The contribution to tidal asymmetry by different ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, C12007, doi:10.1029/2011JC007270, 2011

The contribution to tidal asymmetry by different combinations of tidal constituents Dehai Song,1,2 Xiao Hua Wang,1 Andrew E. Kiss,1 and Xianwen Bao2 Received 5 May 2011; revised 8 September 2011; accepted 22 September 2011; published 7 December 2011.

[1] We provide a general framework for identifying the constituents responsible for asymmetry in any tidal time series, by extending and generalizing the skewness-based approach of Nidzieko (2010) to include any number of tidal constituents. We show that this statistic has two features which greatly simplify the attribution of asymmetry to particular constituents: (1) only combinations of two or three constituents can contribute to skewness, regardless of how many constituents are significant in the time series and (2) of those combinations, only the few meeting the frequency conditions 2w1 = w2 or w1 + w2 = w3 will give rise to long-term mean asymmetry. It is therefore relatively easy to identify every such combination, even when many constituents are present. We then go on to show how the relative contribution of each such combination can be measured and compared, based on the amplitudes, frequencies and relative phases of the constituents. We also show that there is an upper bound to the skewness generated by any such combination. The metrics are applied to data from 335 worldwide sea level stations and from a global ocean tidal model based on TOPEX/POSEIDON altimetry. Global maps are made of the patterns of tidal skewness. We identify the combinations of astronomical tides that dominate skewness in different tidal regimes and geographic locations, and explain the dependence of skewness on tidal form number. Citation: Song, D., X. H. Wang, A. E. Kiss, and X. Bao (2011), The contribution to tidal asymmetry by different combinations of tidal constituents, J. Geophys. Res., 116, C12007, doi:10.1029/2011JC007270.

1. Introduction [2] Tidal asymmetry refers to a difference between the durations of rising and falling tides, as well as a difference in the duration and magnitude of flooding and ebbing currents [Speer et al., 1991]. The generation of M4 and its addition to the M2 constituent is often the main cause of tidal asymmetry in shallow water [Aubrey and Speer, 1985; Speer and Aubrey, 1985], but several other combinations of constituents can also produce an asymmetrical tide. In the semi-diurnal regime, these include the overtide constituent M6 (M2 + M4 + M6) [Blanton et al., 2002] and the compound constituents MS4 and 2MS6 (M2 + S2 + M4 + MS4 and M2 + S2 + M6 + 2MS6) [Byun and Cho, 2006]. For diurnal tides, the O1, K1, and M2 constituents were pointed out in particular to give an asymmetrical wave pattern [Ranasinghe and Pattiaratchi, 2000; Hoitink et al., 2003]. [3] Asymmetry has traditionally been quantified by using the relative phase between constituents (e.g., 28M2 − 8M4) to indicate the direction of tidal asymmetry, and using the ratio

1

School of Physical, Environmental and Mathematical Sciences, University of New South Wales at the Australian Defence Force Academy, Canberra, Australian Capital Territory, Australia. 2 Key Laboratory of Physical Oceanography, Ministry of Education, Ocean University of China, Qingdao, China. Copyright 2011 by the American Geophysical Union. 0148-0227/11/2011JC007270

of constituent amplitudes (e.g., aM2/aM4) to reflect the degree of distortion [e.g., Friedrichs and Aubrey, 1988]. These traditional metrics require the identification of the constituent combination chiefly responsible for asymmetry, and make it difficult to compare locations at which asymmetry involves different constituent combinations. During the last decade, probability distribution functions (pdfs) of tidal elevations have increasingly been used to learn the role of tidal constituents in contributing to asymmetries [e.g., Pugh, 2004; Woodworth et al., 2005; Castanedo et al., 2007]. However, interpretation is hampered by the difficulty of analytically deriving pdfs if a large number of tidal constituents are considered [Hoitink et al., 2006], and consequently a systematic mathematical treatment is still missing [Woodworth et al., 2005]. Although it is impractical to determine the complete pdf in such situations, moments of the distribution can readily be found and related to constituents. Nidzieko [2010] showed that skewness of the time derivative of tidal elevation conveniently captures the amplitude and phase information embodied in the abovementioned traditional metrics and proposed that tidal asymmetry be quantified via skewness. Skewness has the advantage that it can be applied to any time series, regardless of which constituents are responsible for the asymmetry. Skewness can also be used in more complex situations in which several different combinations of constituents simultaneously contribute to the overall asymmetry. [4] Nidzieko [2010] showed how a pair of harmonically related constituents affects the skewness. In this paper, we

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extend Nidzieko’s analysis to show how skewness is determined by the combination of an arbitrarily large number of constituents. Woodworth et al. [2005] and Hoitink et al. [2006] called for a systematic mathematical treatment to include a large number of tidal constituents based on probability distribution functions, but we find skewness is more effective and easier to handle. Our analysis has two key results: we (1) identify the sets of constituents whose combination can lead to mean skewness, and (2) provide a formula to quantify the relative contribution made by each such combination to the overall skewness. We then apply this method to analyze tidal records from 335 stations around the world and from a global ocean tidal model. [5] In this paper, we follow Nidzieko [2010] in analyzing the tidal elevation time derivative, as tidal elevation data has the advantage of being much more readily available than long-term tidal current data. Nevertheless, this method we introduce applies to other tidal time series, such as tidal currents (which reveals asymmetries in direction as well as strength) and asymmetric elevation, which is important for characterizing extreme water levels [Woodworth et al., 2005]. The next section derives the metrics used to quantify tidal asymmetry, based on the skewness of the tidal elevation time derivative. In section 3 the metrics are applied to observed tidal parameters in semi-diurnal, diurnal and mixed tidal regimes, and then global patterns of tidal asymmetry and some important contributors are given according to TOPEX/POSEIDON altimetry data. The final section offers a conclusion of this paper.

positive skew (g > 0) usually indicates a tide that rises more quickly than it falls. Duration asymmetry is a relatively crude metric that discards amplitude information and only considers the sign of z, whereas skewness takes amplitude into account and provides a better characterization of tidal asymmetry (e.g., m3 is proportional to residual bed load sediment transport in situations where tidal currents and elevation are in quadrature [Nidzieko, 2010]). Moreover, a given ΔT represents a greater asymmetry in a semidiurnal tide than a diurnal tide, making interpretation problematic. [7] Assuming the time series is sufficiently long that z → 0, equation (2) can be approximated as h i E ðz Þ3 g≈ h i3=2 : E ðz Þ2

z ¼ z1 þ z2 þ z3 þ ⋯ þ zN

can be expressed respectively as the expectation values of z 3 ¼ ðz 1 þ z 2 þ z 3 þ ⋯ þ z N Þ3 ¼

n¼1

hn ¼

N X n¼1

an cosðwn t − 8n Þ;

ð1Þ

where an is the amplitude, wn = 2p/Tn is the frequency, Tn is the period and 8n is the phase of constituent n. The skewness of the tidal elevation time derivative (z ≡ ∂h/∂t, note that the notation differs from Nidzieko [2010]) was proposed as a measure of asymmetry by Nidzieko [2010], which we express in terms of expectation values (denoted E), rather than sample skewness h i z 3 E z − m g ≡ 33 ¼ h 2 i3=2 ; s E z − z

ð2Þ

where m3 is the third moment about the mean z and s is the standard deviation (normalization by s3 makes g a dimensionless measure of asymmetry which is independent of tidal range; our main conclusions also apply to the nonnormalized metric m3). As discussed by Nidzieko [2010], the skewness g reflects the duration asymmetry DT = E(Tfall−Trise) in the rise (z > 0) and fall (z < 0) of water level. Although they measure somewhat different things, we find that the signs of these two metrics are nearly always the same, especially when asymmetry is large. If a tide spends more time rising than falling, positive values of z are more probable than negative values of z and so skewness is typically negative (g < 0). Conversely, a

N N N X X X z i3 þ 3 z i2 z j þ 6 zizjzk i; j i≠j

i

i; j;k i≠j≠k

ð4Þ

and z 2 ¼ ðz 1 þ z 2 þ z 3 þ ⋯ þ z N Þ2 ¼

[6] Generally, the observed tidal elevation h can be generated by the summation of N individual constituents (hn) as N X

ð3Þ

[8] The third moment and second moment for the addition of all tidal constituents such as

2. Derivation of Metrics

h ðt Þ ¼

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N X i

z 2i þ 2

N X i;j i≠j

zizj;

ð5Þ

where z i ≡ ∂hi/∂t. Notice that only combinations of three or fewer constituents occur in equation (4); this is simply a consequence of the third power in the definition of g. The terms in equations (4) and (5) can be computed, respectively, as 1 z 3i ¼ a3i w3i ½ sinð3wi t − 38i Þ − 3 sinðwi t − 8i Þ; 4 n h i 1 z 2i z j ¼ a2i w2i aj wj sin 2wi þ wj t − 28i − 8j 4 h i o − sin 2wi − wj t − 28i þ 8j − 2 sin wj t − 8j ;

ð6Þ

ð7Þ

n h i 1 z i z j z k ¼ ai wi aj wj ak wk sin wi − wj − wk t − 8i − 8j − 8k 4 h i þ sin wj − wk − wi t − 8j − 8k − 8i h i þ sin wk − wi − wj t − 8k − 8i − 8j h io ; ð8Þ þ sin wi þ wj þ wk t − 8i þ 8j þ 8k 1 z i 2 ¼ a2i w2i ½1 − cosð2wi t − 28i Þ; 2 n h i 1 z i z j ¼ ai wi aj wj cos wi − wj t − 8i þ 8j 2 h io − cos wi þ wj t − 8i − 8j :

2 of 12

ð9Þ

ð10Þ


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[9] Most of the terms in equations (6)–(10) are oscillatory about zero and have an expectation value of zero over a long time interval. However, the first term in equation (9) will be non-oscillatory and hence give nonzero skewness over a long time interval. So do some other terms when specific frequency relationships are met. For instance the second term in equation (7) is constant when 2wi = wj; this also applies to any of the first three terms in equation (8) when e.g., wi + wj = wk. Retaining only such nonzero terms, the expectations of equations (6)–(10) can be written as equations (11)–(15), respectively, E z 3i ¼ 0;

ð12Þ

1 E z i z j z k ¼ ai wi aj wj ak wk sin 8i þ 8j − 8k when wi þ wj ¼ wk ; 4 ð13Þ 1 E z 2i ¼ a2i w2i ; 2

ð14Þ

E z i z j ¼ 0:

ð15Þ

[10] Thus the nonzero expectations of equations (12) and (13) can only be produced by combination of two constituents or three, with a frequency relationship of 2w1 = w2 or w1 + w2 = w3. The fact that only combinations of two or three constituents can contribute to the skewness follows directly from the definition of the skewness, which is related to the third moment. Using these expectations, the skewness can be obtained as 2 E z3 g N ≈ 3=2 ¼ E z2

6 E4

respectively, where 2w1 = w2 in equation (17) and w1 + w2 = w3 in equation (18). Equation (17) can be used to quantify tidal asymmetry caused by one constituent and its first harmonic constituent, e.g., M2 and M4 in shallow water. Substituting 2w1 for w2 in equation (17), the frequency terms are eliminated and we obtain the formula given by Nidzieko [2010] 3 2 a1 a2 sinð281 − 82 Þ g 2 ¼ 2 : 3=2 1 2 2 a þ 4a2 2 1

ð11Þ

1 E z 2i z j ¼ a2i w2i aj wj sin 28i − 8j when 2wi ¼ wj ; 4

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ð19Þ

[12] The calculation of g 2 relies on the amplitudes and phases of two frequency-related constituents; however, it is independent of tidal constituents’ frequencies. The tidal asymmetry caused by triple constituents (e.g., O1, K1, and M2) can be quantified by equation (18). This depends on amplitudes and phases, as well as frequencies, which can no longer be removed. The relative phase 81 + 82 − 83 is the only phase information retained in equation (18) and determines the direction of asymmetry. Typically, the fall-tide duration is shorter for g 3 < 0 and rise-tide duration is shorter for g 3 > 0. Thus, a distorted, but symmetric tide has a relative phase 81 + 82 − 83 of 0° or 180°. If 0 < 81 + 82 − 83 < 180°, then g 3 > 0 and the distorted composite tide typically has a greater duration of the falling tide. If 180° < 81 + 82 − 83 < 360°, the relationship is reversed, typically resulting in a fast-falling tide. [13] The symbol b is introduced here to represent the contribution to tidal skewness by a particular combination of two or three constituents, which allows a comparison of the

3 N P i

zi3 þ 3

0 2

N P i;j i≠j

zi2zj þ 6

N P

7 zizjzk 5

i;j;k i≠j≠k

313=2

N N P B 6P 7C @ E 4 z i 2 þ 2 z i z j 5A i

i;j i≠j

X 3 X 3 ai wi aj wj ak wk sin 8i þ 8j − 8k þ a2i w2i aj wj sin 28i − 8j 2 4 wi þwj ¼wk 2wi ¼wj ¼ : !3=2 N 1X a2 w2 2 i¼1 i i ð16Þ

[11] As special cases, the skewness created by the addition of two (N = 2) or three (N = 3) frequency-related constituents are 3 2 2 a1 w1 a2 w2 sinð281 − 82 Þ g2 ¼ 4 3=2 1 2 2 a1 w1 þ a22 w22 2

0 13=2 3 2 2 B a1 w1 a2 w2 sinð281 − 82 Þ 2 2 2 2C Ba w þ a2 w2 C b¼4 ¼ g 2 ⋅B 1 N1 C !3=2 @ P 2 2 A N 1X a i wi a2i w2i i¼1 2 i¼1

ð17Þ

ð20Þ

for pairs or

and 3 a1 w1 a2 w2 a3 w3 sinð81 þ 82 − 83 Þ g3 ¼ 2 ; 3=2 1 2 2 2 2 2 2 a w þ a2 w2 þ a3 w3 2 1 1

role of different constituent combinations in contributing to the total tidal skewness. It can be obtained as, for example,

ð18Þ

0 13=2 3 B2 2 C a1 w1 a2 w2 a3 w3 sinð81 þ 82 − 83 Þ Ba1 w1 þ a22 w22 þ a23 w23C C b¼2 ¼ g 3 ⋅B !3=2 B C N X N @ A 2 2 1X 2 2 a w i i aw 2 i¼1 i i i¼1

ð21Þ 3 of 12


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Figure 1. Scatterplot of the skewness g N approximated with equation (16) versus the skewness g measured by equation (2) at all 335 stations. for triplets. Therefore, the total tidal skewness (equation (16)) is equal to the summation of individual b g N ¼ ∑b:

ð22Þ

[14] To summarize, we find that tidal asymmetry as indicated by the skewness metric can only be produced by combination of two constituents or three, with a frequency relationship of 2w1 = w2 or w1 + w2 = w3. The direction of asymmetry is exactly decided by the summation of all individual asymmetries (b) contributed by different combinations of constituents. Each such b might be positive or negative. Thus, the observed asymmetry is the net effect of these contributions. This formulation allows the identification of constituent combinations which contribute to asymmetry, and the quantification (via b) of the contribution of each combination to the overall asymmetry. We note that this metric characterizes the long-term average asymmetry of a time series; this may differ significantly from the asymmetry found at specific times with shorter tidal records, for example at different phases of the spring-neap cycle. Its usefulness as a metric applicable in all tidal regimes is demonstrated in the next section.

3. Applications 3.1. Results From Sea Level Station Data [15] To examine the role of constituents in generating tidal asymmetry, we collected sea level data sets of good quality at 335 stations globally, which were all recorded for one year at hourly intervals. When the Robust T_TIDE Harmonic Analysis Program [Pawlowicz et al., 2002; Leffler and Jay, 2009] is applied to a 365-day tidal elevation record, N = 59 tidal constituents can typically be resolved. Since only pairs and triples of constituents contribute to skewness, the number of possible combinations is narrowed down to N(N − 1)/2 = 1711 unique pairs and

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N(N − 1)(N − 2)/6 = 32,509 unique triples. The required frequency relationships (2w1 = w2 or w1 + w2 = w3) then reduce this total by two orders of magnitude, leaving only 291 combinations which can produce skewness (such combinations can be readily identified from their Doodson numbers; for example, see Table 2). Although the number of triples scales as N3, the restrictive frequency relationships keep the problem from growing this rapidly; for example, we resolved 184 constituents in a 40-year record, giving 1,038,220 unique pairs and triples, which the selection rules reduced to 1364. [16] A subtle point arises concerning the 18.61-year nodal modulation, which produces fine splitting of spectral peaks by Doodson’s f5 (see Table 2). If the time series is sufficiently long to resolve these split frequencies and allow oscillations at their difference frequency f5 to average to zero, the selection rules (2w1 = w2 or w1 + w2 = w3) will apply as before. In such cases equation (16) gives the mean skewness over this long record. For shorter records, the split frequencies are not resolved and are instead represented by a single wave whose amplitude and phase may vary; this nodal modulation will lead to variation in the skewness, also with a long-term cycle. Skewness variation on the springneap timescale was also discussed by Nidzieko [2010]. [17] To filter out unresolved satellite components as well as tidal residuals, we constructed yearlong synthetic tidal records via equation (1) with nodal corrected harmonic constituents from the results of T_TIDE. Then, we calculated g from these synthetic tidal records via equation (2) and calculated g N by using nodal corrected harmonic constants via equation (16) and b of all 291 combinations via equation (22); the largest three combinations are denoted by b1, b2, and b 3 in order of decreasing magnitude. We only used constituents with signal/noise ratio larger than 2.0 (based on the square of the ratio of amplitude to amplitude error arising from unresolved noise components, for detail see Pawlowicz et al. [2002]). Nodal modulation differed among the records since they were from different years and different locations. The root mean square (RMS) difference between the g N calculated with and without nodal corrected harmonic constants is small (0.030). The skewness g N approximated with equation (16) is nearly identical (RMS difference = 0.0108) to the skewness g measured by equation (2) (Figure 1), showing that it is reasonable to neglect the mean z and oscillatory terms. Of the 335 stations, for brevity we show in Table 1 some of the subset with jg Nj > 0.1 or jb1j > 0.1 in different tidal regimes, i.e., pronounced asymmetry. If the major b values differ in sign, the total asymmetry g N will be reduced and may even become insignificant (e.g., Stations G194, H341, and G188). Otherwise, the total asymmetry will be enhanced slightly or greatly (e.g., Stations H186, G216, and H387). [18] One hundred ten of these stations are in semi-diurnal regimes (F < 0.25, where F = (aO1 + aK1)/(aM2 + aS2) is the form number [National Ocean Service, 2000]), 163 stations are in mixed, mainly semi-diurnal regimes (0.25 < F < 1.5, hereafter MS regimes), 37 stations are in mixed, mainly diurnal regimes (1.5 < F < 3.0, hereafter MD regimes), and 25 stations are in diurnal regimes (F > 3.0). Tidal asymmetry may originate from nonlinear tidal interactions, or may be due to astronomical tides alone. Pronounced asymmetry in

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Table 1. The Total Asymmetry via Skewness, the Three Major Combinations and Their Contributions to g N, Calculated From a 1-Year Record of Hourly Elevation Dataa Station b

G170 H186b G024c H753b H750b G194b H385b H744b H743b G120c H358b H064b H285b G216c G173c G205b G056c H113b G088b H263b H578b G052b H383b H552b G089b H389b G107c H319b H219b G074b H341b G326b H340b H339b G070b H759b G188b G303b H543b H386b H767b G217b H770b G065c H066b G302c G022c H277b H255b H387b G328b GT03d H637b H769b

Longitude

Latitude

Year

F

gN

Combination 1

b1

Combination 2

b2

Combination 3

b3

077°06.0′W 023°02.0′E 077°35.0′E 081°25.9′W 077°57.2′W 047°55.5′W 117°53.0′E 072°05.2′W 070°05.8′W 134°27.8′E 131°41.0′E 147°08.4′E 058°30.0′W 081°48.5′W 077°09.0′W 005°21.0′E 147°56.0′E 058°52.3′E 140°44.0′E 067°09.8′W 118°30.0′W 113°39.0′E 107°04.3′E 155°50.0′W 144°23.0′E 118°04.0′E 166°17.4′W 111°22.0′W 065°18.1′W 165°25.8′W 121°44.7′E 132°04.0′E 120°16.8′E 147°19.8′E 121°00.0′E 082°37.6′W 064°16.0′W 173°12.0′E 123°22.0′W 116°04.0′E 094°47.4′W 094°47.6′W 097°13.0′W 152°10.5′E 145°48.0′E 176°38.1′W 062°53.0′E 097°47.7′W 064°55.2′W 113°04.0′E 066°56.0′W 109°07.0′E 108°37.0′E 097°02.8′W

03°54.0′N 32°02.0′S 38°43.0′S 30°23.7′N 34°13.6′N 25°01.0′S 04°14.0′N 41°21.3′N 41°17.1′N 07°19.8′N 32°25.0′N 09°28.7′S 34°40.0′S 24°33.2′N 12°03.0′S 43°18.0′N 42°33.0′S 20°41.2′N 41°47.0′N 18°27.4′N 34°00.5′N 24°54.0′S 10°20.4′N 20°02.0′N 42°58.0′N 05°48.6′N 23°52.0′N 26°01.0′N 18°18.0′N 64°30.0′N 25°09.4′N 34°54.0′N 22°36.9′N 42°53.2′S 06°04.0′N 27°45.6′N 65°15.0′S 52°50.0′N 48°25.0′N 05°59.0′N 29°17.2′N 29°18.6′N 27°34.8′N 04°12.0′S 05°12.0′S 51°51.5′N 67°36.0′S 22°15.7′N 18°20.1′N 03°13.0′N 10°37.0′N 21°01.0′N 19°06.0′N 28°01.3′N

1999 1978 1997 1997 2008 2000 2005 2007 2008 1998 1964 1993 1961 1997 1993 2007 2001 2007 2007 2008 2008 2005 2002 2008 2007 2005 1997 1978 2008 2006 2006 2007 2006 2003 1994 2007 2003 1958 1964 2005 2006 2007 1998 1995 1990 2000 1999 2005 2008 2005 1990 2009 1994 2007

0.07 0.09 0.14 0.15 0.23 0.27 0.31 0.32 0.40 0.40 0.51 0.52 0.57 0.64 0.67 0.67 0.71 0.84 0.85 0.88 0.90 0.93 0.98 1.06 1.21 1.23 1.31 1.32 1.35 1.40 1.46 1.56 1.58 1.59 1.62 1.75 1.87 1.93 2.10 2.28 2.35 2.57 2.60 2.63 2.76 3.03 3.04 3.20 3.34 3.44 3.66 4.25 4.41 7.21

−0.241 −0.135 0.110 −0.140 0.395 −0.089 −0.306 0.172 −0.268 0.222 0.187 0.146 0.643 −0.449 0.165 0.540 −0.138 0.266 −0.096 0.221 −0.249 −0.262 0.117 0.126 −0.145 −0.619 0.474 −0.692 0.417 −0.009 −0.015 −0.239 −0.347 −0.328 −0.144 −0.445 −0.037 −0.203 −0.287 −0.732 0.432 0.192 0.592 −0.452 −0.283 −0.291 −0.218 0.373 0.834 0.361 −0.489 −0.522 0.249 −0.289

M2/M4 M2/S2/MS4 M2/S2/MS4 M2/M4 M2/M4 M2/S2/MS4 M2/S2/MS4 M2/M4 M2/M4 M2/M4 M2/M4 O1/K1/M2 M2/M4 O1/K1/M2 O1/K1/M2 M2/M4 O1/K1/M2 M2/M4 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 K1/M2/MK3 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 K1/M2/MK3 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 K1/M2/MK3 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2 P1/K1/S2 P1/K1/S2 O1/K1/M2 P1/K1/S2 O1/K1/M2 O1/K1/M2 O1/M2/MO3 O1/K1/M2 O1/K1/M2 O1/K1/M2 O1/K1/M2

−0.146 −0.064 0.042 −0.111 0.222 −0.154 −0.153 0.162 −0.200 0.100 0.077 0.077 0.244 −0.178 0.110 0.240 −0.116 0.073 −0.129 0.130 −0.176 −0.183 0.030 0.076 −0.126 −0.274 0.301 −0.327 0.302 0.108 0.213 −0.232 −0.343 −0.302 −0.037 −0.438 −0.299 −0.181 −0.181 −0.569 0.412 0.173 0.431 −0.259 −0.159 −0.255 −0.127 0.309 0.607 0.152 −0.430 −0.498 0.228 −0.377

M2/S2/MS4 M2/M4 M2/M4 M2/S2/MS4 M2/M4/M6 M2/M4 M2/M4 M2/N2/MN4 M2/N2/MN4 M2/S2/MS4 M2/S2/MS4 P1/K1/S2 M2/N2/MN4 M2/M4 P1/K1/S2 O1/K1/M2 Q1/K1/N2 M2/S2/MS4 M2/M4 M2/M4 P1/K1/S2 M2/M4 M2/M4 P1/K1/S2 M2/M4 P1/K1/S2 P1/K1/S2 P1/K1/S2 M2/M4 M2/M4 M2/M4 P1/K1/S2 P1/K1/S2 Q1/K1/N2 P1/K1/S2 P1/K1/S2 P1/K1/S2 P1/K1/S2 K1/M2/MK3 P1/K1/S2 P1/K1/S2 M2/M4 P1/K1/S2 K1/K2 K1/K2 Q1/K1/N2 O1/K1/M2 P1/K1/S2 P1/K1/S2 K1/M2/MK3 Q1/K1/N2 K1/K2 K1/K2 K1/M2/MK3

−0.058 −0.058 0.036 −0.014 0.054 −0.081 −0.097 0.037 −0.035 0.085 0.055 0.024 0.110 0.140 0.028 0.084 −0.007 0.036 0.034 0.080 −0.040 −0.048 0.023 0.018 0.011 −0.061 0.063 −0.159 0.055 −0.108 −0.137 −0.027 −0.029 −0.022 −0.032 −0.046 0.125 0.040 −0.036 −0.101 0.025 −0.032 0.036 0.124 −0.051 −0.031 −0.057 0.041 0.065 0.150 −0.020 −0.060 0.025 0.075

M2/N2/MN4 S2/S4 O1/K1/M2 M2/N2/MN4 M2/N2/MN4 M2/N2/MN4 O1/K1/M2 M2/M4/M6 O1/K1/M2 O1/K1/M2 K1/M2/MK3 M2/S2/MS4 O1/K1/M2 M2/S2/MS4 K1/K2 M2/S2/MS4 P1/K1/S2 K1/M2/MK3 M2/S2/MS4 M2/M4/M6 K1/K2 M2/S2/MS4 M2/S2/MS4 K1/K2 K1/K2 M2/M4 K1/K2 K1/K2 P1/K1/S2 O1/K1/M2 M2/S2/MS4 M2/M4 O1/M2/MO3 M2/M4 O1/M2/MO3 K1/K2 K1/K2 Q1/K1/N2 P1/K1/S2 K1/K2 Q1/K1/N2 K1/K2 M2/M4 O1/K1/M2 O1/K1/M2 P1/K1/S2 K1/K2 K1/M2/MK3 M2/M4 P1/K1/S2 M2/N2/MN4 K1/M2/MK3 O1/M2/MO3 O1/M2/MO3

−0.028 −0.005 0.009 −0.014 0.034 0.046 −0.011 −0.013 −0.028 −0.035 0.019 0.014 0.061 −0.042 0.016 0.081 −0.006 0.034 0.012 −0.008 −0.018 −0.032 0.020 0.016 −0.009 −0.055 0.036 −0.069 0.019 −0.033 −0.043 0.025 0.019 0.017 −0.020 −0.039 0.055 −0.025 −0.034 −0.044 0.023 0.017 0.031 −0.076 −0.044 0.024 −0.055 −0.024 0.040 −0.063 0.019 0.048 −0.018 0.058

a Skewness is represented by g N and the contributions to the three major combinations are represented by b 1, b 2, and b 3. The stations are arranged in ascending order of the tidal form number (F). Negative values in g N, b1, b 2, and b 3 mean a fast-falling tide. b Data obtained from the University of Hawaii Sea Level Center. c Data obtained from the British Oceanographic Data Centre. d Data obtained from the Ocean University of China.

16 of the 18 semi-diurnal stations is dominated by the M2/M4 combination; asymmetry in the others is dominated by M2, S2, and MS4. The combinations of S2/S4, M2/N2/ MN4, and M2/M4/M6 play secondary or tertiary roles in producing asymmetry. The tidal wave distortion mainly originates from nonlinearity, as overtides or compound tides are involved in the abovementioned combinations.

[19] In MS regimes, there are 96 stations with jg Nj > 0.1 or jb1j > 0.1, twenty of which are dominated by M2/M4, four by M2/S2/MS4, two by K1/M2/MK3, and the remainder by O1/K1/M2. At stations where asymmetry is dominated by astronomical constituents, P1/K1/S2 usually ranks as the second major contributor to tidal asymmetry. In addition, K1 and K2 can also generate a notably asymmetrical wave.

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Figure 2. The distribution of all 335 sea level stations and their tidal skewness (g N) classified by b 1. Those where the skewness is primarily caused by nonlinear tidal interactions (including M2/M4, M2/N2/MN4, M2/S2/MS4, S2/M4/2MS6, K1/M2/MK3, and O1/M2/MO3) are shown in triangles, while those in which it is primarily caused by astronomical tides alone (including O1/K1/M2 and P1/K1/S2) are shown in squares. Filled color gives the degree of skewness (g N). Coastlines are plotted from World Coastlines and Lakes (NOAA).

Figure 3. Scatterplot of the comparison between b M2+M4 and bO1+K1+M2 versus the tidal form number F at 288 stations, where M2/M4 or O1/K1/M2 contributes most to the tidal asymmetry. A regression line is also plotted to show their connection. Color indicates the degree of tidal asymmetry (g N). The dashed lines divide it into semi-diurnal (S), mixed, mainly semi-diurnal (MS), mixed, mainly diurnal (MD), and diurnal (D) regimes. 6 of 12

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Figure 4. Solid lines give the degree of tidal asymmetry caused (a) by M2/M4 via equation (17) and (b) by O1/K1/M2 via equation (18), where the extreme value of the sine function (+1) is used. Points are located by harmonic analysis at all 335 stations and filled color indicates the tidal regimes (log10 F). The color bar shows the range of semi-diurnal (S), mixed, mainly semi-diurnal (MS), mixed, mainly diurnal (MD), and diurnal (D) regimes.

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Table 2. Six Fundamental Astronomical Frequency (degh−1) and Doodson Numbers of Tidal Constituents From TPXO7-ATLASa f f1 Frequency Name Mf Mm Q1 O1 P1 K1 N2 M2 S2 K2 MN4 M4 MS4

14.492 n1 0 0 1 1 1 1 2 2 2 2 4 4 4

f2 5.490 10 n2 2 1 −2 −1 1 1 −1 0 2 2 −1 0 2

f3 −1

4.107 10 n3 0 0 0 0 −2 0 0 0 −2 0 0 0 −2

f4 −2

4.644 10 n4 0 −1 1 0 0 0 1 0 0 0 1 0 0

f5 −3

f6 −3

2.206 10 n5 0 0 0 0 0 0 0 0 0 0 0 0 0

1.962 10−6 n6 0 0 0 0 0 0 0 0 0 0 0 0 0

a Using Doodson’s expansion [Doodson, 1921], each tidal constituent has a frequency w = n1 f1 + n2 f2 + n3 f3 + n4 f4 + n5 f5 + n6 f6, where f1-f6 are fundamental astronomical frequency (degh−1) and n1-n6 are the Doodson numbers modified from Pugh [1987]. Two pairs (K1/K2 and M2/M4) and ten triplets (M2/Mf/K2, N2/Mm/M2, O1/Mf/K1, Q1/Mm/O1, Mm/MN4/M4, O1/K1/M2, P1/K1/S2, Q1/K1/N2, M2/N2/MN4, and M2/S2/MS4) of tidal constituents from TPXO7-ATLAS are determined by frequency relationship 2w1 = w2 or w1 + w2 = w3.

[20] In MD regimes and diurnal regimes, tidal skewness is dominated by the astronomical triplet of O1, K1, and M2 at most of the stations. However, there are six stations where the tidal wave distortion generated by P1/K1/S2 is much greater than that produced by O1/K1/M2 (at Station G065, the tidal asymmetry generated by O1/K1/M2 is even weaker than K1/K2). Besides those, MK3 and its addition to the related astronomical constituents K1 and M2 are predominant in all the other triplets at Station G070. Similarly, MO3/O1/M2 is the major source of asymmetry at Station H387. Compared to the combination of astronomical tides, the nonlinear effect between diurnal and semi-diurnal constituents is the

dominant cause of asymmetry at these two stations. Additionally, we also find the combination of Q1, K1, and N2 plays a minor role in producing an asymmetrical tide. [21] Figure 2 gives the distribution of these 335 sea level stations and the origins of tidal asymmetry at each station. We find that most Pacific Ocean stations are dominated by the joint action of astronomical tides, whereas most Atlantic Ocean stations are dominated by nonlinear effects. This is mainly related to the classification of tidal regimes [see Hoitink et al., 2003, Figure 7], especially for those in the diurnal regime (e.g., the Gulf of Mexico and the Gulf of Tonkin).

Figure 5. The tidal skewness (g N) of the global open oceans, computed according to the tidal constants derived from TPXO7-ATLAS. 8 of 12

Figure 6. The contribution (b) of (a) O1/K1/M2, (b) P1/K1/S2, (c) K1/K2, and (d) Q1/K1/N2 to the total tidal skewness in the open oceans around the world, computed according to the tidal constants derived from TPXO7-ATLAS. Note that the color scales differ.

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Figure 7. The largest contributor to the tidal skewness in the open oceans around the world, classified according to the contributions among K1/K2, O1/K1/M2, P1/K1/S2, and Q1/K1/N2. The tidal constants are derived from TPXO7-ATLAS. [22] In most previous studies, these two causes of asymmetry are usually discussed separately. Here, the pair of M2 and M4 and the triplet of O1, K1 and M2 are chosen to compare the contributions of nonlinearity and astronomical tides. At each station, bM2+M4 and bO1+K1+M2 are computed from equations (20) and (21), respectively and the ratio of them is plotted in Figure 3. To avoid noise caused by the ratio of two insignificant values, we omit points with b 1 ≠ bM2+M4 and b 1 ≠ b O1+K1+M2 in Figure 3, leaving 288 stations. This shows that as F increases, the astronomical triplet of O1/K1/M2 generally becomes more important than M2/M4; this trend is mainly due to the decreasing prevalence of M2 over K1 and O1 as the latter rise. A fit to the power law

bM 2þM4

b

O1þK1þM2

¼ cF b

ð23Þ

yields c = 0.140 and b = −1.793, although R2 of this regression is only 0.623. Nevertheless, for those stations with strong asymmetry, the asymmetry is dominated by nonlinear tidal interaction in the semi-diurnal regime (log10 F < −0.602), but led by the astronomical triplets in MD regimes and diurnal regimes (log10 F > 0.176). [23] As shown in Figure 3, we also find the largest tidal asymmetry happens mainly in MD regimes or its adjacent diurnal regimes. We show in p theffiffiffiffiffiffiffi Appendix that both g 2 and ffi g 3 have maximum values of 2=3 ≈ 0.8165, which occurs when a2/a1 = (1/2)3/2 for pairs of constituents or when a1w1 = a2w2 = a3w3 for triplets; thus both pairs and triplets have the same power to generate asymmetric tides. Applying equations (17) and (18) with the extreme value of the sine function (+1) to M2/M4 and O1/K1/M2 respectively, we obtain Figures 4a and 4b. As shown in Figure 4a, g M2+M4 increases as aM4/aM2 approaches (1/2)3/2 and

then drops as aM4/aM2 rises. Figure 4b gives the same trend for g O1+K1+M2 with aO1/aM2 and aK1/aM2. Additionally, g O1+K1+M2 is slightly more sensitive to aK1/aM2 than to aO1/aM2 due to larger wK1/wM2 than wO1/wM2. The definition of tidal form number F is only associated with astronomical tides. Thus, there is no obvious link between tidal regime and tidal asymmetry caused by nonlinear tidal interaction (Figure 4a). In contrast, the form number strongly correlates with the degree of tidal asymmetry produced by astronomical tides alone (Figure 4b). The distribution of g O1+K1+M2 in Figure 4b illustrates that the largest tidal asymmetry happens mainly in MD regimes or its adjacent diurnal regimes as shown in Figure 3, where the amplitude ratios among O1, K1, and M2 are optimal. 3.2. Results From TOPEX/POSEIDON Global Inverse Solution [24] Shallow water tides are a nonlinear effect caused by friction or topography and the asymmetries associated with them develop over relatively short distances in coastal, estuarine and river environments [Hoitink et al., 2006]. The degree of tidal asymmetry at a site may not be representative of a large region. Conversely, the asymmetry caused by astronomical tides alone may remain unchanged over a large region. By using the abovementioned equations, we construct global maps of the contributions to tidal asymmetry. [25] We use the OSU TOPEX/POSEIDON Global Inverse Solution (TPXO, http://volkov.oce.orst.edu/tides/global.html), which is a global model of ocean tides. It best fits, in a least squares sense, the Laplace tidal equations and along track averaged data from TOPEX/POSEIDON and Jason (on TOPEX/POSEIDON tracks since 2002). The methods used in the model are described in detail by Egbert et al. [1994] and further by Egbert and Erofeeva [2002]. The TPXO7ATLAS solutions are provided as complex amplitudes of

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earth-relative sea-surface elevation for eight primary (M2, S2, N2, K2, K1, O1, P1, Q1), two long period (Mf, Mm) and three nonlinear (M4, MS4, MN4) nodal-corrected harmonic constituents, on a 1/4 degree resolution full global grid (details in http://volkov.oce.orst.edu/tides/atlas.html). [26] Twelve triplets or pairs are found to contribute to tidal asymmetry (listed in Table 2). The tidal skewness (g N) is plotted in a global map (Figure 5), as well as four contributors from primary astronomical tides alone (Figure 6). The contribution to the tidal asymmetry by the shallow water tides (e.g., M2/M4) is not included in Figure 6 as they are rather weak in the open oceans and cannot be simulated accurately by the global model in the coastal areas (more discussion below). The dipolar patterns appear due to the distribution of relative phase around amphidromic points. Comparing the scales in Figures 6a–6d, we find the triplet of O1, K1, and M2 contributes most to the total asymmetry, following by P1/K1/S2, K1/K2, and Q1/K1/N2. Hence the pattern of O1/K1/M2’s contribution is very similar to that of g N. The contributions from those four combinations are compared in Figure 7. It shows O1/K1/M2 predominates in most open oceans around the world, and P1/K1/S2 dominates in some narrow regions where phase relationships reduce the contribution of O1/K1/M2. The regions led by K1/K2 or Q1/K1/N2 are scattered in the global map. In semi-diurnal regions (e.g., the central parts of Pacific Ocean, Atlantic Ocean and Indian Ocean) all the primary-tide-generated asymmetries are rather weak. As we point out above, asymmetry is maximized under the condition a1w1 = a2w2 = a3w3. For O1/K1/M2, that requires aK1 to be close to 1.927aM2 and aO1 close to 2.079aM2. Clearly, this is contrary to the definition of the semi-diurnal regime, where the amplitudes of M2 are much larger than those of O1 and K1. [27] We note that at some stations shown in Figure 2, g N is inconsistent with that shown in Figure 5. This is mainly due to the coarse resolution of the global tide model, which is unable to capture the complex topography in coastal areas. It also indicates the potential development of the shallow water tides, e.g., Station H285 located in Buenos Aires, Argentina. This station is slightly inland and has stronger asymmetrical tide due to relatively larger M4/M2 ratio than nearby coastal stations. The same situation also happens in Station H186 at Knysna, South Africa, which is located in the middle of an estuary. There the shallow water interaction of M2/S2/MS4 is stronger than at nearby stations located in coastal embayments.

4. Conclusions [28] Tidal asymmetry can be quantified as the skewness of the tidal elevation time derivative, which was proposed by Nidzieko [2010] and reflects the asymmetry in the rise and fall of water level. We generalize his analysis to determine the impact on skewness of the addition of any number of tidal constituents. We show that skewness can only be generated via combinations of two or three frequency-related constituents (2w1 = w2 or w1 + w2 = w3). In such combinations, the direction of asymmetry is determined by the relative phase 281 − 82 for the addition of two constituents or by 81 + 82 − 83 for triple constituents. We show that the total asymmetry is the sum of the asymmetries contributed by

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each such combination, each of which is quantified by the metric b. Note that other measures of asymmetry (such as rise/fall duration difference) may not have this convenient property, which is an advantage of using skewness as a metric. These results generalize readily to the non-normalized third moment m3 as well. [29] These tools can be used to understand and compare the asymmetries of tidal records at different stations, regardless of the tidal regime (diurnal, mixed, semidiurnal) or the constituents that dominate the asymmetry. Applying these metrics to 335 global sea level stations, we find the combinations of M2/M4 and M2/S2/MS4 are dominant in semidiurnal regimes, while O1/K1/M2 and P1/K1/S2 dominate the diurnal and mixed, mainly diurnal regimes. The nonlinearity-induced compound tides (e.g., MK3 and MO3) can also be predominant in some diurnal tide dominated areas. Moreover, we find some combinations such as S2/S4, M2/N2/MN4, and Q1/K1/N2 play minor roles in contributing to tidal asymmetry, which is a result that has not been reported before. [30] Furthermore, we show there is a maximum possible value of the tidal skewness generated by any individual combination. For pairs, the largest skewness occurs when the amplitude ratio between harmonic overtide and its parent tide reaches (1/2)3/2 and the relative phase equals to 90° or 270°. For triplets, the largest asymmetry occurs when the frequencies and amplitudes satisfy a1w1 = a2w2 = a3w3 and the relative phase equals 90° or 270°. Pairs and triplets have the same capability to produce distorted waves. [31] There is no evident connection between tidal form number F and tidal asymmetry caused by shallow waters, because the definition of F is only associated with astronomical tides. In contrast, F correlates strongly with the degree of tidal asymmetry caused by astronomical tides alone. The above analysis explains why the largest tidal asymmetry happens mainly in mixed, mainly diurnal and its adjacent diurnal regions. [32] Our study also shows that the global patterns of tidal asymmetry and its contributors can be made easily through our metrics. Based on 13 tidal constituents derived from TPXO7-ATLAS, the tidal skewness is computed via equation (16) and plotted with its important contributors. Results show that large asymmetries usually appear as positive‐negative pairs and the triplet of O1/K1/M2 dominates most of the open oceans around the world. In addition, in semi-diurnal tidal regimes, asymmetries generated by astronomical tides alone are rather weak, which is due to the amplitude and frequency relationship between semi-diurnal tides and diurnal tides. [33] For future work, we note that almost the same metrics could be used to analyze asymmetry in tidal height (rather than rate of change) and asymmetry in tidal current components.

Appendix A [34] Here we derive the conditions under which g 2 and g 3 are maximized. If we define

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M¼

a2 w2 2=3 ; a1 w1


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then equation (17) can be written as g2 ¼

pffiffiffi 3= 2 ðM −1 þ M 2 Þ3=2

;

ðA1Þ

where we have taken the sine function to have its maximum value. The denominator has its minimum value when 3/2 when M = 2−1/3, i.e., p ffiffiffiffiffiffiffiffi a2/a1 = (1/2) , which gives the maximum g 2 = 2=3. [35] Similarly, if we define M as before and also P¼

a3 w3 2=3 a1 w1

[36] Equation (18) can be written as g3 ¼

pffiffiffi 3 2 ðM −1 P−1 þ M 2 P−1 þ M −1 P2 Þ3=2

ðA2Þ

where again the maximum of the sine function is assumed. g 3 is maximized when the partial derivatives of its denominator with respect to both M and P are zero, i.e., when

1=3 1 þ P3 M¼ 2

and P¼

1=3 1 þ M3 : 2

[37] Combining these we have M = P = 1, i.e., the maximum occurs pffiffiffiffiffiffiffiffiwhen a1w1 = a2w2 = a3w3. The maximum of g 3 is also 2=3. [38] Note that this maximum value is only valid for g 2 and g 3, but not g N or b as it is difficult to estimate the combined effects of pairs and triplets of tidal constituents in the denominator. [39] Acknowledgments. D. Song has been supported by the China Scholarship Council since August 2009 for his Ph.D. study in Australia. X. H. Wang was supported by 2009 UNSW Special Study Program. This study uses international sea level station data provided by the British Oceanographic Data Centre, the University of Hawaii Sea Level Center, and Ocean University of China.

References Aubrey, D. G., and P. E. Speer (1985), A study of non-linear tidal propagation in shallow inlet/estuarine systems. Part I: Observations, Estuarine Coastal Shelf Sci., 21(2), 185–205, doi:10.1016/0272-7714(85)90096-4. Blanton, J. O., G. Lin, and S. A. Elston (2002), Tidal current asymmetry in shallow estuaries and tidal creeks, Cont. Shelf Res., 22(11–13), 1731–1743, doi:10.1016/S0278-4343(02)00035-3.

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Byun, D.-S., and Y.-K. Cho (2006), Double peak-flood current asymmetry in a shallow-water-constituent dominated embayment with a macro-tidal flat, Geophys. Res. Lett., 33, L16613, doi:10.1029/2006GL026967. Castanedo, S., F. J. Mendez, R. Medina, and A. J. Abascal (2007), Longterm tidal level distribution using a wave-by-wave approach, Adv. Water Resour., 30, 2271–2282, doi:10.1016/j.advwatres.2007.05.005. Doodson, A. T. (1921), The harmonic development of the tide-generating potential, Proc. R. Soc. London, Ser. A, 100, 305–329, doi:10.1098/ rspa.1921.0088. Egbert, G. D., and S. Y. Erofeeva (2002), Efficient inverse modeling of barotropic ocean tides, J. Atmos. Oceanic Technol., 19(2), 183–204, doi:10.1175/1520-0426(2002)0192.0.CO;2. Egbert, G. D., A. F. Bennett, and M. G. G. Foreman (1994), TOPEX/ POSEIDON tides estimated using a global inverse model, J. Geophys. Res., 99(C12), 24,821–24,852, doi:10.1029/94JC01894. Friedrichs, C. T., and D. G. Aubrey (1988), Non-linear tidal distortion in shallow well-mixed estuaries: A synthesis, Estuarine Coastal Shelf Sci., 27(5), 521–545, doi:10.1016/0272-7714(88)90082-0. Hoitink, A. J. F., P. Hoekstra, and D. S. van Maren (2003), Flow asymmetry associated with astronomical tides: Implications for the residual transport of sediment, J. Geophys. Res., 108(C10), 3315, doi:10.1029/ 2002JC001539. Hoitink, A. J. F., P. Hoekstra, and D. S. van Maren (2006), Comment on “On the role of diurnal tides in contributing to asymmetries in tidal probability distribution functions in areas of predominantly semi-diurnal tide” by P. L. Woodworth, D. L. Blackman, D. T. Pugh and J. M. Vassie, Estuarine Coastal Shelf Sci., 67(1–2), 340–341, doi:10.1016/j.ecss.2005. 10.008. Leffler, K. E., and D. A. Jay (2009), Enhancing tidal harmonic analysis: Robust (hybrid L1/L2) solutions, Cont. Shelf Res., 29(1), 78–88, doi:10.1016/j. csr.2008.04.011. National Ocean Service (2000), Tide and Current Glossary, NOAA, Silver Spring, Md. Nidzieko, N. J. (2010), Tidal asymmetry in estuaries with mixed semidiurnal/ diurnal tides, J. Geophys. Res., 115, C08006, doi:10.1029/2009JC005864. Pawlowicz, R., B. Beardsley, and S. Lentz (2002), Classical tidal harmonic analysis including error estimates in MATLAB using T_TIDE, Comput. Geosci., 28(8), 929–937, doi:10.1016/S0098-3004(02)00013-4. Pugh, D. T. (1987), Tides, Surges and Mean Sea-Level, Wiley, New York. Pugh, D. T. (2004), Changing Sea Levels: Effects of Tides, Weather, and Climate, Cambridge Univ. Press, New York. Ranasinghe, R., and C. Pattiaratchi (2000), Tidal inlet velocity asymmetry in diurnal regimes, Cont. Shelf Res., 20(17), 2347–2366, doi:10.1016/ S0278-4343(99)00064-3. Speer, P. E., and D. G. Aubrey (1985), A study of non-linear tidal propagation in shallow inlet/estuarine systems. Part II: Theory, Estuarine Coastal Shelf Sci., 21(2), 207–224, doi:10.1016/0272-7714(85)90097-6. Speer, P. E., D. G. Aubrey, and C. T. Friedrichs (1991), Nonlinear hydrodynamics of shallow tidal inlet/estuary systems, in Tidal Hydrodynamics, edited by B. B. Parker, pp. 319–339, Wiley, New York. Woodworth, P. L., D. L. Blackman, D. T. Pugh, and J. M. Vassie (2005), On the role of diurnal tides in contributing to asymmetries in tidal probability distribution functions in areas of predominantly semi-diurnal tide, Estuarine Coastal Shelf Sci., 64(2–3), 235–240, doi:10.1016/j.ecss. 2005.02.014. X. Bao, Key Laboratory of Physical Oceanography, Ministry of Education, Ocean University of China, 238 Songling Rd., Qingdao 266100, China. A. E. Kiss, D. Song, and X. H. Wang, School of Physical, Environmental and Mathematical Sciences, University of New South Wales at the Australian Defence Force Academy, Northcott Drive, Canberra, ACT 2600, Australia. ([email protected])

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