To make solving eq 22 easier, R92 assumed that

To convert the total drag coefficient *C*Dh to its

best fit to the data that he tested his model

10-m value, we need to know the wind speed

against. With this assumption, eq 22 reduces to

profile law. When a surface is covered with

roughness elements that are much larger than *z*0,

[C

]

^ (Φ) exp-*c*λ(Φ) *U*h . (23)

τ(Φ) =

ρ*U*h

+ λ(Φ)*C*R

2

it is often necessary to include a displacement

Sh

height *d *(Thom 1971) in the semi-logarithmic

Notice

wind profile. That is

τ(Φ)

= ln

1

(29)

= * ≡ *C*Dh ≡ 2

(24)

ρ*U*h Uh

2

2

*

where *U*(*z*) is the average wind speed at height *z*.

is the drag coefficient for the total surface stress

But near the tops of the roughness elements,

referenced to height *h*. On inserting this defini-

eq 29 cannot be accurate because of the enhanced

tion and rearranging terms in eq 23, we get

turbulence there created by flow around these

-*c*λ(Φ)r

exp

2

2

*z *- *d *

+ ψ

= ln

(30)

[

]

-1/ 2

*z*w - *d *

^

=

(25)

.

*

2

where ψ is a profile influence function similar to

This has the form

the stability corrections required by Monin-

Obukhov similarity. At *z*w and above, ψ is zero.

(26)

R92 went on to show that

where *a *is a constant for a given value of Φ.

*z*-*d *

*z *-*d *

I solved eq 26 for *X *using Newton's method;

(31)

ψ

+ ln w

=

*z*-*d *

*z*w - *d *

the solution required only two to three iterations.

There are multiple solutions to eq 26, however;

and that

we must select the right one. As R92 explained,

eq 26 has the following properties:

(32)

For 0 < a < e1, eq 26 has two solutions, *X*1 and

where *c*w is a constant. He also defined

For *a *= e1, *X *= 1.

For *a *> e1, there is no solution to eq 26.

*h*-*d *

= ln(cw ) + *c*w - 1

ψ h ≡ ψ

1

(33)

The only nontrivial case is the first of these; and

*z*w - *d *

proach zero as either λ(Φ) or *a *approaches zero.

which will be useful shortly. Notice, there are er-

Thus, to start Newton's method, my first guess

rors in two of R92's equations (eq 29 and 31). In

was *X *= *a*.

an unpublished correction to his paper dated 23

Solving eq 26 yields *X*, which, in turn, is relat-

October 1992, Raupach explained that, in light of

ed to the total drag coefficient we seek

these errors, *c*w is approximately 4 rather than

1.5, the value he originally gave. Hence,

ψh = 0.64, a value independent of surface proper-

2*X*

/

.

(27)

ties.

Equation 32 implies that

Notice, to check this solution, we can make an

asymptotic approximation in eq 23. When λ(Φ) is

(34)

near zero

With *c*w = 4 and the assumption that *z*w > *h*, eq 34

implies that *d *< *h*: The displacement height is al-

^

(28)

ways less than the height of the roughness ele-

ments.

6