GUIDE WAVE ANALYSIS AND FORECASTING - WMO

30
a general sense, and can be applied to the atmosphere.
These concepts have been developed to formulate diagnostic
models that allow us to deduce surface turbulent
flow fields from the free atmospheric flow.
The marine boundary layer itself can be separated
into two regimes: the constant flux (or constant stress)
layer (from the surface to about 50 m) and the Ekman
layer (from about 50 m up to the free atmosphere
≈ 1 km). In the surface layer it can be assumed that the
frictional forces contributing to turbulence are constant
with height, and the effects of the Coriolis and pressure
gradient forces, as well as the horizontal gradient of
turbulent fluxes, are negligible. The wind direction is
consequently constant with height. Using the mixing
length theory developed by Prandtl, it can be shown
that the flow in the constant flux layer (or Prandtl layer)
depends only on the surface roughness length.
2.4.1 Constant flux layer
Under neutral conditions, Prandtl’s solution shows that
the horizontal flow over the ocean surface follows the
well known “log” (logarithmic) profile in the vertical
direction,
u = u *
κ ln
(2.8)
where κ is the von Kármán constant, z 0 is the constant of
integration, known as the roughness length, and u * is the
friction velocity, which has magnitude:
u * =
⎛ z ⎞
⎜ ⎟
⎝ ⎠
,
(2.9)
τ is the magnitude of the surface stress, and ρa the
density of air. u * can be thought of as a proxy for the
surface stress.
It is common also to express stress (τ) through the
bulk transfer relation:
τ = ρa Cd u2 , (2.10)
where Cd is the drag coefficient. In general, Cd and u are
both functions of height. Determining Cd has been the
objective of many field research programmes over the
years.
One of the problems of specifying the wind in the
turbulent layer near the ocean is the formulation of z0 and its relationship to u *. Using dimensional argument,
Charnock (1955) related the roughness length of the sea
surface to the friction velocity of the wind as follows:
z0 = (2.11)
where α is the Charnock constant, with a value as determined
by Wu (1980) of 0.0185 , and g is the acceleration
by gravity.
If the boundary layer is in a state of neutral stratification,
the drag coefficient, which is a function of
height, can be expressed as:
αu * 2
,
g
τ
ρ a
z 0
,
GUIDE TO WAVE ANALYSIS AND FORECASTING
or
C z
d( )=
κ 2
Cd ( z)
=
⎛ ⎡ z ⎤⎞
/ ⎜ln
⎢ ⎥
⎝ ⎣ z
⎟ ,
0 ⎦⎠
⎛ u * ⎞
⎜ ⎟
⎝ uz⎠
( )
(2.12)
(2.13)
However, the boundary layer over the ocean is not
necessarily neutral. A stability dependence was originally
derived by Monin-Obukhov (1954) from profile
similarity theory. This is used to modify the simple log
relation provided above:
u = (2.14)
u *
κ ln
⎡ ⎛ z ⎞
⎜ ⎟ – ψ
⎛ z ⎞ ⎤
⎢
⎝ z0 ⎠ ⎝ L ⎠
⎥
⎣⎢
⎦⎥
⎛ ⎡ ⎛ z ⎞
C z
⎛ z ⎞⎤⎞
d( )= ⎜κ
/ ⎢ln
⎜ ⎟ – ψ
(2.15)
⎝ z ⎠ ⎝ L⎠
⎥⎟
.
⎝ ⎣⎢
0 ⎦⎥
⎠
The function ψ has been derived for both stable and
unstable conditions. L is the Monin-Obukhov mixing
length. For neutral conditions ψ (z/L) = 0. Businger et al.
(1971) proposed functional relationships between the
non-dimensional wind shear and z/L which can be used
to determine the stability function ψ (z/L) in Equation
2.14.
2.4.2 The use of the drag coefficient
In the above discussion, we developed the concept of the
drag coefficient, as defined in Equation 2.10. There have
been many studies to determine Cd under varying wind
speeds and stabilities and a summary can be found in
Roll (1965). More recently, Wu (1980, 1982) has shown
through empirical studies that the drag coefficient at a
given height depends linearly on the wind speed, and
that the following formulation holds for a wide range of
winds under near neutrally stable conditions:
103 C10 = (0.8 + 0.65 U10) (2.16)
where: C10 = Drag coefficient at the 10 m level; and
U10 = Wind speed (m/s) at the 10 m level.
This simple linear empirical relationship, however,
needs to be modified when temperature stratification is
present. Stratification affects turbulent momentum transport,
thereby causing the wind profile to deviate from the
logarithmic form.
Schwab (1978) determined Cd over water for a
wide range of wind speeds and atmospheric stabilities.
Figure 2.9 shows the results of his calculation. A crucial
issue to be addressed at this juncture concerns the effect
of changing stability and wind stress on the prediction
of wave growth. It can be inferred from Figure 2.9 that,
for a given 10 m level wind speed, unstable conditions
result in higher drag coefficients (or surface stress) and
hence larger wave growth than stable conditions. Liu et
al. (1979) have developed a set of equations which
compute the surface variables u * , z0 and the boundarylayer
stability length (L), so that the wind profile of
2
.
2
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