The design of the icosahedral dynamical core DYNAMICO is presented. DYNAMICO solves the multi-layer rotating shallow-water equations, a compressible variant of the same equivalent to a discretization of the hydrostatic primitive equations in a Lagrangian vertical coordinate, and the primitive equations in a hybrid mass-based vertical coordinate. The common Hamiltonian structure of these sets of equations is exploited to formulate energy-conserving spatial discretizations in a unified way.

The horizontal mesh is a quasi-uniform icosahedral C-grid obtained by subdivision of a regular icosahedron. Control volumes for mass, tracers and entropy/potential temperature are the hexagonal cells of the Voronoi mesh to avoid the fast numerical modes of the triangular C-grid. The horizontal discretization is that of Ringler et al. (2010), whose discrete quasi-Hamiltonian structure is identified. The prognostic variables are arranged vertically on a Lorenz grid with all thermodynamical variables collocated with mass. The vertical discretization is obtained from the three-dimensional Hamiltonian formulation. Tracers are transported using a second-order finite-volume scheme with slope limiting for positivity. Explicit Runge–Kutta time integration is used for dynamics, and forward-in-time integration with horizontal/vertical splitting is used for tracers. Most of the model code is common to the three sets of equations solved, making it easier to develop and validate each piece of the model separately.

Representative three-dimensional test cases are run and analyzed, showing correctness of the model. The design permits to consider several extensions in the near future, from higher-order transport to more general dynamics, especially deep-atmosphere and non-hydrostatic equations.

In the last 2 decades, a number of groups have explored the potential of
quasi-uniform grids for overcoming well-known deficiencies of the
latitude–longitude mesh applied to atmospheric general circulation modelling

Since one reason for using quasi-uniform grids is the capability of
benefiting
from the computing power of massively parallel supercomputers, many groups
have set high-resolution modelling as a primary target. For the dynamical
core, which solves the fluid dynamical equations of motion, this generally
implies solving a non-hydrostatic set of equations. Indeed, the hydrostatic
primitive equations commonly used in climate-oriented global circulation models (GCMs) assume that the
modelled motions have horizontal scales much larger than the scale height,
typically about 10 km on Earth. Some hydrostatic models on quasi-uniform
grids have been developed but essentially as a milestone towards a
non-hydrostatic model

In fact in many areas of climate research high-resolution modelling can still
be hydrostatic. For instance palaeo-climate modelling must sacrifice
atmospheric resolution for simulation length, so that horizontal resolutions
typical of the Coupled Model Intercomparison Project (CMIP)-style climate modelling are so far beyond reach, and would
definitely qualify as high-resolution for multi-millenial-scale simulations.
Similarly, three-dimensional modelling of giant planets is so far unexplored
since resolving their small Rossby radius requires resolutions of a fraction
of a degree. Modelling at Institut Pierre Simon Laplace (IPSL) focusses to a
large extent on climate timescales and has diverse interests ranging from
palaeo-climate to modern climate and planetology. When IPSL embarked in 2009 in
an effort to develop a new dynamical core alongside Laboratoire de Météorologie Dynamique – Zoom
(LMD-Z)

By versatility we mean the ability to relax in the dynamical core certain
classical assumptions that are accurate for the Earth atmosphere but not
necessarily for planetary atmospheres, or may have small but interesting
effects on Earth. For instance in LMD-Z it is possible to assume for dry air
a non-ideal perfect gas with temperature-dependent thermal capacities and
this feature is used to model Venus

LMD-Z is a finite-difference dynamical core but the kinematic equations
(transport of mass, entropy/potential temperature, chemical species) are discretized
in flux form, leading to the exact discrete conservation of total mass, total
entropy/potential temperature and species content. Upwind-biased
reconstructions and slope limiters are used for the transport of species,
which is consistent with mass transport and monotonic

Pursuing both objectives of consistency and versatility (as defined above)
implies that generic approaches must be found, rather than solutions tailored
to a specific equation set. For instance the equivalence of mass and
pressure, the proportionality of potential and internal energies are valid
only for the hydrostatic primitive equations and cease to be valid in a
deep-atmosphere geometry, or even in a shallow-atmosphere geometry with a
complete Coriolis force

In addition to the above approach, building blocks for DYNAMICO include a
positive-definite finite-volume transport scheme

The present paper is organized as follows. Section

In this section we describe how the transport of mass, potential temperature
and other tracers is handled by DYNAMICO, using mass fluxes computed by the
dynamics as described in Sect.

The following subsections describe the grid, indexing conventions, the discrete mass and potential temperature budgets, and finally the positive-definite finite-volume scheme used for additional tracers.

The mesh is based on a tessellation of the unit sphere

Staggering and location of key prognostic and diagnostic variables.

Following the spirit of discrete exterior calculus (DEC; see, e.g.

Averages and finite differences are decorated with the location of the
result, i.e.

The discrete mass and potential temperature budgets are written in
flux form:

Either a Lagrangian vertical coordinate or a mass-based vertical coordinate
can be used. In the former case

Equations (

The vertical reconstruction is one-dimensional, piecewise linear,
slope limited, and identical to Van Leer's scheme I

We now turn to the discretization of the momentum budget. A Hamiltonian
formulation of the hydrostatic primitive equations in a generalized
vertical coordinate is used

An ideal perfect gas with

We work within the shallow-atmosphere and spherical geopotential
approximation, so that gravity

Discretizing Hamiltonian Eq. (

In order to reduce Eq. (

We now discretize horizontally the Hamiltonians
(Eqs.

Comparing Eqs. (

We now write the equations of motion corresponding to the discrete
Hamiltonians. First, mass fluxes must be computed for use by kinematics. They are
computed as

Next hydrostatic balance is expressed as

On the other hand, for the incompressible Hamiltonian
Eq. (

Finally, the horizontal momentum balance is written in vector-invariant form.
When

On the other hand, using the incompressible Hamiltonian
Eq. (

When

After spatial discretization one obtains a large set of ordinary algebraic
equations

Two-stage Runge–Kutta schemes of the order of 2 are unconditionally unstable for
imaginary eigenvalues and ruled out. All explicit

Furthermore, the last step is similar to an Euler step; hence,

At the beginning of this computation

From

At the beginning of this computation

From

Centred schemes need stabilization to counteract the generation of
grid-scale features in the flow. Linear sources of grid-scale noise, e.g.
dispersive numerical errors, may be handled by filters, e.g. upwinding or
hyperviscosity. Other sources are genuinely non-linear, e.g. the downward
cascade of energy or enstrophy. Here we handle these sources through
hyperviscosity as well, rather than with a proper turbulence model, e.g.

For this purpose hyper-diffusion is applied every

In addition to its aesthetic appeal, discrete conservation of energy has
practical consequences in terms of numerical stability, which we discuss here
using arguments similar to energy-Casimir stability theory

Assuming a Lagrangian vertical coordinate, the additional integral quantities
conserved by the discrete equations of motion are, for each layer, the
horizontally integrated mass and potential temperature

The above reasoning shows that linearization of the discrete equations of
motion around a steady state making

With a mass-based vertical coordinate, the exchange of mass between layers
reduces the set of discrete Casimir invariants to total mass and potential
temperature

We now proceed to derive the discrete energy budgets corresponding to a Lagrangian and a mass-based vertical coordinate. In these calculations only the adiabatic terms are considered, and the effect of the hyperviscous filters is omitted.

When

More generally, similar calculations yield the temporal evolution of an
arbitrary quantity

In the simplest case of a single layer without topography
(

In

When a mass-based coordinate is used instead of a Lagrangian vertical
coordinate, additional terms proportional to the vertical mass flux

Latitude–altitude plot of advected tracer profile at the mid-time
(

So far we see no indication that this would damage long-duration simulations
(see numerical results in Sect.

At this point some important differences with respect to the approach of

Second,

In a Eulerian formulation of the non-hydrostatic Euler equations, prognosing
covariant components would have the drawback that the no-flux lower boundary
condition involves a linear combination of all three covariant components,
which on a staggered mesh may be difficult to discretize in an
energy-conserving way. This may be a reason why

In this section, the correctness of DYNAMICO is checked using a few idealized
test cases. Horizontal resolutions of

Since our horizontal advection scheme is very similar to one scheme
studied by

This test case consist of a single layer of tracer, which deforms over the
duration of simulation. The flow field is prescribed so that the deformed
filament returns to its initial position in the end of simulation. We used
resolutions

As expected from two-dimensional test cases

Global error norms for Hadley-like meridional circulation test case.
Horizontal resolution is defined as

The baroclinic instability benchmark of

Even without the overlaid zonal wind perturbation, the initial state would
not be perfectly zonally symmetric because the icosahedral grid, as other
quasi-uniform grids, is not zonally symmetric. Therefore, the initial state
possesses, in addition to the explicit perturbation, numerical deviations
from zonal symmetry. This initial error, as well as truncation errors made at
each time step by the numerical scheme, is not homogeneous but reflects the
non-homogeneity of the grid. It nevertheless has the same symmetry as the
grid, here wave number 5 symmetry. Due to the dynamical instability of the
initial flow, the initial error is expected to trigger a wave number 5 mode of
instability (provided such an unstable mode with that zonal wave number
exists). Depending on the amplitude of the initial truncation error, this
mode can become visible, a case of grid imprinting

Figure

The left column shows surface pressure at day 12, after the baroclinic wave
has broken, letting time for grid imprinting to develop. Grid imprinting in
the Southern Hemisphere, measured quantitatively as in

Dry baroclinic instability test case

Figure

Time-zonal statistics of

Time-zonal statistics of

A number of building blocks of DYNAMICO are either directly found in the
literature or are adaptations of standard methods: explicit Runge–Kutta time
stepping, mimetic horizontal finite-difference operators

The first contribution is to separate kinematics from dynamics as strictly as
possible. This separation means that the transport equations for mass,
scalars and entropy use no information regarding the specific momentum
equation being solved. This includes the equation of state as well as any
metric information, which is factored into the prognosed degrees of freedom
and into the quantities derived from them (especially the mass flux). Metric
information is not used to prognose tracer, mass and potential temperature.
It is confined in a few operations computing the mass flux, Bernoulli
function and Exner function from the prognostic variables. This formulation
is in line with more general lines of thought known as physics-preserving
discretizations

The second contribution is to combine this kinematics–dynamics separation
with a Hamiltonian formulation of the equations of motion to achieve
energetic consistency. This approach extends the work of

These two advances yield our design goals: consistency and versatility. The
desired ability to solve different equation sets is currently limited to the
hydrostatic primitive equations and the multi-layer Saint-Venant or Ripa
equations, but little work is required to solve other similar equations like
the recently derived non-traditional spherical shallow-water equations

We would also like to emphasize what the Hamiltonian approach does

A Lagrangian vertical coordinate is currently available as an option. In the
absence of the vertical remapping that must necessarily take place
occasionally in order to prevent Lagrangian surfaces to fold or cross each
other, this option can not be used over meaningful time intervals. However, it
is convenient for development purposes since it allows one to investigate
separately issues related to the vertical and horizontal discretizations.
Nevertheless, a future implementation of vertical remapping would be a useful
addition. There is room for improvement on other points. In particular, it may be
worth improving the accuracy of the transport scheme, especially for water
vapour and other chemically or radiatively active species. Regarding potential
temperature,

The Hamiltonian framework leaves a complete freedom with respect to the
choice of a discrete Hamiltonian. Here the simplest possible second-order
accurate approximation is used, but other forms may yield additional
properties, such as a more accurate computation of the geopotential. Ongoing
work suggests that it is possible to design a Hamiltonian such that certain
hydrostatic equilibria are exactly preserved in the presence of arbitrary
topography. Such a property is sometimes achieved by finite-volume schemes

DYNAMICO is stabilized by (bi)harmonic operators of which we refer as filters
rather than dissipation. Indeed, they are numerical devices aimed at
stabilizing the model rather than physically based turbulence models such as
nonlinear viscosity

Coupling DYNAMICO to the LMD-Z terrestrial physics package suite is ongoing.
For planetary applications, it will be important to also check the discrete
angular momentum budget

In the near future DYNAMICO should become able to solve richer,
quasi-hydrostatic equations

Among various possible ways of generating the triangular mesh, we follow the
method of

The hexagonal mesh is constructed as the Voronoi diagram of the triangular
mesh

Numerical errors can be reduced by various optimization methods (e.g.

Although round-off errors may not be an urgent concern with double-precision computations at presently common resolutions, it may become if formulae with high round-off error are used in sequence, if single precision is used for speed, or at high resolutions. In this Appendix we describe geometric primitives that are not sensitive to round-off error, or more precisely that are not more sensitive to round-off errors than equivalent planar primitives. These primitives are required in the grid generation and optimization process and compute centroids, circumcenters and spherical areas.

Let

Error in circumcenter calculation using direct
formula (

Regarding the spherical centre of mass (Eq.

Finally, computing the area

Results presented in this article are based on release r339 of DYNAMICO.
Instructions to download, compile, and run the code are provided at

The authors would like to thank two anonymous referees for their detailed and constructive criticism. S. Dubey has benefitted from the support of the Indo-French Centre for the Promotion of Advanced Research (IFCPRA/CEFIPRA) under the sponsored project number 4107-1. T. Dubos, Y. Meurdesoif and F. Hourdin acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-14-CE23-0010-01. Edited by: A. Sandu