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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
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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!!
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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!!
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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 dd 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
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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 dd dt with the total jacobian J = L + N eq with N linearized around eq
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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!
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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.
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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
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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”
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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)
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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’
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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
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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
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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
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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
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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:
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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
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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!!
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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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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 )
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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.
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