MAY 2005 NOTES AND CORRESPONDENCE 897

NOTES AND CORRESPONDENCE

On Deep-Water Renewals in Indian Arm, : Sensitivity to the Production of Turbulent Kinetic Energy Caused by Horizontal Variations in the Flow Field

MICHAEL W. STACEY Department of Physics, Royal Military College of , Kingston, Ontario, Canada

S. POND Department of Earth and Ocean Sciences, University of British Columbia, , British Columbia, Canada

(Manuscript received 24 September 2003, in final form 22 September 2004)

ABSTRACT

A two-dimensional (i.e., laterally averaged) numerical model of the circulation in and Indian Arm near British Columbia, Canada, is used to examine the sensitivity of deep-water renewal events in Indian Arm to the turbulent mixing in the lee of the narrow sills in Burrard Inlet. Horizontal variations in the flow field can have an important influence on the production of turbulent kinetic energy near the sills and therefore also on the renewal events in Indian Arm. An ad hoc modification to the expression for the production of turbulent kinetic energy, required to obtain an acceptable simulation downstream of Second Narrows in Burrard Inlet, also results in a reasonable simulation of the observed circulation in Indian Arm. The modified laterally averaged model can reproduce the main features of the circulation away from the narrow sills. However, it seems that a three-dimensional model will be required if the circulation is to be simulated with greater accuracy and without the ad hoc modification, which has a free parameter.

1. Introduction tidal mixing was less intense then than during spring tides and therefore the bottom waters in Burrard Inlet Together, Indian Arm and Burrard Inlet near British were denser. Columbia, Canada, (Fig. 1) are a for which Indian Simulation of this deep-water exchange provided the Arm is the basin and Burrard Inlet is the long, shallow motivation for our development of laterally averaged sill. Both Burrard Inlet and Indian Arm are about 20 fjord models. We were unsuccessful for a long time km long; the maximum depth is about 220 m for Indian until we had sufficient computing power to allow ad- Arm and about 65 m for Burrard Inlet in two limited equate horizontal and vertical resolution. Meanwhile, regions. At First and Second Narrows, Burrard Inlet is we also attempted simulations of the circulation in 15 and 19 m below datum, respectively. Knight Inlet, and Stacey et al. (1995) successfully simu- Water entering Indian Arm from outside the fjord lated the observations of Baker and Pond (1995). The system is subjected to considerable mixing during its simulations included periods of low and high runoff; passage through Burrard Inlet, where the tidal currents forcing due to tides, winds, runoff, and deep-water re- Ϫ1 are up to 2 m s or so in First and Second Narrows. newal were all important and significantly coupled. Observations of deep-water renewal in Indian Arm (de Stacey et al. (2002) successfully simulated the obser- Young and Pond 1988) showed that the renewal events vations of de Young and Pond (1988) with a two- were strongly modulated by the spring–neap cycle. The dimensional numerical model. A level-2 turbulence clo- renewals were enhanced during neap tides because sure scheme was used (Mellor and Yamada 1982) to parameterize the vertical mixing coefficients in the model. That is, all the turbulent motions were assumed to be locally generated and dissipated. The parameter- Corresponding author address: Dr. Michael W. Stacey, Depart- ment of Physics, Royal Military College of Canada, P.O. Box ization of the mixing coefficients depends in part on the 17 000, Station Forces, Kingston, ON K7K 7B4, Canada. expression for the shear production of turbulent kinetic E-mail: [email protected] energy Ps, where

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FIG. 1. Plan view of Burrard Inlet and Indian Arm (modified from deYoung and Pond 1988, and similar to Fig. 1 of Stacey et al. 2002). The solid circles (●) show the locations of CTD stations. The cross (ϫ) shows the location of the cyclesonde station. The dashed line shows the location of the Indian Arm sill.

ѨU ѨU lation in Burrard Inlet was examined by Isachsen and P ϭϪuw Ϫ u2 , ͑1͒ S Ѩz Ѩx Pond (2000), who from March until June 1995 main- tained three moorings; one outside First Narrows and uw and u2 are the ensemble averages of the vertical one each in the deepest holes just up-inlet of First and w(x, z, t) and along-channel u(x, z, t) velocity fluctua- Second Narrows. They found that the spring–neap tions, and U(x, z, t) is the along-channel velocity minus cycle had a very significant influence on the circulation. the fluctuation u(x, z, t). Normally when parameteriz- During neap tides there was significantly less temporal ing the mixing coefficients, only the first term on the variability in the density in the holes and the currents right-hand side of (1) is used because the vertical varia- there were significantly less intense. During spring tides tions in U(x, z, t) are typically much larger than those in there were current pulses, about 1 m sϪ1 in magnitude, the horizontal. However, Stacey et al. (2002) found that into the holes and the average density was lower than inclusion of the second term on the right-hand side no- during neap tides. ticeably improved their simulation of the renewal Stacey and Pond (2003), using the same laterally av- events. The subtidal near-bottom increase in density eraged numerical model as Stacey et al. (2002), were with time in Indian Arm, caused by the renewal events, able to successfully simulate much of what Isachsen and was better simulated [the near-bottom density in- Pond observed up-inlet of Second Narrows but only as creased less over time when the second term in (1) was long as the second term in (1) was included. In fact, the included], and so was the near-surface outflow in In- simulation was noticeably improved when the second dian Arm (the second term made the return flow stron- term was augmented by multiplying it by a factor of 5. ger). These improvements occurred primarily because A plausible justification for the augmentation is that the second term in (1) is significant near First and Sec- the laterally averaged model may underestimate the ond Narrows. Even when the second term was in- maximum currents over First and Second Narrows, and cluded, however, the simulated increase in the near- therefore when unmodified may underestimate the rate bottom density was still somewhat too large, and the of mixing there. Tinis and Pond (2001) found that the simulated near-surface outflow was still somewhat too maximum current velocities over the sill of small. were about 4 times the cross-channel average. Since the The influence of the spring–neap cycle on the circu- second term in (1) depends on a horizontal velocity

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FIG. 3. As for Fig. 2, but for the simulation in which the second term in the rhs of (1) has been multiplied by a factor of 5. FIG. 2. Velocity at C3. The thin, solid lines are for the simulation in which the second term in the rhs of (1) is not augmented, and the thick, solid lines are for the observations. Yearday 1 is 1 Jan 1984 (same as Fig. 5 of Stacey et al. 2002). Figure 2 shows the observed and simulated currents at C3 (see Fig. 1) when the second term of (1) is in- cluded (same as Fig. 5 of Stacey et al. 2002). Figure 3 1/3 ≅ scale cubed, a velocity scaling factor of 5 1.7 would shows the observed and simulated currents at C3 when account for the factor of 5. That is, the factor of 5 is the second term in (1) is multiplied by a factor of 5. plausible, given the result from Sechelt Inlet. Stacey Comparing Figs. 2 and 3 , one sees that augmenting the and Pond showed that a factor of 10 is too high. A second term changes the simulated renewal. Before factor of 5 gives reasonable results, but a somewhat about day 400, the simulation of the currents at 175 m smaller factor would also give reasonable results. are closer to the observations when the second term is Since augmenting the second term in (1) resulted in augmented (Fig. 3). In Fig. 2, the first simulated pulse at an improved simulation of the circulation near Second 175 m at about day 350 is not observed. Also, the first Narrows, it seemed worthwhile to investigate if a simi- observed “small” pulse at about day 370 is closer in lar improvement can be obtained in the simulation of magnitude to the simulated pulse in Fig. 3. The three the renewal events observed by de Young and Pond big pulses between about days 380 and 410 are very well (1988). Also, it is perhaps even more important to simulated when the second term is not augmented (Fig. document if augmenting the second term in (1) de- 2), while the simulation with the augmented term (Fig. grades the simulation of the renewal events, for if it 3) overestimates the strength of the currents at 175 m does significantly degrade the simulation it then be- during the time intervals between the pulses. comes harder to justify augmenting this term in the first After about day 450, when there are observations place to better simulate the circulation near Second again, the observed pulses disappear completely. The Narrows. simulated pulses also disappear when the second term in (1) is augmented. They do not disappear when the 2. Results term is not augmented. At 135-m depth, the simulation may be somewhat The only modification made to the model of Stacey better when the second term in (1) is not augmented, et al. (2002) for the new simulation is that the second and at 105 m neither simulation can reproduce the ob- term in (1) has been multiplied by a factor of 5 (or 2; see served down-inlet pulses that occur between about days Fig. 4). All of the observed and simulated time series 390 and 410. Stacey et al. (2002) noted that, if one have been averaged using a 24-h running mean in order assumes there is no cross-channel variability, the veloc- to remove the tidal signal at diurnal and higher fre- ity observations during the bottom inflows show more quencies. water actually flows out of the basin than flows in. The

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FIG. 4. Simulated and observed sigma-t at C3 (175 m). The solid dots indicate sigma-t values obtained from CTD measurements. The thin solid line is for the simulation in which the second term in the rhs of (1) is not augmented. The dashed (dotted) line is for the simulation in which the second term in the rhs of (1) is multiplied by a factor of 2 (5). outward pulses observed at around 105 m cannot be and simulated density at C3 (both when the term is representative of the laterally averaged flow. augmented by factors of 2 and 5 and when it is not). At 65-m depth, the simulation is better when the Stacey et al. (2002) found that the simulated density term is augmented, producing overall a larger simu- increased more slowly overall with time when the sec- lated outflow. At 35 and 15 m, neither simulation gives ond term in (1) was included, and was therefore in an obviously better result. closer agreement with the observations even though the Simulations have also been run using the factors of 2, simulated density still increased more than the ob- 3, and 4. The simulated velocities show improvements served density after about day 420. When the second at some depths and poorer results at others. term is augmented, the simulated density increases Figure 4 shows the near-bottom subtidal observed even more slowly, and in fact increases more slowly

Unauthenticated | Downloaded 09/25/21 03:14 AM UTC MAY 2005 NOTES AND CORRESPONDENCE 901 than the observed density when the term is augmented (2000), deep-water renewal was not occurring and an by a factor of 5. augmentation factor of 5 gave a reasonable simulation. As noted by Stacey et al. (2002), the initial rises of During the observations of deYoung and Pond (1988), the observed density from the cyclesondes, the heavy a factor of 1 works well in the early stages, about 2 in solid lines in Fig. 4, for the last two records (after day the middle stage, and perhaps up to 5 when deep-water 450) may be an artifact of poor flushing of the conduc- replacement has stopped; that is, a variable factor in- tivity cells. The CTD, water sample data and cy- creasing with time would allow for a rather good match. clesonde velocity data (Fig. 2 or 3) suggest that there was little inflow at depth after day 450. If these initial 4. Conclusions rises were discarded and the remainder readjusted to the CTD casts there would be a rise of about 0.03 for For any fjord system that has fairly shallow narrows the fourth record and a slow decrease for the last one. with strong tidal currents, turbulence production terms in addition to the first term on the right-hand side of (1) are likely to be significant. 3. Discussion To get the details in regions such as Second Narrows and to get rid of the rather arbitrary augmentation fac- Stacey et al. (2002) showed that including the second tor, three-dimensional simulations are desirable. How- term in (1) improved the simulations of the deep-water ever, two-dimensional simulations are much easier and exchange in Indian Arm in 1984/85. Augmenting this can give useful results in inlets with less complicated term somewhat can improve the simulation of the deep- geometry, such as Knight Inlet. Even in systems such as water density increase. The velocity simulations are not that of Burrard Inlet and Indian Arm, the laterally av- changed a great deal because they are driven by the eraged model can give reasonable and useful results. density change from one renewal event to the next. Regardless of the augmentation factor, the phase (tim- Acknowledgments. This work was supported by ing) of the renewal events and their amplitudes remain DND Academic Research Program (ARP) grants to M. consistent with the observations. Stacey and by Natural Sciences and Engineering Coun- Stacey and Pond (2003) showed that an augmenta- cil (NSERC) grants to S. Pond. We thank the officers tion factor of about 5 gave reasonable simulations of and crew of the research vessels of the Institute of the results of Isachsen and Pond (2000) for the deep Ocean Sciences for their assistance with data collection. hole inside Second Narrows. However, based on the comparison in Fig. 4, an augmentation of 5 is too large REFERENCES for the Indian Arm simulation. Baker, P. D., and S. Pond, 1995: Low-frequency residual circula- In a laterally averaged model the second term of (1) tion in Knight Inlet, British Columbia. J. Phys. Oceanogr., 25, is the only way to represent the horizontal variations in 747–763. the flow field, which are in fact present in the real three- de Young, B., and S. Pond, 1988: The deepwater exchange cycle in dimensional system. Visual observations downstream Indian Arm, British Columbia. Estuarine Coastal Shelf Sci., of Second Narrows show back eddies on both flood and 26, 285–308. Isachsen, P. E., and S. Pond, 2000: The influence of the spring- ebb tides. During spring tides on large floods there is neap tidal cycle on currents and density in Burrard Inlet, surface evidence of turbulence (vortices and eddies of British Columbia, Canada. Estuarine Coastal Shelf Sci., 51, scale a few centimeters to many tens of meters) a few 317–330. hundred meters downstream of Second Narrows. Mellor, G., and T. Yamada, 1982: Development of a turbulent closure model for geophysical fluid problems. Rev. Geophys. ADCP data (Isachsen and Pond 2000) show strong sub- Space Phys., 20, 851–875. surface turbulence. Stacey, M. W., and S. Pond, 2003: Dependence of currents and In three dimensions there are seven additional pro- density on the spring–neap cycle and the diurnal inequality in duction terms on the right-hand side of (1). The second Burrard Inlet, British Columbia: Simulations and observa- term in (1) is thus a rough ad hoc attempt to represent tions. J. Phys. Oceanogr., 33, 2366–2374. ——, ——, and Z. P. Nowak, 1995: A numerical model of the eight terms. The proportionality between the second circulation in Knight Inlet, British Columbia, Canada. J. term in (1) and these eight terms is likely to vary with Phys. Oceanogr., 25, 1037–1062. geometry, flow strength, and stratification. Thus the ——, R. Pieters, and S. Pond, 2002: The simulation of deep water values may differ for the two narrows and for flood and exchange in a fjord: Indian Arm, British Columbia, Canada. J. Phys. Oceanogr., 32, 2753–2765. ebb tides, and indeed could vary over the course of an Tinis, S. W., and S. Pond, 2001: Tidal energy dissipation at the sill individual flood or ebb. of Sechelt Inlet, British Columbia. J. Phys. Oceanogr., 31, During the observations by Isachsen and Pond 3365–3373.

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