On the influence of rough topography in barotropic tide models

Generated using the official AMS L A T EX template v6.1 On the influence of rough topography in barotropic tide models1 Callum J. Shakepearea 2 aResearch School of Earth Sciences, The Australian National University, Canberra, Australia3 Corresponding author: Callum J. Shakepeare, callum.shak[email protected]4 1 ABSTRACT: Barotropic tide models are single-layer shallow water models forced by an oscillatory body force at the tidal frequency. They are widely used in a variety of applications at increasingly high spatial resolutions to take advantage of new high-resolution bathymetric datasets. Here we investigate the impact of the small-scale roughness present in such bathymetry on the solutions obtained by these models. It is shown that rough topography induces a form stress that scales with the depth variance relative to the mean depth, ℎ′2/¯ ℎ2, and is out-of-phase with the tidal flow. As such, this force extracts no energy from the tide, but can significantly damp or amplify tidal elevations depending on the resonant properties of the basin. Due to these effects, barotropic tidal models only converge with increasing resolution once the variance of the topography is fully represented. These results have important implications for improving the accuracy of barotropic tide models. 5 6 7 8 9 10 11 12 13 14 15 2 1. Introduction16 Single-layer (barotropic) shallow water models are routinely used to represent the ocean surface17 tide, and depending on the application, may be preferred over more realistic multi-layer (baroclinic)18 models. Example applications include inverse modelling of the surface tide (e.g., Egbert and19 Erofeeva 2002), investigation of future tides associated with rising mean sea levels (e.g., Pickering20 et al. 2017; Schindelegger et al. 2018), and deep-time tidal evolution studies (e.g., Green et al. 2017;21 Daher et al. 2021). The first barotropic tide models (Pekeris and Accad 1969; Hendershott and Munk22 1970; Accad and Pekeris 1978) were run at relatively low horizontal resolutions of approximately23 2◦, commensurate with the computational power and limited bathymetric data available at that24 time. However, modern barotropic tide models are now regularly run at resolutions of 0.1◦or25 higher (e.g., Stammer et al. 2014) with model topographies constructed from bathymetric datasets26 with nominal resolutions of up to 15 arc seconds (or 1/240◦; e.g., GEBCO 2025). As such, these27 models explicitly resolve not only the ocean basins themselves (as in early models) but also a large28 portion of the small-scale roughness associated with the ridges, hills and seamounts that are widely29 distributed over the seafloor. Here we develop a theory to describe the impact of including this30 small-scale roughness in barotropic tide models.31 The accuracy of barotropic tide models has generally increased with higher horizontal resolution32 (e.g., Egbert et al. 2004; M¨uller 2011; Schindelegger et al. 2018; Huang et al. 2022; Wang et al. 2024;33 Yang et al. in rev.), although these improvements have now largely stalled. Tidal elevations, energy34 dissipation and elevation errors in forward models tend to converge with increasing resolution at35 around 1/12◦(Egbert et al. 2004; Schindelegger et al. 2018; Yang et al. in rev.), with a non-negligible36 error always present despite significant model tuning. The improvement with resolution is attributed37 to some combination of better represented ocean basin geometry, shoreline configuration, and rough38 bathymetry. The basin geometry is critical because the tides are a resonant phenomena, such that39 small changes in basin widths and shapes can have a significant impact on tidal amplitudes globally40 (e.g., Clarke 1991; Green 2010; Green et al. 2017). The shoreline configuration relates to the41 resolved coastline shape and the representation of important connections with semi-enclosed seas,42 such as the Hudson Strait (e.g., Arbic et al. 2007; Webb 2014). Such areas can act as resonators43 which amplify the open ocean tide, with impacts felt globally (Arbic et al. 2009; Wang et al. 2024).44 More accurate representation of shorelines and basin geometry through better horizontal model45 3 resolution therefore leads to more accurate tidal solutions. It is also argued that the inclusion of46 more accurate rough bathymetry at higher resolution improves barotropic tide model solutions47 (e.g., Egbert et al. 2004; Huang et al. 2022; Wang et al. 2024), but the evidence in this case is less48 clear. By ‘rough bathymetry’, here we mean topographic roughness away from coastlines and of49 scales much smaller than the basin, such that it does not substantially impact either basin shape50 or coastline configuration. Egbert et al. (2004) added randomised small-scale roughness to the51 topography used in their barotropic tide model in order to illustrate its significant impact on the52 solution. However, while the impact is clear, whether the inclusion of higher resolution rough53 bathymetry improves the barotropic solution is less so.54 It is well known that small-scale roughness significantly impacts the surface tide via baroclinic —55 as opposed to barotropic — effects. Indeed, the primary energy sink for the tide is via near-bottom56 turbulence driven by strong tidal flows on shallow continental shelves, often represented in models57 via a linear or quadratic bottom drag (e.g., Dong et al. 2023). In addition, there is a substantial58 body of work investigating the topographic stress associated with internal tide generation, although59 most studies have focused on the associated energy flux rather than the stress itself (e.g., Bell60 1975; Baines 1982; Llewellyn Smith and Young 2002; Khatiwala 2003; Nycander 2005, 2006;61 Shakespeare et al. 2020, 2021a,b). In radiating wave regimes (where tidal frequency 𝜔exceeds the62 local inertial frequency 𝑓) the oscillatory topographic stress is (at least partly) in-phase with the63 tidal flow and removes energy from the tide. However, in non-radiating regimes (𝜔 < 𝑓 ), or where64 wave reflections occur, the stress may have out-of-phase components which remove no energy from65 the tide, but nonetheless significantly modify its amplitude (Shakespeare et al. 2020, 2021b). In66 barotropic tide models these baroclinic wave dynamics are not directly resolved and (in forward67 ‘free-running’ models) are routinely parameterised via a ‘wave drag’, the coefficient of which68 is typically tuned to achieve a solution that best matches observations (e.g., Egbert et al. 2004;69 Green and Nycander 2013; Buijsman et al. 2015; Arbic et al. 2018). The wave drag is associated70 with topographic scales that produce internal tides, which equate to roughly ∼100-200 km (for71 mode-1 internal tides in the open ocean; e.g., Zhao et al. 2016) and smaller. However, topography72 at these scales is also being directly resolved in modern high-resolution barotropic tide models,73 where it will exhibit barotropic dynamics, despite its scale being such that we would expect (and74 indeed are parameterising) baroclinic wave dynamics. In this sense, modern barotropic tide models75 4 suffer from a resolution mismatch with high horizontal resolution capable of resolving internal76 tide-generating topography, but very low vertical resolution (i.e., 1 layer) incapable of representing77 the actual generation of waves. Permitting baroclinicity in a model (i.e., through the addition of78 a second vertical layer) improves the barotropic tide solution (Arbic et al. 2004), but the details79 of the dynamics at play remain uncertain. Therefore, here we seek to understand nature of the80 (unphysical) barotropic dynamics induced by sub-mode-1 topography in barotropic tide models81 and its influence on the tidal solution obtained.82 The paper is laid out as follows. In §2a we derive an expression for the drag coefficient associated83 with tidal flow over small-scale rough topography in a barotropic ocean and compare with previous84 expressions for the baroclinic stresses. In §2b we demonstrate the significant impact of this85 barotropic stress on the tide in situations of gradually increasing complexity. Lastly, in §3 we86 summarise and discuss the application of our results for improving the representation of tides in87 barotropic models.88 2. Results89 a. Theory90 As a simple model for barotropic tides, here we consider the linearised shallow-water equations91 in one spatial dimension (𝑥) but permitting flow in the orthogonal direction (𝑦):92 𝜕𝑈 𝜕𝑡 −𝑓 𝑉 =−𝑔ℎ 𝜕𝜂 𝜕𝑥 +ℎ 𝐹𝑥−𝜏𝑥,(1a) 𝜕𝑉 𝜕𝑡 +𝑓 𝑈 =−𝜏𝑦,(1b) 𝜕𝜂 𝜕𝑡 +𝜕𝑈 𝜕𝑥 =0.(1c) Here (𝑈,𝑉)=(𝑢ℎ, 𝑣ℎ)are the tidal volume fluxes in the (𝑥, 𝑦)Cartesian directions, 𝐹𝑥a tidal body93 force, 𝑓is the Coriolis parameter, 𝜂the free surface elevation, 𝑔the acceleration due to gravity, and94 ℎthe ocean depth. For completeness, we have also included a generic stress (𝜏𝑥, 𝜏𝑦), which will95 be used in the subsequent section. The linearisation assumption made in these equations is valid96 for 𝜂≪ℎ. The topographic stresses on the fluid, which are the focus of this work, correspond to97 the integration of the term −𝑔ℎ 𝜕𝜂 𝜕𝑥 over space; that is, they occur due to correlations of topographic98 5 Fig. 1. Schematic of a tidal channel defined by basin-scale topography ¯ ℎon which is superimposed small-scale roughness ℎ′(shown in grey). This roughness induces small-scale variations in free surface height 𝜂′, leading to mean stresses 𝜏=𝑔 ¯ ℎℎ′𝜕𝜂′ 𝜕𝑥 when integrated over the topography (𝑥0−𝑙 2< 𝑥 < 𝑥0+𝑙 2) which modify the basin-scale free surface height ¯𝜂. 103 104 105 106 and free surface height variations. For convenience, here we will assume a tidal body force 𝐹𝑥 99 that is constant in space (and periodic in time). While both the assumptions of 1D geometry and100 spatially-uniform 𝐹𝑥may be avoided, they simplify the analysis below and are sufficient for our101 purposes of elucidating the key dynamics of the problem.102 We now define 𝑈=ℜ[b 𝑈 𝑒−𝑖𝜔𝑡]with 𝑖=√−1 and similarly for the other variables. We also define107 b𝜏𝑥=𝑟𝑥b 𝑈and b𝜏𝑦=𝑟𝑦b 𝑉, where 𝑟𝑥and 𝑟𝑦are the (potentially complex valued) linear drag coefficients108 in each spatial direction. With these definitions, (1) may be reduced to109 𝑔ℎ𝑖𝜔 −𝑟𝑦 𝑖𝜔 𝜕2b 𝑈 𝜕𝑥2+((𝑖𝜔 −𝑟𝑦)(𝑟𝑥−𝑖𝜔)− 𝑓2)b 𝑈=(𝑖𝜔 −𝑟𝑦)ℎb 𝐹𝑥,(2) with b𝜂then derivable from b 𝑈via110 b𝜂= 1 𝑖𝜔 𝜕b 𝑈 𝜕𝑥 .(3) While it is trivial to solve (2) numerically (see §2b), here we instead wish to seek an expression for111 the stresses induced by small-scale rough topography and evaluate their impact on the basin-scale112 flow. To ease the derivation, in this section we will set 𝑟𝑥=𝑟𝑦=0 such that (2) simplifies to113 𝑔ℎ 𝜔2−𝑓2 𝜕2b 𝑈 𝜕𝑥2+b 𝑈= 𝑖𝜔 𝜔2−𝑓2ℎb 𝐹𝑥.(4) 6 As illustrated in Fig. 1, we consider a domain of zonal width 𝐿𝑥, defined by a basin-scale (mean)114 topography which we denote ¯ ℎ. Superimposed on this mean we consider topographic variations115 ℎ′of much smaller scale, such that the total ocean depth is ℎ=¯ ℎ+ℎ′. Formally we can define116 the overbar operator as a spatial average over widths 𝑙that are large compared to the small-scale117 variations in topography,118 ¯ ℎ(𝑥0)=∫𝑥0+𝑙/2 𝑥0−𝑙/2 ℎ(𝑥)𝑑𝑥. (5) The small-scale topography will induce variations in fluxes (and free surface elevation) which may119 be similarly defined; e.g., 𝑈=¯ 𝑈+𝑈′. Substituting these decompositions into (4) yields120 𝑔 𝜔2−𝑓2 ¯ ℎ𝜕2b¯ 𝑈 𝜕𝑥2+ℎ′𝜕2c 𝑈′ 𝜕𝑥2+¯ ℎ𝜕2b 𝑈′ 𝜕𝑥2+ℎ′𝜕2b¯ 𝑈 𝜕𝑥2!+b¯ 𝑈+c 𝑈′= 𝑖𝜔 b 𝐹𝑥 𝜔2−𝑓2(¯ ℎ+ℎ′),(6) which, after applying the spatial average ( ¯ ) becomes121 𝑔 𝜔2−𝑓2©« ¯ ℎ𝜕2b¯ 𝑈 𝜕𝑥2+ℎ′𝜕2c 𝑈′ 𝜕𝑥2 | {z } ∗ ª®®®®¬+b¯ 𝑈= 𝑖𝜔 b 𝐹𝑥 𝜔2−𝑓2¯ ℎ. (7) This equation (7) governs the evolution of the basin-scale flow, which is influenced by the small-122 scale topography through the topographic stress (the * term) associated with those scales,123 b𝜏= 𝑔 ¯ ℎℎ′𝜕b 𝜂′ 𝜕𝑥 = 𝑔 𝑖𝜔 ¯ ℎℎ′𝜕2c 𝑈′ 𝜕𝑥2.(8) A general formulation for this stress can be determined by finding an expression for the small-scale124 flux c 𝑈′in terms of the large-scale flux b¯ 𝑈(rather than the forcing b 𝐹𝑥). At this point we need to125 introduce the assumption that the fine scale topographic variations are small in amplitude relative126 to the mean depth, ℎ′≪¯ ℎ. We can then combine (6) and (7) to eliminate b 𝐹𝑥, and drop any terms127 of second or higher order in perturbation quantities. This leaves128 𝜆0 2𝜋2𝜕2c 𝑈′ 𝜕𝑥2+c 𝑈′=b¯ 𝑈ℎ′ ¯ ℎ,(9) 7 where 𝜆0=2𝜋√︁𝑔¯ ℎ/(𝜔2−𝑓2)may be recognised as the wavelength of the barotropic tide, which129 has typical scales of ∼5000 km in the open ocean. As long as the small topographic scales are much130 smaller than 𝜆0, then the second term in (9) is much smaller than the first and may be neglected;131 thus132 𝜕2c 𝑈′ 𝜕𝑥2≃𝜔2−𝑓2 𝑔¯ ℎ2ℎ′b¯ 𝑈. (10) This expression (10) may be substituted into the equation for the stress (8) to give b𝜏=𝜎b¯𝑢=𝜎b¯ 𝑈/¯ ℎ,133 where134 𝜎=−𝑖𝜔2−𝑓2 𝜔 ℎ′2 ¯ ℎ2,(11) is known as the drag coefficient. Therefore, in a barotropic flow the stress due to small-scale135 topography depends only on its variance relative to the mean depth, ℎ′2/¯ ℎ2, but not on the shape136 or scale of that variation. A broad mid-ocean ridge and a sequence of abyssal hills will have the137 same impact on the basin-scale flow so long as their root-mean-square height √︁ℎ′2is the same.138 We further observe that 𝜎in (11) is purely imaginary. This result implies that the topographic139 stress 𝜏is 90◦out-of-phase with tidal flux ¯ 𝑈and therefore extracts no energy from the tidal flow140 in the time mean. In fact, this result follows directly from (1), which says that 𝑈differs by a time141 derivative from 𝜂(1c) — implying it is 90◦out of phase — and therefore the topographic stress142 −𝑔ℎ 𝜕𝜂 𝜕𝑥 term in (1a), which is in phase with 𝜂, does zero work. That is, averaging 𝑊=−𝑔ℎ 𝜕𝜂 𝜕𝑥 𝑈143 over a tidal period yields identically zero. Similar imaginary drag coefficients have previously been144 derived by Shakespeare et al. (2020) in the context of internal tide reflections (in baroclinic radiating145 wave regimes) and evanescent internal waves (in baroclinic non-radiating regimes). Regardless of146 its phasing, the presence of this topographic stress is critical to setting the tidal amplitude in the147 presence of small-scale roughness, as we show in the next section.148 b. Solutions in a tidal channel149 In this section we demonstrate the importance of the out-of-phase barotropic stress associated150 with small-scale topography via explicit solutions in a tidal channel (as drawn in Fig. 1), which can151 be thought of as a zonal transect of an ocean basin. The governing equation (2) is solved subject152 to boundary conditions of b 𝑈=0 at the edges of the channel, 𝑥=0 and 𝑥=𝐿𝑥. This equation can153 either be solved using the full topography ℎwith purely real drag coefficients 𝑟𝑥=𝑟𝑦=𝛼to obtain154 8 the full velocity b 𝑈, or else with ℎ=¯ ℎ,𝑟𝑦=𝛼, and 𝑟𝑥=𝛼+𝜎to obtain the basin-scale flow b¯ 𝑈with155 small-scales parameterised through the imaginary drag coefficient 𝜎as per (11)156 Before proceeding with numerical solutions for more complex cases (where ℎand 𝜎vary in157 space), we first consider the simplest case of a uniform depth basin (¯ ℎ=const.) with uniform158 (parameterised) small-scale roughness (ℎ′2=const.) where an analytic solution is available. In this159 case, (2) becomes160 𝑔¯ ℎ𝑖𝜔 −𝛼 𝑖𝜔 𝜕2b¯ 𝑈 𝜕𝑥2+((𝑖𝜔 −𝛼)(𝛼+𝜎−𝑖𝜔)− 𝑓2)b¯ 𝑈=(𝑖𝜔 −𝛼)¯ ℎb 𝐹𝑥,(12) which may be solved using a Fourier series approach, the details of which are given in the Appendix.161 Here we simply quote the result for 𝛼=0:162 b𝜂= 4b 𝐹𝑥¯ ℎ 𝐿𝑥(𝜔2−𝑓2) ∞ ∑︁ 𝑛=1"1−(2𝑛−1)𝜆0 2𝐿𝑥2 +ℎ′2 ¯ ℎ2#−1 cos (2𝑛−1)𝜋 𝐿𝑥 𝑥.(13) The maximum (𝑥=0) tidal amplitude of this solution is illustrated in Fig. 2a, along with solutions163 for cases with 𝛼 > 0 (see Appendix). The solution exhibits resonance when the width 𝐿𝑥is close to164 an odd multiple of half-wavelengths of the barotropic tide, 𝜆0, with the strength of these resonances165 modulated by the (real) drag 𝛼. The effect of adding small-scale roughness is to shift the resonances166 to slightly smaller basin widths 𝐿𝑥. Thus, as shown in Fig. 2b, the addition of roughness increases167 the tidal amplitude when the basin width is less than the resonant value (𝐿𝑥≲𝜆0/2) and reduces the168 tidal amplitude when the basin width is greater than the resonant value (𝐿𝑥≳𝜆0/2). This pattern169 repeats for subsequent resonances at larger multiples of the barotropic half-wavelength. The impact170 of the small-scale roughness is significant, especially at near-resonant conditions; however, even171 well away from resonance (e.g., 2𝐿𝑥/𝜆0=1.5) a roughness of 30% of the mean depth causes a172 20% change in the tidal amplitude.173 We now turn our attention to a more realistic case where both the largeand small-scale to-180 pographies are spatially varying, and a numerical solution to (2) is required. The intent of181 this example is to demonstrate that the theoretical description of the stresses induced by rough182 topography presented above — and the associated resonant dynamics — is accurate, even183 in the case of more realistic and complex topography. Fig. 3a displays the three topogra-184 9 offset to account for the unphysical (but directly resolved) barotropic stresses studied in this work.291 This tuning approach will be tested in a future work.292 In summary, the present work has quantified the stresses induced by small-scale topography in293 barotropic tide models. While the theory was limited to 1D channel geometry for convenience, it294 can be readily extended to 2D basin geometry if desired; the fundamental nature of the barotropic295 topographic stresses and resonant behaviour will not change. A remaining theoretical question296 requiring further investigation is precisely how the barotropic stresses described here transition297 to the baroclinic stresses described in previous studies as lengthscales reduce and/or stratification298 increases.299 16 Acknowledgments. The author acknowledges funding from the Australian Research Council Dis-300 covery Project scheme (DP230101836).301 Data availability statement. No new data was created by this work. The topographic dataset used302 herein is publicly available (GEBCO 2025).303 APPENDIX304 Here we give further details of the Fourier series solution for a tidal channel of constant mean305 depth ¯ ℎand uniform small-scale roughness parameterised by a constant 𝜎, the governing equation306 for which is (12). The solution proceeds through a sine decomposition for tidal flux b¯ 𝑈:307 b¯ 𝑈= ∞ ∑︁ 𝑛=1b¯ 𝑈𝑛sin 𝑛𝜋 𝐿𝑥 𝑥,(A1) to satisfy the boundary conditions of zero flow at 𝑥=0 and 𝑥=𝐿𝑥. In addition, the right hand side308 (forcing b 𝐹𝑥) of (12) may be decomposed using309 1=∑︁ 𝑜𝑑𝑑 𝑛 4 𝑛𝜋 sin 𝑛𝜋 𝐿𝑥 𝑥.(A2) Substituting these expressions (A1, A2) into (12) leads to an equation for the velocity amplitude310 of each horizontal mode b¯ 𝑈𝑛,311 −𝑔¯ ℎ(𝑖𝜔 −𝛼) 𝑖𝜔 𝑛2𝜋2 𝐿2 𝑥+𝜔2−𝑓2+2𝛼𝑖𝜔 +𝜎(𝑖𝜔 −𝛼)−𝛼2b¯ 𝑈𝑛=(𝑖𝜔 −𝛼)¯ ℎb 𝐹𝑥 4 𝑛𝜋 ,(A3) for odd 𝑛, and b¯ 𝑈𝑛=0 zero otherwise. Substituting the expression for 𝜎(11) into (A3) and312 rearranging yields313 b¯ 𝑈𝑛= 𝑖𝜔4b 𝐹𝑥¯ ℎ 𝑛𝜋(𝜔2−𝑓2)1+𝑖𝛼 𝜔"1−𝑛𝜆0 2𝐿𝑥2 +ℎ′2 ¯ ℎ2−𝛼2 𝜔2−𝑓2+𝑖𝛼 𝜔 −𝑛𝜆0 2𝐿𝑥2 +ℎ′2 ¯ ℎ2+2𝜔2 𝜔2−𝑓2!#−1 (A4) 17 for odd 𝑛, where 𝜆0=2𝜋√︁𝑔¯ ℎ/(𝜔2−𝑓2)is the wavelength of the barotropic tide. To obtain an314 expression for free surface elevation b¯𝜂we substitute (A1) into (3) to find315 b¯𝜂= ∞ ∑︁ 𝑛=1 𝑛𝜋 𝑖𝜔𝐿𝑥b¯ 𝑈𝑛cos 𝑛𝜋 𝐿𝑥 𝑥.(A5) The solution for the free surface height is thus:316 b¯𝜂= 4b 𝐹𝑥¯ ℎ 𝐿𝑥(𝜔2−𝑓2)1+𝑖𝛼 𝜔∞ ∑︁ 𝑛=1"1−(2𝑛−1)𝜆0 2𝐿𝑥2 +ℎ′2 ¯ ℎ2−𝛼2 𝜔2−𝑓2 +𝑖𝛼 𝜔 −(2𝑛−1)𝜆0 2𝐿𝑥2 +ℎ′2 ¯ ℎ2+2𝜔2 𝜔2−𝑓2!#−1 cos (2𝑛−1)𝜋 𝐿𝑥 𝑥.(A6) This is the equation used to make the plots in Fig. 2. The 𝛼=0 limit of (A6) is presented in the317 main text as (13).318 References319 Accad, Y., and C. L. Pekeris, 1978: Solution of the tidal equations for the m2 and s2 tides in the320 world oceans from a knowledge of the tidal potential alone. Philosophical Transactions of the321 Royal Society of London. Series A, Mathematical and Physical Sciences,290 (1368), 235–266.322 Arbic, B. K., S. T. Garner, R. W. Hallberg, and H. L. Simmons, 2004: The accuracy of surface323 elevations in forward global barotropic and baroclinic tide models. Deep-Sea Res.,51 (25-26),324 3069–3101.325 Arbic, B. K., R. H. 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