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

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 omsk, Russia, 3-13 July 2011

Coupled Climate Model of INM RAS INMCM atmospheric atmosphere ocean chemistry 5x4 o , 39 σ-levels up to 90 km or high mesosphere chemical transport Volodin E.M.et al. 2010 reactions Galin V.Ya. et al. 2007 http://ksv.inm.ras.ru diffusion advection to be changed

CABARET advection scheme in 1D case CABARET – Compact Accurately Boundary-Adjusting high-REsolution T echnique (upwind leapfrog) Goloviznin et al., 2003 1D Flux form equation uτ uτ for conservative variables n + 1 n + 1 ( u ρ) 2 −( u ρ) 2 n + 1 −ρ i n i + 1 i − 1 ρ i 2 2 + = 0 τ Δ x i i-1/2 i i+1/2 i+1 Interpolation of flux variables n + 1 n − 1 n 2 +ρ 2 = 2 ρ i ρ x x i + 1 i − 1 2 2

CABARET monotonization procedure Goloviznin et al., 2003,, n+1 x n i-1/2 i+1/2 d ρ dt = f ( x ,t ) n + 1 =ρ o t n + 1 n + ∫ t n + 1 / 2 ) ,t n + 1 / 2 )τ n + f ( x ( t ρ i + 1 / 2 f ( x ( t ) ,t ) dt ≈ρ o n n + 1 / 2 )τ , M i n + 1 / 2 )τ n =ρ i − 1 / 2 n n =ρ i + 1 / 2 n m i + f ( x i ,t + f ( x i + 1 / 2 ,t n + 1 ≤ max ( m i , M i ) min ( m i , M i )≤ρ i + 1 / 2

CABARET advection scheme in 2D case ∂ρ ∂ t + ∂ u ρ ∂ x +∂ v ρ ∂ y = 0 j+1 conservative variables j+1/2 x flux variables Karabasov, Goloviznin 2007 C-grid j 2D i i+1/2 i+1 Kostrykin 2010 B-grid

CABARET advection scheme in 2D case I. Update conservative variables n + 1 / 2 n − 1 / 2 n n n n ρ i + 1 / 2, j + 1 / 2 =ρ i + 1 / 2, j + 1 / 2 − ̃ u (ρ i + 1, j + 1 / 2 −ρ i , j + 1 / 2 )−̃ v (ρ i + 1 / 2, j + 1 −ρ i + 1 / 2, j ) II. Update flux variables n + 1 n + 1 / 2 n n + 1 n + 1 / 2 n ρ i + 1, j + 1 / 2 ̂ = 2 ρ i + 1 / 2, j + 1 / 2 −ρ i , j + 1 / 2 , ̂ ρ i + 1 / 2, j + 1 = 2 ρ i + 1 / 2, j + 1 / 2 −ρ i + 1 / 2, j III. Flux variables correction n n n n m i + 1, j + 1 / 2 = min (ρ i + 1, j + 1 / 2 , ρ i , j + 1 / 2 ) , M i + 1, j + 1 / 2 = max (ρ i + 1, j + 1 / 2 , ρ i , j + 1 / 2 ) , n n n n m i + 1 / 2, j + 1 = min (ρ i + 1 / 2, j + 1 , ρ i + 1 / 2, j ) , M i + 1 / 2, j + 1 = max (ρ i + 1 / 2, j + 1 , ρ i + 1 / 2, j ) , n + 1 n + 1 = max ( min (̂ ρ i + 1, j + 1 / 2 ρ i + 1, j + 1 / 2 ,M i + 1, j + 1 / 2 ) ,m i + 1, j + 1 / 2 ) , n + 1 n + 1 ρ i + 1 / 2, j + 1 = max ( min (̂ ρ i + 1 / 2, j + 1 , M i + 1 / 2, j + 1 ) ,m i + 1 / 2, j + 1 )

Monotonicity of the CABARET scheme 1D Ostapenko, 2009 2D,3D ? IIIa n n n n m i + 1, j + 1 / 2 = min (ρ i + 1, j + 1 / 2 , ρ i , j + 1 / 2 ) , M i + 1, j + 1 / 2 = max (ρ i + 1, j + 1 / 2 , ρ i , j + 1 / 2 ) , n n n n m i + 1 / 2, j + 1 = min (ρ i + 1 / 2, j + 1 , ρ i + 1 / 2, j ) , M i + 1 / 2, j + 1 = max (ρ i + 1 / 2, j + 1 , ρ i + 1 / 2, j ) , n + 1 n + 1 = max ( 0. ,max ( min (̂ ρ i + 1, j + 1 / 2 ρ i + 1, j + 1 / 2 , M i + 1, j + 1 / 2 ) ,m i + 1, j + 1 / 2 )) , n + 1 n + 1 = max ( 0. , max ( min (̂ ρ i + 1 / 2, j + 1 ρ i + 1 / 2, j + 1 , M i + 1 / 2, j + 1 ) ,m i + 1 / 2, j + 1 )) u ≤ 1 v ≤ 1 Statement . Under condition scheme I, II, IIIa is positive. 4 , ̃ ̃ 4 Proof n + 1 / 2 n − 1 / 2 n n ρ i + 1 / 2, j + 1 / 2 ≥ρ i + 1 / 2, j + 1 / 2 − u ρ i + 1, j + 1 / 2 − v ρ i + 1 / 2, j + 1 = 1 n − 1 / 2 n − 1 n ρ i + 1 / 2, j + 1 / 2 2 (ρ i , j + 1 / 2 +ρ i + 1, j + 1 / 2 ) , = 1 n − 1 / 2 n − 1 n ρ i + 1 / 2, j + 1 / 2 2 (ρ i + 1 / 2, j +ρ i + 1 / 2, j + 1 ) , ≥( 1 +( 1 n + 1 / 2 n − 1 n − 1 n n ρ i + 1 / 2, j + 1 / 2 4 −̃ u )ρ i + 1, j + 1 / 2 +̃ u ρ i , j + 1 / 2 4 −̃ v )ρ i + 1 / 2, j + 1 + ̃ v ρ i + 1 / 2, j .

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

Problem of pressure-tracer inconsistency ∂ q ∂ t  div  q u = 0,  1  ∂ ∂ t  div  u = 0,  2  ∂ 1 div 2  u  d = 0  3  ∂ t  ∫ 0 n  1 − q  n  q   F n  1 / 2  q u = 0,  4   t n  1 − n  n  1 / 2  u  ˙ n  1 / 2 = 0,  5   F 2    t n  1 − n  N  ∑ i = 1 n  1 / 2  u i = 0.  6  F 2  t 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

Seasonal cycle of total ozone column CABARET Leapfrog Observations

Annual distribution of Ozone (ppmv) a P h , e r CABARET u s s e r P a P h , e r u leapfrog s s e r P a P h , e HALOE r u s s e r P latitude

Vertical profiles ozone

Zonally mean annual ozone profile leapfrog CABARET

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

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Annual distribution of HCl (ppbv) a P h , e r CABARET u s s e r P a P h , e r u s s leapfrog e r P a P h , e r u HALOE s s e r P latitude