We use OTIS to simulate the effects of SLR on the tides around Australia. OTIS is a portable, dedicated, numerical shallow water tidal model which has been used extensively for both global and regional modelling of past, present, and future ocean tides e.g.. It is highly accurate both in the open ocean and in coastal regions , and it is computationally efficient. The model solves the linearized shallow-water equations e.g. given by ∂U∂t+f×U=-gH∇(ζ-ζEQ-ζSAL)-F∂ζ∂t=-∇⋅U, where U is the depth integrated volume transport, which is calculated as tidal current velocity u multiplied by water depth H. f is the Coriolis vector, g denotes the gravitational constant, ζ stands for the tidal elevation with respect to the moving seabed, ζSAL denotes the tidal elevation due to ocean self-attraction and loading (SAL), and ζEQ is the equilibrium tidal elevation. F represents energy losses due to bed friction and barotropic–baroclinic conversion at steep topography. The former is represented by the standard quadratic law: FB=Cdu|u|, where Cd=0.003 is a non-dimensional drag coefficient, and u is the total velocity vector for all the tidal constituents. The parameterization for internal tide drag, Fw=C|U|, includes a conversion coefficient C, which is defined as C(x,y)=γ(∇H)2NbN‾8π2ω. Here, γ=50 is a scaling factor, Nb is the buoyancy frequency evaluated at the seabed, N‾ is the vertical average of the buoyancy frequency, and ω is the frequency of the tidal constituent under evaluation. Values of both Nb and N‾ follow from the prescription of horizontally uniform stratification N(z)=N0exp⁡(-z/1300), where N0=5.24×10-3 s-1 has been obtained from a least-squares fit to present-day climatological hydrography .

The model solves Eqs. ()–() using forcing from the astronomical tide-generating potential only (represented by ζEQ in Eq. ), with the SAL term being derived from TPXO8 updated version. An initial spin-up from rest over 7 days is followed by a further 15 days of simulation time, on which harmonic analysis is performed to obtain the tidal elevations and transport. Here, we investigate the two dominating semi-diurnal and diurnal tidal constituents, M2 and K1, respectively. The model bathymetry comes from the ETOPO1 dataset see Fig.  for the present domain, which was averaged to 1/20∘ horizontal resolution. For the control run with present-day water depths, the domain model heights at the open boundaries were constrained to elevation data from a coarser-resolution global OTIS run, taken from . TPXO8 was used to validate the model (alongside tide gauge data at 24 locations); see Sect. .

The computation of tidal dissipation rates, D, was done following : D=W-∇⋅P.

Here, W is the work done by the tide-generating force and P is the energy flux given by W=gρ〈U⋅∇(ηSAL+ηEQ)〉P=g〈ηU〉, where the angular brackets mark time averages over a tidal period.

The model runs are split into two sets. In the first, we allow low-lying grid cells to flood as sea level rises, whereas in the second set we introduce vertical walls at the present-day coastline. Following , we denote these sets “flood” (FL) and “no flood” (NFL), respectively. A range of SLR scenarios corresponding to predicted global mean sea-level increases from large-scale ice sheet collapses are investigated in both sets. This is done via the implementation of a uniform depth increase across the entire domain of 1, 3, 5, 7, and 12 m. Boundary conditions for each of the SLR scenarios are generated from global simulations. The 5, 7, and 12 m SLR NFL runs were taken directly from and the remaining global simulations were carried out following the methodology outlined in but with varying global sea-level changes and allowing for inundation in the FL runs. Additionally, we simulated tidal responses to future changes in water depth extrapolated from geocentric sea-level trend patterns as observed by satellite altimetry see. While such projections contain not only the actual long-term trend of sea level but also significant (sub-)decadal variability, little difference was found for tidal perturbations with respect to our uniform SLR scenarios (see the Supplement). It is also unknown how the magnitude and spatial variation of the trend pattern may change over the period of time required to equate a uniform SLR, especially with the larger scenarios considered here. Hence, in the following, the focus is on the uniform SLR scenarios. We choose to show results for the 1 and 7 m SLR simulations because they best exemplify the changes to tidal characteristics across the domain and also correspond to a high but probable level for the end of this century and an extreme case in which large levels of ice sheet collapse has occurred, respectively e.g.. The simulations with 12 m of SLR have little physical justification, but they allow us to assess how any tidal trends seen up to 7 m, if they appear, may evolve for even higher levels of SLR.

For validation of our numerical experiments, time series of hourly sea-level data from 24 stations around Australasia were obtained from the Global Extreme Sea Level Analysis version 2 GESLA-2; and the University of Hawaii Sea Level Center UHSLC;; see Table 1. With few exceptions, record lengths are short, but all time series span at least 28 years to allow for an appropriate representation of the 18.61-year nodal cycle in lunar tidal constituents . Upon removal of years missing more than 25 % of hourly observations, a three-tiered least-squares fitting procedure was applied to (i) extract mean M2 and K1 tidal constants of amplitude H and phase lag G, (ii) deduce linear trends in H and G for both constituents over the complete time series at each station, and (iii) estimate the corresponding long-term trend in MSL. A total of 10 out of 24 tide gauge stations yielded statistically insignificant M2 amplitude trends at the 95 % confidence level and were thus excluded from the model trend validation below; see Fig.  for a graphical illustration.

The processing protocol, essentially taken from , is based on a separation of tidal and non-tidal residuals from the longer-term MSL component through application of a 4-day moving average with Gaussian weighting. The high-frequency filter residuals obtained were then harmonically analysed for 68 tidal constituents using the Matlab^® UTide software package , with analysis windows either set to the entire time series (step i) or shifted on an annual basis (step ii). In both cases, we configured UTide for standard least-squares and a white-noise approach in the computation of confidence intervals. Subsequent regressions of annual M2 tidal constants were performed with a functional model composed of a linear trend, a lag 1-year autocorrelation, AR(1), and sinusoids to account for nodal modulations. Trends in MSL were likewise determined through regression under AR(1) assumptions, upon a priori reduction of the influence of the 18.61-year equilibrium tide.

A regression of the constituent amplitudes calculated from the OTIS control simulation for the present-day bathymetry against the M2 and K1 amplitudes from harmonic analysis of the tide gauge data (Fig. ) reveals that the model performs well at all sample sites except at stations Williamstown and Geelong (see Fig.  and Table 1). Here, simulated amplitudes are overestimated by a factor of 5 (M2) and 2 (K1), respectively. Both of stations lie on the coast of Port Phillip Bay, a shallow bay isolated from the Bass Strait by two peninsulas. The eastern peninsula is long and narrow, and not resolved in the model bathymetry due to the spatial averaging performed on the source ETOPO1 dataset. As such, the tidal wave from the Bass Strait penetrates into the bay and builds up in amplitude through reflection at lateral boundaries. Excluding Williamstown and Geelong from the comparison, we obtain a root mean square (rms) error for the constituent amplitudes of 17 cm for M2 and 2 cm for K1, while correlation coefficients are at 0.97 and 0.99, respectively. Estimates of the median absolute difference between model and test stations can be used to quantify possible biases in a robust way ; corresponding values are 8 cm for M2 and 1 cm for K1 irrespective of whether stations in Port Phillip Bay are included or not. Median phase differences with respect to the tide gauge estimates are 21∘ for M2 and 3∘ for K1. These statistics give confidence in the model's accuracy and in the assertion that areas where the disagreement between model and tide gauge data is largest are where the resolution of the model has not resolved fine-structure coastal features.

Additional comparisons with gridded M2 data were performed using the TPXO8 inverse solution, linearly interpolated to the grid of the model domain (see http://volkov.oce.orst.edu/tides, last access: 17 April 2018, for details). The rms difference between the model and the TPXO8 data is 7 cm for M2 and 2 cm for K1. The variance capture (VC) of the control was also calculated to see how well the overall character of the tidal constituents was represented e.g.: VC=1001-RMSDS2, where RMSD is the rms difference between the control simulation and TPXO8, and S denotes the rms standard deviation of the TPXO8 amplitudes. The VC is above 96 % for M2 and 97 % for K1.

For validation of the simulated M2 changes under SLR, we followed and condensed measured M2 trends (∂H, ∂G) and MSL rates (∂s) to response coefficients in amplitude (rH=∂H/∂s) and phase (rG=∂G/∂s). Simulated amplitude and phase changes from the 1 m FL run at the location of 14 tide gauges were interpolated from nearest-neighbour cells and also converted to ratios of rH and rG. Graphical comparisons in Fig.  indicate that the model captures the sign of the observed M2 amplitude response in 10 out of 14 cases and reproduces large fractions of the in situ variability at approximately half of the analysed stations (e.g. Booby Island, Brisbane, Geelong, Port Hedland, Broome). A similar figure for K1 can be found in the supporting information. Model-to-data disparities on the northeastern seaboard (Hay Point, Gladstone) are markedly reduced in comparison to their Fig. 7 due to the higher horizontal resolution of our setup in a region of ragged coastline features. Neither the increase of M2 amplitudes at Townsville nor the pronounced reduction of the tide at Fort Denison (Sydney Harbour) can be explained by SLR perturbations in the tidal model; both signals may be the effect of periodic dredging to maintain acceptable water depths for port operations; see for a similar analysis. The decrease of the small M2 tide at Spring Bay, Tasmania (20 cm m-1 of SLR), remains puzzling though, given that the gauge is open to the sea and unaffected by harbour activities or variable river discharge rates. Mechanisms other than SLR, such as modulation of the internal tide due to stratification changes along its path of propagation or variations in barotropic transport induced by thermocline deepening , need to be thoroughly addressed to complete the picture of secular changes in the surface tide. Furthermore, uncertainty in the bathymetric data due to sparsity of sounding observations may induce additional errors. Despite these limitations, our 1 m FL simulation captures good portions of the patterns of M2 amplitude changes seen in tide gauge records around Australasia. Moreover, on timescales of centuries, water column depth changes due to SLR will outweigh tidal perturbations from other physical mechanisms. Hence, we conclude that our simulations lend themselves well for analysis of probable future tidal changes over a wider area.

SLR brings some fairly significant changes to the M2 tidal systems around Australia. Many of the regions which stand out as having high M2 amplitude at present (see Fig. b) suffer a marked reduction in amplitude with increasing SLR (Fig. a and c). To the north-west of Australia, amplitudes decrease in regions centred around King Sound, around Joseph Bonaparte Gulf, within Van Diemen Gulf, and across the Timor Sea. The reduction in amplitude to the north-east of Van Diemen Gulf comes alongside the formation of a new amphidrome; the phase lines that run north-south in the Arafura Sea (Fig. b) move closer together before coalescing to form an amphidromic point. A further amphidromic point emerges in the south-east of the Gulf of Carpentaria, around the Wellesley Islands, when a virtual amphidromic point (an amphidrome over land) moves north to become real (an amphidrome over the ocean). Both these points form some time between 3 and 5 m SLR (not shown). The amphidrome that sits north of the Gulf of Carpentaria (Fig. a) moves northwards with SLR in both the FL and NFL scenarios, eventually becoming virtual at the higher SLR scenarios by moving over Yos Sudarso Island. This movement is therefore associated with the change in the propagation properties of the incoming tidal wave, rather than the inundation of Yos Sudarso Island which occurs at 5 m SLR and above.

The Torres Strait features a sharp divide between a large positive and small or weakly negative amplitude difference on its west and east sides, respectively. The strait is shallow and dotted with islands, restricting the flow of the tides. Absolute amplitudes on the east side are initially elevated compared to the west (Fig. b) but this difference is mitigated with increasing SLR, as the larger volume of the channel allows for enhanced tidal transport across the strait (Fig. a and c).

In the remaining part of the domain, the M2 amplitudes around the coast of New Zealand and along the east coast of Australia, particularly between Hay Point and Gladstone (Fig. a, points 3 and 4), both increase with SLR. In the Bass Strait, shown in Fig. , the amplitude also increases along with SLR; however, there is a slight drop in amplitude at 1 m SLR at the eastern entrance of the strait and within Port Phillip Bay (the location of the Geelong and Williamstown tide gauges). These decreases quickly disappear at higher SLR. Along the south coast of Australia, the amplitudes also increase with SLR, but the simulated perturbations there are generally smaller in magnitude than in the north see also. Increased tidal amplitudes at the head and mouth of Spencer Gulf (Fig. a) are associated with a slight weakening of a standing-wave-like pattern where M2 amplitudes increase from the sea towards inland (Fig. b). However, this feature does not evolve proportionally with the imposed level of SLR (see Fig. c for the 7 m case). To the east, in Gulf St. Vincent, there is a similar, if stronger, standing-wave-like pattern (Fig. b). Here, the elevated amplitudes proximity to the coast increase in line with SLR (Fig. c). Moving further west along the south coast of Australia, the amphidromic point off the south-west coast moves southwards with increasing SLR (Fig. a and b), with a faster progression in the NFL than in the FL runs. Note that, almost universally across the domain, the amplitudes seen in the NFL runs are higher than those in the FL runs (Figs. d and d). It is clear that allowing flooding to occur suppresses the magnitude of any SLR effect upon tidal amplitude.

Additionally, we have repeated our control and future simulations using boundary conditions taken from TPXO8, i.e. using present-day boundary conditions, instead of those from global SLR simulations. In these runs, the magnitude and spatial distribution of the amplitude changes are vastly different from the results presented above. Figure  demonstrates, and allows for a comparison of, the pattern of amplitude changes seen for M2 north of Australia in these runs. Especially striking is the difference in magnitude of tidal amplitude increases along the east coast of Australia. Applying present-day boundary conditions leads to a very strong underestimation of the amplitude changes in this area (the differences exceed 40 cm in some locations). Similarly pronounced is the difference on the north-west coast where the run with present-day boundary conditions overestimates the amplitude decreases. In large parts of the Arafura Sea, the tidal amplitude responses are of opposite sign in response to the different boundary conditions. This is a key result and it highlights the importance of applying correct boundary conditions for regional simulations which take into account the far-field change occurring outside the model domain.

Figure a displays the dissipation associated with the M2 tidal constituent for the control simulation. It is evident that a majority of the energy is lost at the coast and on the shelf, especially around sharp and shallow bathymetric features. There is a remarkable reduction in dissipation along the north-west coast of Australia with SLR, matching areas with a marked decrease in tidal amplitude (Fig. b). There are small pockets of dissipation increases in proximity to the coast in the north-west, associated with areas of enhanced flooding. Much of the dissipation increases within the domain come from the undersea ridges and trenches to the north and south of New Zealand, as well as the east side of the Torres Strait, the area between Hay Point and Gladstone on the east coast of Australia (at the southern extent of the Great Barrier Reef), and both ends of the Bass Strait. All of these areas are associated with an increase in M2 amplitude with SLR.

The impact upon the total dissipation across the domain by allowing flooding is illustrated in Fig. . In both the NFL and FL runs, dissipation increases with SLR; the dissipation generated by areas with increased amplitudes outweighs the loss of dissipation in areas of decreased amplitude. With amplitudes in the NFL runs being much higher than in the FL runs, the corresponding dissipation is higher, and the gap in dissipation between the NFL and FL simulations widens with increasing SLR. At 12 m SLR, the NFL dissipation begins to plateau, whilst the FL dissipation continues to rise. Overall, dissipation on the shelf (nominally above the 200 m isobath, accounting for approximately 7 % of the water-covered area of the domain) amounts to about half of the energy dissipated in deeper water.

The changes to K1 are relatively small in comparison to the changes seen for the semi-diurnal constituent. The main point of interest is the amphidromic system in the Gulf of Carpentaria. Figure  shows an initial negative amplitude response in the north and south of the Gulf, while there are slight increases in amplitude north of Yos Sudarso Island and in the Java Sea. The position of the amphidromic system is relatively stable with SLR, as evident from the constituent phase plots in Fig. c and d, and it is evident that K1 suffers a decrease in amplitude around the Gulf of Carpentaria with SLR. Comparing Fig. c and d, a majority of the amplitude difference occurring in the FL scenario is negated when impermeable coastlines are implemented. The dissipation for which K1 is responsible in the domain is approximately 7 times smaller than that of M2 and therefore not discussed further.

One of the most striking characteristics of the response of the semi-diurnal constituents to SLR is the drop in amplitude along the north-west coast of Australia, an area where a large amount of dissipation occurs in the control. We associate this feature with the altered propagation properties of the incoming tidal wave, as the imposed sea-level change in our simulations represents a significant fraction of the average water depth across the shelf; for instance, the 7 m SLR scenario increases the depth of the Gulf of Carpentaria by ∼10 %. As the tidal system goes from hosting a single wave propagating across the Arafura Sea to something more complex with multiple amphidromes within the basin, the semi-diurnal resonant response on the shelf is interrupted. This dynamical change may explain the drop in amplitudes to the north-west of Australia.