Atmosphere. Korean Meteorological Society Vol. 21, No. 2 (2011) pp. 163-172 mw» w d ½ 1)Á½x 1), *Á 2)Á x 2)Á 2) 1) w» w,» d y 2)» ( : 2011 3 25, : 2011 6 2, y : 2011 6 19 ) Development of Tools for calculating Forecast Sensitivities to the Initial Condition in the Korea Meteorological Administration (KMA) Unified Model (UM) Sung-Min Kim 1), Hyun Mee Kim 1), *, Sang-Won Joo 2), Hyun-Cheol Shin 2), and DukJin Won 2) 1) Atmospheric Predictability and Data Assimilation Laboratory, Department of Atmospheric Sciences, Yonsei University 2) Korea Meteorological Administration (Received : 25 March 2011, Revised : 2 June 2011, Accepted : 19 June 2011) Abstract Numerical forecasting depends on the initial condition error strongly because numerical model is a chaotic system. To calculate the sensitivity of some forecast aspects to the initial condition in the Korea Meteorological Administration (KMA) Unified Model (UM) which is originated from United Kingdom (UK) Meteorological Office (MO), an algorithm to calculate adjoint sensitivities is developed by modifying the adjoint perturbation forecast model in the KMA UM. Then the new algorithm is used to calculate adjoint sensitivity distributions for typhoon DIANMU (201004). Major initial adjoint sensitivities calculated for the 48 h forecast error are located horizontally in the rear right quadrant relative to the typhoon motion, which is related with the inflow regions of the environmental flow into the typhoon, similar to the sensitive structures in the previous studies. Because of the upward wave energy propagation, the major sensitivities at the initial time located in the low to mid- troposphere propagate upward to the upper troposphere where the maximum of the forecast error is located. The kinetic energy is dominant for both the initial adjoint sensitivity and forecast error of the typhoon DIANMU. The horizontal and vertical energy distributions of the adjoint sensitivity for the typhoon DIANMU are consistent with those for other typhoons using other models, indicating that the tools for calculating the adjoint sensitivity in the KMA UM is credible. Key words: adjoint sensitivity, sensitive regions, typhoon forecast, KMA UM, 4DVAR 1.» x y w x w ùkùš, d e» y w ƒ s e l» e» ƒ Corresponding Author: Hyun Mee Kim, Department of Atmospheric Sciences, Yonsei University, Seoul, Republic of Korea Phone : +82-2-2123-5683, Fax : +82-2-365-5163 E-mail : khm@yonsei.ac.kr w y. w d» ù w y w w d w (Errico 1997), d ƒ j d š w (Kim and Jung 2006). d Áœ yw, d» d p yw». w de y w, x w 163
164 mw» w d.» x w ƒ w de,,, y (Kim and Beare 2011).»,,, d ƒ w» w (Errico, 1997). s, s w» d ƒ w,, d w (Kleist and Morgan 2005, Ancell and Mass 2008, Jung and Kim 2009). w d ƒ w ww ƒ d t d ww d k. w j ùkù w ƒ w z» ù (e.g., Klinker et al. 1998, Laroche et al. 2002, Kim and Jung 2006, Kim et al. 2008). w d ƒ e w w» w, y w, d w d wš d œ ùkù d l w» š š (Cardinali 2009, Gelaro et al. 2009, Langland and Baker, 2004; Liu and Kalnay, 2008). ù w» w s, d f w, PSU/NCAR (Pennsylvania State University/National Center for Atmospheric Research) MM5 (fifth Generation) Adjoint Modeling System (Zou et al. 1997) w kt RUSA (200215) d w Kim and Jung (2006), y x w Kim et al. (2008), ½ x (2010), x w Jung and Kim (2009).» w, ù l t d l w qw» l w w. p š l q d w» w w ƒ.» w w w w» w ùkù»zw r» ƒ š ùy w» l f w w. w d œ d w» w w w,» ƒ t d sww Ÿ w ƒ w d z l w. w» wz» 21«2y (2011) x» (Korea Meteorological Administration: KMA)» (United Kingdom Meteorological Office: UKMO) l w mw (UM: Unified Model) Áz l» e l w x wš. wz t d d ƒ d e w w sƒ w Õ v w w» w, KMA UM w xk 4 y (four-dimensional variational data assimilation: 4DVAR) l g w mw l w w. d, t d w, wz KMA UM d w d x l w w. wš, w» w kt DIANMU (201004) w d w. 2 w w š, 3 w w kt DIANMU w d r. 4 w. 2. w» w ƒ š de w w. p deƒ» y w w, p de w w w, p de w (response function) š w. w, p v, (½x 2007), x w w w w. MM5 ³ w kt w w Kim and Jung (2006), kt ( ) w w. Kim and Jung (2006) (2.1) w xk w.», y w w. w ƒ w w» mw w, ƒ»» w š ƒ w. R = 1 2 -- e t f ( ) ( ), Ce t f (2.1)
» R w š, < > ü, C Clayton (2004) s³ norm, e(t f ) t f, ùkü š ƒ wš ùkü 1). w R y l l w. δr =< t f R, δx(t f )> (2.2)» t f R t f w w š (2.1) w w. X state vector š, t f X y δx(t f )» X y δx(t 0 ) l δx(t f ) MδX(t 0 ) w. M x x. (2.2) ùký. δr =<e(t f ), MδX(t 0 )>=<Μ T e(t f ), δx(t 0 )> =< t 0 R, δx(t 0 )> (2.3)» t 0 R» w w ( ) š, M T x M. (2.3) w l» w w t 0 R( d ) w w ( ) M T w z w : t 0 R = Μ T e(t f ) (2.4) w d (2.2)» y (» ) ü w w y ( y) w. j»» d ƒ j w» w ƒ j w» d w e w. w w R mw p w s³ š,. R = 1 2 -- < e ( t f ), Ce( t f ) > = 1 --ρ 2 2 2 p ( u' + v' + w' ) 2 -------------- θ' 1 2 + + --------p' dxdydη, 2 2 2γp 2κθ ref ½ Á½x Á Á x Á 165 (2.5)» u', v', w', θ ', p' ƒƒ 3,,», κ = R/C p 0.286, γ = C p /C v 1.4, θ ref = 300 K w. š ρ p ƒƒ» d». w w n w (Buizza 1994) w p w w, š w. 3. 3.1 KMA UM» w w w š w. xk 4DVAR (Coutier et al. 1994, Clayton 2004) w g w x, w :, d» d w ( d ) 4DVAR w., KMA UM k 4DVAR p PF_Forecast_Adj 2) š g w. xk 4DVAR w» w x w j w. Rabier (2005) 4DVAR x k w w. š KMA UM w xk 4DVAR p PF_Forecast_Adj g, p sw. Fig. 1 KMA UM xk 4DVAR p ƒ ù. PF_Forecast_Adj d w ( d ) w True/False w ForecastOnly, Clayton (2004) w vl w True/False w SingleDF ( DoubleDF), (» ) q w LS, š Á w (PF_hat). 4DVAR d sw 1) state vector 3 œ (x, y, z) t l yw. x, y, z w, 56,246,505,300 1. 2) KMA UM xk 4DVAR p bold t w. Atmosphere, Vol. 21, No. 2. (2011)
166 mw» w d Fig. 1. Major subroutines of VarProg_AnalysePF (main code) in the UM 4DVAR algorithm and modified parts to calculate the adjoint sensitivity in the VarProg_AnalysePF. Fig. 2. Flow chart of the modified code to calculate the adjoint sensitivity in PF_Forecast_Adj.f90. j» ForecastOnly = True w. PF_Forecast_Adj timestep w v (Fig. 2). v timestep LastTimestep_local l FirstTimestep_local ¾, v PF_atmstep_Adj PF_hat w timestep z. w» w timestep w v» w w ƒw š, timestep w v ü PF_atmstep_Adj w» w z ( (2.5)) w» w g ƒw. Var_FieldOutput w PF_atmstep_Adj w w (PF_hat PF_hat l w ) w. w ƒw Table 1. (2.4) (e(t f )). w g z z w.» 24 w g š w. 24 (00 h) l (24 h)¾ 25 ( ) v w. 00h_ls, 01h_ls, Ã, 24h_ls txwš 24h_anal txw. 26 q (25 +1 ) Fig. 1 ù Var_ReadALLNL, 00h_ls, 01h_ls, Ã, 23h_ls, 24h_anal, 24h_lsƒ š, LS x LS % state(*) 1 l 26¾. 25 26 state w w LS l w z (Table 1), w. (PF_hat: R) w xk w tx. x k w ( ) R txwš l TE txw. w» wz» 21«2y (2011)
p 1 R =[ρu', ρv', ρw', -----------θ ', ---- p' ] (3.1) 2 γp κθ ref 1 TE = ρ(u' 2 + v' 2 + w' 2 p -- ) + --------------θ ' 2 + -------- 1 p' 2 (3.2) 2 2 2κθ 2γp ref Calc_energy ( (2.5)) w p (ρ),» (p) ƒ (κ, θ ref, γ) (u', v', w', θ ', p') w w w w. Calc_ energy w (3.1) wš PF_hat w w. TE Calc_energy w w w š : ρu' w w, ρu' AAu w z Calc_energy p e ρ 2 u'.» ρ 2 u' AAu (ρu') ù ρ WT (ρ) w (Table 1). š 0.5 (AAu (ρu')) 2 Ý WT (ρ) e 1/2ρu' 2 xk [J m 3 ]. ù w w. l PF_atmstep_Adj w z, Var_FieldOutput w w. w timestep LastTimestep_local w» w PF_atmstep_Adj w» (timestep = LastTimestep_local) Var_FieldOutput w w w (Fig. 2). TE w w. w p w w» w n w (Buizza 1994) w ½ Á½x Á Á x Á 167 Table 1. Variables newly added in PF_Forecast_Adj.f90 to calculate the adjoint sensitivity in KMA UM. y k fcst_u(213,163,50), fcst_v(213,162,50), fcst_w(213,163,50), real fcst_theta(213,163,50), fcst_p(213,163,50) ( w ) anl_u(213,163,50), anl_v(213,162,50), anl_w(213,163,50), real anl_theta(213,163,50), anl_p(213,163,50) ( w ) en_u(213,163,50), en_v(213,162,50), en_w(213,163,50), real xk w en_theta(213,163,50), en_p(213,163,50) (PF_hat ) Pu(213,163,50), Pv(213,162,50), Pw(213,163,50), real Ptheta(213,163,50), Pp(213,163,50) k l w n w AAu(213,163,50), AAv(213,162,50), AAw(213,163,50), real AAtheta(213,163,50), AAp(213,163,50) ( ) real AU, AV, AW, AT, AP, WT ( ) real ilat, flat, ilon, flon integer inx, fnx, iny, fny real dlon = 1.666667, dlat = 1.111107 n w. Pu, Pv, Pw, Pp, Ptheta n w, p 1.0 wš, 0.0 w. PF_hat n w w PF_hat û w. Á» t Cartesian t y w Á» n w w. 3.2 x w mw l kt DIANMU (201004) w w. 2010 8 8 06 UTC w DIANMU 2010 8 10 12 UTC w. z wá wx m w 2010 8 12 18 UTC w» (Fig. 3). 3.1 KMA UM wì xk 4DVAR. xk 4DVAR (Perturbation Forecast: PF) (Adjoint PF: APF) swwš, PF 2 x, APF w.» KMA UM ( 40 km ). e x x UM s 40 km w 50 eta-height hybrid d w š, PF APF s 110 km w x 50 eta-height hybrid d w, d x, PF, APF w t l 63 km š ¾ w. APF w Atmosphere, Vol. 21, No. 2. (2011)
168 mw» w d Fig. 3. Regional Specialized meteorological Center (RSMC) Tokyo-Typhoon Center best track (black line) and UM forecast track (red line) of typhoon DIANMU (201004).» ( ) UM x l œ. x y Edwards-Slingo radiation (Edwards and Slingo 1996), mixed phase precipitation (Wilson and Ballard 1999), Met Office surface exchange scheme (Essery et al. 2001), non-local boundary layer (Lock et al. 2000), new GWD scheme (Webster et al. 2003), mass flux convection scheme (Kershaw and Gregory 1997, Gregory et al. 1997). x (PF, APF ) UM x y mixed phase precipitation mass flux convection scheme y, w d UM x non-local boundary layer fixed boundary layer w 3). 2010 8 8 12 UTC l 2010 8 10 12 UTC ¾ 48 w. 3.3 Fig. 3 Regional Specialized meteorological Center (RSMC) Tokyo-Typhoon Center œw kt d (best track) e x ùkü. kt d w e w š. w kt d w, 8 9 3) Non-local boundary layer PF, APF UM version 24.3 y. 12 UTC kt ƒ d w 6 š, w z 10 12 UTC 18 (Fig. 3). Fig. 4 w ùkü s³ w» wì. kt 48 (Fig. 4e) w» Fig. 4a ù kù. Fig. 4 t n w w.» kt w w d z e w (Fig. 4a), ƒ kt w w ƒ (e.g., Peng and Reynolds 2005, 2006; Kim and Jung 2006, 2009a; Reynolds et al. 2009; Wu et al. 2009; Chen et al. 2009). š» ks š» w» kt» ƒ w œ ƒ j œ. kt l 6000 km ùkù (Fig. 4a) d (Peng and Reynolds 2005, 2006; Kim and Jung 2006, 2009a; Wu et al. 2009). 2010 8 9 00 UTC l 2010 8 10 00 UTC kt ƒ z w kt w w d z w (Figs. 4b, c, d). Fig. 5 8 8 12 UTC l 8 10 12 UTC ¾ 12 w ( : 80 o S~ 80 o N, : 0 o E~360 o E) w s ùkü. 8 10 12 UTC w s» d (Fig. 5e)» Áwd» (Fig. 5a)ƒ w qw q w q (Morgan 2001; Kim and Jung 2009a). w» š (Fig. 5e),»» wd ƒ, d ƒ j ùkù (Fig. 5a). w s q s, norm w,» kt w p l (singular vector) ùkù p (Peng and Reynolds 2005, 2006; Kim and Jung, 2006, 2009a; Wu et al. 2009; Chen et al. 2009). ù p l y norm w w» (Kim and w» wz» 21«2y (2011)
½ Á½x Á Á x Á 169 Fig. 4. Vertically integrated energy distribution of the adjoint sensitivity (J m 3, shaded) superposed on mean sea level pressure (contour interval of 4 hpa) at (a) 0 h (12 UTC 08 August), (b) 12 h (00 UTC 09 August), (c) 24 h (12 UTC 09 August), (d) 36 h (00 UTC 10 August), and (e) 48 h (12 UTC 10 August). The box denotes a geographic region for defining a response function at 48 h. Fig. 5. Vertical energy distribution of the adjoint sensitivity (J m 3, total energy;, kinetic energy;, potential energy; ) at (a) 0 h (12 UTC 08 August), (b) 12 h (00 UTC 09 August), (c) 24 h (12 UTC 09 August), (d) 36 h (00 UTC 10 August), and (e) forecast error (J m 3, total energy;, kinetic energy;, potential energy; ) at 48 h (12 UTC 10 August). The ordinate is height [km]. Atmosphere, Vol. 21, No. 2. (2011)
170 mw» w d Jung 2009b), mw y w wz w v ƒ. w d kt w ƒ d w» w, d s kt w w d z (Fig. 4a), «wd (Fig. 5a) ewš š, d» ƒ j, w d w». d ƒ j w kt w de y k. w w t d ù w ƒ w z» ù. d kt j w kt w w d z sw Wu et al. (2009) ƒ š. 4. KMA UM w x k 4 y (four-dimensional variational data assimilation: 4DVAR) l g, wš š ƒw mw l w wš, w kt DIANMU (201004) w w w. KMA UM l w» w g w : 4DVAR d w ( d ) v» Var_TotalPenAndGrad p PF_Forecast_Adj ForecastOnly True w. š KMA UM k 4DVAR p PF_Forecast_Adj g. Calc_energy w w xk w w š, Var_FieldOutput w k l w. w kt DIANMU w» w w. w s kt w w z ew, ƒ kt w w ƒ.» d» Áwd» ƒ w qw, q w q. kt DIANMU w s s ³ ³ w» (e.g., Morgan 2001; Peng and Reynolds 2005, 2006; Kim and Jung 2006, 2009a; Reynolds et al. 2009; Wu et al. 2009; Chen et al. 2009) w ùkù, w UM l ƒ š q w. w d ƒ e w w w š š (Cardinali 2009; Gelaro et al. 2009; Langland and Baker 2004; Liu and Kalnay 2008). d w d wš d œ ùkù d l w» d ƒ d e w w m d l x (observation system experiment: OSE) w d z. d w d w» w» w d w w. wz y w d w d KMA UM w, x wš d œ ùkù d l ƒ d ƒ e w w» t w.»»» (CATER 2011-2211)» e w. š x ½x,,,, ½,, 2007: t d w w» l, 62pp. ½x,, 2010: 2007 5 6-8 y x d,», 20(4), 399-414. Ancell, B. C., and C. F. Mass, 2008: The variability of adjoint sensitivity with respect to model physics and basic-state trajectory. Mon. Wea. Rev., 136, 4612-4628. Buizza, R., 1994: Localization of optimal perturbations using a projection operator. Quart. J. Roy. Meteor. Soc., 120, w» wz» 21«2y (2011)
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