The estimation of vertical eddy diffusivity in estuaries

Transcription

1 The estimation of vertical eddy diffusivity in estuaries A. ~temad-~hahidi' & J. 1rnberger2 l Dept of Civil Eng, Iran University of Science and Technology, Iran. 2 Dept of Env. Eng., University of Western Australia, Nedlands, Australia. Abstract The prediction of pollutant transport in natural water bodies is becoming an essential task. The water quality prediction models basically simulate the physical process and their performance relies on the estimation of parameters such as vertical eddy diffusivity. An aim of this study was to compare the performance of different methods of estimation of the vertical eddy diffusivity in a coastal water body. The field study was carried out in the Swan River Estuary, Australia. The turbulent properties were measured by using a microstructure probe. Field measurements indicated that the different methods might result in different estimation of the vertical eddy diffusivity. Therefore, these traditional methods which are successful in some cases in the ocean, can not be used in all estuaries. 1 Introduction Coastal waters such as estuaries play an important role in human life. These water bodies are used for recreational activities, fisheries and most importantly they provide habitat for wildlife. Anthropogenical activities change the quality and the quantity of inflows into these systems and therefore, affect their water quality. Stratification and turbulence control horizontal and more importantly vertical transport of incoming materials such as nutrients into these water bodies. In numerical models used for natural water bodies, usually the concept of vertical eddy diffusivity, k,,, is used to predict the vertical transport of mass. It is

2 assumed that in analogy to laminar flow, the (turbulent) mass transport is equal to kv times the gradient of mass. It should be mentioned that in contrast to the molecular diffusivity, k, is mostly a property of flow not fluid. It can be shown (see background) that kv is a function of turbulence activity, background stratification and more importantly, the efficiency of turbulence to produce mixing. In spite of the importance of the estuaries in our life, only a few microstructure measurements have been reported in the literature and most of the field experiments have been conducted in the oceanic thermocline to estimate mixing efficiency. The aim of this work is to study the interaction of turbulence and stratification in a salinity stratified estuary. This will be investigated by comparing the estimates of turbulent properties such as mixing efficiency and k, obtained from different methods. 2 Background Turbulence in estuaries can be driven by shear instability [l], internal waves [2] and Bed generation [3]. In the field, turbulence is studied mostly using vertical profilers [4] or submarines [5]. In contrast to the non-field experiments, the flow is generally unsteady and the stratification and shear are not uniform. Furthermore, the turbulence is generally patchy and intermittent in stratified natural water bodies [4]. Vertical profiles, which are assumed to be snapshots of turbulence, are normally segmented into stationary parts or turbulent patches. The turbulent properties are then, calculated for each patch by using frozen flow hypothesis. One major goal of field studies is to estimate the vertical eddy diffusivity, which is directly related to the buoyancy flux. Since the measurement of the buoyancy flux is very difficult and complicated, it is usually estimated indirectly from other properties of turbulence. The common methods for indirect estimation of the buoyancy flux and the vertical eddy diffusivity are briefly mentioned in the following section. 2.1 Osborn Method The turbulent kinetic equation for an incompressible fluid is:

3 where, q2=u is twice the turbulent kinetic energy (TKE,, is the velocity 2J J fluctuation and U. is the mean velocity. p and are the pressure and density J fluctuations respectively. Ivey and Imberger [6] wrote this eqn. as: m=b+~ (2) where, m is the net rate of the mechanical energy, b is the buoyancy flux and is the rate of dissipation. In this form, the mixing efficiency can be written as [6]: b R -- /-b+& The mixing efficiency can be related to the vertical eddy diffusivity as: (3) where, ~2 is the buoyancy frequency squared. Assuming a steady condition and neglecting the divergence terms eqn (l) reduces to: Osborn [7] assumed an upper bound of 0.15 for R which resulted in to a f' buoyancy flux estimation as: bos (6) This upper limit is commonly used to (indirectly) estimate the buoyancy flux and vertical eddy diffusivity in the field studies. 2.2 Osborn-Cox method In this method which is based on the conservation of temperature fluctuations, the adiabatic lapse rate is neglected. The temperature conservation equation for the temperature variance becomes: where, TI is the temperature fluctuation, K is the molecular thermal diffusivity and X = ~ K(Z' is the rate of diffusive destruction of the temperature fluctuations. For a vertically stratified flow which is steady and laterally homogenous, eqn (7) reduces to [8]. In the dropsonde measurements, it is assumed that the gradients of temperature fluctuation in lateral directions are related to the vertical gradient of temperature and. Finally, the buoyancy flux in this method is estimated as:

4 458 Water Pollution \.I 2.3 Ivey-Imberger Method In an attempt to predict the buoyancy flux and the mixing efficiency, [6] reexamined the previous laboratory experiments and using scaling argument showed that the mixing efficiency was function of turbulent Froude no.,,crp turbulent Reynolds no., R,~, and buoyancy Reynolds no., F, defined as: g Re, = (E"' L~~~~ / v)~" (11) 2 1/2 Frg =(E/ vn ) (12) where, LE is the Ellison scale, which characterizes the size of the overturns. By conducting further numerical experiments, Ivey [9] modified these results and proposed equations for different regimes of turbulence as below: Table 1. Regimes of mixing efficiency based on Ivey-Imberger method, Regime RII Re, ( ~r,~ 0.28 ( Frg - 'l5 ) Re, -&

5 3 Study area, instrumentation and data processing 3.1 Study area The Swan River estuary is located in Southwest of Australia and is surrounded by the city of Perth. The river has a permanent connection to the Indian ocean and its freshwater inflow is highly variable in different seasons. The vertical stratification varies from a homogeneous state in summer to a strongly stratified state in winter. The width of the connection to the Indian ocean is about 300 m and it has been dredged to 13 m. At Blackwall Reach, about 5 km upstream of the mouth, the estuary becomes straight and reaches to a maximum depth of 17 m (Figure 1). Figure 1 : The map of the Swan River estuary and the Bathymetry of Blackwall Reach. The station is marked by a full circle. Field experiment was conducted during _2-~th of March, 1996 at a fixed station in Blackwall Reach (Figure 1). The straight channel in this part of estuary had a depth of 15 m and a unidirectional mean flow. During the experiment the weather was calm and sky was cloudy. The tide during the experiment was a

6 combination of diurnal and semidiurnal tides with a maximum amplitude of only 0.3 m. 3.2 Instrumentation and data reduction The turbulence activity was documented by a microstructure probe called PFP[10]. The PFP is a nearly buoyant vertical profiler with a weight of 25 kg and a length of 1.5 m. With a fall rate of 0.1 ms-1 and sampling frequency of 100 Hz, the vertical resolution was about m. The temperature microstructure was measured by a FP07 fast thermistor along the longitudinal axis of the PFP. The PFP was also equipped with a 4- electrode micro-conductivity sensor. The salinity and background density profile was derived by matching the temperature and conductivity signals. The PFP had two orthogonally mounted inclinometers and a flux gate to monitor the orientation and tilt of the probe. The pressure sensor was a Keller PAA-10 type, which was used to measure the depth. During the experiment 189 microstructure profiles were collected at the Blackwall Reach station. These profiles were divided into stationary segments and a total of 165 stationary turbulent patches with an average length 0.8 m were selected for further data processing. The rate of the dissipation in each patch was determined by using the temperature microstructure spectral roll-off. The value of the dissipation, E was estimated by fitting the Batchelor spectrum [l l]. 4 Results In order to compare the different methods for estimation of vertical eddy diffusivity and buoyancy flux, the mixing efficiency estimated from different methods were calculated and compared for each segment. Figure 2 displays the distribution of This histogram shows that about a quarter of the turbulent patches had a negligible RII and most of the turbulent patches had a less than Both the mean and the (length) weighted average values of RII were 0.07, which indicated that the was independent of the turbulent patches length. Clearly, the findings indicate that was less than the traditional value of 0.15 commonly used in the ocean [7]. These values of RI/ were then used to estimate the rate of vertical eddy diffusivity The obtained mean value of k, was about 9x 10-5 m-2 S-' which was more than ten times smaller than the mean value derived for other estuarine study [e.g This difference is mainly due to the low energy tide observed in this study compared to the energetic turbulence driven by hydraulic jump in Knight inlet [l 21. Estimates of mixing efficiency using the method of Osborn & Cox [S], R O~, were also obtained for the turbulent patches. Figure 3 shows the histogram of ROC The distribution shows a wide range of values with a mean value of 0.22,

7 which is different to that of This method, however, yielded results similar to that of oceanic studies of [l21 and the upper limit suggested by Osborn [7]. Again using in estimation of k,, resulted in a value of 3x 10-4m-2 S-' which is still lower than the value obtained from study of [12], using a similar mixing efficiency. R,, Figure 2: Histogram of for the turbulent patches, displaying a skewed distribution with a wide range of variability.

8 462 Water- PoUutiorz! ' L R~~ Figure 3: Histogram of for the turbulent patches, showing a skewed distribution with a very wide range of variability. 5 Discussion In an attempt to estimate vertical transport of mass, we made measurements of turbulent activity in a unidirectional geophysical flow. A new profiler capable of recording density microstructure was used in a salinity stratified water column. The obtained profiles showed that the turbulent events were patchy and intermittent. Using different indirect methods we estimated the mixing efficiency and compared it to the traditional value of about Both RII and showed a skewed distribution. The difference in the magnitude and range of RII and RO~,, were significant. The estimates of RII were less than This can be explained by considering the assumptions and limitations of these methods such as steadiness of the flow. Furthermore, these methods are derived from different approaches. The turbulence in natural water bodies is intermittent and unsteady which is not considered in any of these methods. It is not possible to say clearly which indirect method gives a better estimate of k,.. The estimate of ROC is in agreement with the indirect estimates using the same method in the ocean, while the estimate of RII is supported by laboratory and numerical experiments. Significantly, more experiments are required to increase the statistical confidence of the results. The indirect methods outlined in this study should be

9 replicated simultaneously with the direct measurement of buoyancy flux. In this way the applicability of these methods can be verified. References [l] Geyer, W.R. & Smith, J.D..Shear instability in a highly stratified estuary, J, Phys. Oceanogr., 27(6), PP , [2] New, A.L. &Dyer, K.R. Internal waves and mixing in stratified estuarine flows, physical processes in estuaries, ed. J. Dronkers and W.V. Leussen, Springer-Verlag:pp , [3] Abraham, G. Turbulence and mixing in stratified flows, Physical processes in estuaries, ed J. Dronkers and W.V. Leussen, Springer-Ver1ag:pp , [4] Gregg, M.C., Microstructures patches in the thermocline, J, phys, Oceanogr,, 10(6), pp , [5] Gargett, A. E. & Holloway, G., Dissipation and diffusion by internal waves breaking, J, Mar. Res.,42, PP , [6] Ivey, G. & Imberger, J., On the nature of turbulence in a stratified fluid. part lthe energetics of mixing, J, phys, Oceanogr., 21(3), pp , [7] Osborn, T. R., Estimation of the local rate of the vertical diffusion from dissipation measurements, J, phys. Oceanogr,, 1 ql), PP , [8] Osborn,.R. & Cox, C.S., Oceanic fine structure. Geophys, jquid Dyn,, (3), pp , [9] Ivey, G., Imberger, J. and Kossef, J. R., Buoyancy fluxes in a stratified fluid, IUTAM sym. on Physical Limnology, ed. J. Imberger, AGU, pp , [l01 Imberger, J. & Head, R., Measurement of turbulent properties in a natural system, Fundamentals and advancements in hydraulic measurements and experimentations, ASCE, PP. 1-20, [l liluketina, D. & Imberger, J., Turbulence and entrainment in a buoyant surface plume, J, Geophys Res.,94 (12) PP , [l21 Peters, H. & and Gregg, M.C., Some dynamical and statistical properties of equatorial turbulence, Small scale turbulence and mixing in the ocean, eds. J.C. Nihoui & B.M. Jamarat, Elsevier, pp , 1988.