# Presentation: 889527

## Development and testing of a global forecast model... configured on a horizontally icosahedral, vertically quasi-material (“flow-following”) grid

- Today’s presenters:
**Jin Lee and Rainer Bleck**
- NOAA Earth System Research Lab, Boulder, Colorado

##     Earth System Research Laboratory

-     ***Earth System Research Laboratory***

**NOAA/ESRL**
**F****low-following- finite-volume    **
**I****cosahedral **
**M****odel**
**          ****FIM**
- Sandy MacDonald
- Rainer Bleck
- Stan Benjamin
- Jian-Wen Bao
- John M. Brown
- Jacques Middlecoff
- Jin-luen Lee

## Topics covered:

- Introduction (Rainer)
- Horizontal discretization: the icosahedral grid; 2-D results (Jin)
- Vertical discretization: the hybrid-isentropic grid (Rainer)
- 3-D Results, conclusions, and outlook (Jin)

## Topics covered:

- Introduction (Rainer)
- Horizontal discretization: the icosahedral grid; 2-D results (Jin)
- Vertical discretization: the hybrid-isentropic grid (Rainer)
- 3-D Results, conclusions, and outlook (Jin)

## ESRL Flow-following, finite volume Icosahedral Model  (FIM)

- Icosahedral grid, with spring dynamics implementation
- Finite volume, flux form equations in horizontal (planned - Piecewise Parabolic Method)
- Hybrid isentropic-sigma ALE vertical coordinate 		(arbitrary Lagrangian-Eulerian)
- Nonhydrostatic (not initially)
- Earth System Modeling Framework

## 1) Icosahedral + FV approach provides conservation.
2) Icos quasi-uniform grid is free of pole problems.
3) Legendre polynomials become inefficient at high resol.
4) Spectral models require global communication which is inefficient on MPP with distributed memory.
5) Spectral models tend to generate noisy tracer transport.

Since Icosahedral models  are based on a local numerical scheme, they are free of above problems 3 – 5.

- 2) Icos quasi-uniform grid is free of pole problems.
- 3) Legendre polynomials become inefficient at high resol.
- 4) Spectral models require global communication which is inefficient on MPP with distributed memory.
- 5) Spectral models tend to generate noisy tracer transport.
- Since Icosahedral models  are based on a local numerical scheme, they are free of above problems 3 – 5.

**     ****Why Icosahedral Finite-Volume (FV) model ?**

## Icosahedral Grid Generation

**Icosahedral Grid Generation**

- N=((2**n)**2)*10 + 2 ; 5th level – n=5  N=10242 ~ 240km;  max(d)/min(d)~1.2
- 6th level – n=6 N= N=40962 ~ 120km;  7th level  – n=7N=163842    ~60km
- 8th level – n=8N=655,362   ~30km;     9th level – n=9N=2,621,442  ~15km

##  high granularity

- N=(m**2)*10 + 2
- “m” is any integer ratio between  arc(AB~8000 km) and target resolution.
- e.g., for dx~20 km, then m=8000/20=400
- N=(400**2)*10+2~1.6 million points.

- Sadourny,  Arakawa, Mintz, MWR (1968)

****** ****high granularity**
**possible with icosa-hedral model**

## Slide 9

## Finite-Volume operators including
  (i) Vorticity operator based on Stoke theorem,
  (ii) Divergence operator based on Gauss theorem,
  (iii) Gradient operator based on Green’s theorem.
Explicit 3rd-order Adams-Bashforth time differencing.
Icosahedral grid is optimized with spring dynamics.

-   (i) Vorticity operator based on Stoke theorem,
-   (ii) Divergence operator based on Gauss theorem,
-   (iii) Gradient operator based on Green’s theorem.
- Explicit 3rd-order Adams-Bashforth time differencing.
- Icosahedral grid is optimized with spring dynamics.

**         ****Numerics of the Icosahedral SWE **

## Finite volume flux computation:

- - flux into each cell from surrounding donor cells

## I) Shallow-water dynamics are evaluated with the standard tests of Williamson et. al. (1992) including:  .Advection of cosine bell over poles.Steady state nonlinear geostrophic flow.Forced  nonlinear translating.Zonal flow over an isolated mountain.Rossby-Haurwitz solution  II) Monotonicity and positive-definiteness achieved by.Zalesak (1979) flux corrected transport (FCT).Demonstrated with emerging seamount experiment III) Tracer eqns are solved by same FCT routine.tracer transport is tested with pot.vort. advection

## The following 2 tests are run with level  5 grid on 10242 grid points, i.e., dx~240 km.     .Rossby-Haurwitz wave     .Emerging seamount

##        Case VI: Rossby-Haurwitz Wave

-        Case VI: Rossby-Haurwitz Wave

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##        Mass conservation in      Rossby-Haurwitz solution

- Time integration

## Emerging seamount experiment to test FCT in the limit of zero layer thickness

## 90N

- Eq
- 90S

- Shallow water equations, uniform density, water depth 3000m. Initial conditions: state of rest.
- Seamount at 30 S growing to 3500 m in 24 hrs. Zero thickness (dry land) after day 3

## Shallow water equations, uniform density, water depth 3000m. Initial conditions: state of rest.

- Seamount at 30 S growing to 3500 m in 24 hrs. Zero thickness (dry land) after day 3

## Shallow water equations, uniform density, water depth 3000m. Initial conditions: state of rest.

- Seamount at 30 S growing to 3500 m in 24 hrs. Zero thickness (dry land) after day 3

## Shallow water equations, uniform density, water depth 3000m. Initial conditions: state of rest.

- Seamount at 30 S growing to 3500 m in 24 hrs. Zero thickness (dry land) after day 3

## Shallow water equations, uniform density, water depth 3000m. Initial conditions: state of rest.

- Seamount at 30 S growing to 3500 m in 24 hrs. Zero thickness (dry land) after day 3

## Topics covered:

- Introduction (Rainer)
- Horizontal discretization: the icosahedral grid; 2-D results (Jin)
- Vertical discretization: the hybrid-isentropic grid (Rainer)
- 3-D Results, conclusions, and outlook (Jin)

## Lagrangian vertical coordinate:Pros and Cons(“Lagrangian” = isentropic in atmospheric applications)

- Major Pros:
- No uncontrolled diabatic **mixing** (in the vertical **and** horizontal)
- Numerical dispersion errors associated with vertical **transport** are minimized
- Optimal finite-difference representation of frontal zones & frontogenesis

- Major Cons:
- Coordinate-ground intersections are inevitable (atmosphere doesn’t fit snugly into  *x,y,***  grid box)
- Poor vertical resolution in weakly stratified regions
- Elaborate transport operators needed to achieve conservation

## The  x,y,  grid box

- north

- east

- .

- 

- _The  __*x,y,*__**__  grid box_

## Major Cons:
Coordinate-ground intersections are inevitable (atmosphere doesn’t fit snugly into  x,y, grid box)
Poor vertical resolution in weakly stratified regions

- Coordinate-ground intersections are inevitable (atmosphere doesn’t fit snugly into  *x,y,*** grid box)
- Poor vertical resolution in weakly stratified regions

- Fixes:
- Reassign grid points from underground portion of *x,y,*** grid box to above-ground “*s”* surfaces
- Low stratification => large portion of  *x,y,***  grid box is underground =>  no shortage of grid points available for re-deployment as *s* points

**=> A “hybrid” grid appears to have distinct advantages ****– **both** **from a grid-economy and a vertical resolution perspective

## ALE maximizes size of isentropic subdomain.

- "Hybrid" means different things to different people:
- 	- linear combination of 2 or more conventional	   		coordinates (examples: p+sigma,				p+theta+sigma)
- 	- ALE (Arbitrary Lagrangian-Eulerian) 				coordinate

## ALE: “Arbitrary Lagrangian-Eulerian” coordinate

- Original concept (Hirt et al., 1974): maintain Lagrangian character of coordinate but “re-grid” intermittently to keep grid points from fusing.
- In RUC, FIM, and HYCOM, we apply ALE in the vertical only and re-grid for 2 reasons:
- (1) to maintain minimum layer thickness;
- (2) to nudge an entropy-related thermodynamic variable toward a prescribed layer-specific “target” value by importing mass from above or below.
- Process (2) renders the grid _**quasi-isentropic**_

- Main design element of layer models: Height (alias layer thickness) is treated as dependent variable.
-  needed: a new independent variable capable of representing 3rd (vertical) model dimension. Call this variable “*s*”.
- Having increased the number of unknowns by 1 (layer thickness), we need 1 additional equation. The logical choice is an equation linking “*s**”* to other variables.
-  popular example:   *s* = potential temperature
- Hence ....

- Principal design element of isentropic models: Height and (potential) temperature trade places as dependent / independent variables
- 		- same number of unknowns, same 	  	number of (prognostic) equations, but 	very different numerical properties
- Driving force for isentropic model development: _**genetic diversity**_

## Continuity equation in generalized (“s”) coordinates

**(zero in fixed**** ****grids)**

**(zero in material coord.)**

**(known)**

## Staggering of variables in layer or stacked shallow-water models:

## 

- 

- 

- 3

- 

- Stairstep profile of  versus pressure

- Layer 1

- Layer 4

- Layer 2

- Layer 3

## 

- 

- 

- 3

- 

- Blue arrows indicate diabatic heating

## 

- 

- 

- 3

- 

- The “regridding” step: find new interface pressure

## 

- 

- 

- 3

- 

- The “regridding” step: find new interface pressure

- equal areas

## 

- 

- 

- 3

- 

- Repeat in all layers

- equal areas

## 

- 

- 

- 3

- 

- Final outcome: diabatic heating translated into interface movement

## Topics covered:

- Introduction (Rainer)
- Horizontal discretization: the icosahedral grid; 2-D results (Jin)
- Vertical discretization: the hybrid-isentropic grid (Rainer)
- 3-D Results, conclusions, and outlook (Jin)

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## Baroclinic instability reduces slope of isentropes

**Baroclinic instability reduces slope of isentropes**

- Low lat.

- High lat.

- Vertical-meridional slice through 2-layer atmosphere

**Thermal restoration (counter-acting effect of baroclinic instability)**

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## 3-D Baroclinic wave 
100-day simulations

**100-day simulations**

## Following movie shows 

**Layer thickness superimposed by**
**surface pressure**

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## Following movie shows 

**Layer thickness superimposed by**
**Rain water**

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## Runs with idealized  mountains 

**Layer thickness +surface pressure**

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##  Future Work

-    (I) yields stable solutions without dissipation,
-    (II) maintains monotonicity and positive-definiteness with mass conservation,
-    (III) successfully combines a hybrid             coordinate with a finite volume icosahedral shallow-water model.
-    (Iv) presently uses a simple GFS-like cloud removal scheme
** ****Future Work**
-  Incorporate full GFS with EMC numerical  framework.
- Real data tests of FIM with GFS physics at end of FY06.
- Continue partnership between NCEP/EMC and ESRL/GSD.

**Final Remarks**

## Implementation of  GFS physics

- The first-order non-local turbulence and surface-layer scheme
- The 4-layer Noah soil model with Zobler soil type
- The simple cloud scheme plus simplified Arakawa-Schubert convective scheme
- The Chou SW, RRTM LW schemes interacting with diagnosed cloud water and RH clouds
- 	(Using the GFS initial condition and static fields including Reynolds SST , NESDIS snow cover, USAF snow depth, NESDIS ice analysis , GCIP vegetation type, NESDIS vegetation fraction)