#### 2.1. Wind speed and air–sea turbulent fluxes

Direct measurements of the turbulent heat fluxes are difficult and expensive to make, requiring continuous high frequency observations (Large and Pond, 1981) together with detailed information on the motion of the observing platform (Edson *et al*., 1998) and the flow distortion over the platform (Yelland *et al*., 2002). Long-term direct measurements of turbulent air–sea fluxes (Prytherch *et al*., 2010) are rare. The difficulty of making these measurements means that the global ocean is extremely sparsely sampled with direct heat flux measurements and we must therefore appeal to flux parameterizations to produce global flux fields based on more frequently observed parameters. These parameters include wind speed, air–sea temperature difference, air–sea humidity difference and pressure. Other parameters such as sea state are also sometimes included (Bourassa *et al*., 1999). Even these basic parameters are not uniformly sampled over the ocean and the use of satellite-derived wind speeds, which are relatively well sampled from space, is highly desirable.

Flux parameterizations, known as bulk formulae, model the turbulent exchange as a function of the surface atmospheric and oceanic states. On the basis of direct measurements of the fluxes made during dedicated research cruises, together with observations of the bulk meteorological variables, many different parameterizations have been developed for the turbulent fluxes (Liu *et al*., 1979; Smith, 1980, 1988; Clayson *et al*., 1996; Fairall *et al*., 2003). The bulk formulae can be written:

- (1)

- (2)

- (3)

- (4)

where *τ* is the wind stress (N m^{−2}), *ρ*_{0} is the density of air (kg m^{−3}), *C*_{D} is the drag coefficient, *u*_{z} is the wind speed (ms^{−1}) at the observation height *z* (m), *u*_{0} is the current speed in the direction of the wind vector (ms^{−1}), *H*_{s} is the sensible heat flux (W m^{−2}), *c*_{p} is the specific heat capacity of air at constant pressure (J kg K^{−1}), *C*_{H} is the transfer coefficient for sensible heat, *T*_{z} is the air temperature at the observation height (K), *T*_{0} is the sea surface temperature (SST K), *H*_{l} is the latent heat flux (W m^{−2}), *L* is the latent heat of vaporization (J kg^{−1}), *C*_{E} is the transfer coefficient for latent heat, *q*_{z} is the specific humidity at the observation height (kg kg^{−1}), *q*_{0} is the specific humidity just above the sea surface (usually taken as the 98% saturation value at the SST, kg kg^{−1}). *F*_{gas} is gas flux from ocean to atmosphere, *s* is the solubility of the gas in seawater at temperature *T*_{0} and salinity *S*, *k* is the transfer velocity across the interface given in units of centimetres per hour and *pX*_{w} and *pX*_{a} are the partial pressures of gas on the sea and air side of the interface, respectively.

The transfer coefficients (*C*_{D}, *C*_{H} and *C*_{E}, Equations (1)(2)(3)) are all functions of the atmospheric stability and several iterations are required to obtain the wind stress and heat fluxes. The bulk formulae differ primarily in the form that is assumed for the neutral transfer coefficients and for the dimensionless stability parameters (WGASF, 2000; Brunke *et al*., 2003). The transfer coefficients are strongly dependent on the atmospheric stability leading to a wide spread in the coefficients at low wind speeds. There is fairly good agreement about the form that the stability adjustments take, with most formulations varying only at the extremes of the stability range. There is less agreement about the neutral values of the transfer coefficients, especially at high wind speeds when direct flux measurements are rare (Brunke *et al*., 2003; Prytherch *et al*., 2010).

Parameterizations of gas transfer velocity of CO_{2} commonly rely on wind speed (Wanninkhof, 1992; Nightingale *et al*., 2000; Equation (4)) and vary between square and cubic functions of wind speed so nonlinear effects are particularly important (Wanninkhof *et al*., 2002; Fangohr *et al*., 2008). In an attempt to parameterise those surface processes that do not scale with wind speed, some more recent theoretical parameterizations of *k* also include a dependence on remotely sensed mean square slope, friction velocity and whitecapping (Glover *et al*., 2002; Woolf, 2005; Fangohr and Woolf, 2007). However, lack of observations of these parameters collocated with direct flux measurements mean that wind-speed-based parameterizations remain in use (Prytherch *et al*., 2010; Fairall *et al*., 2011). A recent overview is given by Wanninkhof *et al*. (2009).

Ideally global flux fields would be calculated from high accuracy, high resolution, collocated observations of each of the input parameters to the bulk formulae. High quality collocated observations of the surface atmospheric and oceanic states are not frequent enough to allow the construction of global fields. It is obvious that biased input variables will lead to biased fluxes, but even normally distributed random errors with zero mean bias may lead to biased estimates of the fluxes due to the nonlinearity of the bulk formulae (Berry, 2009). This is most problematic for observations containing large random uncertainties. If fluxes are calculated from gridded estimates of the surface atmospheric and oceanic states then biases can arise if synoptic scale correlations between the different variables are not captured (Ledvina *et al*., 1993; Josey *et al*., 1995; Gulev, 1997). If the gridded estimates are derived from data sources that are not collocated then there is the potential for bias if the input fields for different variables are mismatched in time or resolve different scales of variability.

#### 2.2. The adjustment of wind speed to a reference height and to neutral conditions

Near the surface of the ocean the wind speed typically decreases with height because of the drag exerted on the atmosphere by the ocean. The bulk formulae therefore require known height of measurement of wind speed and also of temperature and humidity that are needed to estimate the stability of the atmosphere near the ocean surface. An output from the flux calculation (Section 2.1, Equations (1)(2)(3)) is therefore the wind speed, temperature and humidity referenced to a standard height, usually 10 m. The gradient of wind speed with height depends on the atmospheric stability and is greater under stable conditions than unstable conditions. In the terminology used here winds as measured are ‘stability-dependent’. Calculation of the height adjustment requires knowledge of the atmospheric stability to provide a stability-dependent estimate of 10 m wind speed (the wind speed that would have been measured if the sensor was actually at 10 m height). If the atmospheric stability is unknown (because collocated observations of *T*_{0}, *T*_{z} or *q*_{z} are missing) an approximate adjustment can be made assuming that the stratification of the atmosphere is neutral. This approach will provide an approximation to the stability-dependent 10 m wind speed, the bias in which will depend on the difference between the neutral and actual wind profile.

A further adjustment that can be applied is to calculate the equivalent neutral wind speed, again usually referenced to 10 m height. The equivalent neutral wind is the wind speed that in a neutral atmosphere would give rise to the same surface stress as the observed stability-dependent wind speed. Conceptually this means calculating the surface stress using the full stability-dependent calculation and then using this stress to calculate the wind that would have produced this particular value under neutral conditions. Because the near surface gradients are stronger under neutral conditions rather than unstable conditions normally observed over the ocean, the adjustment from stability-dependent to neutral equivalent wind speed is positive (neutral winds are stronger than measured wind speeds in unstable conditions). In the less-common stable conditions neutral winds are weaker than measured winds.

The adjustment of wind speeds for height and to give a neutral equivalent wind speed requires assumptions to be made about the expected structure of the near surface atmosphere. The framework for these assumptions is Monin-Obukhov similarity theory (MOST) that underpins the bulk formulae. Each different version of the bulk formula will therefore give different results due the different assumptions each makes about the relationships among surface roughness, stress and atmospheric stability. This is discussed further in Section 4.1. Details on how to make these adjustments can be found in Chapter 7 of WGASF (2000).