## 1. Streamflow Tidal Variations

[2] Research on the lunar tidal oscillations for streams has strongly focused on near-coastal estuaries with the implied assumption that only streams, or portions of streams, close to coastal areas will experience strong, identifiable tidal activity. If tidal influence can be identified far from the ocean, the periodicities inherent in tidal forcing may aid in the forecasting of flooding as well as in water management and power generation issues. This supposition that only near-coastal streams demonstrate strong tidal responses can be tested through study of a comprehensive database of all stream gauges across the conterminous United States. The United States Geological Survey Surface-Water Data for the Nation [*U.S. Geological Survey*, 2001] includes daily values for 24,964 sites. This is the same type of data from which a study demonstrated a strong diurnal signal in streamflow for rivers in the western United States linked primarily to diurnal snowmelt [*Lundquist and Cayan*, 2002]. From this extensive streamflow dataset, we extracted daily values for streamflow in cubic meters per second across the conterminous United States for the length of their record. We limited the analysis to streamflow gauges classified as being inland river stream gauges, not tidal stream gauges, that is, these stations were not located at the estuary of a river and thereby not experiencing direct oceanic tidal forcing.

[3] From this long record across the conterminous United States, we standardized each station’s daily streamflow values to its computed long-term mean and standard deviation. We selected only stations in which at least thirty years of daily data from 1900 to 2009 exist, for a total of 10,994 stations in eighteen hydrological basins (Figure 1). We then calculated the lunar phase for each day coinciding with the streamflow record by determining the average angular difference between the apparent longitudes of the moon and sun for that day [e.g., *Meeus*, 1991]. The cycle of lunar-phase magnitudes varies over the lunar synodic cycle of 29.53 days. We subdivided the phase data into ten classes, hereafter termed lunar synodic decimals as denoted in previous research [*Bradley et al.*, 1962; *Brier and Bradley*, 1964; *Hanson et al.*, 1987]. The lunar synodic decimal advances about 0.03 per day. A ten-unit moving total of a distribution within successive classes, each 0.1 in width, therefore, equates roughly to a 3-day moving total [*Bradley et al.*, 1962].

[4] Each standardized daily streamflow value was classified by the associated lunar synodic decimal value. The composited average result of all stations across the conterminous United States along with the associated standard deviation of the mean is shown in Figure 2. The interesting and unexpected result is that a clear tidal response is evident such that the biggest tidal impact on streamflow across the United States is evident during and just after the quarter moon (halfway between full and new moons). When harmonic analysis is applied to the lunar-categorized data, the variance explained by the second harmonic (a two-peak distribution over the course of a lunar month) is r^{2} = 0.5546 (p < 0.01). The total variance explained by the first and second harmonics is r^{2} = 0.8924 (p < 0.001).

[5] For the periods of maxima in Figure 2 (lunar synodic decimal classes 2,3,7,8 and 9) and minima (lunar synodic decimal classes 1,4,5,6, and 10), the means of the standardized streamflow values were compared via a two-sample t-test. Additionally, the significance of the coincidence between above median streamflow values and the maxima was determined through a 2 × 2 chi-square test for independence [*Hollander and Wolfe*, 1999]. The results from both tests were extremely significant with *χ*^{2} = 635.1 and a t-value of 29.6. As both these statistical tests require independence of observations (likely not the case for the streamflow data due to spatial autocorrelation) we also calculated two-sided pseudo-significance values through Monte Carlo techniques similar to *Wolter et al.* [1999] by performing the above tests 9,999 times, each with a random assignment of the ten lunar decimal classes. Significance was determined from these empirical probability density functions as (K + 1)/1000 where K is the number of random trials producing test statistics more extreme than those originating from the actual lunar synodic decimal class assignments. Five runs of the Monte Carlo experiments yielded pseudo p-values of 0.0466 to 0.0496 for the chi-square test and from 0.0390 to 0.0414 for the t-test suggesting that streamflow values for the United States are highest during and just after the quarter moon (halfway between full and new moons).

[6] The strength of the inland tidal influence on streamflow can be demonstrated visually (Figure 3) by showing the amount of variance (r^{2}) explained by the second harmonic for the lunar synodic month. A large number (2590, or 23.6 percent of all stations) of inland USGS streamgauge stations—some as far inland as the upper Midwest—display explained variances by a second-order harmonic for the synodic month which are statistically significant at the 0.05 confidence level. This reflects a double-peaked cycle with maxima in both the period between new and full moons and the period between full and new moons. The question can immediately be raised as to why such a lunar phase signal is evident in inland streamflow data.