A Multistep Flux-Corrected Transport (MFCT)

Jin Lee Earth System Research Laboratory (ESRL)

NOAA

Continuity equation Monotonic advection

Conservative and monotonic tracer transport scheme

Conservation & Monotonicity

VDt

D

V

tDt

D

Dt

Dq ;0

onconservaticer tra0

onconservati mass 0

qVt

q

Vt

• Montonoticity: no new maximum/ no new minimum• Consistency: q=1, then tracer eq -> continuity eq.

Two approaches for monotonic transports :

• Flux limiters:

- which constraint the fluxes based on the monotonic subgrid distributions.

• Flux Corrected transports:

- which include the diffusive and antidiffusive stages designed to limit fluxes in strong gradient regions and over smooth parts of solution, the maximum allowable high-order flux is restored to the low-order solution without violating the monotonicity.

Two-time level .vs. Multistep schemes :

12

53

;12

162

;12

231

)2(3

)1(2

)(1

1

)()2(3

)()1(2

)()(1

1

) AB3 :schemeorder 3 Bashforth-Adams (e.g.,

Flux Massand eqs.,-tracer-,continuity

)(1

)()(1

: Flux Massand eqs.,-tracer-,continuity

rd

Multistep

level time-Two

ttt

qVnk

F

qVnk

FqVnk

Fn

kqn

kq

Vnk

F

Vnk

FVnk

Fn

kn

k

qVnk

Fn

kqn

kq

Vnk

Fn

kn

k

Brief review of Zalesk two-time level FCT scheme :

eq continuity becomes eq- tracer1 :Note

solorder -low monotone diffusivea is

)()(~

~1

:scheme FCTlevel time- two(1979)Zalesak

)()(1

core dynamical level time- twoin eq-Continuity

q

Lkq

qVnLk

FqVnk

Fqadk

F

qVadk

FLkq

nkq

qVnLk

FnkqL

kq

Vnk

Fn

kn

k

qVnLk

F

qVnk

FqVnk

FqVnk

Fqadk

F

qVadk

FLkq

nkq

qVnLk

FnkqL

kq

Vnk

FVnk

FVnk

Fnk

nk

)2(3

)1(2

)(1

)(~

~1

: (MFCT) FCTmultistep AB3

23

12

1

)1(

: eq continuity multistep AB3

Extension of Zalesak FCT to AB3-MFCT scheme

Low-order monotone sol: discontinuous function / strong gradients.High-order AB3-sol: smooth function.

FIM icos plotting – Ning Wang – 240km res

domain. thein

edges allover

Compute fluxesorder low flxl

domain. thein

edges allover

Compute fluxesorder highflxh

cell-i

domain. thein

edges allover

Define fluxes diffusive-antiflxlflxhfad

blue). (inoutflux

and red) (inInflux

asfluxes iveantidiffus Group

splitting) ldirectiona no n,formulatio D-(2

.undershoot avoid to(flux)outflux clip

overshoot, avoid to(red)influx clip

.undershoot-or-over avoid to

fluxes iveantidiffus enough"just "Clip

Analytical test of “Cosine Bell Advection over poles” in one of Williamson et. al. (1992) standard test cases (dx~240 km).

“Slotted-Cylinder Advection over poles” (dx~120 km)

Cosine Bell Advection:(dx~120 km)Curve A: analyticCurve B: MFCT

Slotted cylinder:(dx~120 km)Curve A: analyticCurve B: MFCT

km 240~x

km 120~x

km 60~x

No-FCT MFCTSmooth Functon

km 240~x

km 120~x

km 60~x

Discontinuous Functon No-FCT MFCT

FIM model/ system

-Contributors

FIM DESIGN

Jin Lee

Sandy MacDonald

Rainer Bleck

Jian-Wen Bao

John M. Brown

Jacques Middlecoff

Ning Wang

Stan Benjamin

Tom Henderson

Chris Harrop

Bill Moninger Brian JamisonSusan SahmEd Szoke

Georg Grell

Jeff Whitaker,

SOS project:Bob LipschutzDavid HimesBeth RussellSteve AlbersTom Kent+ dozens more…

ESMF, Subversion

WorkFlow Manager, xml real-time scripts

Verification,Web page,Evaluation

FIM- chemistry, aerosols

EnKF data assimilation

Results of FIM medium-range weather forecasts

FIM real time forecasts initialized with GFS initial condition at 00 and 12 UTC

(comparisons of FIM and GFS 500 mb ACC)

• Interpolate GFS initial data to Icosahedral grid.

• Perform hydrostatic initialization.

• Perform 7-day fcst with dx~30km.

• Same initial condition, terrain & sfc parameters, physics package runing at similar model resolutions (dx~30 km)

Same IC, terrain & sfc parameters, physics package, runing at similar model resolutions (dx~30 km)

Effect of different numerics, e.g., spectral v.s. finite-volume, on tracer transports.

•A general MFCT tracer algorithm is developed to achieve conservative and monotonic transports in any multistep dynamical cores which provide mass fluxes by which tracers are transported.

•AB3-MFCT has been implemented on our Icosahedral SWM and tested with continuous and discontinuous analytical test cases as well as real time medium range weather forecasts.

•The advection accuracy to the monotonic constraint ?

Final remarks

or