Seiching in Cockburn Sound

Transcription

1 Seiching in Cockburn Sound by Emma Molloy Department of Environmental Engineering University of Western Australia November, 2001

2 Abstract A seiche is the oscillating response of an enclosed or semi-enclosed water body to external forcing, in order to return the system to equilibrium. The disturbances that cause seiches can be a variety of atmospheric and water-based mechanisms, including wind, air pressure, waves and tsunamis. In semi-enclosed micro-tidal water bodies such as Cockburn Sound, the prime mechanisms driving seiches are changes in wind speed and direction (Luettich et al, 2000). Cockburn Sound is a north-south oriented oval-shaped harbour system located 35 km south of Perth, Western Australia. It is enclosed on all sides by land, with a large opening on the north-east side. This opening facilitates water exchange to the north of Cockburn Sound. However, to the west there are reefs restricting flow. Therefore, the seiches oscillate between Mangles Bay in the south of Cockburn Sound and Fremantle. This is considered to be a semi-enclosed ( open ) system. Cockburn Sound houses a large variety of industry, as well as marine and bird life, and is used by the public for recreation. The seiches in Cockburn Sound have an impact on all of these areas. The primary effect of seiches is in influencing the mixing and flushing of the system, especially of contaminants released by the industries. Water level data were collected from Mangles Bay at the southern end of Cockburn Sound during May Water level data were also available from previous studies in this area. Meteorological data, including wind speed and direction and atmospheric pressure, were obtained from the Bureau of Meteorology. Spectral plots of the water level data from all the locations indicated peak at hours, assumed to be due to seiches. The water level and meteorological data were closely examined to determine the mechanisms responsible for causing seiches and changing the seiches that were already present. The general patterns that were observed are that: The wind direction changing is the primary cause of seiches in Cockburn Sound. Various factors, such as the wind speed, atmospheric pressure, and total water level, will influence the magnitude of the seiches generated. A change in wind speed of a least 2 m/s may also change the water level oscillations. During spring and summer, when there is a diurnal wind pattern, the seiches vary diurnally in amplitude. They may also be influenced by the diurnal variation in total water level associated with tides.

3 Data from other locations in the Perth area indicated that the seiches generated propagate up the coast from Mangles Bay at least to Two Rocks Marina, and up the Swan River to at least Barrack Street Jetty. The celerity of the seiche was calculated to be 9.5 m/s. The seiche was expected to have a node at Fremantle. However, this was not observed. Further work needs to be done in this area to determine the mode and limits of the seiche that travels along the west coast of Western Australia. Also, other forcing mechanisms that may have an influence on the seiches in this area should be studied.

4 Table of Contents Literature Review and Background Information 1 1 Introduction to Seiches What Are Seiches? How Seiches Form The Theoretical Study of Seiches The Effects of Seiches Observing Seiches 8 2 Introduction to Cockburn Sound Forcing Mechanisms in Cockburn Sound Remote Mechanisms Local Mechanisms The Local Climate Other Important Influences The Importance of Understanding Seiches in This Area 17

5 3 Methodology Field Work Data Collection Data Transformation Data Analysis 25 4 Results and Discussion Spectral Analysis Time Series Analysis Correlations Between Data Sets James Point 2000 Seiches Mangles Bay 2001 Seiches Wind Direction Changing from South to North Wind Direction Changing from North to South Wind Direction Changing between East and West Wind Speed Changes Estimating the Friction in this System Data Seiches Cross-Spectral Analysis 69 5 Conclusions 73 6 Recommendations for Further Work 75 References 77

6 Literature Review and Background Information 1 Introduction to Seiches 1.1 What Are Seiches? A seiche is the oscillating response of an enclosed or semi-enclosed water body to external forcing. A free seiche occurs when there is an initial force that disturbs the water level in a system. When the initial force stops, the water level changes in the opposite direction to that imposed by the initial force, in an attempt to return the system to equilibrium. However, the inertia of the water carries the system past equilibrium. The water level then continues to oscillate about the equilibrium level at a period characteristic of the system (Sorensen, 1978). This natural period depends on the length and depth of the basin (Sorensen, 1978). The magnitude of the initial force only influences the magnitude of the oscillations (Sorensen, 1978). Free seiches decay exponentially due to friction, if the forcing is not repeated (Sorensen, 1978). A forced seiche occurs when the forcing event is cyclic, but with a period different to the natural period of the system (Sorensen, 1978). This causes the water level to oscillate at periods that are closer to the period of the forcing than to the natural period of the system. There is resistance to oscillation at these periods, so work must be done to maintain a forced seiche (Wilson, 1972). The oscillation of a seiche is a special case of a standing wave, which may be considered as the addition of two identical waves travelling in opposite directions. The interactions of these waves result in nodes and antinodes. Nodes are points where there is no vertical movement of the water surface, but there is maximum horizontal movement. Nodes occur at openings to basins (van Rijn, 1994). Antinodes are points where there is maximal vertical movement of the water surface, but there is no horizontal movement. Anti-nodes occur at points where the seiche is reflected off a surface (van Rijn, 1994). For example, anti-nodes are present at the closed end of a basin. Seiches can occur in various wavelengths for each water body. This results in different numbers and locations of nodes and antinodes (figure 1). Page 1

7 Literature Review and Background Information Figure 1.1: Different modes of oscillation of seiches in a rectangular basin of uniform depth. (a) uninodal; (b) binodal; (c) trinodal; (d) quadrinodal; (e) quinquinodal; (f) sextinodal; (g) septuanodal; (h) plan view (from Wilson, 1972). 1.2 How Seiches Form Seiches can only form in systems where the standing wave can reflect off of something at each end. A typical example of this is a closed basin such as a lake. Seiches can also occur in semi-enclosed systems such as harbours that are mostly enclosed, with an opening at one end. A semi-enclosed system may have an abrupt change in bathymetry that the seiche is reflected off. The causes of seiches are various. In both lake and harbour systems, seiches can be caused by: 1. The passage of small barometric fluctuations, associated with the general system of isobars, with a period close to the natural seiche period of the system. Page 2

8 Literature Review and Background Information 2. A rapid change in air pressure due to a squall. 3. The impacts of wind gusts on the water surface. 4. A lapse in strong onshore winds, causing pent-up water at the shore to be released. 5. Heavy rain, snow or hail over a portion of the water body. 6. Flood discharge from a river at one end of the system. 7. Tilting or movement of the lake or sea bed resulting from seismic movement of the earth due to earthquakes. (Chrystal, ) For both coastal and lake seiches, meteorological causes are the most important (Wilson, 1972). However, coastal seiches have other causes as well. The first of these is long-period ocean waves. A harbour will respond to long-period waves that have the same period as the natural resonance of the harbour (Wilson, 1972). However, seiches are also proposed to respond to long waves of very low height, which are hard to detect (Wilson, 1972). This response is not well understood. Long-period ocean waves can be generated in three ways. The first is a combination of variable wind stress and atmospheric pressure fluctuations. It is theoretically possible for long-period waves to be generated by moving pressure disturbances. However, the identification of waves generated this way is difficult due to the various wave spectra close to the shore, including surf beat and seiches. (Wilson, 1972). Long-period ocean waves can also be generated by surf beat. These long-period waves are predominantly in shallow water, moving shoreward. They are produced by the radiation stress resulting from ordinary waves breaking on the beach. Seiches also occur during periods of high swell. These seiches may be caused by long-period waves induced by the swell, which cannot otherwise be seen. (Wilson, 1972). The third way in which long-period ocean waves are generated is by seismic disturbances of the ocean bed (Wilson, 1972). These long-period waves are known as tsunamis. Another cause of coastal seiches is deep-sea internal waves generated by tides. The generation of seiches by this method is dependent on a large tidal range. The depth of the mixed layer and/or the strength of the relevant oceanic current may also play an important role. Seiches may be caused by tide-generated deep-sea internal waves at sites on the shores of marginal seas, where seiches have a distinctly seasonal distribution. (Giese et al, 1990). Page 3

9 Literature Review and Background Information Coastal seiches may also be induced by shear flow. This results from the instability caused by the interaction between harbour flows and a sea current with a horizontal shear passing the harbour mouth. (Fabrikant, 1995). Eddies generated by the currents moving past a harbour opening may also cause seiches inside the harbour (Sorensen, 1978). Atmospheric pressure is an important force involved in generating seiches. However, as areas of different atmospheric pressure move, and have various scales, it is unlikely that they would directly cause seiches. It is believed that the atmospheric pressure changes induce an oceanic wave, which is the intermediate mechanism responsible for seiche generation. (Gomis et al, 1993). Additional causes of seiches include strong wind stress events and edge waves travelling on the continental shelf. (Gomis et al, 1993). 1.3 The Theoretical Study of Seiches In order to simplify the theoretical study of seiches, various assumptions have been made. These are (Miles, 1974): 1. The seiches occur in a perfect, incompressible fluid. This means that compression (sound) waves, and viscous and capillary effects may be neglected. This is a safe assumption for studies in the real world. 2. Only small displacements occur due to the seiches. This is adequate for the calculation of motion within a harbour. However, there are some limitations (see Miles, 1974). 3. The water surface is considered to be a plane level surface in an inertial reference frame. Hence, the effects of the earth s rotation are only important for fluctuations with periods of more than twelve hours. 4. The perturbation pressure is assumed to be proportional to the vertical displacement of the free surface. That is, if the fluid is homogeneous then the hydrostatic pressure will be in equilibrium vertically. This excludes internal gravity waves associated with stratification. 5. The seiches may be described using the theory of simple harmonic motion. 6. The basin is of uniform depth. This assumption is often unsuitable for harbours and the open ocean, especially if several modes of oscillation are being considered. Page 4

10 Literature Review and Background Information Seiches can occur in both closed and semi-enclosed ( open ) water bodies. However, some semienclosed water bodies may act more like closed systems. A semi-enclosed system may be treated as a closed system if: 1. The seiches are transverse. 2. The seiches are longitudinal, but the harbour has a narrow entrance. 3. The seiches are longitudinal, but there is a shelf restricting the flow of water out of the harbour, and reflecting the seiches back. A narrow entrance is when the width of the entrance is much less than the width of the basin (van Rijn, 1994). The most important characteristic of a seiche is its mode. The mode of a seiche is the number of nodes it has within the system (figure 1.1). The period of a seiche with n nodes is given by Merian s formula. 2L T n = 1 n( gh)2 This assumes that the basin is rectangular, with a uniform depth. For closed systems (Wilson, 1972): For open systems (Giese et al, 1990): T n = n 4L 1 ( gh)2 Where: Tn is the period of an n th mode seiche, L is the wavelength of the seiche (the length of the basin), n is the number of nodes/ mode of the seiche, g is acceleration due to gravity, 9.81 m/s 2, and h is average water depth. The different seiche modes are not mutually exclusive. Seiches with various different modes can occur together in a system. However, the uninodal (fundamental) oscillation is usually dominant (Wilson, 1972). For closed systems, there is an anti-node at each end. At the fundamental mode, there is a single node in the middle of the basin. The period derived from Merian s formula is the time it takes for the water to Page 5

11 Literature Review and Background Information oscillate from one end of the basin to the other and back. That is, to travel a distance of twice the basin length. For open systems, there is an anti-node at the closed end of the basin, and a node at the open end of the basin. The period derived from Merian s formula is still the time it takes for a complete oscillation. However, only half of a wave is contained within the length of the basin. Therefore, for a complete oscillation the wave has to travel four times the length of the basin. The topography of a closed system is important in determining the likelihood of seiches occurring. If the topography is relatively regular, then there is less damping of the oscillation. Hence, a small disturbance will produce a relatively strong oscillating response. In contrast, if the topography is rough then there will be heavy damping of the oscillations. Therefore, a relatively weaker oscillating response is expected. (Wilson, 1972). In a semi-enclosed system, the topography is not as critical. This is because there is a large spectrum of disturbances impacting on the system from the ocean. Hence, the right frequency disturbance to induce natural resonance is often present. Disturbances that can induce forced oscillations at non-resonant frequencies are also often present. (Wilson, 1972). The damping of seiches is primarily due to bottom friction. If data are available that indicate a seiche reducing due to friction, then the bottom friction factor can be calculated according to the following formula (van Rijn, 1994): ( t + T ) h() t m h T 2 = e Where: h(t) is the total height of the oscillation at time t in metres, h(t + T) is the total height of the oscillation at time t + T, T is the period of the oscillation in seconds, and m is the bottom friction factor. That is, the ratio of the consecutive heights of the oscillations is a constant that is proportional to the bottom friction. Page 6

12 Literature Review and Background Information The damping ratio is the ratio of the bottom friction to the frequency of the oscillation measured in rad/s (van Rijn, 1994). Depending on the damping ratio and the ratio of frequency times basin length to wave speed (ω l / Co ), the amplification factor, which is related to the friction in the system, can be determined according to figure 1.3. Figure 1.3: Ratio of water level amplitude at the end and at the entrance of an open basin (van Rijn, 1994). 1.4 Effects of Seiches There are both beneficial and detrimental effects of seiches. Seiches have a large mixing effect at the edges of systems, or in between layers if the system is stratified and the seiche is internal. This mixing effect controls the position and strength of a salt wedge, if it is present, and the thickness of the diffusive bottom boundary layer, where exchange occurs between the sediments and the water column (Luettich et al, 2000). In turn, the distributions of heat, salinity, dissolved oxygen and nutrients are influenced by the oscillation of the water (Wilson, 1972). This has a strong influence on the ecology. In general, the ecology benefits due to the increased availability of dissolved oxygen and nutrients. The changed water circulation resulting from seiches may also influence the residence time of water in a system (Luettich et al, 2000). The currents produced by seiches are quite significant. At the opening of Page 7

13 Literature Review and Background Information the system, these currents transport water out. If the water inside the system is the same as the water outside of the system, it is likely that the water transported out with one oscillation will come back in at the next oscillation. However, if there is a density difference between the outside water and the inside water, the water that is transported out may be replaced by fresh water. This would increase the flushing of the system. Hence, seiches have beneficial effects due to the increased mixing and flushing that they cause in closed or semi-enclosed basins. On the other hand, the currents induced by seiches may also cause destruction or erosion of structures. For example, a dolphin enclosure at Two Rocks Marina (Western Australia) was destroyed by the extreme seiche currents in the area (Gwynne, 1993). Longer period seiches in harbours also produce strong reversible currents at the entrance to harbours (Sorensen, 1978). This hampers the navigation of ships. Shorter period seiches cause problems due to both the vertical and horizontal oscillation of the water. This can hamper loading or unloading of ships (Fabrikant, 1995), or cause problems for moored ships. In extreme oscillating conditions, ships may break their moorings and cause damage to each other or dock structures (Wilson, 1972). Nevertheless, the most severe effects of seiches are those that threaten human lives. For example, a squall line that passed over Lake Michigan in Chicago in 1954 produced large seiches. The floods induced by this event caused seven drownings (Wilson, 1972). These effects, both good and bad, illustrate the importance of understanding the factors that lead to the formation of seiches. These factors include the forcing mechanisms, as well as the natural resonance period of a system. Seiches can then be predicted, and the detrimental effects can be minimised or eliminated, while the beneficial effects can be used for the most good. 1.5 Observing Seiches There are two important aspects to observing seiches. Firstly, data have to be collected the measurement of the seiche. Secondly, the data have to be analysed. Page 8

14 Literature Review and Background Information Two methods of measuring seiches are by measuring the pressure due to changing water level, or by measuring the currents. The pressure is measured by pressure sensors, usually near the bottom of the water column. These pressure sensors may be placed in different areas of the water body to observe the influence of seiches in different areas, or the interactions between water levels in different areas. (Okihiro et al, 1993). An Acoustic Doppler Current Profiler (ADCP) or similar instrument is used to provide a profile of the currents throughout the water column. Seiches produce barotropic currents, which can be easily distinguished from other, baroclinic currents (Abraham, 1997). The motivation for using currents to measure seiche activity is that seiches may cause only small changes in the water surface elevation (measured by pressure), but still produce observable currents (Abraham, 1997). Therefore, in some cases analysing only the pressure changes may result in overlooking a seiche that has significant effects. For both of these methods of observing seiches, the data that are collected also contain a lot of nonseiche data. For example, currents or changes in pressure due to tides and waves. Therefore, the data that are collected need to be analysed to extract the relevant information. This may be done in various ways, including wavelet analysis, detrending or filtering the data, simple observation of the data or elimination of irrelevant information. Wavelet analysis is suitable for isolating periodic signals in non-stationary data, which is required when analysing for seiches. In wavelet analysis, a localised phase and amplitude is generated for each frequency component in the time series. For more information on the analysis of data for seiches using wavelet analysis, see Luettich et al (2000). The time-series data collected may be detrended to suppress fluctuations that have periods outside the range of interest. For example, Okihiro et al (1993) detrended their data with a cubic polynomial to suppress motions with periods longer that the record length of 4.6 hours. Gwynne (1993) performed a similar data transformation. She filtered her data using a Butterworth filter in order to remove periods outside the range she was interested in (Gwynne, 1993). Data collected with an ADCP may be analysed by simple observation. Seiches produce currents that are approximately uniform throughout the depth of the water column, in contrast to other currents. Hence, the seiches produce vertical bands in the ADCP data. Page 9

15 Literature Review and Background Information The final method for analysing data for seiches involves elimination of irrelevant data. For example, the mechanisms causing tides are well known, so tides can be predicted. Therefore, the influence of tides on the water level changes can be subtracted from the data (Wilson, 1972). This will leave a much simpler data set. Any other known influences can also be subtracted. Page 10

16 Literature Review and Background Information 2 Introduction to Cockburn Sound Cockburn Sound is a harbour system located about 35 km south of Perth, Western Australia. It is approximately oval shaped, oriented north-south (see figure 3.1.1). The main basin is approximately 16km long and 7 km wide, with a maximum depth of 22 m (Steedman and Craig, 1983). The mean depth is 12 m (Chiffings, 2000). The east and south sides are completely enclosed by the mainland, while most of the west side is bound by Garden Island. A causeway connects the mainland and Garden Island at the south end of the western side of the Sound. The causeway is built on a rock-filled base, with only two openings: one is 300 m wide and 2.8 metres deep, the other is 600 m wide and 4.5 m deep (Steedman and Craig, 1983). At the northern end of the Sound is a sill called Parmelia Bank. It ranges in depth from 2 m to 5 m, and covers most of the northern opening (Steedman and Craig, 1983). The northern opening is 7 km wide. The width of the basin is also approximately 7 km. Obviously, the width of the opening in not much less than the width of the basin. Therefore, Cockburn Sound is an open system. As Cockburn Sound is essentially an open system, the tides are very similar to those of the open ocean (DEP, 1996). The maximum spring tide has a range of less than 0.9 metres (DEP, 1996). This is a micro tidal system. As Cockburn Sound is an open basin, the water freely flows out of the opening. However, north of Cockburn Sound to Fremantle, the west side is bound by a submerged reef system (see figure 3.1.1). Therefore, after leaving Cockburn Sound the water can only flow north to Fremantle. When the water reaches Fremantle, the westerly border is no longer as restrictive and the water can flow both westerly and northerly. Therefore, between Mangles Bay in the south of Cockburn Sound and Fremantle, there is a natural basin that is open at the northern end (Wright, 2000). It is this basin in which the seiches oscillate. Again, the width of the opening is not much less than the width of the basin. Therefore, this is an open system. The length of this natural basin is approximately 24.5 km. The mean depth is approximately 10 m. Therefore, using Merian s equation for an open basin, the fundamental frequency is calculated to be 2.7 hours. This is the natural frequency at which the water level is expected to oscillate after an initial disturbance. As this is an open basin, it is expected that there would be an anti-node at Mangles Bay and a node at the opening at Fremantle. Page 11

17 Literature Review and Background Information 2.1 Forcing Mechanisms in Cockburn Sound Seiches in Cockburn Sound may theoretically be caused by any of the forcing mechanisms described in 1.2. These can be divided into remote or local forcing mechanisms. The remote forcing mechanisms include: long period ocean waves, which may be generated by remote atmospheric pressure changes, surf beat, swell, or remote seismic disturbances edge waves on the continental shelf. The local forcing mechanisms include: heavy rain, snow or hail over a portion of the water body flood discharge at one end of the water body tilting of the sea bed due to seismic activity deep-sea internal waves generated by tides shear flow across the mouth of the harbour the effects of local winds and local atmospheric pressure changes, including the general system of isobars, and squalls. Each of these mechanisms will be discussed to determine the most likely forces that are causing the seiches in Cockburn Sound Remote Mechanisms Long period ocean waves generated by remote mechanisms may also cause seiches. The first mechanism that may generate long period ocean waves is a change in atmospheric pressure at a remote location. However, this mechanism has not been observed. It is simply a proposal at this stage (Gomis et al, 1993). Therefore, this mechanism will not be directly studied in Cockburn Sound. Long period ocean waves may also be generated by the action of surf beat and swell (Wilson, 1972). However, Cockburn Sound is sufficiently sheltered that surf beat and swell do not have an impact (Treloar et al, 1989). Page 12

18 Literature Review and Background Information The final mechanism for generation of long period ocean waves is remote seismic activity. This is only relevant for Cockburn Sound if there is seismic activity in an area bordering the Indian Ocean. It is assumed that tsunamis are not an issue in Cockburn Sound. Edge waves on the continental shelf may also cause seiches. Continental shelf waves are often caused by tropical cyclones (Pattiaratchi, 2001a). However, there is only limited understanding of continental shelf waves on the West Australian coast (DEP, 1996) Local Mechanisms Heavy precipitation over one end of a water body may increase the pressure enough to disturb the water level (Chrystal, ). However, it is difficult to separate the pressure effect of the rainfall from the disturbance created by the barometric pressure (Chrystal, ). Therefore, precipitation as a forcing mechanism will not be considered separately for Cockburn Sound. Flood discharge at one end of a system will increase the water level initially at that end (Chrystal, ). This may cause seiches in the water body. However, this is unlikely to be a forcing mechanism in Cockburn Sound for two reasons. Firstly, there are no rivers within the Cockburn Sound catchment to discharge high water flows into the Sound. Secondly, the longitudinal axis of Cockburn Sound is aligned with the coast (figure 3.1.1). Therefore, any large water flows would increase the water level along the length of Cockburn Sound. This would not generate longitudinal seiches. Tilting of the sea bed due to local seismic activity would increase the water level in one section of a system, while reducing the water level at another area in the system. This would be an effective mechanism for generating seiches. However, no seismic activity of magnitude greater than 2 on the Richter scale have occurred in the Perth metropolitan area (Department of Geology and Geophysics, UWA, 2001). Also, all of the seismic activity near Perth is east of the city (Department of Geology and Geophysics, UWA, 2001). Therefore, tilting of the sea bed will not be considered as a forcing mechanism for seiches in Cockburn Sound. Deep-sea internal waves generated by tides are a forcing mechanism for seiches in some harbours. However, large tides are required to generate seiches by this mechanism (Giese et al, 1990). The tides in Cockburn Sound are small (Steedman and Craig, 1983). Therefore, deep-sea internal waves generated by tides will not be considered as a forcing mechanism for seiches in Cockburn Sound. Page 13

19 Literature Review and Background Information Shear horizontal flow across a harbour mouth generated by currents may induce seiches. The Leeuwin Current is the most significant current in the Perth area. However, this is about 200 km offshore. Therefore, it is not expected that the shear horizontal flow across the harbour mouth cause seiches in Cockburn Sound. Local winds may induce seiches in two ways. Firstly, a prevailing wind from one direction will cause a build-up of water against the shore. When the wind stops, the built-up water will cause seiche oscillations in order to return the water level to equilibrium (Chrystal, ). If the water is deep enough, return flows will prevent the build-up of water against a shore (Chrystal, ). However, Cockburn Sound is not very deep. Therefore, return flows do not transport built-up water. This seems to be a likely mechanism by which seiches may be generated in Cockburn Sound. Secondly, wind gusts may have an impact on the water level of a system. A wind velocity of 16 km/hr produces a pressure equivalent to 1.5 mm of water, when it acts directly on a small area (Chrystal, ). This pressure will then cause a decrease in the water level in this area, which may induce seiches to return the system to equilibrium. However, there are problems in obtaining data on seiches caused by this phenomenon (Chrystal, ). Also, the wind data that are available are only documented for every half hour the peak gusts are documented, but not the time within the half hour when they occurred. The wind data were also obtained from Swanbourne. The gusts that were measured in Swanbourne may not be closely related to the gusts that occurred in Cockburn Sound. Therefore, this mechanism will not be considered in Cockburn Sound. Local atmospheric pressure changes are an important cause of seiches. Again, they may cause seiches in two main ways. Firstly, the progression of the general system of isobars, if they have a period similar to the natural resonance of the system, may theoretically induce seiches (Chrystal, ). However, Chrystal (1908-9) considered an extreme case and concluded that the maximum change in the range of a seiche resulting from this cause was mm. Therefore, the natural progression of isobars will not be considered as a cause of seiches in Cockburn Sound. Secondly, local atmospheric pressure changes caused by squalls may effect seiches. A squall is a sudden increase in the mean wind speed, lasting for at least several minutes (Bureau of Meteorology, 2001). This increase in wind speed leads to a decrease in atmospheric pressure (Chrystal, ). If the decrease in pressure is only on one end of a water body, it can induce a seiche (Chrystal, ). This is a possible mechanism for inducing seiches in Cockburn Sound. Also, if the squall lasts for a reasonable time, say an hour, then the low pressure will cause the water level to become elevated. The Page 14

20 Literature Review and Background Information cessation of the squall will then release the elevated water and may cause a seiche. This mechanism may also induce seiches in Cockburn Sound. 2.2 The Local Climate The climate in Perth is described as Mediterranean, with hot dry summers and cool wet winters. Summer is from December to February. The wind during this period has a typical pattern of warm easterlies in the morning, at about 10 knots, and cool sea breezes from the south-west at knots in the afternoon (Bureau of Meteorology, 1993). Winter is from June to August. During this period, the wind is from the east to north-east in the morning, at about 10 knots (Bureau of Meteorology, 1993). In the afternoon, the wind is from the north to southwest at about 10 knots (Bureau of Meteorology, 1993). Cold fronts also pass through this area from the west approximately every 7-10 days during winter (Bureau of Meteorology, 1993). These fronts bring strong winds that blow towards the south-east (Wright, 2000). 2.3 Other Important Influences Three other forces have been identified that may be important influences on the seiches in this area. These include the Coriolis force, the inverse barometric effect, and the interactions of water depth and wind in causing water level set-up. The Coriolis force deflects water motion at 90 to the acting force, due to the rotation of the earth. In the southern hemisphere, the Coriolis force deflects motion to the left of the acting force. The Coriolis force is proportional to latitude. For a latitude of 32, which corresponds to James Point in Cockburn Sound (figure 3.1.1), the Coriolis force is 7.73 x 10-5 (Wright, 2000). The importance of the Coriolis force (f) within a system also depends on the characteristic velocity (U) and length scales (L) operating in the system. The non-dimensional Rossby radius quantifies the importance of the Coriolis force within a particular system: Page 15

21 R = 0 U fl Literature Review and Background Information For seiches in Cockburn Sound, the characteristic length scale is the distance between Mangles Bay and Fremantle, which is m. The characteristic velocity scale is assumed to be of the order of 10 m/s. This gives a Rossby radius of A Rossby radius of less than one means that the Coriolis force has a significant effect in that system. Therefore, the Coriolis force does not appear to be significant within Cockburn Sound. In order for the Coriolis force to have a significant effect, the characteristic length scale would have to be about 130 km. The second force to be considered is atmospheric pressure. When the atmospheric pressure is higher, this restricts the ability of the water level to rise. This is known as the inverse barometric effect. It is quantified by the sea level response, which indicates that a change in the atmospheric pressure by 1 hpa will induce a change in the water level by 1 cm (Pattiaratchi, 2001). This is considered to be important when determining the magnitude of an oscillating water level response to an imposed force. The third force to be considered is water level set-up by wind, and the role that water depth plays in this occurrence. The change in water level associated with wind stress is quantified by: ƒη C = ƒx D ρ W A ρgh 2 10 Where: CD is the drag coefficient, ρa is the air density, W10 is the wind speed 10 m above the water surface, ρ is the water density, g is acceleration due to gravity = 9.81 ms -2, and h is the water depth. It is obvious that a higher wind speed will induce a higher increase in the water level. It can also be seen from this equation that there is an inverse relationship between the water level response and the water depth. That is, if all other conditions are the same, water that is deeper will have a smaller water level set-up than water that is shallower. The influence of wind set-up of water level, and the constraints of water depth in this response, are considered to be important when analysing the seiches in Cockburn Sound. Page 16

22 Literature Review and Background Information 2.4 The Importance of Understanding Seiches in This Area The land bordering Cockburn Sound, and the water body itself, are hosts to various, often conflicting, parties. These include (Montague et al, 1999): The major heavy industry area of Perth, including an oil refinery, a fertiliser plant, an alumina mill, steelworks and a power station. A ferry and ship building industry, which is worth billions of dollars and exports worldwide. The Royal Australian Navy s Fleet Base West, on Garden Island. A water skiing club, pleasure boat club, and several beaches. 700 ha of seagrass meadows, birds, fish, dolphins and crustaceans living in the water. Public and ecosystem health are the main victims of development around Cockburn Sound. TBT used in ship paints is a danger to public health. Algal blooms have increased since the construction of breakwaters for the boat club. Also, seagrass meadows have reduced by 82% since industrial development began in the 1950 s. (Montague et al, 1999) The poor water circulation in Cockburn Sound means that pollutants can accumulate in the water body. Hence, the water circulation is of interest in Cockburn Sound. As discussed in section 1.4, seiches may have an impact on the mixing and flushing of semi-enclosed systems. Therefore, it is important to understand the conditions under which seiches are prominent, in order to be able to predict the mixing and flushing of the water in Cockburn Sound. This may be especially important is a chemical spill or algal bloom occurs. Page 17

23 Literature Review and Background Information Page 18

24 Methodology 3 Methodology 3.1 Field Work Data Collection The water level data were collected using a pressure sensor. The instrument used was an FSI 1D Wave and Tide Sensor. This instrument is made by Falmouth Scientific, Inc. The full-scale pressure is measured with an accuracy of ±0.01%. The wave and tide sensor was programmed to sample at a frequency of 1 Hz. The pressure was measured every second for 30 seconds, and the average of these values was logged. Nothing was measured for the next 30 seconds, then the cycle repeated. Hence, there is a pressure reading every minute. The wave and tide sensor was put out on the 11 th of May, 2001 (day 131 of the year). This was the end of autumn. The instrument was tied to a ladder at the end of the jetty on the corner of Esplanade and Fisher St, Rockingham, in Mangles Bay (point 1 in figure 3.1.1). The instrument was about ten metres offshore. The wave and tide sensor was collected on the 7 th of June (day 158). However, the battery had run out before then, so data were only logged from day 131 to day 148 (28 th of May). This provided almost 18 days of data. These data are referred to as the Mangles data, as they were collected in Mangles Bay. Three other data sets were obtained from various sources. The Garden Island data were collected at the same time as the Mangles data, but in a different location using a different instrument. The Garden Island data were collected from just south of Garden Island, on the western side of the larger Causeway opening, at point 2 in figure These data were collected using an Interocean S4DW Current Meter. An S4DW Current Meter measures currents by creating an electromagnetic field through which the water flows. The water then produces a voltage that is proportional to the magnitude of the water velocity. This instrument measured the currents twice a second for one minute, then recorded the average of these data, then rested for a minute. This resulted in data points every two minutes. Data were also collected by unknown means from James Point (point 3 in figure 3.1.1). The sampling frequency was approximately once every two minutes, for almost eight days the 18 th of May to the 3 rd of June, This illustrates conditions typical of autumn. Page 19

25 Methodology Furthermore, five data sets were collected simultaneously in 1995 by unknown means. These data were collected in Mangles Bay, Fremantle, Barrack Street Jetty, Hillarys Boat Harbour and Two Rocks Marina (see figure 3.1.2). The sampling frequency was once every 15 minutes. The data were collected from the first of September to the 31 st of December. This shows conditions typical of spring and summer. Wind and air pressure data were obtained from the Bureau of Meteorology. The wind data were from Swanbourne, and were comprised of the wind speed and wind gusts in knots and the direction the wind was going to in degrees, with north as zero degrees. The air pressure data were from Rottnest, and the air pressure was measured in hpa. All meteorological data were recorded every half hour. Page 20

26 Methodology Figure 3.1.1: Map showing approximate location of data collection points from Cockburn Sound in 2000 and 2001(adapted from Transport WA, 1998). Scale approximately 1: Page 21

27 Methodology Two Rocks Hillarys Barrack Street Jetty Fremantle Mangles Bay Figure 3.1.2: Map showing approximate location of data collection points for 1995 data (adapted from West Australian Newspapers Ltd., 1997). Scale approximately 1: Page 22

28 Methodology 3.2 Data transformation The water level data were filtered to remove low frequency fluctuations, such as tides. This was done firstly using the buttord function in Matlab. This function takes inputs of Wp, Ws, Rp and Rs and creates outputs that are used in the butter function in Matlab. The meaning of the inputs can be seen in figure below. Rp Rs Wp Ws Figure 3.2.1: Inputs used to design Butterworth filters (adapted from Gwynne, 1993). The butter function uses these inputs to create a lowpass Butterworth digital filter, and then output the coefficients of the filter. These coefficients can be used by the freqz function to plot the filter. For example, the filter for the Mangles Bay 1995 data is plotted in figure below. Page 23

29 Methodology Figure 3.2.2: An example of a low pass Butterworth filter. This one was used to filter the 1995 Mangles data. However, it is not necessary to plot the filter. The function filtfilt takes inputs of the filter coefficients obtained from butter and the raw data, and filters the data according to the defined filter. The output of filtfilt is then subtracted from the raw data to obtain the filtered data. The inputs to the buttord function for each of the data sets are given in table 3.2 below. The first two inputs vary according to the sampling frequency of the data. Table 3.2: Inputs for the buttord function for each of the data sets. Data Set Wp Ws Rp Rs Mangles Garden James Pt All 1995 data The wind data were converted from knots to m/s by multiplying the value in knots by The wind speed and wind direction data were used to derive the north-south and east-west components of the wind speed, also in m/s. The north-south component of wind speed is positive when the wind is travelling to the north, Page 24

30 Methodology and negative when the wind is travelling to the south. The east-west component of wind speed is positive when the wind is travelling to the east, and negative when the wind is travelling to the west. 3.2 Data Analysis The data analysis involved a spectral analysis of the data, plotting the data against time, calculating the correlation between different variables in the data, and closely looking at the data to deduce the patterns evident in it. Using the Mangles 2001 data, the bottom friction factor and damping ratio for the system were calculated. A cross-spectral analysis of the 1995 data was also calculated. Using information from the cross-spectral analysis, the celerity of the seiche was calculated. The spectral analysis was performed using the oppsd2 function in Matlab, developed by Bendat and Piersol (1986). This function determines a suitable length for the data, detrends the data so that it is stationary, applies a taping function to the data to minimise leakage in the analysis, and minimises variance of the spectral estimates. This makes the data suitable for spectral analysis using Fast Fourier Transforms. The power spectral density of the data is then plotted, with lines on either side representing the 95% confidence interval. The plot is on a log-log scale, so that the lower frequency peaks (below one hour) can be more easily identified. All of the water level and meteorological data were plotted against time (decimal days) on separate plots. This was also done in Matlab, using the subplot command to plot multiple graphs on one page. The northsouth and east-west components of the wind speed were plotted against each other for each day using quiver. This made it easier to see which direction the wind was going in, and when the wind changed. The correlation coefficients for the wind and water level data were calculated in Matlab using corrcoef. Before this could be done, the water level data needed to be manipulated so that it had values only every half hour, corresponding to the wind data. This was done by creating a new matrix made up of every 30 th water level value corresponding to every half hour. The correlation between variables was also calculated with the water level data lagging behind the wind data, to see if there was a strong lag response to the wind forcing. The lags were calculated for the more interesting half hours for the mangles data. Page 25

31 Methodology In order to deduce the patterns in the data, and explain what was causing the changes in water level, the data had to be looked at closely. The zoom tool in Matlab graphics was used to zoom in on consecutive sections of the time series plots of the Mangles data. Firstly, the Mangles data were analysed from the beginning to the end. For each change in the oscillations of the water level, the wind speed, wind direction and any other relevant information were noted down. From this information, it appeared that the wind direction was the primary forcing for the changes in water level, followed by the wind speed. The atmospheric pressure and total water level also appeared to have some influence. Therefore, the time series plots were again investigated in detail. For each change in the wind direction, the wind speed, atmospheric pressure, total water level and filtered water level response were noted down in a table, along with the wind direction change. The same was done for each time there was a significant change in wind speed. A significant change in wind speed was taken to be about 2 m/s, based on when the wind speed changes had an observed effect on the water level. Both of these tables of data were then analysed to construe any patterns. For the wind direction changes, the data were grouped according to the change in wind direction north to south, south to north, west to east and east to west. Within these groups, the data were grouped according to the wind speed and atmospheric pressure. The predominant filtered water level response in each of these groups and subgroups was then obvious. For the wind speed changes, the data were grouped according to whether the wind speed was increasing or decreasing, and the wind direction at the time. These variables appeared to be the controlling factors when the wind speed changes were responsible for the filtered water level changes. Again, after grouping the data together, the patterns in filtered water level response were obvious. The bottom friction factor and damping ratio were calculated using the Mangles Bay data from In these data, an example of a seiche with a relatively uniform decay rate was identified. The heights of the seiche at each oscillation were determined. The ratio h( t + T) h( t) was calculated, where h is the seiche height, t is time and T is the seiche period. That is, this is the ratio between the seiche heights at consecutive oscillations. The average of these ratios was determined, and the bottom friction factor m was calculated using the following equation: Page 26

32 Methodology ( t + T ) h() t m h T 2 = e The damping ratio was calculated as m / ω, where ω is the frequency of the seiche in rad/s. The 1995 data were analysed mostly by looking at the time series plots. Also, an extract of the 1995 data from Mangles Bay was looked at more closely to determine the important mechanisms operating in Cockburn Sound in spring and summer. A cross-spectral analysis was carried out on the 1995 data. This was done in order to identify the lag between the peaks of the seiches in the different areas. The analysis was performed using spectrum in Matlab to calculate a cross spectral density and corresponding phase difference. The phase difference was obtained for the frequency of interest. This frequency (in Hz) corresponded to the period of the seiches. The phase difference is in degrees. This was converted to minutes using the following formula: T ( hrs) * phase(deg) Phase (min) = *60 o 360 where T is the period of the seiches in hours. Using the phase difference between Mangles Bay and Two Rocks, the celerity of the seiche was calculated. The phase difference indicated the time the seiche took to travel from Mangles Bay to Two Rocks. The celerity is then simply the distance travelled divided by the time taken. Page 27

33

34 Results and Discussion 4 Results and Discussion 4.1 Spectral Analysis A spectral analysis was carried out on all of the data sets. This was done primarily to identify the peak in the spectrum corresponding to seiches, and the period of this peak. However, other important peaks can also be identified from a spectral analysis. The y-axis, spectral density, represents the momentum corresponding to the specific frequencies. A higher momentum or spectral density corresponds to a larger amplitude oscillation. The spectral analysis of the James Point data is shown in figure There is a strong peak at 2.8 hours, which is due to seiches. This peak is very close to the fundamental frequency predicted using Merian s formula. Hence, the seiche proposed, with a node at Fremantle and an anti-node at Mangles Bay, appears to be present. There are also two strong peaks at the diurnal and semi-diurnal tidal frequencies. The diurnal tide has significantly larger amplitude than the semi-diurnal tide. Hence, there are two distinct peaks in the spectrum corresponding to the periods of the two tidal components. The peak at the tidal frequencies is higher than the peak due to seiches (figure 4.1.1). This indicates that the amplitude of a tide is higher than the amplitude of a seiche. This is expected, as seiches typically have an amplitude of less than 20 cm, whereas tides are generally more than 0.5 m. Page 29

35 Results and Discussion Figure 4.1.1: Spectral analysis of James Point data, with the periods of the peaks indicated in hours. The spectral analysis of the Mangles data is shown in figure The period of 2.8 hours is indicated with a line. However, there is not a significant peak at this period. There does not appear to be any distinct peak corresponding to the seiche period. This is because the seiches have variable amplitudes. There are some sections with amplitudes of up to 0.1 m at the beginning of the study period, whereas for half of the study period the amplitudes were only 0.01 m (figure 4.2.2). This variation in amplitudes results in a range of momentums resulting from seiches. Therefore, the period corresponding to the seiche is spread over a range of spectral densities. Also, because the seiche amplitudes are only small for most of the study period there is not a great deal of momentum associated with the seiches. There is no peak at the period of 12 hours, corresponding to a semi-diurnal tide. This is because no semidiurnal tide is evident during this data set (figure 4.2.2). In contrast, there are two peaks at about 24 hours, corresponding to a diurnal tide. This is because of the variation in the tide over the study period. Both a neap tide and a spring tide were observed in the data that were collected. The periods of these two types of tides are slightly different, resulting in the two peaks on the spectral density plot. Page 30

36 Results and Discussion The peaks at five days, eleven days and approximately thirty days indicate low frequency water movements. For example, continental shelf waves have a period of approximately ten days (Pattiaratchi, 2001). Smaller tidal constituents also have periods of several days (Pattiaratchi, 2001). Figure 4.1.2: Spectral analysis of Mangles data, with lines indicating peaks or expected peaks in hours or days (d). A spectral analysis of the 1995 data was also performed to determine the period of the seiches. The plot of the Mangles Bay 1995 data is shown in figure This shows a peak at a period of 2.7 hours corresponding to seiches. This gives further support to the proposal of a fundamental mode seiche between Mangles Bay and Fremantle. There are also distinct peaks at 24 hours and 12 hours representing the diurnal and semi-diurnal tides. The peaks at 7 days, 12 days and 24 days may represent continental shelf waves and longer tidal constituents. For example, the peak at 24 days may represent the spring/ neap tide cycle. Page 31

37 Results and Discussion Figure 4.1.3: Spectral analysis of 1995 data collected from Mangles Bay. The lines indicate periods corresponding to the peaks, in hours or days (d). The spectral plots of the 1995 data collected from Fremantle, Barrack Street, Hillarys and Two Rocks are very similar to the spectral plot of the Mangles Bay data. They all have peaks at 24 days, 12 days and 7 days, corresponding to low frequency water level disturbances such as smaller tidal components and continental shelf waves. The other four plots also have peaks at 24 hours and 12 hours corresponding to the diurnal and semi-diurnal tides. It is expected that the five data sets would have similar peaks at these periods, as these water level disturbances affect a large area. As the data for these five areas were collected at the same time, and they are relatively close to each other, they experienced the same tides and other long period water level fluctuations during the sampling period. The Fremantle spectral analysis indicated a peak at 2.8 hours relating to seiches. This is 0.1 hour (6 minutes) more than that observed in Mangles Bay. However, this variation is not considered to be significant. Fremantle and Mangles Bay are the two ends of the basin in which the seiches in Cockburn Sound oscillate. Hence, they have seiches at approximately the same period. Page 32

38 Results and Discussion The periods of the seiches observed in Two Rocks, Barrack Street and Hillarys were all 3 hours. This is only 0.2 hours (12 minutes) different from the period observed at Mangles Bay. Also, the natural frequencies within Two Rocks Marina are of the order of minutes (Gwynne, 1993). This suggests that the observed seiches are not a similar response to a forcing occurring within the separate systems. Rather, the seiche appears to be the same one travelling between all of these systems. That is to say, at least from Two Rocks to Mangles Bay. The fundamental period of a seiche between Two Rocks and Mangles Bay can be calculated using Merian s formula. The distance between Two Rocks and Mangles Bay is about 87.5 km. The mean depth can be assumed to be between 10 and 15 metres. This gives a fundamental period of between 8 and 9.8 hours. Obviously, this is much too big to be the seiches observed in 1995, which have a period of about 3 hours. Hence, a seiche with a higher mode must be present. Between Cockburn Sound and Fremantle there is a single mode seiche, with an anti-node at Mangles and a node at the opening at Fremantle. The distance between a node and an anti-node for this seiche is therefore equal to the distance between Mangles Bay and Fremantle, which is 24.5 km. Between Mangles Bay and Two Rocks, there are 3.57 node to anti-node distances. That is, a quarter of a wavelength between Mangles Bay and Fremantle, half a wavelength (a complete sinusoidal) between Fremantle and somewhere 49 km further north up the coast, and Two Rocks is about an eighth of a wavelength (14 km) further up the coast. This is presented diagrammatically in figure As Two Rocks is not at a node or anti-node, and there is no obvious obstruction to water movement further north, it is proposed that the seiche continues further up the coast. Using Merian s formula, the average depth of the water can be calculated. Using the equation for open basins, the seiche between Two Rocks and Mangles Bay has a mode of The length of the basin is metres. The period is approximately 2.8 hours. This gives an average depth of 9.6 metres, which seems reasonable for a near-coastal zone in the Perth area. The spectral analysis of the data collected in Cockburn Sound in 2000 and 2001 indicates that the predominant period of the seiches is 2.8 hours. This is consistent with seiches travelling in the semi-enclosed system between Fremantle and Mangles Bay. The length of this system is 24.5 km, and the average depth is 10 m. Moreover, spectral analysis of data collected at various locations between Two Rocks and Mangles Bay in 1995 indicates that there is a higher mode seiche travelling at least between Two Rocks and Mangles Bay, with a period of about hours. The length the seiche would travel between Two Rocks and Mangles Bay is 87.5 km. For a mode of 3.57 between Two Rocks and Mangles Bay, the average depth, calculated using Merian s formula, is 9.6 m. Page 33

39 Results and Discussion Page 34

40 Results and Discussion Figure 4.1.4: The predicted locations of nodes and anti-nodes of a seiche propagating north up the coast of Western Australia from Mangles Bay. Scale approximately 1: , but magnitude of oscillations (horizontally in this figure) not to scale. Page 35

41 Results and Discussion

42 Results and Discussion 4.2 Time Series Analysis All of the water and meteorological data were all plotted against time. These plots are presented in figures to The purpose of these initial plots was to get a feeling for the data a general idea of what they looked like, the range of the data, and any obvious patterns. The meteorological data are only considered for the relevant time period corresponding to the water level data. The James Point data have a daily pattern in wind speed and direction for the first six days (figure 4.2.1). In the mornings, the wind was stronger (about 5 m/s), with the wind direction easterly. In the afternoons, the wind was generally weaker (about 3 m/s). For the first three days, the afternoon wind direction was southeasterly, and for the next three days the afternoon wind direction was south-westerly. This daily variation is due to the sea breeze. The James Point data also have daily variation in the total water level, due to tides. These tides have a maximum amplitude of about 0.3 m. Again, the variation is not large, due to the micro tidal environment of Perth waters. The spring tides occurred just before and just after the data were collected, with an amplitude of about 0.3 m. The neap tide occurred at about day 147, with an amplitude of about 0.1 m. The filtered water level data again indicated that seiches were present throughout the study time. The predominant period of these seiches was 2.8 hours (figure 4.1.1). There were two times of seiches with larger amplitudes, at the beginning and end of the data. These seiches had an average period of about 2.5 cm. For the rest of the data, the average amplitude was about 1 cm. Page 37

43 Results and Discussion Figure 4.2.1: Time series plot of wind speed (m/s), wind direction (degrees from north that the wind is going to) and total and filtered water level (m) measured at James Point. The Mangles data had no general pattern in the wind speed and direction (figure 4.2.2). However, there was a pattern in the atmospheric pressure data (figure 4.2.2). The atmospheric pressure increased by about 15 hpa in waves that lasted for four to six days, during the sampling period. These waves of atmospheric pressure variation are due to the passage of high pressure cells (Bureau of Meteorology, 1993). The Mangles total water level data had daily fluctuations due to the tides (figure 4.2.2). These fluctuations were large for the first day (11 th May) and for the last six days (from the 23 rd May), with total height variations of 0.6 metres. These correspond to spring tides on the 7 th and 23 rd of May. In contrast, the tidal fluctuations for the 11 th to the 22 nd of May, in the middle section of the data collection, were less than 0.2 metres. This corresponds to a neap tide on about the 15 th pf May. Hence, the variation in the tidal amplitude is due to lunar fortnightly cycle of tides resulting in spring peaks and neap lows. The filtered water level data indicated that seiches were present throughout the study time (figure 4.2.2). However, for days 134 to 142, the seiches were small, with a total height of less than 0.05 metres. For the Page 38

44 Results and Discussion two ends of the data, the seiches are large, with a total height of about 0.2 metres (figure 4.2.2). The predominant period is 2.8 hours (figure 4.2.2). Figure 4.2.2: Time series plot of wind components (m/s) (positive on y-axis indicates wind blowing to the north, positive on x-axis indicates wind blowing to the east), wind speed (m/s), filtered and total water level (m) measured at Mangles Bay in 2001, and air pressure (hpa). The Garden Island data were measured at the same time as the Mangles data, so the meteorological information is the same. However, the water level data are different. The total water level data for Garden Island has the same diurnal tidal fluctuations as the Mangles data (figure 4.2.3). However, the tide appears to be large for the first day or two, with height of 0.6 metres. It then reduces to a height of 0.4 metres for five days, until day 140. The tidal amplitude then increases, until by day 145 it is at 0.7 metres. The tidal amplitude then decreased from this peak to 0.3 metres by the end of the recording time. The filtered Garden Island data still have a lot of noise considerably more than the Mangles data (figure 4.2.3). This excessive noise is because the Garden Island site was not as sheltered as the Mangles site (figure 3.1.1). The filtered water level data for Garden Island also indicates that seiches are always present. Again, the predominant period of the seiches is 2.8 hours. Page 39

45 Results and Discussion Figure 4.2.3: Time series plot of meteorological data and filtered and total water level (m) data measured outside of the Causeway near Garden Island. An extract of the time series plot of the 1995 data collected from Mangles Bay is shown in figure Time series plots of the other 1995 data will not be analysed, as they have the same meteorological data and similar tidal and other water level responses. Only an extract of the Mangles data is shown, as the data set was four months long, so no medium-term patterns can be seen if the entire time series plot is presented. The extract that is presented is 20 days long, from the 7 th of October (day 310) to the 27 th of October (day 330). These data show patterns typical of spring and summer in the Perth region. The wind data show a diurnal pattern for most of the extract presented. In the morning the wind is going to the west, and in the afternoon the wind is going to the east a sea breeze (figure 4.2.4). This is consistent with the typical summer wind patterns for Perth. Also, the sea breeze in the afternoon is typically stronger than the morning wind going to the west (figure 4.2.4). The total water level shows the tidal fluctuations clearly. There seems to be a neap tide on about day 323, with a tidal amplitude of less than 10 cm (figure 4.2.4). The spring tides are on either side of the neap tide, 14 Page 40

46 Results and Discussion days before and after. These days are not included in this extract. However, the maximum tidal amplitude, which would be close to spring tide, is about 40 cm on day 328 (figure 4.2.4). The changes in the diurnal and semi-diurnal tidal components are also evident in the total water level plot in this extract (figure 4.2.4). Days 310 to 314 and 325 to 328 have a strong diurnal component and a weaker but still obvious semi-diurnal component. In contrast, days 314 to 320 and 328 to 330 have a very strong diurnal component and an insignificant semi-diurnal component. On days 322 to 324, the semi-diurnal and diurnal components are of similar amplitudes. The variation in semi-diurnal and diurnal tidal amplitudes is due to declination effects, which vary throughout the year (Pattiaratchi, 2001). The filtered water level data indicate that seiches are evident throughout the study period. The average amplitude of the seiches is about 5 cm (figure 4.2.4). The peak during day 319 was the highest peak during the study period, with an amplitude of about 20 cm. The seiches appear to have a diurnal fluctuation in their magnitude, which may be related to the diurnal wind forcing. Figure 4.2.4: Extract of time series plot of data collected from Mangles Bay in The time series analysis of the data sets has given an indication of the medium-term patterns in meteorological and water level data, including seiches. These patterns may be responsible for causing some of the observed patterns in the filtered water level data. Page 41

47 Results and Discussion

48 Results and Discussion 4.3 Correlations Between Data Sets Firstly, a correlation was calculated between the Mangles and the Garden Island data, to see how closely related the data were. The correlation coefficient between the filtered water level data for Mangles and Garden Island was 1. This indicates that the variation in the filtered water level for Garden Island can be fully explained by the variation in the Mangles filtered water level. Hence, only the Mangles filtered water level data will be analysed beyond this point. Correlation coefficients were calculated between the filtered water level and the meteorological data. The aim was to discover which of the meteorological variables had the strongest influence on the filtered water level. The correlation coefficients that were calculated are shown in table 4.3. Where there are blank spaces, there was no relevant meteorological data available to correlate with the water level. Table 4.3: Correlation coefficients between James Point and Mangles filtered water level data and their related meteorological data. Mangles James Pt. Decimal day (days) Wind speed (m/s) Wind direction ( from N) North-south wind speed component (m/s) East-west wind speed component (m/s) Air pressure (hpa) None of the correlations are very high. However, they will give an indication of what to look at as the first influence on changes in filtered water level, and what to consider only as a last resort. As expected, the decimal day, that is the day and the time of day, have little correlation with the filtered water level data. This is because the forcing mechanisms for changes in water level, such as wind, occur at all hours of the day, and do not conform to our segmentation of the day into hours. Page 43

49 Results and Discussion The highest correlations for each water level data set are highlighted in bold. This indicates that the wind direction has the highest influence on the James Point data. This is a negative correlation (table 4.3). That is, a decrease in wind direction corresponds to an increase in water level. For example, the wind direction may decrease from westerly (270 - going to the west) to southerly (180 ), corresponding to an increase in water level. A wind going to the west would push water away from James Point, on the east side of Cockburn Sound, decreasing the water level. On the other hand, a wind going to the south may cause water to build up in the southern end of Cockburn Sound, near James Point. The highest correlation for the Mangles data was with the north-south component of wind speed. This correlation was negative, indicating that when the wind is travelling to the north (positive), the filtered water level is low, but when the wind is travelling to the south (negative), the filtered water level is high. This is as expected, as the Mangles data were collected from the south of Cockburn Sound. Hence, when the wind is going to the south, the water level at Mangles Bay will be increased. The correlation coefficients that were calculated indicate that the strongest influence on filtered water level changes is wind direction, and in particular for Mangles Bay, the north-south component of wind direction. Page 44

50 Results and Discussion 4.4 James Point 2000 Seiches The seiches in the James Point data have a maximum amplitude of about 0.1m, with an average amplitude of about 0.04 m (figure 4.4). For the first seven days (139 to 146), the seiches appear to be larger at low tide. However, the importance of this correlation is that the low tide appears to coincide with the afternoon sea breeze (figure 4.4). Therefore, the change in wind direction from east to west and the increase in wind speed, which changes the water level set-up, is the main driving force for seiches in this data set. Figure 4.4: Time series plot of meteorological and water level data collected from James Point in The lines indicate strong wind changes, which correlate with large changes in filtered water level. For the next five days (days 146 to 151) the seiches are more equal throughout the day. There is also not a strong diurnal change in wind direction and wind speed (figure 4.4). The wind direction is predominantly easterly for this period. However, the seiches have a larger magnitude in this section than in the last section. There are no changes in wind direction to drive these seiches. Therefore, the large and relatively frequent changes in wind speed may be responsible for initiating the seiches. Page 45

51 Results and Discussion During day 152 there is a large and fast change in wind direction. This induces a large change in the water level set-up, which causes a large seiche (figure 4.4). After day 152 the seiches again appear to have a diurnal pattern in amplitude, with the larger seiches occurring at low tide. This is presumed to reflect when the larger changes in wind direction occur (figure 4.4). It is possible that the low water level at low tide enables the seiches to reach a higher amplitude, and therefore increases the effect of the forcing mechanisms. This theory is supported due to the slight diurnal pattern of the seiches from days 146 to 151. During these days, the forcing mechanism is proposed to be changes in wind speed. These changes in wind speed do not have a daily pattern. However, the seiches still appear to be larger at low tide. Therefore, it seems that the seiches can reach a higher magnitude at low tide. An analysis of the filtered water level measured at James Point in Cockburn Sound indicates that the seiches are driven by changes in wind direction and speed. Diurnal wind direction changes associated with the sea breeze cause diurnal variation in the magnitude of the seiches. However, without diurnal variation in the forcing mechanism the seiches still have an extent of diurnal variation. This is due to the decreased total water level associated with low tide, which is proposed to enable the seiches to reach a higher magnitude. Page 46

52 Results and Discussion 4.5 Mangles Bay 2001 Seiches The seiches in the Mangles data vary in size throughout the study period (figure 4.5). They are largest for the first two days, with an amplitude of 0.07 m on average, and some peaks up to 0.17 m. The seiches then diminish to an amplitude of about 0.02 m for about nine days. The seiches subsequently increase to an amplitude of 0.05 m for the next six days. There are several reasons for the variation in seiche amplitude. The larger seiche oscillations correspond to when the tidal oscillations have larger amplitude, due to the spring tide (figure 4.5). This is probably coincidental, as the small tidal fluctuations in Cockburn Sound will not contribute to seiches. The lower magnitude of the seiches from day 134 to 142 may be due to influences such as atmospheric pressure, the presence of forcing events, or other factors that were not measured during this study. The highest atmospheric pressure (1025 hpa) is present during the period of lowest water level oscillations (figure 4.5). It is likely that the seiches are restricted in amplitude due to the inverse barometric effect associated with the high atmospheric pressure. Additionally, there are not as many forcing events during the periods of smaller seiches. The wind direction remains constant for four days, and the wind speed is below 5 m/s from day 134 to day 144 (figure 4.5). This encompasses the period of low seiche amplitude. Page 47

53 Results and Discussion Figure 4.5: Time series plot of meteorological and water level data collected from Mangles Bay in The Mangles data were examined to determine the forcing mechanisms for each seiche or change in water level oscillations. These changes occurred several times a day. As the data set is 18 days long, this is a large number of changes in the water level oscillations. In general, changes in wind direction and wind speed were found to be responsible for causing or changing the water level oscillations. A summary of these changes, and the meteorological conditions present at the time, are presented in tables and The last column of the table (#) is the number of observations that concur with the response to the described conditions. Page 48

54 Table 4.5.1: Changes in wind direction driving seiches, influenced by other meteorological factors. Results and Discussion Wind Wind speed Atmospheric Total Filtered water level response # direction (m/s) pressure ((hpa) water level (m) a. S! N Decrease Water level decreases by 0.05 m. 3 by > 4 m/s b. S! N Low, 2 3 Low, Water level decreases by 0.1 m. 2 m/s c. S! N Low, 2 4 m/s Water level decreases by 0.01 m. 1 d. S! N Low High, Less than m change in water 6 level e. S! N Increase by High, 1020 Low, 6 Water level decreases by 0.01 m. 1 4 m/s f. N! S Increase by Water level unchanged 2 > 5 m/s g. N! S Increase by Low, Water level increases by 0.1 m. 1 2 m/s h. N! S Increase by Water level increases by m/s m. i. N! S Decrease High, Water level unchanged. 2 by > 1 m/s j. N! S Low, 2 m/s Water level increases by 0.02 m. 1 k. N! S Low, 2 m/s Water level increases by 0.06 m. 1 l. N! S Low, 2 m/s High, Water level increases by m. 1 m N! S 4-5 m/s Water level unchanged. 1. n. W! E Decrease Water level unchanged. 2 o. W! E Low, 2 4 m/s Water level decreases by m. 2 p. E! W Decrease Water level increases by 0.01 m. 1 Page 49

55 Table 4.5.2: Changes in wind speed driving seiches, influenced by other meteorological factors. Results and Discussion Wind Wind Atmospheric Total Filtered water level response # speed (m/s) direction pressure ((hpa) water level (m) q. Increase South Water level increases by m. 5 r. Increase North Water level decreases by m. 6 s. Increase West Water level increases by 0.04 m. 1 t. Decrease South Water level decreases by m. 5 u. Decrease Southeast Water level increases by 0.02 m. 1 v. Decrease North Water level increases by m. 10 The results in the above tables are discussed in the next four sections. Various pictorial examples of some of the items in the tables are also presented. These are labelled according to the letters in the above tables. In the discussion of the results, it must be remembered that the water level was measured in Mangles Bay, which is in the southern end of Cockburn Sound. Page 50

56 Results and Discussion 4.6 Wind Direction Changing from South to North Firstly, when the wind direction changes from south to north, it is expected that more water will be transported north, decreasing the water level at the south. From table 4.5.1, items a to e, it is evident that this does occur. However, the extent of the decrease in water level at the south end of Cockburn Sound is determined by the wind speed and atmospheric pressure. If the atmospheric pressure is higher, then the change in water level will be restricted due to the inverse barometric effect. For example, items b and c in table have the same wind direction and wind speed forcing. However, the atmospheric pressure for item c is about 7 hpa higher than for item b, leading to a change in water level that is about 0.09 m (9 cm) less. This complies with the inverse barometric effect. The influence of wind speed is seen when comparing items d and e in table These items have the same wind direction change (south to north) and atmospheric pressure (~1020). However, the wind speed for item d is constant, whereas the wind speed for item e is increasing. It is expected that the change in wind direction would have a stronger effect on the water level when the wind speed is increasing. This is satisfied, as the water level decreases by 0.01 m for item e, whereas the water level only decreases by m for item d (table 4.5.1). Two examples of a change in wind direction from the south to the north causing a change in water level oscillations are presented below. Firstly, in figure below, there is a change in wind direction at This decreases the expected height of the seiche peak at by 0.1 m. This is an example of item b in table Page 51

57 Results and Discussion Figure 4.6.1: b. Change in wind direction from south to north, constant low wind speed, low atmospheric pressure. Secondly, in figure below, there is a change in wind direction from south-east to north at , corresponding with an increase in wind speed. This causes the water level to decrease by about 0.01 m, initiating a seiche. The change in water level is relatively small (only 1 cm) due to the high atmospheric pressure. This is an example of item e in table

58 Results and Discussion Figure 4.6.2: e. Change in wind direction from south to north, increase in wind speed, large atmospheric pressure. When the wind direction changes from south to north, the water level decreases in Mangles Bay, setting up a seiche. The magnitude of this decrease in water level, and the resulting seiche, is influenced by the atmospheric pressure and the wind speed.

59 Results and Discussion

60 Results and Discussion 4.7 Wind Direction Change from North to South The second forcing mechanism that was considered was a change in wind direction from the north to the south. This was expected to cause an increase in the water level measured at Mangles Bay. Items f to m in table indicate that this was the case. The magnitude of the change in water level is influenced by the regional atmospheric pressure. For example, a comparison between items j, k and l demonstrates the effect of atmospheric pressure. The wind speed for these three items is low. The atmospheric pressure consecutively increases for each item, whereas the water level change consecutively decreases for each item (table 4.5.1). This illustrates the inverse barometric effect. The influence of atmospheric pressure can be seen when comparing figures and below. In both figures there is a similar increase in wind speed. In figure 4.7.1, the wind speed increases from eight to ten m/s, whereas in figure the wind speed increases from three to six m/s. Therefore, due to the change in water level set-up caused by the increase in wind speed it would be expected that the increase in water level would be similar for the two plots. However, the increase in water level is actually much higher in figure (10 cm) than in figure (1 cm). This is because of the higher total atmospheric pressure that is present in figure The atmospheric pressure is about 1017 hpa in figure 4.7.1, compared to 1005 in figure Therefore, the large total atmospheric pressure restricts the water level response. Figure 4.7.1: g. Wind direction change from north to south, increase in wind speed, low atmospheric pressure. Page 55

61 Results and Discussion Figure 4.7.2: h. Wind direction change from north to south, wind speed increase, high air pressure. A wind direction change from north to south will increase the water level in Mangles Bay in the southern end of Cockburn Sound. This can generate or influence ongoing seiches. The magnitude of this influence is related to the regional atmospheric pressure. Page 56

62 Results and Discussion 4.8 Wind Direction Change between East and West A change in wind direction between east and west will also drive an oscillating water level response. There were not as many changes in wind direction between east and west as there were between north and south during the data collected from Mangles Bay in However, the results that were obtained reflect what was found for the north-south changes in wind direction. All of the changes in wind direction between east and west occurred when the atmospheric pressure was between 1012 and 1017 hpa. Therefore, no results about the influence on atmospheric pressure on the water level changes were found. The primary factor influencing the magnitude of the change in water level oscillations was the wind speed. The size of Cockburn Sound is small enough that coriolis does not have a significant effect. Therefore, the change in water level set-up is the mechanism that is responsible for the change in seiches induced by a change in wind direction between east and west Wind Direction Change from West to East Firstly, a change in wind direction from west to east induces a decrease in the water level at Mangles Bay. The wind speed primarily determines the magnitude of the water level change. For a decreasing wind speed, the water level change is negligible (see item n in table 4.5.1). This is demonstrated in figure There may be no effect from the change in wind direction due to the decreasing wind speed. Page 57

63 Results and Discussion Figure 4.8.1: n. Wind direction change from west to east and wind speed decrease from 7 to 3 m/s Wind Direction Change from East to West There was only one instance of a change in wind direction from east to west. This coincided with a decrease in wind speed from 4 to 2 m/s and an atmospheric pressure of 1017 hpa. The result was an increase in the water level oscillations by 0.01 m. The influence of regional air pressure and wind speed on the magnitude of the water level change cannot be determined, as no comparative results are available. A change in wind direction between east and west changes the water level set-up. This influences the water level oscillations in a similar way to a change in wind direction between north and south. The regional atmospheric pressure and wind speed are thought to have a similar influence on the magnitude of the change in filtered water level. Page 58

64 Results and Discussion 4.9 Wind Speed Changes The second major forcing mechanism that changes the water level oscillations in Cockburn Sound is a change in wind speed. A change in wind speed of 2 m/s is a common threshold observed in the Mangles Bay data to drive a change in water level oscillations. A summary of these results is included in table From this table, it is obvious that the regional air pressure does not appear to have an effect on the magnitude of the water level oscillations induced. The magnitude of the wind speed change also did not appear to have an influence on the size of the water level response. The wind direction is the main influence on the magnitude and direction of the water level response. For an increase in wind direction, results were available for a wind direction of southerly, northerly and westerly (items q to s in table 4.5.2). When the wind direction is southerly, an increase in wind speed causes an increase in water level (figure 4.9.1). This is because the wind causes a build-up of water level in the south of Cockburn Sound, where the water level was measured. This changes the water level set-up, which induces seiches. Similarly, an increase in wind speed to the west causes a build-up of water in the west (figure 4.9.2). This change in the water level set-up induced an increase in water level in the south, similar to what happened when there was an increase in wind speed to the south. In contrast, an increase in wind speed to the north causes a decrease in water level to the south, due to the build-up of water in the north (figure 4.9.3). All of these forcing mechanisms and the water level responses are illustrated in figures to Page 59

65 Results and Discussion Figure 4.9.1: q. Two cases of an increase in wind speed to the south causing an increase in water level. Figure 4.9.2: s. An increase in wind speed to the west inducing an increase in water level. Page 60

66 Results and Discussion Figure 4.9.3: r. An increase in wind speed to the north causing a decrease in water level in Mangles Bay. The other wind speed forcing mechanism is a decrease in wind speed. Results were obtained for a decrease in wind speed when the wind direction was southerly, south-easterly and northerly (items t, u and v in table 4.5.2). A decrease in wind speed when the wind is blowing to the south causes a decrease in the built-up water to the south. Hence, the water level decreases. In contrast, a decrease in the wind speed when the wind is blowing to the south-east and the north causes an increase in the filtered water level in the south. When the wind speed decreases and the wind is blowing to the north, this causes a decrease in water level build-up to the north, and a corresponding increase in the water level at the south. When the wind speed decreases and the wind is blowing to the south-east, it is expected that the southerly component of the wind speed would dominate, due to the low influence of coriolis forces. This should cause a decrease in the build-up of water level in the south. However, the results show that the water level in the south increases when the wind speed to the south-east decreases (figure 4.9.4). This is due to a change in the water level set-up, but the actual mechanism causing an increase in water level to the south is not understood. Page 61

67 Results and Discussion Figure 4.9.4: u. A decrease in wind speed when the wind is blowing to the south-east causing an increase in water level in southern Cockburn Sound. A change in wind speed will influence seiches if there is not already a seiche with a large amplitude present. The direction and magnitude of this influence are dependent on the wind direction when the wind speed changes. The atmospheric pressure does not appear to have a large influence. Changes in wind direction, and to a lesser extent changes in wind speed, are the main driving forces that change water level oscillations in Cockburn Sound. The magnitude of the changes induced by these mechanisms is dependent on the atmospheric pressure and the wind characteristics. Page 62

68 Results and Discussion 4.10 Estimating the Friction in this System The friction in the Mangles Bay to Fremantle system can be determined by looking at examples of seiches that have a relatively uniform decay. One of these was identified in the data collected from Mangles Bay in This is shown in figure 4.10 below. Figure 4.10: An extract of filtered water level data from Mangles Bay, 2001, with lines indicating relatively uniform damping of the seiches. The average ratio of amplitudes, h( t + T) h( t), was calculated to be using the heights from both the actual data and the idealised line. This resulted in a bottom friction factor, m, of 1.78 x The frequency of the seiches was calculated to be 6.5 x 10-4 rad/s using a period of 2.7 hours. This gave a damping ratio (friction factor frequency (rad/s)) of For ωl / Co of 1.47 (ω = 6.5 x 10-4 rad/s, l = m, and C o = g h = = m/s), there is low friction in the system (van Rijn, 1994). This is expected, as Cockburn Sound has a relatively uniform width and depth. The bottom friction factor was calculated to be 1.78 x This resulted in a damping ratio of 0.027, which indicates that there is low friction in this system. Page 63

69 Results and Discussion

70 Results and Discussion Data Seiches All of the data collected in 1995 is presented in figure below. There are two main things to notice about these data the similarities and the differences between the data sets. Figure : Filtered water level data (cm) measured at various locations in 1995 and wind speed and direction indicated in panel 1 in m/s. The Mangles Bay and Fremantle data sets are very similar almost exactly the same. The amplitudes are equivalent at about 5 cm, and the seiches have the same amplitudes in the same places. This is to be expected, as Mangles Bay and Fremantle are two ends of a basin in which seiches are occurring. However, this system is an open system, so there should be an anti-node at Mangles Bay and a node, with no vertical water movement, at Fremantle. There are two possible explanations for why there is vertical water movement at Fremantle of a similar order of magnitude to Mangles Bay. Firstly, it is possible that although this is an open system, there is an anti-node at Fremantle, with a node somewhere between Fremantle and Mangles Bay. For example, where the water level decreases at the entrance to Cockburn Sound there may be a node. Page 65

71 Results and Discussion Secondly, if the seiche system is big enough, coriolis forces may have an effect. Whether this would be a big enough effect to make a node look like an anti-node is unknown. Also, the system would have to be about 130 km long for Coriolis to be significant. Therefore, it is more likely that a higher mode seiche is present, resulting in an anti-node at Fremantle. There are similarities between the other data sets, also. In many cases all the data sets appear to respond with a relatively similar magnitude at the same time. For example, at about day 320, all of the data sets have their largest seiche response. The reason for this unusually large response is unknown. As can be seen from the wind data, there are changes in the wind direction and wind speed at this time. However, there are similar changes in wind direction and wind speed at other times that did not induce a response this big. There are two possible explanations for this variation in response to the same wind forcing. Firstly, there may be some mechanism operating most of the time that reduces the ability of the water to respond to the wind forcing. For example, there may be generally high atmospheric pressure during the sampling period that restricts the water level response, due to the inverse barometric effect. If this limiting factor were not present on day 320, then the water level would have been free to have a larger response to the wind forcing. Secondly, there may have been some additional forcing on day 320 that was not present on the other days. This would have caused a larger water level response because of the additional forcing. For example, there may have been continental edge waves at a particular period and coincident with the wind forcing that induced a higher seiching response in these water bodies. Another similarity between the data sets is that the seiches all have the same period, as discussed in section 4.1. These similarities lead to the conclusion that the seiche is not restricted to between Mangles Bay and Fremantle, but it propagates up the coast. Nevertheless, there are differences between the seiches observed in the different water bodies. For example, the amplitudes of the seiches are lower at Hillarys and Two Rocks, at about 3 cm compared to 5 cm at Mangles Bay and Fremantle. The amplitudes of the seiches are extremely low at Barrack Street, at about 1 cm. The variation in amplitudes is predicted to be due to the amount of friction along the path that the seiche travels to reach each location. Page 66

72 Results and Discussion The path from Fremantle to Barrack Street is non-uniform, up the Swan River. There is assumed to be more friction in this path, so the amplitude of the seiche would be reduced. Presumably, this is also why the seiche is smaller at Hillarys and Two Rocks than at Fremantle the friction associated with travelling that distance would reduce the amplitude of the seiche, and probably slow it down. There are also differences in the relative magnitudes of the seiches at different locations. For example, during days 273 to 283 the seiche has a smaller magnitude than expected at Hillarys. Also, during days 340 to 350 the seiche is smaller than expected at Two Rocks. These variations are expected to be due to unknown location conditions that are affecting the seiches and the water level response. An analysis of the seiches measured in various locations in the Perth area in 1995 supports the theory that the seiches in the Cockburn Sound area are propagating up the coast to at least Two Rocks. The amplitudes of the seiches in the different locations are influenced by the friction the seiche is exposed to on the journey to the location and unknown local effects. An extract of the Mangles Bay data from 1995 is presented below (figure ), to get a closer look at the seiches. As mentioned in section 4.2, the seiches appear to have a diurnal variation in amplitude. The lines on the figure indicate most of the clear changes in wind direction resulting from the sea breeze. That is, a change in the wind direction from going to the west to going to the east. These changes in wind direction often coincide with the ebb tide or low tide. It appears that the seiches increase in magnitude after an indicated change in wind direction. This is expected, as the change in wind direction would change the water level setup, which would cause the water level to oscillate in a seiche in an attempt to return to equilibrium. The decrease in the amplitude of the seiches that is experienced every day may be due to friction overcoming the inertia in the water. Page 67

73 Results and Discussion Figure : Extract of data collected from Mangles Bay in The lines indicate changes in wind direction due to the sea breeze. However, the total water level also has an influence on the magnitude of the change in water level. This is an inverse relationship, so that a deeper water level means a reduced change in water level response for the same change in wind stress. Hence, when it is low tide the seiches are expected to be larger for the same change in wind direction. Also, the magnitude of the seiches may increase with the decrease in water level associated with ebb tide. This appears to be the case during days 322 to 324. As the total water level decreases, the amplitude of the seiches increases slightly. However, it is hard to separate this effect from the influence of the wind. A closer look at an extract of data from Mangles Bay in 1995 indicates diurnal variation in winds, tides and seiches. Both the diurnal wind direction change associated with the sea breeze and the diurnal low tides are predicted to increase the magnitude of the seiches diurnally and concurrently. Page 68

74 Results and Discussion 4.12 Cross-Spectral Analysis The spectrums of the data from 1995 were correlated to determine the lag between the peaks of the seiches in the different areas. The results are shown in table 4.12 below. The location that is first in each pair is the location of the data that were first in the spectral analysis. The period used to obtain these results was 2.7 hours. Table 4.12: Phase difference between two data sets from 1995, using a seiche period of 2.7 hours. Pair of locations Phase difference Degrees Minutes Mangles Two Rocks Mangles Hillarys Mangles Fremantle Mangles Barrack Street Hillarys Two Rocks Fremantle Hillarys Fremantle Two Rocks When the phase difference is positive, the first location listed has a peak before the second location listed. Conversely, when the phase difference is negative, the first location has a peak after the second location. A phase difference of 180 indicates that the two locations are perfectly opposite in phase. That is, when there is a peak at one location, there is a trough at the other location. In an analysis of these results, it must be remembered that the phase difference indicates only a difference between when the waves are at each point. It doesn t take into account that the waves may be different waves, due to the progression of the seiche up the coast. That is, there may be a peak at Mangles Bay at the same time as a peak at Two Rocks, but these peaks are not the same one. A simplistic diagram of the proposed seiche and the phase differences is presented in figure Each colour represents the water level that would be recorded at each location. Along the vertical axis is time. The different heights of the water level in the different colours at each time indicate the water level measurement Page 69

75 Results and Discussion that would be recorded at that location at that time. This figure was drawn next to the map so that the relative length of the oscillations and the distances between the points could be compared. This figure indicates that the measured water levels at Mangles Bay, Fremantle, Hillarys and Two Rocks are almost in phase. That is, when there is a peak at Mangles Bay it is close to a peak at Fremantle, Hillarys and Two Rocks. This supports the notion that the seiche has a higher mode, resulting in an anti-node at Fremantle rather than a node. However, if the mode between Fremantle and Mangles Bay were two instead of one, this would result in a period that was half as long. The period of the seiches was clearly defined on the spectral analysis. Therefore, this analysis will continue assuming that the seiche between Fremantle and Mangles Bay is unimodal, in agreement with the results of the spectral analysis. The speed of the seiche can be determined simply by dividing the distance the seiche travels by the time it takes to travel it. The distance between Two Rocks and Mangles Bay is m. The period of the seiche was calculated using Merian s formula for open basins, as about 2.8 hours. This is the time it takes the seiche to travel four times the length of the basin, or m. This gives: time =? time = = 2.5hours This implies that there should be a lag between the Two Rocks data and the Mangles Bay data of 0.3 hours, or 18 minutes. This agrees closely with the observed lag. The observed lag will be used to calculate the travel time, as this is what is actually happening: time = 2.8 ( ) = 2.57hours = 9258sec onds Therefore, the celerity of the seiche is = 9. 5 m/s. This seems to be a suitable celerity in a water depth of about 9.6 metres. The cross-spectral analysis supports the suggestion that there is a seiche with a mode of 3.57 occurring between Mangles Bay and Two Rocks. This seiche may propagate further up the coast, as there are no obstacles to prevent it. Page 70

76 Results and Discussion time Key: Mangles Bay; Fremantle; Barrack Street Jetty; Hillarys Boat Harbour; Two Rocks Marina. Figure 4.12: Relative phase lag between measured water level data from five locations in Scale approximately 1: , but magnitude of oscillations (horizontally on this figure) not to scale. Page 71