# A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models

## Abstract

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 (2Δ*x*) modes. Furthermore, high-order accuracy also eliminates the need for a reference state to maintain hydrostatic balance. In this work we demonstrate the use of high vertical order as a means of improving simulation quality at relatively coarse resolution. We choose a test case suite that spans the range of atmospheric flows from predominantly hydrostatic to nonlinear in the large-eddy regime. Lastly, our results show that there is a distinct benefit in using the high-order vertical coordinate at low resolutions with the same robust properties as the low-order alternative.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of California, Davis, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1255170

- Alternate Identifier(s):
- OSTI ID: 1268264

- Grant/Contract Number:
- SC0014669

- Resource Type:
- Published Article

- Journal Name:
- Geoscientific Model Development (Online)

- Additional Journal Information:
- Journal Name: Geoscientific Model Development (Online) Journal Volume: 9 Journal Issue: 5; Journal ID: ISSN 1991-9603

- Publisher:
- Copernicus GmbH

- Country of Publication:
- Germany

- Language:
- English

- Subject:
- 54 ENVIRONMENTAL SCIENCES; step integration methods; navier-stokes equations; shallow-water equations; optimal representation; spectral element; prediction

### Citation Formats

```
Guerra, Jorge E., and Ullrich, Paul A. A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models. Germany: N. p., 2016.
Web. doi:10.5194/gmd-9-2007-2016.
```

```
Guerra, Jorge E., & Ullrich, Paul A. A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models. Germany. doi:10.5194/gmd-9-2007-2016.
```

```
Guerra, Jorge E., and Ullrich, Paul A. Wed .
"A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models". Germany. doi:10.5194/gmd-9-2007-2016.
```

```
@article{osti_1255170,
```

title = {A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models},

author = {Guerra, Jorge E. and Ullrich, Paul A.},

abstractNote = {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 (2Δx) modes. Furthermore, high-order accuracy also eliminates the need for a reference state to maintain hydrostatic balance. In this work we demonstrate the use of high vertical order as a means of improving simulation quality at relatively coarse resolution. We choose a test case suite that spans the range of atmospheric flows from predominantly hydrostatic to nonlinear in the large-eddy regime. Lastly, our results show that there is a distinct benefit in using the high-order vertical coordinate at low resolutions with the same robust properties as the low-order alternative.},

doi = {10.5194/gmd-9-2007-2016},

journal = {Geoscientific Model Development (Online)},

number = 5,

volume = 9,

place = {Germany},

year = {2016},

month = {6}

}

DOI: 10.5194/gmd-9-2007-2016

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