Tidal and internal tidal impacts in the Tasman Sea

Abstract

Barotropic and baroclinic tides were simulated for the Coral and Tasman Seas off eastern Australia and compared to a simulation without tides. Both simulations included geostrophic currents and replicated a current analogous to the East Australian Current (EAC). Tides and tidal currents in most of this region are weak, generally 1–2 cm s⁻¹, with the exception of the far northwest portion of the Coral Sea. Even these weak tides were found to impact the mean EAC-like current transport, enhancing it by 1–4 Sv in some areas. Southward flow increased over the continental shelf. Tides did not appear to impact eddy formation, size, or rotational speed; however, they affected eddy propagation. Cyclonic eddies propagated northward faster with tides than without tides. Tides impacted cross-shelf transport of colder water, with significantly more on-shore transport occurring with tides, particularly equatorward of the diurnal critical latitudes. Cross-shelf transport of nutrient rich water onto the shelf is important in this oligotrophic region. Although the prime source of vertical shear and mixing were mean currents and eddies, tides played a secondary role. Tides influenced mixing by increasing vertical temperature diffusivities to 10⁻⁴ to 10⁻³ m² s⁻¹ over portions of the continental slope and over rough topography, particularly in regions near the diurnal critical latitudes (27°–30°). In conclusion, even small tides can significantly impact the circulation through their effects on the mean currents, eddy rotation velocities, eddy propagation, and mixing.

Introduction

Tides influence many processes involved in ocean circulation and mixing, impacting currents, heat transfers, distribution of chemicals, biological productivity, sediment redistribution, sea ice rafting and ridging, ice shelf melt, and others. The primary mechanisms by which tides impact these processes are associated with: the mean flow, boundaries, tidal fluctuations, and mixing. These tidal impacts originate from both the barotropic and baroclinic tides. Barotropic impacts are those connected with elevation changes and the movement of the entire water column as a vertical unit. Baroclinic effects are primarily tied to internal or baroclinic tides, which are internal waves at the tidal frequencies generated through the interaction of barotropic tides with abrupt topographic features, such as seamounts, ridges, or the continental shelf break. Furthermore, internal tides become resonant at their critical latitudes, amplifying the effects of many of these mechanisms. (The critical latitude is generally defined as the latitude, where the inertial frequency equals the tidal frequency.) Finally, these mechanisms do not occur in isolation, but typically, multiple mechanisms occur simultaneously, interacting either constructively or destructively.

The primary ways tides affect mean currents are through residual velocities and increased friction. Tidal residual velocities are mean flows generated by horizontal gradients in the tidal velocities. The ellipse followed by a particle during a tidal cycle is shorter on one side than the other and the ellipse does not close, yielding a current in the direction of the longer side. Tidal residual currents can either enhance or reduce other existing currents. Another way tides impact the mean flow is through increased bottom friction. Frictional force increases as a square of the velocity, so even if the net tidal velocity is zero over a tidal period, the sum of the squares of the velocities is larger than it would be without tides, increasing frictional losses. This effect can significantly decelerate mean flows. For example, tides reduced the mean flow in the Weddell Sea by 30% (Robertson et al. 1998). The overall tidal effect on the mean currents is the combination of these two opposing effects.

There are also fluctuating effects through the generation of internal tides, internal waves, topographically trapped waves, and near the critical latitude, inertial oscillations. Many of these effects are modulated by the variation in the strength of tidal currents and/or mixing through the daily and spring-neap tidal cycles. One of these effects is the variation in the boundary layer thickness associated with the cycling of the barotropic tide (Lamb 1993). The critical latitude also plays a role here increasing the boundary layer thickness near the critical latitude (Furevik and Foldvik 1996).

Tides are believed to be one of the prominent mixing mechanisms in the ocean. The energy from tides has been estimated at 2 TW (1 TW = 10¹² W), with half of the energy going into ocean mixing by internal tides (Egbert and Ray 2000). Internal tides generate shear instabilities, which can lead to turbulence and mixing, if the stabilizing effect of the stratification is less than the destabilization of the water column by the shear. In shallow coastal waters, tidal mixing can extend throughout the entire water column, homogenizing the water column, and generating a tidal front between the mixed region and the deeper, stratified waters. Again, critical latitude effects increase the generation of internal tides near their respective critical latitude. Tidal mixing redistributes nutrients, larvae (Stevens et al. 2012), and sediments (Hosegood et al. 2004), affecting biological productivity and fisheries. In addition, internal tides impart strong currents in the water column, with opposing flows at different depths. These currents induce forces on man-made structures, such as pipe lines and oil rigs.

Internal tidal effects are amplified near the tidal critical latitudes. In general, the critical latitude is accepted to occur at the exact latitude, where the tidal frequencies equals inertial frequency from planetary vorticity. However, there is also an effective critical latitude, where the tidal frequency equals the sum of the inertial frequency and any relative vorticity that is present from other processes (Kunze and Toole 1997). Relative vorticity can come from eddies, the edges of mean currents, the interactions of currents with topography, wind, inertial oscillations, and other processes. The effective critical has been shown to have significant effects even in regions with small tides (Kunze and Toole 1997; Robertson 2013; Robertson et al. 2017; Dong et al. 2019); however, critical latitude effects for this region are confined to within 3° of the diurnal critical latitudes (Robertson et al. 2017).

Vorticity is important when considering tidal interactions with eddies and the critical latitude, since the interactions often occur through vorticity. Three factors associated with tides will impact the vorticity: (1) both directional changes in the mean currents and the decrease in magnitude at their edges induce vorticity, although it can be positive or negative; (2) Internal tides and waves generated over rough topography by barotropic tides add vorticity; (3) Tides rotate in counterclockwise ellipses in the Southern hemisphere, which will increase vorticity.

These physical mechanisms also impact biological organisms, sea ice, and ice shelves. Besides redistribution of nutrients and larvae, internal tides affect photosynthetic organisms by shifting isopycnals vertically in the water column, changing their depth and the amount of available light for photosynthesis (Stevens et al. 2012). Tides induce divergence in the surface velocities, particularly in regions of changing topography. In the polar regions, this divergence causes lead formation, which was first noticed in the Arctic by Nansen on the Fram (1898). In the western Weddell Sea, tidal frequencies were found to be the main contributors to sea ice deformation (Geiger 1998) with the variability of sea ice over continental slope controlled by spatial inhomogenity of tides (Kottmeier and Sellman 1996; Kottmeier et al. 2005). Furthermore, internal tides were found to increase ice shelf melting by 20–50% in the Amundsen Sea (Robertson 2013).

The Tasman Sea off the east coast of Australia has interesting dynamics with a strong western boundary current, the East Australian Current (EAC), an undercurrent, coastal currents, and a very active eddy field. This region also has not been frequently simulated. Wilkin and Zhang (2007) simulated the EAC, but did not include tides. Their simulation had peak velocities exceeding observations by 20%. Since tides can reduce mean currents, it is possible that including tides would reduce this overestimate. Another simulation was performed by (Zavala-Garay et al. 2012) and they improved the subsurface performance in an inverse simulation by including XBT and SynCTD data; however, they also did not include tides or look at tidal impacts.

Tidal impacts on the circulation, mixing, and biology along Australia’s east coast are not well-known. Although no impacts on sea ice or ice shelves are expected, many other impacts are expected, such as the formation of tidal fronts, the interactions of tides with the EAC and the eddies it spawns, cross-shelf flows, and mixing. The model used for the simulations along with the optimal parameters for internal tides (Robertson and Dong 2019) is described “Model description” Section. The effects of tides and internal tides on the dynamics of the region are investigated, starting with tidal effects on the along-slope transport and a strong current analogous to the East Australian Current (EAC), in “North-South and along-slope transports” Section. The impacts of tides on cross-shelf transports, particularly of colder shelf water, which like upwelling brings nutrient rich waters onto the continental shelf, are detailed in “East-West and across-slope transports” Section. Eddies generated by the EAC are another key feature of the region and are common in the waters off eastern Australia (Roughan et al. 2017). The interactions of tides with eddies are described in “Interactions of tides with eddies” Section. Estimates of tidal mixing from the model are compared to observations in “Tidal mixing” Section. “Discussion” Section offers an overall discussion of these impacts. Finally, a summary is provided in “Summary” Section. Additional file plots have been given a prefix S-#, where # is the Additional file figure number.

Model description

Model

The simulations were performed with Version 3.4 of the Regional Ocean Model System (ROMS). ROMS is a terrain-following, primitive-equation model developed by the Ocean Modeling group at Rutgers University (http://www.ocean-modeling.org/index.php?page=models&model=ROMS). ROMS uses split 2-D (barotropic) and 3-D (baroclinic) modes to obtain the required temporal resolution. Third order upstream differencing was used for horizontal advection, Laplacian lateral diffusion was determined along geopotential surfaces (10 m² s⁻¹) (Shchepetkin and McWilliams 2004), and the splines density Jacobian scheme was used for the baroclinic pressure gradient calculation. Vertical mixing was determined according to the modified Mellor-Yamada 2.5 level closure scheme of Nakanishi and Niino (2009). These options were selected based on the results of sensitivity studies on model operating parameters for a baroclinic tidal simulation of the region around Fieberling Guyot (Robertson 2006; Robertson and Dong 2019). ROMS has successfully been utilized for barotropic and baroclinic tidal modeling (Robertson et al. 2003, 2017; Robertson 2001, 2005a, 2005b, 2010, 2011, 2013; Robertson and Ffield 2008).

Domain

Bathymetry for the simulations was taken from Geoscience Australia (Fig. 1) (Whiteway 2009). A simple averaging filter was used to downsample the bathymetry down to a 4 km grid cell resolution resulting in a grid 315 by 405 cells in the East–West and North–South directions, respectively. Regions shallower than 50 m were deepened to 50 m. The model was first run with constant temperature and salinity to characterize errors induced by steep bathymetry. Bathymetry was only smoothed for regions, where steep topography in that simulation generated large errors (> 0.01 m s⁻¹), using a set of linear programming routines (Sikirić et al. 2009). Most of the smoothing occurred at the base of Sydney canyon,

Fig. 1

Model domain with bathymetry with Integrated Marine Observing System (IMOS) moorings indicated as yellow crosses

Initial and boundary conditions

Initial hydrography, consisting of potential temperature, θ, and salinity, S, fields, was obtained from the World Ocean Atlas (Levitus, 2013). Boundary conditions were set for elevation, velocities, temperature, and salinity. With velocity, the barotropic mode velocities used Flather radiative boundary conditions in the normal direction (Flather and Proctor 1983) and advective conditions in the tangential directions. The baroclinic velocities used flow relaxation boundary conditions to initial conditions over four cells as described by Martinsen and Engedahl (1987). Similarly, for the temperature and salinity, the boundary values were relaxed over four cells. Surface and bottom fluxes for temperature and salinity, including solar radiation and precipitation, were zero and the volume was unconstrained. Mean currents were not prescribed along open boundaries.

Forcing

Tidal forcing was implemented through the elevations along the boundary updated at every 2-D time step, with complex coefficients taken from a 2-D model, Egbert et al.’s global inverse tidal model TPXO8.0 (Egbert and Erofeeva 2002). Eight major tidal constituents were simulated, four semi-diurnals, M₂, S₂, N₂, and K₂, and four diurnals, K₁, O₁, P₁, and Q₁. Tidal elevations were updated every 2-D time step, which was implemented by Robertson in 2001, but not adapted by the ROMS model in general (2001). For the non-tidal simulations, the tidal elevations and velocities for all constituents were set to zero along all boundaries.

Other operational considerations

The 2-D (barotropic) and 3-D (baroclinic) mode time steps for the base case were 8 s and 240 s, respectively. All simulations were run for 60 days, with hourly data used for analysis. Mean kinetic and potential energies stabilized around 5 days, thus the first 5 days of simulated data was discarded; however, mean elevations took longer to stabilize in some locations. Analysis was primarily performed on the last 30 days of data (days 30–60); however, to show the evolution of some of the parameters, days 15–60 are shown.

Analysis

Pawlowicz et al.’s (2002) Matlab^® tidal analysis routines were used to analyze the elevation and velocity fields, respectively, and generate the tidal ellipse parameters for the four primary constituents, M₂, S₂, K₁, and O₁. North–South (East–West) transports were determined by integrating the product of the North–South (East–West) velocities, the appropriate grid cell width, and the grid cell height at each time step for each grid cell and vertical level. In addition, the northward and southward flows (eastward and westward) were calculated, using the positive and negative North–South (East–West) velocity values at each time step for each grid cell and vertical level. Since there is interest in colder, deep, nutrient rich water transports on and off the continental shelf, transports were determined in the same way for waters with temperatures below 18 °C, which is below the mixed layer and seasonal thermocline at any of these latitudes and roughly the upper 200 m of the water column, which is the approximate depth of the continental shelf (Additional file 1: Fig. S1). To determine the influence of tides on the along-shelf (cross-shelf) flow at the continental shelf break, the along-slope (cross-slope) transports were also calculated along the tangent (perpendicular) to the horizontal angle of the continental shelf break at the 200 m isobath for each latitude. These transport time series were then integrated in time and the respective longitude ranges to yield transport values for each latitude. To identify eddies, the eddy detection algorithm of Xu (2020), which is based on sea level anomalies, was used. The resultant eddy positions were verified against maps of surface velocity magnitude, temperatures, and vorticity.

North–South and along-slope transports

Mean current field

Although the simulations did not include wind, they did have realistic potential temperatures and salinities taken from observations. The potential temperature and salinity fields generated density fields with horizontal gradients, which induced geostrophic currents. At the end of the spin-up, a strong current, peaking at ~ 1.7 m s⁻¹, ran south roughly along the continental shelf break from north of 24^°S to around 29^°S (Fig. 2a, b), where it separated from the coast and headed out into the Tasman Sea, in simulations both with and without tides (Fig. 2c, d). This current behaved basically analogous to the East Australian Current (EAC). It transported roughly 40–60 Sv southward (Fig. 3f). There was also a deep northward flow of 10–30 Sv between 24° and 29^°S (Fig. 3g) similar to the East Australian Undercurrent (EUC). Combined, this yielded a net southward flow of ~ 40 Sv (Fig. 3e). The circulation pattern included a portion recirculating northward offshore at 157^°E roughly between 30 and 25^°S (Fig. 2), analogous to the recirculation cell that feeds back into the EAC and often observed in this latitude range. Between 30 and 33^o S, an anti-cyclonic eddy was present (Figs. 2 and 4). South of 33^°S, the flow along the shelf break was also southward with a net transport of ~ 20 Sv (Fig. 3a). The region south of 33^°S was heavily influenced by eddies, as can be seen by the large southward (Fig. 3b) and northward (Fig. 3c) transports. Along with eddies, these currents formed the predominant North–South and along-slope flows.

Fig. 2

Mean barotropic transports over time in the a North–South and c East–West velocities with tides in the simulation and b and d without tides, respectively

Fig. 3

Integrated North–South transports west of 155^°E for water depths greater than 200m for the a net transport, b southward transport, and c northward transports for the tidal (red) and non-tidal (black) simulations South of 30^°S. d The integrated North–South net transport west of 155^°E at the continental shelf break for the tidal (red) and non-tidal (black) simulations south of 30^°S. e–h The same parameters as a–d North of 30^o S, respectively

Fig. 4

Surface temperatures and velocity vectors at 30 days from simulations a without and c with tides and at 60 days b without and d with tides. The three eddies discussed in the text are indicated with a yellow circle for their edge and labeled A, B, and C in each panel

Observations along the eastern Australian coast have shown the presence of topographically trapped or continental shelf waves along the continental shelf/slope break (Woodham et al. 2013). These waves are Rossby waves and are driven by events that occur at locations along the western Australian coast and propagate around the continent. These simulations do not include this forcing mechanism or these Rossby waves.

To identify the tidal influences on these flows, the differences between tidal and non-tidal simulations were determined for the North–South transport. The predominant differences in the mean North–South transports due to tides occurred west of 155^°E, primarily along the continental shelf/slope break, the area, where the EAC-like current flows east, and where eddies were present (Fig. 5a). In Fig. 5a, positive (red) value indicates a stronger northward flow or weaker southward flow with tides. To separate these, the differences associated with southward (northward) flows are shown in Fig. 5b, d. The largest differences in the northward flows occur between 30° and 33^°S in the zone of strong separation of the current from the coast (Fig. 5d). The strongest southward differences also occurred in this latitude range, slightly inshore of the northward differences (Fig. 5b). Differences in the southward flow also occurred along the continental shelf break south of 31^°S (Fig. 5b). Most of these differences were associated colder water below 200 m (Fig. 5c).

Fig. 5

Differences in the mean barotropic North–South transports a for all flow, b for southward flows, c for flow of waters 18 °C or below, and d) for northward flows

Tides increased the net southward transport by 1–4 Sv between 24 and 26^°S and either did not change or reduced net southward transport by 0–1 Sv between 27° and 29^°S (Fig. 3e). From 24° to 26^°S, there was a result of an increase in the southward flow (Fig. 3f) and at a few latitudes a decrease in the northward flow (Fig. 3g). The strengthening of the current from 24°to 26^° S is attributed to tidal residual currents and the weakening from 27° to 29^o S to tidal friction. The continental shelf between 27° and 29^°S is much wider and the bottom topography slope less steep than it is either side of this latitude range (Fig. 1). Stronger tidal residual currents are generated by narrower, steeper topography. Thus, the enhancement of the EAC-like flow by tidal residuals was weakened between 27^°30ʹ and 29^°S. At the same location, frictional influences increased due to the shallower water, resulting in a decrease in the net transports. This is supported by the changes in the southward flow (Fig. 3f). South of the separation of the current from the coast, tides increased the southward flow from 30^° 30ʹ to  32^o S (Fig. 3a) and increased the northward flow between 31^°40ʹ and 34^o S (Fig. 3c), resulting in a northward shift of the southward net transport (Fig. 3a). The change here was associated with differences in location and propagation of a cyclonic eddy. The cyclonic eddy propagated northward faster with tides than without (See “Interactions of tides with eddies” Section), resulting in an increase in the northward flow with tides. South of 34^°S, tidal residuals again enhanced the southward flow (Fig. 3a), although eddies complicated the dynamics.

Alongshore transports over the continental shelf (depths <  = 200 m) were small, ~ 1 Sv; however, in these shallow waters, 1 Sv differences are associated with significant velocities, 0.8 m s⁻¹ or greater. There was a definite difference in the coastal flows with the tides. Net transports were northward without tides and southward with tides (Fig. 3d) North of 28^°S. South of 31^°30ʹS, net transports were southward both with and without tides, but a stronger southward flow was associated with tides.

The aforementioned tidal impacts have been separated by latitude. The tidal impacts are clearly latitude dependent. Inspection of the differences in the alongshore transports at the continental shelf break with time clearly show this latitude dependence (Fig. 6). It should be noted in Fig. 6, that until day 30, the model elevation is still stabilizing the tidal inflow and outflow from the domain, so the elevation is increasing at this location until day 30. Days 15–30 are shown to indicate the evolution of the parameters. The behavior of the differences in the alongshore transports can be broken into three latitude zones, with the northernmost and southernmost 1° latitude bands ignored due to boundary influences. The three zones are: the diurnal critical latitude zone (27^o30ʹ–30° S), equatorward/north of this zone (23°–27^° S) and poleward/south of this zone (31°–37^o S). Separation of the EAC-like current occurs in the diurnal critical latitude zone. Tidal impacts to the alongshore flow at the continental shelf break are generally weaker in this band (Fig. 6b), especially for waters below 18 °C (Fig. 6c), with the exception of one large negative anomaly during spring tide during days 50–56 (Fig. 6a). Two positive anomalies also occur further north in this band, one during neap tide and one during spring tide. There is no clear correlation between the spring-neap tidal cycle and these differences (Fig. 6a, b). North of the critical latitude band, the daily tidal cycle is apparent in the differences, particularly north of 25^° S. The warm surface waters contribute to the differences (Fig. 6b). South of the critical latitude band, alternating bands of positive and negative differences propagate northward at speeds of 0.2–0.4 m s⁻¹, with the faster speeds occurring later in the simulation. The bands are separated by ~ 40 km early in the simulation and by ~ 100 km later in the simulation. These differences are not due to continental shelf waves or topographically trapped waves, which move much faster 2.0–3.6 m s⁻¹ (Woodham et al. 2013). Inspection indicated a succession of small eddies or billows, 20–40 km in diameter, propagating northward along the continental shelf/slope break during this period, both with and without tides. These billows took ~ 2 days to pass a point in latitude. Their northward propagation was basically blocked by the separation region. Differences in the propagation speed of these billows due to tides induced this propagating signal.

Fig. 6

a Time series of the elevation at 30° S 154° 11’E indicating the daily and spring-neap tidal cycles b Difference in flow tangent to the continental shelf break at 200 m between the tidal and non-tidal simulations. c Same as b for water colder than 18 °C. Negative values indicate either an increase in a southward flow or a decrease in a northward flow

East–West and across-slope transports

Mean currents

The predominant East–West transports were associated with the recirculation cell and eddies (Fig. 2c, d). The basic structure of the East–West transports was very similar between the simulations with (Fig. 2c) and without (Fig. 2d) tides.

Tidal impacts

The differences in the East–West transports due to tides were primarily associated with the separation of the current from the coast, an eddy at 31–33^o S, and a patch of eastward flow at 34^°S (Fig. 7a). Most of the differences occurred in waters with potential temperatures below 18 °C (Fig. 7c). Since they are associated with the eddy, the differences occurred in both the westward (Fig. 7b) and eastward (Fig. 7d) transports, with the westward transports more affected than the eastward. It should be noted that differences in the position of an eddy in the simulations with and without tides will result in a pair of positive and negative transports, such as these. Like with the North–South velocities, tides essentially slowed rotational transports in the eddy at 31°–33^o S (eddy B in Fig. 7). The pattern of differences is consistent with the eddy B propagating north faster with tides than without tides and will be more fully discussed in “Interactions of tides with eddies” Section.

Fig. 7

Differences in the mean East–West transports a for all flow, b for southward flows, c for flow of waters 18 °C or below, and d for northward flows

The behavior of the across-slope transports across the continental shelf break also falls into the same three latitude bands as the along-slope transports (Fig. 8). Most of the across-slope transports at the continental shelf break occurred in the waters below 18 °C (Fig. 8c). North of the critical latitude band (23°–27^°S), across-slope transports showed a stronger response to the daily tidal cycle (Fig. 8b). One patch of strong off-shelf transport occurred during spring tide, but again there is no clear correlation between the spring-neap cycle and the across-shelf transports. Differences in the across-slope transports at the continental shelf break in the critical latitude band (27^°30ʹ to 30^° S) were smaller than in the other bands, with the exception of one patch of off-shelf transport during spring tide on days 50–56. South of the critical latitude zone (31°–37^°S), differences in the across-slope transports were banded like the differences in the along-slope transports (Fig. 8b). Like the along-slope transport differences, these across-slope differences are attributed to the billows propagating northward along the continental slope.

Fig. 8

a Time series of the elevation at 30° S 154° 11’E indicating the daily and spring-neap tidal cycles. b Difference in flow normal to the continental shelf/slope break at 200 m between the tidal and non-tidal simulations. c Same as b for water colder than 18 °C. Negative values indicate either an increase in a westward flow or a decrease in an eastward flow

Interactions of tides with eddies

Cyclonic and anti-cyclonic eddies ranging in size from 40 to ~ 300 km in diameter were present during the simulations both with and without tides (Table 1). Three eddies were chosen as examples for this study: a large, ~ 230 km, cyclonic, cold-core eddy present from before day 15 through day 60 (Fig. 4A), a ~ 170 km, cyclonic, cold-core eddy present at day 15, but diminished by day 45 (Fig. 4B), and a large, ~ 300 km, anti-cyclonic, warm-core eddy present from before day 15 through day 60 (Fig. 4C). A multitude of small eddies were generated in both simulations, more than can be discussed here. (See Additional file 1: Figs. S2–S15 for a fuller range of snapshots of surface velocities and temperatures, and of surface differences in temperature, velocities, and vorticity. Some eddies observed in this region have significant vertical tilt (Roughan et al. 2017); however, this did not occur in the model results for the eddies investigated, probably due to the model’s horizontal resolution.

Table 1 Eddy center locations, North–South distance traveled, eddy diameter, and rotational speed between days 15 and 60 for the three main eddies at 15, 45, and 60 days

Since vorticity is key near the critical latitude and eddies are vortical motions, the vorticity fields over the domain and the differences between the tidal and non-tidal simulations were determined, and are shown in Additional file 1: Figs S2–S15. The most striking difference between the tidal and non-tidal simulations was the increase in vorticity in the western part of the domain away from the boundaries and the EAC separation zone, basically east of 157^o E in the tidal simulation (Additional file 1: Figs. S2–S15). Without tides, this region had very little vorticity at 30 days and less at 60 days. Spectra of the vorticity differences between the tidal and non-tidal simulations indicated the differences appeared at the semidiurnal and diurnal tidal frequencies and at a half cycle per day (not shown). Thus, the vorticity differences were associated with tides and the 2 day period that is roughly equivalent to the rotational speeds of the billows with diameters of 25–50 km.

At day 30, the three eddies appear similar in size and location between the tidal and non-tidal simulations as seen in surface temperature and velocity fields (Fig. 4a, c). Eddy A is further south with tides by ~ 7 km and the other two eddies are slightly north with tides by ~ 3 km, but these differences are quite small. By 45 days, the higher velocities around Eddy B have disappeared, less than 0.50 m s⁻¹, although the temperature signal appears to remain (Fig. 4b, d). By 60 days, differences between the eddy fields from the tidal and non-tidal simulations emerged due to eddy propagation. Eddy A traveled further north with tides by ~ 90 km by day 60 (Table 1). Differences in eddy propagation for eddies B and C were very small, ~ 10 km. Furthermore, surface temperature differences developed between the tidal and non-tidal simulations between 30 and 60 days (Fig. 10a, b). At 30 days, the surface temperature differences were primarily confined to the EAC separation zone, the waters near the continent, and the boundaries (Fig. 9a), basically, where eddies were present. There were only a few small patches with differences of 1 °C or more at 30 days. By 60 days, surface temperature differences between the two simulations were spread throughout the domain and there were many large patches with differences exceeding 1 °C (Fig. 9b). Most of these large potential temperature differences were associated with eddies and can easily result from minor shifts in the eddy position.

Fig. 9

Surface differences in temperature at a 30 and b 60 days and vorticity at c 30 and d 60 days. Approximate eddy edge locations are indicated by yellow circles

The large, cold-core, cyclonic eddy (A) did not appreciably differ in size between the tidal and non-tidal simulations (Table 1 and Fig. 4) or in the depth of the eddy velocities in transects through the eddy (Fig. 10a, b, e and f). Note, the East Australian Undercurrent is apparent in the transects (red deep area in Fig. 10a–d.) In both simulations, eddy A moved over 1000 km north, and it moved ~ 90 km further north in the tidal simulation than in the non-tidal simulation by 60 days (Table 1 and Fig. 11). Tides did not affect the rotational velocity of the eddy; however, the eddy slowed its rotation with time from 1.2 to 1.0 m s⁻¹ between 15 and 60 days (Table 1).

Fig. 10

Vertical transects in longitude at 26^oS of the North–South velocities for the a non-tidal and b tidal simulations at 30 days and c and d at 60 days, respectively, and of the East–West for the e non-tidal and f tidal simulations at 30 days and g and h at 60 days, respectively

Fig. 11

Tracks of eddies A, B, and C, for the simulation without tides (magenta) and with tides (yellow) at 7.5 day intervals after 15 days. The location at day 15 is labeled for each of the eddies and indicated with a 15

The large, cold-core, cyclonic eddy (B), which was mentioned in “North-South and along-slope transports” Section, was similar for the tidal and non-tidal simulations from days 15 to 30 (Figs. 4a, c and 11). In both simulations, it traveled north into the EAC region and weakened to less than 0.5 m s⁻¹ by 45 days, although it appears to have a surface temperature signal at 60 days (Fig. 4 and Table 1). Tides did not affect the rotational speed in this eddy, but it decreased from 1.3 to 0.5 m s⁻¹ over time (Table 1). Tides slightly promoted propagation of the eddy northward, with the eddy ~ 10 km further north at 37.5 days with tides (Table 1).

The large, warm-core, anti-cyclonic eddy (C in Fig. 4) had the strongest rotational speed, ~ 1.8 m s⁻¹, and it was not tidally dependent. This also decreased over time from 1.8 to 1.3 m s⁻¹ by 60 days (Table 1) There were slight differences in eddy shape, and it shifted less southward by ~ 15 km with tides (Figs. 4 and 11), inducing large potential temperature differences occurred on the southern side of the eddy at 60 days (Fig. 9b). There were no appreciable differences in the vertical velocity structure at 30 days (Fig. 12a, b and e–f), but were in both the North–South and East–West velocities at 60 days due to the decreased southward propagation with tides (Fig. 12c, d, g, h, respectively).

Fig. 12

Vertical transects in longitude at 35^o15’S of the North–South velocities for the a non-tidal and b tidal simulations at 30 days and c and d at 60 days, respectively, and of the East–West velocities for the e non-tidal and f tidal simulations at 30 days and g and h at 60 days, respectively

South of 33^o S, a series of small eddies or billows, 20–40 km in diameter, formed along the continental shelf/slope break and propagated north at a rapid rate, 2–3 m s⁻¹ (Fig. 4). These eddies formed both with and without tides, but they propagated north faster by 0.2–0.4 m s⁻¹ with tides than without tides. They induced a net southward flow. Differences associated with these billows increased with time (Additional file 1: Fig. S16).

Tidal mixing

Tides are considered one of the major mixing mechanisms, particularly deeper in the water column away from influence of surface gravity waves in the upper ocean. Here, mixing is investigated though inspection of the vertical temperature diffusivities. Vertical diffusivities of momentum follow a similar pattern as those of temperature, so only the former are shown and discussed. The model uses the vertical shear in the horizontal velocities and the stratification to determine vertical diffusivities for both momentum and temperature according to Nakanishi-Niino [2009] vertical mixing parameterization scheme. Since diffusivities are always positive and generally have a large percentage of essentially background values, 60–90% depending on the location and mixing processes, a bimodal distribution results with the major peak roughly at the background value. Consequently, their mean or standard deviation do not reflect either the typical value or the fluctuations, respectively. Furthermore, interest in mixing is usually on bursts of high values, not single values, such as a maximum. Thus, the usual statistics based on a Gaussian or normal distribution are not well-suited for descriptions of diffusivities. Even the standard deviation is misleading as the bimodal distribution is one-sided. Furthermore, diffusivity standard deviations typically exceed mean values, often by 1–2 orders of magnitude, which is again misleading. Since the interest was mixing hotspots that exceed the background diffusivity, a significant diffusivity was defined as the average of the largest one-third of the diffusivity values, as done previously by Robertson and Dong [2019]. Investigation of the distribution of the diffusivity values from the simulations confirmed the bimodal nature with predominantly (~ 60–70%) background diffusivity values, 1 × 10⁻⁶ m² s⁻¹and a smaller number (~ 30–40%) of very high values resulting in small means and standard deviations exceeding the means by 1–2 orders of magnitude (not shown). Note, that if the significant diffusivity is used to estimate total mixing for the simulation over a time period, it should only be multiplied by one-third of the time period, since it is based on the highest third of the values.

Model diffusivity estimates

Model estimates of the significant vertical temperature diffusivity are shown along four longitudinal cuts at 27^o15ʹS, 30^o16ʹS, 33^o59ʹS, and 36^o9ʹS (Fig. 13). Significant vertical temperature diffusivities were elevated in the bottom 1000 m over much of the region (Fig. 13). High temperature diffusivities occurred throughout most of the water column over the continental slope and large peaks reaching high in the water column occurred over rough topography in the two northernmost transects (Fig. 13). Significant vertical diffusivities exceeded 10⁻² m² s⁻¹ in some locations. This is an extremely high value and an indication that the model overestimates diffusivities, which has been determined in other studies (Robertson and Dong 2019). Typical vertical temperature diffusivities in the ocean for mixing events are 10⁻⁴ to 10⁻³ m² s⁻¹, although values exceeding 10⁻² m² s⁻¹ have been observed in very active mixing environments (Koch-Larrouy et al 2015). Background diffusivities for the ocean are estimated at 10⁻⁶ to 10⁻⁵ m² s⁻¹ (Polzin et al. 1997; Kunze et al. 2006), although 10⁻⁶ m² s⁻¹ was used here both in the model parameters and as a threshold in analysis.

Fig. 13

Log of the significant vertical temperature diffusivity between the tidal and non-tidal simulations along four latitude transects: a 27° 15’ S off Brisbane, b 30° 16’ S off Coffs Harbour, c 33° 59’ S off Sydney, and a 36° 9’ S off Bateman’s Bay

Much of the diffusivity was due to mean currents and eddies, and not tides. To identify changes in the diffusivities associated with tides, the differences of the mean significant temperature diffusivities with and without tides were calculated, regardless of whether the difference in diffusivities originated from mean currents, eddies, internal tides, internal waves, or other sources (Fig. 14). In the color scale for Fig. 14, it should be noted that the positive values do not represent the magnitude of the difference, but rather its sign. Figure 14 shows the log of the difference of the significant temperature diffusivities, with a positive, blue and green colors, (negative, red and purple colors) exponent sign indicating an increase (decrease) with tides. Differences less than 10⁻⁵ m² s⁻¹ are in gray and the brighter colors, red or blue, indicate the largest differences. These differences are both positive and negative and are quite noisy, since small shifts in the location of a high diffusivity value can easily result in adjacent large positive and negative values. Tides increased the benthic mean vertical temperature diffusivity differences by 10⁻⁴ m² s⁻¹ over much of the domain, based on the mean of the differences (not shown). The maximum vertical temperature diffusivity differences were all positive (not shown). However, over the continental slope, tides generally diminished the vertical diffusivities by 10⁻⁵ to 10⁻² m² s⁻¹ (Fig. 14). This was attributed to the aforementioned retardation of the mean flow by tides with the increased friction decreasing both the flow and mixing. The two northernmost transects, which fall near the diurnal tidal critical latitudes, have larger mid-water column differences than transects poleward of the diurnal critical latitudes, reflecting tidal critical latitude effects. For these two transects, tides increased the significant vertical diffusivity by 10⁻³ m² s⁻¹ at ~ 155° E for the transect at 27^o15’S (Additional file 1: Fig. S15a) and at ~ 156^o15ʹ E for the transect at 30^o16ʹS (Fig. 15b). Both of these locations are over rough topography and the increase was attributed to shears from internal tides and diurnal critical latitude effects. The two transects poleward of the diurnal critical latitudes do not have increased vertical temperature diffusivities in the mid-water column, reflecting the lack of diurnal internal tides, since they are poleward of the diurnal critical latitudes.

Fig. 14

Log of the difference of the significant vertical temperature diffusivity between the tidal and non-tidal simulations along four latitude transects: a 27° 15’ S off Brisbane, b 30° 16’ S off Coffs Harbour, c 33° 59’ S off Sydney, and a 36° 9’ S off Bateman’s Bay

Fig. 15

Log of the estimate of the temperature diffusivity from the CTD/LADCP observations from the first pass by the mooring locations according to a shear formulation and b Thorpe displacement. The log of the c shear and d Thorpe estimates of the vertical temperature diffusivity for the third pass by the mooring locations. Log of the e mean and f significant vertical temperature diffusivity from the tidal simulation along the observational transect

Observational data

To determine if the model was overestimating the vertical temperature diffusivities, diffusivities were estimated from observational data. Using Conductivity, Temperature and Depth (CTD) data and Lowered Acoustic Doppler Current Profiler (LADCP) data kindly provided by Dr. Bernadette Sloyan of the Commonwealth Scientific and Industrial Research Organization (CSIRO) and available on the Australian Ocean Data Network (AODN), vertical diffusivities were calculated following the methodology developed by Kunze et al. (2006). Sloyan’s voyage basically made three transects past the mooring locations (yellow crosses in Fig. 1) as they deployed new instruments to replace earlier moorings. The diffusivities from the observational data are basically snapshots in time. Consequently, the transects are an amalgam of profiles from different phases of both the spring-neap and daily tidal cycles, along with the presence of eddies and shifts in the EAC position. Two estimates of the diffusivities were made: one from the density profiles based on Thorpe displacements and another based on velocity shear from the LADCP profiles. The two methods yielded roughly equivalent results for both transects past the mooring locations, although the Thorpe estimates were lower than those from shear (Fig. 15a–d). Both the mean and the significant vertical temperature diffusivities from the model (Fig. 15e, f) were higher than the observed estimates and the observations did not have any values exceeding 10⁻² m² s⁻¹, which the model estimates did (Fig. 15a–d). Although the observed estimates fell within 1 standard deviation of the mean estimate (not shown), the model probably overestimates the vertical diffusivities.

Discussion

Even though the tides off eastern Australia were small, there were clear tidal effects on the circulation and mixing and these effects were often latitude dependent. The most prominent effects were: (1) enhancement of the EAC-like current north of 27^o S, (2) enhancement of both the propagation of billows north and the southward net transport they induced, (3) increases in cross-slope and along-slope transports of deeper, colder water onto the continental shelf, (4) modification of the eddy rotation and propagation rates, (5) increased vorticity, and (6) increased vertical diffusivities. Several of these behaviors were latitude dependent, with three basic latitude zones: the diurnal critical latitude zone (27^o30ʹ–30° S), equatorward/north of this zone (23°–27^o S), and poleward/south of this zone (31°–37^o S). In this simulation, the EAC-like current separated from the coast in the critical latitude zone. The latitude independent effects could be attributed to barotropic tides and semidiurnal internal tides and the latitude dependent effects to diurnal internal tides, although the diurnal tides impact the semidiurnal tides and high frequencies near the diurnal critical latitude (Robertson et al. 2017).

Residual tidal current effects were latitude independent, but primarily due to topographic differences not the latitude. They contributed to the enhanced southward transport of the EAC-like current and the extension by the billows. Enhancement occurred whenever the residual tidal current exceeded the retardation of the flow by increased friction due to tides, typically over the continental shelf/slope break and slope. This is likely to also be a tidal effect in other western boundary currents, especially those along steep slopes, such as the Kuroshiro, and in eastern boundary currents, such as the California Current.

Tides and eddies are well-known to interact, (Takeoka and Murao 1993; Sutyrin et al. 2003; Callendar et al. 2011). The most common interaction is the generation of eddies through the interaction of tidal currents with topography. Eddies have been found to impact tides through vorticity (Takeoka and Murao 1993; Sutyrin et al. 2003; Callendar et al. 2011). Although not previously noted, it is not a stretch to for tides to impact eddies through vorticity. Since eddy propagation is governed by the β effect (Sutyrin et al. 2003), additional vorticity will impact their propagation as was observed in the model runs. Unfortunately, it is not possible to stop the tides and verify this with observations.

The other tidal effects were latitude dependent with different behavior between the three zones, primarily due to two factors: tidal-eddy interactions and internal tides. Internal tides and critical latitude effects clearly enhanced tidal mixing particularly near the diurnal critical latitudes, both planetary and effective. Not only do the diurnal tides become resonant in this region, but diurnal tides have been shown to transfer energy to the semidiurnal frequencies in this region, increasing the baroclinic response of the semidiurnal internal tides (Robertson et al. 2017). The result is enhanced vertical diffusivities and vertical tidal mixing, particularly over rough topography in this region. Comparisons with observational data indicate diffusivities were overestimated by the model. Tidal effects on across-shelf and along shelf transports at the continental shelf/slope break were also latitude dependent, with heightened transports poleward of the critical latitude zone due to a series of small billows located at the edge of the continental shelf. These transports were primarily of waters colder than 18 °C. Transport of colder, nutrient rich waters onto the continental shelf is of interest due to its potential to increase primary productivity. In the critical latitude zone, instances of increased on-shore transport of cold waters associated with tides occurred, but it was not linked to the spring-neap tidal cycle. The sporadic nature of these differences points to a link between the passage of an eddy and the tidal effects on eddy propagation.

Tides affected cyclonic eddy propagation at all latitudes increasing the speed of propagation, although the propagation direction varied with latitude. This is primarily true for the northern cyclonic eddy and the cyclonic billows. The anti-cyclonic eddy did not move appreciably either with or without tides. The cyclonic eddy and billows moved north. The exact mechanism is unknown, but is believed to be linked to vorticity.

Summary

Barotropic and baroclinic tides were simulated for the waters off eastern Australia and compared to a simulation without tides to evaluate the impact of tides on the circulation and mixing in the region. Even though the tides in the region were small, they had significant impacts on the region, affecting the mean currents, eddies, and mixing. The most noticeable effect was tides strengthened the mean southward current by 1–4 Sv, due to contributions of the residual tidal current overwhelming the retardation of the current by increased friction due to tides. In a region with a gentle continental slope, frictional effects were stronger and totally offset the residual tidal current, resulting in a reduction of the mean current. Eddies were generated both with and without tides. Tides did not noticeably change the generation, size or rotational speed of the eddies; however, they did slightly accelerate the northward propagation of cyclonic eddies. Despite the predominant sources of vertical mixing being mean currents and eddies, tides increased vertical mixing. Estimates of vertical temperature diffusivities changed by 10⁻⁴ to 10⁻² m² s⁻¹ in some places when tides were present. Over the continental slope, there was primarily a decrease due to the increased friction from tides. However, over rough topography, there was an increase and occurred near the diurnal critical latitudes (27–30°), but not poleward of their band. It should be noted, that when compared to observations, the model appeared to overestimate vertical mixing.

In the future, it is planned to include wind and solar radiation forcing and to put in flow through the eastern boundary to better represent the EAC. I would also like to investigate the influence of different stratification conditions though the annual seasonal cycle on baroclinic tidal effects. Since the mean currents also vary during the year, the interactions between mean currents and internal tides will be impacted. Finally, the complex vorticity interactions between the planetary vorticity and relative vorticity generated by horizontal current shears, eddies, tides, Rossby waves, winds, and other factors beg for further investigation for the interactions between tides and eddies.