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Schemes for the incompressible Navier-Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit-Explicit Runge-Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed–element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.

Haley, P.J., Jr., A. Agarwal, P.F.J. Lermusiaux, 2015. *Optimizing Velocities and Transports for Complex Coastal Regions and Archipelagos*. Ocean Modeling, 89, 1-28. doi:10.1016/j.ocemod.2015.02.005

We derive and apply a methodology for the initialization of velocity and transport fields in complex multiply-connected regions with multiscale dynamics. The result is initial fields that are consistent with observations, complex geometry and dynamics, and that can simulate the evolution of ocean processes without large spurious initial transients. A class of constrained weighted least squares optimizations is defined to best fit first-guess velocities while satisfying the complex bathymetry, coastline and divergence strong constraints. A weak constraint towards the minimum inter-island transports that are in accord with the first-guess velocities provides important velocity corrections in complex archipelagos. In the optimization weights, the minimum distance and vertical area between pairs of coasts are computed using a Fast Marching Method. Additional information on velocity and transports are included as strong or weak constraints. We apply our methodology around the Hawaiian islands of Kauai/Niihau, in the Taiwan/Kuroshio region and in the Philippines Archipelago. Comparisons with other common initialization strategies, among hindcasts from these initial conditions (ICs), and with independent in situ observations show that our optimization corrects transports, satisfies boundary conditions and redirects currents. Differences between the hindcasts from these different ICs are found to grow for at least 2-3 weeks. When compared to independent in situ observations, simulations from our optimized ICs are shown to have the smallest errors.

Duda T.F., W.G. Zhang, K.R. Helfrich, A.E. Newhall, Y.-T. Lin, J.F. Lynch, P.F.J. Lermusiaux, P.J. Haley Jr., J. Wilkin, 2014. *Issues and Progress in the Prediction of Ocean Submesoscale Features and Internal Waves*. OCEANS'14 MTS/IEEE.

Data-constrained dynamical ocean modeling for the purpose of detailed forecasting and prediction continues to evolve and improve in quality. Modeling methods and computational capabilities have each improved. The result is that mesoscale phenomena can be modeled with skill, given sufficient data. However, many submesoscale features are less well modeled and remain largely unpredicted from a deterministic event standpoint, and possibly also from a statistical property standpoint. A multi-institution project is underway with goals of uncovering more of the details of a few submesoscale processes, working toward better predictions of their occurrence and their variability. A further component of our project is application of the new ocean models to ocean acoustic modeling and prediction. This paper focuses on one portion of the ongoing work: Efforts to link nonhydrostatic-physics models of continental-shelf nonlinear internal wave evolution to data-driven regional models. Ocean front-related effects are also touched on.

De Dominicis M., S. Falchetti, F. Trotta, N. Pinardi, L. Giacomelli, E. Napolitano, L. Fazioli, R. Sorgente, P.J. Haley Jr., P.F.J. Lermusiaux, F. Martins and M. Cocco, 2014. *A Relocatable Ocean Model in support of environmental emergencies - The Costa Concordia emergency case*. Ocean Dynamics,
64, 5:667–688. DOI 10.1007/s10236-014-0705-x

In contemporary ocean science, modeling systems that integrate understanding of complex multiscale phenomena and utilize efficient numerics are paramount. Many of today’s fundamental ocean science questions involve multiple scales and multiple dynamics. A new generation of modeling systems would allow to study such questions quantitatively, by being less restrictive dynamically and more efficient numerically than more traditional systems.
Such multiscale ocean modeling is the theme of this topical issue. Two large international workshops were organized on this theme, one in Cambridge, USA (IMUM2010), and one in Bremerhaven, Germany (IMUM2011). Contributions from the scientific community were encouraged on all aspects of multiscale ocean modeling, from ocean science and dynamics to the development of new computational methods and systems. Building on previous meetings (e.g. Deleersnijder and Lermusiaux, 2008; Deleersnijder et al., 2010), the workshop discussions and the final contributions to the topical issue are summarized next.
The scientific application domains discussed and presented ranged from estuaries to the global ocean, including coastal regions and shelf seas. Multi-resolution modeling of physical, biological, chemical, and sea ice processes as well as air-sea interactions were described. The multiscale dynamics considered involved hydrostatic, non-hydrostatic, turbulent and sea surface processes.
Computational results and discussions emphasized multi-resolution simulations using unstructured and structured meshes, aiming to widen the range of resolved scales in space and time. They included finite volume and finite element spatial-discretizations, high-order schemes, preconditioners, solver issues, grid generation, adaptive modeling, data assimilation, coupling with atmospheric or biogeochemical models, and distributed computing. The advantages of using unstructured meshes and related approaches, in particular multi-grid embedding, nesting systems, wavelets and other multi-scale decompositions were discussed. Techniques for the study of multi-resolution results, visualization, optimization, model evaluations, and uncertainty quantification were also examined.

A fundamental requirement in realistic ocean simulations and dynamical studies
is the optimal estimation of gridded fields from the spatially irregular and multivariate
data sets that are collected by varied platforms. In this work, we derive
and utilize new schemes for the mapping and dynamical inference of ocean fields
in complex multiply-connected domains and study the computational properties
of these schemes. Specifically, we extend a Bayesian-based multiscale Objective
Analysis (OA) approach to complex coastal regions and archipelagos. Such OAs
commonly require an estimate of the distances between data and model points,
without going across complex landforms. New OA schemes that estimate the
length of shortest sea paths using the Level Set Method (LSM) and Fast Marching
Method (FMM) are thus derived, implemented and utilized in idealized and
realistic ocean cases. An FMM-based methodology for the estimation of total velocity
under geostrophic balance in complex domains is also presented. Comparisons
with other OA approaches are provided, including those using stochastically
forced partial differential equations (SPDEs). We find that the FMM-based OA
scheme is the most efficient and accurate. The FMM-based field maps do not
require postprocessing (smoothing). Mathematical and computational properties
of our new OA schemes are studied in detail, using fundamental theorems and illustrations.
We find that higher-order FMM’s schemes improve accuracy and that
a multi-order scheme is efficient. We also provide solutions that ensure the use of
positive-definite covariances, even in complex multiply-connected domains.

Accurate numerical modeling of biogeochemical ocean dynamics is essential for numerous applications, including coastal ecosystem science, environmental management and energy, and climate dynamics. Evaluating computational requirements for such often highly nonlinear and multiscale dynamics is critical. To do so, we complete comprehensive numerical analyses, comparing low- to high-order discretization schemes, both in time and space, employing standard and hybrid discontinuous Galerkin finite element methods, on both straight and new curved elements. Our analyses and syntheses focus
on nutrient-phytoplankton-zooplankton dynamics under
advection and diffusion within an ocean strait or
sill, in an idealized 2D geometry. For the dynamics,
we investigate three biological regimes, one with single
stable points at all depths and two with stable
limit cycles. We also examine interactions that are
dominated by the biology, by the advection, or that
are balanced. For these regimes and interactions, we study the sensitivity to multiple numerical parameters
including quadrature-free and quadrature-based
discretizations of the source terms, order of the spatial
discretizations of advection and diffusion operators,
order of the temporal discretization in explicit
schemes, and resolution of the spatial mesh, with and
without curved elements. A first finding is that both
quadrature-based and quadrature-free discretizations
give accurate results in well-resolved regions, but the
quadrature-based scheme has smaller errors in underresolved
regions. We show that low-order temporal
discretizations allow rapidly growing numerical errors
in biological fields. We find that if a spatial discretization
(mesh resolution and polynomial degree) does not
resolve the solution, oscillations due to discontinuities
in tracer fields can be locally significant for both lowand
high-order discretizations. When the solution is
sufficiently resolved, higher-order schemes on coarser
grids perform better (higher accuracy, less dissipative)
for the same cost than lower-order scheme on finer
grids. This result applies to both passive and reactive
tracers and is confirmed by quantitative analyses of
truncation errors and smoothness of solution fields. To
reduce oscillations in un-resolved regions, we develop
a numerical filter that is active only when and where
the solution is not smooth locally. Finally, we consider
idealized simulations of biological patchiness. Results
reveal that higher-order numerical schemes can maintain
patches for long-term integrations while lowerorder
schemes are much too dissipative and cannot,
even at very high resolutions. Implications for the use
of simulations to better understand biological blooms,
patchiness, and other nonlinear reactive dynamics in
coastal regions with complex bathymetric features are
considerable.

We derive conservative time-dependent
structured discretizations and two-way embedded
(nested) schemes for multiscale ocean dynamics
governed by primitive equations (PEs) with a nonlinear
free surface. Our multiscale goal is to resolve tidalto-
mesoscale processes and interactions over large
multiresolution telescoping domains with complex
geometries including shallow seas with strong tides,
steep shelfbreaks, and deep ocean interactions. We
first provide an implicit time-stepping algorithm for
the nonlinear free-surface PEs and then derive a
consistent time-dependent spatial discretization with
a generalized vertical grid. This leads to a novel timedependent
finite volume formulation for structured
grids on spherical or Cartesian coordinates, second
order in time and space, which preserves mass and
tracers in the presence of a time-varying free surface.
We then introduce the concept of two-way nesting,
implicit in space and time, which exchanges all of the
updated fields values across grids, as soon as they become available. A class of such powerful nesting
schemes applicable to telescoping grids of PE models
with a nonlinear free surface is derived. The schemes
mainly differ in the fine-to-coarse scale transfers and
in the interpolations and numerical filtering, specifically
for the barotropic velocity and surface pressure
components of the two-way exchanges. Our scheme
comparisons show that for nesting with free surfaces,
the most accurate scheme has the strongest implicit
couplings among grids. We complete a theoretical
truncation error analysis to confirm and mathematically
explain findings. Results of our discretizations and
two-way nesting are presented in realistic multiscale
simulations with data assimilation for the middle
Atlantic Bight shelfbreak region off the east coast of
the USA, the Philippine archipelago, and the Taiwan-Kuroshio region. Multiscale modeling with two-way
nesting enables an easy use of different sub-gridscale
parameterizations in each nested domain. The new
developments drastically enhance the predictive capability
and robustness of our predictions, both qualitatively
and quantitatively. Without them, our multiscale
multiprocess simulations either were not possible or
did not match ocean data.

Methods for widening the range of resolved scales (i.e.
performing multi-scale simulations) in ocean sciences and
engineering are developing rapidly, now allowing multiscale
ocean dynamics studies. Having recourse to grid
nesting has been and still is a popular method for increasing
marine models’ resolution when and where needed and for
easily allowing the use of different dynamics at different
resolution. However, this is not the only way to achieve this
goal. Various techniques for modifying locally the grid
resolution or dealing with complex-geometry domains are
available. For instance, composite, structured grids and
unstructured meshes offer an almost infinite geometrical
flexibility.
This special issue focuses on multi-scale modelling
of coastal, shelf and global ocean dynamics, including the
development of new methodologies and schemes and their
applications to ocean process studies. Several articles focus
on numerical aspects of unstructured mesh space discretisation.
Danilov (2010) shows that the noise developing on
triangular meshes on which the location of the variables is
inspired by Arakawa’s C-grid is the largest for regimes
close to geostrophic balance. The noise can be reduced by
specific operators but cannot be entirely suppressed,
“making the triangular C-grid a suboptimal choice for
large-scale ocean modelling”. Then, the companion articles
of Blaise et al. (2010) and Comblen et al. (2010) describe
the space and time discretisation of a three-dimensional,
baroclinic, finite element model based on the discontinuous
Galerkin (DG) technique. This is a significant step forward
in the field of finite element ocean modelling, though this
model cannot yet be regarded as suitable for tackling
realistic applications. Ueckermann and Lermusiaux (2010)
also consider DG finite element techniques, focusing on
biological-physical dynamics in regions with complex
bathymetric features. They compare low- to high-order
discretisations, both in time and space, for regimes in which
biology dominates, advection dominates or terms are
balanced. They find that higher-order schemes on relatively
coarse grids generally perform better than low-order
schemes on fine grids. Kleptsova et al. (2010) assess
various advection schemes for z-coordinate, threedimensional
models in which flooding and drying is taken
into account. In this study, the ability to conserve
momentum is regarded as the main criterion for selecting
a suitable method. On the other hand, Massmann (2010) assesses automatic differentiation for obtaining the adjoint
of an unstructured mesh, tidal model of the European
continental shelf.
Two articles deal with grid nesting. Nash and Hartnett
(2010) introduce a flooding and drying method that can be
used in structured, nested grid systems. This can be
regarded as an alternative to flooding and drying techniques
that are being developed for unstructured mesh models (e.g.
Karna et al. 2010). Then, Haley and Lermusiaux (2010)
derive conservative time-dependent structured finite volume
discretisations and implicit two-way embedded
schemes for primitive equations with the intent to resolve
tidal-to-mesoscale processes over large multi-resolution
telescoping domains with complex geometries including
shallow seas with strong tides, steep shelf breaks and deep
ocean interactions. The authors present realistic simulations
with data assimilation in three regions with diverse
dynamics and show that their developments enhance the
predictive capability, leading to better match with ocean
data.
Various multi-scale, realistic simulations are presented.
Using a finite element ice model and a slab ocean as in
Lietaer et al. (2008), Terwisscha van Scheltinga et al.
(2010) model the Canadian Arctic Archipelago, focusing on
the pathways for freshwater and sea-ice transport from the
Arctic Ocean to the Labrador Sea and the Atlantic Ocean.
The unstructured mesh can represent the complex geometry
and narrow straits at high resolution and allows improving
transports of water masses and sea ice. Walters et al. (2010)
have recourse to an unstructured mesh model to study tides
and current in Greater Cook Strait (New Zealand). They
identify the mechanisms causing residual currents. By
means of the unstructured mesh Finite Volume Coastal
Ocean Model (FVCOM), Wang et al. (2010) study the
hydrodynamics of the Bohai Sea. Xu et al. (2010) simulate
coastal and urban inundation due to storm surges along US
East and Gulf Coasts. A sensitivity analysis reveals the
importance of precise topographic data and the need for a
bottom drag coefficient accounting for the presence of
mangroves. Finally, Yang and Khangaonkar (2010) resort to
FVCOM to simulate the three-dimensional circulation of
Puget Sound, a large complex estuary system in the Pacific
Northwest coastal ocean, including variable forcing from
tides, the atmosphere and river inflows. Comparisons of
model estimates with measurements for tidal elevation,
velocity, temperature and salinity are deemed to be
promising, from larger-scale circulation features to nearshore
tide flats.
This special issue suggests that numerical techniques for
multi-scale space discretisation are progressively becoming
mature. One direction for future progress lies in the
improvement of time discretisation methods for the new
generation models, so that they can successfully compete
with finite difference, structured mesh models based on
(almost) constant resolution grids that have been developed
and used over the past 40 years (e.g. Griffies et al. 2009).

In 1969, the Journal of Computational Physics published a
seminal article by K. Bryan presenting the first ocean
general circulation model. Since then, many numerical
studies of the World Ocean, as well as regional or coastal
flows, used models directly or indirectly inspired by the
work of Bryan and his colleagues. A number of these
models have evolved into highly modular and versatile
computational systems, including multiple physical modules
and options as well as varied biogeochemical,
ecosystem and acoustics modeling capabilities. Several
modeling systems are now well-documented tools, which
are widely used in research institutions and various
organizations around the world. The list of such modeling
systems is large and too long to be summarized in this
editorial.
Over the last three decades, significant progress has been
made in the parameterization of subgrid-scale processes, in
data assimilation methodologies and in boundary condition
schemes, as well as in the efficient implementation of
algorithms on fast vector and subsequently parallel computers,
allowing higher and higher resolution in space and
time. However, many of today’s popular modeling systems
can still be regarded as members of the first generation of
ocean models: at their core, rather similar geophysical fluid
dynamics equations are solved numerically using a conservative
finite-difference method on a structured grid.
Today, several aspects of structured-grid models could
benefit from significant upgrades, learning from major
advances in computational fluid dynamics. In particular, the
use of a structured grid limits the flexibility in the spatial
resolution and does not allow one to take full advantage of
numerical algorithms such as finite volumes and finite
elements, which can achieve their best performance when
implemented on unstructured meshes.
Even though many of today’s complex marine modeling
and data assimilation systems have evolved significantly
since Bryan’s prototype, it would be challenging to modify
them step-by-step from a structured-grid approach to an
unstructured-grid one. Therefore, novel marine model
design research is underway, paving the way for the second
generation of ocean modeling systems. It is difficult to
predict today if this new generation of ocean models will
achieve its chief objective: widening the range of resolved
scales of motion with increased efficiencies and accuracies,
possibly allowing multi-resolution, multi-scale, and multidynamics
numerical simulations of marine flows, all
occurring seamlessly within distributed computing environments.
In fact, hybrid approaches merging the advantages
of structured and unstructured-grid modeling may be the
way forward.
Whether or not unstructured mesh approaches will
prevail is all the more difficult to predict now that
structured mesh modelers have developed powerful solutions
for increasing the resolution when and where
needed. For instance, grid embedding is still a popular
and useful method for enhancing model resolution. It can
involve multiply nested domains and allows the relatively straightforward use of different dynamics or models in each
domain. Research is also underway for developing multigrid,
wavelet, and other multi-scale decompositions for the
numerical solution of dynamical equations but also for the
study of results, model evaluation or data assimilation.
This special issue presents a number of examples of the
abovementioned developments. Ringler et al. examine the
potential of spherical centroidal Voronoi tessellations for
performing multi-resolution simulations; they apply this
method to the Greenland ice sheet and the North Atlantic
Ocean. Lambrechts et al. present a triangular mesh
generation system and its applications to the World Ocean
and various shelf seas, including the Great Barrier Reef,
Australia. Finite element models on unstructured grids are
described and utilized in several manuscripts. Bellafiore et
al. study the Adriatic Sea and the Lagoon of Venice, while
Jones and Davies simulate tides and storm surges along the
western coast of Britain. Danilov et al. assess two finite
element discretizations, i.e., a continuous element and a
non-conforming one, and compare the results of these
discretizations with those of a finite-difference model. In
Harig et al., the tsunami generated by the great Sumatra-Andaman earthquake of 26 December 2004 is simulated by
means of a finite element model. Comparisons are carried
out with various types of data as well as with the results of
a structured mesh model using a nested structured-grid
system. A nested-grid ocean circulation model is also
employed by Yang and Sheng to carry out a process study
on the Inner Scotian Shelf, Canada, focusing on the
circulation induced by a tropical storm. Debreu and Blayo
present a detailed review of two-way embedding algorithms
for structured-grid models. Finally, Logutov develops a
multi-scale assimilation scheme for tidal data within the
framework of a multiply nested structured-grid barotropic
tidal modeling approach.
As illustrated by these manuscripts, the next generation
of ocean modelers is motivated by a wide range of research
opportunities over a rich spectrum of needs. Future progress
will involve fundamental and applied numerical and
computational research as well as new multi-scale geophysical
fluid modeling. Domains of ongoing interest range
from estuaries to the global ocean, including coastal regions
and shelf seas. New multi-scale modeling of physical as
well as biological, chemical or interdisciplinary processes
will flourish in the coming decades.
We are grateful to the authors for their contributions and
to the chief-editor for his support in this endeavor. We are
thankful to the reviewers for their time and help in assessing
the manuscripts submitted to this special issue. Eric
Deleersnijder is a Research associate with the Belgian
National Fund for Scientific Research (FNRS); he is
indebted to the Communaut Francaise de Belgique for its
support through contract ARC 04/09-316. Pierre Lermusiaux
is grateful to the Office of Naval Research for support under
grant N00014-08-1-1097 to the Massachusetts Institute of
Technology.

Correct representation of tidal processes in regional ocean models is contingent on the accurate specification of open boundary conditions. This paper describes a new inverse scheme for the assimilation of
observational data into a depth-integrated spectral shallow water tidal model and the numerical implementation of this scheme into a stand-alone computational system for regional tidal prediction. A novel
aspect is a specific implementation of the inverse which does not require an adjoint model. An optimization is carried out in the open boundary condition space rather than in the observational space or model
state space. Our approach reflects the specifics of regional tidal modeling applications in which open
boundary conditions (OBCs) typically constitute a significant source of uncertainty. Regional tidal models
rely predominantly on global tidal estimates for open boundary conditions. As the resolution of global
tidal models is insufficient to fully resolve regional topographic and coastal features, the a priori OBC estimates potentially contain an error. It is, therefore, desirable to correct these OBCs by finding an inverse
OBC estimate that is fitted to the regional observations, in accord with the regional dynamics and respective error estimates. The data assimilation strategy presented in this paper provides a consistent and
practical estimation scheme for littoral ocean science and applications where tidal effects are significant.
Illustrations of our methodological and computational results are presented in the area of Dabob Bay and
Hood Canal, WA, which is a region connected to the open Pacific ocean through a series of inland
waterways and complex shorelines and bathymetry.

The Harvard Ocean Prediction System (HOPS) provides real-time and hindcast, multi-scale oceanic field estimates for Maritime
Rapid Environmental Assessment (MREA). Results of aspects of the validation, calibration and verification of HOPS for MREA03
and MREA04 are presented, with implications for future MREA exercises. A new method of model training, via bias correction
through the use of limited data, was applied to MREA03 and shown to produce significant forecast improvement while reducing
computational requirements. Advances in, and the demand for, adaptive modeling, require that aspects of validation, calibration
and verification be carried out in real-time in order to expand the usage and relevance of dynamical forecast-based MREA tactical
decision aids

Lermusiaux, P.F.J. and F. Lekien, 2005. *Dynamics and Lagrangian Coherent Structures in the Ocean and their Uncertainty.* Extended Abstract in report of the "Dynamical System Methods in Fluid Dynamics" Oberwolfach Workshop. Jerrold E. Marsden and Jurgen Scheurle (Eds.), Mathematisches Forschungsinstitut Oberwolfach, July 31st - August 6th, 2005, Germany. 2pp.

We discuss the concepts involved in the evaluation and quantitative verification of ocean forecasts and present two predictive skill experiments to develop and research these concepts, carried out in the North Atlantic and Mediterranean Sea in 2001 and 2002. Ocean forecasting involves complex ocean observing and prediction systems for ocean regions with multi-scale interdisciplinary dynamical processes and strong, intermittent events. Now that ocean forecasting is becoming more common, it is critically important to interpret and evaluate regional forecasts in order to establish their usefulness to the scientific and applied communities.
The Assessment of Skill for Coastal Ocean Transients (ASCOT) project is a series of real-time Coastal Predictive Skill (CPSE) and Rapid Environmental Assessment (REA) experiments and simulations focused on quantitative skill evaluation, carried out by the Harvard Ocean Prediction System group in collaboration with the NATO SACLANT Undersea Research Centre. ASCOT-01 was carried out in Massachusetts Bay and the Gulf of Maine in June 2001. ASCOT-02 took place in May 2002 in the Corsican Channel near the island of Elba in the Mediterranean Sea. Results from the ASCOT exercises highlight the dual use of data for skill evaluation and assimilation, real-time adaptive sampling and skill optimization and present both real-time and a posteriori evaluations of predictive skill and predictive capability.