6. Discussion

#### 6.1 Tidal asymmetry

The fact that higher high water always precedes lower low water in Elkhorn Slough is one reason that ebb current speeds exceed flood current speeds. The advance of the flood tide and the retreat of the ebb tide are retarded by frictional effects. The mud flats and Salicornia marsh areas are large compared to the channel area and they contribute to these frictional effects and thus to the retardation. The retardation, however, is apparently greater on the incoming tide than it is on the outgoing tide and we will say more about this in what follows. According to Wong [13], this retardation contributes to the tidal asymmetry where the duration of the ebb tide is reduced relative to that of the flood tide.

Because the asymmetry between flood and ebb currents is cumulative, the effect on tidal height becomes more pronounced towards the head of the Slough. Distortion of the incoming tidal wave also occurs as a result of the frictional effects associated with the bottom and lateral boundaries and is compounded by the effects of decreasing bottom depth, narrowness of the entrance, and decrease in channel cross section toward the head of the Slough. This distortion produces a number of new shallow water tidal constituents including overtides and compound tides. Overtides arise as the incoming tidal wave runs into shallow water where the trough is retarded more than the crest due to bottom friction and thus the wave loses its simple harmonic form [26]. These frictional effects lead to the production of compound tides which also occur in ES. The existence of overtides and compound tides is clearly evident from the tidal records at Kirby Park, approximately 7 km from the harbor entrance. Finally, because tidal heights and currents have often been found to be roughly 90° out of phase, the tides in ES may approximate a standing wave system.

#### 6.2 Generation of overtides and compound tides

Observations presented in Section 3 revealed the presence of higher frequency tidal constituents in ES. Because of tidal transformations, the tidal regime in ES differs from that of Monterey Bay. Tidal periods of 8.4, 8.3, 8.2, 6.2, and 4.1 h were found that correspond to the 2MK3, M3, MK3, M4, and M6 tidal constituents, respectively. The 8.3-h M3 lunar terdiurnal tide is not classified as a shallow water constituent [26] and may originate outside the Slough. The 6.2-h M4 and the 4.1-h

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

M6 constituents are overtides that represent the first harmonic of the primary M2 tide, and the sixth-diurnal tides, respectively. The 2MK3 (8.4 h) and the MK3 (8.2 h) constituents, or terdiurnal components, are compound tides that correspond to the sums (MK3 = M2 + K1), or differences (2MK3 = 2M2 K1) of the primary M2 and K1 constituents. We find the amplitudes of these constituents to increase monotonically with distance inland. However, the amplitudes of the primary constituents do not decrease significantly with distance up the Slough. M2 increases by 40 mm and K1 is nearly constant. That the M2 amplitude inside ESNERR is only slightly smaller than at the entrance to Parsons Slough clearly suggests that the return flow during ebb at this location is not at the present time choked or partially choked [18].

According to Wong [13] and Clark [14], the time of maximum ebb flow occurred slightly later than or midway through the ebb tide, while the time of maximum flood tide occurs slightly later than midway through the flood tide, towards the time of high water. They attributed this delay during the flood tide to the volume of water that must be transported across the tidal flats which may also contribute to the observed overtides in ES [30]. We have examined this process using a month-long current record from January 2002 (Figure 6, bottom panel). From a cross-correlation analysis between the mean corrected pressure (i.e., tidal elevation) and the along-channel current speed, we found that the maximum lag was 3.24 h, which is very close to quadrature for the dominant M2 tide (12.42 h). Harmonic analysis of these data showed that the phase angle between the dominant semidiurnal and diurnal harmonic constituents for tidal elevation and current speed (M2 and K1) were 84° and 88°, respectively, consistent with Clark and Wong's standing wave description of the tidal regime in ES.

Considerable research has been conducted on tidal transformations in wellmixed estuaries, primarily along the U.S. East Coast [27–30]. In many respects, their results should be generally applicable to any well-mixed estuary. However, there is one important difference between the tides along the East Coast and West Coast of the U.S. Along the East Coast, they are semidiurnal, whereas along the West Coast, they are mixed, mainly semidiurnal, and the greatest tidal range occurs from higher high water to lower low water. This characteristic produces initial conditions for tides entering shallow embayments along the West Coast that clearly favor ebb domination prior to any tidal transformation. Once the tide has entered the estuary, shallow water effects produce overtides and compound tides which experience down-channel evolution in amplitude and remain phase-locked to the parent tides throughout the estuary [29]. According to Parker [31], the increase in amplitude of the overtides and compound tides with distance up the Slough occurs at the expense of the fundamental constituents to which they are harmonically related through the nonlinear transfer of momentum and energy. However, we note that our results are not necessarily consistent with Parker's explanation since we found that although the amplitudes of the overtides and compound tides did increase with distance inland, the amplitudes of the primary constituents did not decrease significantly over the same distance.

According to Boon and Byrne [32], distortion of the incoming tide leads to temporal asymmetries in the rise and fall of the surface tide. These, in turn, result in temporal and amplitude asymmetries in the velocity field. Further, these asymmetries cause estuaries to be either flood- or ebb-dominant. According to Friedrichs and Aubrey [30], non-linear tidal distortion has two principal causes, frictional interaction between the tide and the channel bottom which leads to shorter flood tides, and intertidal storage which causes the ebb tides to be shorter. Based on the work of Friedrichs and Aubrey, intertidal storage due to the presence

of the extensive Salicornia marsh and mud flats in ES may be a principal factor that contributes to the dominance of the ebb tide in ES.

Basic mathematics can be used to illustrate how certain estuarine tidal transformations arise. First, we show one simple approach that illustrates how both overtides and compound tides can be generated. We assume that shallow-water effects are proportional to the square (or higher power) of tidal sea level, following Pugh [33]. Take two primary constituents such as M2 and K1, whose frequencies are ω<sup>2</sup> and ω1, form their sum, and square the result,

$$\begin{array}{c} \left[\eta\_{\text{M2}}\cos 2\alpha\_{2}t + \eta\_{\text{K1}}\cos 2\alpha\_{1}t\right]^{2} = \mathbf{1}/2\left(\eta\_{\text{M2}}\,^{2} + \eta\_{\text{K1}}\,^{2}\right) + \mathbf{1}/2\left(\eta\_{\text{M2}}\,^{2}\right)\cos 4\alpha\_{2}t\\ \qquad + \mathbf{1}/2\left(\eta\_{\text{K1}}\,^{2}\right)\cos 4\alpha\_{1}t + \eta\_{\text{M2}}\eta\_{\text{K1}}\cos 2(\alpha\_{1} + \alpha\_{2})t\\ \qquad + \eta\_{\text{M2}}\eta\_{\text{K1}}\cos 2(\alpha\_{1} - \alpha\_{2})t, \end{array} \tag{4}$$

where η is the free surface elevation, t is time, ω = 2π/T, and T is the constituent period. This expansion contains additional harmonics with frequencies 4ω<sup>2</sup> and 4ω<sup>1</sup> which represent the overtides.

The last two terms contain the sum and difference frequencies for ω<sup>2</sup> and ω<sup>1</sup> which represent the compound tides. Also, the first term contains the sum, 1/2(ηM2 <sup>2</sup> + ηK1 2 ), which corresponds to an increase in mean sea level. Observations in many estuaries have shown an increase in mean sea level for the incoming tide as the head of the estuary is approached [33]. Closer at hand, the NOS tide survey in 1976 [12] shows that mean sea level may increase by as much as 30 mm (0.1 ft) between the H1B and Hudson's Landing.

To illustrate an alternate approach for the generation of overtides, we employ the one-dimensional equations of motion and continuity to illustrate how the relevant hydrodynamics apply to tidal transformations. For the along-channel ("x") momentum equation, we have the following,

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial \mathbf{x}} = -\mathbf{g} \frac{\partial \eta}{\partial \mathbf{x}} \tag{5}$$

where x, t, η, and g are distance along the x-axis, time, surface elevation, and the acceleration of gravity, respectively, and u is the along-channel current velocity. The governing equation for continuity which employs the same variables with the addition of H, the bottom depth, can be expressed as,

$$(H+\eta)\,\frac{\partial u}{\partial \mathfrak{x}} + u\frac{\partial \eta}{\partial \mathfrak{x}} = -\partial \eta/\partial \mathfrak{t}.\tag{6}$$

This equation has been modified slightly to take into account the geometry that we often apply to estuaries where the rates of mass transport into and out of a vertical column are equated [34]. These two simultaneous nonlinear differential equations can be solved, as shown in Officer [34], to obtain a solution for η(x,t), of the following form,

$$\eta(\mathbf{x},t) = \eta\_{\rm o} \cos\left(k\mathbf{x} + at\right) - \left(\Im g k \eta\_{\rm o}{}^2 \alpha / 4c^3\right) \mathbf{x} \ \sin\left\{ 2(k\mathbf{x} + at) \right\} \tag{7}$$

where only terms of order η<sup>o</sup> <sup>2</sup> or lower have been retained. η<sup>o</sup> is the tidal amplitude at an arbitrary point of origin for x and t, and k is the wave number, 2π/L, where L is the shallow water wavelength, g is the acceleration of gravity, and c = gH, the shallow water wave velocity. The second term in (7) captures the essence of overtide generation where η(x,t) is seen to be directly proportional to x, the distance in the upestuary direction. As x increases, H generally decreases which also contributes to

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

increasing amplitude as the tidal wave propagates inland. The increase in η(x,t) clearly reflects the increasing distortion experienced by the incoming tidal wave as it propagates in the up-slough direction. Finally, our tidal observations presented in Table A1 show a monotonic increase in the M4 and M6 overtides and in the MK3, 2MK3, and SO3 compound tides between the H1B and the head of ES.

With respect to the tides in estuaries, the magnitude and phase of each constituent reflect the hydrodynamic processes that are important. For estuaries where the M2 tide is dominant, its first harmonic, the M4 tide, is usually the dominant overtide, as in ES. The phase relationship between the M2 and M4 components determines the direction and magnitude of the tidal asymmetry [29]. Ebbdominance is further enhanced by inefficient water exchange around high water in estuaries with relatively deep channels and extensive intertidal water storage. Low water velocities in intertidal marshes and mud flats cause the high tide to propagate slower than the low tide [30]. At low tide, the marshes and flats are empty while the channels serve to accelerate the flow in the down-channel direction. The extensive mud flats and marshes in ES contribute to weak or sluggish water exchange around the time of high tide, as indicated by the results of Wong [13].

To expand on this topic slightly, we refer back to the idealized cross-section for ES (Figure 12). One aspect of this model cross-section is that the mud flats slope downward toward the main channel. We expect these slopes to retard rising water levels on the incoming tide, and accelerate flow back into the main channel on the return tide. This mechanism should contribute to the asymmetry of the tides in ES depending on the magnitude of the slopes and their extent, and, most importantly, helps to explain why the incoming tide experiences greater retardation than the outgoing tide. Thus, intertidal water storage due to sloping muds flats may also be a significant factor that contributes to ebb-dominance in ES.
