Examples Of Tropical And Equatorial Tide

Periodic movements which are directly related in amplitude and phase to some periodic geophysical force are defined a tides and astronomic tides are the most widely recognised phenomena affecting water levels (Pugh 1987). These tides are the harmonic fluctuations of water level developed through the gravitational attraction from astronomic bodies (mainly the sun and moon). In majority of the world's coastlines there are two tidal cycles per day (i.e. two high and low waters per day and these are termed semi-diurnal (twice-daily) tides with a tidal period of 12.24 h. In a few locations (e.g. Gulf of Mexico, Gulf of Thailand), there is only one high and low waters per day and are known as diurnal (daily) tides and have periods ~24 h. Spring tides are periods of increased tidal range and occur when the Earth, the Sun, and the Moon are along the same axis such that the gravitational forces of the Moon and the Sun both contribute to the tides. Spring tides occur immediately after full moon and the new moon. Neap tides periods of low tidal range which occur when the gravitational forces of the Moon and the Sun are perpendicular to one another (with respect to the Earth). Neap tides occur during 1st and 3rd quarter of the moons.

As the tide generating forces are related to the periodic gravitational forces of the Moon and the Sun, there are specific periods which may be identified from the equilibrium tide (Pugh 1987). For example, these include the 12.24 h and the 24 h for the main semi-diurnal and diurnal periods; the lunar month (29.5 days); the period between two successive full or new moons; the annual cycle due to the changes in the earth's orbit around the Sun. In the longer term, changes in the orbits of the moon and the sun provide 4.45 year and 18.6 year variations in the tides and these are discussed in Sect. 7.9.

The dynamic theory of tides which governs the tidal characteristics of the ocean basins considers the configuration of the ocean basins (width, length, and depth), frictional forces, Coriolis force, convergence and resonance, and many other variables (Boon 2004). As a result, tides are considered as a series of Amphidromic systems consisting of rotating (Kelvin) waves which rotate around a point where the amplitude of the tide is zero, defined as an Amphidromic point. Due to the influence of the Coriolis force, the Amphidromic systems rotate clockwise in the southern hemisphere and counterclockwise in the northern hemisphere. Close to an Amp-hidromic point the tidal range is zero and the range increases away from the point (Boon 2004). The tides at Fremantle, which generally are representative of the tides experienced along south-western Australia, are classified as diurnal (Ranasinghe and Pattiaratchi 2000). This is due to the location of a semi-diurnal amphidromic system close to the coast and the diurnal amphidromic system located off the coast of South Africa. The four largest tidal constituents (Pugh 1987) are associated with diurnal and semi-diurnal effects of the sun and moon (Fig. 7.5; Table 7.3).

Along south-western Australia, the tide's diurnal component has a range of 0.6 m, and the semidiurnal tide has a range of only 0.2 m. The semidiurnal tidal range is related to the lunar cycle, with the maximum tidal range occurring close

Il.5 el

1996 1996.1 1996.2 1996.3 1996.4 1996.5 1996.6 1996.7 1996.8 1996.9 1997

1996 1996.1 1996.2 1996.3 1996.4 1996.5 1996.6 1996.7 1996.8 1996.9 1997

Semi-diurnal Diurnal b

1996.1 1996.2 1996.3 1996.4 1996.5 1996.6 1996.7 1996.8 1996.9 1997 Time (year)

Fig. 7.5 a Time series of water level record at Fremantle for 1996 and b Morlet wavelet analysis of Fremantle tide record showing the alignment of diurnal and semi-diurnal energy during the equinox and out of phase during the solstice

Semi-diurnal Diurnal b

1996.1 1996.2 1996.3 1996.4 1996.5 1996.6 1996.7 1996.8 1996.9 1997 Time (year)

Fig. 7.5 a Time series of water level record at Fremantle for 1996 and b Morlet wavelet analysis of Fremantle tide record showing the alignment of diurnal and semi-diurnal energy during the equinox and out of phase during the solstice

Table 7.3 Principal tidal constituents for Fremantle

Constituent Amplitude Period Description

Kj 0.165 m 23.93 h Principal Lunar Diurnal

O1 0.118 m 25.82 h Principal Solar Diurnal

M2 0.052 m 12.42 h Principal Lunar Semi-diurnal

S2 0.047 m 12.00 h Principal Solar Semi-diurnal to the full and new moons, and minimum tidal ranges occurring close to the lunar cycle's first and last quarter's—the spring-neap cycle (see above). Diurnal tides are related to the declination angle of the moon's orbital plane. Therefore, the terminology of spring and neap tides is inaccurate in diurnal systems and are defined as tropic and equatorial tides. For tropic tides (analogous to spring tides for semidiurnal systems) the tidal range is a maximum when the declination of the moon is a maximum north or south of the equator. For equatorial tides (analogous to neap tides for semi-diurnal systems) the moon is directly above the equator resulting in a low tidal range. The diurnal and semidiurnal tides oscillate at a frequency of 13.63 and 14.77 days, respectively. This phase difference, of 1.14 days, between the two tidal signals modulates the resultant tide over an annual cycle, causing the diurnal and semidiurnal tides that are in phase during the solstice (resulting in a maximum tropic tidal range) and out of phase at the equinox (resulting in a minimum tropic tidal range). This process is illustrated in Figs. 7.5 and 7.6. This means the highest tropic tidal range does not always correspond with the full/new moon cycle with the daily tidal range varying biannually, with solstice tidal peaks (December-January and June-July) producing a tidal range that is about 20% higher than during equinoctial troughs (February-March and September-October).

Oceanography 1940

273 294 315 336 357

b Year day (2001)

Fig. 7.6 a Diurnal and semi-diurnal components of the tide at Fremantle from day 273 (October 1) to day 365 (December 31) in 2001; b water level from the summation of the diurnal and semidiurnal constituents. The moon phases are shown at the top of the Figure. (From O'Callaghan et al. 2010)

273 294 315 336 357

b Year day (2001)

Fig. 7.6 a Diurnal and semi-diurnal components of the tide at Fremantle from day 273 (October 1) to day 365 (December 31) in 2001; b water level from the summation of the diurnal and semidiurnal constituents. The moon phases are shown at the top of the Figure. (From O'Callaghan et al. 2010)

During the solstice, when the diurnal and semidiurnal tides are in phase, the maximum tidal range corresponds with the full/new moon cycle; during the equinox, the maximum tidal range does not correspond with the full/new moon cycle. Mixed tides occur during equatorial tides closest to the equinox, with two high and low waters commonly observed over a tidal cycle. Hence, in a diurnal tidal system, such as along south-west Australia, definitions such as spring and neap tides do not always relate to phases of the moon, as is the case for semidiurnal tides.

Another consequence of the diurnal tides is the seasonal change in the times of high/low water. During the summer, along the south-west Australian coast, low water generally occurs between 4 a.m. and 12 p.m., depending on the phase of the moon, with high water in the evening. As summer progresses, the low water occurs earlier; as winter starts, the low water occurs later at night, becoming progressively earlier in the evening (with high water occurring in the morning).

7.6 Coastal-Trapped Waves

The power spectra of sea level (Fig. 7.2) indicates a broad peak in energy in the 'weather' band (5-20 days) and these are generally due to atmospheric effects. Closer examination and comparison of the tidal residuals with local meteorological data revealed that a number of significant tidal residuals that were not fully explained by local synoptic conditions but was a combination of locally generated and remotely generated signals, the former through local changes in atmospheric pressure and local wind. The remote signal is characteristic of a long period coastally trapped shelf wave, travelling anti-clockwise relative to the Australian coast.

A coastally trapped wave is defined as a wave that travels parallel to the coast, with maximum amplitude at the coast and decreasing offshore. Examples of these waves include continental shelf waves (CSWs) and internal Kelvin waves (Le Blond and Mysak 1978), which are governed through vorticity conservation (Huyer 1990). Coastally trapped waves need a shallowing interface and may develop a range of modes according to the shelf structure (Tang and Grimshaw 1995). They travel with the coast to the left (right) in the southern (northern) hemisphere. Along the Australian coast, shelf waves propagate anti-clockwise relative to the landmass. The governing equations (neglecting advection and friction) are (Huyer 1990):

where, u and v are the velocities in the x (east) and y (north) directions; n is the displacement of the sea surface and f is the Coriolis parameter. The solutions for

Eqs. 7.2 and 7.3 (together with the continuity equation and appropriate boundary conditions), along a boundary oriented east-west are given by (Huyer 1990):

where, no is the maximum amplitude at the shoreline, h is the water depth, k and a> are the wave number and frequency, respectively. This is an equation of a Kelvin wave, propagating along the coastal boundary, with the wave signal reducing in amplitude exponentially with distance offshore. Continental shelf waves (CSWs) depend on only the cross-shelf bathymetry profile and the vertical density profile controls the structure of an internal Kelvin wave (Huyer 1990). The alongshore component of wind stress usually generates CSWs, which are active along the Western Australian coast, first reported by Hamon (1966). Provis and Radok (1979) demonstrated that these waves propagate anti-clockwise along the south coast of the Australian continent over a maximum distance of 4000 km at speeds of 5-7 ms-1 (see also Eliot and Pattiaratchi 2010).

Along the west Australian coastline, the continental shelf waves are generated through the passage of mid-latitude low-pressure systems and tropical cyclones. The continental shelf waves can be identified from the sea level records by low-pass filtering (i.e. removal of the tidal component). An example is shown on Fig. 7.7 for tidal records from Geraldton, Fremantle and Albany (Fig. 7.1). Several CSWs with amplitudes ranging from 0.1 to 0.5 m can be identified. For example, between days 290 and 295, an increase of ~0.5 m in the sub-tidal water level was observed at Geraldton. The same variation in water level signal was seen at Fremantle and Albany,

«

1

1 1 1 t L I I

1 1 1 1 1 I 1

i c

1

320 330

Year day (in 2001)

350 360

2BO 290

320 330

Year day (in 2001)

350 360

Fig. 7.7 Low-frequency water levels at a Geraldton, b Fremantle, and c Albany for days 275-365 in 2001 showing the presence of continental shelf waves. (From O'Callaghan et al. 2007)

and could be attributed to the passage of a CSW. The correlation coefficients between sub-tidal water levels at these three locations were all greater than 0.8, despite observations being several hundred kilometers apart. The propagation time of the CSW between Geraldton and Fremantle was 23 h, and between Fremantle and Albany it was 17 h, yielding a mean propagation speed of ~500 km day1 (~6 ms-1). The period of the continental shelf wave range between 3-10 days and corresponds to the passage of synoptic systems from west to east across the west Australian coastline.

Tropical cyclones are intense low pressure systems which form over warm ocean waters at low latitudes and are associated with strong winds, torrential rain and storm surges (in coastal areas). They may cause extensive damage as a result of strong winds and flooding (caused by either heavy rainfall and/or coastal storm surges). The impacts of tropical cyclones on the North-West region of Australia are well known with several severe cyclones impacting this region over the past few years. The most noticeable impacts of these cyclones are normally restricted to the region of impact of the cyclone, and hence the direct effect of cyclones on southwestern Australia is rare. Fandry et al. (1984) identified 1 to 2 m amplitude peaks in sea level propagating southwards with speeds ranging between 400-600 km day-1. These were associated with tropical cyclones travelling southward and were attributed to a resonance phenomenon when speeds of the southward component of the cyclone speeds were close to the southward propagating continental shelf wave.

Sea level records at Fremantle indicate remote forcing due to tropical cyclones. Comparison between the low frequency component of sea level records along the west and south coasts of Western Australia with the occurrence of tropical cyclones in the North-West shelf region has revealed that every tropical cyclone, irrespective of its severity and path, generated a southward propagating sea level signal or a continental shelf wave (Eliot and Pattiaratchi 2010). The wave can be identified in the coastal sea level records, initially as a decrease in water level, 1-2 days after the passage of the cyclone and has a period of about 10 days. As an example, water level record at Fremantle for the period 1-19 December 1995 is shown on Fig. 7.8. Tropical cyclone Frank was declared as a category 1 cyclone on 7 December and

Fig. 7.8 Sea level record at Fremantle (thin black line) during December 1995 showing the low-frequency water level variation (thick-line) induced by Tropical Cyclone Frank

developed into a category 4 cyclone by 11 December and crossed the coastline near Carnarvon on 12 December. The evidence of the continental shelf wave becomes evident on 8 December when the water level starts to decrease and reaches a minimum level on 10 December and a maximum peak on 14 December. The wave height (trough to crest) was 0.55 m, higher than the tidal range during this time (Fig. 7.8).

7.7 Seasonal Changes

Mean sea level varies in an annual cycle averaging 0.22 m with water levels reaching a maximum in May-June and minimum October-November (Fig. 7.9). This variation is attributed to changes in the strength of the major ocean current in the region, the Leeuwin Current (Thompson 1984; Pattiaratchi and Buchan 1991; Feng et al. 2004).

The Leeuwin Current is a shallow (<300 m), narrow (<100 km wide) poleward boundary current flowing off the West Australian coast. It transports relatively warm, lower salinity water of tropical origin southward generally along the 200 m depth contour (Pattiaratchi and Woo 2009). During October to March the Current is weaker as it flows against the maximum southerly winds, whereas between April and August the Current is stronger as the southerly winds are weaker (Godfrey and Ridgway 1985). The Leeuwin Current is driven by the large-scale density field in the eastern Indian Ocean and is in geostrophic balance (Woo and Pattiaratchi 2008) and hence, along the Western Australian coast, a southward flow generates onshore motion. This onshore motion, which is dependent on the strength of the current, creates a set-up of the water level at the coast. This channels the flow along the shelf edge, with a sea surface gradient balancing the tendency for shoreward motion. Thus sea level is higher when the Leeuwin Current is stronger (April to August due to lower southerly wind stress) and lower between October and January when the Current is weaker (higher southerly wind stress).

Flow Motion Tides

Jun Aug

Time (months)

Fig. 7.9 Mean monthly sea levels at Fremantle for the period 1943-1988

Fig. 7.10 Time series of mean annual sea levels at Fremantle for the period 1960-1990

1945 1950 1955 1960 1965 1970 1975 1980 1985 Time (years)

Fig. 7.10 Time series of mean annual sea levels at Fremantle for the period 1960-1990

7.8 Inter-Annual Changes

Inter-annual changes in sea level, with amplitudes up to 20 cm (Fig. 7.10), are also linked to the strength of the Leeuwin Current (Sect. 7.7). During La Nina events the Leeuwin current is stronger (higher sea level) whilst during El Nino events the Current is weaker (lower sea level). This also implies a strong correlation between mean sea level and the Southern Oscillation Index (SOI), an index reflecting El Nino/La Nina events (Pattiaratchi and Buchan 1991; Feng et al. 2004). Annual and inter-annual variability is mainly due to changes in volume transport of oceanic current systems (the Leeuwin Current) and to the El Nino Southern Oscillation (ENSO). The relationship between the annual mean sea level and the Southern Oscillation Index (SOI), a measure of ENSO, varies over time. From 1989 to 1998, the sea level and SOI signals were virtually identical in relative amplitude and phase, with a 1 unit change in SOI representing a 13 mm change in mean sea level. The relationship is less clear during the period 1920-1940 which exhibit a poor correlation between SOI and mean annual sea level. This period corresponds to a period where the SOI was almost invariant and but experienced the highest changes in mean water level over the past 100 years. This indicates processes other than the SOI signal is contributing to the variability in mean sea level.

7.9 Decadal Variations Due to Tides

Tides are modulated by variations in the amplitude of the diurnal or semi-diurnal tide, associated with longer-period relative motions of the earth, moon and sun (Pugh 1987). The effects of long-term tidal modulation have been identified from different regions with the two main signals being the 18.61-year lunar nodical cycle and the 8.85-year cycle of lunar perigee (Boon 2004; Shaw and Tsimplis 2010). Although there are fluctuations in gravitational potential associated with these motions, the direct tidal response to forcing at these time scales is theoretically small and is of the order of 4% for the semi-diurnal tide (Pugh 1987). Higher tidal modulation at the

Fig. 7.11 The 99% (grey) and 95% storm surge exceedance curves showing the 18.61 nodal cycle. (Modified from Eliot (2010))

18.6-year cycle has been identified in diurnal regions, ranging between ±15% and ±20% of the tidal constituent (Pugh 1987). Thus at Fremantle, located in a diurnal tidal regime, the influence of the 18.6-year cycle could be an important component of the modulation of the tide over this time scale.

The 18.61-year lunar nodical cycle arises from the variation in the moon's orbit. The moon, in making a revolution around the earth once each month, passes from a position of maximum angular distance north (23.5° ± 5°) of the equator to a position of maximum angular distance south (23.5° ± 5°) of the equator during each half month. This is termed a tropical month and has a period of 27.32 days (Pugh 1987). This angular distance is defined as the lunar declination and twice a month the moon crosses the equator. The cycle of variation from 18.5° (23.5° - 5°) to 28.5° (23.5° + 5°) is defined as the nodal cycle and has a period of 18.61 years. This cycle modulates the tide generating forces and in particular influences the diurnal tides. Analysis of the tidal record from Fremantle indicates that the lunar nodal cycle has a range ~15 cm in the region (Fig. 7.11) which comparable to the a number of other processes discussed in this paper (Table 7.1) and is ~25% of the mean tidal range. Thus it forms a significant component of the sea level variability in decadal terms. The cycle most recently peaked in 2007 (Fig. 7.11) and thus the region will experience a decreasing effect from this process until 2016-2017. The increase in the magnitude of the nodal tides in the region has been attributed to the dominant diurnal tides in the region (Eliot 2010). These decadal changes in tidal modulations have a significant influence on coastal flooding and management.

7.10 Global Mean Sea Level Processes

Relevant global sea level processes can be considered from two time-scales:

(1) inferred from geological evidence, particularly over the last 20,000 years; and,

(2) the historic record, largely determined from coastal tide gauge measurements. Sea level rise in the west Australian region over recent geological time frames has been inferred from geological records (Wyrwoll et al. 1995). This behaviour

Fig. 7.12 Sea level event data for Western Australia. (From Wyrwoll et al. 1995)

Present Sea Level

+ Huon coral reef data • Morley core data ■ Suomi core data □ Swan River data o Abrolhos outcrop data

-3000

largely corresponds to global analysis of sea level records with rapid sea level rise subsequent to the last Ice Age, reaching present levels approximately 6,000 years before present, then subsequently staying largely constant with the mean sea level has increasing more than 120 m since the last glacial maximum (Fig. 7.12).

As a result of global warning due to the enhanced greenhouse effect the mean sea level has been increasing over the past few decades. For example, the mean global sea level rise over the twentieth century is recorded to be 1.1-1.9 mm year-1 whilst the rate of increase since 1993 is of the order of 3 mm year-1 (Church et al. 2004). Majority of this increase has resulted due to global warning with a contribution from melting glaciers.

Sea level has been recorded at Fremantle continuously since 1897 and is the longest sea level data record in the southern hemisphere. This record indicates that there has been a mean rate of sea level rise of 1.54 mm per annum (Fig. 7.13). This rate of increase is similar to that observed globally, which has been estimated to range between 1.1-1.9 mm per annum (Douglas 2001; Church et al. 2004). Al

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Fig. 7.13 Time series of Fremantle sea level (one year running mean) with the linear trend of 1.54 mm per annum shown with dashed line

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Time (years)

Fig. 7.13 Time series of Fremantle sea level (one year running mean) with the linear trend of 1.54 mm per annum shown with dashed line though there has been an increasing trend over the past 100 years, there have been periods, which are revealed when the linear trend is removed, where the rate of mean sea level change varied with time. These variations were dominated by the inter-annual variability of sea level linked to the ENSO phenomenon. From 1900 to 1952 there were cyclic periods of sea level increase and decrease ranging between 10-14 years. Between 1952 and 1991, there was a decreasing trend, but in combination with the mean sea level rise resulted in almost constant mean sea level. A reversal of this trend occurred between 1991 and 2004, producing an apparent rapid mean sea level rise at a rate of 5 mm per annum—a rate more than 3 times the trend over the previous 100 years (Pattiaratchi and Eliot 2005). This resulted in Fremantle recording maximum sea levels in 2003 and 2004.

Acknowledgements The authors acknowledge the contributions from Mathew Eliot and Ivan Haigh to this contribution; Tony Lamberto and Reena Lowry from Department for Transport (WA), for the provision of water level data.

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