# Reduced order ocean model using proper orthogonal decomposition

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2 186 D.A. SALAS-DE-LEÓN, M.A. MONREAL-GÓMEZ, E. AN-DE-EN, S. WEILAND, AND D. SALAS-MONREAL FIGRE 1. Geometry of te model.. Description of te barotropic ocean model Te numerical model used in tis paper is te Salas-de-León and Monreal-Gómez 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 non-linear 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 dept-averaged 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 nort-sout 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 co-oscillations. In addition, te cosen model allows ig frequencies waves to be excited by te applied wind forcing [1]. System (1) was semi-implicit finite differences approximate in order to solve it numerically in a modified staggered stencil of te Arakawa-C 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. SALAS-DE-LEÓN, M.A. MONREAL-GÓMEZ, E. AN-DE-EN, S. WEILAND, AND D. SALAS-MONREAL 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 no-normal flow and no-slip 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 temporal-spatial 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 tidally-driven 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 complex-valued 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 two-point 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 n-t order truncation Spectral Decomposition is based on Fourier expansion []: (p, t) = a uj (t) ϕ uj (p); j=1 and te approximation based on n-t 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 Karunen-Loève 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 Karunen-Loève 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 time-variant matrices faster is based on pre-known 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. SALAS-DE-LEÓN, M.A. MONREAL-GÓMEZ, E. AN-DE-EN, S. WEILAND, AND D. SALAS-MONREAL 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 low-dimensional 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 large-scale 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 large-scale variability of te wind stress and M tidal component obtained by te original model can be captured well by a low-dimensional 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 follow-up 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. Garcia-Hernadez, 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. Balsa-Canto, 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. Salas-de-León and M.A. Monreal-Gómez, Revista Geofísica, 58 (003) S. Pond, and G.L. Pickard, Introductory dynamical oceanograpy (Pergamon Press, Oxford, 1983) p D.A. Salas-de-León, Modelisation de la maree M et de la circulation residuel dans le Gulf du Mexique (PD Tesis Liege niversity, Belgium, 1986), p X. Ciappa-Carrara, L. Sanvicente-Añorve, M.A. Monreal- Gómez, and D.A. Salas-de-León, 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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