*3.2. Spectral Behaviour of the Time Series*

The normalized FFT power spectra of the complete time series of fluctuations provide information on the occurrence and the strength of the diurnal and semi-diurnal cycles with respect to the annual oscillation. Only in the case of ILW, the semi-annual oscillation is stronger than the annual oscillation, and so we normalize ILW by the power of the semi-annual oscillation. The FFT power spectra mainly provide information on the phase-locked fluctuations, e.g., phase-locked to the daily or annual cycle of solar radiation. Thus, many intermittent short-term fluctuations may average out by taking the spectrum over the time interval 2004–2016. This is the reason why we show next the amplitude spectrum obtained by a wavelet-method.

Figure 3 shows the normalized FFT power spectra in a blue colour for the fluctuations of IWV, ILW, CF1, CF2, CF3 and CF4 above Bern from 2004 to 2016. The annual oscillation is the blue spike at 1/(365 day), which is near 0 cycles/day. Compared to the power of the annual oscillation in IWV, the diurnal cycle in IWV power is about 400 times smaller. This means that the amplitude of the diurnal cycle is about 0.4 kg/m2, which is 20 times smaller than the annual cycle of IWV, which is about 8 kg/m<sup>2</sup> in Figure 2. Furthermore, the amplitude of the diurnal cycle is about five times larger than the amplitude of the semi-diurnal cycle in IWV. The magnitudes of the annual and diurnal cycle in IWV are consistent with those of [25].

The amplitudes of the diurnal and semidiurnal oscillation in IWV are known to 5 *σ*-level confidence. This is indicated by the red line, which lies at a 5 *σ* distance above the yellow line, which is the mean power of the blue spectra sampled over 1000 frequency grid points. Here, we assumed that the mean power is equal to the noise *σ*.

**Figure 3.** Fast Fourier Transform (FFT) power spectra (blue line) of the temporal fluctuations of IWV, ILW and CF in Bern for the time interval from 2004 to 2016. The spectra are normalized by the power of the maximum (power of the annual or semi-annual oscillation). The yellow line is the mean of the spectrum averaged over 1000 frequency grid points. The red line is the five sigma level of confidence. A significant diurnal cycle shows up as a blue spike at 1 cycle/day in each parameter. A significant semidiurnal cycle (at 2 cycles/day) is present for IWV and CF4.

In the case of ILW, we see a peak of the diurnal cycles that is rather close to those of the semi-annual oscillation, which is the dominant oscillation. There are several other significant oscillations with peaks closely above the red line, e.g., the semi-diurnal oscillation. In the cases of CF1, CF2, CF3 and CF4, we find significant diurnal cycles. The semi-diurnal cycle is well present for CF2 and CF4.

As a supplement to the FFT power spectra, we derive amplitude spectra by means of the fast response bandpass filter that takes care of phase-unlocked oscillations, which may persist only about time intervals of a few wave periods. Figure 4 shows the short-term variability of IWV, ILW, CF1, CF2, CF3 and CF4. An unresolved small peak is seen at the position of the diurnal cycle, which is marked by the red vertical line. The amplitude of the diurnal cycle in IWV is about 1 kg/m2, which is larger than those obtained by the FFT power spectrum. This is reasonable since the phase of the diurnal cycle may change in time, and phase-unlocked intermittent oscillations with a one day period may occur as well. In case of the thin liquid water clouds (CF1), we can see that the maximal amplitudes are reached for periods smaller than one day. This is likely since small and thin clouds have a short life time and short horizontal scales. In the cases of IWV, ILW, CF3 and CF4, there are relatively strong oscillations with periods from two to ten days, which might be related to changes in synoptic weather patterns. The peak amplitude of IWV is around the seven-day period, whereas, for clouds, the peak is for shorter periods.

**Figure 4.** Amplitude spectra of IWV, ILW and CF over Bern for the time interval 2004–2016 obtained by a fast response bandpass filter. The red line marks the position of the diurnal cycle. The short-term variability of fluctuations with periods <10 days is high for ILW and CF.

#### *3.3. Seasonal Dependence and Annual Mean of the Diurnal Cycle in IWV*

Figure 5 shows the absolute and the relative diurnal cycles in IWV with respect to the monthly mean IWV. The subtraction of the monthly mean is necessary for the intercomparison of the seasonal curves since the monthly mean strongly varies from 8 kg/m2 in winter to 24 kg/m2 in summer. The absolute diurnal cycle is shown in the upper panel and is defined as ΔIWV = IWV − IWV. The relative diurnal cycle is shown in the lower panel and is defined as ΔIWV = (IWV− IWV)/ IWV. The seasonal curves of the diurnal cycles are given in color while the black dots denote the annual mean of the diurnal cycle. The amplitude of the mean diurnal cycle is 0.41 kg/m2. Morland, J. et al. [25] obtained a quite similar curve for the mean diurnal cycle in IWV using measurements of TROWARA from 2003 to 2007. They found a mean amplitude of 0.32 kg/m2. The phase of the diurnal cycle with a maximum around 19:00 LT in Figure 5 was also found by [25]. In addition, they compared the TROWARA results with the mean diurnal cycle of a GPS station in Bern, and they found an excellent agreement for the amplitude and the phase of the diurnal cycle.

Morland, J. et al. [25] suggested that evaporation of soil moisture into the atmosphere may explain the shape of the diurnal cycle in IWV. Accumulation of the evaporated water in the atmosphere during the daytime leads to the maximum of IWV in the evening while accumulated condensation of water vapour during the nighttime induces the IWV minimum in the morning hours. The hydrological atlas of Switzerland shows that the daily evaporation rate ranges from about 0.1 kg/m2 per day in winter to about 3.8 kg/m2 per day in summer in the Swiss plateau, which is dominated by agriculture [34]. Since the surroundings of Bern also have forests and a river, we expect higher evaporation rates of about 0.5 kg/m2 per day in winter [34]. The annual mean of the rate of change of IWV in Figure 5 is about 0.82 kg/m<sup>2</sup> from the morning to the evening, which lies within the range of values of the evaporation rate in Bern. Thus, we regard the diurnal cycle of insolation, evaporation and condensation as the main reason for the observed diurnal cycle in IWV. Other factors such as diurnal variations of the surface wind vector and the vertical mixing rate of moisture lead to additional modifications of the diurnal cycle in IWV.

**Figure 5.** Seasonal dependence of the diurnal cycle in ΔIWV as a function of local time over Bern for the time interval 2004–2016. The upper panel shows the absolute diurnal cycle (ΔIWV = IWV − monthly mean of IWV) while the lower panel shows the relative diurnal cycle with respect to the monthly mean and in percent. The black dots indicate the annual mean of the seasonal curves.

Analyzing a GPS ground station network in Spain, Ortiz de Galisteo, J.P. et al. [35] found an amplitude of about 0.35 kg/m2 for the mean diurnal cycle in IWV of all stations. The time of the maximum is at 18:30 UTC (which is close to local time in Spain). These values agree well with TROWARA in Bern. However, we find a significant difference if we look at the time of the minimum of IWV. It is about 10:00 LT for TROWARA in Bern while the IWV minimum is about 05:00 LT for the GPS network in Spain.

Ortiz de Galisteo, J.P. et al. [35] found a semi-diurnal oscillation in IWV with an amplitude of 0.13 kg/m2, which is consistent with our FFT power spectra in Figure 3. Although the mean diurnal cycle of the Spanish stations agrees quite well with TROWARA in Bern, the Spanish stations showed a remarkable variation amongst themselves. Ortiz de Galisteo, J.P. et al. [35] explained that the shape of the diurnal cycle depends on factors that cause condensation, evaporation and moisture transport. The diurnal cycle in solar radiation is possibly most important for the observed diurnal cycle in IWV since evaporation of water from the surface, increase of air temperature and the growth of the planetary boundary layer during the daytime are closely related to absorption of solar radiation. On the other hand, the decrease of air temperature during the night leads to enhanced condensation of water vapour and to the observed minimum of IWV in the early morning hours. The time lag of the IWV minimum and maximum with respect to those of the air temperature might be related to the slow accumulation processes of liquid water on the ground and water vapour in the air.

Dai, A. et al. [36] retrieved diurnal cycles in IWV from a GPS station network in North America and found amplitudes of 1.0–1.8 kg/m<sup>2</sup> in the summer season and weaker in other seasons, which is greater than the 0.8 kg/m<sup>2</sup> June and August amplitudes of TROWARA in Figure 5. There is a large variability in the diurnal IWV cycles of the North American stations, but there are several stations that show an increase in IWV during the afternoon hours and a maximum around 16:00–19:00 LT, which is similar to that of TROWARA in Bern. The time of the IWV minimum is often late in the morning hours from 07:00–10:00 LT. Generally, Dai, A. et al. [36] reported relative diurnal IWV cycles with amplitudes less than 5%, while Figure 5 shows maximal relative amplitudes of 4%. In the percentage scale, the seasonal curves come closer together and the strong absolute diurnal cycles of the summer months do not differ much from those of March, April and May. The smallest relative amplitudes (<2%) occur in winter from November to January in Bern.
