Prospects for improving the representation of coastal and shelf seas in global ocean models

Abstract

Accurately representing coastal and shelf seas in global ocean models represents one of the grand challenges of Earth system science. They are regions of immense societal importance through the goods and services they provide, hazards they pose and their role in global-scale processes and cycles, e.g. carbon fluxes and dense water formation. However, they are poorly represented in the current generation of global ocean models. In this contribution, we aim to briefly characterise the problem, and then to identify the important physical processes, and their scales, needed to address this issue in the context of the options available to resolve these scales globally and the evolving computational landscape.We find barotropic and topographic scales are well resolved by the current state-of-the-art model resolutions, e.g. nominal 1/12[deg], and still reasonably well resolved at 1/4[deg]; here, the focus is on process representation. We identify tides, vertical coordinates, river inflows and mixing schemes as four areas where modelling approaches can readily be transferred from regional to global modelling with substantial benefit. In terms of finer-scale processes, we find that a 1/12[deg] global model resolves the first baroclinic Rossby radius for only ∼ 8% of regions < 500m deep, but this increases to ∼ 70% for a 1/72[deg] model, so resolving scales globally requires substantially finer resolution than the current state of the art.We quantify the benefit of improved resolution and process representation using 1/12[deg] global- and basin-scale northern North Atlantic nucleus for a European model of the ocean (NEMO) simulations; the latter includes tides and a k-[straight epsilon] vertical mixing scheme. These are compared with global stratification observations and 19 models from CMIP5. In terms of correlation and basin-wide rms error, the high-resolution models outperform all these CMIP5 models. The model with tides shows improved seasonal cycles compared to the high-resolution model without tides. The benefits of resolution are particularly apparent in eastern boundary upwelling zones.To explore the balance between the size of a globally refined model and that of multiscale modelling options (e.g. finite element, finite volume or a two-way nesting approach), we consider a simple scale analysis and a conceptual grid refining approach. We put this analysis in the context of evolving computer systems, discussing model turnaround time, scalability and resource costs. Using a simple cost model compared to a reference configuration (taken to be a 1/4[deg] global model in 2011) and the increasing performance of the UK Research Councils' computer facility, we estimate an unstructured mesh multiscale approach, resolving process scales down to 1.5km, would use a comparable share of the computer resource by 2021, the two-way nested multiscale approach by 2022, and a 1/72[deg] global model by 2026. However, we also note that a 1/12[deg] global model would not have a comparable computational cost to a 1[deg] global model in 2017 until 2027. Hence, we conclude that for computationally expensive models (e.g. for oceanographic research or operational oceanography), resolving scales to ∼ 1.5km would be routinely practical in about a decade given substantial effort on numerical and computational development. For complex Earth system models, this extends to about 2 decades, suggesting the focus here needs to be on improved process parameterisation to meet these challenges.

Geosci. Model Dev., 10, 499523, 2017 www.geosci-model-dev.net/10/499/2017/ doi:10.5194/gmd-10-499-2017 Author(s) 2017. CC Attribution 3.0 License.

Jason Holt1, Patrick Hyder2, Mike Ashworth3, James Harle1, Helene T. Hewitt2, Hedong Liu1, Adrian L. New4, Stephen Pickles3, Andrew Porter3, Ekaterina Popova4, J. Icarus Allen5, John Siddorn2, and Richard Wood2

1National Oceanography Centre, 6 Brownlow Street, Liverpool, L3 5DA, UK

2Met Ofce Hadley Centre, FitzRoy Rd, Exeter, EX1 3PB, UK

3STFC Daresbury Laboratory, Sci-Tech Daresbury, Warrington, WA4 4AD, UK

4National Oceanography Centre, European Way, Southampton, SO14 3ZH, UK

5Plymouth Marine Laboratory, Prospect Place, Plymouth, PL1 3DH, UK

Correspondence to: Jason Holt ([email protected])

Received: 7 June 2016 Published in Geosci. Model Dev. Discuss.: 6 July 2016 Revised: 8 December 2016 Accepted: 27 December 2016 Published: 1 February 2017

Abstract. Accurately representing coastal and shelf seas in global ocean models represents one of the grand challenges of Earth system science. They are regions of immense societal importance through the goods and services they provide, hazards they pose and their role in global-scale processes and cycles, e.g. carbon uxes and dense water formation. However, they are poorly represented in the current generation of global ocean models. In this contribution, we aim to briey characterise the problem, and then to identify the important physical processes, and their scales, needed to address this issue in the context of the options available to resolve these scales globally and the evolving computational landscape.

We nd barotropic and topographic scales are well resolved by the current state-of-the-art model resolutions, e.g. nominal 1/12 , and still reasonably well resolved at 1/4 ;

here, the focus is on process representation. We identify tides, vertical coordinates, river inows and mixing schemes as four areas where modelling approaches can readily be transferred from regional to global modelling with substantial benet. In terms of ner-scale processes, we nd that a 1/12 global model resolves the rst baroclinic Rossby radius for only 8 % of regions < 500 m deep, but this in

creases to 70 % for a 1/72 model, so resolving scales

globally requires substantially ner resolution than the current state of the art.

We quantify the benet of improved resolution and process representation using 1/12 global- and basin-scale northern

North Atlantic nucleus for a European model of the ocean

Prospects for improving the representation of coastal and shelf seas in global ocean models

(NEMO) simulations; the latter includes tides and a k-" vertical mixing scheme. These are compared with global stratication observations and 19 models from CMIP5. In terms of correlation and basin-wide rms error, the high-resolution models outperform all these CMIP5 models. The model with tides shows improved seasonal cycles compared to the high-resolution model without tides. The benets of resolution are particularly apparent in eastern boundary upwelling zones.

To explore the balance between the size of a globally rened model and that of multiscale modelling options (e.g. nite element, nite volume or a two-way nesting approach), we consider a simple scale analysis and a conceptual grid rening approach. We put this analysis in the context of evolving computer systems, discussing model turnaround time, scalability and resource costs. Using a simple cost model compared to a reference conguration (taken to be a 1/4

global model in 2011) and the increasing performance of the UK Research Councils computer facility, we estimate an unstructured mesh multiscale approach, resolving process scales down to 1.5 km, would use a comparable share of the computer resource by 2021, the two-way nested multiscale approach by 2022, and a 1/72 global model by 2026. However, we also note that a 1/12 global model would not have a comparable computational cost to a 1 global model in 2017 until 2027. Hence, we conclude that for computationally expensive models (e.g. for oceanographic research or operational oceanography), resolving scales to 1.5 km would be

routinely practical in about a decade given substantial effort

Published by Copernicus Publications on behalf of the European Geosciences Union.

500 J. Holt et al.: Prospects for improving the representation of coastal and shelf seas

on numerical and computational development. For complex Earth system models, this extends to about 2 decades, suggesting the focus here needs to be on improved process parameterisation to meet these challenges.

1 Introduction

Improving the representation of coastal and shelf seas in global models is one of the grand challenges in ocean modelling and Earth system science. Global ocean models often have poor representation of coastal and shelf seas (Renner et al., 2009; Holt et al., 2010), further quantied below, due to both their coarse resolution and their lack of coastal ocean process representation. See Grifes and Treguier (2013) for a recent review of the state of the art in global ocean modelling.In this paper, we aim to identify the relevant physical processes, quantify the horizontal scales needed to resolve these processes and explore the approaches that could be employed to realise an improvement. In particular, we compare the relative merits of a continued renement of quasi-uniform structured grids with multiscale approaches, which would allow increased resolution where required. The multiscale approach could, for example, use unstructured meshes or multiple two-way nested grids. There have been other previous explorations of the scales important in shelf sea models (Greenberg et al., 2007; Legrand et al., 2007). These have tended to focus on specic numerical methods and approaches, largely around triangular unstructured meshes. Here, we step back from a detailed analysis of the numerics and consider, in general terms, what is likely to be practical to achieve improved coastal and shelf sea modelling on a global scale, on what timescales and what the ways forward may be. We primarily draw on experience with the nucleus for a European model of the ocean (NEMO; Madec, 2008) to provide a specic context, but expect the conclusions drawn to be generic.

The remainder of this section describes the background and motivation. Coastal ocean processes and scales and their relation to global quasi-uniform model grids are described in Sect. 2. Section 3 considers modelling approaches that might address coastal ocean process representation and resolution, drawing on the CMIP5 coupled oceanatmosphere climate models (Taylor et al., 2012) and two 1/12 NEMO congu-rations in comparison with EN4 prole observations (Good et al., 2013) to provide quantitative examples. These considerations are related to changing computer architectures and issues of model performance in Sect. 4, to estimate when they may be practical. The paper ends with conclusions in Sect. 5.

Background and motivation

Coastal and shelf seas represent a small fraction of the area of the global ocean (9.7 % of the global ocean is < 500 m deep and 7.6 % < 200 m) but have a disproportionately large

impact on many aspects of the marine environment and human activities. While our focus here is on modelling physical ocean processes, facets of marine biogeochemistry and ecosystems, and the climate system often provide the underlying motivation. These seas are the most highly productive regions of the world ocean, providing a diverse range of resources (e.g. food, renewable energy, transport) and services (e.g. carbon and nutrient cycling and biodiversity), and also expose human activity to hazards such as ooding and coastal erosion.

The geography of these seas is very varied, including semi-enclosed seas, broad open shelves, narrow shelves exposed to the open ocean and coastal seas behind barrier islands. Rather than adopt a typological approach (e.g. Liu et al., 2010), we focus on generic physical processes described by some straightforward spatially varying properties, as is appropriate for the global case; regional model studies would go beyond this to consider the detailed conditions specic to the region and tailor the model accordingly. While many of the largest shelf seas are in polar regions, we limit our investigation here to liquid water modelling and leave considerations of sea-ice modelling in this context to further work.

The study of coastal and shelf seas in a global context involves both upscaling (small scales inuencing large) and downscaling (large scales inuencing small) considerations, alongside the internal dynamics. Both dynamics and biogeochemistry provide motivations to studying the inuence of coastal ocean processes on a global scale. A particularly important dynamical feature is the formation of dense water on Arctic and Antarctic (Orsi, 2010; Orsi et al., 1999) shelves and its subsequent downslope transport and mixing to form deep water masses through the cascading process, thereby contributing to the global thermohaline circulation.Similarly, coastal upwelling is both an important control of airsea heat ux with implications for regional climate (e.g. in the southeast Pacic; Lin, 2007) and a key process in global marine ecosystems. The coastal ocean plays a key role in global biogeochemical cycles, for example, through the drawdown of carbon in highly productive shelf seas and its transport either to on-shelf sediments or off-shelf to the deep ocean, where it is isolated from atmospheric exchange (Bauer et al., 2013; Chen and Borges, 2009). Shelf seas are also a source of potent greenhouse gases, such as nitrous oxide (Seitzinger and Kroeze, 1998) and methane release from hydrates (Shakhova et al., 2010). The coastal ocean is the rst point of entry for all material of terrestrial origin entering the marine environment, for example, freshwater from rivers and ice sheets/shelves, inorganic nutrients, organic material and anthropogenic pollutants and this material can be substantially modied as it is transported across the coastal ocean (Barrn and Duarte, 2015). Hence, the coastal and open ocean biogeochemical cycles are intimately coupled.There is still substantial uncertainty in their role and feedbacks with the wider climate system, and making progress on this is largely dependent on the accurate simulation of the

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J. Holt et al.: Prospects for improving the representation of coastal and shelf seas 501

physical environment in the coupled coastalopen ocean system.

Investigating the large-scale impacts on smaller-scale processes in the coastal ocean can often be successfully treated by (one-way or two-way) nested regional studies, focusing on an area of interest ranging from local (e.g. Zhang et al., 2009) to regional (e.g. Wakelin et al., 2009) to basin (e.g. Holt et al., 2014; Curchitser et al., 2005) scale. There are, however, occasions where a global or quasi-global approach is appropriate. These relate to cases where it is important to consider impacts on human systems of global relevance. Examples include global food security and the role of living marine resources in ensuring this (Merino et al., 2012; Barange et al., 2014), and quantifying global vulnerability to sea level rise and coastal ooding (e.g. Nicholls, 2004). Moreover, cases where basin-scale oceanic processes directly inuence the coastal ocean are best considered on a global scale (Popova et al., 2016), as regional simulations may be compromised by errors propagating from simplied boundary condition approaches (see below). Coastal up-welling (Popova et al., 2016; Hobday and Pecl, 2014) and impacts of changes in western boundary currents (Wu et al., 2012) are notable examples.

While regional or local models often provide the optimal solution for many coastal ocean questions there is a signicant overhead in their deployment. A global model with improved representation of the coastal ocean opens up the opportunity to provide rapid and cost-effective information in a particular region for either scientic or operational use, without needing to congure a new domain. A particular example here is the European Copernicus Service (http://marine.copernicus.eu

Web End =ma http://marine.copernicus.eu

Web End =rine.copernicus.eu ). In this multi-million Euro investment, operational forecast and reanalysis products are provided to a range of users, from bespoke models of several European regions. If the global model in this service had improved coastal ocean representation, then a similar, but not optimal, range of information could be provided for a much wider range applications around the world, notably where this level of investment is not available.

Hence, we dene the context of the present study to be the improvement of the representation of the coastal oceans in four classes/uses of global ocean models: (i) global climate models, (ii) global Earth systems models, (iii) global models used as a resource for regional scientic studies and(iv) global models providing regional operational information.

2 Coastal ocean processes and scales

The distinct physical characteristics of the coastal ocean, in comparison to other oceanic regions, are largely determined by their shallow depth and proximity to land. This has several implications for the dynamics:

The water depth is generally similar to or not much greater than the surface or seabed boundary layers, so turbulence and mixing are invariably important.

Extreme variations in topography (compared with the water depth) are a dening feature.

Incident waves grow in amplitude in shoaling water to conserve energy ux, so, for example, these can be regions of large tides.

The inertia (thermal and mechanical) of shelf seas is small, so they are highly constrained by external forcing.

The horizontal length scales of the dominant physical processes decrease with decreasing depth (see below) and so are generally much smaller than in the deep ocean.

Rivers and glacial melt provide a source of buoyant fresher water that forms coastal currents and impacts stratication and mixing near the coast.

In polar regions, land provides both a point of attachment (land fast ice) and a source of divergence (polynyas) for sea ice.

Alongside these internal dynamics, coastalopen ocean coupling is of critical importance to the considerations here. At ocean margins, currents tend to follow contours of the Coriolis parameter divided by water depth (f/h), and so coastal regions are largely isolated from the large-scale geostrophic circulation. Physical processes at the shelf break mediate the transfer of material across this barrier (Huthnance, 1995), e.g. the Ekman drain within the bottom boundary layer, eddies and internal waves; these tend to be of ne scale and high frequency.

While there are numerous physical processes at work in shelf and coastal seas, the underlying principles and equations are the same as in the open ocean, and many features noted above are represented in the current generation of global models. Their relative importance and scale differ signicantly in the two cases, and so does how they are treated in numerical models. The processes are reviewed by Robinson and Brink (1998) and Huthnance (1995), so we do not discuss the dynamics in any detail here; we are primarily concerned with their characteristic horizontal scales (Table 1).

Ocean tides make a substantial contribution to the mixing and transport in most coastal ocean regions. For example, the mean M2 semi-major axis tidal current speed is0.29 m s1 for water shallower than 500 m, compared with a global mean of 0.06 m s1 (based on data from the TPXO inverse tidal model; Egbert and Erofeeva, 2002). Only 8 % of the area of these shallow regions have tides < 0.12 m s1(i.e. weak tides). The barotropic tide propagates on-shelf as a

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502 J. Holt et al.: Prospects for improving the representation of coastal and shelf seas

Table 1. Physical process horizontal scales in coastal and shelf seas.

Process Horizontal scale Reference

Barotropic tide Lbt pgH/max(f,!) Huthnance (1995)

Tidal excursion Le UT /! Polton (2014) Topographic steered barotropic current LT H [notdef] (rH)

1 Greenberg et al. (2007)

Front/frontal jet, coastal upwelling L1 Ciw/f Huthnance (1995)

Baroclinic eddy LE 2L1 Pedlosky (1987) Internal wave/tide Liw Ciw/! Huthnance (1995) Coastal current/river plume Lr (2Qf/g[prime])1/2 (rH)

1 Yankovsky and Chapman (1997), Avicola and Huq (2002)

Here, UT is tidal current, ! is frequency, H is water depth, g is gravitational acceleration, f is the Coriolis parameter, Ciw is rst-mode internal wave phase speed and Q is riverine volume ux.

(2Qg[prime])0.25/f 0.75

coastal trapped wave (CTW), amplifying and transferring energy to higher harmonics as the water depth shoals; the scale (Lbt) is characterised by either their wavelength or Kelvin wave scale. The (substantially ner) scale of rectication of tidal currents around topography and the periodic mixing and stratication at fronts is set by the tidal excursion (Le; e.g.

Polton, 2014). Topographic steering of currents is a characteristic feature of shelf seas and ocean margins (e.g. the Dooley Current in the North Sea), with a barotropic scale of the water depth divided by the slope (LT) (Greenberg et al., 2007). Other topographic scales, such as the size of individual features, will be locally relevant but are not considered here.

The annual stratication cycle is a key feature of many shelf seas that are shallower than the winter ambient open ocean mixed-layer depth. This is well described by a balance between surface heating and mixing (Simpson and Hunter, 1974) and the general spatial pattern is then set by the propagation of tides across the shelf and the topography (i.e. the barotropic scales; Lbt, LT). Mixed and seasonally stratied waters are bounded by sharp mixing fronts. These provide effective barriers to lateral transport and drive baroclinic frontal jets (Hill et al., 2008) at a scale characterised by the rst baroclinic Rossby radius (L1; Table 1). While mesoscale eddies are present in shelf seas (e.g. Badin et al., 2009), their importance in dynamics and transport on shelf is much less clear than in the open ocean (Hecht and Smith, 2008) or for oceanshelf transport (e.g. Zhang and Gawarkiewicz, 2015). Coastal upwelling, and consequent frontal jets and laments (Peliz et al., 2002) also scale with the Rossby radius.

Tidal ow over topography in a density-stratied ocean excites internal waves at tidal frequencies (Baines, 1982), and their role in mixing at the shelf break (Rippeth and Inall, 2002) and in the vicinity of banks (Palmer et al., 2013) is now well established. Much of the energy resides (at least initially) in the rst mode, so their scale (Liw; Table 1) is closely related to L1. Hence, we see that resolving Lbt, LT and L1 is crucial for a wide range of coastal-process representation.

Riverine and glacial freshwater inputs form buoyant coastal currents that can form a substantial part of the coastal ocean circulation and are an important control mediating the transport of terrestrial material, notably by inhibiting its direct off-shore transport. Their scale is difcult to quantify in general terms on a theoretical basis. To characterise how well riverine coastal currents are modelled, we consider the minimum of two scales (Lr): the horizontal scale characteristic of seabed frontal trapping, dened as the depth of trapping (Yankovsky and Chapman, 1997) divided by the local slope and the inow Rossby radius (Avicola and Huq, 2002) (Table 1).

2.1 Coastal ocean process scales in a global context

To put the scales described above and listed in Table 1 into a global context, we calculate values using the global ORCA12 NEMO model (set up by the DRAKKAR group; e.g. Marzocchi et al., 2015; Duchez et al., 2014) as a reference grid and bathymetry. This is a tri-polar grid with a coarsest resolution of 9.3 km, but decreasing to minimum values of1.8 km in the Southern Ocean and 1.3 km in the Canadian archipelago. The median scale is 6.3 km. The bathymetry is a combination of GEBCO and ETOPO2. The process scales are themselves very much dependent on the scale of the information used to calculate them (e.g. the level of detail in topographic roughness used in calculating LT), so a high-resolution model grid used in practice is a good starting point, although the results presented below are not generally dependent on this grid choice. Figure 1 shows values of the barotropic (Lbt) and rst baroclinic Rossby radii (L1), the topographic length scale (LT) and the tidal excursion (Le); see gure caption for further details of the calculation.

The barotropic Rossby radius is, as expected, large (> 1000 km) even at high latitudes, except in shelf seas and near the coast, e.g. in 20 m water depth at midlatitudes, Lbt 100 km. For LT, values < 100 km are widely dis

tributed across the ocean, reecting features such as ridges and sills. Values < 10 km are, however, restricted to the slopes at the ocean margins between the deep ocean and ei-

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Figure 1. Global scales: (a) rst baroclinic Rossby radius; the maximum value calculated from monthly ORCA12 density proles (each month being an average from 1981 to 2010) following Nurser and Bacon (2014), using the model run described by Marzochhi et al. (2015);

(b) barotropic Rossby radius calculated from ORCA12 bathymetry; (c) topographic scale calculated from ORCA12 bathymetry and mesh;(d) tidal excursion, calculated from TPXO barotropic tidal currents (Egbert and Erofeeva, 2002).

ther the continents or the continental shelves. For the baro-clinic Rossby radius (L1), values < 10 km occur in high-latitude oceans, whereas values < 6 km are limited to continental shelves. The tidal excursion (Le) is much smaller, generally < 10 km. It shows an opposite pattern to the baro-clinic Rossby radius, being largest at the coast. Where it is very small (e.g. in the open ocean) so is the tidal velocity and it is of minimal importance. Where it is large, however, it can make a signicant contribution to local water column mixing/stability and ne-scale residual transport.

To assess how model resolution compares with these scales, we dene a parameter,

e = Lx/(max([Delta1]x,[Delta1]y) [notdef] E), (1) at each model grid cell (size [Delta1]x, [Delta1]y) of the ORCA12 mesh,i.e. the number of cells per length scale for process x. We multiply the size of each cell of the original grid ([Delta1]x, [Delta1]y) by a factor, E, to approximate other grid resolutions (without needing to generate the grids; e.g. E = 3 for a nomi

nal 1/4 resolution). We focus on the barotropic and baro-clinic Rossby radii and the topographic scale. We do not consider the tidal excursion further in this context, as resolving it is only benecial in regions where the tide is large.

Here, we consider the nominal model resolutions listed in

Table 2, along with example applications for the global and coastal ocean cases. It is worth noting the current generation of forced, high-resolution global models are of similar resolution to many historic and on-going shelf sea simulations (see references in Table 2). The cumulative distributions of e (Fig. 2), weighted by the area of each grid cell, then show the fraction of the model area at a particular resolution that resolves scale Lx with e or more grid cells. This gure also shows the distribution calculated just for water depth < 500 m, i.e. the coastal ocean. What constitutes adequate resolution then depends on the process in question. Hallberg (2013) suggest two grid cells per baroclinic Rossby radius gives a good representation of eddy uxes, so we take e > 2 to be eddy resolving. If eddies have a characteristic size of 2 L1 (i.e. half the wavelength of the fast growing

baroclinically unstable mode; Pedlosky, 1987) then we take e < 1 to be eddy excluding (i.e. a full parameterisation of eddy effects would need to be included in the model) and between these to be eddy permitting. For the barotropic Rossby radius (Lbt), we take the limits on excluding and resolving to be e < 2 and e > 10, on the basis that this scale needs to be well resolved to capture many coastal ocean processes (as

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504 J. Holt et al.: Prospects for improving the representation of coastal and shelf seas

discussed above). For the topographic scale, we set the limits at one and three grid cells, respectively, since at least three cells are required to represent a topographically constrained jet.

We can therefore demonstrate that a 1/4 global model is eddy resolving for 27 % of the globe; this increases to 52 % for 1/12 , 77 % for 1/36 and 91 % for 1/72 . The fraction of the coastal ocean that is eddy resolving is signicantly less: 8 % at 1/12 ; 34 % at 1/36 ; a 1/72 res

olution is needed to be eddy resolving over 70 % of the

coastal ocean. The topographic scale is much more promising. A 1/12 model has e > 3 for 90 % of the global and 70 % coastal ocean case. Resolving the barotropic Rossby radius is a somewhat more stringent criterion than resolving the topographic scale in the coastal ocean at 1/4 or coarser resolution.

To explore the ability of models of different resolution to represent river plumes on a global scale, Fig. 3 shows the cumulative distribution of the number of rivers (out of the 925 largest by volume ux; Dai et al., 2009) where the scale, Lr, is resolved to level e. This suggests modelling riverine coastal currents is an extreme challenge for structured grid global models. Using the same criteria limits as for LT, at 1/12 , only 38 of the largest 925 rivers meet the permitting criteria. This implies that, while the freshwater balance is correct, its dispersion and transport properties will be limited. This number increases to 165 at 1/72 .

We see from this scale analysis that 1/72 ( 1.5 km)

might be taken as a good target for resolving many small-scale coastal ocean processes such as eddies, upwelling and the largest river plumes. We would also expect it to be adequate for resolving tidal excursions (where important) and internal tides. However, it is important to consider these results in the context of coastal ocean dynamics and previous modelling experience. Very few regional coastal ocean modelling studies have been conducted at eddy-permitting resolution, yet signicant progress in our understanding of the dynamics of these regions has still been achieved. Hence, while 1.5 km might be seen as an aspiration, the practicalities of eddy resolving on shelf (when/how this can be reached are discussed below) should not be seen as a particular obstacle to making shorter-term progress in modelling the coastal ocean on a global scale, for example, by using 1/36 as a compromise resolution (as in many coastal ocean studies; e.g. Maraldi et al., 2012) or focusing on process representation (e.g. Luneva et al., 2015). Some features with scales of the Rossby radius, such as coastal upwelling, river plumes and frontal jets will still be present in models that do not resolve this scale; they will just not be particularly well represented. For example, continuity will lead to upwelling in a model of any resolution; its horizontal scale will be determined by the grid and numerics rather than the physics. Internal waves and eddies, in contrast, will simply be absent, and so need to be parameterised. The barotropic and topographic scales are vitally im-

portant for the accurate modelling of coastal ocean dynamics, but can be reached at more modest global resolutions.

3 The modelling approaches

Here we consider, in general terms, how the processes considered above are represented by the model dynamical equations or specic parameterisations and those that are resolved by the model grid. The inadequacy of global climate models in the coastal ocean is frequently quoted but rarely quantied. Thus, we start this section with a consideration of how well the CMIP5 generation of climate models (Taylor et al., 2012) performs in these regions. We focus on the potential energy anomaly (PEA) as a useful measure of water column stratication. The PEA is dened by

' =

0

g

h

[integraldisplay]

z=h

z((T ,S) ( [notdef]T, [notdef]S))dz, (2)

where h is the water depth (here, the integration is limited to 200 m), g is the gravitational acceleration, the density and z the (positive upwards) vertical coordinate. An overbar indicates an average over the same depth as the integration.This represents the energy (per depth) needed to mix the water column. It is a commonly used metric for stratication since it is an integral quantity that does not relate to a particular vertical structure or threshold and connects with simple theories of stratication evolution (Sharples and Simpson, 1996; Simpson and Hunter, 1974). Using the historical period (19702005) of 19 CMIP5 models, we calculate mean PEA for each month and average over these 36 years to give a mean annual cycle, interpolated onto the northern North Atlantic 1/12 NEMO model grid (see below). These models were selected because they all simulate aspects of marine biogeochemistry. We also calculate the PEA for each prole in the EN4 conductivitytemperaturedepth (CTD) prole dataset (Good et al., 2013), and average these onto the same model grid to give an observed mean annual cycle on a common grid. The model and observed values are then compared to give rms error (RMSE) and correlation statistics, here mixing spatial and seasonal variability. Figure 4 shows these values calculated for the whole northern North Atlantic and only where depths are less than 500 m (approximately the coastal ocean). This shows the general picture that the performance of these models is degraded in the coastal ocean (RMSEs are higher, correlations are lower). This is the case for all models for RMSE except one (marginally), and 11 out of 19 for correlation; for the models lying above the line, the correlations are either small or sit very close to the line. This gure also shows that all the higher-resolution models (0.5

or ner) perform well, but there is not a clear resolution dependence; e.g. some coarser-resolution model performs similarly well.

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J. Holt et al.: Prospects for improving the representation of coastal and shelf seas 505

Table 2. A selection of current model grids.

Nominal Scale at the Global Coastal ocean Examples resolution Equator (km) application application

1 111 Typical of Earth system models

CMIP4 and 5

n/a HadGEM3 (Hewitt et al., 2011),HadCM3 (Gordon et al., 2000)

1/4 25 CMIP6 ESMs Historical HadGEM3 (Williams et al., 2015) 1/12 9.3 Next-generation coupled Shelf scale/ocean margin ORCA12 (Marzocchi et al., 2015)

AMM7 (ODea et al., 2012)

AMM12 (Wakelin et al., 2009) 1/36 3.5 Next-generation forced Shelf scale IBI (Maraldi et al., 2012)

ECOSMO (Daewel and Schrum, 2013)

1/72 1.5 n/a High-res. shelf/coastal HRCS (Holt and Proctor, 2008) AMM60 (Guihou et al., 2017)

n/a not applicable

Figure 2. Cumulative distribution of the fraction of global (top) and coastal (bottom) ocean, resolving L1, LT and Lbt for different global model resolutions.

3.1 Process representation

The representation of coastal processes in global ocean models is straightforward, at least in concept. For example, the NEMO model from v3.2 onwards has the capability of simulating both open ocean and shelf sea cases (ODea et al., 2012; Maraldi et al., 2012), with capability improving in later versions. This essentially allows these processes to be included by conguration selection as the global model resolution is rened. The open question is whether features

pertinent to the coastal ocean can be introduced into global models without degrading the solution in the open ocean or signicantly increasing their computational cost. The model development process is largely focussed around reconciling the differences between coastal ocean and global ocean approaches; a good guiding principle could well be to minimise the changes needed in the global modelling approach, on the basis that these choices are well suited for the majority of open ocean processes.

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Figure 3. Cumulative distribution of number of rivers where the scale Lr is resolved at a particular level (e). Based on ow data from the 925 largest ocean-owing rivers globally (Dai et al., 2009).

3.1.1 Tides

The representation of tides in global models is the natural starting point. There are two approaches that can be considered: direct simulation and parameterisation. Along with tide-generating forces, self-attraction, loading and solid-earth tides need to be represented to achieve an accurate tidal simulation (e.g. Stepanov and Hughes, 2004). In addition, the correct energy dissipation through bottom friction and internal tide generation is required. Baroclinic global tidal models with prognostic temperature and salinity (e.g. HYCOM; Arbic et al., 2012) can directly simulate the internal tide eld. However, low-mode internal tides can propagate large distances from their generation region, making their impact (e.g. on mixing) hard to adequately parameterise (Simmons et al., 2004). As identied above, global models at resolutions ner than 1/4 permit low-mode inter

nal tides in the open ocean, but not higher modes or wave numbers or internal tides in the coastal ocean. For example, Niwa and Hibiya (2011) nd a strong resolution dependence of barotropic to baroclinic tidal energy conversion with no convergence even at 1/15 . Hence, some form of wave drag parameterisation may be required. Arbic et al. (2012) found a carefully tuned wave drag parameterisation is necessary to accurately simulate tides in the isopycnal HYCOM model, whereas Mller et al. (2010) found that a wave drag scheme was not required in the geopotential (z) level Max Planck Institute Ocean Model (MPI-OM) model.

Introducing tides into a global model requires changes to a number of model formulations. These include the accurate representation of the bottom boundary layer by ne near-bed vertical resolution, e.g. through terrain following (s) or arbitrary LagrangianEulerian (ALE) coordinates (Petersen et

Figure 4. The RMSE and correlation for PEA of 19 CMIP5 models compared with PEA calculated from EN4 prole data (1970 2014) in the North Atlantic. In both model and observations, a mean annual cycle is calculated, and then error statistics calculated, with both being interpolated onto the 1/12 northern North Atlantic (NNA) model grid. Values for the full region are compared with data only at water depths < 500 m. Values listed by the model names are the inverse mean meridional resolutions of each model. Also shown are results from the global ORCA12 model and from NNA NEMO model (a regional extraction from the ORCA12 grid, with identical vertical coordinates and forced by this at the boundaries) including tides and k-" (GLS) mixing (both for 19852003, DFS forcing).

al., 2015). Also required are turbulence models suited for multiple boundary layers (Burchard et al., 2008) and a sophisticated representation of bottom friction. This can be achieved using a quadratic friction with a log-layer formulation (Blumberg and Mellor, 1987) and a semi-implicit bed stress implementation for numerical stability, given the large stresses and thin vertical layers. For conservation reasons (Campin et al., 2004), most global ocean models are now moving towards using a non-linear free surface, as is also required to represent large tidal amplitudes. Tidal dynamics are most accurately represented with a mode-split time-stepping approach (e.g. Shchepetkin and McWilliams, 2005) rather

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than a fully implicit solution; this is also a trend in recent global ocean model development.

The high frequency cross-coordinate surface vertical displacement of isopycnals arising from an energetic internal tide eld in an s or z coordinate model, but not in an isopycnal model, might be expected to lead to increased spurious mixing, unless accompanied by methods to control it. For example, this motivated the development of the z tilde coordinate in NEMO (Leclair and Madec, 2011). A recent review of spurious numerical mixing, focusing on global ocean models with energetic eddies, is provided by Grifes and Treguier (2013). With energetic eddying or tidal ows, the non-linear advection of momentum becomes more important, which poses a challenge for the numerical methods of momentum advection. This ultimately results in spurious dia-neutral tracer transports since, even with very accurate tracer advection schemes (Colella and Woodward, 1984; Prather, 1986) or adaptive vertical coordinate systems (Leclair and Madec, 2011; Grwe et al., 2015), this will produce spurious transport and/or dispersion errors if the velocity eld contains too much energy near the grid scale (Ilicak et al., 2012). They show the spurious dianeutral transport is proportional to the grid-scale Reynolds number, dened as Re[Delta1] =

min([Delta1]x, [Delta1]y)U/KH, where KH is the Laplacian viscosity that dissipates the mechanical energy. Re[Delta1] should be maintained below a value of 2 to minimise this spurious transport (Grifes and Treguier, 2013; Ilicak et al., 2012). As U increases with the inclusion of tides and other high frequency processes, maintaining Re[Delta1] below this limit becomes more problematic. Beyond this simple criteria, quantifying spurious numerical mixing remains challenging, with a number of different methods having been proposed, each with different assumptions and applicability (Grifes and Treguier, 2013, and references therein; Burchard, 2012; Klingbeil et al., 2014).

When changes to the underlying numerics or rening the grid to at least resolve the barotropic and topographic scales is not practical (e.g. for an Earth system model), or if numerical mixing remains an issue, making direct tidal modelling undesirable, then the alternative is to use tidal mixing parameterisations, which can be adjusted not to overmix in the deep ocean. These make use of the increasingly ne-resolution tidal information available from altimetry-constrained models, e.g. TPX08 at 1/30 (Egbert and Erofeeva, 2002). The parameterisations should include benthic and under-ice mixing (Luneva et al., 2015), and mixing by baroclinic tides (St. Laurent et al., 2002). Simmons et al. (2004) consider the application of an internal tide energy ux parameterisation, driven by a barotropic tidal model (St. Laurent et al., 2002; Jayne and St. Laurent, 2001), and how to translate this to an interior vertical diffusivity for implementation in a coarse-resolution ocean circulation model. In contrast, Allen et al. (2010) explore using a 1-D mixing model (GOTM) driven at each horizontal grid cell by imposed sea-surface slopes to estimate the vertical proles of tidal shear. This has

the advantage that it can accurately account for the interaction of tidal boundary layers and stratication, which is seen to be important, for example, in the Arctic (Luneva et al., 2015), but does not account for internal tide mixing. Transport by tidal rectication is less easy to parameterise, but is expected to be secondary to the mixing effects on a global scale.

3.1.2 Vertical coordinates

Vertical coordinates are a key consideration when modelling the coastal ocean; the bathymetry necessarily varies substantially at the transition from open ocean to shelf sea and from coastal seas to the land. As noted above, mixing processes require the accurate resolution of the benthic boundary layer, as do downslope ows such as cascades and Ekman drains. Moreover, bottom boundary mixing and freshwater input lead to exceptionally sharp pycnoclines. For example, an analysis of CMIP5 models by Heuz et al. (2013) showed that those (few) models that correctly produced Antarctic Bottom Water on the shelves were unable to cascade this water downslope to the deeper ocean.

This need to increase resolution in shoaling water, alongside the need for smoothly represented across-isobath ows, has led to a prevalence of s coordinates in coastal ocean models, accepting some exceptions (Maraldi et al., 2012; Daewel and Schrum, 2013) that have used z coordinates. The large majority of global ocean models use z or isopycnal coordinates. The reasons behind the lack of global s coordinate models are the well-documented issues of calculating horizontal pressure gradient and diffusion terms on sloping coordinate surfaces.

The requirement for tidal simulations to employ a nonlinear free surface and sophisticated vertical grid leads to time-varying vertical coordinates with large slopes. This requires the use of complex schemes to derive the horizontal pressure gradient term in order to avoid spurious currents at steep topography (e.g. Shchepetkin and McWilliams, 2003) and an unrealistically energetic inverse energy cascade (e.g.Holt and James, 2006). As with numerical diffusion, accurately diagnosing this issue in realistic model simulations is problematic, so recourse is usually made to theoretical constrains such as the hydrostatic consistency condition to dene limits on the steepness of coordinate surfaces. Substantial progress has been made in addressing this issue through advanced numerics (e.g. Shchepetkin and McWilliams, 2003) or hybrid coordinate approaches (e.g. Siddorn and Furner, 2013); bathymetric smoothing is the last resort, but is still required in some cases. Given the principle of minimising the changes to the model in the open ocean, the natural choice (for a z coordinate model) is to move to a hybrid system with z coordinates in waters greater than a certain depth, transitioning to terrain-following coordinates in shallower water (Shapiro et al., 2013; Luneva et al., 2015; Zhang and Baptista, 2008). These can be formulated to match the original

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Quasi-horizontal mixing approaches suitable for both open and coastal ocean require careful consideration. These schemes play two distinct roles: rst, to represent the effect of unresolved eddies in transport, and second, to complete the cascade of energy to unresolved scales. The former is particularly important in non-eddying open ocean models (Gent and McWilliams, 1990) but is not generally required in coastal ocean models. The latter is common across both types of model, and is often treated as a stabilisation term without reference to specic physical principles. Both shelf and global ocean models tend to employ a combination of Laplacian and/or bi-Laplacian mixing for momentum and tracers.Mixing of temperature and salinity usually takes place along isoneutral, rather than geopotential, surfaces and this requires careful implementation in the case of sloping vertical coordinates and at fronts where isopycnals intersect the sea surface and bed. The sloping coordinate systems require the use of rotation operators for lateral tracer mixing and this can prove less accurate or more challenging for time-varying and highly sloping coordinates (compared to z level models since isopycnal slopes tend to be fairly close to horizontal), owing to the small-slope approximation (Beckers et al., 2000).This issue has received signicantly less attention than similar considerations with regard to the horizontal pressure gradient calculation, but is explored in this context by Lemari et al. (2012).

As we see above, global models span a wide range of dynamic scales and this is exacerbated when shelf seas are considered in detail; a quasi-uniform-resolution model generally includes both eddying and non-eddying regions. Hence, any model that aims to accurately cross these scales needs to account for the qualitatively and quantitatively changing nature of sub-grid-scale processes. This requires scale-selective approaches to determining sub-grid-scale diffusivities and viscosities (or other forms of closure). The simplest are just depth dependence (Wakelin et al., 2009) or based on horizontal shear (Smagorinsky, 1963). Combination of these with water column density structure (Hallberg, 2013) are likely to be most appropriate, but have not yet been tested in both the open and coastal ocean contexts.

3.1.4 Coastal boundary conditions and rivers

A key feature of the coastal ocean that needs to be considered is coastlines and related bathymetry (e.g. restricting exchange between regional basins). The treatment of the coastal topology is very much dependent on the horizontal gridding approach. Quadrilateral meshes approximate coastlines by a blocked mask and the resulting representation of the coast is highly resolution dependent and leads to two specic issues. First, the detailed representation of coastal features, e.g. at an inlet or a strait, is limited by this resolution; there is some limited scope to alleviate this through mesh distortion. Second, the staircase representation of a straight coastline impacts the fundamental numerical prop-

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models vertical coordinate system at the transition depth.

Such an approach does require the use of a sophisticated horizontal pressure gradient calculation, but minimises the effect of any residual error from this term in the low dissipative open ocean region where it is likely to be most harmful (e.g. in feeding spurious energy into the inverse energy cascade).This approach has potentially substantial benets for mixing and downslope ows. For example, Wobus et al. (2013) have shown some success with a mixed zs coordinate model to facilitate the cascading downslope near Svalbard. It could also be used to facilitate accurate cross-basin transports at deep sills.

The issues with using s coordinates in climate models are described by Lemari et al. (2012). These include spurious mixing through diffusion associated with the advection scheme. This is particularly problematic as it occurs on steep slopes where physical mixing from, e.g. internal tides, may be prevalent. A solution to this is to use a non-diffusive advection scheme coupled with a rotated biharmonic diffusion scheme (Marchesiello et al., 2009). Another issue is the need for vertical mixing schemes that can accommodate wide variations in layer thickness and still retain low mixing in the ocean interior.

3.1.3 Vertical and horizontal mixing parameterisations

Surface mixing processes of wind stress, convection and wave effects are common to open and coastal oceans, and so the primary consideration for vertical mixing schemes in the coastal ocean that differ from the open ocean is the need to accurately model mixing at the benthic boundary layer. Two equation turbulence models (e.g. k-") readily accommodate this and by using the generic length-scale approach (Umlauf and Burchard, 2003) these can be exibly incorporated in a global model. While these approaches give good results in shelf seas (Holt and Umlauf, 2008), they differ substantially from schemes used in global models (e.g. the turbulent kinetic energy TKE scheme in NEMO and the K-prole parameterisation KPP scheme in MOM5). The implications for global ocean simulations, e.g. deep water mass preservation properties and maintenance of the meridional overturning circulation, have yet to be established. Particularly the issue of the k-" models performance at low vertical resolution needs to be established. With the length-scale limiter that is usually used with this model, it reduces to a background value inversely proportional to the buoyancy frequency in strongly stratied, weakly turbulent regimes (Holt and Umlauf, 2008; Eq. 6 therein). This is broadly consistent with the behaviour of ocean interior internal wave mixing (e.g. Gargett, 1984), so might be expected to give good results with careful parameter selection. This issue has been explored for the KPP model with terrain-following coordinates by Lemari et al. (2012) and modications proposed in the context of these coordinates.

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erties of the model; notably, the propagation of Kelvin waves is retarded (Greenberg et al., 2007) and the accuracy of solution is reduced (e.g. from second to rst order; Grifths, 2013), even for coasts very closely aligned with the mesh, with only an occasional step. This can be seen as a special case of the stepped representation of topography by z coordinate models; the issue of bottom topography representation is alleviated by using terrain-following coordinates.Available solutions at the coastline for quadrilateral meshes are through shaved cell (Adcroft et al., 1997; Ingram et al., 2003) or immersed boundary (Tseng and Ferziger, 2003) approaches for high-resolution models, or porous barriers (Ad-croft, 2013) for coarser-resolution models. Triangular mesh models, when paired with terrain following coordinates, do not encounter these issues: they can t the coastline with an arbitrary degree of accuracy limited by the minimum acceptable scale and accuracy of the geographic information.The representation of the details of the coastline is a key advantage of triangular mesh models. However, even with the highest-resolution models being considered, the accurate details of coastlines will not be well represented, particularly in bays, estuaries, fjords, etc., and these must be left to local-scale models (often with a resolution of a few hundred metres), which include more detailed processes such as the capability to wet and dry with the tide. Similarly for the resolutions considered here, parameterisation of riverine effects is still required for an accurate representation of their transport processes. This can be achieved by, for example, box modelling approaches, as currently being tested in the Community Earth System Model (Bryan et al., 2015).

3.2 An example of improved resolution and process representation in comparison with CMIP5 models

To illustrate what might be achieved through higher resolution, introducing tides and sophisticated mixing schemes in the context of the coastal ocean, we consider runs of the global ORCA12 (Marzocchi et al., 2015; Duchez et al., 2014) and the northern North Atlantic (NNA; Holt et al., 2014)NEMO models. Both use the same NEMO code base (v3.5) and have 75 z partial step layers in the vertical. The ORCA12 model uses a TKE vertical mixing scheme, a ltered free-surface formulation and does not include tides. The NNA model is an extraction of the grid and bathymetry from ORCA12 that includes tides, a k-" vertical mixing scheme (implemented by the GLS approach; Umlauf and Burchard, 2003), with log-layer bottom friction and a mode-split explicit free surface with variable volume. Both use Drakkar forcing sets (DFS) surface forcing (Brodeau et al., 2010) and NNA takes lateral boundary conditions from ORCA12. For brevity, here we focus on the PEA (Eq. 2) as a measure of upper ocean stratication, and Fig. 4 shows that both these models perform substantially better than the CMIP5 models in both RMSE and correlation across the whole domain. The same is true for the correlation in the coastal ocean, but there

Figure 5. PEA (Eq. 2) for July (mean 19852003; note log scale) for the northern North Atlantic (NNA) NEMO conguration (top: including tides and k-" (GLS) mixing) and global ORCA12 model (bottom: with the TKE mixing scheme and no tides). Also shown are regions 110 used for a seasonal analysis (Fig. 6).

are two 0.5 CMIP5 models (NorESM-ME and CNRMCM5) of comparable performance in terms of RMSE. No signicant difference between NNA and ORCA12 is apparent in these overall statistics.

Figure 5 shows the mean July PEA for these two models to examine the differences in stratication in more detail. It shows that while the differences across the whole region are comparatively minor, the differences in some localised regions are very marked. A particular example is in the southern North Sea, English Channel and Irish Sea, where the expected well mixed regions are much clear in NNA than ORCA12, and more in accord with observations (Holt and Umlauf, 2008).

To explore how well these models reproduce the seasonal cycle in stratication, the mean annual cycles of PEA for 10 regions (see Fig. 5) are shown in Fig. 6; this is limited to

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water depth < 500 m to focus the comparison on the coastal ocean. Each region is selected to cover sufcient data to give a reasonably smooth mean seasonal cycle in the observations, but the results are inevitably dependent on the details of this choice. Moreover, because EN4 is not a systematic dataset, deriving a mean annual cycle in this way potentially mixes inter-annual and spatial variability (for further discussion on this, see Holt et al., 2012). Nonetheless, it provides a useful guide to model performance. Alongside NNA and ORCA12, results are shown for the same 19 CMIP5 model as in Fig. 4 and the overall RMSE, the median and minimum RMSE values for CMIP5 and which model is the lowest. We see that NNA has a lower RMSE than ORCA12 in all regions except 1 and 9 (Norwegian Sea and Georges Bank, where both models exhibit no signicant improvement on CMIP5) and region 4 (southern North Sea, where both have small errors). This demonstrates a clear advantage of this combination of process representation (i.e. including tides and a two-equation turbulence closure scheme). Apart from in regions 1 and 9, the NNA and ORCA12 models improve on the median error from the CMIP5 models; this is not a remarkable result and it would be worrying if it were not the case that high-resolution, reanalysis-forced models could not outperform coarser-resolution coupled models. However, what is more interesting is that in most regions the best of these CMIP5 models outperforms or is very close to either NNA or ORCA12. This may well happen by chance: there is a broad spread of CMIP5 models results here and 6 different models are the highest performers. However, some of the higher-resolution models (CNRM-CM5, 0.5 ; MPI-ESM-MR, 0.3 ;

NorESM1-ME, 0.45 ) lead (among this CMIP5 ensemble) in 6 out of 10 regions, which deserves further investigation.For example, the CNRM-CM5 (Voldoire et al., 2013) and NorESM1-ME (Bentsen et al., 2013) both include the tidal mixing parameterisation of Simmons et al. (2004).

Some other aspects are clear from this comparison. These CMIP5 models generally overestimate the PEA and its annual cycle. This is particularly apparent in the eastern boundary upwelling regions (5 and 6) where the observed annual cycle is very small. Such biases are not apparent in NNA or ORCA12.

While a much more comprehensive assessment is required to inform the appropriate aspects of model development, this does demonstrate some clear advantages to improved resolution and process representation. It also identies some areas for further investigation, notably the biases on the eastern US coast.

3.3 Resolving the pertinent scales

The most signicant challenge in representing the coastal ocean in global models relates to the small scales needed to represent the processes and geography (coastline, bathymetry, straits) of these seas. There are essentially two options for achieving a rened horizontal resolution: either

increase the quasi-uniform resolution of the whole grid or introduce a multiscale capability that allows renement in specic locations. We consider briey what these capabilities might be below, but rst explore the balance between these two options if we desire to resolve a particular set of processes globally, rening the model locally to achieve this.We quantify this conceptually, with no consideration of mesh structure, by building on the scale analysis above and use

Nx = [Sigma1]F 2 = [Sigma1](n/e)2 = [Sigma1](nmax([Delta1]X,[Delta1]Y ) [notdef] E/Lx)2 (3) to dene the global sum of the number of cells needed in each global model grid cell to resolve a process, characterised by length scale Lx at a particular level (n). Following the discussion in Sect. 2.1, we take n = 10, 3, 2 for Lbt, LT and L1,

respectively. A constraint is imposed on this:

Lmin < Lx/n < max([Delta1]X,[Delta1]Y ) [notdef] E. (4)

The upper limit species a base resolution, i.e. a multiple (E) of the global ORCA12 grid (resolution: [Delta1]X, [Delta1]Y ) that is being rened. The lower limit, Lmin, acknowledges that there are limits to how ne a resolution is desirable, particularly in the case of scales that tend to zero with the water depth, and with respect to time step constraints.

As an example, Fig. 6 shows how a 1/12 ORCA tripolar grid might be rened to a minimum scale of 1.5 km ( 1/72 ) as required by the above criterion, with Lx be

ing the smaller of the baroclinic, barotropic and topographic scales. Values at each cell range from F 2 = (n/e)2 = 1 (no

renement) to (max([Delta1]X,[Delta1]Y )[notdef]E/Lmin)2 = 36 in this case.

Midlatitude and arctic shelves require modest renement ([notdef] 1015 extra cells); the reduced based mesh size of the

ORCA grid counters the reduced Rossby radius here (noting the absolute values of F are dependent on this grid structure). In some very shallow tropical regions, the number is at or close to the maximum value, indicating that the desired level of process resolution is not always achieved. The accuracy of this estimate is limited by the underlying information (notably the bathymetry and the Rossby radius) and no consideration of the renement needed to resolve the coastline is made. Nonetheless, this still provides a useful guide in terms of the relative cost of multiscale and globally rened resolution approaches. Here, we compare this calculation with the total number of grid cells in the globally rened case (at Lmin). Because the minimum scale is the same for both, no timescale factor is needed. This approach takes no account of the mesh structure needed. In particular, there will be limits on how quickly scales can be allowed to vary on an unstructured mesh (see, for example, Fig. 6; the lower panels show the change in resolution needed can be locally very abrupt) and so this puts a lower bound on the number of cells needed in the multiscale case.

We consider three values of Lmin: 9.3 km ( 1/12 ),

3.5 km ( 1/36 ) and 1.5 km ( 1/72 ) (cf. Table 2) in

Fig. 8. So, for example, a 1/4 global model rened where

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Figure 6. Mean seasonal cycle of PEA averaged over the 10 regions shown in Fig. 5 for water depth < 500 m. Results for 19 CMIP5 models (light lines) are shown along with the ORCA12, NNA (heavy lines) and EN4 observations (lines and circles). The numbers refer to RMSE compared with EN4 showing the minimum of all these CMIP5 models (and the corresponding model number from the list in Fig. 4), the median CMIP5 value and the values for NNA and ORCA12.

necessary to resolve the smallest of these scales down to a minimum scale of 9.3 km (left panel) requires about 0.25 the number of grid points of a full 1/12 grid, or a saving of about a factor of 4. As the minimum scale decreases to3.1 km (middle panel) and 1.5 km (right panel), the saving increases to 0.095 (10 times fewer points) and 0.046 (factor of 22), compared to the full global grid at the minimum resolution. Similarly, a 1/12 model rened to a minimum scale of 1.5 km has 0.06 (factor of 17) times fewer cells than a 1/72 global model. The limiting behaviour evident from

these plots arises because at coarse base resolution most of the grid is rened to meet the criteria (i.e. the base resolution becomes irrelevant), while at a ne base resolution this meets the criteria in many regions anyway and the renement becomes less relevant. These results are considered in terms of what may be computationally practical in Sect. 4.

3.4 Options for multiscale modelling

There is already a substantial literature on multiscale modelling and we do not attempt to review this here. Unstructured

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(Grger et al., 2013) to focus resolution on European seas.

While this can address the downscaling issue for a single region, it does not help with the upscaling question.

Nesting is the most common approach to multiscale modelling. In its simplest form, boundary conditions for a ne-resolution regional model are taken from a previous run of a larger area ocean model. It has the signicant advantage that the global model does not have to be rerun for each regional simulation. There is, however, the practical consideration of the effort required to set up and test a new regional conguration for each new area of interest. Nesting remains an important approach for investigating of regional systems, and providing ne-scale information, e.g. for operational or research purposes. The general downside to nesting is the accuracy at which information can be exchanged between the two domains and the degradation of the solution at the boundary; it is usual to linearise the boundary conditions and to only exchange a limited subset of information at lower frequency than the model time step. That said, there has been extensive work on regional model boundary conditions (e.g. Marsaleix et al., 2006; Mason et al., 2010) and by using a careful combination of active and passive approaches good solutions can be obtained. One-way nesting can be extended to a global scale using multiple regional nests (Holt et al., 2009). The problem is simply one of standardising the conguration procedure and of managing workow. However, one of the key advantages of regional models, that they can be tailored to the specic conditions of a region, is generally lost in automatically congured domains. The underlying assumption to such one-way nesting is that feedbacks between the regional and global simulations are small, at least on the timescales of interest, and again it only addresses the downscaling question.

A natural extension of the nesting approach, which allows for upscaling, is two-way global-scale nesting. The AGRIF tool (Debreu et al., 2012) provides a capability to automatically generate nests, which has been utilised in both the ROMS and NEMO systems (e.g. Biastoch et al., 2008), generally with individual regions being rened with one or more nests. In theory, this is extendable to the global scale, with multiple nests placed to locally resolve coastal ocean processes. Several approaches exist to couple the two grids, reviewed by Debreu and Blayo (2008). Because this occurs in memory, these can be substantially more sophisticated than ofine nesting by le exchange, and essentially aim to link solution approaches in the two grids, coupling at the time steps of the respective grid. This means that as well as having two-way interaction, many of the issues associated with ofine boundary conditions noted above are alleviated, although noise and wave reection are two issues that require particular attention. An issue with this approach in the global context is the restriction (for AGRIF) to rectangular domains (in model coordinate space; see below). This is somewhat inefcient and inexible, and the coupling between neigh-bouring rened regions, with potentially different levels of

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Figure 7. An example of how a 1/12 global grid (a) might conceptually be rened to resolve the dominant scales. The parameter shown is the number of cells needed in each global grid cell to resolve these scales down to a minimum scale of 1.5 km, so ranges from 1 (no renement) to (72/12)2 = 36. Below are two examples

in more detail for (b) east Asia and (c) northwest Africa.

mesh approaches generally focus on triangular mesh models using a nite volume approach, e.g. FVCOM (Chen et al., 2003); FESOM2 (Danilov et al., 2016) or a nite element approach, e.g. FESOM1.4 (Wang et al., 2014), SELFE (Zhang and Baptista, 2008) and SCHISM (Zhang et al., 2016). In contrast, MPAS (Ringler et al., 2013) is based on hexagonal meshes using a nite volume approach. Danilov (2013) provides an account of the issues of unstructured mesh modelling, and what is clear from that review is that selecting a solution approach or grid arrangement, for example, on the basis of a lack of computational modes or formal accuracy is far from straightforward, and must be left to detailed investigations in idealised and realistic cases.

Structured grid models have scope for multiscale capability by distorting their horizontal coordinates and through nesting. Coordinate transformations generally limit the renement to a single region of interest. An example to facilitate regional impact studies is the use of a rotated polar grid

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Figure 8. Number of grid cells to achieve process representation in shelf seas with a multiscale approach, relative to a rening global reference resolution of 1/12, 1/36 and 1/72 and down to a minimum scale set by this global reference.

renement needs to be considered. Large, irregularly shaped nests (e.g. Holt et al., 2009) would be good option, not over-rening in the open ocean and limiting the number of grids and connections between them. This would require substantial development to AGRIF or an alternative approach. An approach that has yet to be thoroughly explored is using model couplers (e.g. OASIS3-MCT) as a two-way down-scaling tool. This would allow complete exibility between nests, e.g. a different executable can be run in each nest, but whether the coupling system is sufciently efcient to permit coupling at the model time steps is unclear.

A key limitation to multiscale models is time stepping, which is closely related to the scalability of the models, discussed below. The trade-off is between explicit models, which are computationally efcient with MPI parallelism, but have a time step limited by the CourantFriedrichsLewy (CFL) condition and implicit models, which are not limited by the CFL condition, but are less computationally efcient with MPI parallelism, due to the need for global matrix inversions. Currently, the balance is towards explicit time-stepping models (e.g. with time splitting between barotropic and baroclinic modes), given the use of global models on many thousands of processor cores. This has implications for multiscale approaches: the approach must either accept the limitation of the smallest scale in the grid, introduce some level of implicit time stepping, with consequent implications for scalability, or else introduce a locally rened time-stepping approach, whereby different time steps are used in

different regions. The latter is natural for the multi-blocking approach (and is assumed in the analysis below) but is highly complex for unstructured mesh multiscale approaches (e.g. Dawson et al., 2013). A move to implicit time stepping, aside from any accuracy and diffusion issues, requires the development and use of very efcient numerical solvers.

To put global nesting in the same context as the above scale analysis, we consider a multi-block approach (accepting the limitation to rectangular domains for now), and consider the global ORCA12 grid divided into 15 [notdef] 15 blocks. Each

of these is then given a renement level F 2 ranging from 1 to 36, as above. To provide a representative maximum value (but not set by a very few large grid point values), this is taken to be the 95th percentile of the grid cells in each block. To mimic the AGRIF renement process, each block takes an integer value: (int(F ))2. This example leads to 194 out of 344 cells requiring renement (Fig. 9). Such a set-up would be a challenging computational engineering effort and certainly less elegant than an unstructured mesh approach, but may be more efcient (quantied below) and is available as an evolution of the structured mesh approach, common to most of the current climate-scale global modelling effort (and so building on the expertise therein), rather than a move to a radically different approach. Whether it is more or less accurate than a comparable nite volume or element unstructured mesh approach must be left for future investigation.

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514 J. Holt et al.: Prospects for improving the representation of coastal and shelf seas

Figure 9. Renement of 15 [notdef] 15 blocks to the 95th percentile of

the distribution of F 2 in each block. Set to (int(F ))2 to approximate renement by an approach such as AGRIF.

4 Utilising the computational resources

4.1 Trends in high-performance computing

Ocean modelling has beneted from the general exponential growth in high-performance computing (HPC) capability, with the largest machines approximately doubling in performance every 18 months since 1993 (TOP5001 list). There are two technology drivers for this: rstly, increases in clock speed and improvements in architecture (particularly instruction-level parallelism) and secondly massive increases in parallelism. In 1993, the TOP500 list still contained machines with only one processor; in June 2016, the smallest system had 5310 cores and the largest had over 10 million cores. The rst of these drivers has largely stalled as clock speeds have peaked at around 23 GHz due to power density limitations. Instruction-level parallelism has also peaked at around 48 instructions per clock cycle; memories are not fast enough to provide enough operands to justify greater values. Further performance increase into the future is therefore expected to be driven solely by an increase in parallelism, through a larger and larger number of processor cores.

Continuing the current exponential growth towards exaop performance (1018 operations per second) specically requires a substantial reduction in power consumption (by

100-fold) to keep the power costs of HPC systems within reasonable limits. If these power efciency constraints are lifted to achieve exascale systems, there are two major impacts for ocean modelling. First is the prospect of a single ocean model running at exascale performance levels on, e.g. 100 million cores. Alongside this there would be a knock-

1http://www.top500.org/

Web End =www.top500.org/

Figure 10. The UK research computer facility peak performance and memory per core. Also shown are two projected possible future machines.

on impact on smaller systems as petascale systems become available with about 100 000 cores in a single rack, consuming only 100 kW, and so accessible by the modelling com

munity at an institutional level.

To use the UK research community perspective as a practical example, Fig. 10 shows the increase in the peak performance of the UK Research Councils (RCUK) HPC facility, from HPCx in 2006, through the four phases of Hector to the current machine, Archer2. The peak performance of this facility has increased exponentially over the past 10 years,

although the general trend has attened off since the rapid increase between HPCx and Hector Phase 2a. A conservative estimate is to extrapolate the trend from Phase 2a to Archer Phase 2. This gives a peak performance of 13 times Archer

Phase 2 by 2019 (32 Pop s1) and 745 times by 2023

(610 Pop s1). This closely follows the TOP500 trends, and predicts the UK will maintain a performance about a factor of 10 lower than the US at any one time (or lags by 3 4 years). There are of course many unknowns in this projection, such as the size of the overall research community and share of the resource which the marine science sector may receive. Nonetheless, this usefully quanties the often-quoted remarks around continually increasing computer power and puts bounds on what may be expected.

In terms of ocean model design, to effectively utilise large numbers of cores, codes will have to extract very high degrees of parallelism from the underlying numerical algorithms. This requires at least three-way nested parallelism with high-level coarse-grained parallelism at the node level probably using MPI, multithreading on a node using OpenMP or OpenACC, and ne-grained parallelism within a core, e.g. vectorisation at the loop level. Memory manage-

2http://www.archer.ac.uk

Web End =www.archer.ac.uk

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J. Holt et al.: Prospects for improving the representation of coastal and shelf seas 515

ment will become increasingly important. The size of memory cannot increase to match the numbers of cores, on ground of cost and power, and the amount of memory per core is expected to reduce signicantly (although memory per core is still relatively stable in the example presented here; Fig. 10).Memory bandwidth per core and interconnect speed per core is also expected to drop. Algorithm design must therefore focus on management and movement of data in memory and between nodes.

4.2 Scalability and efciency of ocean models

An important distinction needs to be made between computational resource (CPU hours) and model simulation time (simulated years per day; SYPD). The available computer resource is generally increasing through increased parallelism (more cores per chips, more CPUs and novel, accelerator-based architectures), and is what is metered and limited by computer centres. It is, however, the SYPD that limits the science that can be done with a particular model (assuming the resource is available). For example, Table 3 lists anecdotal reports of turnaround time (SYPD) for three global ocean models (MOM6 GFDL CM2.6, NEMO ORCA12 and FESOMV2 Glob15) and a high-resolution coastal ocean model (NEMO AMM60); also shown is this value scaled with the grid cells (horizontal and vertical) per core and the time step to give a rough comparison of the efciency. The two structured grid global models have a comparable efciency (2.35 and 3.5 SYPD; 207 and 223 kTimestep grid cells s1; kT

GPS), whereas the unstructured mesh model is somewhat more efcient (17 SYPD and 396 kTGPS; discussed further below) but is run on considerably fewer processors.

Currently, the minimum efcient size of model run by each MPI process is about 20 [notdef] 20 or 400 grid cells per core.

As the resolution reduces, then, for an explicit time-stepping model the CFL stability criteria requires the time step to reduce, and the model runs more slowly (reduced SYPD), irrespective of the increased resource. For example, the 1/60

resolution NEMO model of northwest European continental shelf achieves only 1 SYPD, but is somewhat more

efcient at 245 kTGPS than the global NEMO model. For large-scale climate and Earth system model (ESM) simulations, with substantial resources available but a requirement to complete many centuries of simulation in a restricted time period, this is the key limitation. When running large numbers of shorter simulations in research mode (e.g. by a whole research community), the resource (CPU hours) itself provides the limit. Given static CPU speeds, there are two options to mitigate this reduction in SYPD: modify the model numerics (with respect to time stepping; see above) and/or improve the parallel scalability. The latter can occur in two ways. Firstly, by reducing the size of the subdomain within an MPI process that can be used efciently, essentially by reducing the ratio of communication costs to computation costs, e.g. by using larger halos to increase message size

Table3.Reportedturnaroundtimeofthreeglobalmodels:theMOM6andNEMOstructuredgridandtheFESOMnite-volumetriangularmeshmodels,andacoastaloceanNEMO

modelofthenorthwestEuropeanshelf,AMM60,shownassimulatedyearsperday(SYPD).AlsoshownisanoverallefciencyscalingfactorinkTimestep.Gridcellspersecondare

indicatedaskTGPS.

Model/NominalVerticalCoresforGridcells/TimestepSYPDkTGPSReference

cong.resolutionlevelssimulationcore(s)

MOM6CM2.61/10 50 10 000 972 300 3.5 207 Dr M. Ward (Australian National University, personal communication, 2016)

NEMOORCA121/12 50 8972 1040 300 2.3 223 Dr A. Coward (NOC; personal communication, 2016)

NEMOAMM601/60 75 2000 806 60 1.0 245 Dr J. Polton (NOC; personal communication, 2016)

FESOMV2Glob1515km461728115090017364Danilovetal.(2016)

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516 J. Holt et al.: Prospects for improving the representation of coastal and shelf seas

and reduce latency effects. Secondly, by introducing alternative levels of parallelism, so that more cores can be used efciently by a single MPI process, e.g. using multithreading (loop-level parallelism) with OpenMP or OpenACC. Another possibility, which is being explored, is parallelisation of the time domain, especially attractive for long-time duration, low-resolution climate runs. Parallel-in-time methods offer the prospect of another 2 orders of magnitude in concurrency (Haut and Wingate, 2014) but are still at the early research stage.

An important and complex question is whether unstructured mesh models are inherently more computationally expensive than structured grid ones. They are certainly more complex, with more oating point operations (ops) required per degree of freedom and also require indirect memory addressing (only in the horizontal, assuming a structured vertical data structure). However, we are moving to a computational situation where ops form a minor part of the cost, and the cost of indirect addressing can be effectively hidden by sufcient vertical computation. Triangular meshes require more edges to span a particular domain with comparable resolution to quadrilaterals, but equally can have fewer edges if areas can be identied where reduced resolution is required. They have an advantage over structured mesh models in that computation is only over sea points, but this is a marginal advantage with a large number of cores when land-only cores are excluded and/or sophisticated load-balancing algorithms for domain decomposition are used (e.g. k partitioning; Ashworth et al., 2004). There is no a priori reason why either class of model is more or less scalable than the other, simply on the basis of grid and data structure: the answer in practice lies in the details of numerics, e.g. choice between implicit and explicit time stepping and the number of halo exchanges needed for high-order advection schemes, and the success of the optimisation in a specic context.The higher computation per degree of freedom may weigh in favour of unstructured grid models in terms of relative scalability. Historically, there has been a substantially computational penalty, e.g. the nite element model FESOM1.4 was reported to be 10 times slower than comparable structured

grid models (Wang et al., 2014) and experience with the FVCOM in the northwest European shelf suggest this model is

5 times slower than NEMO. More recent work suggests a very different picture: MPAS quotes a penalty of 3.4 compared with POP (Ringler et al., 2013) and the nite volume FESOM2 (Danilov et al., 2016) code reports a through-put 5 times faster than FESOM1.4. The comparison of this model with two ner-resolution structured grid models in Table 3 suggests this model is, if anything, more efcient. This may be because the time step ratio (of 3) between this model and the NEMO and MOM6 cases is substantially larger than the ratio of nominal grid resolutions (15 km / 9.3 km = 1.6; cf.

the ratio of efciencies in Table 3: 1.6); i.e. the unstructured mesh model achieves a longer time step maybe because it can have a more uniform grid (recalling the nest cell in

the ORCA12 grid is 1.3 km). So these anecdotal results (accepting different problem sizes, runs with different processor counts and on different computers are being compared here) suggest parity in resource cost and turnaround time between present-day structured and unstructured mesh models is a realistic prospect. However, the structured grid models are themselves being continually and extensively optimised (e.g. for OpenMP parallelisation), so there is also the possibility that a gap similar to the MPAS-POP comparison persists.

4.3 Exploiting future HPC architectures

Exploiting petascale or exascale levels of performance will require substantial algorithmic development to achieve the required level of concurrency. Many researchers have been looking at ways to improve the parallel scalability of ocean models on massively parallel architectures. Within the context of NEMO, there has been work looking at hybrid MPI/OpenMP parallelisation strategies (Epicoco et al., 2017) and a port of the code using OpenACC directives targeting accelerator-based architectures (Milakov et al., 2013). The layered approach to software design (Ford et al., 2017) provides one way to achieve this, while retaining code that can still be straightforwardly developed by an ocean modeller.The key idea in this approach is the PSyKAl (parallel system, kernel and algorithm) separation of concerns. The ocean modeller should not have to be concerned about the (ever-increasing) complexity of the underlying computer and the computational scientist who optimises the code should not have to understand ocean processes. In practice, the separation is achieved by using domain-specic knowledge about the type of problem being solved (in particular, the fact that the majority of the available parallelism comes about through performing the same computations at each point in the model mesh). In PSyKAl, the ocean modeller is responsible for writing the algorithm and kernel layers while all performance optimisation (including all code related to parallelism) is restricted to the PSy layer. The algorithm describes the model computation in terms of logically global elds (e.g. add eld1 to eld2) while kernels implement the actual computation to be performed at a grid point, and the PSy layer distributes this across the model domain with a particular parallel optimisation approach.

This approach is in use for the new, nite-element LFRic atmosphere model being developed by the UK Met Ofce.It has also been applied to two different, nite-difference shallow water models. The rst, shallow, is a benchmark code originally developed by Paul Swarztrauber of NCAR.The second consists of (only) the free-surface component of NEMO and is therefore named NEMOLite2D. It has been demonstrated that any loss in performance resulting from the PSyKAl restructuring of these codes can be regained by optimising the PSy layer (Porter et al., 2016). The middle, PSy, layer of such models can be automatically generated from

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J. Holt et al.: Prospects for improving the representation of coastal and shelf seas 517

a knowledge of the kernels (as described in metadata) and the algorithm. Since this generation can be tuned to match a particular computer architecture (e.g. CPU or GPU) as well as being used to support different forms of parallel execution (e.g. distributed versus shared memory), the aim is that the model as a whole will be performance portable while requiring no changes to the natural science parts of the code.There are also substantial software engineering benets to this approach in that scientists no longer have to worry about writing correct parallel code (e.g. the Do I need to do a halo swap? problem) and optimisation experts are protected from introducing errors in the science.

4.4 The comparative cost of ocean models

To link the scale analysis and the computational issues, Table 4 lists a set of possible future model congurations, and an estimate of their relative resource cost with and without a time step penalty. The relative resource cost is calculated by

C =

taking the 1/4 model in 2011 as a baseline (Y0) for a routine high-performance global physical oceanography research model. From Fig. 10, P = 0.258 yr1 (i.e. doubling

every 1.2 years). So, for example, in 2017, a 1/12 model

uses a comparable fraction of the total computer resource available as a 1/4 model in 2011. There are many caveats to these estimates, not least the scientic development time needed to achieve the various stages, but they do serve as a reasonable guide to either encourage or constrain aspirations.

A key milestone in this growth is a 1/12 global model rened to 1/72 to resolve coastal ocean processes. This represents the amalgamation of the current state of the art of global- and regional-scale coastal ocean modelling. When this would be comparable to a 1/4 model in 2011 for an unstructured mesh multiscale approach depends critically on the efciency in the unstructured modelling technology. If these achieve parity with present-day structured grid models (S = 1), then this point is reached in 2021; if factors sim

ilar to the present-day MPAS experience persist, then this date becomes 2023. The estimate for the block-rened multi-scale approach is 2022. All of these are sufciently ahead of the gure of 2026 for a 1/72 global model, assuming static computational efciency for the structured grid model (i.e. the development effort is primarily toward scalability and reducing SYPD rather than reducing resource requirement).This sets a clear challenge for ocean model developers and computer scientists to develop an efcient and accurate multiscale approach by this date.

The considerations above have focused on high-resolution physical ocean models, e.g. as part of a coupled climate model or an operational forecast system. For Earth system models with complex marine and land surface ecosystem and atmospheric chemistry components, we must accept that the routine model of today (2016) is a 1 resolution ocean. The scaling then suggests that a 1/12 global model and a 1/12 global model rened to 1/72 would not have a comparable computational cost to a nominal 1 ocean model until 2027 and 2035, respectively. This suggests options to improve the coastal ocean in centennial-scale ESM simulations (e.g. for fully coupled carbon cycle simulations) will remain highly parameterised for at least the next decade, and at least 2 decades for ne-scale processes.

5 Conclusions

The analysis and investigation presented here suggest the prospects for improving the representation of the coastal ocean in global models are now promising. We can identify three concurrent avenues of development to achieve this.Firstly, global models are now routinely run at the horizontal resolution of past shelf sea model simulations that capture many of the pertinent scales, and with dynamics that allow the representation of relevant processes, such as split-explicit time stepping rather than long wave-ltered or implicit ap-

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Lmin25

N N25

Lmin S, (5)

where N is the total number of grid cells in a conguration dened by Lmin, and N25 and Lmin25 are the reference values for nominal 1/4 global NEMO conguration, ORCA025. The inverse ratio of length scales is included in Eq. (5) to introduce a global time-stepping penalty on the assumption that local time-stepping approaches have not been implemented (except in the block-rened approach). S is a factor for models that need unstructured meshes, which, following the discussion above, we take to range from 3.4 (e.g. Ringler et al., 2013) to 1 (parity between structured and unstructured cases). Here, we focus on resource rather than turnaround time (SYPD) and, without the scalability and time-stepping improvements identied above, an increased resource may only be utilisable by waiting longer for a ner-resolution model to nish. Moreover, this simplistic cost model ignores all the real-world issues that would have to be faced, notably the changing balance between computation, memory access and communication, and also all arising data handling and storage issues.

Three quasi-uniform structured meshes, three unstructured mesh multiscale options and an example of a block-rened multiscale case are considered. For the block-rened approach, we assume a variable time step and assign a time step penalty (reducing it by 1/F ) for each block independently. This assumes the model is load balanced and optimised for the nest meshes, so again the measure here is the resource used rather than time to completion. To estimate when these could become routine models, an exponential t to the growth of RCUK computer peak performance (Fig. 10) is used so that

Y = int(log10 (C)/P + Y0), (6)

518 J. Holt et al.: Prospects for improving the representation of coastal and shelf seas

Table 4. Possible model grids, their costs (Eq. 5) and when they might be computationally equivalent to ORCA025 model (nominal 1/4 ) in 2011 based on Eq. (6), from Fig. 10. Unstructured grids are rened to resolve the minimum of L1, Lbt, LT according to Eqs. (3) and (4).

The block-rened approach allows the time step to vary between blocks; in other cases, it is limited by the global minimum scale. S is cost penalty for unstructured grid models.

S/US Vertical Size Cost vs. ORCA025 When routine physics model

Global scale (k cells) No time step, S = 1 S = 1 S = 1 S = 3.4 1/4 S 75 905 1 1 20111/12 S 75 8149 9 27 20171/36 S 100 73 342 108 972 20231/72 S 100 293 370 432 7776 20261/4 + 1/12 US 100 2037 3 9 2015 2017

1/12 + 36 US 100 10 910 16 145 2019 2021

1/12 + 1/72 US 100 17 409 26 461 2021 2023

1/12 + 1/72 BL 100 45 558 50 593 2022

S: structured; US: unstructured; BL: blocked-rened, e.g. using AGRIF.

proaches. In this case, some (comparatively) straightforward developments can be included in the simulations to signicantly improve the representation of the coastal ocean. These are (i) including tides, their generating forces, self-attraction and loading and wave drag effects; (ii) using vertical coordinate systems that retain resolution in shallow water, resolve the benthic boundary and allow smooth ow over steep topography; (iii) adopting vertical mixing schemes that represent mixing at the surface, pycnocline and benthic boundary layers. These are all existing features of regional ocean models and the general challenge here is ensuring the introduction of these features does not compromise the deep and open ocean simulation, or signicantly increase the computational costs; the single example in Table 3 suggests this would not be the case. Further developments to achieve this are likely, for example, through non-diffusive advection schemes and quasi-isopycnal vertical coordinates. We quantify the benet of improved process representation within the context of the current state of the art in global resolution. This shows substantial benets in including tides in terms of reproducing the seasonal stratication cycle, although interestingly, two of the CMIP5 models (including tidal mixing parameterisations) perform particularly well.

The second area of development is the continued renement of horizontal resolution to the point that the pertinent scales are well resolved (estimated to be 1.5 km). This

is the case in the current generation of region models, and the analysis presented here suggests it would be computationally practical in about a decades time. The options considered here, in very general terms, are a continued renement of the quasi-uniform structured mesh, some form of unstructured mesh (presumed to be either nite element or volume), or else a multi-blocking renement (whereby rectangular regions are rened to a fraction of the parent mesh and two-way coupled to it). The block renement and unstructured mesh approaches show signicant advantages over

the rened structured mesh using the objective renement criteria and very simple cost model considered here. The block-rened approach uses 13 times less computational

resource. The resources needed for the unstructured mesh approach depend critically on the relative performance of this class of model; here, we estimate 517 times less resources depending on how close to parity with structured grid models the unstructured models can achieve.

These results need to be seen alongside the needs of the open ocean model. For example, Grifes et al. (2009) note (in the context of mesoscale eddies) There is no obvious place where grid resolution is unimportant. The renement criteria we have considered here, while chosen for coastal ocean processes, have been applied globally. It is apparent that a modest-resolution model rened to the level of current high-resolution global models offers only marginal benets when an objective renement approach is used. For example, a 1/4 model rened to 1/12 only differs from a 1/12

global model by a factor of 4 fewer grid cells: much of the ocean is rened to meet, e.g. the Rossby radius criteria, globally. If the criteria was extended to include additional aspects, e.g. ocean variability (Sein et al., 2016), then this factor will reduce, and the benets of the multiscale approach become less apparent. If, however, ne-resolution process representation is desirable, then the scaling clearly favours multiscale modelling, and if we are sufciently condent in the renement criteria to use a coarser base resolution than would be otherwise be chosen (i.e. to allow a degree of coarsening from a contemporary high-resolution model), then the multiscale approach can achieve a substantial reduction in the resource needed.

The nal area of development, and by no means the least important, is the improved representation of the coastal oceans through improved process parameterisation. This essentially uses fundamental theoretical and empirical understanding to make up for deciencies in the dynamical ap-

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J. Holt et al.: Prospects for improving the representation of coastal and shelf seas 519

proach and the computational resource. This covers both processes that would not be resolved by any scales considered here and the cases where signicant horizontal renement is not practical (e.g. centennial-scale ESMs). Particular areas that deserve attention are tidal mixing, topography and coastlines, horizontal mixing schemes that account for the large change in scales at the ocean margins, and river plumes.Given that the scale analysis presented here suggests we may be 1 or 2 decades away from a well-resolved coastal ocean routinely run in fully coupled complex ESMs, these parameterisations are paramount.

This conclusion describes three complementary strands of work, which together have the potential to make substantial progress on our ability to model the coastal ocean at a global scale, and thus our ability to simulate global change and its impact on the societally pressing questions.

6 Code and data availability

NEMO model code used to run the northern North Atlantic Model conguration can be obtained from http://forge.ipsl.jussieu.fr/ipsl/forge/projets/nemo/svn/branches/NERC/dev_r3874_FASTNEt

Web End =forge.ipsl.jussieu.fr/ipsl/forge/projets/nemo/svn/branches/ http://forge.ipsl.jussieu.fr/ipsl/forge/projets/nemo/svn/branches/NERC/dev_r3874_FASTNEt

Web End =NERC/dev_r3874_FASTNEt .

Data used to prepare Figs. 1ac, 2, 4, 5, 6, 7, and 8 are provided at ftp://ftp.nerc-liv.ac.uk/pub/general/jth/GMD_Holt_GloabalCoasts/

Web End =ftp://ftp.nerc-liv.ac.uk/pub/general/jth/GMD_Holt_ GloabalCoasts/.

CMIP5 data are available from http://pcmdi.llnl.gov/search/cmip5/

Web End =pcmdi.llnl.gov/search/ http://pcmdi.llnl.gov/search/cmip5/

Web End =cmip5/ .

Data used for Fig. 1d are from http://volkov.oce.orst.edu/tides/TPXO7.2.html

Web End =volkov.oce.orst.edu/tides/

http://volkov.oce.orst.edu/tides/TPXO7.2.html

Web End =TPXO7.2.html , and for Fig. 3 from http://www.cgd.ucar.edu/cas/catalog/surface/dai-runoff/coastal-stns-Vol-monthly.Constructed.wateryr-v2-updated-oct2007.nc

Web End =www.cgd.ucar.edu/ http://www.cgd.ucar.edu/cas/catalog/surface/dai-runoff/coastal-stns-Vol-monthly.Constructed.wateryr-v2-updated-oct2007.nc

Web End =cas/catalog/surface/dai-runoff/coastal-stns-Vol-monthly. http://www.cgd.ucar.edu/cas/catalog/surface/dai-runoff/coastal-stns-Vol-monthly.Constructed.wateryr-v2-updated-oct2007.nc

Web End =Constructed.wateryr-v2-updated-oct2007.nc http://www.cgd.ucar.edu/cas/catalog/surface/dai-runoff/coastal-stns-Vol-monthly.Constructed.wateryr-v2-updated-oct2007.nc

Web End = .

Figures 4 and 6 uses EN4.0.2 prole data from http://www.metoffice.gov.uk/hadobs/en4/download-en4-0-2.html

Web End =www. http://www.metoffice.gov.uk/hadobs/en4/download-en4-0-2.html

Web End =metofce.gov.uk/hadobs/en4/download-en4-0-2.html http://www.metoffice.gov.uk/hadobs/en4/download-en4-0-2.html

Web End = .

Information for Fig. 10 was obtained from http://www.hpcx.ac.uk/services/hardware/

Web End =www.hpcx.ac. http://www.hpcx.ac.uk/services/hardware/

Web End =uk/services/hardware/ , http://www.hector.ac.uk/service/hardware/

Web End =www.hector.ac.uk/service/hardware/ and www.archer.ac.uk/.

Competing interests. The authors declare that they have no conict of interest.

Acknowledgements. This work was supported by the NERC Next Generation Ocean Dynamical Core Roadmap Project, and the National Capability programme in ocean modelling at NOC and PML. J. I. Allen is supported by the NERC Integrated Modelling for Shelf Seas Biogeochemistry Programme (NE/K001876/1). H. T.Hewitt and R. Wood were supported by the Joint UK DECC/Defra Met Ofce Hadley Centre Climate Programme (GA01101). J. Harle is supported by the NERC RECICLE project (NE/M003477/1).

Edited by: R. MarshReviewed by: S. Danilov and R. Proctor

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