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

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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

NOAA/ESRL

Flow-following- finite-volume

Icosahedral

Model

FIM

Sandy MacDonald

Rainer Bleck

Stan Benjamin

Jian-Wen Bao

John M. Brown

Jacques Middlecoff

Jin-luen Lee

Earth System Research Laboratory

- 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)

- 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.

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=7N=163842 ~60km

8th level – n=8N=655,362 ~30km; 9th level – n=9N=2,621,442 ~15km

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.

high granularity

possible with icosa-hedral model

Sadourny, Arakawa, Mintz, MWR (1968)

- 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 springdynamics.

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 FCTroutine.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

Mass conservation in Rossby-Haurwitz solution

Time integration

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

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

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

- 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)

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,q grid box)

Poor vertical resolution in weakly stratified regions

Elaborate transport operators needed to achieve conservation

The x,y,q grid box

north

east

q

.

Major Cons:

Coordinate-ground intersections are inevitable (atmosphere doesn’t fit snugly into x,y,q grid box)

Poor vertical resolution in weakly stratified regions

Fixes:

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

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

"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 maximizes size of isentropic subdomain.

- 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

(zero in fixedgrids)

(zero in material coord.)

(known)

Layer 4

Layer 3

Layer 2

Stairstep profile of q versus pressure

Layer 1

q1

q2

q3

q4

Blue arrows indicate diabatic heating

q1

q2

q3

q4

The “regridding” step: find new interface pressure

q1

q2

q3

q4

The “regridding” step: find new interface pressure

equal areas

q1

q2

q3

q4

Repeat in all layers

equal areas

q1

q2

q3

q4

Final outcome: diabatic heating translated into interface movement

q1

q2

q3

q4

- 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)

Solution order in FIM

Shallow-water 2-D transport of u,v,Dp,T,q, other tracers

(Intermediate prognostic variables are used in subsequent Physics)

Physics (determine *, Dp* and source/sink tendencies)

Vertical regridding and remapping of prognostic variables

based on (*, Dp* ). This includes “vertical transport” which

is applied to all prognostic variables at same time to

achieve conservation in an environment of changing Dp*

Compute diagnostic variables from prognostic variables

Vertical-meridional slice through 2-layer atmosphere

Thermal restoration (counter-acting effect of baroclinic instability)

Baroclinic instability reduces slope of isentropes

High lat.

Low lat.

3-D Baroclinic wave

100-day simulations

Following movie shows

Layer thickness superimposed by

surface pressure

Following movie shows

Layer thickness superimposed by

Rain water

Runs with idealized mountains

Layer thickness +surface pressure

Final Remarks

- A finite-volume icosahedral hybrid model has been developed. This model is free of pole problems and
- (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.

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)