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Paul van der Vaart 1 , Henk Schuttelaars 1,2 , Daniel Calvete 3 and Henk Dijkstra 1

Challenges in the use of model reduction techniques in bifurcation analysis (with an application to wind-driven ocean gyres). 1: Institute for Marine and Atmospheric research, Utrecht University, Utrecht, The Netherlands 2: Faculty of Civil Engineering and Geosciences, TU Delft, The Netherlands

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Paul van der Vaart 1 , Henk Schuttelaars 1,2 , Daniel Calvete 3 and Henk Dijkstra 1

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  1. Challenges in the use of model reduction techniques in bifurcation analysis(with an application to wind-driven ocean gyres) 1: Institute for Marine and Atmospheric research, Utrecht University, Utrecht, The Netherlands 2: Faculty of Civil Engineering and Geosciences, TU Delft, The Netherlands 3: Department Fisica Aplicada, UPC, Barcelona, Spain Paul van der Vaart1, HenkSchuttelaars1,2, Daniel Calvete3 and Henk Dijkstra1 Multipass image of sea surface temperature field of the Gulf Stream region. Photo obtained from http://fermi.jhuapl.edu/avhrr/gallery/sst/stream.html

  2. Introduction • From observations in: • meteorology • ocean dynamics • morphodynamics • … Warm eddy, moving to the West Dynamics seems to be governed by only a few patterns Often strongly nonlinear!! Wadden Sea

  3. Research Questions: model understand predict Can we the observed dynamical behaviour? Model Approach: reduced dynamical models, deterministic! • Based on a few physically relevant patterns • physically interpretable patterns • Can be analysed with well-known mathematical techniques Choice of patterns!!

  4. Construction of reduced models Define: state vector F = (…), i.e. velocity fields, bed level,… parameter vector l = (…), i.e. friction strength, basin geometry Dynamics of F: • coupled system of nonlinear ordinary and • partial differential equations • usuallyNOT SELF-ADJOINT dF M+ L(l) F + N(l,F) = F dt Where • M : mass matrix, a linear operator. • In many problems M is singular • L : linear operator • N : nonlinear operator • F : forcing vector

  5. Step 1: identify a steady state solution Feq for a certain l. L(l) Feq + N(l,Feq) = F Step 2: investigate the linear stability of Feq. Write F = Feq + f and linearize the eqn’s: df M+ J(l) f = 0 dt withthe total jacobian J = L (l) + N(l,f,Feq) with N linearized around Feq

  6. This generalized eigenvalue-problem (usually solved numerically) gives: • Eigenvectors rk • Adjoint eigenvectors lk These sets of eigenfunctions satisfy: • < J rk, lk > = sk • < M rk , lm > = dkm with <.>: inner product sk : eigenvalue Note: if Mis singular, the eigenfunctions do not span the complete function space!

  7. Step 3: model reduction by Galerkin projection on eigenfunctions. • Expand f in a FINITE number of eigenfunctions: N f = Srj aj(t) j=1 • Insert F = Feq + f in the equations. • Project on the adjoint eigenfunctions evolution equations • for the amplitudes aj(t): N N N aj,t - Sbjk ak + S S cjkl ak al = 0, for j = 1...N k=1 k=1 l=1 system of nonlinear PDE’s reduced to a system of coupled ODE’s.

  8. Open questions w.r.t. the method of model reduction: • Which eigenfunctions should be used? • How many eigenfunctions should be used in the expansion? • How ‘good’ is the reduced model? To focus on these research questions, the problem must satisfy the following conditions: • not self-adjoint • validation of reduced model results with • full model results must be possible • no nonlinear algebraic equations

  9. Example: ocean gyres Gulf stream: resulting from two gyres Subpolar Gyre • Not steady: • Temporal variability on many timescales • Results in low frequency signals in the climate system Subtropical Gyre “Western Intensification”

  10. Temporal behaviour of gulf stream from observations from state-of-the-art models Two distinct energy states (low frequency signal) Oscillation with 9-month timescale (After Schmeits, 2001)

  11. One layer QG model Step ‘0’ • Geometry: square basin of 1000 by 1000 km. • Forcing: symmetric, time-independent wind stress • Equations: + appropriate b.c. • Critical parameter is the Reynolds number R: • High friction (low R): stationary • Low friction (high R): chaotic Route to chaos

  12. Step 1 Bifurcation diagram resulting from full model (with 104 degrees of freedom): • R<82: steady state • R=82: Hopf bifurcation • R=105: Naimark-Sacker • bifurcation Steady state: pattern of stream function near R = 82 (steady sol’n)

  13. Step 2 At R=82 this steady state becomes unstable. A linear stability analysis results in the following spectrum: QUESTION: which modes to select? • Most unstable ones • Most unstable ones + • steady modes • Use full model results and • projections

  14. Step 3 Example: take the first 20 eigenfunctions to construct reduced model. Time series from amplitudes of eigenfunctions in reduced model Black: Rossby basin mode (1st Hopf) Red + Orange: Gyre modes (Naimark-Sacker) Blue: Mode number 19 • Quasi-periodic behaviour at R =120: Neimark-Sacker bifurcation • Good correspondence with full model results

  15. Another selection ofeigenfunctions to construct reduced model. • Mode 19 essential • Choice only possible • with information of • full model Rectification in full model Mode #19

  16. Conlusions w.r.t. reduced models of one layer QG-model: • More modes do not necessarily improve the results: • Mode # 19 is essential: this mode is necessary to stabilize. • physical mechanism! • Modes can be compensated by clusters of modes deep in • the spectrum (both physical and numerical modes) • By non-selfadjointness, these modes do get finite • amplitudes Low frequency behaviour:

  17. Two layer QG model Instead of one layer, a second, active layer is introduced allows for an extra instability by vertical shear (baroclinic) • Bifurcation diagram from • full model: again a Hopf • and N-S bifurcation. • In reduced model (after • arbitrary # of modes), • a N-S bif. is observed: N-S Reduced model • Different R • Different frequency

  18. Linear spectrum looks like the spectrum from 1 layer QG model. • Use basis of eigenfunctions calculated at R=17.9 (1st Hopf bif) • and increase the number of e.f. for projection: || ffull – fproj|| || ffull|| • E = • Some modes are • active (clusters). • Which modes • depends on R • Note weakly • nonlinear beha- • viour!! E =

  19. Conclusions: • Possible to construct ‘correct’ reduced model • Insight in underlying physics • Full model results selection of eigenfunctions Challenge: To construct a reduced model without a priori knowledge of the underlying system’s behaviour in a systematic way Apart from the problems mentioned above (mode selection, ..), this method should work for coupled systems of nonlinear ‘algebraic’ equations and PDE’s as well.

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