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

NOAA/ESRL

Flow-following- finite-volume

Icosahedral

Model

FIMSandy MacDonaldRainer BleckStan BenjaminJian-Wen BaoJohn M. BrownJacques MiddlecoffJin-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.

Why Icosahedral Finite-Volume (FV) model ?

Icosahedral Grid Generation

N=((2**n)**2)*10 + 2 ; 5th level – n=5 N=10242 ~ 240km; max(d)/min(d)~1.26th level – n=6 N= N=40962 ~ 120km; 7th level – n=7N=163842 ~60km8th 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=400N=(400**2)*10+2~1.6 million points.

Sadourny, Arakawa, Mintz, MWR (1968)

high granularitypossible with icosa-hedral model

Sourceqsps

sq

sp

mVm

spq

ppcwhereM

TCH

sp

sps

ssp

mVm

sp

sps

ssp

mVm

sp

ym

yEMm

pv

spsuv

xm

xEMm

pu

spsvu

hs

t

cRp

p

hs

t

hs

t

t

t

p

2

/0

2

2

/

0

• 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.

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

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

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

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

"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.

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

divergencefluxmass

horizontalintegratedvertically

surfacethroughmotionvertical

surfaceof

motionvertical

ss

(zero in fixed

grids)

(zero in material coord.)

(known)

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

)/(, spsp

)/(, spsp

)/(, spsp

)/(, spsp

tracersotherqvu ,,,,

tracersotherqvu ,,,,

tracersotherqvu ,,,,

3

Stairstep profile of versus pressure

Laye

r 1

Laye

r 4

Laye

r 2

Laye

r 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)

Compute diagnostic variables from prognostic variables

Solution order in FIM

Shallow-water 2-D transport of u,v,p,T,q, other tracers(Intermediate prognostic variables are used in subsequent Physics)

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

Vertical regridding and remapping of prognostic variablesbased on (*, p* ). This includes “vertical transport” which is applied to all prognostic variables at same time toachieve conservation in an environment of changing p*

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)

3-D Baroclinic wave 100-day simulations

Following movie shows Layer thickness superimposed bysurface pressure

Following movie shows Layer thickness superimposed by

Rain water

Runs with idealized mountains Layer thickness +surface pressure

• 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.

Final Remarks

Implementation of GFS physics1. The first-order non-local turbulence and surface-

layer scheme

2. The 4-layer Noah soil model with Zobler soil type

3. The simple cloud scheme plus simplified Arakawa-Schubert convective scheme

4. 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)