ISBN: 978-1-925627-03-9
Dynamics of a tidal estuary with two inlets: Lagrangian drifter observation of dispersion in Pumicestone Passage, Australia Callum Tyler1, Kabir Suara1, Adrian McCallum2, Roy Sidle2, Helen Fairweather2 and Richard Brown1. 1Science
and Engineering Faculty, Queensland University of Technology, Brisbane, Australia E-mail:
[email protected]
2Environmental
Engineering School, University of the Sunshine Coast, Sippy Downs, Australia
Abstract Understanding the properties of surface dispersion within estuaries provides an insight into important environmental processes that govern pollutant transport and water quality issues. Simultaneous field observation of dispersion measured by clusters of GPS-tracked drifters were used to examine the dispersive behaviours at the ends of an estuary with two tidal inlets. The field study was conducted in Pumicestone Passage- an estuary which extends over 45 km from the southern to the northern inlets and covers ~60 km2 with a width >500 m and depth 2 - 3 m mid-estuary. We examine the difference in the dynamics of the two inlets through applying pair-particle statistics, a metric closely tied to mixing processes. Average diffusivity values indicated that larger diffusivity exists at the southern inlet, with values of 𝐾𝑁𝑠 = 0.244 𝑚2 𝑠 −1 , 𝐾𝑁𝑛 = 0.146 𝑚2 𝑠 −1 , 𝐾𝑆𝑛 = 0.831 𝑚2 𝑠 −1 and 𝐾𝑁𝑛 = 0.878 𝑚2 𝑠 −1 , for the northern and southern inlets, respectively. Additionally power relations were developed for diffusivities with separation scales less than 5 m.
1. INTRODUCTION Understanding the properties of surface dispersion within estuaries provides an insight into important environmental processes. Issues such as water quality and the transport of nutrients, pollutant, sediment and larvae are closely linked to dispersivity and diffusivity in water bodies. Previously, spreading of radioactive and coloured dye tracers and Eulerian devices were only available for investigating mixing and dispersion processes, (Okubo 1971; Fischer et al. 1979). With the recent advancements in the development of GPS-tracked drifters for shallow estuaries, Lagrangian transport can now be studied, (Tinka et al. 2013; Suara et al. 2015a). Drifter is the term for a GPS tracked Lagrangian particle. Drifters have previously been deployed to study the dynamics of ocean currents, rivers, coastal and estuarine systems (LaCasce & Ohlmann 2003; Spydell et al. 2015; Suara et al. 2015a). Drifter trajectories can be used to examine the dynamics and dispersive behaviour of a system through single particle and relative particle statistics. Single particle statistics measures the variance of individual displacements relative to their starting point, as opposed to relative particle which measures the separation between particles relative to their initial separation. Relative dispersion is more closely tied to mixing processes as it reflects both the small and large scale processes (LaCasce 2008). Previous studies in estuarine systems with multiple tidal inlets have focused on applying harmonic analysis of tidal circulation and a numerical simulation of the hydrodynamics (Huang et al. 2002; Salles et al. 2005). Findings have shown that estuaries with multiple tidal inlets are unstable and have non-linear mechanisms (Huang et al. 2002). Systematic experimental observations of the difference in the dynamics and dispersion behaviour of estuarine systems with more than one tidal inlet are rare and this stands as a motivator for research presented in this paper. Another motivator is the need to resolve mixing processes at length scales less than 10 m and time scales less than a tidal period. For
Dynamics of a tidal estuary with two inlets Tyler et al. length scale in the < [O(1)] m in a shallow estuary, eddy diffusivity was similar to Richardson 4/3 power law and the results were comparable in scale to dye tracer diffusivities in larger water bodies, (Okubo 1971; Richardson 1926; Suara et al. 2016b). However diffusivities at larger length scales i.e., > [O(1)] were found to be weaker than Richardson scale and dominated by strain field, (Suara et al. 2017). This paper uses the pair-particle statistics to estimate the relative dispersion behaviours. This technique has previously been used by (Suara et al. 2017) to investigate the relative dispersion of a micro-tidal estuary with a single inlet. In the present work, pair-particle separation and associated diffusivity are estimated for clustered drifters concurrently deployed at both ends of the estuary. In this paper we examine the dispersion regimes and the estuaries at time scale less than a tidal period, with the length scale ranging from 0.1 to few metres.
2. METHOD 2.1.
STUDY AREA
Pumicestone Passage is located in South East Queensland, Australia. It is a two inlet estuary system that separates Bribie Island from mainland Australia. The southern inlet opens widely to Moreton Bay without obstructions, whereas the northern inlet opens over an unstable sand bar to the Pacific Ocean. It is fed by eight separate creeks leading from the Glasshouse Mountains, Beerburrum, Sunshine Coast regions, (Eyre & France 1997). The southern inlet spans a width of ~800 m whereas the northern inlet is 200 m wide at the beach front and opens to ~780 m. Both inlets are separated by approximately 45 km. Figure 1 depicts the location of the passage. The northern inlet experience a tidal change approximately 100 minutes before the southern inlet. The maximum depth of the estuary is no more than 3 m. The coordinates are (-26° 48' 33.48", 153° 8' 5.01") and (-27° 4' 52.05", 153° 9' 2.8") for the northern and southern inlets, respectively.
2.2.
EXPERIMENT
On 6 July 2016, two clusters of low resolution drifters were deployed six times at each inlet over the course of the day. All drifters were deployed simultaneously, with a cluster of ten and eight drifters deployed at the northern and southern inlets, respectively. Each deployment lasted until either the drifter was trapped by the bank or the deployment was interfered with. The longest deployment was ~4500 s and the shortest was ~1200 s for both inlets. Each drifter contained a Holux M241 Figure 1: Pumicestone Passage is the estuary system single point precision GPS unit capable of that separates mainland Australia from Bribie Island. logging the latitude, longitude, altitude and time logged at 1 Hz with a position accuracy of 2-3 m. Suara et al. (2016a) characterised the error associated with the GPS unit while stationary and it was observed that a speed variance of 𝜎 2 = 0.0005 𝑚2 𝑠 −2 existed. The drifters were constructed of PVC cylindrical pipe with 0.04 m diameter and 0.5 m height. The drifters were positively buoyant to ensure constant satellite fixation, with 16𝑚. Method of segmentation was additionally adopted to increase the number of realisations such that drifter pairs were restarted after 500 s segments, (Colin de Verdiere 1983; Suara et al. 2015a). This resulted in increasing the number realisations from 224 to 560 for both inlets. Once categorised the relative dispersion was calculated using mean square separation of the two particles, as follows: 2
2 (𝑡, 𝐷𝑝𝑙 𝑟0𝑙 ) = 〈(𝑟𝑙𝑘 (𝑡) − 𝑟0𝑙𝑘 (𝑡)) 〉 − 〈(𝑟𝑙𝑘 (𝑡) − 𝑟0𝑙𝑘 (𝑡))〉2
(2)
where l represents the directions, k represents the paired realisation, 0 represents the initial separation, 〈 〉 is the ensemble average over all pair realisations at time, t and D2pl represents the relative dispersion. The rate of change of 𝐷𝑝2 indicates the dispersion regimes responsible for particle separation in a turbulent flow field, (LaCasce 2008). The following regimes are commonly observed, 1. 𝐷𝑝2 (𝑡, 𝑟0𝑙 ) ~ 𝑡, diffusive regime where diffusivity is constant; 2. 𝐷𝑝2 (𝑡, 𝑟0𝑙 ) ~ 𝑡 2 , “Ballistic” dispersion regime;
3. 𝐷𝑝2 (𝑡, 𝑟0𝑙 ) ~ 𝑡 3 , Richardson’s power law regime; 4. 𝐷𝑝2 (𝑡, 𝑟0𝑙 ) ~ 𝑒 𝑡 , exponential separation regime.
Relative dispersion was calculated for both the streamwise and cross-stream directions. As dispersion is time dependant, diffusivity is a metric dependent upon the length scale of turbulence associated with the dispersion processes. Therefore it is necessary to calculate the separation length scale alongside the diffusivity, 𝐾𝑐𝑙 (𝑡, 𝑟0𝑙 ) =
2
1 𝜕𝐷𝑝𝑙 4 𝜕𝑡
, 𝑑(𝑡) = √(𝐷𝑝𝑠 ∙ 𝐷𝑝𝑛 )
(3)
As both inlets of Pumicestone Passage are wide we are expecting the important length scale to be the depth, as postulated by Fischer et al. (1979). In estuarine systems, dispersion and mixing processes are dominated by eddy action and strain and shear deformation (Suara et al. 2016b). In an attempt to model the diffusive behaviour of the streamwise and cross-stream directions of both inlets, a power
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law relation was chosen, which relates the separation length scale to the relative diffusivity. This relation has previously been used by Jones & Pattiaratchi (2005) but is a fundamental relationship dating back to studies such as Richardson (1926), Taylors (1921) and some laboratory studies thereafter. Experimental studies that followed have focused upon relating these length and time scales to diffusivity and dispersion - e.g., Okubo’s research (Okubo 1971) on Ocean diffusion diagrams. These relations have been established for not only horizontal diffusion in deep water, but also for diffusion in gulfs, surf zone and estuarine systems (LaCasce 2003; Brown et al. 2009; Suara et al. 2017). In Fischer et al. (1979) a theoretical derivation of streamwise diffusivity coefficient was presented. If the shear velocity was observed, this could be a basis for comparison. The power relationship is defined as, 𝐾𝑐 = 𝑐1 𝑑 𝑐2 (4) Where 𝑐1 and 𝑐2 are the slope and the intercept of the diffusivity axis, d is the separation scale distance and 𝐾𝑐 is the diffusivity. Note that 𝑑 reflects the spatio-temporal growth of a patch because of the original separation, 𝑟0 is removed from 𝐷𝑝 so that the scale dependence of diffusivity is similar to those in literature, where 𝐾𝑝 ~ 𝑙 𝛾 (where, 𝑙 = 𝑑 2 + 𝑟0 ) (Richardson 1926; Brown et al. 2009; Suara et al. 2017).
4. RESULTS & DISCUSSION 4.1.
DISPERSION & DIFFUSIVITY
Figures 3 and 4 show the relative dispersion as a Figure 2: Illustration of the SNU (Stream, function of time for the streamwise and cross-stream Normal, Up) coordinate system. This is the directions for both the northern and southern inlets. coordinate system which is used for the Likewise, Figures 5 and 6 present the relative analysis of particle dispersion in diffusivity as a function of the separation length Pumicestone Passage, Australia. scale. All graphs show the dispersive and diffusive values at different initial separations. An interesting phenomenon at both inlets is that the streamwise dispersion varies little from the cross-stream component, except for large initial separation, 𝑟0 > 16𝑚. This suggests that the dispersion within the period of 500 s is isotropic. Such dispersion was not observed in Eprapah Creek where within the first few minutes the magnitude of dispersion in the streamwise direction grew larger than that in the cross-stream direction (Suara et al. 2017). This difference can be explained by the difference in widths between the two estuaries. Due to the much larger inlet widths of Pumicestone passage, much less shear flow and velocity to induce streamwise flow was observed. Figures 3 and 4a show that while 𝑡 < 25𝑠 the dispersive behaviour follows a 𝐷𝑝2 ~ 𝑡 2 regime. Commonly known as a “Ballistic” regime, this indicates that the paired particles behave as independent particles (Suara et al. 2016b). Thereafter the dispersion follows a diffusive regime. In Figure 4b, dispersion starts to resemble a diffusive regime after 𝑡~5𝑠. Such dispersive behaviour can be attributed to very large initial separation that could be a result of large cross flow. In general within the first 50 s the dispersion grows as a power between 2 and 3. Whereas from 50s onwards the dispersion growth slowed and transitioned to a power of 1. This indicated that dispersion in the system was generally weaker than Richardson scale. Additionally, the divergence and convergence at 𝑡~190𝑠 (Figure 4a) may be explained by a reduction of the width and depth of a trench at the southern inlet. A similar event of divergence and convergence can be seen in Figure 3a and 3b. Additionally effects of secondary flow may exist in the system that would contribute to the divergence of the system (Suara et al. 2017). With regards to the centreline of the inlets, the true thalweg was not used due to the complexity of its identification. Due to the complex morphology of both inlets, there did not exist one deep channel that could be used to calculate the thalweg. Instead, as previously mentioned the low tide interface was used to calculate the centreline. Such decisions do not affect the magnitude of the diffusivity.
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Relative diffusivities at large separation of 𝑟0 > 16𝑚, for the southern inlet were significantly affected by the initial separation. This is evidence of a combination of both large and small scale turbulence (Suara et al. 2017). This was evident but less so for the northern inlet – here the effects of initial separation distance were less discernible. For each initial separation category there exists multiple asymptotes. This is explained by there existing multiple scales of turbulence. An asymptote refers to the paired particles separation reaching the limit of the length scale (Yeung 1994).
Figure 3: Relative dispersion of northern inlet, (a) streamwise and (b) normal direction; black slanted lines correspond to the power law relationships in the form 𝑫𝒑 ~ 𝒕𝒙 .
Figure 4: Relative dispersion of southern inlet, (a) streamwise and (b) cross-stream direction; black slanted lines correspond to the power law relationships in the form 𝑫𝒑 ~ 𝒕𝒙 . All diffusivity plots seem to follow a power relation slightly weaker than Richardson’s 4/3 law. The relationships we found appears to be closer to 𝐾𝑐 ~ 𝑑1 . The average diffusivity values for the inlets were calculated and are presented in Table 2. These values indicate that overall there is higher diffusivity at the southern inlet compared to the northern inlet. The lower diffusivity of the north could be attributed to less water being released at the beach due to the constant arrival of waves. Thus, the width of the opening and the shallow sand bars could act as barrier to outgoing water, dissipating the overall energy further upstream and reducing the diffusive behaviour. For the southern inlet, such a barrier do not exist. Instead it opens into Moreton Bay and experiences more effects due to tidal changes. Another important attribute of the southern
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inlet is that it experiences much more marine traffic than the north. This traffic is mostly focused at the opening of the marina to the passage. The trajectories of the drifters did pass close to this opening, possibly influencing the diffusivity values observed due to an increase in energy of the system.
Figure 5; Relative Diffusivity of northern inlet, (a) streamwise and (b) cross-stream direction; black slanted lines correspond to Richardson’s 4/3 power law relations.
Figure 6: Relative Diffusivity of southern inlet, (a) streamwise and (b) cross-stream direction; black slanted lines correspond to Richardson’s 4/3 power law relations. Table 2: Average diffusivity values of the two inlets, averaged over the entire experiment.
4.2.
Inlet
Streamwise (𝒎𝟐 𝒔−𝟏 )(∙ 𝟏𝟎−𝟏 )
Cross-stream (𝒎𝟐 𝒔−𝟏 )(∙ 𝟏𝟎−𝟏 )
North
2.44
1.46
South
8.31
8.78
DIFFUSIVITY SCALING
The relationships between the diffusivities and the associated length scales were derived by fitting a non-linear least squares power relation to the diffusivity-separation length scale data. The diffusivity and length scales were averaged in 1 m bins before plotting, to identify general trends. Table 3
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presents the coefficients, their 95% confidence intervals, regression coefficients and root mean square errors. No relationships were developed for diffusivities with length scales of 𝑑 < 5𝑚 as the deployments were not long enough to allow larger scale processes to be observed. The relationships revealed that at smaller length-scales the diffusivities were similar but generally weaker than the Richardson’s 4/3 power law. Above 𝑑 < 5𝑚, we assume that higher ordered relations exist for the streamwise direction with a constant relation eventually being reached in the cross-stream direction. Further experimental studies are needed to observe these predicted relations. Comparing the relations developed for the diffusivities of the two inlets, we observe that both inlets are subject to similar increases in diffusivity per increase in length scale. It can also be seen that higher values of diffusivity are present at lower length scales in the southern inlet, perhaps being attributed to a larger overall mean velocity in the southern inlet. Table 3: Coefficients of the relationships between relative diffusivity and their associated length scales for both inlets, 𝑲𝑪 = 𝒄𝟏 𝒅𝒄𝟐 . Results are presented in Figure 7. 𝒄𝟏 (∙ 𝟏𝟎−𝟐 )
95% CI of 𝒄𝟏 (∙ 𝟏𝟎−𝟐 )
𝒄𝟐 (∙ 𝟏𝟎−𝟏 )
95% CI of 𝒄𝟐 (∙ 𝟏𝟎−𝟏 )
R2(∙ 𝟏𝟎𝟎 )
RMSE (∙ 𝟏𝟎−𝟐 )
NorthStream
11.2
7.14, 15.3
5.00
2.01, 8.00
0.932
1.88
North Cross
9.44
1.81, 17.1
7.03
0.616, 13.4
0.873
3.57
South Stream
30.2
9.40, 51.0
3.90
-1.94, 9.73
0.720
11.47
South Cross
20.1
15.1, 25.0
50.2
2.98, 7.05
0.967
2.69
Inlet / Direction
Figure 7: Diffusivity relations for separation scale of d
