5.4: Indirect Calculation of Fluxes: Bulk Formulas

The use of gust-probes is very expensive, and radiometers must be carefully maintained. Neither can be used to obtain long-term, global values of fluxes. To calculate these fluxes from practical measurements, we use observed correlations between fluxes and variables that can be measured globally.

For fluxes of sensible and latent heat and momentum, the correlations are called bulk formulas. They are: \[\begin{align} T &= \rho_{a} C_{D} U_{10}^{2} \\ Q_{S} &= \rho_{a} C_{p} C_{S} U_{10} \left(t_{s} - t_{a}\right) \\ Q_{L} &= \rho_{a} L_{E} C_{L} U_{10} \left(q_{s} - q_{a}\right) \end{align} \nonumber \]

Air temperature \(t_{a}\) is measured using thermometers on ships. It cannot be measured from space using satellite instruments. \(t_{s}\) is measured using thermometers on ships or from space using infrared radiometers such as the AVHRR.

The specific humidity of air at 10 m above the sea surface, \(q_{a}\), is calculated from measurements of relative humidity made from ships. Gill (1982: pp: 39– 41, 43–44, & 605–607) describes equations relating water vapor pressure, vapor density, and specific heat capacity of wet air. The specific humidity at the sea surface, \(q_{s}\), is calculated from \(t_{s}\) assuming the air at the surface is saturated with water vapor. \(U_{10}\) is measured or calculated using the instruments or techniques described in Chapter 4. Note that wind stress is a vector with magnitude and direction. It is parallel to the surface in the direction of the wind.

The problem now becomes: How to calculate the fluxes across the sea surface required for studies of ocean dynamics? The fluxes include: 1) stress; 2) solar heating; 3) evaporation; 4) net infrared radiation; 5) rain; 5) sensible heat; and 6) others such as CO₂ and particles (which produce marine aerosols). Furthermore, the fluxes must be accurate. We need an accuracy of approximately \(±15 \ \text{W/m}^{2}\). This is equivalent to the flux of heat which would warm or cool a column of water 100 m deep by roughly \(1^{\circ}\text{C}\) in one year. Table \(\PageIndex{1}\) lists typical accuracies of fluxes measured globally from space. Now, let’s look at each variable.

Table \(\PageIndex{1}\). Accuracy of Wind and Fluxes Observed Globally From Space.

Variable	Accuracy	Comments
Wind Speed	\(\pm 1.5 \ \text{m/s}\) \(\pm 1.5 \ \text{m/s}\)	Instrument Error Sampling Error (Monthly Average)
Wind Stress	\(\pm 10 \%\) \(\pm 14 \ \text{Pa}\)	Drag Coefficient Error Assuming \(10 \ \text{m/s}\) Wind Speed
Insolation	\(\pm 5 \%\) \(\pm 15 \ \text{W/m}^{2}\) \(\pm 10 \%\)	Monthly Average Monthly Average Daily Average
Rain Rate	\(\pm 50 \%\)
Rainfall	\(\pm 10 \%\)	\(5^{\circ} \times 5^{\circ}\) area of TRMM
Net Long Wave Radiation	\(\pm 4-8 \%\) \(\pm 15-27 \ \text{W/m}^{2}\)	Daily Average
Latent Heat Flux	\(\pm 35 \ \text{W/m}^{2}\) \(\pm 15 \ \text{W/m}^{2}\)	Daily Average Monthly Average

Wind Speed and Stress

Stress is calculated from wind observations made from ships at sea and from scatterometers in space as described in the last chapter.

Insolation

Insolation is calculated from cloud observations made from ships and from visible-light radiometers on meteorological satellites. Satellite measurements are far more accurate than the ship data because it’s very hard to measure cloudiness from below the clouds. Satellite measurements processed by the International Satellite Cloud Climatology Project (ISCCP) are the basis for maps of insolation and its variability from month to month (Darnell et al. 1988; Rossow and Schiffer 1991).

The basic idea behind the calculation of insolation is this. Sunlight at the top of the atmosphere is accurately known from the solar constant, latitude, longitude, and time. Sunlight is either reflected back to space by clouds, or it eventually reaches the sea surface. Only a small and nearly constant fraction is absorbed in the atmosphere. But, recent work by Cess et al. (1995) and Ramanathan et al. (1995) suggest that this basic idea may be incomplete, and that atmospheric absorption may be a function of cloudiness. Assuming atmospheric absorption is constant, insolation is calculated from: \[\text{Insolation} = S(1-A) - C \nonumber \]

where \(S = 1365 \ \text{W/m}^{2}\) is the solar constant; \(A\) is albedo, the ratio of incident to reflected sunlight; and \(C\) is a constant which includes absorption by ozone and other atmospheric gases and by cloud droplets. Insolation is calculated from cloud data (which also includes reflection from aerosols) collected from instruments such as the avhrr on meteorological satellites. Ozone and gas absorption are calculated from known distributions of the gases in the atmosphere. QSW is calculated from satellite data with an accuracy of 5–7%.

Water Flux In (Rainfall)

Rain rate is another variable that is very difficult to measure from ships. Rain collected from gauges at different locations on ships and from gauges on nearby docks all differ by more than a factor of two. Rain at sea falls mostly horizontally because of wind, and the ship’s superstructure distorts the paths of raindrops. Rain in many areas falls mostly as drizzle, and it is difficult to detect and measure.

The most accurate measurements of rain rate in the tropics \((\pm 35^{\circ})\) are calculated from microwave radiometers and radar observations of rain at several frequencies using instruments on the Tropical Rain Measuring Mission (TRMM) launched in 1997. Rain for other times and latitudes can be calculated accurately by combining microwave data with infrared observations of the height of cloud tops and with rain gauge data (figure \(\PageIndex{1}\)). Rain is also calculated from the reanalyses weather data by numerical models of the atmospheric circulation (Schubert, Rood, and Pfaendtner, 1993), and by combining ship and satellite observations with analyses from numerical weather-prediction models (Xie and Arkin, 1997).

Figure \(\PageIndex{1}\): Rainfall in m/year calculated from data compiled by the Global Precipitation Climatology Project at NASA’s Goddard Space Flight Center using data from rain gauges, infrared radiometers on geosynchronous meteorological satellites, and the SSM/I. Contour interval is 0.5 m/yr, light shaded areas exceed 2 m/yr, heavy shaded areas exceed 3 m/yr.

The largest source of error is due to conversion of rain rate to cumulative rainfall, a sampling error. Rain is very rare, it is log-normally distributed, and most rain comes from a few storms. Satellites tend to miss storms, and data must be averaged over areas up to 5\(^{\circ}\) on a side to obtain useful values of rainfall.

Net Long-Wave Radiation

Net long-wave radiation is not easily calculated because it depends on the height and thickness of clouds, and the vertical distribution of water vapor in the atmosphere. It is calculated by numerical weather-prediction models or from observations of the vertical structure of the atmosphere from atmospheric sounders.

Water Flux Out (Latent Heat Flux)

Latent heat flux is calculated from ship observations of relative humidity, water temperature, and wind speed using bulk formulas \((\PageIndex{3})\) and ship data accumulated in the ICOADS described below. The fluxes are not calculated from satellite data because satellite instruments are not very sensitive to water vapor close to the sea. Perhaps the best fluxes are those calculated from numerical weather models.

Sensible Heat Flux

Sensible heat flux is calculated from observations of air-sea temperature difference and wind speed made from ships, or by numerical weather models. Sensible fluxes are small almost everywhere except offshore of the east coasts of continents in winter when cold, Arctic air masses extract heat from warm, western, boundary currents. In these areas, numerical models give perhaps the best values of the fluxes. Historical ship report give the long-term mean values of the fluxes.