Tidal Sensitivity

This document outlines the tidal sensitivity to various parameters such as bottom friction and lateral viscosity. The following details are presented: set up of simulations, analysis techniques, and tidal response. The \(M_2\) and \(K_1\) tidal constituents are the main focus.

The modelled tidal response is compared to observations from Foreman et al. (1995), Foreman et al. (2004), and Foreman et al. (2012).

Procedure

Several 5-day long simulations forced with \(M_2\) and \(K_1\) tidal constituents have been completed. A sinusoidal curve fitting technique is applied to the time series of the modelled sea surface height to estimate the \(M_2\) and \(K_1\) amplitude and phase. The fitting curve is defined as follows:

\[f(t) = A_{M_2}\cos(\omega_{M_2}t - \phi_{M_2}) + A_{K_1}\cos(\omega_{K_1}t - \phi_{K_1})\]

where \(A_{M_2}\), \(\phi_{M_2}\), \(A_{K_1}\), and \(\phi_{K_1}\) are the \(M_2\) and \(K_1\) amplitudes and phases determined by the fitting procedure. The \(M_2\) and \(K_1\) frequencies \(\omega_{M_2}\), \(\omega_{K_1}\) are known.

Results

The modelled amplitude and phase will be compared to observations at several points within the domain: Port Renfrew, Point Atkinson, and Yorke Island. This way we can evaluate model performance over different regions. Other locations are available but will not be summarized here.

The one area that the model performs badly is the Gulf/San Juan Islands. The phase change through the Islands to too large. Probably related, around Victoria, there is an M2 amphidrome; the amphidrome is too deep in the model. So one critical area is the difference in phase across these Islands, here measured by considering the full change in phase between Port Renfrew and Port Atkinson. In the observations, this value is 150.1 degrees.

We have calculated the phase difference \(\Delta \phi =\phi_{mod}-\phi_{obs}\) and the amplitude ratio \(R = \frac{A_{mod}}{A_{obs}}\). The results are summarized in the tables below.

\(M_2\) Results

Simulation	Description					Diff. in Phase betw. PR & PA	Port Renfrew		Point Atkinson		Yorke Island
		Bottom Friction	Viscosity	Mean Error (m/deg)	RMS Error (m/deg)	(deg)	\(R\)	\(\Delta \phi\)	\(R\)	\(\Delta \phi\)	\(R\)	\(\Delta \phi\)
tide_flux_M2K1	original tides at west, flux corrected at north	\(0.005\)	20	0.100/9.57	0.13/10.5	158.6	1.01	1.74	0.911	10.4	1.11	-3.91
tide_flux_west	\(M_2\) and \(K_1\) flux increased by 25% at west	\(0.005\)	20	0.061/10.6	0.11/12.2	157.3	1.16	6.06	1.01	13.4	1.15	-4.03
tide_bottom	decreased bottom friction	\(0.003\)	20	0.090/9.31	0.13/11.1	161.0	0.991	-0.944	0.943	9.94	1.13	-5.44
tide_nu15	decreased lateral viscosity	\(0.005\)	15	0.098/9.84	0.13/10.3	158.5	1.00	1.54	0.914	10.2	1.16	-4.08
tide_bottom1e-3	decreased bottom friction	\(0.001\)	20	0.083/11.9	0.14/17.3	164.4	0.965	-5.14	0.993	9.13	1.09	-8.24
tide_K1phase2	\(K_1\) phase decreased 5 deg	\(0.005\)	20	0.100/9.59	0.13/10.5	158.7	1.01	1.70	0.911	10.5	1.16	-3.95
tide_K1amp	above plus \(K_1\) amplitude decreased 15% and b.f. \(1\times 10^{-3}\)	\(0.001\)	20	0.082/12.0	0.14/17.2	164.6	0.965	-5.55	0.998	9.13	1.08	-8.33
tide_M2phase	above plus \(M_2\) phase decreased 9 deg	\(0.001\)	20	0.071/6.9	0.13/13.1	166.0	0.969	-15.2	1.01	0.662	1.08	-6.66
tide_slip	above plus closer to free slip BCs: rn_shlat is 0.1 (from 0.5)	\(0.001\)	20	0.094/8.0	0.13/11.3	170.1	0.963	-20.0	1.07	-0.04	1.03	-13.3
tide_slipH	above plus closer to noslip BCs: rn_shlat is 1.0 (from 0.5)	\(0.001\)	20			161.2	0.998	-8.98	0.929	2.14

Summary

• Decreasing bottom friction increases \(M_2\) amplitude at Point Atkinson but decreases it at Port Renfrew. There is a good match at Point Atkinson when bottom friction is \(1\times10^{-3}\).

• Decreasing bottom friction decreases the \(M_2\) phase difference at all three locations, with the largest response at Port Renfrew and Yorke Island.

• Decreasing bottom friction increases the phase difference between Point Atkinson and Port Renfrew.

• Decreasing the viscosity has little effect at all three stations but does slightly decrease the phase difference between Point Atkinson and Port Renfrew.

• Increasing the flux decreased the phase difference between Point Atkinson and Port Renfrew but increased the amplitude drop.

• Decreasing the \(K_1\) phase has little effect on the \(M_2\) amplitude and phase.

• Point Atkinson \(M_2\) phases are very consistent over all of the iterations, except the last where the \(M_2\) phase has changed in forcing. The phases at Port Renfrew and Yorke Island are more sensitive to changes in bottom friction and viscosity.

• Increasing the slip, increases the phase difference between Point Atkinson and Port Renfrew. Decreasing the slip, decreases the phase difference but increases the amplitude drop.

\(K_1\) Results

Simulation	Description					Port Renfrew		Point Atkinson		Yorke Island
		Bottom Friction	Viscosity	Mean Error (m/deg)	RMS Error (m/deg)	\(R\)	\(\Delta \phi\)	\(R\)	\(\Delta \phi\)	\(R\)	\(\Delta \phi\)
tide_flux_M2K1	original tides at west, flux corrected at north	\(0.005\)	20	0.060/7.14	0.066/7.5	1.07	3.51	1.09	8.46	1.14	-5.78
tide_flux_west	\(M_2\) and \(K_1\) flux increased 25% at west	\(0.005\)	20	0.126/5.75	0.131/6.3	1.19	0.151	1.18	7.11	1.16	-5.09
tide_bottom	decreased bottom friction	\(0.003\)	20	0.079/6.7	0.085/7.2	1.08	6.31	1.11	7.10	1.14	-4.42
tide_nu15	decreased lateral viscosity	\(0.005\)	15	0.059/7.0	0.066/7.4	1.07	3.48	1.08	8.24	1.14	-5.69
tide_bottom1e-3	deceased bottom friction	\(0.001\)	20	0.110/6.4	0.119/7.7	1.10	10.2	1.15	5.27	1.14	-2.22
tide_K1phase2	\(K_1\) phase decreased 5 deg	\(0.005\)	20	0.063/3.5	0.069/4.2	1.07	-1.36	1.09	3.52	1.16	-5.56
tide_K1amp	above plus \(K_1\) amplitude decreased 15% and b.f. \(1\times 10^{-3}\)	\(0.001\)	20	0.030/2.7	0.044/4.5	0.934	5.81	0.984	0.541	1.13	-5.78
tide_M2phase	above plus \(M_2\) phase decreased 9 deg	\(0.005\)	20	0.026/2.7	0.045/4.5	0.955	5.68	1.01	0.594	1.14	-5.24
tide_slip	above plus closer to free slip BCs: rn_shlat is 0.1 (from 0.5)	\(0.001\)	20	0.034/3.1	0.047/3.9	0.971	8.51	1.03	-1.42	1.14	-2.13

Summary

• Decreasing the bottom friction increases the \(K_1\) amplitude at Port Renfrew and Point Atkinson. There is no effect at Yorke Island.

• Decreasing the bottom friction increases the \(K_1\) phase difference at Port Renfrew and Yorke Island, with the largest response at Port Renfrew. The Point Atkinson phase difference has decreased.

• Decreasing the viscosity has little effect on the \(K_1\) amplitude and phase at these three locations.

• Increasing the flux greatly increased the amplitude errors.

Implications

From the M2 results, we decided that the bottom friction should not be reduced and we fixed it at 0.005, that the smaller viscosity was better so we fixed that at 15 \(m^2 s^{-1}\). We increased the Flux by 25% but left the slip at 0.5.

Both amplitude drop and phase increase between Port Renfrew and Point Atkinson are important, but the large phase change causes the largest total errors. So choices were made to reduce the phase increase at the expense of increasing the amplitude drop.

References

• Foreman, M.G.G., R.A. Walters, R.F. Henry, C.P. Keller and A.G. Dolling, 1995. A tidal model for eastern Juan de Fuca Strait and the southern Strait of Georgia, Journal of Geophysical Research, 100, 721-740.

• Foreman, M.G.G., G. Sutherland, and P.F. Cummins, 2004. M2 tidal dissipation around Vancouver Island: an inverse approach. Continental Shelf Research, 24, 2167-2185.

• Foreman, M.G.G., D.J. Stucchi, K.A. Garver, D. Tuele, J. Isaac, T. Grime, M. Guo, and J. Morrison, A Circulation Model for the Discovery Islands, British Columbia, 2012, Atmosphere-Ocean, 50:3, 301-316.