3.2 Wind input and dissipation
Results of the numerical solution of the momentum balance of air flow over growing surface gravity waves have been presented in a series of studies by Janssen (1989), Janssen et al. (1989a) and Janssen (1991). The main conclusion was that the growth rate of the waves generated by wind depends on the ratio of friction velocity and phase speed and on a number of additional factors, such as the atmospheric density stratification, wind gustiness and wave age. So far systematic investigations of the impact of the first additional two effects have not been made, except by Janssen and Komen (1985) and Voorrips et al. (1994). It is known that stratification effects observed in fetch-limited wave growth can be partly accounted for by scaling with 
(which is consistent with theoretical results). The remaining effect is still poorly understood, and is therefore ignored in the standard WAM model. In this section we focus on the dependence of wave growth on wave age, and the related dependence of the aerodynamic drag on the sea state, which effect is fully included in the WAM model.
A realistic parametrization of the interaction between wind and wave was given by Janssen (1991), a summary of which is given below. The basic assumption Janssen (1991) made, which was corroborated by his numerical results of 1989, was that even for young wind sea the wind profile has a logarithmic shape, though with a roughness length that depends on the wave-induced stress. As shown by Miles (1957), the growth rate of gravity waves due to wind then only depends on two parameters, namely
As usual, 
denotes the friction velocity, 
the phase speed of the waves, 
the wind direction and 
the direction in which the waves propagate. The so-called profile parameter 
characterizes the state of the mean air flow through its dependence on the roughness length 
. Thus, through 
the growth rate depends on the roughness of the air flow, which, in its turn, depends on the sea state. A simple parametrization of the growth rate of the waves follows from a fit of numerical results presented in Janssen(1991). One finds
where 
is the growth rate, 
the angular frequency, 
the air-water density ratio and 
the so-called Miles' parameter. In terms of the dimensionless critical height 
(with 
the wavenumber and 
the critical height defined by 
) Miles' parameter becomes
where 
is the von Kármán constant and 
a constant. In terms of wave and wind quantities 
is given as
and the input source term 
of the WAM model is given by
where 
follows from (3.3) and with 
the action density spectrum.
The stress of air flow over sea waves depends on the sea state and from a consideration of the momentum balance of air it is found that the kinematic stress is given as (Janssen, 1991)
where
Here, 
is the mean height above the waves and 
is the stress induced by gravity waves (the `wave stress')
The frequency integral extends to infinity, but in its evaluation only an 
tail of gravity waves is included and the higher level of capillary waves is treated as a background small-scale roughness. In practice, we note that the wave stress points in the wind direction as it is mainly determined by the high-frequency waves which respond quickly to changes in the wind direction.
The relevance of relation (3.8) cannot be overemphasized. It shows that the roughness length is given by a Charnock relation (Charnock, 1955)
However, the dimensionless Charnock parameter 
is not constant but depends on the sea state through the wave-induced stress since
Evidently, whenever 
becomes of the order of the total stress in the surface layer (this happens, for example, for young wind sea) a considerable enhancement of the Charnock parameter is found, resulting in an efficient momentum transfer from air to water. The consequences of this sea-state-dependent momentum transfer will be discussed in Chapter 7.
This finally leaves us with the choice of two unknowns namely 
from (3.11) and 
from (3.4). The constant 
was chosen in such a way that for old wind sea the Charnock parameter 
has the value 0.0185 in agreement with observations collected by Wu (1982) on the drag over sea waves. It should be realised though, that the determination of 
is not a trivial task, as beforehand the ratio of wave-induced stress to total stress is simply not known. It requires the running of a wave model. By trial and error the constant 
was found to be 
.
The constant 
is chosen in such a way that the growth rate 
in (3.3) is in agreement with the numerical results obtained from Miles' growth rate. For 
and a Charnock parameter 
we have shown in Fig. 3.1 the comparison between Miles' theory and (3.3). In addition observations as compiled by Plant (1982) are shown. Realizing that the relative growth rate 
varies by four orders of magnitude it is concluded that there is a fair agreement between our fit (3.3), Miles' theory and observations. We remark that the Snyder et al. (1981) fit to their field observations, which is also shown in Fig. 3.1 , is in perfect accordance with the growth rate of the low-frequency waves although growth rates of the high-frequency waves are underestimated. Since the wave-induced stress is mainly carried by the high-frequency waves an underestimation of the stress in the surface layer would result.
Figure 3.1 Comparison of theortical growth rates with observations by Plant (1982) (full line): Miles' theory; (full dots): parametrization of Miles' theory (3.3); dashed line: the fit by Snyder et al. (1981).
We conclude that our parametrization of the growth rate of the waves is in good agreement with the observations. The next issue to be considered is how well our approximation of the surface stress compares with observed surface stress at sea. Fortunately, during HEXOS (Katsaros et al. 1987) wind speed at 10 m height, 
, surface stress 
and the one-dimensional frequency spectrum were measured simultaneously so that our parametrization of the surface stress may be verified experimentally. For a given observed wind speed and wave spectrum, the surface stress is obtained by solving (3.7) for the stress 
in an iterative fashion as the roughness length 
depends, in a complicated manner, on the stress. Since the surface stress was measured by means of the eddy correlation technique, a direct comparison between observed and modelled stress is possible. The work of Janssen (1992) shows that the agreement is good.
It is, therefore, concluded that the parametrized version of quasi-linear theory gives realistic growth rates of the waves and a realistic surface stress. However, the success of this scheme for wind input critically depends on a proper description of the high-frequency waves. The reason for this is that the wave-induced stress depends in a sensitive manner on the high-frequency part of the spectrum. Noting that for high frequencies the growth rate of the waves (3.3) scales with wavenumber as
and the usual whitecapping dissipation scales as
an imbalance in the high-frequency wave spectrum may be anticipated. Eventually, wind input will dominate dissipation due to wave breaking, resulting in energy levels which are too high when compared with observations. Janssen et al. (1989b) realized that the wave dissipation source function has to be adjusted in order to obtain a proper balance at the high frequencies. The dissipation source term of Hasselmann (1974) is thus extended as follows:
where 
and 
are constants, 
is the total wave variance per square metre, 
the wavenumber and 
and 
are the mean angular frequency and mean wavenumber, respectively. In practice, we take 
and 
. The choice of the above dissipation source term may be justified as follows. In Hasselmann (1974), it is argued that whitecapping is a process that is weak-in-the-mean, therefore, the corresponding dissipation source term is linear in the wave spectrum. Assuming that there is a large separation between the length scale of the waves and the whitecaps, the power of the wavenumber in the dissipation term is found to be equal to one. For the high-frequency part of the spectrum, however, such a large gap between waves and whitecaps may not exist, allowing the possibility of a different dependence of the dissipation on wavenumber.
This concludes the description of the input source term and the dissipation source term due to whitecapping. Although the wind input source function is fairly well-known from direct observations, there is relatively little hard evidence on dissipation. Presently, the only way out of this is to take the functional form for the dissipation in (3.14) for granted and to tune the constants 
and 
in such a way that the action balance equation (2.24) produces results which are in good agreement with data on fetch-limited growth and with data on the dependence of the surface stress on wave age. In addition, a reasonable dissipation of swell should be obtained. It was decided to follow this method and, after an extensive tuning exercise, the constants 
and 
were given the values 4.5 and 0.5 while the constant 
in the Charnock parameter was given the value 0.01.
3.2.1 Wind gustiness and air density
The input source term given in (3.6) and (3.3) assumes homogeneous and steady wind velocity within a model grid-box and during a time-step. Assuming that the wind speed variations with scales much larger than both the spatial resolution and the time step are already resolved by the atmospheric model, we need to include the impact of the wind variability at scales comparable to or lower than the model resolution (which is called wind gustiness). To achieve this, an enhanced input source term with the mean impact of gustiness can be estimated as:
where 
represents the instantaneous (unresolved) wind friction velocity, 
is the standard deviation of the friction velocity and the over-barred quantity represents the mean value of the quantity over the whole grid-box/time-step. Note that this is the (gust-free) value obtained from the atmospheric model. The integral above can be approximated using the Gauss-Hermite quadrature as:
The magnitude of variability can be represented by the standard deviation of the wind speed. To estimate the standard deviation value, one can use the empirical expression proposed by Panofsky et al. (1977) which can be written as:
where 
is the standard deviation of the 10 m wind speed, 
is the height of the lowest inversion, 
is the Monin-Obukhov length, and 
is a constant representing the background gustiness level that exists all the times irrespective of the stability conditions. The quantity 
, which is a measure for the atmospheric stability, is readily available in the atmospheric model. The impact of the background level of gustiness is already included implicitly in the parameterisations of the atmospheric model as well as in the wave model. Therefore, the constant 
value is used as 0 (see Abdalla and Bidlot, 2002).
The growth rate of waves is proportional to the ratio of air to water density, 
, as can be seen in (3.3). Under normal conditions, seawater density varies within a very narrow range and, therefore, it can be assumed to be constant. On the other hand, air density has a wider variability and need to be evaluated for better wave predictions. Based on basic thermodynamic concepts, it is possible to compute the air density using the following formula:
where 
is the atmospheric pressure, 
is a constant (
) defined as 
, with 
the universal gas constant (
) and 
is the molecular weight of the dry air (
) and 
is the virtual temperature. The virtual temperature can be related to the actual air temperature, 
, and the specific humidity, 
, by: 
. In particular, the surface pressure is used for 
, the skin temperature is used for 
, and the humidity at 2 m height is used for 
(see Abdalla and Bidlot, 2002).
3.2.2 Use of neutral winds
The WAM model was developed in term of surface stress as expressed by the friction velocity u*. The relation between u* and the wind speed at a given height (currently 10m) is assumed to be given by the logarithmic profile corresponding to neutral stability condition. The wave model should therefore be forced by surface stresses. However it is usually forced by wind speeds because they are readily available. Hence, these winds should be transformed into their neutral wind counterparts. In the coupled IFS/WAM system, this transformation can easily be achieved on the IFS side by using the atmospheric surface stress and the logarithmic wind profile with the roughness length based on the Charnock parameter (Fig. 3.2 ). This conversion has been successfully tested and implemented in CY28R1.
Figure 3.2 Schematic representation of the interface between the IFS and WAM.
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