Tidal Energy Dissipation at the Sill of Sechelt Inlet, British Columbia

Transcription

1 VOLUME 31 JOURNAL OF PHYSICAL OCEANOGRAPHY DECEMBER 001 Tidal Energy Dissipation at the Sill of Sechelt Inlet, British Columbia SCOTT W. TINIS* AND STEPHEN POND Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada (Manuscript received 3 May 000, in final form 30 March 001) ABSTRACT The energy budget of a tidally active, shallow silled fjord is discussed. Constriction of the flow over the shallow sill causes a reduction in tidal amplitude and a phase lag across the sill. A generalized expression for the total power extracted from the barotropic tide by dissipation at the sill is derived as a function of the tidal amplitude difference and phase lag of the tidal elevation. Using tide gauge data from both sides of the sill at the entrance to Sechelt Inlet, British Columbia, this generalized expression yields estimates for the energy flux of the barotropic tide, which approach 100 MW during spring tides. From direct measurements of the currents, the estimated frictional dissipation is equal to the flux out of the barotropic tide (within experimental error). A small amount of the energy flux (5%) is estimated to go into the generation of a tidal jet, which dissipates within a few kilometers of the sill and contributes to the formation of a mid-water layer. 1. Introduction The southern coast of British Columbia comprises a complex system of inland waterways and fjord estuaries connected to the open ocean through a series of narrow channels. Tidal streams through these narrow channels are highly energetic, characterized by strong currents (often exceeding 3ms 1 ) and treacherous whirlpools. Many of the fjord estuaries are separated from seaward waters by shallow sills that also produce strong tidal streams. Tidal energy dissipation at the sills of tidally energetic fjord estuaries has been linked to changes in tidal phase landward of the sill (Freeland and Farmer 1980; Stacey 1984). Extensive work has been published relating the energy dissipation to the production of internal waves, and in particular, the internal tide (Stacey 1984; Stigebrandt 1976). In general, the net up-inlet energy flux computed from the tidal characteristics is correlated with changes in stratification that promote or suppress the production of internal waves at the sill. The effect of friction in prior studies of British Columbia fjords has usually been small compared to that of other processes (Freeland and Farmer 1980; Stacey 1984; de Young and Pond 1987), and the tidal phase differences *Current affiliation: Department of Fisheries and Oceans, Institute of Ocean Sciences, Sidney, British Columbia, Canada. Corresponding author address: Dr. Scott W. Tinis, Department of Fisheries and Oceans, Institute of Ocean Sciences, P.O. Box 6000, Sidney, BC V8L 4B, Canada. tiniss@dfo-mpo.gc.ca due to dissipation of the tidal wave have been typically less than 10. In 1991, the Oceanography Department of the University of British Columbia embarked on an extensive study of the Sechelt Inlet System (Fig. 1). The passage over the sill of Sechelt Inlet is known as Skookumchuck Narrows and boasts the fastest tidal stream (8 m s 1 peak) on the west coast of British Columbia. The constriction at the narrows is approximately 1000 m long and 500 m wide, with a mean depth of 15 m (Fig. ). Frictional processes dominate over the sill, resulting in marked changes in phase and decreases in amplitude for all tidal constituents (Table 1). Previous analytical expressions for net energy flux were based on phase difference alone since amplitude differences were negligible. A discussion of previous work on tidal power transfer over fjord sills and the derivation of a generalized expression for net energy flux dissipated over a sill are presented in section, followed by a comparison of the theoretical energy dissipation to estimated dissipation by frictional and tidal jet processes in section 3. Results from this study are summarized in section 4.. Power withdrawn from the barotropic tide Interaction of the barotropic tide with the sill of an inlet causes energy to be withdrawn from the tide. There are four sinks for this energy: 1) friction, ) tidal jets, 3) internal waves at tidal frequency, and 4) high-frequency internal waves. In deep-silled inlets, such as Knight and Observatory Inlets, the greatest part of the barotropic energy flux is balanced by the generation of progressive internal tides and internal hydraulic distur- 001 American Meteorological Society 3365

2 3366 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 FIG. 1. Map of the Sechelt Inlet system (a) with locations of CTD stations (solid circles), surface moorings (crossed circles), and subsurface moorings (solid triangles and squares) and (b) the section A B showing the depth along the main channel of the inlet. bances, with only a small amount (5%) by frictional dissipation. Freeland and Farmer (1980) found that as stratification increased so did the power withdrawn from the barotropic tide in Knight Inlet. Stacey (1984) also studied the removal of energy from the barotropic tide by sill interactions in Observatory Inlet, using a progressive internal tide model that was an extension of the linear model presented by Stigebrandt (1976, 1980).

3 DECEMBER 001 TINIS AND POND 3367 TABLE 1. The amplitudes and Greenwich phases for selected tidal constituents from Boom Islet and Porpoise Bay at the head of Sechelt Inlet. Data are courtesy of the Tides and currents section of the Institute of Ocean Sciences, Fisheries and Oceans Canada, Sidney, British Columbia. Frequency (cph) M m MS f O 1 K N M S MK 3 M Boom Islet Constituent Amplitude (m) Phase (deg) Porpoise Bay Amplitude (m) Phase (deg) FIG.. Skookumchuck Narrows. The internal tide was found to be the largest sink, while high-frequency internal waves and internal hydraulic jumps (plus a small amount of friction) accounted for the rest. In the Indian Arm Burrard Inlet system, much of the up-inlet tidal energy flux between the entrance at First Narrows and the sill (10 km up-inlet of the entrance) is dissipated by friction in Vancouver harbor, but internal wave generation is the largest sink landward of the sill itself (de Young and Pond 1987). In contrast to high-frequency internal waves, internal hydraulic processes, and frictional dissipation, which mainly provide energy for mixing near the sill, the breaking of the baroclinic tide on the bottom and along the lateral boundaries of the basin provides the energy for vertical diffusion more directly to the interior of the inlet (Stigebrandt 1976; Stacey 1984). Barotropic flux model The rate at which energy is withdrawn from the barotropic tide may be determined by examining changes in the characteristics of the tidal wave as it interacts with the sill. The method used to calculate the tidal energy flux was first introduced by Freeland and Farmer (1980), then later modified by Stacey (1984) for a more generalized inlet. The net flux of energy across the sill is given by (Garrett 1975) P gu da, (1) A where is the density, g is the acceleration due to gravity, u is the barotropic velocity, is the surface displacement, and A is the cross-sectional area at the point of measurement. The overbar represents an average over a tidal cycle. When power is extracted from the barotropic tide at a sill, u and fall out of quadrature by some phase angle. Substituting 0 sin(t) and u u 0 cos(t ) into (1), and noting that u ½u 0 0 sin, 1 P gu00 sin da. () A Freeland and Farmer (1980) evaluated this integral and assumed that was small, allowing the substitution u 0 0 S/A (by continuity), where S is the surface area up inlet of the measurement point. Equation () then reduces to 1 P g0s sin(). (3) Therefore, by placing a tide gauge and a string of current meters outside of the inlet, P may be computed from measured values of 0 and. However, computing the phase lag is very difficult, even with good vertical current meter coverage, because barotropic velocities are difficult to estimate accurately where baroclinic currents are dominant. Farmer and Freeland (1980) demonstrated that by using two tide gauge stations (one seaward and one landward of the sill) it is possible to eliminate the dependence on the velocity measurements and use only the tidal elevation phase differences to compute P. Using tide gauges at two stations (sections 1 and in Fig. 3a), the tidal elevations and barotropic velocities may be written: 1 ˆ 1 sin(t) (4a) u û cos(t ) 1 1 (4b) u ˆ sin(t ) (4c) 1 u û cos(t ). (4d)

4 3368 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 bodies seaward of their sills, the difference in tidal range between Sechelt Inlet and Jervis Inlet can be over m due to the severe choking of the tide at Skookumchuck Narrows. We rederive (3) without the assumption that ˆ 1 ˆ. From Fig. 3a, the conservation of fluid volume between sections 1 and may be written d d ua 1 1 ua (1S 1) (S). (6) dt dt Using (4a d) and (6), expanding the trigonometric terms and equating coefficients gives FIG. 3. (a) Idealized inlet schematic showing quantities used in the energy flux derivation. (b) Schematic diagram showing the relationship of two ideal tidal curves, one outside (larger) and one inside the sill. The ratio of the amplitudes ( ˆ 1 and ˆ ) and the phase lag are related by ˆ / ˆ 1 cos. Freeland and Farmer (1980) assumed that the tidal elevation and velocity at section were in quadrature (i.e., 1 ). Stacey (1984) generalized the Freeland and Farmer model so that the quadrature assumption was not required. In both cases, however, the difference between the tidal amplitude at section 1 and section was assumed to be negligible. It can be shown (de Young 1986) that using (4a d), and letting 1, (3) becomes 1 P gˆ 1S sin(). (5) The total available power in the oscillating tide is shown by de Young (1986) to be P a ½gˆ 1 S, meaning that the percentage of the energy flux represented by (5) is P/P a sin(). While tidal amplitudes in most British Columbia fjords do not differ perceptibly from connecting water 1 [(ˆ 1S 1) (ˆ S ) ˆ (S S) A1 û ˆ ˆ SS cos ˆ ˆ S (S S) cos / 1 ˆ S(S S) cos( )] (7) ˆ S sin1 ûa sin 1 tan. (8) ˆ 1S1 ˆ S cos1 ûa cos Equations (7) and (8) are both expressed in terms of measurable quantities from tide gauges except for the terms involving in (8). By making the assumption that 1 (which is true where the up-inlet tidal dissipation is small) and noting that at section, û ˆ (S 1 S )/A, gˆ 1 P (ˆ 1S 1) (ˆ S ) ˆ 1ˆ SS 1 cos [ ] 1 1 ˆ S sin 1 sin tan. (9) ˆ S ˆ S cos Equation (9) represents the general expression for the energy flux of the barotropic tide in terms of tidal elevation amplitudes and phases. For comparison purposes let us consider the case where ˆ S k ˆ 1 S 1 (a fairly general assumption, since S k S 1 in most inlets). The energy flux expression reduces to 1 P gˆ 1ˆ S sin(), (10) which is similar to (5) with ˆ 1ˆ in place of ˆ 1. The tidal amplitudes ˆ 1 and ˆ and phase shift are related by ˆ / ˆ cos() (illustrated in Fig. 3b), giving 1 1 sin() P gˆ 1S. (11) For small angles, (11) is equivalent to (5); however, the ratio of energy flux to available power is now sin()/, which means that the maximum rate of removal of energy from the barotropic tide occurs when the phase shift across the sill is /4 and that the maximum flux is equal to one-half of the available power. This situation is analogous to an electronic circuit that has a maximum

5 DECEMBER 001 TINIS AND POND 3369 FIG. 4. Sample time series of the tidal signal measured at Boom Islet (solid) and Rapid Islet (dashed) located on either side of Skookumchuck Narrows. power transfer (50% of the total) to a load whose impedance is matched to that of the generator. Equation (11) can be derived in a more straightforward fashion, as shown by Stigebrandt (1999), by bypassing many of the generalizations made in Eqs. (4) and (6). Stigebrandt (1999) also demonstrated that (11) is valid for a linear drag model applied to tidal exchange in landlocked basins where dissipation depends on frictional and baroclinic wave drag; however, Eq. (11) slightly overestimates the numerically computed maximum dissipation of a quadratic drag model where dissipation depends on both frictional and barotropic form drag. In light of the small differences in dissipation for a given phase lag computed by both the linear and nonlinear models outlined above, Stigebrandt (1999) asserts that (11) may be used in conjunction with phase information to obtain an approximate tidal dissipation estimate for all types of tidal exchange channels, regardless of dissipation process. 3. Energy dissipation a. Friction model Direct measurements of the currents in Skookumchuck Narrows were made in 1983 at the point of strongest flow in the channel by the Tides and Currents section of the Institute of Ocean Sciences (IOS), Sidney, British Columbia, Canada. Together with elevation measurements from the Boom Islet and Rapid Islet stations (Fig. 4), the data were compared to a simple model based on frictionally dominated hydraulic flow. The model results were made available courtesy of M. Woodward (IOS). For the purposes of this study, the current speeds were scaled to reflect cross-channel averages rather than currents at the point of strongest flow. The model balances the pressure gradient across the sill by the frictional bed stress: 1 P g 1 [ u Cu u ]. d (1) x L H The friction is expressed using linear and quadratic terms (u and C d u u ), with a constant term, which arises from the uneven leveling of the two gauges. The pressure head,, and flow speed, u, are fitted in a least squares sense to (1) to determine the values of,, and C d (see Fig. 5); L is the length of the channel. The term affects the fit primarily near the x axis: a strong linear term reduces the slope of the fitted curve near the x intercept. Although the dominant balance for most of the data is between the pressure gradient and friction, acceleration effects become important during periods of weak flow and small pressure head. The acceleration/deceleration of the flow is the source of the hysteresis in the raw data near the origin in Fig. 5a. To correct the data for the acceleration effects, estimates of the velocity before and after an actual velocity observation are made using the pressure data from the tide gauges and Bernoulli s equation (u g ). An estimate of u/t in (13) may then be made, which is less noisy than using direct current measurements. The pressure gradients responsible for the acceleration are subsequently calculated by 1 u. (13) x g t

6 3370 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 FIG. 5. Directly measured current velocity at the fastest point in the tidal stream (as measured by M. Woodward, IOS) vs hydraulic head across Skookumchuck Narrows. Data are shown (a) without and (b) with acceleration terms included to correct for hysteresis in the curve near slack water. The pressure corrections are applied only where the hysteresis is pronounced; for the data in Fig. 5, the correction is applied where 150 u 300 cm s 1 (Fig. 5b). A fit of (1) to the corrected data is made to find the values of the parameters,, and C d. The IOS model is designed to predict the maximum currents expected for a given tidally active channel since that is the property of greatest interest to mariners. In order to analyze the bulk friction effects on the flow in Skookumchuck Narrows, the IOS data were scaled based on the average tidal flow expected over the sill calculated by conservation of volume, u max max S/ A, where u max is the maximum channel-averaged velocity, A is the cross-sectional area of the channel, max is the tidal amplitude inside the inlet, and S is the surface area of the inlet landward of the sill. Using the tide gauge data to obtain max and for several tidal events, the largest value of u max was found to be 185 cm s 1. Because the maximum measured flood current is 770 cm s 1, the currents were scaled by a factor of 0.4. The best fit of (1) to the scaled flood tide data yielded C d 0.08, 0.04 m s 1, and 0.03 m s The values of the drag coefficient, C d, found for the flood and ebb tide may be different, depending on the geometry of the sill. For example, in Skookumchuck Narrows the water flows over a reef south of Boom Islet during flood tide and is convergent near the middle of the stream; the ebb flow, however, is strongest just downstream of the tip of Boom Islet (M. Woodward 1993, personal communication). The maximum channel flows for the flood and ebb tides need to be scaled separately, in order to examine the bulk friction effects. For simplicity, only the flood events are used in the dissipation analysis. In the case where there is dissipation due to boundary friction, the energy loss can be explicitly included in the shallow-water equations by including a stress term: u D(x) g 0 (14) t x h(x) hu 0, (15) t x where D(x) has the dimensions of u (Freeland and Farmer 1980). An energy equation can be obtained by adding hu (14) and g (15): E (ghu) ud 0. (16) t x Integrating over the surface area of the inlet, and averaging over a tidal cycle (overbar), x gwhu w(x)ud dx 0, (17) 0 where w(x) is the width, H h(x ), and W w(x ) [ E/t 0, following Garrett (1975)]. The second term is the dissipation due to friction; note that if u and are in quadrature ( u 0), there is no dissipation. Be- cause friction is only important in the sill region, the limits of integration can be bounded. The dissipation function, D(x), was parameterized by Freeland and Farmer (1980) as D C d u u ; using the form of the friction term given by (1), the expression for the dissipation rate due to friction (assuming a constant width, W, and length, L, in the narrows) becomes 3 Ploss WL( u C d u ). (18) In Fig. 6, the dissipation rates given by (18) are compared to the theoretical estimate given by (11) for each flood event found in six months of tide gauge data from Skookumchuck Narrows. For every flood event, the phase shift in peak surface elevation, tidal period, and tidal amplitudes inside and outside the sill were measured. To a very good approximation, the tidal elevations vary sinusoidally over the flood period, justifying the use of (11).

7 DECEMBER 001 TINIS AND POND 3371 FIG. 6. Estimated frictional energy dissipation for several tidal cycles vs the theoretical tidal energy flux across the sill. Solid line represents the least squares linear regression (slope ). A linear regression of the data in Fig. 6 shows that the dissipation rate due to friction is times that of the estimated barotropic tidal energy flux. Because of the uncertainties in picking values such as average depth and width of the constriction, and the uncertainty in the values of and C d from the simple friction model, the error in the frictional dissipation rate is somewhat higher than the regression would suggest. The importance of the result lies in the fact that the frictional dissipation rate is comparable in magnitude to the tidal energy flux. b. Tidal jet FIG. 7. Three-layer density profile up inlet of the sill. The strong stratification in the upper layer is broken down by the turbulent jet generated at the sill, and a middle layer is formed (seen here between approximately 5 and 15 m depth). Not all of the turbulent energy is dissipated directly over the sill. Some of the energy will be advected into the inlet during a flood tide in the form of a turbulent jet. This jet will be negatively buoyant with respect to surface water and will tend to flow under the surface layer. The advection of mixed water from the sill region and mixing by the jet are responsible for the middle layer in the characteristic three-layer density profiles of inlets with strong mixing at the sill (Fig. 7). Lazier (1963) was the first to propose that the tidal jet in Sechelt Inlet was responsible for the formation of the middle density layer. Direct measurements of the tidal jet were made at the current meter mooring just inside the sill (Fig. 8). The currents are quite strong at 0 m during flood tide, and a strong return flow is observed below the inflow. Stigebrandt and Aure (1989) derived the following

8 337 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 FIG. 8. Alongchannel velocity profile of the tidal jet measured by a cyclesonde profiling current meter approximately after maximum flood in Skookumchuck Narrows (negative velocity indicates flow toward the head of the inlet). The tidal jet is visible at the top of the profile with a strong return flow below. expression for estimating the energy flux of a turbulent tidal jet: Ej * A i f/a s, (19) 4 where * is the average density of the inflow, is the angular frequency of the tide, i is the tidal amplitude in the inlet, A f is the cross-sectional area of the sill, and A s is the surface area landward of the sill. Here, * 1000 kg m 3, A s 10 8 m, and A f 10 4 m. The frequency,, and amplitude, i, are determined from the tide gauge data, and, as with the frictional dissipation across the sill, the kinetic energy flux of the turbulent tidal jet was estimated for each inflow period. In Fig. FIG. 9. Estimated tidal jet energy dissipation for several tidal cycles vs the theoretical tidal energy flux across the sill. Solid line represents the least squares linear regression (slope ).

9 DECEMBER 001 TINIS AND POND , E j is plotted against the theoretical flux estimated by (11). Over the range of measured tidal energy fluxes, the energy flux of the tidal jet accounts for about 5% of the total. 4. Conclusions The barotropic tidal energy flux in Sechelt Inlet is very large with values approaching 100 MW during extreme flood tides. Of the tidal energy that is dissipated, most is dissipated by friction through Skookumchuck Narrows, while a comparatively small amount (5%) is transferred to the kinetic energy flux of the tidal jet near the entrance. An even smaller amount (0.5%) is transferred to the progressive internal tide (Tinis 1995). Energy dissipated by friction over the sill goes into mixing the exchange water through the narrows and is eventually dissipated as heat and as an increase in potential energy through the mixing of stratified water. The intense mixing of the exchange waters is clearly evident by the lack of surface layer stratification near the sill. Energy dissipated by the tidal jet, on the other hand, penetrates deeper to form an intermediate mixed layer. This mixed layer has also been observed in density profiles of other British Columbia inlets where there is strong mixing at the sill. For both the frictional dissipation at the sill and the dissipation by the tidal jet, the effects of the dissipated energy remain local to the mixing region. Evidence for the transfer of energy in the internal tide to the turbulent diffusive processes in the deep water of the basin (which subsequently works to raise the potential energy of the water column) exists and is discussed by Tinis (1995). The energy partition for Sechelt Inlet outlined here, which is dominated by frictional dissipation at the sill, is in contrast to the partition in deep-silled fjords where the majority of the tidal energy flux is dissipated by the generation of progressive internal waves near the sill (Stacey and Zedel 1986; Freeland and Farmer 1980). Acknowledgments. The authors would like to thank M. Woodward at the Institute of Ocean Sciences for the use of both the sill friction model and the direct current measurements in Skookumchuck Narrows. This work was funded by an NSERC strategic grant and support for S. W. Tinis was provided through the NSERC postgraduate scholarship program. REFERENCES de Young, B., 1986: The circulation and internal tide of Indian Arm, B.C. Ph.D. thesis, University of British Columbia, Vancouver, British Columbia, Canada, 175 pp., and S. Pond, 1987: The internal tide and resonance in Indian Arm. J. Geophys. Res., 9, Freeland, H. J., and D. M. Farmer, 1980: Circulation and energetics of a deep, strongly stratified inlet. Can. J. Fish. Aqua. Sci., 37, Garrett, C. J. R., 1975: Tides in gulfs. Deep-Sea Res.,, Lazier, J. R. N., 1963: Some aspects of the oceanographic structure in the Jervis Inlet system. M.Sc. thesis, Institute of Oceanography, University of British Columbia, Vancouver, British Columbia, Canada, 54 pp. Stacey, M. W., 1984: The interaction of tides with the sill of a tidally energetic inlet. J. Phys. Oceanogr., 14, , and L. J. Zedel, 1986: The time-dependent hydraulic flow and dissipation over the sill of Observatory Inlet. J. Phys. Oceanogr., 16, Stigebrandt, A., 1976: Vertical diffusion driven by internal waves in a sill fjord. J. Phys. Oceanogr., 6, , 1980: Some aspects of tidal interaction with fjord constrictions. Estuarine Coastal Mar. Sci., 11, , 1999: Resistance to barotropic tidal flow in straits by baroclinic wave drag. J. Phys. Oceanogr., 9, , and J. Aure, 1989: Vertical mixing in basin waters of fjords. J. Phys. Oceanogr., 19, Tinis, S. W., 1995: The circulation and energetics of the Sechelt Inlet System, British Columbia. Ph.D. thesis, Dept. of Oceanography, University of British Columbia, Vancouver, British Columbia, Canada, 173 pp.