National Center for Atmospheric Research
EOL Integrated Surface Flux Facility

Corrections to Sensible and Latent Heat Flux Measurements

Calculation of Sensible and Latent Heat Fluxes

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 krypton hygrometer measurements of water vapor density kh2o. However, several corrections must be made to these measured quantities. These are:

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. Note that the specific humidity flux correction to <w'tc'> is important for |B|< 1 or |H|<|LE|.

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.

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)rhod .

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 rhod 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 m3/gm-cm for path lengths of 1.3 and 2.6 cm.

Ignoring the contribution of pressure fluctuations to rhod', we find 

  <w'h2o'> = <w'kh2o'> + Cko(rhod/T)<w't'>

           ~ <w'kh2o'>/(1 - 8CkoB)
where

  Cko = (ko/kw)(CoMo/Ma) = 0.23(ko/kw)

For the krypton hygrometer, ko/kw is on the order of 1-2%, so that Cko is on the order of 0.5% or less. Thus the oxygen correction to <w'kh20'> is important for |B|>1 or |H|>|LE|.

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

With correction for oxygen absorption, the kyrpton hygrometer measures 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 = rhod <w'mr'> 

    = rhoa <w'q'>/(1-Q) 

    = (1 + mu*MR)[<w'h2o'> + (rhov/T)<w't'>]

    ~ (1 + mu*MR)<w'h2o'>(1 + 8QB)

where rhoa and rhov are the densities of (moist) air and water vapor, 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|.

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, 2003)

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/fm)/[1 + (f/fm)2] .

Here f is frequency and fm is the frequency at the peak of the frequency-multiplied cospectrum. Horst (1997) compares this cospectral shape to observed cospectra. With this assumption, the correction for flux attenuation is 

  A = exp(2 pi fmS/U)

    = exp(2 pi nmS/z)

where S is the sensor displacement, U is mean wind speed, nm = fmz/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 nm, it can be seen that the spatial separation correction is small for S/z << 1 .

We have fit the Horizontal Array Turbulence Study (HATS) data with the following (preliminary, tentative, etc. as of 10/28/03) empirical formulas for nmx and nmy, the values of nm in the streamwise and crosswind directions, respectively, 

  nmx = 0.065 ,                           z/L < -0.1

      = 2.16 - 2.095/(1.015 + 0.15z/L)2 , z/L > -0.1

and 

  nmy = 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, nmy > nmx 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 nm on wind direction,

  nm = sqrt[(nmx cos(dir))2 + (nmy 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.

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. 2003. Attenuation of scalar fluxes measured with displaced sensors. EGS-AGU-EUG Joint Assembly, April 6-11, 2003, Nice, France.

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'>: 

   <w't'> = [<w'tc'> - Ctc(T/rhoa)A <w'kh2o'>]/
                 [1 + Ctc(Q + Cko(1-Q))]

  <w'mr'> = (1+mu*MR)[(A/rhod)<w'kh2o'> + ((Cko+MR)/T)<w'tc'>]/
                 [1 + Ctc(Q + Cko(1-Q))]
where

  Ctc = 0.51[1 + Q(mu-1)]

Last modified: Tuesday, 28-Oct-2003
© NCAR/Earth Observing Laboratory
This page was prepared by Tom Horst, NCAR ATD Research Technology Facility