An Assessment of Eddy-Covariance-Based Surface Fluxes Above an Evaporating Heated Surface Under Fair-Weather Daytime Conditions

Abstract

Above an evaporating heated surface under fair-weather daytime conditions, the cospectra between the vertical velocity component, temperature, and water-vapour mixing ratio should be positive. We have applied a multi-resolution technique to a 3.64-h long, 10-Hz time series centred at midday for 16 fair-weather days at a mid-latitude site during spring to measure the averaging period τ_c at which the crossover from the domain of the three positive cospectra to a mixed-sign domain occurs. The τ_c values broadly range from 9 to 42 min, with 13 of the 16 days having values less than 30 min. When mesoscale circulations induced by surface heterogeneity are likely to be present, the vertical heat (or moisture) flux computed with the conventional averaging period of 30 min τ₃₀ is as large (or small) as 1.09 (or 0.78) times that using τ_c. However, on 14 (or 13) days, the vertical heat (or moisture) fluxes using the period τ_c are explained by those calculated with the period τ₃₀ within a difference range of ± 1%. The insignificant difference is due to the insensitivity of the fluxes to the averaging period at scales larger than approximately 7 min. Therefore, despite a broad range of τ_c values, the 30-min-averaged surface fluxes can be treated as the required turbulent fluxes. Although this finding is not robust, given that data were collected at one location over 16 days, it supports the use of 30-min-averaged surfaces fluxes, particularly for the composite midday fluxes on fair-weather days.

Appendices

Appendix 1: Non-stationarity Ratio

Some readers may be concerned about the non-stationarity of the 3.64-h long (2¹⁷ 10-Hz data points) time series, since stationarity is a necessary condition to estimate ensemble-averaged fluxes satisfying the surface energy balance. However, we focus on the measurement of the cut-off time scale that separates turbulent eddies from significant non-turbulent physical processes.

For each day, the 2¹⁷ 10-Hz data points are partitioned into 2¹⁵-point (about 0.91 h) records, where the number of records (I) is 2¹⁷⁻¹⁵ = 2² and each record is further divided into J segments. In fact, depending on the segment length τ, the number of segments varies. For example, the number of 2¹⁴-point (about 27.3 min) segments is two in a 2¹⁵-point record.

For each day of the 16 fair-weather days, with four 2¹⁵-point records, we compute the non-stationarity ratio (NR) of Mahrt (1998),

$$ N\!R \equiv \frac{{\sigma_{btw} }}{R\!E}, $$

(17)

and the random error

$$ R\!E = \frac{{\sigma_{wi} }}{\sqrt J }. $$

(18)

In Eq. 18, the within-record standard deviation σ_wi of the flux for the ith record is computed as

$$ \sigma_{wi} \left( i \right) = \sqrt {\frac{1}{J - 1}\mathop \sum \limits_{j = 1}^{J} \left[ {F\left( {i,j} \right) - \bar{F}\left( i \right)} \right]^{2} } , $$

(19)

where F(i, j) is the heat or moisture flux for the jth segment of the ith record, and $\bar{F}\left( i \right)$ is the average of the segment fluxes for the ith record. In Eq. 17, the between-record standard deviation σ_btw of the flux is

$$ \sigma_{btw} = \sqrt {\frac{1}{I - 1}\mathop \sum \limits_{i = 1}^{I} \left( {\bar{F}\left( i \right) - \bar{F}} \right)^{2} } , $$

(20)

where $\bar{F}$ is the segment flux averaged over all of the segments and records.

Figure 6 presents the computed NR values as a function of segment length τ (2¹¹ ≤ τ ≤ 2¹⁴ in 10-Hz data points; $3.4 \le \tau \le 27.3$ min) for each day. The NR value increases to slightly larger than four with τ = 13.7 min on 31 March 2015. However, except for this case, the other values are all smaller than four and further, most values are smaller than two. Mahrt (1998) states that when NR < 2, the residuals of the surface energy budget are less than 20%, but rapidly grow to about 40% and 80% when NR = 3 and 6, respectively.

Fig. 6

Nonstationary ratio (NR) for each day of the 16 fair-weather days with four 0.91-h records obtained from the 3.64-h time series centred at midday

Appendix 2: Cospectra as Functions of the Normalized Frequency

In the literature, numerous studies (e.g., Kaimal 1978; Roth et al. 1989; Kaimal and Finnigan 1994) present cospectra as a function of the normalized frequency f ≡ 2πz/τV, where V is the wind speed, and z is the measurement height, where z = 5 m above ground level here. Our purpose is to investigate whether the 30-min-averaged surface fluxes are appropriate to be treated as turbulent fluxes. Thus, in Fig. 4, the composite cospectra are obtained as a function of the averaging time scale τ.

For the comparison with previous studies, we present the cospectra as a function of the normalized frequency f. As demonstrated in Table 1, the wind speed V averaged over the 3.64-h period centred at the midday varies for each day. With a higher (or lower) mean wind speed, the normalized frequency becomes relatively lower (or higher). For example, on 16 March 2015 when the mean wind speed of 3.82 m s⁻¹ is the strongest, the cospectra are shifted into the lowest frequency range from 8.2 × 10⁻¹ to 1.3 × 10⁻⁵. In contrast, on 11 May 2015, when the mean wind speed of 0.96 m s⁻¹ is the weakest, the cospectra into the highest frequency range from 3.3 × 10⁰ to 5 × 10⁻⁵. Thus, the composite cospectra over the 16 days are obtained from the highest f of 3.3 × 10⁰ on 16 March to the lowest f of 5 × 10⁻⁵ on 11 May. The composite cospectra of wT and wr normalized with the heat and moisture fluxes both have the cospectral peak at f = 1.3 × 10⁻². The spectral peak of the w component is located at 5 × 10⁻², and the peaks of T and r at 5 × 10⁻⁵. Compared with previous studies (e.g., Kaimal 1978; Roth et al. 1989; Kaimal and Finnigan 1994), the cospectral peaks of ww, wT, and wr are located within the reference ranges, although the spectral peak of the w component is slightly shifted to a higher frequency. In addition, in Fig. 7c, the spectra of T and r have the mesoscale spectral peak at the lowest frequency of f = 5 × 10⁻⁵, but the turbulence peak at f = 1 × 10⁻², which also matches results of Kaimal and Finnigan (1994).

Fig. 7

As Fig. 4 but as a function of the normalized frequency f ≡ 2πz/τM, where z is the measurement height of 5 m, M the wind speed averaged over the 3.64-h period centred at the midday for each day of the 16 fair-weather days (Table 1), and τ is the averaging period. At each value of f from $3.3 \times 10^{0} \left( {{\text{the highest }}\,f\, {\text{on }}\,11\, {\text{May}}} \right)$ to $5 \times 10^{ - 5} \left( {{\text{the lowest }}\,f\, {\text{on }}\,16\, {\text{March}}} \right)$, the standard deviation range of the cospectra over the 16 days is marked with colour shading