Reduced order ocean model using proper orthogonal decomposition


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1 INESTIGACIÓN REISTA MEXICANA DE FÍSICA 55 (3) JNIO 009 Reduced order ocean model using proper ortogonal decomposition D.A. SalasdeLeón and M.A. MonrealGómez Instituto de Ciencias del Mar y Limnología, niversidad Nacional Autónoma de México, Circuito Exterior S/N, Cd. niversitaria, D.F., México. E. vandeen and S. Weiland Electrical Engineering Department, Tecnisce niversiteit Eindoven, Den Dolec, 561 AZ Eindoven, Neterlands. D. SalasMonreal Centro de Ecología y Pesquerías, niversidad eracruzana, Calle Hidalgo No. 617, Col. Río Jamapa, CP 9490, Boca del Río, eracruz, México. Recibido el de febrero de 009; aceptado el 15 de abril de 009 Te proper ortogonal decomposition (POD) is sown to be an efficient model reduction tecnique for simulating pysical processes governed by partial differential equations. In tis paper, a POD reduced model of a barotropic ocean circulation for coastal region domains was made. Te POD basis functions and te results from tis POD model were constructed and compared wit tat of te original model. Te main findings were: 1) te variability of te barotropic circulation obtained by te original model is well captured by a low dimensional system of order of, wic is constructed using 15 snapsots and 7 leading POD basis functions; ) te RMS errors for te POD model is of order 10 4 and te correlations between te original results wit tat from te POD model of more tan 0.99; 3) te CP model time solution is reduced is five times less tan te original one; and 4) it is necessary to retain modes tat capture more tan 99% of te energy is necessary in order to construct POD models yielding a ig accuracy. Keywords: POD; reduced order model; PDE; Galerkin metods; EDP. La descomposición ortogonal propia (POD) es una técnica eficiente para la reducción de los modelos que describen procesos físicos gobernados por ecuaciones diferenciales parciales. En este trabajo, se ace una reducción de un modelo barotrópico de circulación costera del océano mediante POD. Se construyen las funciones bases y los resultados de la aplicación de la reducción mediante POD se compara con la solución original del modelo barotrópico. Los principales resultados son: 1) Se reproduce la variabilidad de la circulación barotrópica obtenida con el modelo original con un sistema de baja dimensión de orden, el cual se construye usando 15 aproximaciones y 7 funciones básicas de POD, ) El error (RMS) del modelo mediante POD es del orden de 10 4 y la correlación entre los resultados del modelo original y los obtenidos mediante la aproximación POD es más de 0.99, 3) El tiempo (CP) de obtención de la solución mediante POD es cinco veces menor que con el esquema original de solución y 4) Es necesario construir modelos POD que retengan más del 99% de la energía del sistema para que estos sean aceptables. Descriptores: POD; reducción del orden de modelos; PDE; métodos de Galerkin; EDP. PACS: 90; z; Pq; c; 9.10.Sx 1. Introduction Te proper ortogonal decomposition (POD) is an efficient way to carry out reduced order modelling by identifying te few most energetic modes in a sequence of snapsots from a timedependent system, and providing a means of obtaining a lowdimensional description of te system s dynamics. Since its original introduction by Loeve in 1945 [1] and Karunen in 1946 [], te metod as been extensively used in researc in recent years and successfully applied to a variety of fields. One of tese important applications was te application to spatially organized motions in fluid flows, suc as cylinder flows [3]. POD was also used for identification of coerent structures, signal analysis and pattern recognition [4,5]. Many researcers ave also applied te POD tecnique to optimal control problems. For instance, tis metod as been used for Burger s equation [68], te GinzburgLandau equation and te Bénard convection [9], and in oter fluid control problems [1015]. More recently POD as also been used in inverse problems [16]. In addition, te metod as also been used for industrial applications suc as supersonic jet modelling [17], termal processing of foods [15,18], and studies of te dynamic wind pressures acting on buildings [19], to name but a few. For a compreensive description of POD teory and state of te art POD researc, see Gunzburger [0]. Compared wit te above efforts, little attention was paid to application of POD to geopysical fluid dynamics suc as atmosperic or oceanic systems. In general tese dynamic systems are quite complex and teir discrete models are ard to solve due to teir large dimensions (typically ). In tis study, we make a POD reduced modelling of a barotropic ocean circulation model for coastal region domains. We construct te POD basis functions and te results from tis POD model are compared wit tat of te original barotrópic model.
2 186 D.A. SALASDELEÓN, M.A. MONREALGÓMEZ, E. ANDEEN, S. WEILAND, AND D. SALASMONREAL FIGRE 1. Geometry of te model.. Description of te barotropic ocean model Te numerical model used in tis paper is te SalasdeLeón and MonrealGómez s barotropic model [1] wit variabledept and free surface layer, wic is studying te tidal currents, wind forcing currents, and free surface dynamics in coastal regions. Te model is a nonlinear transport model, consisting of one layer above te maximum dept wit te same constant density in te layer (Fig. 1). Te equations for te deptaveraged transport are [1]: t + ( 1 ) + ˆk f = ρ p + τ s τ b ρ + υ H t + = 0 (1) p ρ = g η were Horizontal transport components of te deptaveraged currents ( = î + ĵ ; = u, = v ) t τ b τ s p H f Time Bottom friction force Wind stress force Pressure Total layer tickness Average dept Coriolis parameter (f = ω sin φ), ω angular velocity of te eart and φ latitude FIGRE. Te study region, left panel is te batymetry (depts) of te region in meters. η Free surface anomaly (η = H) ρ υ H Density of water Horizontal eddy viscosity Rev. Mex. Fís. 55 (3) (009)
3 REDCED ORDER OCEAN MODEL SING PROPER ORTHOGONAL DECOMPOSITION 187 Te bottom friction coefficients ( τ b ) are approximated by [3]: τ b = C b v (u b, v b ) ρh (3) were ρ is te water density, C B te bottom friction coefficient tat depends on te water velocities tat is of te order of 10 3 [3], v te water velocity vector, and (u b, v b ) te components of te water velocity vector. 3. Te study region FIGRE 3. Spatial and temporal scema used in te numerical approximation of te equation system (1). = î x + ĵ Te wind stress is calculated by te aerodynamic bulk formula []: (τ x, τ y ) = ρ a C W v s (u s, v s ) () were ρ a is te density of te air, C W te wind stress drag coefficient tat depends on te wind velocities [], v s te wind speed vector, and (u s, v s ) te components of te wind velocity. Te barotropic model was successfully applied to a coastal lagoon in te Mexican Caribbean in order to depict te current pattern induced by tides and winds [4]. Te coastal lagoon is located in te Sian Ka an biospere reserve in te Mexican Caribbean (Fig. ), and as a nortsout lengt of approximately 88 km, and a maximum widt of 3 km. Te maximum dept is 6 m, wit an average dept of 1.5 m. Te system as two mouts or connections wit te adjacent sea, one to te sout (Boca Grande) and te oter to te nortwest (Boca Paila). Water excanges between te open sea and te lagoon are produced at bot mouts, and are forced by tides and te wind stress Numerical sceme Te dynamical model Eq. (1) are governed by long wave dynamics suc as tides, via ocean cooscillations. In addition, te cosen model allows ig frequencies waves to be excited by te applied wind forcing [1]. System (1) was semiimplicit finite differences approximate in order to solve it numerically in a modified staggered stencil of te ArakawaC sceme [1] (Fig. 3). Te resulting approximation is: First step: n+ 1 + υ H t = n t [ ( )] n t [ x ( ) n + t ( ) τ y n s D t ρ n+ 1 = n t x [( + (f o +βy) t n+ 1 υ H t + ( )] n tg n+ 1 ) n ] n+ 1 t [ ( )] [ n tg ( η n ( ) n+ 1 x + υ H t [ η ] n (f o +βy) t n + υ ( H t ) n x ) n+ 1 ] ( ) n + t ( ) τ x n s D t ρ n n+ 1 (4) η n+ 1 = η n t ( ) n+ 1 ( ) n t x x Rev. Mex. Fís. 55 (3) (009)
4 188 D.A. SALASDELEÓN, M.A. MONREALGÓMEZ, E. ANDEEN, S. WEILAND, AND D. SALASMONREAL Second step: n+1 = n+ 1 + υ H t t [ x ( ( ) n+ 1 x + υ H t )] n+ 1 t [ ( )] n+ 1 [ tg ( ) n+ 1 + t ( ) τ x n s D t ρ ( η )] n+ 1 + (fo + βy) t n+ 1 n+ 1 n+1 = n+ 1 t were + υ H t η n+1 = η n+ 1 x [( )] n+ 1 [ ( ) n+ 1 ] [ t n+1 tg n+ 1 ( ) n+ 1 x + υ ( H t ) n+1 + t ( ) τ y n+ 1 s D t ρ t ( ) n+ 1 ( ) n+1 t x x ( ) ] n+1 η (f o + βy) t n+1 n+ 1 n+1 (5) β = f Conditions at te solid boundaries are nonormal flow and noslip conditions, and at te open boundaries te amplitude and pase of te M tidal signal and in te free surface te wind stress. Te time integration uses a leapfrog sceme. Te spatial interval for te dynamical model was cosen to be 0 m and te time step to be 745. s, wic is 1 / 60 of te M tidal period (1.4 ). Tis temporalspatial resolution will make it possible to resolve te M tidal wave caracteristic and make te model integration numerically stable. It takes about 5 tidal cycles for te model to reac a periodic constant cycle at tat time. Te model was calibrated using current velocities measured in situ wit an acoustic Doopler current profiler (ADCP). Numerically tidallydriven currents during flood and ebb tides are sown in Fig. 4. elocities reac teir igest values near te openings and along te cannels. Results of te model agree well wit observed currents (more tat 0.85 correlation) [4]. 4. Proper ortogonal decomposition 4.1. Fundamentals of te proper ortogonal decomposition For simplicity te proper ortogonal decomposition in te context of scalar fields was introduced: A complexvalued functions defined on an interval Ω of te real line. Te interval migt be te widt of te flow, or te computational domain. We restrict ourselves to te space of functions tat are square integral, or, in pysical terms, fields wit finite kinetic energy on tis interval so we need an inner product given by [5]: (6) and a norm: (f, g) = f (x)g (x) dx (7) Ω f = (f, f) 1 (8) Tat is, we find te member tat maximizes te normalized inner product wit te field v, wic is most nearly parallel in function space. Tis is a classical problem in te calculus of variations were a necessary condition is tat φ be an eigenfunction of te twopoint correlation tensor given by [5]: u (x) u (x ) φ (x ) dx =λφ (x) (9) Te integral is from 1 to infinity. Almost every member, in a measure sense, of te ensemble may be reproduced by a modal decomposition in te eigenfunctions [10]: u (x) = k a k φ k (x) (10) Equation (10) is te proper ortogonal decomposition. 4.. Approximation based on nt order truncation Spectral Decomposition is based on Fourier expansion []: (p, t) = a uj (t) ϕ uj (p); j=1 and te approximation based on nt order truncation is: Rev. Mex. Fís. 55 (3) (009)
5 REDCED ORDER OCEAN MODEL SING PROPER ORTHOGONAL DECOMPOSITION 189 of te ensemble in some sense. Suc a coordinate system is provided by te KarunenLoève expansion. Actually ere te basis functions Φ is a mixture of te snapsots so we take snapsots at appropriate points in time: T (snap u ) (p 1, t 1 ) (p 1, t ) (p 1, t f ) (p, t 1 ) (p, t ) (p, t f ) =... (p n, t 1 ) (p n, t ) (p n, t m ) (1) We denote by (p n, t m ), te set of observations (also called snapsots) of some pysical process taken at appropriate points in time at positions i= 1,, k. In tis section, we consider te discrete KarunenLoève expansion to find an optimal representation of te ensemble of snapsots. In general, eac sample of snapsots (p n, t m ) wic is defined on a set of n m node, were (p n, t m ) represent components of a vector Missing point estimation Te metod to calculate timevariant matrices faster is based on preknown spatial information in te ortogonal bases [6]. Actually ere te basis function Φ is a mixture of te snapsots. Tus, wit te POD mode computed, one must solve an m m eigenvalue problem. For a discretization of an ocean region, te dimension often exceeds, so it is often not feasible for te direct solution of tis eigenvalue problem. Te eigenvalue problem can be transformed into an m = 10 4, m m tat is an n n eigenvalue problem [9]. Te n n eigenvalue problem can be solved wit te metod of snapsots. At tis moment we must calculate error for every point. We select te k out of n points wit te greatest error as [6]: e (X 0 ) = 5. Results and discussion Φ T Φ I FIGRE 4. Classical barotropic ocean circulation numerical model results. a k (t) = N ϕ k (j) A (j, t) ; j= Snapsot creation A { (p, t), (p, t), Z (p, t)} (11) To find an optimal compressed description first we proceed to a series of expansion in terms of a set of basis functions. Intuitively, te basis functions sould represent te members In tis section, we report te results of te numerical computations related to te approaces presented in te previous paragraps. Te POD metod is applied to te above tidal and wind stress model for a coastal tropical lagoon in te Mexican Caribbean. Tis metod can provide a systematic way of creating a reduced basis space wit te state of te system at different time instances and different space locations. As in general reduced order basis metods, one can derive te states from full order numerical computations and sould be sufficiently large so tat te snapsots may contain all te salient features of te dynamics being considered. Terefore, troug a nonlinear Galerkin procedure te POD basis functions wit te original dynamics offer te possibility of acieving a ig fidelity model (albeit) wit a possible large dimension. Rev. Mex. Fís. 55 (3) (009)
6 190 D.A. SALASDELEÓN, M.A. MONREALGÓMEZ, E. ANDEEN, S. WEILAND, AND D. SALASMONREAL To acieve model reduction, we carry out a nonlinear Galerkin procedure wit te set of elements. How to coose te values of te nonlinear Galerkin transformation is a crucial question. Te associated POD eigenvalues sould define a relative information content to coose a lowdimensional basis by neglecting modes corresponding to te small eigenvalues in order to capture most of te energy of te snapsot basis. Here for our case, if te POD is constructed for 5 and a reduced order model wit 3 it yields a ratio of about 0.98; and if is constructed wit 15 it yields a ratio of above 0.99 for te percentage of kinetic energy retained (Fig. 5 and 6). We are now returning to te barotropic tidal and wind stress model for a costal lagoon in te Mexican Caribbean to apply te POD tecnique. Terefore, we solve Eqs. (4)  (6) after 5 tidal cycles of te M armonic. Results using classical model are depicted grapically in Figure 4. Te results of te model using POD are grapically almost te same and will not be sown. FIGRE 5. Ortogonal base and order evolution of te approximations. To quantify te performance of te reduced basis metod, we use two metrics namely te root mean square error (RMSE) and correlation of te difference between te full order and te reduced order simulation. Tis is obtained by first taking te five tidal cycles full order results and te corresponding five tidal cycles reduced order results and computing te error, for example, for te variable u and v components of te velocity vector ( v); te errors are sown in Figs. 6 and 7. Here, if n = 10 basis function, te first four PODs modes (Fig. 6), capture nearly 100%, wile for n = 15 basis function, te first seven POD (Fig. 6) capture nearly 100% wit an error ranging from 10 4 to Modes capture about 99% of te energy. Tus, different POD modes may be used to reconstruct fields respectively. For different numbers of snapsots but for te same energy percentage captured, te RMSE decrease stops at 15 snapsots. Te correlation taking te five tidal cycles full order results and te corresponding five tidal cycles reduced order clearly, wen increasing te POD mode, te correlation increases also for te same snapsots. Tis increase stops at 5 snapsots and te reported best approximation obtained wit 15 snapsots produced a correlation at te same level as te approximation 0 snapsots. However, one must also note tat a simple linear independence is not a sufficient criterion for coosing te POD mode. It only provides one wit some reference. Te error between te full order and te reduced order is displayed in Fig. 7 for a retained energy percentage of 99%. Tere is a little improvement between eiter 10 snapsots or 15 snapsots and 5 snapsots, but tere is almost no difference between 15 snapsots and 30 snapsots. Order approximation may be sufficiently close to te full order approximation. Oter experiments ave also been carried out, wit eiter more or fewer snapsots taken and for different percentages of energy captured, not sown ere. From te computational cost and memory storage aspects, 15 snapsots and te energy captured at 99% level yielded te best results. FIGRE 6. Computed error wit n = 15 basis functions. a) Absolute value of te currents, continuous line u, and dotted line v components of te velocity vector v, and b) absolute error of te current compared wit te classical barotropic ocean circulation numerical model results. FIGRE 7. Computed absolute error in v wit n= 15 base functions and K= 60 at time 140 for all positions in te numerical spatial grid. Rev. Mex. Fís. 55 (3) (009)
7 REDCED ORDER OCEAN MODEL SING PROPER ORTHOGONAL DECOMPOSITION Conclusions We studied problems related to POD reduced modelling of a coastal ocean circulation model in te Mexican Caribbean area. Te largescale variability of te wind stress and M tidal component is first simulated using a barotropic vertically integrated numerical model wit spatial resolution of x = y= 0 m and a time step of t= 745. s. Ten we constructed different POD models wit different coices of snapsots and different numbers of POD basis functions. Te results from tese different POD models are compared wit tose of te original model. Te main conclusions are: 1) te largescale variability of te wind stress and M tidal component obtained by te original model can be captured well by a lowdimensional system of order tat is constructed by 15 snapsots and 7 leading POD basis functions; ) by analysis of RMS errors and correlations, we found tat te modes tat capture 99% of te energy are necessary to construct POD models, 3) RMS errors for te velocity components of te POD model of order is less tan 10 4 order compared wit te original model tat is less tan 1% of (u,v) in te original model; correlations of te original model from te POD model are around 0.99; and 4) compared wit te original model, te velocity fields from te POD model are less accurate tan te free surface oscillation results (not sown because te agreement was more tan 99% between te original model and te POD). Tis remains a problem to be furter explored in fortcoming researc. Our preliminary investigations on te use of POD tidal and wind stress ocean circulation simulation yield encouraging results and sow tat POD can be a powerful tool for various applications suc as fourdimensional variational data assimilation. Tese results will be described in a followup paper. 1. M. Loeve, Compte Rend. Acad. Sci., Paris (1945) 0.. K. Karunen, Ann. Acad. Sci. Fennicae 37 (1946). 3. X. Ma, and G. Karniadaks, J. Fluid Mec. 458 (00) P. Holmes, J.L. Lumley, and G. Berkooz, Cambridge Monograps on Mecanics, (Cambridge niversity Press, 1996). 5. C. Lopez, and E. GarciaHernadez, Pysica A 38 (003) J.A. Atwell, J.T. Borggaard, and B.B. King, Int. J. Appl. Mat. Comput. Sci 11 (001) D.H. Cambers, R.J. Adrian, P. Moin, D.S. Stewart, and H.J. Sung, Pys. Fluids, 31 (1988) K. Kunisc and S. olkwein, J. Optimization Teory Appl. 10 (1989) L. Sirovic, Pysica D 37 (1989) K. Afanasiev and L. Hinze, Lect. Notes Pure Appl. Mat. 16 (001) G.M. Kepler and H.T. Tran, Optimal Control Application & Metods 1 (000) A.K. Bangia, P.F. Batco, I.G. Kevrekidis, and G.E. Karniadakis, SIAM J. Sci. Comput. 18 (1997) G. Berkooz, P. Holmes, and J. Lumley, Ann. Rev. Fluid Mec 5 (1993) H.. Ly and H.T. Tran, Quarterly of Applied Matematics 60 (00) S.S. Ravindran, SIAM Journal on Scientific Computing 3 (00) H.T. Banks, M.L. Joyner, B. Wincesky, and W.P. Winfree, Inverse Problems 16 (000) E. Caraballo, M. Saminny, J. Scott, S. Narayan, and J. Debonis, AIAA J. 41 (003) E. BalsaCanto, A. Alonso, and J. Banga, J. Food Process. Pres. 5 (00) H. Kikuci, Y. Tamura, H. eda, and K. Hibi, J. Wind. Eng. Ind. Aerod. 71 (1997) M.D. Gunzburger, Perspectives in flow control and optimization Society for Industrial and Applied Matematics, (Piladelpia, 003) p D.A. SalasdeLeón and M.A. MonrealGómez, Revista Geofísica, 58 (003) S. Pond, and G.L. Pickard, Introductory dynamical oceanograpy (Pergamon Press, Oxford, 1983) p D.A. SalasdeLeón, Modelisation de la maree M et de la circulation residuel dans le Gulf du Mexique (PD Tesis Liege niversity, Belgium, 1986), p X. CiappaCarrara, L. SanvicenteAñorve, M.A. Monreal Gómez, and D.A. SalasdeLeón, J. Plankton Res. 5 (003) F. van Belzen, S. Weiland, IEEE Trans. Signal Processing 56 (008). 6. P. Astrid, S. Weiland, K. Wilcox, A.C.P.M. Backx, IEEE Trans. Automatic Control, 53 (008). Rev. Mex. Fís. 55 (3) (009)
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