Modified ogive analysis

The ogive analysis was introduced by Desjardins et al. (1989) and Oncley et al.

(1990) for investigating the flux contribution from each frequency range and deter-mining the suitable averaging periods that can capture most of the turbulent fluxes.

The ogive function of the turbulent flux (ogw,c) is defined as the cumulative integral of the cospectrum of the turbulent flux (Cow,c) starting with the highest frequency.

ogw,c(f0) = Z f0

∞

Cow,c(f)df, (4.3)

where w is the vertical wind velocity, c is a horizontal wind velocity or a scalar quantity like temperature and humidity, andf is a frequency, which corresponds to a time period (τ) as

τ = 1

f. (4.4)

This analysis was once applied to data measured over the maize field (A6) of the LITFASS-2003 experiment in Foken et al. (2006), where the ogive function was

calculated from the raw 20 Hz data over 4-hour period and mainly focused on 3 selected days (7-9 June 2003). This article shows that the ogive curves can be classified into three cases. Case 1, where the ogive curve exhibits the asymptotic behavior toward the low frequency within 30-minute period. This indicates that the 30-minute averaging time is sufficient to capture most of the turbulent fluxes. Case 2, in which the ogive curve shows the extreme value (peak) within 30-minute period, this means the total turbulent fluxes has been reached before 30 minutes. Hence the longer averaging time obviously reduce the flux and the period shorter than 30 minutes would be enough to capture most of the turbulent fluxes. Case 3, in which the ogive curve does not converge within 30-minute period. This implies that there is a significant contribution from the low frequency part of the turbulent spectrum and 30-minute averaging time is not enough to capture most of the fluxes.

10⁻⁵

Figure 4.1: The short term average time series can estimate the turbulent flux at a 30-minute period (F³⁰) and its evolution after that (gray solid lines in a gray band). The error band of width 2η (gray band) was defined for identifying the ogive case. See Table 4.1 for ogive case definition.

To apply the ogive analysis to data from all EC towers of the LITFASS-2003 experiment, the raw high-frequency data are required. Nevertheless, this type of data is not available in all EC towers. Only the short-term statistics at every 5 or 10 minutes exists for all sites, in which turbulent fluxes at a longer period can be estimated with Eq. 2.18. In this thesis, the modified ogive analysis (MOG) was developed to deal with this short-term statistics data. According to the spectral analysis, the spectra calculated from high and low frequency data behave similarly in the low frequency region (Kaimal et al., 1972). Hence, the turbulent spectra calculated from the short-term statistics data can be used to estimate the change in

the turbulent fluxes after 30 minutes without any information prior to 30 minutes.

The turbulent cospectra of the short-term average statistics were calculated with a standard Fast Fourier Transform (FFT) method. To avoid influences from the diurnal effect, the time extension was kept up to 4 hours as in Foken et al. (2006).

The turbulent fluxes change after 30 minutes were determined from the cumulative integral of the cospectra starting from the frequency, which corresponds to a period of 30 minutes. Then its maximum value was set to be the maximum flux difference (∆max),

The ogive curves classification was done by comparing ∆max with the turbulent flux at 30 minutes (F30). F30 could be estimated in two different ways. The first ap-proach, time averaged fluxes of each 30-minute block (w^′c^′)j were averaged together as

The second approach,F30 was determined from the difference between a total flux over 4-hour period (F4hr) and the turbulent flux after 30-minute period (Fτ >30),

F30 =F4hrs−Fτ >30. (4.7)

F4hr was calculated from short term average data with the help of Eq. 2.18, while Fτ >30 was calculated from the cumulative integral of the cospectra from the lowest frequency (fmin) to the frequency corresponding to 30-minute period,

Fτ >30 =

Z τ=30

fmin

Cow,c(f)df. (4.8)

Both estimations in Eq. 4.6 and Eq. 4.7 gave quite compatibleF30. The error bar of width 2η was then set around the turbulent flux at 30-minute period (Fig. 4.1). If

∆max was still confined in this band, it indicated that the turbulent flux difference after 30 minutes was not significant, which conformed to case 1 in Foken et al. (2006).

If ∆max exceeded this band, this meant the turbulent flux difference was significant and could be classified into 2 cases depending on the change of turbulent fluxes after 30-minute period. It was equivalent to case 2 in Foken et al. (2006), when the size of turbulent fluxes decreased; and case 3, when the size of turbulent fluxes increased.

The size of an error band η was set to be 10% (or 20%) of the turbulent flux at

30-minute period, which must not be smaller than the measurement errors of each turbulent flux (section 3.2.3). The ogive case definition in analogy to Foken et al.

(2006) is shown in Table 4.1.

Table 4.1: Ogive case definition in analogy to Foken et al. (2006). ∆max is a maximum flux difference after 30-minute averaging time. F30 is the turbulent flux at 30-minute period. η is the width of an error band.

Case Criterion 1 ∆max/F30≤η

2 ∆max/F30> η and ∆max<0 3 ∆max/F30> η and ∆max>0

In this thesis, the MOG was applied to all listed stations in Table 3.1, except A1 and A2, because their data were not always available. The period of investigation also covers the entire period of the LITFASS-2003 experiment (section 3.2.3). For the energy balance component, the MOG was applied to all listed sites. For the CO2flux, the MOG was only applied to the sites with the LI-7500 hygrometer, except the lake which has very low concentration of CO2. Note that none of flux corrections were applied in the MOG. Since each point of the cospectra corresponds the turbulent flux at different duration, the choice of suitable duration for the flux corrections would be ambiguous. According to Mauder and Foken (2006), flux corrections would reduce the residual by 17%. Therefore, this reduction might be assumed to reflect in the increasing of QH and QE.