A description of ISFS calculation of sensible and latent heat flux, including several corrections.

### Introduction

The sensible and latent heat fluxes, `H` and `LE`, can be obtained from sonic anemometer measurements of vertical velocity `w` and temperature `Tc` and fast-response hygrometer measurements of water vapor density `h2o`. However, several corrections must be made to these measured quantities. These are:

- correction of the sonic temperature, derived from the speed of sound, for the effect of moisture
- correction of krypton hygrometer data for UV absorption by oxygen
- application of the WPL correction to the water vapor flux
- correction for spatial separation between the sonic and krypton hygrometers

### Sonic Temperature

The sonic (speed of sound) temperature is, `Tc = T(1 + 0.51Q)`, where `T` is absolute temperature and `Q` is specific humidity. Thus

<w't'> = <w'tc'> - 0.51T <w'q'> ~ <w'tc'>/(1 + 0.06/B)

where `B = H/LE` is the Bowen ratio. Therefore the specific humidity flux correction to `<w'tc'>` is important for `|B|<1` or `|H|<|LE|`.

Note that the following references also discuss a correction to sonic temperature for the wind component normal to the sonic measurement path. We do not include this correction here because most sonic anemometers make this correction to the sonic temperature (or speed-of-sound) data internally.

**References***:*

*Schotanus, P., F.T.M. Nieuwstadt, and H.A.R. DeBruin, 1983. Temperature measurement with a sonic anemometer and its application to heat and moisture fluctuations. Boundary-Layer Meteorol, 26, 81-93.*

*Kaimal, J.C., and J.E. Gaynor, 1991. Another look at sonic thermometry. Boundary-Layer Meteorol, 56, 401-410.*

### Oxygen Correction

The krypton hygrometer, KH2O, measures UV absorption by both water vapor and oxygen. In order to obtain the water vapor density, the KH2O measurements must be corrected for absorption by oxygen,

`h2o = kh2o - (ko/kw)(CoMo/Ma)rho _{d}`

where `ko` and `kw` are the KH2O extinction coefficients for oxygen and water vapor, `Co=`0.21 is the percent concentration of oxygen in the atmosphere, `Mo=`32 and `Ma=`28.97 are the molecular weights of oxygen and dry air, and `rho _{d}` is the density of dry air. Van Dijk et al (2002) have shown that

`ko`depends on the path length of the krypton hygrometer. Their Table 2 provides values of 0.0034 and 0.0013 m

^{3}/gm-cm for path lengths of 1.3 and 2.6 cm.

Ignoring the contribution of pressure fluctuations to `rho _{d}'`, we find

<w'h2o'> = <w'kh2o'> + C_{ko}(rho_{d}/T)<w't'> ~ <w'kh2o'>/(1 - 8C_{ko}B)

where

`C _{ko} = (ko/kw)(CoMo/Ma) = 0.23(ko/kw) `

For the krypton hygrometer, `ko/kw` is on the order of 1-2%, so that `C _{ko}` is on the order of 0.5% or less. Thus the oxygen correction to

`<w'kh20'>`is important for

`|B|>1`or

`|H|>|LE|`.

Note that IR-absorption (e.g. Licor) fast-response hygrometers measure extinction in a wavelength band that is not sensitive to oxygen absorption. The following formulas can be applied to IR hygrometer measurements by simply setting ko (or C_{ko}) equal to zero and ` <w'kh2o'>=<w'h2o'>`.

**Reference***:*

*van Dijk, A., W. Kohsiek, and H.A.R. DeBruin, 2003. Oxygen sensitivity of krypton and Lyman-alpha hygrometers. J. Atmos. Oceanic Tech., 20, 143-151.*

### WPL Correction

The kyrpton (corrected for water vapor absorption) and IR-absorption hygrometers measure water vapor density `h2o`. However, Webb et al (1980) show that the vertical flux of water vapor density is not equal to `<w'h2o'>`, but rather

E = rho_{d}<w'mr'> = rho_{a}<w'q'>/(1-Q) = (1 + mu*MR)[<w'h2o'> + (rho_{v}/T)<w't'>] ~ (1 + mu*MR)<w'h2o'>(1 + 8QB)

where `rho _{a}` and

`rho`are the densities of (moist) air and water vapor,

_{v}`MR`and

`mr'`are the mean value and fluctuations of mixing ratio,

`Q`and

`q'`are the mean value and fluctuations of specific humidity, and

`mu=Md/Mw=`1/0.622 is the ratio of the molecular weights of dry air and water vapor. Since near-surface values of

`Q`rarely reach 50 gm/kg, the WPL correction is generally important only for

`|B|>1`or

`|H|>|LE|`.

**Reference***:*

*Webb, E.K., G.I. Pearman, and R. Leuning, 1980. Correction of flux measurements for density effects due to heat and water vapor transfer. Quart. J. Roy. Meteorol. Soc., 106, 85-100.*

### Spatial Separation of Eddy Covariance Sensors (Horst and Lenschow, 2009)

In order to reduce flow distortion, the krypton hygrometer (or other scalar sensor) is commonly displaced spatially from the sonic anemometer. This spatial displacement reduces the correlation between the measurements of vertical velocity and scalar concentration. Kristensen et al (1997) find that the corresponding reduction of the measured flux is minimized if the scalar sensor is placed below the anemometer. However, Wyngaard (1988) showed that "cross talk" error caused by flow distortion is eliminated if the sensor configuration has reflection symmetry about a horizontal plane through the measuring point, which requires a horizontal displacement of the scalar sensor.

In the case of a horizontal separation, a simple formula for the flux loss can be derived by assuming

that the cospectrum has approximately the form

`fCo(f) = <w'h2o'>(2/pi)(f/f _{m})/[1 + (f/f_{m})^{2}]`

Here `f` is frequency and `f _{m}` is the frequency at the peak of the frequency-multiplied cospectrum. Horst (1997) compares this cospectral shape to observed cospectra, which agree better for stable stratification than for unstable stratification. With this assumption, the

**correction factor**for flux attenuation is

A = exp(2 pi f_{m}S/U) = exp(2 pi n_{m}S/z)

where `S` is the sensor displacement, `U` is mean wind speed, `n _{m} = f_{m}z/U`, and

`z`is the measurement height. This formula provides a reasonable fit to the data of Kristensen et al. (1997), Figure 5, for scalar flux attenuation due to horizontal sensor displacement. With the following formulas for

`n`, it can be seen that the spatial separation correction is small for

_{m}`S/z << 1`.

We have fit the Horizontal Array Turbulence Study (HATS) data with the following empirical formulas for `n _{mx}` and

`n`, the values of

_{my}`n`in the streamwise and crosswind directions, respectively,

_{m}n_{mx}= 0.07, z/L < -0.1 = 2.31 - 2.24/(1.015 + 0.15z/L)^{2}, z/L > -0.1

and

n_{my}= 0.15, z/L < -0.05 = 2.43 - 2.28/(1.01 + 0.2z/L)^{2}, z/L > -0.05

where `L` is the Obukhov length. As might be expected, `n _{my} > n_{mx}` due to the stretching of eddies by vertical shear in the streamwise direction.

We use the suggestion of Lee and Black (1994) for the dependence of `n _{m}` on wind direction,

`n _{m} = sqrt[(n_{mx} cos(dir))^{2} + (n_{my} sin(dir))^{2}]`

where `dir` is the wind direction relative to the direction of sensor separation (`dir =` 0° for wind direction parallel to sensor separation; 90° for wind direction normal to sensor separation). Lee and Black base this formula on the assumption of an elliptical shape for the turbulent eddies in the horizontal plane and show that it is consistent with their data set for unstable stratification.

**References***:*

*Horst, T.W., 1997. A simple formula for attenuation of eddy fluxes measured with first-order response scalar sensors. Boundary-Layer Meteorol., 82, 219-233.*

*Horst, T.W., and D.H. Lenschow, 2009. Attenuation of scalar fluxes measured with spatially-displaced sensors. Boundary-Layer Meteorol., 130, 275-300.*

*Kristensen, L., J. Mann, S.P. Oncley, and J.C. Wyngaard, 1997. How close is close enough when measuring scalar fluxes with displaced sensors? J. Atmos. Oceanic Technol., 14, 814-821.*

*Lee, X., and T.A. Black, 1994. Relating eddy correlation sensible heat flux to horizontal sensor separation in the unstable atmospheric surface layer. J. Geophys. Res., 99(D9), 18,545-18,553.*

*Wyngaard, J.C., 1988. Flow-distortion effects on scalar flux measurements in the surface layer: Implications for sensor design. Boundary-Layer Meteorol., 42, 19-26.*

### Sensible and Latent Heat Fluxes

The sensible and latent heat fluxes can be calculated from the basic measurements of `<w'tc'>` and `<w'kh2o'>` by solving the preceding set of equations for `<w't'>` and `<w'mr'>.` (As noted previously, IR-absorption hygrometers are insensitive to oxygen absorption, so that C_{ko}=0 and ` <w'kh2o'>=<w'h2o'>`).

<w't'> = [<w'tc'> - C_{tc}(T/rho_{a})A <w'kh2o'>] / [1 + C_{tc}(Q + C_{ko}(1-Q))] <w'mr'> = (1+mu*MR)[(A/rho_{d})<w'kh2o'> + ((C_{ko}+MR)/T)<w'tc'>] / [1 + C_{tc}(Q + C_{ko}(1-Q))]

where

`C _{tc} = 0.51[1 + Q(mu-1)]`