Atmospheric dynamical cores are a fundamental component of global atmospheric modeling systems and are responsible for capturing the dynamical behavior of the Earth's atmosphere via numerical integration of the Navier–Stokes equations. These systems have existed in one form or another for over half of a century, with the earliest discretizations having now evolved into a complex ecosystem of algorithms and computational strategies. In essence, no two dynamical cores are alike, and their individual successes suggest that no perfect model exists. To better understand modern dynamical cores, this paper aims to provide a comprehensive review of 11 non-hydrostatic dynamical cores, drawn from modeling centers and groups that participated in the 2016 Dynamical Core Model Intercomparison Project (DCMIP) workshop and summer school. This review includes a choice of model grid, variable placement, vertical coordinate, prognostic equations, temporal discretization, and the diffusion, stabilization, filters, and fixers employed by each system.

The Dynamical Core Model Intercomparison Project (DCMIP) is an ongoing effort targeting the intercomparison of a fundamental component of global atmospheric modeling systems: the dynamical core. Although this component's role is simply to solve the equations of fluid motion governing atmospheric dynamics (the Navier–Stokes equations), there are numerous confounding factors and compromises that arise from making global simulations computationally feasible. These factors include the choice of model grid, variable placement, vertical coordinate, prognostic equations, representation of topography, numerical method, temporal discretization, physics–dynamics coupling frequency, and the manner in which artificial diffusion, stabilization, filters, and/or energy/mass fixers are applied.

To advance the intercomparison project and provide a unique educational opportunity for students, DCMIP hosted a multidisciplinary 2-week summer school and model intercomparison project at the National Center for Atmospheric Research (NCAR) in June 2016, that invited graduate students, postdocs, atmospheric modelers, expert lecturers, and computer specialists to create a stimulating, unique, and hands-on driven learning environment. The 2016 workshop and summer school followed from earlier DCMIP and dynamical core workshops (held in 2012 and 2008, respectively), and other model intercomparison efforts. Its goals were to provide an international forum for discussing outstanding issues in global atmospheric models and provide a unique training experience for the future generation of climate scientists. Special attention was paid to the role of simplified physical parameterizations, physics–dynamics coupling, non-hydrostatic atmospheric modeling, and variable-resolution global modeling. The summer school and model intercomparison project promoted active learning, innovation, discovery, mentorship, and the integration of science and education. Modeling groups were then invited to contribute model descriptions and results to the intercomparison effort for publication.

The summer school directly benefited its participants by providing a unique
educational experience and an opportunity to interact with modeling teams
from around the world. The workshop is expected to have further repercussions
on the development of operational atmospheric modeling systems by allowing
modeling groups to assess their models in the context of the global dynamical
core ecosystem. Past and present intercomparison efforts have been leveraged
by modeling groups to improve their own models, in turn leading to a positive
impact on the quality of weather and climate simulations. The workshop
component of DCMIP has also advanced our knowledge of (1) the relative
behaviors exhibited by atmospheric dynamical cores, (2) differences that
arise among mechanisms for coupling the physical parameterizations and
dynamical core, and (3) the impacts of variable-resolution refinement regions
and transition zones in global atmospheric simulations. Notably, the use of
idealized test cases to isolate specific phenomena gave us a unique
opportunity to assess specific differences that arise due to the choice of
dynamical core. Another important outcome of the workshop was the development
of a standard test case suite and benchmark set of simulations that can be
used for assessment of any future dynamical core. The test cases introduced
in the 2016 workshop build on the previous DCMIP test case suites

This paper is the first in a series of papers documenting the results of this
workshop. Its purpose is two-fold: first, to review the multitude of
technologies and techniques that have been developed for non-hydrostatic
global atmospheric modeling; and second, to provide a mechanism to understand
the differences that arise in the test cases of later papers in this series.
For ease of reference, a list of mathematical symbols that are employed in
this paper (and subsequent DCMIP papers) is given in Table

A standard list of symbols used throughout this paper and in the DCMIP.

This section provides a brief overview of key discretization choices, along
with unique features or design specifications from participating dynamical
cores. Further details on these choices can be found in subsequent sections.
In total, simulation results and model descriptions have been submitted from
11 dynamical cores (see Table

Participating modeling centers and associated dynamical cores that have submitted a model description and/or simulation results.

Details on the prognostic variables and horizontal discretization
for participating dynamical cores. The equation set indicates whether a model is
hydrostatic (H) or non-hydrostatic (NH), and whether the model presently
supports the deep-atmosphere formulation (D). Only three numerical methods
are represented among participating models, namely finite difference (FD),
finite volume (FV), and spectral element (SE). More details on horizontal
staggering can be found in Sect.

Vertical staggering (detailed in Sect.

Principal options for diffusion, stabilization, filters, or fixers
in participating dynamical cores (detailed in Sect.

The Accelerated Climate Model for Energy–Atmosphere (ACME-A) has much in common with the Community Atmosphere Spectral
Element Model (CAM-SE)

The Colorado State University (CSU) model is a finite-volume model using an optimized geodesic grid

DYNAMICO is a mimetic finite-difference/finite-volume model using a geodesic
grid (Sect.

The GFDL Finite-Volume Cubed-Sphere Dynamical Core (FV

The Finite-Volume Module (FVM) of the Integrated Forecasting System (IFS) is
currently under development at ECMWF

The Global Environmental Multiscale (GEM) model

The ICOsahedral Non-hydrostatic (ICON) model

The Model for Prediction Across Scales (MPAS)

Non-hydrostatic ICosahedral Atmospheric Model (NICAM) is a finite-volume model that solves the non-hydrostatic Euler
equations using a geodesic grid (Sect.

Ocean–Land–Atmosphere Model (OLAM)

The Tempest model

The horizontal discretization determines how the atmosphere, which consists
of a set of approximately continuous fields, is mapped into a very limited
and discrete computational space. The horizontal discretization essentially
consists of two major choices: the model grid, which determines the density
and connectivity of discrete regions

The classic latitude–longitude grid is produced by subdividing the sphere
along lines of constant latitude and longitude. The latitude–longitude grid
has the benefits of being globally rectilinear, which simplifies data access
and subdivision of computation across processors, and yields a vector basis
that is locally orthogonal nearly everywhere. This structure accurately
maintains purely zonal flows and simplifies data post-processing for
visualization. Because of the convergence of grid lines near the poles, the
operational use of this grid requires that the associated numerical scheme be
resilient to arbitrarily small Courant numbers, or that polar filtering be
employed to remove unstable computational modes

The equiangular, gnomonic cubed-sphere grid

The icosahedral triangular grid is derived from the spherical icosahedron
that consists of 20 equilateral spherical triangles, 30 great circle edges,
and 12 vertices. These initial triangles are then subdivided repeatedly until
the desired mean resolution is obtained. For a single subdivision, each edge
is divided in

Several methods are available for subdividing the triangular regions. One
such approach is implemented by the ICON grid generator, which allows an
“arbitrary” subdivision factor

The icosahedral (hexagonal) grid, also commonly referred to as the geodesic
grid, is most directly obtained by taking the dual to the icosahedral
(triangular grid) – that is, by replacing grid nodes with spherical
polygons. The resulting grid's cells are hexagonal, except for 12
pentagonal cells. Given an icosahedral–triangular mesh, vertices of the
corresponding icosahedral–hexagonal mesh are then defined as either
circumcenters or barycenters of triangles, leading to either a Voronoi mesh,
used by DYNAMICO (see also Sect.

It is often useful to optimize icosahedral–hexagonal grids as well. DYNAMICO
applies a number of iterations of Lloyd's algorithm

OLAM optimizes by applying the spring dynamics method of

Detail of one step of local mesh refinement used by the OLAM Voronoi mesh. The transition zone is constructed by explicit topological reconnection of the grid lines, which produces pairings of heptagons (red dots) and pentagons (blue dots) along the refinement perimeter.

Given a set of

A constrained centroidal Voronoi tessellation mesh with localized grid density that could be employed in the MPAS model.

As with the classical reduced Gaussian grid of

Locations of the octahedral reduced Gaussian grid nodes

The overset Yin–Yang grid

The Yin–Yang grid is a combination of two limited-domain latitude–longitude grids assembled to provide complete coverage of the sphere.

The horizontal placement of variables impacts a number of properties of the
numerical method, including how energy and enstrophy conservation is managed,
any computational modes that might arise due to differencing, dispersion
properties, and the maximum stable time-step size for explicit time-stepping
schemes

Arguments in favor or against particular staggerings have generally emerged
from linear analyses and typically in the absence of either implicit or
explicit diffusion. In this context, the A grid tends to support large time-step
sizes but produces unphysical phase speeds and negative group
velocities at high wavenumbers, including a stationary

Horizontal staggering options represented among DCMIP models, in
this case depicted on a rectilinear grid and geodesic grid. Here,

Other specialized staggerings have been developed that couple horizontal
staggering with the formulation of the time integrator. In the FV

Because of the vast differences between horizontal and vertical scales in global simulations, most atmospheric models use dimension splitting in order to separate the horizontal discretization from the vertical discretization. In this section, design considerations related to the vertical column are discussed, including the staggering of prognostic and diagnostic variables, and the choice of vertical coordinate.

Along with the choice of prognostic variables, the vertical discretization of
the equations of motion also allows for the staggered placement of prognostic
variables. As with hydrostatic models, certain discretizations give rise to
spurious computational modes that can contaminate the solution

In the context of dimension splitting, the “horizontal” typically refers to
either the contravariant basis, which is perpendicular to the vertical, or
the covariant basis, which is directed along coordinate (e.g.,
terrain-following) surfaces. In contrast, the vertical dimension is strictly
aligned with the radial vector pointing from the center of the Earth.
Vertical position is typically labeled using an arbitrary function

Mass-based coordinates

The vertical coordinate in the GEM model, denoted

In the floating Lagrangian formulation

A pure

The Navier–Stokes equations that govern atmospheric motion can take on many
forms, depending on the choice of prognostic variables and coordinate system.
A derivation of many forms of these equations can be found in Appendix

ACME-A presently solves the compressible shallow-atmosphere equations using a hybrid
terrain-following pressure vertical coordinate

The CSU model uses the vorticity divergence form of the equations of motion,
as described in Sect.

The prognostic equations employed by DYNAMICO are based on a Hamiltonian
formulation

The hydrostatic FV

The FVM formulation is based on conservation laws for dry mass (Eq.

In GEM, the non-hydrostatic equations are written explicitly as deviations
from hydrostatic balance represented by

ICON solves a non-hydrostatic equation set based on

Additional prognostic variables include total air density
(Eq.

The specific heat capacities and ideal gas constant are approximated to be equal to their dry values

In the current implementation, the following simplifications are made with regard to the treatment of moisture:
the atmospheric mass loss/gain due to precipitation/evaporation is neglected in the total mass continuity
Eq. (

The evolution equations used by MPAS are fully described in

NICAM prognoses horizontal and vertical momentum analogous to the approach
described in Sect.

OLAM solves the deep-atmosphere, fully compressible equations in mass- and
momentum-conserving finite-volume form using Eqs. (

Momentum is C-staggered in the horizontal and vertical (Lorenz vertical
staggering is used), meaning that prognosed components live on the grid cell
faces and are each normal to the respective face, and the pressure-gradient
force is evaluated and applied at those locations. However, evaluation of
advective and turbulent momentum transport (as well as the Coriolis force)
involves a diagnostic reconstruction of the total momentum vector at the
centers of scalar grid cells

Tempest is a shallow-atmosphere Eulerian model with terrain-following

Most dynamical cores implement specialized techniques for diffusion or
stabilization (see Table

In both ACME-A and Tempest, scalar hyperviscosity is employed for

The CSU model utilizes an explicit diffusion scheme that consists of
fourth-order hyperdiffusion (

In DYNAMICO, (hyper-)diffusive filters are used to eliminate spurious noise
due to the energy-conservative centered discretization. Filters are applied
every

Explicit dissipation in FV

FV

FV

Within the dynamical core, FVM does not apply any explicit dissipation/diffusion. For the DCMIP test cases, the implicit regularization of the monotonic MPDATA provides sufficient dissipation/diffusion needed to remove excess energy from the finest scales and maintain model stability. An absorbing layer is also available for damping vertically propagating waves near the model top.

An explicit hyperviscosity in GEM is handled via applications of the Laplacian operator
for both wind components and tracers.
A vertical sponge layer, which uses a Laplacian operator,
is employed on wind components and

The ICON model employs damping and diffusion operators for numerical
stabilization and dynamic closure. The details of this scheme appear in
Sect. 2.4 and 2.5 of

ICON also includes Rayleigh damping on

The horizontal diffusion consists of a flow-dependent second-order
Smagorinsky diffusion of velocity (

A fourth-order computational diffusion is also available for vertical wind
speed

The MPAS model applies fourth-order hyperdiffusion and Smagorinsky diffusion

Smagorinsky diffusion, which is often applied in atmospheric models to
parameterize turbulent processes, uses a second-order Laplacian and a
physically motivated eddy viscosity

NICAM implements three types of diffusion: 3-D divergence damping,
fourth-order horizontal hyperdiffusion, and sixth-order vertical
hyperdiffusion, as described in

OLAM requires two types of artificial damping. In the upper layers of the
model, vertical velocity and small-scale horizontal divergence are damped in
order to attenuate gravity waves and thereby mitigate their reflection off
the rigid top boundary of the domain. The damping layer is commonly applied
in the uppermost 10

Temporal discretizations are important for capturing the discrete dynamical
evolution of the global atmosphere. In the past two decades, a variety of new
temporal discretizations have been developed, leaving behind the days when
the leapfrog scheme was ubiquitous across models. This diversity is in part
because of the demands of non-hydrostatic models: unlike their hydrostatic
counterparts, non-hydrostatic atmospheric models must include a mechanism for
dealing with vertically propagating sound waves. These waves are
meteorologically insignificant, but with a vertical grid spacing of 100

Implicit–explicit schemes are a broad category of time-integration schemes
that divide the terms of the prognostic equations into a set of explicitly
integrated terms and implicitly integrated terms. At the very least, terms
associated with vertically propagating sound waves are included among the
implicit terms. For the remaining terms, there is some freedom in choosing
how to integrate terms associated with vertical advection and horizontally
propagating sound waves. Semi-implicit schemes are one such class of schemes
that typically incorporate horizontally propagating sound waves into the
implicit solve and thus rely on a global Helmholtz-type solve. Additive
Runge–Kutta schemes are another mechanism to ensure high-order temporal
accuracy, and many such schemes have been described throughout the literature
(see, for example,

ACME-A and Tempest both use the ARS(2,3,2) scheme described in

CSU uses a semi-implicit time-integration scheme with third-order Adams–Bashforth scheme for explicit integration of the continuity equation, potential temperature equation, and terms related to advection. Since potential temperature is updated prior to the computation of the pressure-gradient force, this term can be thought of as implicit in time. The horizontal wind field is then predicted through integration of the vorticity and divergence of the horizontal wind and a multi-grid method applied to solve a pair of two-dimensional Poisson equations for the stream function and velocity potential, which are then differentiated to obtain the velocity field. Horizontal diffusion is then applied forward in time.

FV

DYNAMICO uses an additive Runge–Kutta time scheme with two Butcher tableaus,
one explicit and one implicit. A Hamiltonian splitting decides which terms of
the equations of motion are treated explicitly or implicitly (Dubos and
Dubey, 2017). As a result, the implicit terms couple the vertical
acceleration due to the pressure gradient and the adiabatic pressure change
due to vertical displacements of fluid parcels. The resulting implicit
problem reduces to independent, scalar, purely vertical, non-linear problems
which are solved to machine precision in two Newton iterations involving one
tridiagonal solve each. The overall time scheme has a HEVI (horizontally
explicit, vertically implicit) structure. Currently, the second-order, three-stage
ARK(2,3,2) scheme is used

ICON consists of a two-time-level predictor–corrector scheme, which is
explicit for all terms except for those describing the vertical propagation
of sound waves. No time splitting is used with respect to sound waves,
because the ratio between the speed of sound and the maximum wind speed in
the mesosphere, which is in part covered by the vertical domain, can be close
to 1. Instead, time splitting is employed to dynamics on the one hand and
horizontal diffusion, tracer transport, and fast physics on the other hand.
Typically, a full time step consists of four or five dynamical substeps in which a
constant forcing originating from the slow physics is applied.
Mass-consistent transport is achieved by passing time-averaged air-mass
fluxes from the dynamical substeps to the transport scheme. The details of
the predictor–corrector scheme, including measures to increase the numerical
efficiency and to optimize the accuracy, are described in Sect. 2.4 of

MPAS and NICAM use a split-explicit formulation

OLAM uses a unique temporal discretization that combines elements of the Adams–Bashforth (AB2) scheme and a Lax–Wendroff formulation for advected quantities. However, instead of extrapolating all prognostic tendencies forward to the half-future time level as in AB2, the horizontal momentum components alone (not their tendencies) are extrapolated in time at the cell boundaries where they reside. The extrapolated momentum provides the time-centered cell-to-cell total mass flux across the grid cell faces that is responsible for advective transport. Advection of all quantities, including all three velocity components that are diagnostically reconstructed at scalar cell centers, and advancement in time from the current to the future time level is based on the time- and space-centered Lax–Wendroff formulation. This scheme is horizontally explicit, but a trapezoidal-implicit formulation is used in the vertical for stable integration of vertically propagating sound waves. A byproduct of the implicit formulation is an implicit time-centered vertical momentum that joins the time-extrapolated horizontal momentum to form a complete set of mass fluxes for advection. The vertical momentum equation is solved first so that the time-centered vertical momentum is available for computing transport of horizontal momentum and all scalar quantities. A time-split scheme is most often used where momentum and potential temperature are updated more frequently than other scalar fields in order to accommodate horizontally propagating sound waves.

A characteristic feature of the FVM (Sect.

The solution procedure of Eq. (

GEM differs from the approaches above by using a semi-Lagrangian advection.
Any model equations, prognostic or diagnostic, are written in the form

There are two intensive calculation sections in this process:
the so-called semi-Lagrangian calculations
(twice estimating departure positions, twice interpolating right-hand side

As discussed earlier, this paper represents the first in a series of papers
documenting the results from the 2016 Dynamical Core Model Intercomparison
Project workshop and summer school. In this paper, we have provided a
description of the differences and similarities between participating models,
including the choice of computational grid, horizontal staggering, vertical
staggering, vertical coordinates, prognostic equations, choice of diffusion,
stabilization, filters and fixers, and temporal discretization. The
literature on dynamical core development is vast, with origins that go back
over half a century. Consequently, the models discussed in this paper only
represent a sample of the many dynamical cores that have been developed for
general circulation modeling. Some of the models that have not been discussed
include

The vast diversity within the modern dynamical core ecosystem suggests that
there is no consensus on a single approach that is intrinsically superior to
other options. Choices made in the dynamical core confer advantages that
include parallel scalability

Information on the availability of source code for the models featured in this paper is tabulated below.

In this Appendix, we provide a detailed derivation of the fluid equations
utilized by non-hydrostatic models. The physical constants which are used
throughout this document are given in Table

A list of physical constants used in this document.

The atmospheric fluid is assumed to be an ideal gas. For moist air, the ideal
gas constant

Relationships between key thermodynamic variables arise from the ideal gas
law, along with definitions of Exner pressure, potential temperature, and
virtual potential temperature:

Note that, as a consequence of Eq. (

In coordinate-invariant form, the prognostic velocity equations may be written
in either the Lagrangian or Eulerian frame as

The equations above still provide some flexibility with regard to the choice
of

Under the orthogonal formulation, projection of a vector field

Thus, the vertical velocity equation, obtained by taking
Eq. (

The dynamical equations are now formulated in terms of the vertical
coordinate

From Eq. (

The generalized velocity

Using Eq. (

The prognostic equations utilizing horizontal kinetic energy

An alternative form of these equation can similarly be obtained in terms of

Note that under the shallow-atmosphere approximation, the first metric terms
(those that include

From Eqs. (

In conjunction with Eq. (

The vorticity divergence form of the dynamical equations in an arbitrary
vertical coordinate predicts the absolute vorticity (

By taking the material derivative (Eq.

By taking the material derivative of Eq. (

The momentum form of the prognostic equations emerges by combining the
prognostic velocity equations with a continuity equation. Essentially, any of
the continuity equations can be chosen, as long as the mass field represented
by the equation is everywhere non-zero. However, the most common options are
moist pseudo-density

Text in this paper describing individual models was provided by the respective modeling teams. Final composition and development of Appendix A was performed by PAU.

The authors declare that they have no conflict of interest.

DCMIP2016 is sponsored by the National Center for Atmospheric Research Computational Information Systems Laboratory, the Department of Energy Office of Science (award no. DE-SC0016015), the National Science Foundation (award no. 1629819), the National Aeronautics and Space Administration (award no. NNX16AK51G), the National Oceanic and Atmospheric Administration Great Lakes Environmental Research Laboratory (award no. NA12OAR4320071), the Office of Naval Research, and CU Boulder Research Computing. This work was made possible with support from our student and postdoctoral participants: Sabina Abba Omar, Scott Bachman, Amanda Back, Tobias Bauer, Vinicius Capistrano, Spencer Clark, Ross Dixon, Christopher Eldred, Robert Fajber, Jared Ferguson, Emily Foshee, Ariane Frassoni, Alexander Goldstein, Jorge Guerra, Chasity Henson, Adam Herrington, Tsung-Lin Hsieh, Dave Lee, Theodore Letcher, Weiwei Li, Laura Mazzaro, Maximo Menchaca, Jonathan Meyer, Farshid Nazari, John O'Brien, Bjarke Tobias Olsen, Hossein Parishani, Charles Pelletier, Thomas Rackow, Kabir Rasouli, Cameron Rencurrel, Koichi Sakaguchi, Gökhan Sever, James Shaw, Konrad Simon, Abhishekh Srivastava, Nicholas Szapiro, Kazushi Takemura, Pushp Raj Tiwari, Chii-Yun Tsai, Richard Urata, Karin van der Wiel, Lei Wang, Eric Wolf, Zheng Wu, Haiyang Yu, Sungduk Yu, and Jiawei Zhuang. We would also like to thank Rich Loft, Cecilia Banner, Kathryn Peczkowicz, and Rory Kelly (NCAR); Perla Dinger, Carmen Ho, and Gina Skyberg (UC Davis); and Kristi Hansen (University of Michigan) for administrative support during the workshop and summer school. Edited by: Julia Hargreaves Reviewed by: Hilary Weller and Thomas Melvin