Estuaries have always attracted human settlements. Sheltered harbours, good fishing grounds, access to transport along rivers have been important reasons why people set up cities along the shores of estuaries for centuries. The various human uses of estuaries impact on the water quality and on the health of the estuarine ecosystem. As the human population grew during the 19th and 20th century and will continue to grow during the first half of the current century, human settlements along estuarine shores increase in size. As a result, a use of an estuary that may have been a small or irrelevant problem a century ago, can turn into a serious threat to entire estuarine systems.
The ocean has always been a convenient place to dispose of unwanted material. Estuaries are always under pressure to serve as waste disposal areas, particularly for fluid waste. Sewage disposal into estuaries has been a practice for centuries without major adverse effects on the ecosystem; it becomes a serious problem when today's megacities continue the practice on a much larger scale. The introduction of new industrial production methods has greatly increased the list of potentially harmful waste products for which estuaries serve as dumping grounds. Managing the health of the estuarine ecosystem has therefore become a necessity.
One of the tools of estuarine management is the flushing time concept. It is often used to determine how much of a potentially harmful substance an estuary can tolerate before its ecosystem is adversely affected to significant degree. While the flushing time concept is a legitimate scientific tool, it has to be understood as exactly that: a scientific aid for decision makers. The basic decsion whether a particular substance should be disposed of and introduced into an estuary has to be made before the flushing time concept comes into play and depends on many factors, such as alternative disposal methods on land, possible recycling options, economics of alternative solutions and others. Once a decision has been made that disposal in an estuary is the correct solution, the flushing time concept is used to evaluate where, how and in what quantities the particular substance can be disposed of.
The motivation for the flushing time concept stems from two problems of modern estuary management. The harmful effects of a potentially harmful substance is usually a function of its concentration, and knowledge of the flushing time can assist in determining allowable disposal loads for a particular estuary. In addition, knowledge of the flushing time can provide some guidance how to handle accidental spills of harmful or toxic material and design emergency procedures for industrial disaster situations.
Figure 15.1
In its simplest form, the flushing time is defined as the time needed to drain a volume $\mathbf{V}$ through an outlet $\mathbf{A}$ with current velocity v (Figure 15.1). More specifically, the flushing time $t_F$ of an estuary can be defined as the time needed to replace its freshwater volume $\mathbf{V_F}$ at the rate of the net flow through the estuary, which is given by the river discharge rate $\mathbf{R}$:
\[ t_F = \frac{V_F}{R} \]
Practical applications of this concept require significant observational information. To keep these requirements down various approximations are introduced in practice.
The most rigourous approach estimates the freshwater volume of the estuary from measurements of salinity. If mixing in the estuary is fully turbulent (ie it can be described mathematically as a linear mixing process), the fresh water fraction $f$ in a water sample is given by
\[ f = \frac{S_0 - S}{S_0} \]
where $S_0$ is the oceanic salinity found outside the estuary mouth. The fresh water volume of the estuary, or of any particular section of the estuary, is the volume integral over $f$:
\[ f^* = \int f\; d\mathbf{V} \]
where $\mathbf{V}$ is the volume of the estuary or of the section under consideration and $f^*$ is the average fresh water fraction found by integration over $(S_0 - S) / S_0$. It follows that the flushing time is given by
\[ t_F = f^* \; \frac{\mathbf{V}}´{\mathbf{R}} \]
Calculation of the flushing time through this method requires the knowledge of the estuary volume (which is acquired through a detailed depth survey that has to be performed only once), measurement of the river discharge rate $\mathbf{R}$ (which can be acquired at a single point at the inner end of the estuary) and a survey of the salinity distribution through the entire estuary.
The flushing time $t_F$ changes with variations in the river discharge, but not in direct proportion to $1 / \mathbf{R}$, because $f^*$ varies with $\mathbf{R}$ to some degree, too. Observations in New York Bight, for example, showed that $f^*$ decreases by less than 50% from 10.6 to 6 days, if the river discharge increases by one order of magnitude from $13-140 \times 10^6 \;m^3/dia$.
The observational requirement of a complete survey of the salinity distribution in the estuary can be demanding in time and financial resources. Efforts to derive flushing times from a smaller observational data base introduce additional assumptions. The "tidal prism" method starts from the concept that a sea water volume $\mathbf{V_T}$ enters the estuary with the rising tide, while a freshwater volume $\mathbf{V_R}$ enters the estuary during a tidal cycle (rising and falling tide). It assumes that the salt water volume $\mathbf{V_T}$ is completely mixed with the fresh water volume $\mathbf{V_R}$ at high tide, and that the combined volume $\mathbf{V_T} + \mathbf{V_R}$ representing the mixture leaves the estuary during the falling tide.
The salinity of the fresh water volume $\mathbf{V_R}$ is zero. If the salinity of the salt water brought in by the rising tide is $S_0$, the salinity $S^*$ of the mixed water in the volume $\mathbf{V_T} + \mathbf{V_R}$ is easily calculated from
\[ (\mathbf{V_T} + \mathbf{V_R})\; S^* = V_T \; S_0 \]
and found to be
\[ S^* = S_0 \; \frac{\mathbf{V_T}}{\mathbf{V_T} +\mathbf{V_R}} \]
This gives the fresh water fraction
\[ f^* = \frac{S_0 - S^*}{S_0} = 1 - \frac{S^*}{S_0} \]
as
\[ f^* = \frac{\mathbf{V_R}}{(\mathbf{V_T} +\mathbf{V_R})}\]
The flushing time was previously defined as
\[ t_F = f^* \; \frac{\mathbf{V}}{\mathbf{R}}\]
where $\mathbf{R}$ is the river discharge rate, or freshwater volume per unit time. In the tidal prism method the unit of time is the tidal period $\mathbf{T}$, so $\mathbf{R = V_R / T}$. Using the result for the fresh water fraction obtained under the assumptions of the tidal prism method, this gives
\[ t_F = \mathbf{T} \; \frac{\mathbf{V}}{\mathbf{V_T} +\mathbf{V_R}}\]
Figure 15.2
The combined volume $\mathbf{V_T} + \mathbf{V_R}$ represents the difference between high water and low water (Figure 15.2); it is therefore often called the tidal prism. It is the only quantity (besides knowledge of the estuary volume) required to calculate the flushing time in this method and can be easily obtained from tide gauge records.
The assumptions of the tidal prism method are never completely met in real estuaries. Mixing of the two volumes $\mathbf{V_T}$ and $\mathbf{V_R}$ is never complete, and some of the mixed water that leaves the estuary with the ebb tide will enter it again with the rising tide. The flushing time derived from the tidal prism method represents the shortest possible time during which the entire fresh water fraction of an estuary can be removed; in other words, it represents a lower limit for any flushing time calculation.
Another estimate of the flushing time can be obtained from the salt balance equations (the Knudsen formula). If $\mathbf{R}$ denotes the river discharge rate as before and $\mathbf{Q_{bottom}}$ and $\mathbf{Q_{top}}$ are the volume transports of the water entering and leaving the estuary at its mouth with respective salinities $S_{bottom}$ and $S_{top}$, the continuity of volume and the continuity of mass (salt) give
\[ \mathbf{Q}_{top} - \mathbf{Q}_{bottom} = \mathbf{R} \]
\[ \mathbf{Q}_{top} S _{top} = \mathbf{Q}_{bottom} S_{bottom}\]
Solving for $\mathbf{Q_{top}}$ and noting that the flushing time is given by the volume $\mathbf{V}$ of the estuary divided by $\mathbf{Q_{top}}$, it follows that
\[\displaystyle t_F = \frac{\mathbf{V} (S_{bottom} - S_{top})}{(S_{bottom} \mathbf{R})} = \frac{\mathbf{V}}{\mathbf{R}} \left(1 - \frac{S_{top}}{S_{bottom}} \right). \]
This determination of the flushing time requires the knowledge of the estuary volume and of the river discharge as before and in addition the measurement of $S_{bottom}$ and $S_{top}$, which can be obtained from a single station at the mouth of the estuary.
The core assumption of the Knudsen formula is that all sea water that enters the estuary leaves it in the upper layer after complete mixing with fresh water. If this is not the case and the interface between the two layers is more diffuse, the upper layer will not maintain zero salinity downstream, and some sea water from the lower layer will mix with water that is already slightly saline. The resulting mixture will have a salinity larger than $S_{top}$. The Knudsen formula therefore underestimates the true flushing time in a similar manner as the tidal prism method, since it also assumes complete mixing of fresh water and oceanic water. In practice, the Knudsen formula appears more appropriate for highly stratified and salt wedge estuaries, whereas the tidal prism method is more suitable for slightly stratified and vertically mixed estuaries. The following table summarises the three methods.
| Input data | Result | Remarks |
|---|---|---|
| \[\displaystyle \mathbf{t_F} = \frac{f^* \mathbf{V}}{\mathbf{R}} \] | |
| \[\displaystyle \mathbf{t_F} = \mathbf{T} \;\left(\frac{\mathbf{V}}{V_{\mathbf{T}} + V_{\mathbf{R}}}\right)\] | tidal prism method, underestimates $t_F$. Suitable for slightly stratified estuaries |
| \[\displaystyle \mathbf{t_F} = \frac{\mathbf{V}}{\mathbf{R}} \left(1 - \frac{S_{bottom}}{S_{top}} \right) \] | Knudsen formula, underestimates $t_F$. Suitable for highly stratified estuaries |
Regardless which method is used to derive a flushing time for a real estuary, there can be no doubt that the derived flushing time is only a very basic description of the flushing process. The next chapter will show that many introduced substances circulate through an estuary in the same way as the fresh water introduced by a river. A single number may give a reasonable estimate of the time it takes to remove the freshwater fraction and as a consequence any introduced substance from an estuary, but it is not necessarily representative for the time required to remove an introduced substance from all parts of the estuary. Many estuaries have a complicated topography, which includes isolated depressions and secluded embayments with little water exchange. While most of a potential pollutant may be flushed from the estuary in the time estimated from any of the three methods, high pollutant levels can remain in pockets of stagnant water for much longer. Proper estuary management requires estimates of flushing times for all parts of the estuary.
Figure 15.3
All three methods can be applied to individual parts of estuaries. The Knudsen formula, for example, can be used with salinity observations from any station in an estuary and will then give an estimate for the flushing time of the part of the estuary between the river input and the station. This increases the amount of data required to derive flushing times.
An alternative method that requires only verification with data from selected locations has become available as a result of the development of numerical modelling techniques. Numerical models derive flushing times for every location in the estuary by calculating the time a water particle needs to reach the estuary mouth. This time is determined by particle tracking. The procedure results in a flushing time map, which gives flushing time as a function of location. Such maps highlight problem areas in estuaries where the circulation stagnates and can be used to improve the water movement through engineering measures. Figure 15.3 shows an example.