Atmospheric modeling systems require economical methods to solve
the non-hydrostatic Euler equations. Two major differences between
hydrostatic models and a full non-hydrostatic description lies in the
vertical velocity tendency and numerical stiffness associated with sound
waves. In this work we introduce a new arbitrary-order vertical
discretization entitled the staggered nodal finite-element method (SNFEM).
Our method uses a generalized discrete derivative that consistently combines
the discontinuous Galerkin and spectral element methods on a staggered grid.
Our combined method leverages the accurate wave propagation and conservation
properties of spectral elements with staggered methods that eliminate
stationary (

The accurate representation of vertical wave motion is essential for models
of the atmosphere. The vertical coordinate for the non-hydrostatic fluid
equations has traditionally been discretized in the Eulerian frame via a
second-order Charney–Phillips

This paper describes a new discretization for the vertical that combines the
accuracy of finite-element methods with the desirable wave propagation
properties of staggered methods. This method of vertical discretization was
originally described in

Composition of interpolation

Our staggered method is similar to the mixed finite-element formulations of

To introduce our approach for the construction of a generalized, staggered, variable order-of-accuracy, finite-element vertical discretization. We emphasize discretization of the non-conservative differential form of the Navier–Stokes equations (in vector invariant or so-called Clark form), which is independent of coordinate system.

To validate the implementation of this discretization within the Tempest framework using a selection of
test cases in Cartesian geometry through a range of horizontal scales from

To determine the qualitative and quantitative effect of vertical order of accuracy on solutions by conducting validation experiments at coarse resolutions relative to finer reference solutions. We consider the effects of Lorenz (LOR) and Charney–Phillips (CPH) staggering both in the interior flow and at the lower boundary.

To determine whether a high-order vertical discretization greatly improves the simulation quality, and consequently to recommend whether there is an optimal order-of-accuracy that provides the best tradeoff between accuracy and computational cost.

To assess the performance of SNFEM, this discretization has been implemented
in the spectral element Tempest model

We will show that a high-order vertical discretization at coarse resolution
more accurately approximates the reference solution relative to the low
vertical-order alternative when total count of degrees of freedom is kept
constant. Since the interpolation and derivative operators in the finite-element approach are easily expressed as linear matrix operators, there is
minimal cost in adjusting the order-of-accuracy. We will present control experiments in Sect.

The remainder of this manuscript is as follows:
Sect.

In an arbitrary coordinate frame

Covariant components can be obtained in terms of contravariant components via
contraction with the covariant metric

The reverse operation uses the contravariant metric

The volume element

Using covariant horizontal velocity components, vertical velocity, potential
temperature

The vertical velocity

The specific Kinetic energy is

Here

Consequently, the rotational terms in the equation of motion take the form

Note that this formulation does not specify a coordinate system.
Consequently, these equations can be used for either Cartesian or spherical
geometry. To account for topography, terrain-following

We note that in this framework, the discretization is decoupled from the grid
definition. As such, Tempest is designed to target flows on the sphere and in
Cartesian domains simultaneously with or without terrain. This is convenient
in the analysis, implementation, and validation of the numerical techniques
that follow. We focus our validation on Cartesian cases and will address test
cases on the sphere in a subsequent publication based on the same
discretization framework. Lastly, derivatives of the vertical coordinate in

The horizontal discretization of
Eqs. (

Each vertical column consists of

For the remainder of this manuscript we will use script

Note that Eqs. (

To define the interpolant from levels to interfaces, a two-step procedure is
employed. Since basis functions on levels are discontinuous, we define the
left and right interpolants at element boundaries as

These interpolation operators can also be obtained from equivalence via the
variational (weak) form. At model interfaces, the accuracy of
Eq. (

Differentiation is required for all combinations of model levels and
interfaces:

A depiction of the derivative operators

Differentiation from interfaces to levels is obtained by simply
differentiating Eq. (

This works in practice as there is an exact mapping from derivatives of the continuous polynomial space (over interfaces) to the discontinuous polynomial space (over levels).

Differentiation from levels to levels is computed via the composed operator

Differentiation from interfaces to interfaces requires averaging the
one-sided derivatives at element interfaces, but is otherwise simply the
derivative of Eq. (

Differentiation from levels to interfaces (

There is some flexibility in the discretization that depends on the specific
choice of flux correction functions.

The second derivative operators are used in viscosity and hyperviscosity
calculations. They are obtained as approximations to the equation

For the viscous operator from interfaces to interfaces, the discretization is
obtained from the variational (weak) formulation. Specifically, from
Eq. (

Then using Eqs. (

For model interfaces on Gauss–Lobatto nodes, the integral is discretized via Gauss–Lobatto quadrature.

The viscous operator from levels to levels is derived in a similar manner,
although the non-differentiability of

For model levels on Gauss nodes, the integral is discretized directly via
Gaussian quadrature. Note that the boundary condition implies that we must
impose

The basic spectral element method is an energy conservative scheme

The interpolation and differentiation operators given in the previous
sections provide a framework for constructing staggered vertical grids in the
context of the nonlinear system
Eqs. (

For example, applying the discrete derivative operators with Lorenz
staggering to Eqs. (

Here the vertical interpolation operators are defined in
Sect.

It is important to note the great deal of flexibility available in the
computation of spatial terms in
Eqs. (

Many options are available for the temporal discretization of the
semi-discrete equations, including several fully explicit and
implicit–explicit schemes

For the first time step, an implicit update is applied,

Explicit terms are evolved using a Runge–Kutta method, which supports a large
stability bound for spatial discretizations with purely imaginary
eigenvalues. This particular scheme is based on

Hyperviscosity is then applied in accordance with

When active, Rayleigh friction is applied via backward Euler to relax all
prognostic variables to a specified reference state,

In accordance with Strang splitting, a final implicit update is applied,

In this section we present a set of test cases with the purpose of
investigating the performance of the SNFEM for mesoscale atmospheric
modeling. Our emphasis is on a wide range of resolutions from the global
scale (200 km) to the large-eddy scale (5 m). These scales transition from
hydrostatic to scales where all nonlinear terms in the
Eqs. (

The horizontal discretization is kept as a standard fourth-order spectral element formulation for all simulations,
as outlined in Sect.

The time integration scheme is based on Strang-split IMplicit EXplicit (IMEX) outlined in Sect.

Vertical terms

Reference solutions employ consistent fourth-order vertical and horizontal discretizations at a resolution at least twice as fine as experiments

The total number of vertical levels in each test is kept constant. Only the vertical order of accuracy is changed and consequently the distribution of grid spacing according to the locations of element nodes.

For these tests, we investigate the effect of a relatively high-order

Reference results are computed with a consistent spatial (horizontal and vertical) discretization or order “O4”. Experiments done at coarser resolutions with varying vertical order of accuracy are titled “VO(no.)”.

Baroclinic wave in a 3-D Cartesian channel at the reference resolution

The first test represents steady-state geostrophically balanced flow in a
channel and is based on a new test case defined by

The simulation is performed for the original

The grid spacing for the reference solution is

Baroclinic wave in a Cartesian channel at vertical orders 2, 4, and 10. Vorticity at 500 m on days 10
and 15 at the resolution

The baroclinic instability is a primary mechanism for the development of
mid-latitude storm systems and so it is important that an atmospheric
modeling platform reproduce these phenomena accurately. We present a
reference solution of the baroclinic wave shown in Fig.

The reference solution for temperature and vorticity at 500 m elevation
shown here can be compared at day 10 with the original results from

The vorticity field at coarse resolution (Fig.

Baroclinic wave in a Cartesian channel at vertical orders 2, 4, and 10. Vertical velocity at 500 m on days
10 and 15 at the resolution

Atmospheric flows are strongly influenced by the lower boundary, where
topographically induced waves transport momentum and energy vertically.

Schär flow at steady-state (10 h) vertical velocity in (m s

To validate that Tempest produces the correct mountain wave response, the
Schär mountain test was performed until

Schär flow at steady state (10 h). Collocated method (all variables on column levels)
results
compared to staggered (Lorenz) solution at the same spatial order and resolution.

As discussed in

Still atmosphere experiment over Schär mountain profile at vertical
orders 2, 4, and 10 showing errors in vertical velocity.

Because our model makes use of a terrain-following coordinate, it is expected
that a hydrostatically balanced rest state is not exactly preserved over
topography. Imbalance will arise as a consequence of inexact cancellation of
the terrain-following and vertical pressure gradient terms in the discrete
equations. Experiments carried out with zero background flow in the presence
of topographic features shown in Fig.

Schär flow at steady-state (10 h) vertical velocity in (m s

Experiments are conducted at vertical order 2, 4, 10, and 40 (in the limit
where the polynomial order is equal to the total number of levels, denoted
ST) at a relatively coarse resolution of

Schär flow steady state (10 h). Vertical velocity difference with respect to the reference solution
(Fig.

Schär mountain vertical profile of momentum flux for all
experiments. The flux profiles are computed by

We further compare the resulting profiles of momentum flux for all
experiments in the Lorenz configuration (Fig.

The density current test case of

The initial state consists of a hydrostatically balanced state with a uniform
potential temperature of

The domain is an enclosed box

For the experiments with vertical flow-dependent hyperviscosity, the viscous
coefficients are given by Eq. (

Straka density current test reference solutions at vertical order 4 in two staggering configurations LOR
and CPH. Converged resolution of

The grid spacing for the reference solution is

For the density current, we emphasize results from the Lorenz (LOR)
staggering. Under CPH staggering, the vertical advection
term for potential temperature (see Table

Straka density current test at vertical order 2, 4, and 10. Coarse, evaluation resolution of

We often desire diffusion to be as weak as possible while still preserving
the stability of the underlying method. However, as can be seen here, the
structure of the density current is also strongly dependent on the
dissipation mechanisms employed in the simulation. Here we present the
reference solution equivalent to

Straka density current test at vertical order 2, 4, and 10. Coarse, evaluation resolution of

Cold wave front position (

From Table

The use of flow-dependent hyperviscosity in second and fourth derivative order
changes the structure of coarse experiments tending toward a three-rotor flow
field shown in the reference solution as shown in
Fig.

Straka density current test at vertical order 2, 4, and 10. Coarse, evaluation resolution of

Moreover, Fig.

Thermal bubble experiments have become a standard in the development of
non-hydrostatic mesoscale modeling systems. At very fine resolutions (

We present two flow scenarios: (a) the bubble rises and is allowed to
interact with the top and lateral boundaries and (b) the so-called Robert
smooth bubble experiment (as outlined in

The background consists of a constant potential temperature
field

Here we choose the amplitude and radius of the perturbation to be

The reference grid spacing is

The fourth-order scalar and vector (vorticity and divergence
separately) diffusion coefficients are given by

Rising thermal bubble potential temperature reference solution at vertical order 4. Reference resolution

Rising Robert bubble potential temperature reference solution at vertical order 4. Reference resolution

Rising bubble experiments show the nonlinear dynamics of dry 2-D convection.
The classic thermal bubble test shown in Fig.

The rising thermal bubble experiment is typically carried out and compared at
700 s precisely before the convective bubble interacts with the top boundary
of the domain. We present this result for comparison with previous results in
Fig.

The Robert smooth bubble experiment extends the vertical domain allowing for
the onset of Kelvin–Helmholtz instabilities in the flow. The solution at the
reference resolution is shown in Fig.

Rising thermal bubble potential temperature at vertical orders 2, 4, and 10. Convection bubbles at 700 and
1200 s. Coarse resolution

Rising Robert bubble potential temperature at vertical orders 2, 4, and 10.
Convection bubbles at 800 at 1200 s. Coarse, evaluation resolution

High-order vertical discretizations are typically associated with strong
oscillations (Gibbs ringing) that can induce perturbations that can amplify
turbulence, particularly if stabilization (such as upwinding or diffusion) is
weak. The net effect is that a high-order vertical discretization, given the
same horizontal discretization, changes the local mixing characteristics of
the flow. This effect is seen clearly in Fig.

Spatial (left) and temporal (right) self convergence at various vertical orders of accuracy.
Thermal bubble test at 200 s. Spatial resolution for temporal convergence is
10 m with reference

We briefly characterize the combined discretization strategy (horizontal
spectral element, vertical SNFEM, and Strang IMEX) described in
Sect.

Numerically computed estimates of the Courant–Friedrichs–Lewy
condition (maximum Courant number) using Thermal bubble tests over a wide
range of horizontal : vertical aspect ratios. The maximum wave speed
corresponds
to the speed of sound given by

A numerically computed estimate of the Courant–Friedrichs–Lewy (CFL) condition (maximum Courant number)
as a function of grid spacing and element aspect ratio is given in
Table

Thermal bubble test

Furthermore, we show preliminary parallel performance scaling in
Table

Plots of the normalized change in mass and energy, along with integrated
zonal and vertical momentum from the Robert smooth bubble test
(Sect.

Further investigation of this issue seems to suggest roots in the way the
stabilization mechanism interacts with the lateral boundaries, since the
purely advective scheme with no stabilization shows nearly flat total energy.
Consequently, we hypothesize this result may be associated with the inverse
energy cascade from 2-D turbulence theory drawing energy from the unresolved
scales in a limited manner. Note that the stabilization mechanisms described
by this work (horizontal and vertical hyperviscosity), which work directly on
the

For a horizontally symmetric test such as the rising thermal bubble
(anti-symmetric in

Observed normalized change in mass and energy (top row), along with zonal and vertical
momentum (bottom row) using the Robert bubble experiment in Sect.

The idea of separating the vertical and horizontal dynamics in atmospheric modeling systems has roots in the scale differences that characterize atmospheric flows. This principle has been fully exploited in the development of global and mesoscale models, along with the application of the hydrostatic approximation. This paper adds to the modern literature on modeling atmospheric dynamics by analyzing a novel discretization technique for achieving high-order accuracy in the vertical while maintaining the desirable properties of staggered methods. We refer to this technique as the staggered nodal finite-element method (SNFEM).

The test suite we present in this work is not exhaustive, but it is intended
to evaluate the performance of the numerical schemes under conditions of near-hydrostatic synoptic-scale flow in
Sect.

However, there are some trade offs when increasing the vertical order:
(1) for a vertically implicit method, fewer high-order elements lead to a
dense matrix structure that is more expensive to invert, (2) the oscillatory
nature of the polynomial functions that make up the interpolants within an
element have physical consequences (involving nonlinear processes) at the
smallest scales, and (3) higher-order spatial discretizations often require
smaller time steps or higher order temporal discretizations.
Figure

The first point can be addressed in the construction of the software where
parallelization and correct use of hardware resources minimizes the dense
operations that high-order elements imply. We saw in
Fig.

Furthermore, when physical instabilities arise, a consistent, high-order, and
scale selective dissipation strategy is necessary. In this regard, finite-element methods allow for the construction of diffusion operators for this
purpose e.g., Sect.

The numerical dissipation strategy implemented here serves two primary goals: (1) stabilization of the computations and (2) as a form of closure for the Euler equations solved on a truncated grid. The methods we employ allow for the construction of derivative operators of various orders in a consistent manner. Tempest features a system that allows for diffusion to be applied in a selective manner on variables that are split according to the time integration scheme.

Further experiments are necessary to test the extent of the third point
above. For this work, we used a second-order Strang time integration scheme
(Sect.

The authors conclude the following based on the experiments conducted and properties of the SNFEM:

Staggering has been generalized to finite-element methods combining continuous and discontinuous formalisms. The result is a method that closely parallels the behavior of staggered finite differences eliminating stationary modes. This is strictly true for the lowest order finite elements and we restrict ourselves to observe that consistent behavior extends to high-order staggered elements pending a formal wave analysis.

Variable order of accuracy is an effective strategy that can compensate for limitations in grid-scale resolution. However, the effects at very high order must be understood and controlled with appropriate stabilization methods. In general, “intermediate” orders (about 4th order) are recommended with consideration for consistency in overall spatial order given an IMEX partitioned architecture

We emphasize that, while the equations are formulated in a coordinate-free
manner, the results given all correspond to regular Cartesian coordinates as
defined by the metrics in Eqs. (

Tempest is constructed to provide a unified multi-scale platform for atmospheric simulation. Experiments can be carried out readily at all scales of importance from long-term climate simulations to high-resolution weather prediction. Development is underway to include moisture transport and phase transformations as well as to further improve time integration performance. Coupled with highly accurate, efficient, and robust methods to compute dynamics, Tempest will evolve to produce reliable precipitation forecasts as well as long-term climate simulations as part of the greater effort to understand the impending challenges brought on by rapid climate change.

The Tempest codebase used to generate the results in this publication are
available through the following Git repository:

The authors would like to thank Hans Johansen, Mark Taylor, and David Hall for their assistance in refining this manuscript. Funding for this project has been provided by the Department of Energy, Office of Science project “A Non-hydrostatic Variable Resolution Atmospheric Model in ACME.”Edited by: S. Marras