1. Challenges in the use of model reduction techniques in bifurcation analysis (with an application to wind-driven ocean gyres) Paul van der Vaart 1, Henk Schuttelaars 1,2, Daniel Calvete 3 and Henk Dijkstra 1 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 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 Wadden Sea Dynamics seems to be governed by only a few patterns Often strongly nonlinear!!

3. Research Questions: model understand predict Can wethe 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  = (…), i.e. velocity fields, bed level,… parameter vector = (…), i.e. friction strength, basin geometry Dynamics of  : M   L  N  F dd dt M : mass matrix, a linear operator. In many problems M is singular L : linear operator N : nonlinear operator F : forcing vector Where coupled system of nonlinear ordinary and partial differential equations usually NOT SELF-ADJOINT

5. Step 1: identify a steady state solution  eq for a certain. L  eq  N  eq  F Step 2: investigate the linear stability of  eq. Write  eq  and linearize the eqn’s: M   J  0 dd dt with the total jacobian J = L  + N  eq  with N linearized around  eq

6. This generalized eigenvalue-problem (usually solved numerically) gives: Eigenvectors r k Adjoint eigenvectors l k These sets of eigenfunctions satisfy: =  k =  km : inner product  k : eigenvalue with Note: if M is singular, the eigenfunctions do not span the complete function space!

7. Step 3: model reduction by Galerkin projection on eigenfunctions. Expand  in a FINITE number of eigenfunctions:  =  r j a j (t) j=1 N Insert  eq  in the equations. Project on the adjoint eigenfunctions evolution equations for the amplitudes a j (t): a j,t -   jk a k +  c jkl a k a l = 0, for j = 1...N l=1 N k=1 N N system of nonlinear PDE’s reduced to a system of coupled ODE’s.

8. Which eigenfunctions should be used? How many eigenfunctions should be used in the expansion? How ‘good’ is the reduced model? Open questions w.r.t. the method of model reduction: 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 Subtropical Gyre Not steady: Temporal variability on many timescales Results in low frequency signals in the climate system “Western Intensification”

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

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

12. Bifurcation diagram resulting from full model (with 10 4 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) Step 1

13. 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 Step 2

14. 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 Step 3

15. Another selection of eigenfunctions 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 (1 st Hopf bif) and increase the number of e.f. for projection: E = ||  full –  proj || ||  full || E = Some modes are active (clusters). Which modes depends on R Note weakly nonlinear beha- viour!!

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.

20. Step 3, Case B: model reduction by Galerkin projection on components of eigenfunctions. This is necessary if M is singular some equations do not depend on time explicitly Expand components of  in the components of a FINITE number of eigenfunctions.   =  r 1j u j (t) j=1 N   =  r 2j h j (t) j=1 N Insert these expansions  in the equations. Example:  = (  1,  2 )

21. Project the equations on the components of the adjoint eigenfunctions algebraic equations + evolution equations -   jk u k +   jkl u k h l = 0 l=1 N k=1 N N h j,t -   jk h k +   jkl u k u l = 0 l=1 N k=1 N N for j = 1...N 1 st eqn: algebraic, nonlinear dependence on amplitudes h k 2 nd eqn: ODE, describing the temporal behaviour of h k system of nonlinear PDE’s reduced to a system of coupled algebraic equations and ODE’s.