4. Estimating the tidal prism

In this section, we address the tidal prism, a crucial factor that essentially determines the tidal currents. The tidal prism is the volume of water exchanged between a bay and its parent body of water over a tidal cycle. Also, the current regime of an estuary is strongly influenced by three factors: morphology, river flow, and tidal forcing. The ratio of river flow to tidal volume characterizes the physical transport of water and other materials through the system. Hence, one of the most important problems is to evaluate the total tidal transport through the Slough.

From the conservation of volume, the average current speed is directly related to the tidal prism and inversely related to the cross-sectional area. This may be expressed as

A 30-Year History of the Tides and Currents in Elkhorn Slough, California DOI: http://dx.doi.org/10.5772/intechopen.88671

$$
\mu\_{\text{max}} = 4/3(rc/2)(\text{A/S})(\text{H/T}),\tag{2}
$$

giving the mid-channel maximum tidal current, umax, in an enclosed basin [20]. A is the surface area of the basin inland from the main tidal channel, S is the crosssectional area through which the current flows, H is the half-tidal range, and T is the half-tidal period. The factor 4/3 represents horizontal shear: the mid-channel current is about 1/3 greater than the cross-sectional mean current, and rc/2 relates the mean velocity for the half-tidal period to the maximum half-tidal velocity. Finally, the tidal prism corresponds to the volume AH.

Where the tides are mainly mixed diurnal and semidiurnal, the tidal prism is often taken to represent that volume of water associated with the change in elevation between MHHW and MLLW, but other tidal transitions could be used. To the extent that other sources and sinks of salt and fresh water contribute to the total volume of water in ES, the tidal prism will depart from the total water flux that is exchanged over a tidal cycle. The tidal prism can be estimated in several ways: by metering water fluxes through a vertical section on successive ebb and flood tides, by mapping sea level in the embayment at various tide levels, or by measuring the surface area and thickness of the discharge plume at an appropriate stage of the tide. The tidal prism for ES has been estimated using the first two methods with varying degrees of success, and the differences obtained from these methods provide at least one measure of the uncertainty in estimating this quantity.

During a 1956 survey of tidal inlets on the Pacific coast, Johnson [21] reported a tidal prism of 2.65 � <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> for ES, but the details of its estimation were not given. Smith [10] constructed a cross-sectional model for ES from which he estimated the tidal prism and the volume of slough waters at different tidal stages. He divided the Slough into three provinces: main channel, mud flats, and marsh. An idealized cross-section for ES, based on this classification scheme, identifies these provinces (Figure 12). He observed a predicted tide height of 0.88 m at the mud flat level and 1.45 m at the Salicornia marsh level. These heights are referenced to the tidal datum: 0.0 m at MLLW. Using Hansen's map (the basis for Figure 1), which was based on aerial photographs, he estimated the areas for each of these surfaces. The volumes obtained using this approach (i.e., the first method above) were then compared with volume transport estimates obtained using the second method, based on Clark's current meter measurements at the H1B. By adjusting the elevations of the mud flats and marsh, Smith showed reasonable agreement between the two methods. Smith's estimate of the tidal prism for HHW to LLW was 4.7 � <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> . Smith used his results to estimate that the tidal prism extended 4.8 km into the Slough, assuming insignificant mixing between the ambient slough water and new

#### Figure 12.

Idealized cross section of ES. Mean sea level occurs near the edge of the mud flats, and the Salicornia marsh is about 1.45 m above MLLW. Adapted from Smith [10]. Labeled quantities are used in Smith's volume continuity model of the currents.

tidal volumes. Of historical note, a foam line produced by waters discharged from the PG&E power plant was often observed 4–5 km inland. Because coolant waters are now discharged directly into Monterey Bay, this foam line is no longer present. Continuing, in 1992, we applied Smith's model to a new aerial photograph that included the ESNERR South Marsh reclamation area, a province of the Slough that did not exist in 1974. The tidal prism from this work was estimated to be 6.2 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> (Figure 13).

Clark [14] used observations of umax at the H1B and H/T to form a regression relationship whose intercept provides an estimate of the non-tidal flow, and whose slope provides a measure of A/S, the ratio of the flooded surface area to the crosssectional area (Figure 14). This regression can also be used to estimate maximum tidal flows. Using comparisons of data from different periods, we can illustrate changes in the tidal flow and thus changes in the tidal prism. Using this approach, Wong found an intercept of about 10 cm/s indicating a net non-tidal flow directed out of the Slough. Because the values were relatively small, however, its significance is uncertain. Several factors most likely accounted for the net seaward flow according to Wong, including recent rainfall, agricultural runoff, and the discharge of cooling water into the lower slough from the PG&E power plant which operated the Slough outfall at that time.

Umax vs. H/T has been plotted for measurements that were acquired between 1971 and 1992 (Figure 14). The increase in regression slopes, based on a linear least squares fit to the data, indicates that maximum currents and thus the tidal prism (by an increase in the surface area, A) have increased by nearly a factor of two since 1971. At least two factors have contributed to increased tidal fluxes. The first was the restoration of formerly diked pasture which led to tidal flooding (Table 1). This sudden increase in surface area clearly contributed to Wong's [13] observed increase in tidal currents. However, erosion continues to enlarge the Slough at rates which have been documented on several occasions, e.g., [3–5], and this process continues to enhance the tidal flow, unabated.

#### Figure 13.

Cumulative volume of water in ES at MLLW and MHHW from Smith [10] and as re-evaluated in 1993 from more recent aerial photographs. The volume between MLLW and MHHW is defined as the tidal prism, and estimated to be 4.7 106 <sup>m</sup><sup>3</sup> in 1973 and 6.2 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> in 1993.

A 30-Year History of the Tides and Currents in Elkhorn Slough, California DOI: http://dx.doi.org/10.5772/intechopen.88671

#### Figure 14.

Summary of current meter observations at the H1B. Y-axis values give smoothed along channel maximum speeds during a single half-tidal cycle. X-axis values are the ratios of the predicted or observed tidal range and half-tidal period (Eq. (2)). The symbols "o" represent Clark's [14] observations, "+" represent Wong's [13] measurements, and "\*" represent our 1992 data. The least squares regression lines show a systematic increase in time which reflects the increase in tidal prism.


Table 1.

Recent additions to the Elkhorn Slough system that contribute to tidal volume increase and salt water incursion [6].

Our newer estimates of the tidal prism and total volume of water in ES also indicate major increases. The tidal prism from these more recent estimates is approximately 6.2 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> (Figure 13), a 32% increase in approximately 20 years. Interpreting recent changes to the water budget in ES is further complicated by the 1989 Loma Prieta earthquake which may have caused subsidence in the upper Slough. In addition to the restoration area discussed earlier, other additions have been made to ES which have increased its volume since the mid-1980s. Man-made alterations to the Slough and the approximate increases in surface area and volume caused by these changes are listed in Table 1 [6]. The Parsons Slough area (Figure 1) is the major contributor to these increases. Also, the bottom depth at the entrance channel to Parsons Slough has increased from about 3 m in 1993 [6] to almost 5 m in August 2002 based on recent measurements.

In addition to water volume, current measurements in ES have been used to estimate several related parameters of interest including non-tidal contributions to the circulation, and the geometry of the Slough. Using Eq. (2), Clark estimated the ratio A/S for Elkhorn Slough. Then, using measurements of the cross-sectional area at the Highway 1 Bridge (Figure 4), he estimated the effective surface area of the Slough to be 1.5 and 1.1 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>2</sup> at high and low tides, respectively. Using a similar approach, Wong [13] obtained estimates of the effective surface area, which were almost twice those obtained by Clark. According to Wong, the increase in surface area was caused primarily by restoration of Parsons Slough, and this lead to accelerated erosion through increased tidal action.

Wong also calculated the tidal volume using the product of the tidal height and the effective surface area. He compared his volume estimates with previous values from Smith [10] who estimated the volume as the product of water height and the areas that covered the channel, the mud flats, and the marsh. Wong found that the mean tidal prism had increased from slightly over 4 <sup>10</sup><sup>6</sup> to almost 7 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> , and that the total water volume at high tide had increased from approximately 9 106 to 10 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> . Although the uncertainty associated with these estimates is high, they show a trend toward higher values which we believe is significant. Wong found that the mean diurnal tide flushes roughly 75% of the total volume of water from the Slough. Based on these results and the assumption that the remaining waters in the Slough do not mix with the incoming waters from Monterey Bay, he estimated that the tidal prism would extend almost 5 km inland from the mouth, only slightly larger than the value of 4.8 km obtained by Smith [10].
