Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 1(6), 16-21, June (2012) Res.J.Recent Sci. International Science Congress Association 16 Convergence of Numerical Solutions of the Data Assimilation Problem for the Atmospheric General Circulation ModelIpatova V.M. Moscow Institute of Physics and Technology, Dolgoprudny 141700, RUSSIA Available online at: www.isca.in (Received 22nd March 2012, revised 24th March 2012, accepted 30th March 2012)Abstract We consider a two-layer quasigeostrophic model of the general atmospheric circulation. It is assumed that there are field measurements of air velocity. These observations are used to find the unknown initial state of the model. The discrepancy between the observed values and the model results is measured by a cost function value. We prove the solvability of the optimization problem for positive values of the regularization parameter. The system of equations is approximated by an explicit spectral-difference scheme. A theorem is proved that the numerical solutions of the data assimilation problem converge to its exact solutions. Keywords: Optimization problem, model of the atmospheric dynamics, spectral-difference scheme. IntroductionA rigorous mathematical analysis and justification of the variational data assimilation procedure includes the study of such issues as the existence of solutions to the optimization problem and the convergence of numerical solutions to exact solutions. In this paper we study these issues in relation to the two-layer baroclinic quasi-geostrophic atmospheric general circulation model. The main variables of the model are barotropic and baroclinic components of the stream function, but the stream function is not one of variables for which in meteorology are carried out the field observations. For this reason, it is assumed that the measurements of the velocity of air are known. The initial state is chosen as the model parameter to be determined because the initialization problem is one of the best known and most commonly solved in practice. Note that the convergence of numerical solutions of the data assimilation problem earlier has been studied for the quasi-geostrophic models in a rectangular region under the assumptions that the equations are approximated by the implicitor semi-explicit finite-difference schemes and the observations on the ocean surface elevation are given . Material and MethodsThe atmospheric general circulation model: Let S be a two-dimensional sphere of radius R , [0,2) qp be the longitude, [ ] /2;/2 jppÎ- be the latitude, W be the the angular velocity of the Earth rotation. By =2sin l j we denote the Coriolis parameter, 22211=coscoscosRR jj jqj  ¶¶¶ D+  ¶¶ ¶  is the Laplace-Beltrami operator, (,)=cos uvvu Juv qjqj j  ¶¶¶¶ -  ¶¶¶¶  is the Jacobian. The atmosphere is divided vertical into two layers, the first layer correspond to the pressure from 0 to 500 mb and the second layer correspond to the pressure from 500 to 1000 mb, 11 =(,,) t yyqj , 22 =(,,) t yyqj are the stream functions within the first and the second layers, 112 =()/2 yy, 221 =()/2 yy are barotropic and baroclinic components of the stream function, 12 =(,) xxx . We consider the atmospheric general circulation model 3 : ()()() 11221121 ,,=, x JxxlJxxxxxf ms¶D+D++DD-D++ (1) ()()()()2112 2121212122 (),,= =,, JxxlJxx xxxJxxxxf a msams ¶D- +D++DD-D++-D++ (2) 0 =0 =. t xx (3) Here 11 ,,,, smsma are positive constants and 12 , ff are given functions. Introduce the real Hilbert space { } 22 =(,)(),=0 LuLSudSqjwith the scalar product (,)= uvuvdS  and the norm 1/2 =(,) uuu  . Associate the operator () -D with the scale of Hilbert spaces =() pp HHS , Î  , by assuming { } 0/2=(,),=()pHuLuuqj Î-D+¥ . For vector functions 12 =(,) xxx we introduce the spaces =()= pp pp VVSHH with Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(6), 16-21, June (2012) Res. J. Recent Sci. International Science Congress Association 17 the norm ( ) 1/2 2212pppxxx, where 00 022 = VLL ´ , xx º  . Let 0 +¥ and =(0,) GST . By (,) G ×× and G ×  we denote the scalar product and the norm in the space () LG respectively. Introduce the real spaces of the functions determined in G : ( ) ( ) ( ) 2322121 =0,;,=0,;,=0,;, XLTVYLTVYLTV () 211 =0,;,=,. x ZLTVWxXY t-  ÎÎ \n ¶ \r We also introduce the bilinear forms (,) auv , (,) auv , (,) buv and the trilinear form (,,) buvw by setting ( ) 1112222 (,)=, S auvuvuvuvdS ÑÑ+ÑÑ+211221221212122(,)={()()()}, auvuvuvuv uuvvuvdSmmss DD+DD+ÑÑ+ +Ñ+Ñ++ [ ] 11122 (,)=(,)(,), S buvJulvJulvdS 2111221122212221(,,)=(,)(,)(,)(,)(,). buvwJwvuJwvuJwvu JwvuJwvudS D+D+D+ +D+The system (1)-(2) can be written as ()()()() 12122 ,,=,,,, ayaxybxybxxyfyyV  ++-"Î where 12 =(,) fff . Further the letter c denotes various positive constants. We will need the following statements. Lemma 1.The inequalities hold 3,4 22222 |(,,)|,,, buvwcuvwuvwV £"Î  1/41/21/22 ()12 2, LS uuuuH Ñ£"Î  2 ()()244|((,),)((,),)| ,,. LSLSJwuvJwvu cuvwuvwH D+D£ £ÑÑ"ÎLemma 2:Let 0 B , 1 B , 2 B be three Banach spaces where 012 BBB ÌÌ, 0 B and 2 B are reflexive, 0 B is compactly embedded into 1 B , and 1 B is continuously embedded into 2 B ; let 0201 =(0,;),(0,;), ppWxLTBLTB  ÎÎ \n ¶ \r where T is finite and 1 kp ¥ , =0,1 . Then the embedding of W into 1 0 (0,;) p LTB is compact 5 Theorem 1 3 : For all 02 xV Î and FZ Î the problem (1)-(3) has the unique solution xW Î , and the estimates hold ( ) 101()max YZ tTxtxcxf££+£+  , ( ) 202()max XZ tTxtxcxf££+£+  , 1 1/Y xtc ¶¶£ , where 1 c is a constant depending on 02 x  and Z f  Write the problem (1)-(3) down briefly as ( ) 0 ()=; xfx . It follows from Theorem 1 that there exists a bounded inverse operator ZVW F´® defined on all 2 ZV ´ . Lemma 3:Let fZ Î , ( ) 1 =; nn xfy , ( ) 1 0 =; xfx and 0 n yx weakly in 2 V . Then xx ® weakly in W Proof. The sequence { } n y is bounded in 2 V . On Theorem 1 the sequence { } n x is bounded in W . Choose a convergent subsequence in it: xz ® weakly in W . By Lemma 2, then xz ® strongly in Y . The space [ ] ( ) 2 0,; CTV is continuously embedded 6 into W , so (,,0) yzqj weakly in 2 V . Using the estimates of Lemma 1, we see that ( ) ( ) 22 ,,,, nn bxxybzzy weakly in ( ) 2 0, LT for all 2 yV Î . Taking the limit as n ®¥ , we find that z is a solution of (1)-(3), i.e. = zx . The data assimilation problem: Let us assume that we know observation data for the velocity vector of air in the first and the second layers given by functions 0 k u , 0 k v , =1,2 on some measurable subset GG Ì . Denote by c the characteristic function of 0 G and extend 0 k u , 0 k v onto 0 \ GG by zero. We associate to each solution of (1)-(3) the functions 112 ()= xxx y - , 212 ()= xxx y + , () ()= x ux j , () ()=cos x vx jq , =1,2 . Define the following cost functional on W : () 22001112112200322422=()()()()GGGGIxmuxumvxvmuxumvxvcccc -+-+ +-+-where 1 m , 2 m , 3 m , 4 m are non-negative weight coefficients. Assume that the external forcing f in the model is known and the observation data should be used for determination of the initial state 0 x . Define on 2 V the functional () () ( ) 0000=; JxxxIfx -+F (4) where 0 l ³ is a regularization parameter, 02 a xV Î is a priori known approximate value of 0 x . Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(6), 16-21, June (2012) Res. J. Recent Sci. International Science Congress Association 18 Consider the following data assimilation problem: given an external action fZ Î , determine 02 xV Î so that ( ) ( ) { } 02 =inf|. JxJyyV ll (5) Sufficient conditions for its solvability gives the theorem. Theorem 2:If �0 l and functions 0 k u , 0 k v , =1,2 belong to ( ) 2 LG , then problem (5) has a solution.Proof: Denote ( ) { } 2 =inf| mJyyV and consider a sequence { } n y minimizing the functional J l , that is ( ) lim= nn Jym ®¥. If �0 l then { } n y is bounded in 2 V . Choose a subsequence 0 n yx weakly convergent in 2 V . Define ( ) 1 =; nn zfy and ( ) 1 0 =; xfx . By Lemma 3 we have a convergence zx ® weakly in W . It follows from Lemma 2 that zx ® strongly in Y . Then ()() knk uzux and ()() knk vzvx , =1,2 , strongly in ( ) 2 LG . Thus, ( ) ( ) lim= IzIx ®¥. By the property of weak lower semicontinuity of norms we have 000 22 limsupaa xxyx ®¥-£-, therefore ( ) 0 Jxm l £ . Taking into account the definition of m , we conclude that 0 x is a solution of (5). Since ( ) 0 = Jxm , then 202 n yx ®  so 0 n yx strongly in 2 V . The approximate data assimilation problem: Now consider a discrete method for determination of approximate solutions to the problem (5). Let n  be the eigensubspace of the Laplace-Beltrami operator corresponding to the eigenvalue =(1) nn L+ and spanned onto spherical harmonics (,) mn qj , || mn £ . Denote =1NN nn È  , NNN X´  and also denote the operator of the orthogonal projection onto N  by N P . Let =/ TK be the grid time step, tk t , =0, kK , k x is the approximate solution in the layer = k tt . Further we assume that for varying t and N the inequality 2 =(1)() NN ttmnm - L+£- (6) holds with some constant (0,) nm . Approximate the problem (1)-(3) by the explicit spectral-difference scheme: 111221211221121212212122/(,)(,)()=,()/(,)(,) (,)()=, ,=0,,=,kkkkkNNkkkkNkkkkkNN kkkkkkkkN kNNDPJxxlPJxxxxxqDPJxxlPJxxPJxxxxxxxqxkKxsmatasmmsD+D++D+D+-DÎD-+D++D-+D+-D+D-ÎÎXÎX  (7) where kkk jjj Dxx + - , =1,2 . Write down system (7) in a brief form ()=(;) Fxq r , where the operator F depends on t and N , but for the sake of brevity, we omit this dependence. Equations (7) form a system linear with respect to 1 k x + with a nondegenerate matrix. Therefore, the operator F is uniquely invertible on the whole ( ) K NN X´X . In order to extend the grid function =0 ={} kKxx onto the whole time segment [0,] T , we associate it with the function of a continuous argument 11 1 ()(,,)=(,)(,)for[,]. kkkkkkttttAxtxxtttqjqjqjtt--+ÎWe define on N X the cost functional similar to the functional (4) by setting () () ( ) ( ) =;SxIAFq rlrr -+, where the external influence ( ) K ÎX is considered to be known and fixed. Consider the following discrete data assimilation problem: given an external action ( ) K ÎX, determine the initial function N r ÎX so that ( ) { } =inf(). SSyyllÎX (8) Note that this problem is the approximate finite-dimensional analogue of the optimization problem (5). For the time-dependent functions we define the projection operator on the grid h P by the formula 11 =() hN PfPftdt . Introduce the norms 1/2 =1xx    , 1/2 =0qq     1 []=maxkK xx ££  , =[] WhX hh xxx +  . We will need the following statement. Theorem 3 7 : Let X and Y be Banach spaces, XY ®  be a Frechet-differentiable operator and:) (0)=0  ;2) the Lipschitz inequality is valid for its derivative 121212 ()(),(0) XYXyyLyyyyB¢¢-£-"Î  where (0)={} rX ByXyr Σ  , =()�0 LLr is a constant depending on r ; 3) the operator (0) ¢  is closed and has the continuous inverse operator 1 ((0)) - ¢  determined on the whole Y . Then for any qY Î such that /() Y qML g£  , where 01 g , =((0)) YX , there exists a unique element x being the solution to the equation ()= xq  and satisfying the estimate /() X xML g£  . Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(6), 16-21, June (2012) Res. J. Recent Sci. International Science Congress Association 19 Denote by () Fy ¢ the derivative of F and consider the equation ()=(;) Fyxq r ¢ (9) which is a system 1122 11221211 (,)(,) (,)(,)()=, kkkkNN kkkkkkkk NNPJxylPJxy PJyxPJyxxxxq sm+D++D++D+D+D+-D (10) 2112211212 1212212122 ()(,)(,)(,)(,)(,) (,)()=, kkkkNNkkkkkkNNN kkkkkkkk PJxylPJxyPJyxPJyxPJyx PJxyxxxxxq asmmsD-+D++D++D+D---+D+-D+D- (11) 0 ,=0,,=. kNN xkKx ÎXÎX Theorem 4: If (6) is valid, then the solution of equation (9) satisfies ( ) 1/2 2221WZhhxcq£+, where �0 depends on W h y  only. Proof: By taking the inner product of (10), (11) by 1 k x t + in 00 22 LL ´ , we have 12212212 11222 221121122112212112211212112222 =(,)((,) 22(,)(,)(,),)((,)(,)(,)(,)(,kkkkkkkkkkkkkkkkkkkkkkkkkkkxDxDxxxqxJxylJxyJyxJyxxJxylJxyJyxJyxJyxaatmtt++++++++-+D++D+D+D+D++D+D+D-  1222111121211122122)(,),)(,)((),)(,)(,).kkkkkkkkkkkkkkJxyxDxxxxxxxxxtmtstmts++++++D+D+++D-Using (6), Lemma 1 and Young inequality, we estimate the following quantities: 21122212 1212 21212|(,)|(), 4 kkkkkkNkkDxDxDxDxtmtmtmtmn+++D£L£+L£+-   ( ) () 11 22221122()2() 4411/211/21/41/2122122212()12224221212()(,)(,),664kkkkkkkk LSLS kkkkkNLSkkkkLSJxyJyxxcxxycxxxDyxDcxytttn++++D+D£ÑÑ£+LÑ£++ÑBy applying the similar arguments we obtain the inequality 121212122 22222 121212 (1(1))kkkkkkkkxxxcyyxxcqatntat+++ - ++££++++   which implies the estimate ( ) ( ) 2422expWWZ hhh xcyq£+  . Lemma 4: For F ¢ the Lipschitz inequality is valid ()() WZVW hhh FyFzLyz ®´ ¢¢ -£-  where L is a positive constant not depending on t and N . Proof: Set = syz - . For every 1 () NK + ÎX we have the equality ()()=(;0) FyxFzx x ¢¢ where 111221122 =((,)(,)(,)(,)), kkkkkkkkkPJxsJxsJsxJsxD+D+D+D 221122112 1212 =((,)(,)(,)(,) (,)(,)). kkkkkkkkk kkkk PJxsJxsJsxJsx JsxJxsaaD+D+D+D--Let () NK ÎX. Using the estimates such as ( ) 11111121()1() 441/21/21/21/21211121112(,)(,),kkkkkkkk LSLS kkkkkJxsJsxrcrsxcrssxxD+D£ÑÑwe get the inequality 1/211 21/21/21/21/2 =0=0 (,)[][]. KKkkkhXhX hh kkrLrssxxtxt--  Now verify the following assertions of the stability and the convergence of scheme (7). Theorem 5: If (6) is valid and x is the solution to the equation ()=(;) Fxq r and y is the solution to the equation ()=(;) Fyqdqd rr ++ then for any �0 e there exists �0 d depending on e and W h x  only and such that xy e -£  for ( ) 1/222ddq rd +£ Proof: Denote = zyx - and consider the operator ()=()() zFxzFx +-  acting from h W into 1 h ZV ´ . By Lemma 4 the derivative ()=() zFxz ¢¢ +  satisfies the Lipschitz inequality and virtue of Theorem 4 the norm of the inverse operator 1 ((0)) - ¢  satisfies the estimate 2 ((0))ZVWhh c ´® £   . Thus,  satisfies all the conditions of Theorem 3. Since the solution to (7) is unique, then for completing the proof it is sufficient to assume =(1)/() cL dgg, where =min{,1/2} cLge. Theorem 6: Let (6) be valid, 02 xV Î , fZ Î , a function xW Î be the solution to problem (1)-(3), =() k Nk wPxt , =0, kK , a grid function y be the solution to the equation ( ) 0 ()=;hN FyPfPx . Then we have the convergence 0 yw -®  for 0 t ® , ®¥ . If in addition 0 j u , 0 j v , =1,2 , belong to () LG , then (())() IAyIx as 0 t ® , ®¥ . Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(6), 16-21, June (2012) Res. J. Recent Sci. International Science Congress Association 20 Proof: Applying the operator h P to both sides of (1) and (2), we obtain the equation 0 ()=(;) hN FwPfdqPx . Denote zPx , =t z z t ¶ ¶ , =t x x t ¶ ¶ , 2 0 []=max() tT xxt ££  and estimate the typical terms in the residual dq : 122211 =(())=(), tkkkkttk dztwdtttzdt tt+D-D-D 1/2 111212111 =()(), tttkNkkktktttttkkkdttzdtttzdtxdttt+++-++  -£-£   so 1 ZtY dcx  . Further we estimate () =(,)(,)=, tk kkkkk PJwwJxxdt bh D-D+ () ()111 =(,)(,)=(,)(,), tkNtk PJzzJxxdtPJzxxJzzxdt tt+D-D-D+D-D 1/2 12222222 , tktkccxxzdtxxzdt+  £-£-    1 [][]. ZNYX cxPxxxx £-£For the second term included in k d we have () 1 11(,) =(,)(,)=(), tkkkkkNtkJzz PJwwJzzdtttPdt tt+¶DD-D- 12122121()()tNkktkttk ttzzdtttxzdt tt+-++£-£- 1/22221 ,[]. NtZtY cxxdtcxx tht£L£Applying the similar arguments we see that 0 dq ®  for 0 t ® , ®¥ . It is not difficult to see that WW h wcx £  . By Theorem 5 we have the convergence 0 yw -®  for 0 t ® , ®¥ . Now we estimate 1111 ()()()() YYYY AyxAyAwAwzzx -£-+-+-  , where ()() YW h AyAwcyw -£-  , 11 ()=(()())(()())[,], kkkkkktttt Awzztztztztforttt tt+-----+-Î 111221() (), ttttkkkttttttkkkktt Awzdtczdtdtczdt +++-££ 111 (),. YtYYNX Awzcxzxxtt-£-£L  Since () Ayx ® strongly in 1 Y , then (())() IAyIx as 0 t ® , ®¥ . Results and DiscussionThe main result of the paper is the following theorem on the convergence of numerical solutions to the data assimilation problem. Theorem 7: Let the data 0 j u , 0 j v , =1,2 , belong to () LG and the sequence of functions n r is such that: 1) n r is the solution to data assimilation problem (8) with qPf , n NN , grid time step = n tt , and the regularization parameter =0 ll ³ ; 2) 0 nt ® , ®¥ , �0 llfor n ®¥ , and (6) holds. Then n r contains a subsequence converging strongly in 2 V to the solution of problem (5) with the same data and 0 = ll . Proof: Denote ( ) { } 2 =inf| mJyyV ll, ( ) { } =inf| N sSyyll ÎX . We show that for any 0 l ³ the following inequality holds 0 0,, . limsup sm ll tll®®¥® (12) Indeed, by the definition of the infimum, for any �0 e there exists a vector function 2 yV Î such that 00 ()/2 Jymll e £+ . Due to Theorem 6, there exist �0 , Î  , and �0 such that 00()()/2SPyJymlll ee £+£+ for all 0 tt £ , 0 NN , and || d ll -£ . Then ()sSPymlll e ££+ , which gives (12). For �0 the sequence n r is bounded in 2 V . Select from it a subsequence 0 n x converging weakly in 2 V and strongly in 1 V . Denote 11100 =(;),=(;),=(;). nhNnhnxfxyFPfPxzFPf r --- By Theorem 6 we have (())()forIAyIxn ®®¥ (13) and |||||||| nWW h ycx £ for all sufficiently large n . Since 01 ||||0 NnPx r -® , then Theorem 5 implies the convergence ||||0 nnWyz -® as n ®¥ . Consequently, ()()0 nnAyAz -® strongly in 21 (0,;) LTV , then (())(())0 nnIAyIAz -® . Taking into account (13), we see that 000 ()()for. SJxn ®®¥ (14) A weak convergence of n r to 0 x in 2 V implies that 02002 |||||||| liminfaa xxx ®¥-³-. Taking into account (14) and the convergence of 0 n ll , we get Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(6), 16-21, June (2012) Res. J. Recent Sci. International Science Congress Association 21 0 0 ()(). liminf SJx ll®¥ (15) But n r are the solutions of (8), so ()= nn Ss ll r . From (12) we have 0 00 ()() limsup SmJx lll®¥££. Comparing (15) with the last inequality we see that 00 ()=()=, lim SJxm lll®¥ (16) that is 0 x is the solution of data assimilation problem (5). In addition, from (14) and (16) we find that 22 020002 limaann xxx lrl®¥--  , then 22 02002 limaaxxx®¥--  , so 0 n x strongly in 2 V Notice that the arguments of Theorem 7 imply that if =0 and the sequence n r is bounded in 2 V , then n r contains a subsequence weakly converging in 2 V to the solution to problem (5) with the same data and =0 l . ConclusionIn this paper we have considered a method of approximate solution of the data assimilation problem for the two-layer quasigeostrophic atmospheric general circulation model and have proved the convergence of numerical solutions to the exact solutions of the optimization problem. One can hope that in future data assimilation techniques will find application in various branches of science 810 - . AcknowledgementThe work was supported by the Federal Target Program "Scientific and scientific-educational stuff of innovative Russia" for years 2009-2013. References1.Agoshkov V.I. and Ipatova V.M., Convergence of solutions to the problem of data assimilation for a multilayer quasigeostrophic model of ocean dynamics,Russ. J. Numer. Anal. Math. Modelling, 25(2), 105-115 (2010)2.Ipatova V.M., Convergence of the numerical solution of the variational data mastery problem for altimetry data in the quasigeostrophic model of ocean circulation, Diff. 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