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Ocean Circulation Modeling for Operational Oceanography: Current Status and Future Challenges

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Ocean Circulation Modeling for
Operational Oceanography: Current
Status and Future Challenges
Julien Le Sommer, Eric P. Chassignet and Alan J. Wallcraft
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without prior written permission from GODAE OceanView.
Julien Le Sommer1, Eric P. Chassignet2, and Alan J. Wallcraft2
1Univ. Grenoble Alpes, CNRS, IRD, IGE, Grenoble, France; 2Center for Ocean-Atmospheric Prediction
Studies (COAPS), Florida State University, Tallahassee, FL, USA
This chapter focuses on ocean circulation models used in operational oceanography, physical oceanography
and climate science. Ocean circulation models are a particular branch of ocean numerical modeling that
focuses on the representation of ocean physical properties over spatial scales ranging from the global scale
to less than a kilometer and time scales ranging from hours to decades. As such, they are an essential build-
ing block for operational oceanography systems and their design receives a lot of attention from operational
and research centers.
cean modeling is an important branch of operational oceanography. Ocean numerical
models are an essential building block of global and regional operational oceanography
systems. In this context, ocean circulation models are used in conjunction with data
assimilation for extrapolating both in space and in time the available satellite and in situ oceanic
observations in order to build a physically consistent estimate of the ocean state and its evolution.
Ocean modeling is a relatively recent discipline in the field of oceanography. The underlying
principles of the algorithmic formulation of ocean circulation models were first proposed in the
1960s by Bryan (1969; see also McWilliams [1996] for an historical review). Since then, the
continuous increase in computing power has allowed ocean circulation models to provide ever more
meaningful and comprehensive descriptions of the ocean circulation. Ocean modeling is now
recognized as an essential supplement to more traditional scientific methodologies in oceanography.
The variety of oceanic physical processes accounted for in ocean circulation models has also notably
broadened over past decades. The original scope of ocean circulation models was to describe
oceanic properties and physical processes at scales significantly larger than the mesoscale
(horizontal scales on the order of 100 km and time scales on the order of three months). But the
increase in computing power and the improved physical consistency of their formulation now allow
ocean circulation models to resolve routinely oceanic flows down to the submesoscale (horizontal
Le Sommer, J., E.P. Chassignet, and A.J. Wallcraft, 2018: Ocean circulation modeling for operational oceanography: Current
status and future challenges. In "New Frontiers in Operational Oceanography", E. Chassignet, A. Pascual, J. Tintoré, and J.
Verron, Eds., GODAE OceanView, 289-306, doi:10.17125/gov2018.ch12.
scales on the order of 10 km; Chassignet and Xu [2017]) and to describe internal wave and internal
tides at a global scale (Shriver et al., 2012).
The range of uses and applications of ocean circulation models has diversified over the past
several decades. Ocean circulation models are now used standalone for simulating ocean
circulation; when integrated in a data assimilation framework, they may be used for producing
short-range ocean forecasts and for ocean reanalysis; coupled to atmospheric circulation models,
they may be used for seasonal to decadal forecasts; and fully integrated into Earth System Models,
they are crucial to climate modeling. Scientists also use ocean circulation models as the
experimental tool of choice for improving our mechanistic understanding of the ocean.
The variety of oceanic physical processes accounted for in ocean circulation models has also
broadened in recent decades. Ocean circulation models were initially used to describe oceanic
properties and physical processes at scales significantly larger than the mesoscale (horizontal scales
on the order of 100 km and time scales on the order of three months). But the increase in computing
power and improved physical consistency of their formulation now allow ocean circulation models
to resolve routinely oceanic flows down to the submesoscale (horizontal scales on the order of 10
km; Chassignet and Xu [2017]) and to describe internal wave and internal tides at the global scale
(Shriver et al., 2012; Savage et al., 2017).
However, despite the maturity and broad range of applications of ocean circulation models
achieved to date, ocean modelers still face notable challenges. Running ocean circulation models at
increased model resolution does not solve the problem of subgrid scale closures (unresolved
processes), and may even question some of the underlying assumptions and algorithms of current
ocean circulation models (hydrostatic vs. non hydrostatic, for example). The modularity of modern
geoscientific models requires the design of robust and physically rational approaches for coupling
ocean circulation models with other model components. Most importantly, there is a growing
concern about the necessity to describe explicitly how model uncertainty propagates in geoscientific
modeling systems. This is why ocean circulation model design is still a very active field of research
and will likely remain so well into the future.
In this chapter, we aim to: (i) briefly describe the principles that underpin the formulation of
ocean circulation models, (ii) review current skills of ocean circulation models, and (iii) present
what we believe are the new frontiers in ocean circulation model design. Obviously, given the
complexity of the issues at stake and the amount of energy that is put into ocean model development,
this chapter will only briefly touch upon the above objectives. Our goal is therefore not to cover
these questions thoroughly, but rather to provide an entry point to the science of ocean models. The
second section of this chapter focuses on the science and applications of ocean circulation models;
the third section describes ongoing research avenues geared toward improving the representation of
physical processes in ocean circulation models; and the fourth section describes the ongoing
paradigm shift in ocean modeling towards a more probabilistic description of oceanic flows. More
general considerations regarding the future of ocean modeling are presented in the conclusion.
Ocean modeling is a branch of numerical modeling that focuses on representation of the physical
mechanisms governing the evolution of ocean physical properties, namely T, S, u, v, and w, where
T is temperature, S is salinity, and u, v, and w are the horizontal and vertical components of the
velocity V. A good understanding of a range of physical processes is required in order to build a
numerical model that is capable of faithfully representing the ocean circulation. Without this
physical understanding, it is easy to derive erroneous conclusions since a numerical model is
constructed using discretized equations of motion. Indeed, direct numerical simulations (DNS) of
the ocean (i.e., numerical representation of the smallest turbulent scale, i.e. the Kolmogorov length
scale, which is on the order of 1 cm (Smyth et al., 2001)), is not possible with present-day
computers. The largest simulation achievable today is on scales of 10 m (Yeung et al., 2015).
At present, we cannot represent these small scales; we can only achieve a truncated
representation of the ocean and this will remain the case for the foreseeable future. The spatial and
temporal scales that one can currently represent strongly depends on the model configuration and
application. High resolution operational oceanography requires accurate depiction of upper ocean
structure and mesoscale features such as eddies and meandering fronts. Accurate sea level
representation is crucial for coastal models (response to wind, tides, and surface pressure) and
seasonal-to-interannual forecasts require a good representation of the upper ocean mass field and
the coupling to an active atmosphere. On global and basin scales, high horizontal resolution (1/10o
to 1/25o and very rarely to 1/50o; Chassignet and Xu [2017]) is mostly used for ocean “weather”
and seasonal-to-decadal variability (Fig. 12.1). The emphasis is on short integrations (years to
decades) and most models of this class are coupled to a sea-ice model but are stand-alone and use
prescribed atmospheric fields. Coarser resolution (1/4o to 1o) is principally used for climate
applications (Griffies et al., 2000). The emphasis for this class of models is on long integrations
with fully coupled ocean-ice-atmosphere models.
Figure 12.1. Example of submesoscale permitting modeling – March 1st surface vorticity over the Gulf Stream
in a1/50° North Atlantic domain (~1.5 km horizontal grid spacing). Adapted from Chassignet and Xu (2017).
In practice, ocean models consist of a numerical solution to a set of partial differential equations
(PDEs) describing the ocean dynamics. These PDEs are based on an approximated version of
Navier-Stokes equations adapted to our regimes of interest (see the review papers by Griffies [2004]
and Griffies and Treguier [2013] for details). An ocean numerical model typically consists of 20,000
to 200,000 lines of code, usually written in FORTRAN, and can easily take up to 10 years of
community development to be fully functional.
The first ocean numerical model was put forward by Bryan (1969) following in the footsteps of
numerical weather prediction. The first discretized equations for an atmospheric application were
solved manually by Richardson (1922) and numerically by Charney et al. (1950). The latter
consisted of a 15 by 18 grid (∆x = 736 km) and demonstrated that numerical weather prediction was
possible. The latest configurations use horizontal resolution x on the order of 1 km (Chassignet
and Xu, 2017). The sheer size of the ocean problem is such that it will always require the latest
generation of supercomputers, and ocean/climate models have traditionally been one of the biggest
users of computer resources. This means that close collaborations with computer scientists are
essential to ensure that the numerical codes run efficiently and take full advantage of the computing
architecture, as this changes constantly and ocean modelers need to be flexible and ready to adopt
new approaches. The main limitation is not computational speed of the processors, but access to
memory and latency in reading/writing on disk drives (I/O). This limitation is not likely to change
in the near future since supercomputer development is closely linked to the performance of
commodity chips, which are not well-adapted to ocean applications (i.e., GPUs) and do not facilitate
memory access.
There is a wide range of applications for ocean models. Without data assimilation, they are
mostly used to scientifically rationalize the observed ocean by testing mechanisms underlying
observations via idealized or realistic configurations. With data assimilation, they are used to
perform hindcasts in order to understand past evolution and to perform forecasts that can be used
for societal applications (Chassignet and Verron, 2006; Dombrowsky et al., 2009; Schiller and
Brassington, 2011; Bell et al., 2015). Observational data via data assimilation sets the stage for
model state estimates and forecasts (Chassignet et al., 2009). The quality of the estimates and the
forecast will depend on the ability of an ocean numerical model to faithfully represent the resolved
dynamics of the ocean and the parameterized subgrid scale physics. It is therefore important to
realize that even using an infinite amount of data to constrain the initial conditions of an ocean
model will not necessarily improve the forecast when using a poorly performing ocean numerical
It is also important to realize that while numerical models are necessary to understand ocean
dynamics, they cannot represent reality because of limitations in computational power, incomplete
understanding of subgrid scale parameterizations, poorly known forcing fields, and poorly
understood interactions with other components of the earth’s system such as the atmosphere and
sea ice. Observations are also not an accurate representation of reality because of the many space
and time gaps in the observations that give us only limited information about the ocean’s state, its
variability, trends, and possible instabilities and regime shifts.
In summary, while numerical models allow for hypothesis testing and experimentation, one
needs to understand their limitations in order to use them to their full potential. Good modelers are
aware of the strengths AND weaknesses of their models, and the main difficulty is the quantification
of the truncation errors introduced by the discretization of the Navier-Stokes equations.
Truncation errors arise from the discretization of PDEs derived from the Navier-Stokes
equations, which usually assume that the ocean is incompressible and make the spherical
approximation (assumes the earth is a sphere), the thin-shell approximation (allows to neglect
variations of the local rotation rate with respect to depth, therefore simplifying the treatment of the
Coriolis acceleration), the Boussinesq approximation (neglects variation of relative density in the
horizontal momentum equations), and the hydrostatic approximation (allows to neglect the vertical
The primitive Navier-Stokes equations are
with Uh is the horizontal velocity vector (u,v); U, the three-dimensional velocity (u,v,w); f, the
Coriolis parameter; p, the pressure; g, the gravity; , the density; T, the temperature; S, the salinity;
D, the dissipation; and F, the forcing. Temperature and salinity here implicitly refer to conservative
temperature and absolute salinity as defined in the Thermodynamic Equation Of Seawater - 2010
(TEOS-10; McDougall and Barker, 2011). The equation of seawater relates density to temperature,
salinity and pressure:
Several other equations are required for setting boundary conditions at the ocean surface and at the
seafloor. For instance, ocean circulation models enforce the following conditions at the ocean
- Kinematic boundary condition
- Surface pressure condition
- Air-sea fluxes of freshwater and heat
- Air-sea fluxes of momentum
where  is the sea surface height, E, the evaporation; P, the precipitation; , the wind stress; K, the
diffusivity; A, the viscosity; and Q, the heat flux. Finally, one needs to use the turbulent closure
hypothesis to close the system by assuming that the nonlinear terms involving correlations of
variables at unresolved small scales can be fully prescribed from knowledge of the large scales.
In this section, we review some of the issues that arise as we strive to improve the representation of
key physical processes in the ocean model component of operational systems. This includes
broadening the spectrum of explicitly represented processes and improving the representation of
unresolved processes in ocean circulation models.
As discussed in the previous section, ocean circulation models do not account explicitly for the
entire range of scale interactions that control ocean circulation (Fig. 12.2). Because of the turbulent
nature of oceanic flows, ocean circulation at a given scale is indeed fundamentally dependent on
oceanic motions at scales ranging from global (of order 10,000 km) to dissipative (of order 1 cm;
Fig. 12.3). But the finite grid resolution of a particular ocean model configuration constrains the
spectrum of scales of motions that are explicitly represented in the model solution. Consider, for
instance, ocean mesoscale eddies, which are known to play a fundamental role in shaping ocean
circulation (McWilliams, 2008) and their representation in ocean models, which has motivated a
large number of studies. Even the most high-end global ocean circulation models with grid
resolutions down to just a few kilometers (Chassignet and Xu, 2017) are still not able to fully capture
the dominant length scales of mesoscale variability at high latitudes (Hallberg, 2013). Scale
interactions involving oceanic mesoscale eddies are therefore not explicitly represented at high
latitudes in these models (Fig. 12.4).
Figure 12.2. Range of ocean physical processes (adapted from Griffies and Treguier [2013]).
Figure 12.3. Space and time scale of ocean physical processes.
Therefore, a practical challenge for most applications of ocean circulation models is to describe
the broadest possible spectrum of oceanic scales of motion at a given computational cost. The
overall computational budget of an application is constrained by the available computing resources
and by the expected time-to-solution, the latter constraint being particularly limiting in operational
applications that deliver near real-time products. The model grid of ocean model components of
operational systems is also usually constrained by other practical factors. The same model grid is
often used for several years because of the effort required to tune the physical model configuration,
calibrate the data assimilation components, and prepare the downstream data production chains.
Given these constraints, the optimal design of ocean model components in operational systems
requires a robust and rational understanding of a priori what controls the spectrum of resolved
process in ocean circulation models besides the model grid resolution. Developing this
understanding of what controls the spectrum of resolved scales in ocean model should rely on
idealized process studies, model inter-comparison exercises and confrontation of model solutions
with observations.
A commonly accepted practical criteria for deciding whether a flow feature is resolved in an
ocean model is that there should be no less than 2 grid points spanning the feature (see, for instance,
Griffies and Treguier [2013]). This is supported by the notion of effective resolution, defined by
Skamarock (2004) as the smallest resolved scale that is not significantly affected by numerical
dissipation due to discretization errors. The effective resolution of an ocean circulation model is
about 6 to 10x, where x is the horizontal grid resolution, but the validity of the concept of
effective resolution is generally restricted to model configurations with small enough grid
resolutions and using high order discretization schemes (Soufflet et al., 2016).
Although effective resolution allows us to characterize the smallest scale of motion that could
ideally be described without numerical error in an ocean model, it is generally impossible to
distinguish unambiguously between resolved and unresolved physical processes in ocean
circulation models. Indeed, only the physical processes with dynamics that do not depend on scales
smaller than the effective resolution are unambiguously represented in a given ocean model
configuration. In practice, a lot of physical processes of interest involve scales between the model
grid scale and the model effective resolution. In such cases, although the model is a priori unable to
fully account for the dynamics, the model solutions may still exhibit some properties of the
misrepresented physical process. A good example of such a situation is discussed in a recent study
by Uchida et al. (2017), who found signatures of submesoscale mixed layer instabilities in their
1/10 resolution model although their model grid resolution is a priori not able to adequately capture
these dynamics.
Using computationally efficient numerical discretization schemes with good mimetic properties is
a first approach for broadening the spectrum of resolved processes in ocean circulation models at a
given computational cost. The notion of computational efficiency refers to the accuracy of model
solution relative to the computational cost of numerical integration. The notion of mimetic
properties refers to the ability of a discrete model to emulate the properties of the underlying
continuous mathematical model, for instance as conservation laws and geometrical symmetries.
Ocean circulation models with grid resolutions approaching 1 km can be sensitive to numerical
discretization schemes (Ducousso et al., 2018). This is because ocean model solutions are generally
more nonlinear and less controlled by diffusion in this range of resolution. The current trend in
ocean circulation modeling is, therefore, to use high order discretization schemes in order to
increase model effective resolution (Soufflet et al., 2016).
Multiscale modeling approaches coming from coastal ocean modeling also provide a framework
for locally increasing the effective resolution according to a particular scientific or operational
objective and at a given computational cost. Multiscale modeling methods, which allow us to locally
refine the grid mesh, are now commonly used in large-scale ocean circulation modeling.
Applications of multiscale approaches in ocean circulation modeling can be classified according to
their grid mesh topology into structured mesh methods (for instance as orthogonal curvilinear grids
used by most ocean circulation models), unstructured mesh methods (e.g., see Piggott et al., 2008;
Danilov et al., 2017; Ringler et al., 2013) and block structured mesh methods (see Blayo and
Debreu, 1999; Jablonowski et al., 2006). Although many research efforts have focused on
unstructured mesh ocean circulation modeling over recent years, the design of robust discretization
schemes for large-scale ocean models on triangular or hexagonal meshes remains challenging
(Danilov, 2013). And it should also be noted that it is still unclear how to define objective criteria
for adjusting meshes and, therefore, optimally use the potential for mesh refinement and mesh
adaptability of multiscale modeling approaches in large-scale ocean circulation models.
Improving the performance of ocean circulation models on modern high performance
computing (HPC) platforms is also essential for broadening the spectrum of resolved scales. The
current trend in HPC is toward massively parallel machines with heterogeneous multicore
architectures (Giles and Reguly, 2014). However, while modern HPC platforms can deliver a peak
performance in the Petaflop/s range, existing ocean circulation models are unable to exploit this
potential. The computational intensity, i.e., floating point operations per memory access, of ocean
models is very low because they are dominated by stencil operations (typical of discretized PDEs).
So, they typically run at ~5% of the system’s peak speed. Also, ocean models have a very small
vertical dimension, typically O(10), so they scale more like 2-D domains than 3-D domains. In
practice, 1/12o and 1/25o global ocean models might scale well to 8,000 and 30,000 cores,
respectively. Scalability is eventually limited by the communication overhead, load imbalance, and
latency inherent to spatial 2-D domain decomposition and by I/O overhead. Single processor
performance depends on computational intensity, which can be improved by using higher order
discretization schemes; but scalability can actually be reduced by better single processor
performance because relatively more time is spent in communications.
In the future, overcoming single processor and scalability bottlenecks will require sustained
collaborations between ocean modelers and computer scientists. Practical approaches will likely
involve an increase in hybrid parallel programming in order to more efficiently exploit memory
hierarchy and innovative algorithms for solving the set of PDEs that govern ocean dynamics, as for
instance parallelization in time in addition to spatial domain decomposition (Schreiber et al., 2017).
Looking ahead, because of the multiscale nature of oceanic flows and the needs of end-users of
operational oceanography, it is necessary for ocean circulation models to account explicitly for the
broadest range of physical scales possible. But this broadening of the spectrum of resolved scales
will not simply be a consequence of an increase in computing power. How efficiently ocean
circulation models will use computing resources will depend on the efficiency of the algorithms
used for translating ocean dynamical equations into practical computation. And foreseeing what
approach will be most used in the future for broadening the spectrum of resolved scales is not
straightforward, but it is arguable that HPC and algorithmic aspects will be crucial in future ocean
model development. New advances will most likely rely on a high level of collaboration between
ocean modelers, applied mathematicians, and computer scientists.
Figure 12.4. Grid resolution required to resolve the first baroclinic radius with 10 grid points. The figure shows
the local grid resolution (in kilometers) divided by 10 as an estimate the grid resolution required for a model
to effectively capture structures with wavelength close to the first baroclinic Rossby radius. The first baroclinic
Rossby radius is estimated from a CMEMS 1/12° global reanalysis. The estimate of the number of points
required for the effective resolution to match the first baroclinic Rossby radius is slightly more restrictive than
that proposed by Hallberg (2013) following the work by Soufflet et al. (2016).
Ocean circulation models need to allow for the important physical processes that are not accounted
for in their discrete approximation to the ocean dynamical equations. A common approach to the
representation of unresolved processes in ocean circulation models is to use a subgrid scale (SGS)
closure (aka physical parameterization). Typically, these do not focus on particular nonlinear terms
in the primitive equations, but rather on particular physical processes (that may affect several terms).
A rough classification of unresolved physical processes would include: (i) SGS closures for
balanced turbulence (eddy closures), (ii) SGS closures for processes in the ocean surface boundary
layer, (iii) SGS closures for interior mixing processes, and (iv) SGS closures for bottom boundary
layer processes. The design of SGS closures associated with physical processes usually involves
some combination of idealized process studies with models, realistic process studies with models,
targeted field experiments at sea, and (more rarely) lab experiments. Importantly, a SGS closure
should always target a particular ocean circulation model resolution because the information at the
resolved scale depends on the model resolution.
Depending upon the available information from the scales assumed to be resolved in the target
model, two categories of closures can be distinguished. The first is a RANS-type (cf Reynolds
average) closure, implicitly based on some sort of ensemble average (in practice emulated through
time average); for example, the Gent-McWilliams closure for mesoscale eddy transport (Gent,
2011). The other is a LES type closure (by analogy to Large Eddy Simulation) implicitly based on
some spatial average; for example, Leith’s (1968) momentum closure. LES-type subgrid closures
can further be subdivided into functional and structural closures. Functional closures consider the
bulk action of the subgrid terms on the resolved scales (e.g., Smagorinsly momentum closures
[Griffies and Hallberg, 2000]). Structural closures aim at estimating the best local approximation
of the unknown SGS term by constructing it from the known small-scale stuctures (e.g., closures
defined by Taylor series expansion). A recent trend in the design of SGS closures in ocean
circulation modeling deals with how to adapt them to the resolved physical scales of motions in the
model solution (Bachman et al., 2017). There is also a growing concern in ocean models design
regarding the complex interplay between numerical discretization errors and SGS closures,
especially in the finest resolution applications (Campin et al., 2011; Lemarie et al., 2012). Another
emerging approach to the representation of unresolved processes in ocean circulation models is to
couple ocean circulation models with third-party code components, which are specifically designed
to describe the evolution of a particular class of processes. Physical processes that are not explicitly
accounted for in ocean models have been extensively studied (e.g., wave modelling) and estimates
of the subgrid states associated with these unresolved processes must take into account non-local
effects in space and time (e.g., propagation and dissipation of internal waves, cf. IDEMIX [Olbers
and Eden, 2013]). It may also make sense to modularize, as much as possible, a complex closure so
that it can be used in different ocean models and thus become an independent model library or
component (e.g., How to fully integrate multiple code components,
each describing different aspect of the dynamics, in operational systems with data assimilation is
likely to become a key research question in the future. A particularly important concern is associated
with the representation of the fine-scale feedbacks between the atmospheric and oceanic boundary
Classical approaches in the design of SGS closures for ocean circulation models assumes that the
bulk effect of SGS fluctuations can be deterministically predicted from the resolved scales of
motions. A number of recent works have started to reconsider this assumption. These more recent
studies allow the closure to be only weakly constrained by the resolved scales. Carefully designed
stochastic parameterizations appear to be a promising approach to the representation of a large class
of unresolved SGS processes in ocean circulation models (Berloff, 2005; Palmer and Williams,
2010; Williams, 2012; Grooms and Majda, 2013; Porta Mana and Zanna, 2014; Cooper and Zanna,
2015; Brankart et al., 2015; Andrejczuk et al., 2016). A certain amount of randomness is introduced
with a stochastic procedure to account for the possible variability of fluxes at macroscales.
Stochastic parameterizations seem particularly well-suited to the representation of the cross-scale
exchanges of energy and momentum for SGS-balanced turbulence (in particular, energy
backscatter). Notable improvement of Gulf stream dynamics has also been obtained through the
stochastic representation of upscaling due to the nonlinear nature of the equation of the state of sea-
water (Brankart, 2013). More generally, the concept of introducing random perturbations in ocean
models is also supported by another rationale coming from the theory of non-linear systems. The
topology of the phase space of non-linear systems can indeed be rather complicated, showing
multiple possible local energy minima. Adding perturbations could allow a system to escape from
local potential wells and, therefore, explore more reliably the range of possible states.
The paradigm underpinning this idea of stochastic parameterization is actually not restricted to
the design of SGS closures and could also allow us to account for more general sources of
uncertainty in ocean circulation models. These include errors in their initial condition (particularly
at depth), errors in the forcing function (uncertain atmospheric fields, uncertain parameter in bulk
formula. etcetera), errors associated with subgrid closures (closures are imperfect models, with
usually weakly constrained parameters), errors from the physical approximation use in for their
continuous formulation (e.g., non-traditional Coriolis terms), and numerical discretization errors
that tend to accumulate over time (even high order schemes have non-zero errors). Overall,
uncertainty is a major property of ocean circulation models that physicists often tend to neglect,
although some modeling frameworks are now tackling this problem more explicitly (e.g., see
Brankart et al., 2015). At this point, it is arguable that further developing stochastic
parameterizations for ocean circulation models will require a better understanding of their impact
on the resolved scales of motion.
Ensemble modeling is a now common approach in operational forecast and climate modeling for
accounting for the inherent uncertainty of geoscientific model solutions. Ensembles are, for
instance, routinely used in operational systems to account for uncertainty in initial states, model
formulation, or forcing. Because they sample the space of possible states given such uncertainty,
ensemble approaches allow for a more objective comparison of model solutions with observations,
which is key to operational systems. More recently, ensemble modeling has also become an
experimental tool of choice for investigating the sources of oceanic variability in eddying regimes
(Sérazin et al., 2017). It should be stressed that ensemble modelling is a reasonable strategy in terms
of high performance computing, because in most cases ensemble runs are entirely independent of
each other and therefore 100% scalable.
The combination of ensemble and stochastic methods allows us to deal more objectively with
uncertainty in ocean models (Toth et al., 2003; Palmer, 2012; Brunton et al., 2016; Bessieres et al.,
2017). The purpose of ongoing efforts is to attempt to explicitly sample the probability distribution
of possible states given explicitly formulated uncertainties following a Bayesian approach. This
shift towards probabilistic ocean circulation modeling is arguably a paradigm transition in our field.
But it is still unclear how to deal with the daemon of dimensionality. Indeed, ensemble sizes are
usually constrained to several tenth of members for practical reasons while the parameter space they
are sampling is orders of magnitude larger. The emergence of probabilistic ocean circulation
modeling could bring several benefits to ocean model design and usage in the future. But
probabilistic ocean modeling also raises several technical and scientific challenges that ought to be
addressed in the future.
In this chapter, we have briefly described the fundamental principles that underpin the formulation
of modern ocean circulation models, shown some of their recent achievements, and discussed what
we believe are the future frontiers in ocean circulation model development. This chapter has, in
particular, illustrated how a modeling strategy proposed in the 1960s for solving the primitive
equations has yielded ocean circulation models that are now used for a wide range of applications
and that form a building block of modern operational oceanography. We have also presented the
view that future frontiers in ocean circulation modeling will depend upon (i) the computational
performance of ocean models, (ii) their ability to represent scale interactions either explicitly or
though parameterizations, and (iii) the representation of model uncertainty with stochastic and
ensemble approaches.
This chapter illustrates how the field of ocean circulation model design has reached its maturity
and now involves strong collaboration between different fields of expertise. Ocean circulation
model design is a very active field of research with entire scientific teams dedicated to developing
or improving ocean circulation models. The maturity of the field is arguably a consequence of the
high level of collaboration and merging of efforts among different groups involved in different
applications and aspects of ocean circulation models. A key driver for this collaborative approach
to ocean model development is this notion of seamless geoscientific modeling, which suggests that
the same numerical code can actually be used for a range of different applications covering a range
of different resolutions and dynamical regimes (Hurrell et al., 2009). Overall, this approach has
improved the robustness of ocean circulation models and the sustainability of the ocean model
development process over recent decades.
Although not been discussed much in this chapter, we would like to stress the importance of
ocean observing networks for improving ocean circulation models. Sustained ocean observations
from satellites and in situ networks are critical for routinely assessing the skills and limitations of
circulation ocean models over different timescales ranging from days to decades. There is also mu ch
to be learned from targeted field observations aimed at documenting specific oceanic processes in
order to improve their representation in models. For instance, recent experiments documenting fine-
scale ocean processes in the ocean surface boundary layer provided a wealth of information that can
be used for improving surface processes in ocean circulation models (Shcherbina et al., 2015;
Buckingham et al., 2016).
An aspect that we believe has also been critical in the continuous improvement of ocean
circulation models over recent decades is the shared vision in the ocean modeling community that
open source is the only sustainable approach to geoscientific model development. Ocean model
developers have generally been early adopters of modern practices in software development (e.g.,
version control, unit testing, continuous integration). All the major ocean circulation codes are also
distributed under open source licenses and therefore exposed to the scrutiny of other research
groups. Arguably, workflows in ocean modeling could be improved and made more reproducible,
but it is fair to recognize that this community has long been concerned with these issues. Building
more open and transparent data processing chains for pre- and post-processing is probably an
important next step to making ocean circulation modeling more robust and reproducible in the future
(Stodden et al., 2016).
In conclusion, we would like to raise what is likely to become a key issue in the future
development of ocean circulation models. With a broadening of scope and an increase in the number
of users, ocean circulation models have also grown in complexity. But the actual size of the
community of ocean model developers is still rather small. Furthermore, most of the research
questions that have been raised in this chapter would require strong and sometimes new
collaborations among ocean modelers, field oceanographers, process-oriented oceanographers,
applied mathematicians, and computer scientists. Sustained and proactive initiatives from major
funding agencies (one good example of such an initiative being the US CLIVAR Climate Process
Team scheme) will certainly be vital to the success of these interdisciplinary collaborations and to
bringing more early career scientists in to contribute to the science of ocean circulation models.
EPC and AJW are supported by the Office of Naval Research and the Naval Research Laboratory.
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... In particular, evolving the core models to address adequate scales in space and time, accurately representing physical processes, and running fastly is the baseline for improving predictability, as well as past reconstruction of the blue, green and white ocean. These are the challenges that have to be tackled for the improvement of future OOFS. 5 Le Sommer et al. (2018) showed that the evolution in ocean modelling for operational oceanography is strictly connected to resolve physical processes down to the submesoscale (Chassignet and Xu, 2021) and to describe internal wave and internal tides at a global scale thanks to increase in computer power and improved physical parameterization (Shriver et al., 2012). Increasing resolution in space and time is not the only way to address high quality operational products: modularity of modern geoscientific models is key for addressing modelling complexity (Le Sommer et al., 2018). ...
... These are the challenges that have to be tackled for the improvement of future OOFS. 5 Le Sommer et al. (2018) showed that the evolution in ocean modelling for operational oceanography is strictly connected to resolve physical processes down to the submesoscale (Chassignet and Xu, 2021) and to describe internal wave and internal tides at a global scale thanks to increase in computer power and improved physical parameterization (Shriver et al., 2012). Increasing resolution in space and time is not the only way to address high quality operational products: modularity of modern geoscientific models is key for addressing modelling complexity (Le Sommer et al., 2018). ...
... Evolutions in High Performance Computing is then another pillar on which establishing OOFS; scientific questions to be solved require also to face technological challenges. Le Sommer et al. (2018) highlighted how the main current limitations in the modelling framework capacity is not due to computational speed of the 5. pdf/2020-10/CMEMS-requirements-satellites.pdf ...
... In particular, evolving the core models to address adequate scales in space and time, accurately representing physical processes, and running fastly is the baseline for improving predictability, as well as past reconstruction of the blue, green and white ocean. These are the challenges that have to be tackled for the improvement of future OOFS. 5 Le Sommer et al. (2018) showed that the evolution in ocean modelling for operational oceanography is strictly connected to resolve physical processes down to the submesoscale (Chassignet and Xu, 2021) and to describe internal wave and internal tides at a global scale thanks to increase in computer power and improved physical parameterization (Shriver et al., 2012). Increasing resolution in space and time is not the only way to address high quality operational products: modularity of modern geoscientific models is key for addressing modelling complexity (Le Sommer et al., 2018). ...
... These are the challenges that have to be tackled for the improvement of future OOFS. 5 Le Sommer et al. (2018) showed that the evolution in ocean modelling for operational oceanography is strictly connected to resolve physical processes down to the submesoscale (Chassignet and Xu, 2021) and to describe internal wave and internal tides at a global scale thanks to increase in computer power and improved physical parameterization (Shriver et al., 2012). Increasing resolution in space and time is not the only way to address high quality operational products: modularity of modern geoscientific models is key for addressing modelling complexity (Le Sommer et al., 2018). ...
... Evolutions in High Performance Computing is then another pillar on which establishing OOFS; scientific questions to be solved require also to face technological challenges. Le Sommer et al. (2018) highlighted how the main current limitations in the modelling framework capacity is not due to computational speed of the 5. pdf/2020-10/CMEMS-requirements-satellites.pdf ...
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The architecture of ocean monitoring and forecasting systems
... We may categorize interpolation methods according to the prior they assume to represent sea surface dynamics. While the operational SOTA product [4] exploits an optimal interpolation (OI) based on a covariance-based prior, assimilation-based products [5] relies on the assimilation of ocean circulation models [6]. Recently data-driven approaches, especially neural networks [7], have shown some success for the interpolation of sea surface fields from satellite-derived observation data. ...
... where δ t is the time step. A key issue in 4DVar frameworks is the parameterization of dynamical model M. General ocean circulation models [6] lead to the inversion of the full ocean state dynamics, which may be highly-complex and unstable when considering only sea surface observation. One may also consider dynamical representation of sea surface dynamics. ...
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Satellite radar altimeters are a key source of observation of ocean surface dynamics. However, current sensor technology and mapping techniques do not yet allow to systematically resolve scales smaller than 100km. With their new sensors, upcoming wide-swath altimeter missions such as SWOT should help resolve finer scales. Current mapping techniques rely on the quality of the input data, which is why the raw data go through multiple preprocessing stages before being used. Those calibration stages are improved and refined over many years and represent a challenge when a new type of sensor start acquiring data. Here we show how a data-driven variational data assimilation framework could be used to jointly learn a calibration operator and an interpolator from non-calibrated data . The proposed framework significantly outperforms the operational state-of-the-art mapping pipeline and truly benefits from wide-swath data to resolve finer scales on the global map as well as in the SWOT sensor geometry.
... As stated by Le Sommer et al. (2018), there is a wide range of uses and applications of ocean circulation models. Ocean circulation models are first and foremost used in idealized and realistic configurations to test hypotheses for any mechanisms underlying oceanic observations. ...
... Presently, supercomputer development is closely linked to the performance of commodity chips (i.e., GPUs), and because of their reduced memory access, these are not welladapted to ocean applications. The main limitation is, therefore, not just the computational speed of the processors but also, as stated by Le Sommer et al. (2018), access to memory and latency in reading/writing on disk drives (see Wang et al. (2020) for an application of a GPU-based version of LICOM3). ...
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Eddying global ocean models are now routinely used for ocean prediction, and the value-added of a better representation of the observed ocean variability and western boundary currents at that resolution is currently being evaluated in climate models. This overview article begins with a brief summary of the impact on ocean model biases of resolving eddies in several global ocean-sea ice numerical simulations. Then, a series of North and Equatorial Atlantic configurations are used to show that an increase of the horizontal resolution from eddy-resolving to submesoscale-enabled together with the inclusion of high-resolution bathymetry and tides significantly improve the models’ abilities to represent the observed ocean variability and western boundary currents. However, the computational cost of these simulations is extremely large, and for these simulations to become routine, close collaborations with computer scientists are essential to ensure that numerical codes can take full advantage of the latest computing architecture.
... The most prominent feature to occur over the course of the saildrone cruise was the salinity minimum experienced during [16][17][18][19] February 2020. This sharp drop in salinity was spatially visualized in Figure 6 to examine how well the SMAP products reproduce the saildrone data. ...
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The 2020 ‘Elucidating the role of clouds-circulation coupling in climate-Ocean-Atmosphere’ (EUREC4A-OA) and the ‘Atlantic Tradewind Ocean-Atmosphere Mesoscale Interaction Campaign’ (ATOMIC) campaigns focused on improving our understanding of the interaction between clouds, convection and circulation and their function in our changing climate. The campaign utilized many data collection technologies, some of which are relatively new. In this study, we used saildrone uncrewed surface vehicles, one of the newer cutting edge technologies available for marine data collection, to validate Level 2 and Level 3 Soil Moisture Active Passive (SMAP) satellite and Hybrid Coordinate Ocean Model (HYCOM) sea surface salinity (SSS) products in the Western Tropical Atlantic. The saildrones observed fine-scale salinity variability not present in the lower-spatial resolution satellite and model products. In regions that lacked significant small-scale salinity variability, the satellite and model salinities performed well. However, SMAP Remote Sensing Systems (RSS) 70 km generally outperformed its counterparts outside of areas with submesoscale SSS variation, whereas RSS 40 km performed better within freshening events such as a fresh tongue. HYCOM failed to detect the fresh tongue. These results will allow researchers to make informed decisions regarding the most ideal product and its drawbacks for their applications in this region and aid in the improvement of mesoscale and submesoscale SSS products, which can lead to the refinement of numerical weather prediction (NWP) and climate models.
... Therefore, it has become the major tool that has facilitated the development of operational oceanography and its applications (Le Traon, 2018). Ocean modeling has also recorded similar progress and it has become a vital aspect of operational oceanography over the last 60 years (Le Sommer et al., 2018). Kim et al. (2013) demonstrated that the integration of both data sources could enhance the quality and reliability of the last products. ...
Scientific and technological progresses have introduced diverse data sources for seawater temperature over broad temporal and spatial ranges. Here, we investigated the performance of satellite and model-based seawater temperature data for different temporal composites and depths. We applied an in-situ temperature time-series obtained in a coastal bottom in the Aegean Sea over three years, as the reference. Both datasets showed largely significant relationships based on cross-correlation analyses and presented descriptive properties of the in-situ conditions at corresponding depths. Based on the results of analyses, the modeling datasets presented more reliable results and representations of in-situ conditions than the datasets obtained from satellite for the coastal region. However, the datasets obtained from the satellite also provided reliable data for all time frames investigated, particularly in the mixed surface layer. Monthly datasets were more effective in providing descriptive values in long term studies. This is the first detailed study to explore the descriptive capacities of modeling for water temperature in coastal environments. According to the results, the selection of a dataset as a proxy for seawater temperature requires careful consideration. The present study provides an extensive baseline for evaluating the suitability of the application of specific datasets as proxies in coastal ecosystems.
... Another SWH observation method is modeling. A model tends to use a mathematical approach and assumptions in the predictions so that the results tend to be different from reality [8]. In regard to the use of the model, the Marine Meteorological Center (MMC) at the Meteorological, Climatological and Geophysical Agency (BMKG) currently has a marine weather model (OFS-models) implemented since the end of 2016, including the SWH predictions. ...
Computational Oceanography is the study of ocean phenomena by numerical simulation, especially dynamical and physical phenomena. Progress in information technology has driven exponential growth in the number of global ocean observations and the fidelity of numerical simulations of the ocean in the past few decades. The growth has been exponentially faster for ocean simulations, however. We argue that this faster growth is shifting the importance of field measurements and numerical simulations for oceanographic research. It is leading to the maturation of Computational Oceanography as a branch of marine science on par with observational oceanography. One implication is that ultra-resolved ocean simulations are only loosely constrained by observations. Another implication is that barriers to analyzing the output of such simulations should be removed. Although some specific limits and challenges exist, many opportunities are identified for the future of Computational Oceanography. Most important is the prospect of hybrid computational and observational approaches to advance understanding of the ocean.
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Sea Surface Height (SSH) observations describing scales in the range 10 - 100 km are crucial to better understand energy transfers across scales in the open ocean and to quantify vertical exchanges of heat and biogeochemical tracers. The Surface Water Ocean Topography (SWOT) mission is a new wide-swath altimetric satellite which is planned to be launched in 2022. SWOT will provide information on SSH at a kilometric resolution, but uncertainties due to various sources of errors will challenge our capacity to extract the physical signal of structures below a few tens of kilometers. Filtering SWOT noise and errors is a key step towards an optimal interpretation of the data.The aim of this study is to explore image de-noising techniques to assess the capabilities of the future SWOT data to resolve the oceanic fine scales. Pseudo-SWOT data are generated with the SWOT simulator for Ocean Science, which uses as input the SSH outputs from high-resolution Ocean General Circulation Models (OGCMs). Several de-noising techniques are tested, to find the one that renders the most accurate SSH and its derivatives fields while preserving the magnitude and shape of the oceanic features present. The techniques are evaluated based on the root mean square error, spectra and other diagnostics.In Chapter 3, the pseudo-SWOT data for the Science phase is analyzed to assess the capabilities of SWOT to resolve the meso- and submesoscale in the western Mediterranean. A Laplacian diffusion de-noising technique is implemented allowing to recover SSH, geostrophic velocity and relative vorticity down to 40 - 60 km. This first step allowed to adequately observe the mesoscale, but space is left for improvement at the submesoscale, specially in better preserving the intensity of the SSH signal.In Chapter 4, another de-noising technique is explored and implemented in the same region for the satellite's fast-sampling phase. This technique is motivated by recent advances in data assimilation techniques to remove spatially correlated errors based on SSH and its derivatives. It aims at retrieving accurate SSH derivatives, by recovering their structure and preserving their magnitude. A variational method is implemented which can penalize the SSH derivatives of first, second, third order or a combination of them. We find that the best parameterization is based on a second order penalization, and find the optimal parameters of this setup. Thanks to this technique the wavelengths resolved by SWOT in this region are reduced by a factor of 2, whilst preserving the magnitude of the SSH fields and its derivatives.In Chapter 5, we investigate the finest spatial scale that SWOT could resolve after de-noising in several regions, seasons and using different OGCMs. Our study focuses on different regions and seasons in order to document the variety of regimes that SWOT will sample. The de-noising algorithm performs well even in the presence of intense unbalanced motions, and it systematically reduces the smallest resolvable wavelength. Advanced de-noising algorithms also allow to reliably reconstruct SSH gradients (related to geostrophic velocities) and second order derivatives (related to geostrophic vorticity). Our results also show that a significant uncertainty remains about SWOT's finest resolved scale in a given region and season because of the large spread in the level of variance predicted among our high-resolution ocean model simulations.The de-noising technique developed, implemented and tested in this doctoral thesis allows to recover, in some cases, SWOT spatial scales as low as 15 km. This method is a very useful contribution to achieving the objectives of the SWOT mission. The results found will help better understand the ocean's dynamics and oceanic features and their role in the climate system.
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The impact of horizontal resolution (1/12° to 1/50°; 6 to 1.5 km at midlatitudes) on Gulf Stream separation, penetration, and variability is quantified in a series of identical North Atlantic experiments. The questions the authors seek to address are twofold: 1) Is the realism of the modeled solution increased as resolution is increased? 2) How robust is the modeled mesoscale and submesoscale eddy activity as a function of grid spacing and how representative is it of interior quasigeostrophic (QG) or surface quasigeostrophic (SQG) turbulence? This study shows that (i) the representation of Gulf Stream penetration and associated recirculating gyres shifts from unrealistic to realistic when the resolution is increased to 1/50° and when the nonlinear effects of the submesoscale eddies intensifies the midlatitude jet and increases its penetration eastward, (ii) the penetration into the deep ocean drastically increases with resolution and closely resembles the observations, and (iii) surface power spectra in the 70-250-km mesoscale range are independent of the horizontal resolution and of the latitude and are representative of 2D QG and SQG turbulence.
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A global 1/4∘ ocean/sea-ice 50-member ensemble simulation is used to disentangle the low-frequency imprints of the atmospherically-forced oceanic variability and of the chaotic intrinsic oceanic variability (IOV) on the large-scale (10° x10° ) ocean heat content (OHC) between 1980 and 2010. The IOV explains most of the interannual-to-decadal large-scale OHC variance over substantial fractions of the global ocean area that increase with depth: 9%, 22%, and 31% in the 0-700m, 700-2000m and 2000m-bottom layers, respectively. Such areas concern principally eddy-active regions, mostly found in the Southern Ocean and in western boundary current extensions, but also concern the subtropical gyres at intermediate and deep levels. The oceanic chaos may also induce random multidecadal fluctuations so that large-scale regional OHC trends computed on the 1980-2010 period cannot be unambiguously attributed to the atmospheric forcing in several oceanic basins at various depths. These results are likely to raise detection and attribution issues from real observations.
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This paper presents the technical implementation of a new, probabilistic version of the NEMO ocean–sea-ice modelling system. Ensemble simulations with N members running simultaneously within a single executable, and interacting mutually if needed, are made possible through an enhanced message-passing interface (MPI) strategy including a double parallelization in the spatial and ensemble dimensions. An example application is then given to illustrate the implementation, performances, and potential use of this novel probabilistic modelling tool. A large ensemble of 50 global ocean–sea-ice hindcasts has been performed over the period 1960–2015 at eddy-permitting resolution (1∕4°) for the OCCIPUT (oceanic chaos – impacts, structure, predictability) project. This application aims to simultaneously simulate the intrinsic/chaotic and the atmospherically forced contributions to the ocean variability, from mesoscale turbulence to interannual-to-multidecadal timescales. Such an ensemble indeed provides a unique way to disentangle and study both contributions, as the forced variability may be estimated through the ensemble mean, and the intrinsic chaotic variability may be estimated through the ensemble spread.
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Version 2 of the unstructured-mesh Finite-Element Sea ice–Ocean circulation Model (FESOM) is presented. It builds upon FESOM1.4 (Wang et al., 2014) but differs by its dynamical core (finite volumes instead of finite elements), and is formulated using the arbitrary Lagrangian Eulerian (ALE) vertical coordinate, which increases model flexibility. The model inherits the framework and sea ice model from the previous version, which minimizes the efforts needed from a user to switch from one version to the other. The ocean states simulated with FESOM1.4 and FESOM2.0 driven by CORE-II forcing are compared on a mesh used for the CORE-II intercomparison project. Additionally, the performance on an eddy-permitting mesh with uniform resolution is discussed. The new version improves the numerical efficiency of FESOM in terms of CPU time by at least 3 times while retaining its fidelity in simulating sea ice and the ocean. From this it is argued that FESOM2.0 provides a major step forward in establishing unstructured-mesh models as valuable tools in climate research.
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In this paper, a generic implementation approach is presented, with the aim of transforming a deterministic ocean model (like NEMO) into a probabilistic model. With this approach, several kinds of stochastic parameterizations are implemented to simulate the non-deterministic effect of unresolved processes, unresolved scales and unresolved diversity. The method is illustrated with three applications, showing that uncertainties can produce a major effect in the circulation model, in the ecosystem model, and in the sea ice model. These examples show that uncertainties can produce an important effect in the simulations, strongly modifying the dynamical behaviour of these three components of ocean systems.
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A signature of submesoscale flows in the upper ocean is skewness in the distribution of relative vorticity. Expected to result for high Rossby-number flows, such skewness has implications for mixing, dissipation and stratification within the upper ocean. An array of moorings deployed in the Northeast Atlantic for one year as part of the OSMOSIS experiment reveals that relative vorticity is positively skewed during winter even though the scale of the Rossby number is less than 0.5. Furthermore, this skewness is reduced to zero during spring and autumn. There is also evidence of modest seasonal variations in the gradient Rossby number. The proposed mechanism by which relative vorticity is skewed is that the ratio of lateral to vertical buoyancy gradients, as summarized by the inverse gradient Richardson number, restricts its range during winter but less so at other times of the year. These results support recent observations and model simulations suggesting the upper ocean is host to a seasonal cycle in submesoscale turbulence.
This paper introduces two methods for dynamically prescribing eddy-induced diffusivity, advection, and viscosity appropriate for primitive equation models with resolutions permitting the forward potential enstrophy cascade of quasigeostrophic dynamics, such as operational ocean models and high-resolution climate models with km horizontal resolution and finer. Where quasigeostrophic dynamics fail (e.g., the equator, boundary layers, deep convection), the method reverts to scalings based on a matched two-dimensional enstrophy cascade. A principle advantage is that these subgrid models are scale-aware, meaning that the model is suitable over a range of grid resolutions: from mesoscale grids that just permit baroclinic instabilities to grids below the submesoscale where ageostrophic effects dominate. Two approaches are presented here using Large Eddy Simulation (LES) techniques adapted for three-dimensional rotating, stratified turbulence. The simpler approach has one non-dimensional parameter, Λ, which has an optimal value near 1. The second approach dynamically optimizes Λ during simulation using a test filter. The new methods are tested in an idealized scenario by varying the grid resolution, and their use improves the spectra of potential enstrophy and energy in comparison to extant schemes. The new methods keep the gridscale Reynolds and Péclet numbers near one throughout the domain, which confers robust numerical stability and minimal spurious diapycnal mixing. Although there are no explicit parameters in the dynamic approach, there is strong sensitivity to the choice of test filter. Designing test filters for heterogeneous ocean turbulence adds cost and uncertainty, and we find the dynamic method does not noticeably improve over setting Λ = 1. This article is protected by copyright. All rights reserved.
This paper presents, discusses and analyses a massively parallel-in-time solver for linear oscillatory partial differential equations, which is a key numerical component for evolving weather, ocean, climate and seismic models. The time parallelization in this solver allows us to significantly exceed the computing resources used by parallelization-in-space methods and results in a correspondingly significantly reduced wall-clock time. One of the major difficulties of achieving Exascale performance for weather prediction is that the strong scaling limit – the parallel performance for a fixed problem size with an increasing number of processors – saturates. A main avenue to circumvent this problem is to introduce new numerical techniques that take advantage of time parallelism. In this paper, we use a time-parallel approximation that retains the frequency information of oscillatory problems. This approximation is based on (a) reformulating the original problem into a large set of independent terms and (b) solving each of these terms independently of each other which can now be accomplished on a large number of high-performance computing resources. Our results are conducted on up to 3586 cores for problem sizes with the parallelization-in-space scalability limited already on a single node. We gain significant reductions in the time-to-solution of 118.3× for spectral methods and 1503.0× for finite-difference methods with the parallelization-in-time approach. A developed and calibrated performance model gives the scalability limitations a priori for this new approach and allows us to extrapolate the performance of the method towards large-scale systems. This work has the potential to contribute as a basic building block of parallelization-in-time approaches, with possible major implications in applied areas modelling oscillatory dominated problems.
This paper discusses a numerical closure, motivated from the ideas of Smagorinsky, for use with a biharmonic operator. The result is a highly scale-selective, state-dependent friction operator for use in eddy-permitting geophysical fluid models. This friction should prove most useful for large-scale ocean models in which there are multiple regimes of geostrophic turbulence. Examples are provided from primitive equation geopotential and isopycnal-coordinate ocean models. 1.