases we take

A = 0.0004 Ug/f .

The value of the parameter A has been discussed by Taylor (1969b).If we combine Equations (10) to (14), we can reduce the problem to the solution

of the pair of simultaneous second-order ordinary differential equations:

d2 U dK dU

Kd2 dz dzd2 V dK dV (16)

K dz2 + dz d = f (U- Ug)

where K is given by Equation (14).

113

P.A.TAYLOR AND YVES DELAGE

The boundary conditions are

U=V=0 on z=0U = Ug, V=0 on z = ZH, or ideally as z o.

If we are using a finite-difference method we must apply the upper boundarycondition at some predetermined height zH, which must have a value greater than the'actual' height of the boundary-layer for a given set of parameters. Typically we take

zH near 1500 m.We will consider finite-difference schemes corresponding to central differences in

several different transformed coordinates and also the finite-difference scheme usedby Estoque and Bhumralkar (1970). Using the coordinate transformation (7), Equation(16) becomes

[2 d2U dUl + j2 K dU

· ,~ d2V 2dKdV (18)K[F2 d V F I F - =f (U-U9 )

dC d dC dCwhere dF

F =dz

We solve Equations (18) together with the companion equations for K and dK/dCby replacing dU/dC and d2 U/dC2 by the usual central-difference formulae and thensolving the resulting (2N-2) non-linear algebraic equations iteratively.

The choice of coordinate transformation (C =F(z)) is governed by a desire to reducethe number of grid points used whilst retaining as much accuracy as possible. For theplanetary boundary-layer equations, the coordinate transformation (8) will give goodresolution close to the ground but will lead to poor results in the upper part of thelayer (500 to 1000 m) where there will be very few grid points (see Table II). Taylor(1969c) used a finite-difference mesh that was based on this coordinate transformationfor 0 z/zo < 5000 and a constant grid spacing in z above this. The method workedsatisfactorily but was somewhat inelegant.

A method which is apparently 'well known' but whose originator is unknown,according to Perov et al. (1967) (see also Gutman, 1969, p. 268), is to make a trans-formation

C=Kj(- (19)

0

where K-* K, as z - oo.This method is perfectly satisfactory if the eddy viscosity K is specified as a function

of z and does not involve a U/&z. In our case this is not true but we can modify Equation(19) and propose the transformation

z

C = Cf () (20)0

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A NOTE ON FINITE-DIFFERENCE SCHEMES

where C is a constant which is to be chosen for convenience. Within a constantshear-stress layer with neutral thermal stratification, we would then have a solutionUoc which would again be given exactly by our central-difference scheme. Clearly theplanetary boundary layer is not a constant shear stress layer, but it is approximatelyso near the ground, where most of the difficulties arise. With the transformation (20),U() should be a slowly varying function which will be adequately represented byfinite differences. This may not be true in non-neutral conditions where Equation (19)could be written

i = c'f 1ji dz (21)0

where OM is the non-dimensional wind shear and C' is an arbitrary constant. In theouter regions of the boundary layer, the removal of the PM term from Equation (21)in stable conditions can lead to the construction of too coarse a grid and somemodifications to Equation (20) may be required. There should be no problems in thecase of unstable thermal stratification and we would recommend the use of the trans-formation (20) in neutral and unstable conditions.

TABLE II

Grid points z(I) with different coordinate transformations, zH = 1200 m, zo = 1 cm

N N N zw=2m zw=2m zw=40m zw=40mlinear log log-linear log log-linear log log-linear

Equation (30)I N=20 N=20 N=20 N=20 N=20 N=12 N=12

(a) (b) (c) (d) (e) (f) (g)

I 0 0 0 1.4 0.74 27 11.82 63 0.008 0.02 2.0 2.0 40 403 126 0.023 0.11 2.9 5.4 56 994 189 0.049 0.41 4.1 13.7 79 1865 252 0.095 1.44 5.8 32 111 2916 316 0.180 4.9 8.3 64 156 4087 379 0.332 15.2 12 111 219 5318 442 0.61 41 17 170 308 6609 505 1.1 89 24 239 433 792

10 568 2.0 157 34 314 608 92611 632 3.6 239 49 393 854 106212 695 6.5 331 70 476 1200 120013 758 11.9 430 100 56214 821 21 533 143 65015 884 38 640 203 73816 948 70 748 290 82917 1011 125 859 414 92018 1074 225 972 590 101319 1137 410 1087 841 110620 1200 1200 1200 1200 1200

N - No wall layer

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P. A. TAYLOR AND YVES DELAGE

With the particular form of 1(4) chosen here (Equation 15), we can integrate (20)directly to give

+o 1 1= A - In + z (22)

Lk zo i-kzo

where A is a constant which may be chosen for convenience and, in most instances,kzo < A and may be neglected in the second term.

A finite-difference scheme with no wall layer (see Section 3) could be used in con-junction with the transformation (20). For most purposes, however, we would re-commend the use of a wall layer to reduce the total number of grid points whilstretaining a reasonable degree of resolution in the outer part of the region. If we areintending to use these vertical finite-difference spacings in the numerical solution of atime-dependent and/or horizontally inhomogeneous problem using an explicit finite-difference scheme, then the use of a wall layer will enable us to use larger time (or Jx)steps.

It is perhaps instructive to consider examples of the values of z corresponding togrid points, equally spaced in r, after applying different transformations as in Table II.Here z(1) is zero if no wall layer is used and, if there is a wall layer, then z(2) =zw.

Linear spacing, (a), is inadequate at the lower boundary while logarithmic spacing,(b, d andf ), is inadequate in the upper region whether or not a wall layer is used.

It is however worth noting that solutions obtained with scheme (g) are moreaccurate than those obtained with scheme (f). The transformation (22) gives a log-linear spacing which provides adequate values of z(I) throughout the boundary layerwith or without a wall layer (c, e, g).

5. A Comparison of Results Using Different Grid Spacings

To illustrate the behaviour of finite-difference solutions for the neutral boundarylayer, we have tried different grid spacings in solving basically the same problem, de-fined by Equations (14) to (17) with the parameters,

Ug = 10 m s - 'f = 10- 4 s-1zo = 1 cm (some cases involve z, = 10- 3 cm and 0 for comparison).

We set z, equal to 1.2 km for most of the runs. Increasing zH beyond this height madelittle or no difference to the profiles. The other finite-difference scheme used for com-parison is taken from Estoque and Bhumralkar (1970) (hereafter referred to as E andB). Let us emphasise at this point that this particular finite-difference scheme is chosenfor comparison somewhat arbitrarily since it was published in a recent issue of thisjournal; it is by no means unique in giving results with relatively large errors. In theirscheme (16) is represented by

2K,+1/2 (U+_ - U) 2K,_ 1/2 (U,- W _,)V (Z+- Z1) (ZI+1 - Z- 1 ) (Z - Zi, 1 ) (Z1,+1 - Z- 1 ) (23)

116

A NOTE ON FINITE-DIFFERENCE SCHEMES

with a similar expression for the second equation. In the calculations here, Equation(14) is used for K. The grid points are (in meters) z=0, 0.25, 0.5, 1, 2, 6, 16, 32, 64,100, 200, 300, 600, 1000 and 1500 m. The last grid point was added to E and B's gridto make z, compatible with the present input parameters. We will refer to Equation(23) together with these grid points as E and B's scheme. It will sometimes be usedwith a wall layer between the ground and z =0.5 m.

The results given by the log-linear spacing using 39 grid points and no wall layerwere found to be virtually identical to those obtained by Taylor (1969b) using a shoot-ing method. These are shown as a wind hodograph in Figure 3 and also as curve A inFigure 4 which shows the U-component of the wind as a function of height. In orderto illustrate the accuracy of the log-linear spacing, different numbers of grid pointswere used. Results are presented in Table III. The numbers of grid points were chosen

V (m sec-')2.5 - 887

23920 -152

.5 - 1.4

1.0 -. 11

L5 - L}76391876'

0 - I I I I I I I 108 5,8 -59

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0U (m sec-')

Fig. 3. Wind hodograph for basic test case, f= 10 - 4 s- 1, Ug 10 m/s, Vg 0, zo = 1 cm using alog-linear spacing and 39 grid points. Heights are in metres.

- . U (m sec')-

Fig. 4. U - component of velocity in basic test case; A - Log-linear spacing, no wall layer; B - Es-toque and Bhumralkar's scheme, 0.5 m wall layer; C - Log-linear spacing, 40 m wall layer. D - Es-

toque and Bhumralkar's scheme, no wall layer.

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P. A. TAYLOR AND YVES DELAGE

TABLE III

Log-linear spacings; comparisons with different numbers of grid points

Z(m) U(m s- 1) V(m s- 1) T1/2(m s- )

N=39 N=20 N=ll N=39 N=20 N=ll N=39 N=20 N=11

0.111 1.92 1.92 1.92 0.65 0.65 0.65 0.325 0.325 0.3251.44 3.84 3.84 3.85 1.30 1.30 1.29 0.324 0.324 0.324

15.2 5.76 5.76 5.77 1.89 1.89 1.89 0.318 0.317 0.31688.7 7.63 7.64 7.66 2.22 2.22 2.24 0.286 0.285 0.281

239 9.22 9.23 9.27 1.99 1.99 1.99 0.230 0.229 0.224430 10.23 10.24 10.27 1.28 1.28 1.25 0.170 0.169 0.162639 10.54 10.54 10.54 0.46 0.45 0.42 0.116 0.114 0.107859 10.34 10.33 10.29 -0.06 - 0.06 - 0.08 0.069 0.067 0.059

1085 10.05 10.04 10.01 -0.10 -0.10 -0.09 0.037 0.034 0.024a 18.8 18.8 18.6

a is the angle between the surface and geostrophic wind directions.

so that values of U, V and T/ 2 at the same height could be compared directly. The

differences amongst the cases for N= 11, 20 and 39 could not be properly resolved in

a diagram of the scale of Figure 3 or 4. The same is true for a case where a wall layer

2 m deep was introduced with a scheme of 20 grid points. All these cases can be re-

presented by curve A in Figure 4. A deepening of the wall layer increases the inaccuracy

200

100

I10

.5

1-0 2.0 4.0U (m sec-

1)

Fig. 5. U - component of velocity, U, = 10 m/s, f= 10- 4 S-i; A - Log-linear spacing, 0.5-m walllayer, zo = 0.001 cm; B - Estoque and Bhumralkar scheme, 0.5-m wall layer, zo = 0.001 cm; C - Es-toque and Bhumralkar scheme, no wall layer, zo =0; D - Estoque and Bhumralkar scheme, no

wall layer, zo = 1 cm.

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A NOTE ON FINITE-DIFFERENCE SCHEMES

TABLE IV

Values of shear stress, r, and of the angle between the surface and geostrophic wind directions,a, for the cases illustrated in Figures 4 and 5.

Figure Curve Description TI2wL T/2FD a(when applicable) m s-1 (deg)m s- 1

4 A Log-linear spacing 0.325 18.8No wall layerzo= 1 cm

4 A Taylor's 1969 results - 0.325 18.8No wall layer, shooting methodzo = 1 cm

4 A Log-linear spacing 0.324 0.324 18.5Wall layer, 2 mzo = 1 cm

4 B E and B's scheme 0.326 0.338 19.0Wall layer, 0.5 mzo = 1 cm

4 C Log-linear spacing 0.313 0.308 16.1Wall layer, 40 mzo =1 cm

4-5 D E and B's scheme - 0.370 21.6No wall layerzo = 1 cm

5 C E and B's scheme - 0.365 21.3No wall layerZo =0

5 B E and B's scheme 0.220 0.228 11.04Wall layer, 0.5 mzo = 0.001 cm

5 A Log-linear spacing 0.220 0.219 10.89Wall layer, 0.5 mzo = 0.001 cm

since a constant shear stress is assumed in this layer. When a wall layer of 40 m is used,

the same scheme gives curve C in Figure 4. The remaining curves in Figure 4 are

results obtained using E and B's scheme. With a wall layer 0.5 m deep we obtain curve

B, which shows virtually no error below 2 m and good overall accuracy. When no wall

layer is used (as is the case in E and B's paper), we get curve D, which is not a very

accurate solution. Moreover the results are virtually independent of zo when z is

small (< 5 cm). A clear illustration of this is given in Figure 5, where curve D is the

same as in Figure 4, i.e., E and B's scheme (no wall layer) with z = 1 cm. Curve C

results from the same scheme with zo =0 (the limiting case) for which the 'exact'

solution is U=10 m s - ' for z>0 and U=O at z=0. There is very little change fromD to C. For comparison, curves A and B (Figure 5) are solutions using log-linear

spacing and E and B's scheme, respectively, with a wall-layer 0.5 m deep and with zo =0.001 cm. These results demonstrate the necessity of using a wall layer when the grid

spacing is not chosen suitably. Table IV gives additional information on the cases

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P. A. TAYLOR AND YVES DELAGE

illustrated in Figures 4 and 5. We tabulate the value of the shear stress at the groundand the angle () between the surface and geostrophic wind directions. Here /L isthe square root of the shear stress given by the wall-layer solution while 4;/2 iscalculated using finite differences. Notice that for the log-linear scheme, values of

1/2 and of z4/2 are virtually equal, except when the wall layer is too deep. The use ofE and B's finite-difference scheme introduces slight differences between the two values.

6. Conclusions

The idea for this note originated when the authors discovered errors in a section ofan earlier paper. We soon realised that we were not alone in our careless use of finite-

difference schemes in the planetary boundary layer and we were able to find manypapers where the results obtained were highly dependent upon the grid spacings used.

In terms of designing finite-difference schemes for use in the planetary boundarylayer, we find that the transformation (Equation (20)) of the height coordinate basedon the mixing length, together with central differences in the new variable, give goodresults in neutral conditions with relatively few discretization points. We are confidentthat it will also give good results for unstable conditions. We would also recommendthe use of a 'wall layer' in which constant fluxes are assumed.

Acknowledgements

This work has been supported by the Meteorological Service of Canada. One of us(Y.D.) wishes to thank them for granting him Ph. D leave and the other (P.A.T.) fora research grant.

References

Blackadar, A. K.: 1962, 'The Vertical Distribution of Wind and Turbulent Exchange in a NeutralAtmosphere', J. Geophys. Res. 67, 3095-102.

Blackadar, A. K.: 1965, 'A Single Layer Theory of the Vertical Distribution of Wind in a BaroclinicNeutral Atmospheric Boundary-Layer', Final report AFCRL-65-351, Dept. of Meteorology,Pennsylvania State University.

Delage, Y. and Taylor, P. A.: 1970, 'Numerical Studies of Heat Island Circulations', Boundary-LayerMeteorol. 1, 201-26.

Estoque, M. A. and Bhumralkar, C. M.: 1970, 'A Method for Solving the Planetary Boundary-LayerEquations', Boundary-Layer Meteorol. 1, 169-94.

Gutman, L. N.: 1969, Introduction to the Non-Linear Theory of Mesometeorological Processes (inRussian), Hydrometeorological Publishing House, Leningrad.

Keller, H. B.: 1968, Numerical Methods for Two-Point Boundary Value Problems, Blaisdell, Waltham,Mass.

Luther, F. M.: 1970, 'A Numerical Model of the Energy Transfer Processes in the Lower Atmosphere',Ph.D. Thesis, Contributions in Atmospheric Science No 2, University of California-Davis.

Pandolofo, J. P.: 1971, 'Numerical Experiments with Alternative Boundary-Layer FormulationsUsing BOMEX Data', Boundary-Layer Meteorol. 1, 277-89.

Perov, V. L., Mal'bakhov, V. M., and Gutman, L. N.: 1967, 'Non-Linear, Non-Stationary Model ofSlope Winds', Izv. Atmospheric Oceanic Phys. 3, 1179-86 (pp. 694-98 in English translation).

Taylor, P. A.: 1967, 'On Turbulent Wall Flows above a Change in Surface Roughness', Ph.D. Thesis,University of Bristol.

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A NOTE ON FINITE-DIFFERENCE SCHEMES 121

Taylor, P. A.: 1969a, 'On Wind and Shear Stress Profiles Above a Change in Surface Roughness',Quart. J. Roy. Meteorol. Soc. 95, 77-91.

Taylor, P. A.: 1969b, 'On Planetary Boundary-Layer Flow under Conditions of Neutral ThermalStability', J. Atmospheric Sci. 26, 427-31.

Taylor, P. A.: 1969c, 'The Planetary Boundary-Layer above a Change in Surface Roughness', J.Atmospheric Sci. 26, 432-40.