Modification and implementation of the CABARET scheme in the Coupled - - PowerPoint PPT Presentation

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Modification and implementation of the CABARET scheme in the Coupled Climate Model INM RAS

Kostrykin S.V.

Institute of Numerical Mathematics RAS Moscow, Russia

Email:kostr@inm.ras.ru CITES-2011, T

• msk, Russia, 3-13 July 2011

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• cean

atmospheric chemistry atmosphere transport chemical reactions advection diffusion to be changed

Volodin E.M.et al. 2010 Galin V.Ya. et al. 2007

Coupled Climate Model of INM RAS INMCM

5x4o, 39 σ-levels up to 90 km or high mesosphere http://ksv.inm.ras.ru

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CABARET advection scheme in 1D case

ρ

i+1 2 n+1 2+ρ i−1 2 n−1 2=2ρi n

ρi

n+1−ρi n

τ + (uρ)

i+1 2 n+ 1 2−(uρ) i− 1 2 n+1 2

Δ xi =0

i-1/2 i i+1/2 i+1

x x uτ uτ

Flux form equation for conservative variables Interpolation of flux variables

Goloviznin et al., 2003

CABARET – Compact Accurately Boundary-Adjusting high-REsolution T echnique 1D (upwind leapfrog)

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CABARET monotonization procedure

Goloviznin et al., 2003,,

d ρ dt = f (x ,t) ρi+1/ 2

n+1 =ρo n+∫t

n

t n+1

f (x(t),t)dt≈ρo

n+ f (x(t n+1/ 2),t n+1/ 2)τ

mi

n=ρi−1/2 n

+ f (xi ,t

n+1/2)τ , M i n=ρi+1/ 2 n

+ f (xi+1/2 ,t

n+1/ 2)τ

min(mi , M i)≤ρi+1/2

n+1 ≤max(mi , M i)

n n+1 i-1/2 i+1/2

x

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x

i+1/2 i+1 i j j+1/2 j+1

CABARET advection scheme in 2D case

conservative variables flux variables

∂ρ ∂ t + ∂uρ ∂ x +∂ v ρ ∂ y =0

B-grid Kostrykin 2010 C-grid Karabasov, Goloviznin 2007 2D

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ρi+1/2, j+1/2

n+1/2

=ρi+1/2, j+1/2

n−1/2

− ̃ u(ρi+1, j+1/2

n

−ρi , j+1/2

n

)−̃ v(ρi+1/2, j+1

n

−ρi+1/2, j

n

) ̂ ρi+1, j+1/2

n+1

=2ρi+1/2, j+1/2

n+1/2

−ρi , j+1/2

n

, ̂ ρi+1/2, j+1

n+1

=2ρi+1/2, j+1/2

n+1/2

−ρi+1/2, j

n

mi+1, j+1/2=min(ρi+1, j+1/2

n

,ρi , j+1/2

n

), M i+1, j+1/2=max(ρi+1, j+1/2

n

,ρi , j+1/2

n

), mi+1/2, j+1=min(ρi+1/2, j+1

n

,ρi+1/2, j

n

), M i+1/2, j+1=max(ρi+1/2, j+1

n

,ρi+1/2, j

n

), ρi+1, j+1/2

n+1

=max(min(̂ ρi+1, j+1/2

n+1

,M i+1, j+1/2),mi+1, j+1/2), ρi+1/2, j+1

n+1

=max(min(̂ ρi+1/2, j+1

n+1

, M i+1/2, j+1),mi+1/2, j+1)

CABARET advection scheme in 2D case

• I. Update conservative variables
• II. Update flux variables
• III. Flux variables correction

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• Statement. Under condition scheme I, II, IIIa is positive.

ρi+1/2, j+1/2

n+1/2

≥ρi+1/2, j+1/2

n−1/2

−uρi+1, j+1/2

n

−vρi+1/2, j+1

n

ρi+1/2, j+1/2

n−1/2

=1 2 (ρi , j+1/2

n−1

+ρi+1, j+1/2

n

), ρi+1/2, j+1/2

n−1/2

=1 2 (ρi+1/2, j

n−1

+ρi+1/2, j+1

n

), ρi+1/2, j+1/2

n+1/2

≥(1 4−̃ u)ρi+1, j+1/2

n

+̃ uρi , j+1/2

n−1

+( 1 4−̃ v)ρi+1/2, j+1

n

+ ̃ vρi+1/2, j

n−1

. ̃ u≤1 4 , ̃ v≤1 4 mi+1, j+1/2=min(ρi+1, j+1/2

n

,ρi , j+1/2

n

), M i+1, j+1/2=max(ρi+1, j+1/2

n

,ρi , j+1/2

n

), mi+1/2, j+1=min(ρi+1/2, j+1

n

,ρi+1/2, j

n

), M i+1/2, j+1=max(ρi+1/2, j+1

n

,ρi+1/2, j

n

), ρi+1, j+1/2

n+1

=max(0. ,max(min(̂ ρi+1, j+1/2

n+1

, M i+1, j+1/2),mi+1, j+1/2)), ρi+1/2, j+1

n+1

=max(0., max(min(̂ ρi+1/2, j+1

n+1

, M i+1/2, j+1),mi+1/2, j+1))

Monotonicity of the CABARET scheme Ostapenko, 2009 1D 2D,3D ?

IIIa Proof

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S-N E-W Results of the solid-body rotation test on the sphere = 2

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Problem of pressure-tracer inconsistency

∂ q ∂t div qu=0,1 ∂ ∂t div u=0,2 ∂ ∂t ∫0

1 div2 ud =03

 q

n1− q n

 t F n1/ 2 qu=0,4 

n1− n

 t F 2

n1/2 u ˙



n1/2=0,5



n1− n

 t ∑i=1

N

F 2

n1/2 ui=0.6

1) For global conservation one should know the pressure field on the next time step before advection 2) Inconsistency in calculation of advection of tracer and pressure could lead to the non-monotonicity of the tracer distribution

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Observations CABARET Leapfrog

Seasonal cycle of total ozone column

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Annual distribution of Ozone (ppmv)

CABARET leapfrog HALOE

P r e s s u r e , h P a latitude P r e s s u r e , h P a P r e s s u r e , h P a

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Vertical profiles ozone

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Zonally mean annual ozone profile

CABARET leapfrog

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Conclusions: 1) A positive multidimensional version of CABARET scheme is proposed 2) The implementation of the new advection schemes into chemical transport model leads to:

• improvement of ozone climatology near stratopause

(disappear artificial maximum) and near stratospheric

• zone maxima (decreasing)
• improvement in the description of total ozone column at

high and middle latitudes SH (deeper ozone holes, weaker midlatitude maxima)

• at tropics a total ozone column becomes larger than in
• bservations

3) Elapsed time for CTM increases on 30%. Memory storage increases in 4 times for each tracer.

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Thanks for your attention!

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HALOE

latitude

CABARET leapfrog

P r e s s u r e , h P a P r e s s u r e , h P a P r e s s u r e , h P a

Annual distribution of HCl (ppbv)