# Presentation: 484536

## Euler Solutions using Flux-based Wave Decomposition

**Nash’at Ahmad**
**Center for Atmospheric Physics**
**Science Applications International Corporation**
**ahmadn@saic.com**

## Overview

**Background**
**High-resolution Wave Propagation Methods**
**f-wave Riemann Solver**
**Implementation within CLAWPACK**
**Simulations & Comparisons with WRF**
**Non-hydrostatic Inertia Gravity Waves**
**Convection in Neutral Atmosphere**
**Non-linear Density Currents**
**Summary/Future Work**

## Godunov-Type Schemes

**Advantages**
**Conservative**
**Ability to resolve regions of steep gradients**
**Minimal dispersion and phase errors**
**Positivity of scalars ensured by applying the TVD condition appropriately**
**No grid-staggering or explicit diffusion required for stability**
**Can be implemented on unstructured adaptive grids**
**Disadvantage(s)**
**Numerically diffusive**

## Godunov-Type Schemes

**Previous Work (Atmospheric Flows)**
**Carpenter, R. L., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, “Application of the Piecewise Parabolic Method (PPM) to Meteorological Modeling”, *****MWR*****, Vol. 118, 1990, pp. 586-612.**
**Botta, N., R. Klein, S. Langenberg, and S. Lutzenkirchen, “Well balanced finite volume methods for nearly hydrostatic flows”, *****JCP*****, Vol. 196, 2004, pp. 539-565.**
**Ahmad, N., Z. Boybeyi, R. Löhner, and A. Sarma, “A Godunov-Type Finite Volume Scheme for Flows on Meso- and Micro-scales”.  AIAA Paper 2005-1234.**** **

## CLAWPACK

**Implements high-resolution Godunov-type methods for hyperbolic PDEs**
**Randall J. LeVeque, University of Washington, Seattle**
**Open Source, General purpose**
**User has to supply the Riemann solver**
**rpn.f**** – Riemann solver for the normal direction**
**rpt.f**** – Riemann solver for the transverse direction**
**Second-order accuracy achieved by adding a correction term**
**Wave limiters for enforcing the TVD condition**
**Diffusion can be added as a source term**
**AMR and MPI capabilities**

## Noye-Tan Test

**Noye and Tan (IJNMF, 1989, 9: 75-98)**
**Scalar advection-diffusion equation **
**Domain: [0:2m] x [0:2m]**
**Final time = 1.25s**
**kx = ky = 0.01**
**Initial u-velocity = v-velocity = 0.8 m/s**
**mx = my = 100**

## Governing Equations

**Q**** denotes the source terms and ****D**** is the sub-grid scale diffusion. In this study:**
**Atmosphere is assumed to be dry and the only source term is gravity.**
**Subgrid-scale diffusion is ignored and only Euler solutions are considered.**

## Numerical Scheme – (1)

**U****n****ij**** is the cell average over the cell (i,j) at time = t****n**

**x****i-1/2**

**x****i+1/2**

**y****i-1/2**

**y****i+1/2**

- Uij

## Numerical Scheme – (2)

**Consider the linear system of equations:**

**Flux-differencing form of the Lax-Wendroff Method can be written for F****i-1/2,j**** as:**

## Numerical Scheme (3)

**The jump in ****U**** at cell centers is usually decomposed as a linear combination of **
**the eigenvectors, to obtain the waves:**

**Conservation is enforced by the following condition:**

## f-waves – LeVeque (1)

**Use flux-based wave decomposition, in which the flux differences are written **
**directly as a linear combination of the right eigenvectors:**

**Advantages:**
**No need for the explicit definition of Roe matrix, which does not exist**
**for the hyperbolic PDEs governing atmospheric flows.**
**Since we are dealing directly with fluxes, the discretization errors**
**by including the source term due to gravity are minimal. **

## f-waves – LeVeque (2)

**The update of U can now be written as:**

**S****p****i-1/2**** are the wave speeds given by the eigenvalues of the hyperbolic system**

**Fluctuations which contribute to the cell-averaged quantity U****i**** due to the wave propagation across cell interfaces:**

## f-waves – LeVeque (3)

**Second-order accuracy achieved by adding a correction term:**

**Where:**

## Eigenspace – 2D Euler Eq.

**From the Jacobian corresponding to F(U):**

**From the Jacobian corresponding to G(U):**

## Eigenspace – 2D Euler Eq.

**Matrix of right eigenvectors and its inverse:**

## Eigenspace – 2D Euler Eq.

**Beta coefficients:**

**Jumps in flux functions are given by:**

## Inertia-Gravity Waves

**Skamarock & Klemp (MWR, 1994, 122: 2623-2630)**
** **
**Domain: [0:300km] x [0:10km]**
**Final time = 3000s**
**Theta initialized by a constant Brunt-V****ä****is****ä****l****ä**** frequency = 10****-2**** s****-1**
**Initial u-velocity = 20 m/s and v-velocity set to zero**
**Periodic boundary conditions in the lateral**
**Solid wall at top and bottom boundaries**
**Ran simulations for different mesh resolutions**

**Initial theta**
**perturbation**
**max = 10****-2**** (K)**

## Inertia-Gravity Waves

## Inertia-Gravity Waves

- Theta perturbation (K)

- v-velocity (m/s)

## Buoyant Thermal

**Wicker and Skamarock (MWR, 1998, 126: 1992-1999)**
** **
**Domain: [0:20km] x [0:10km]**
**Final time = 1020s = 17 minutes**
**Neutral atmosphere in hydrostatic balance**
**Initial u-velocity and v-velocity set to zero**
**Outflow/farfield boundary conditions in the lateral**
**Solid wall at top and bottom boundaries**
**dx = dy = 125 m**

**v-vel (m/s) @ t=1020s**

**u-vel (m/s) @ t=1020s**

## Buoyant Thermal

## Buoyant Thermal

## Buoyant Thermal

**Carpenter et al. (MWR, 1990, 118: 586-612)**
** **
**Case 1 – Domain: [0:5km] x [0:10km] **
**Case 2 – Domain: [0:5km] x [0:4km]**
**Final time = 1020s = 17 minutes**
**Neutral atmosphere in hydrostatic balance**
**Initial u-velocity and v-velocity set to zero**
**Initial Theta Perturbation Max = 2K**
**Left boundary = Axi-symmetric**
**Right boundary = Outflow/farfield**
**Solid wall at top and bottom boundaries**
**dx = dy = 20 m**

## Buoyant Thermal

**Case 1**

**Case 2**

## Non-linear Density Currents

**Droegemeier and Wilhelmson **
**(JAS, 1987, 44:1180-1210)**
** **
**Domain: [0:25km] x [0:10km]**
**Final time = 900s**
**Neutral atmosphere**
**Initial u- and v-velocity set to zero**
**Left lateral boundary set to wall**
**Outflow/farfield conditions for the right lateral boundary**
**Solid wall at top and bottom boundaries**
**dx = dy = 50 m**

**Modified Straka Benchmark**
**(IJNMF, 1993, 17:1180-1210)**
** **
**Domain: [-26.5:26.5km] x [0:6.4km]**
**Final time = 900s**
**Neutral atmosphere**
**No diffusion added**
**Initial u- and v-velocity set to zero**
**Outflow/farfield boundary conditions in the lateral**
**Solid wall at top and bottom boundaries**
**dx = dy = 50 m**

## Non-linear Density Currents

- theta (K)

## Non-linear Density Currents

- theta (K)

## Non-linear Density Currents

**Straka et al. (IJNMF, 1993, 17:1180-1210)**
** **
**Domain: [-20.0:20.0km] x [0:6.4km]**
**Final time = 900s**
**Neutral atmosphere**
**Diffusion added (K****m****= K****h****= 75m****2****/s)**
**Initial u- and v-velocity set to zero**
**Outflow/farfield boundary conditions in the lateral**
**Solid wall at top and bottom boundaries**
**dx = dy = 50 m**

## Non-linear Density Currents

## Non-linear Density Currents

| Variable | fwave | REFC | REFS | REFQ |
| --- | --- | --- | --- | --- |
| p ́max (mb) | 1.26 | 2.87 | 2.49 | 1.74 |
| p ́min (mb) | 6.26 | -5.14 | -5.55 | -5.21 |
|   ́max (K) | 8.92E-03 | 0.00 | 0.00 | 0.00 |
|   ́min (K) | -9.82 | -9.77 | -9.77 | -10.00 |
| umax (m/s) | 34.44 | 36.46 | 35.02 | 34.72 |
| umin (m/s) | -15.75 | -15.19 | -16.32 | -15.31 |
| vmax (m/s) | 13.63 | 12.93 | 13.28 | 13.04 |
| vmin (m/s) | -16.37 | -15.95 | -16.11 | -16.89 |
| front location (m) | 15525.00 | 15537.44 | N/A | 15509.17 |

- REFC, REFS and REFQ are the three reference solutions reported in Straka et al. (1993).
- All reference solutions are on a 25m grid.

## Computational Cost

**From Carpenter et al. (1990)**
**2D simulation comparisons of PPM with a finite difference code in terms of CPU time**
**492s for PPM vs. 553s for the FD code with no time-splitting**
**492s for PPM vs. 46s for the FD code with time-splitting**
**Need to conduct CLAWPACK vs. WRF time comparisons**

## Summary

**Designed and implemented a Godunov-type scheme for atmospheric flow simulations**
**Conservative, stable, able to resolve regions of steep gradients**
**Compared with WRF model with favorable results**
**Well-suited for simulating non-hydrostatic flows on the meso-, micro-, and urban scales**

## Future Work

**Currently working on implementing the scheme on unstructured adaptive grids**

## Acknowledgement

**The proposed Riemann solver has been implemented in CLAWPACK which was developed by Randall J. LeVeque at the University of Washington, Seattle. For details see: ****LeVeque, R. J., 2002: Finite Volume Methods for Hyperbolic Problems. Cambridge Press, 558 pp.**
**ARW-WRF simulations for validation and verification were performed by John Lindeman (School of Computational Sciences, George Mason University).**