To obtain CDN10 from CDh, we must match the
Notice, from eq 19, 20 and 23
profile laws, eq 29 and 30, at zw. From eq 29 and
λ(Φ)CR (Φ)
^
τR
32, we have the following results for z ≥ zw
=
.
(44)
CSh + λ(Φ)CR (Φ)
τ
^
U(10) 1 10 - d
CD1N2 0 ≡
= ln
/
(35)
1
k z0
u
R92 then related dR to h. Following his argu-
*
ments but applying them to the specific geome-
and
try of my model, I get
U(zw ) 1 h - d
[
]1/ 2
+ ln(cw ) (36)
dR = h - cdCD1h 2 A(Φ)/ λ(Φ)
CD1zw ≡
= ln
/
/2
(45)
k z0
u
*
in which cd is a constant that R92 took to be 0.6.
where h, d, z0 and zw must be in meters. Subtract-
Inserting eq 5 in this and combining it with eq 43
ing eq 35 from 36, we get
yields
[
].
1 h-d
d = h(τR / τ) 1 - cdCD1h 2 (mn / 2γ )
ln 10 - d + ln(cw ) .
1/ 2
CD1zw = CD1N2 0 +
/
/2
/
(37)
(46)
k
1
Thus, it is easy to find τR/τ from eq 44; and once
From eq 3033, for h ≤ z ≤ zw
my model yields CDh, it is simple to compute d.
Using this value in eq 41 finally gives the quan-
Uh 1 h - d
tity we seek, CDN10.
+ ψh
CD1h 2 ≡
= ln
/
k z0
(38)
u
I pointed out in the previous section that we
*
need to convert CS10 to CSh before we can begin
and
computations. Equation 41 also makes this con-
version. Unfortunately, we need CSh before we
U(zw ) 1 h - d
+ ln(cw ) .
have obtained d. I could handle this problem by
CD1zw =
= ln
/2
k z0
(39)
u
iterating the entire set of equations on d. But, as I
*
will show in the next section, d/h ≈ 0.3, a result
Subtracting eq 38 from 39 yields
also consistent with eq 46. At this step in the
computations, this simple approximation is rea-
[
]
1
ln(cw ) - ψ h .
sonable--especially in light of uncertainties in
CD1zw = CD1h 2 +
/2
/
(40)
k
the other model parameters--because CDN10 is
not very sensitive to CS10 for the range of values
On equating eq 37 and 40, we finally can relate
that this parameter can realistically assume.
CDN10 to CDh
1 10 - d
RESULTS
CD1N2 0 = CD1h 2 +
- ψh .
ln
/
/
(41)
k h-d
1
I have not yet discussed the value of γ, the
fractional coverage of sastrugi-like roughness el-
We still do not know the displacement height d,
ements. Vladimir Churun* tried to quantify the
however.
roughness of the ISW floe using the radar on the
Thom (1971) identified d as the effective level
Akademik Fedorov at the time the station was de-
at which the roughness elements absorb the mo-
ployed. His survey suggested that hummock
mentum being transferred to the surface. R92
coverage varied over the floe from 10 to 30%.
used this definition to derive d. That is, if dR is the
While hummock coverage is not the same as sas-
centroid of the form drag on the roughness ele-
trugi coverage, my personal experience on ISW
ments and if dS is the centroid at which the skin
suggests that 1030% sastrugi coverage is also
friction acts
about right. Notice, because of the geometry of
the sastrugi that I am modeling (Fig. 3), the tight-
dτ = dR τR + dS τS = dR τR
(42)
est possible packing of roughness elements will
yield a γ value of only 0.5. Thus, my first guess,
because dS = 0 by definition. Thus
d = dR (τR / τ).
* Personal communication (Arctic and Antarctic Research
(43)
Institute, St. Petersburg, Russia).
7
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