Challenges in the use of model reduction techniques in bifurcation analysis

(with an application to wind-driven ocean gyres)

Paul van der Vaart1, Henk Schuttelaars1,2, Daniel Calvete3 and Henk Dijkstra1

1: Institute for Marine and Atmospheric research, Utrecht University, Utrecht, The Netherlands2: Faculty of Civil Engineering and Geosciences, TU Delft, The Netherlands3: 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

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!!

Research Questions:

modelunderstandpredict

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!!

Construction of reduced models

Define: state vector = (…), i.e. velocity fields, bed level,… parameter vector = (…), i.e. friction strength, basin geometry

Dynamics of :

M LNFddt

•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

Step 1: identify a steady state solution eq for a certain .

LeqNeqF

Step 2: investigate the linear stability of eq.

Writeeqand linearize the eqn’s:

M J0ddt

with the total jacobian J = L + N eq with N linearized around eq

This generalized eigenvalue-problem (usually solved numerically) gives: •Eigenvectors rk

•Adjoint eigenvectors lk

These sets of eigenfunctions satisfy:

•< J rk, lk > = k

•< M rk , lm > = km <.>: inner product k : eigenvalue

with

Note: if M is singular, the eigenfunctions do not span the complete function space!

Step 3: model reduction by Galerkin projection on eigenfunctions.

•Expand in a FINITE number of eigenfunctions:

= rj aj(t)j=1

N

•Insert eqin the equations.•Project on the adjoint eigenfunctions evolution equations for the amplitudes aj(t):

aj,t - jk ak + cjkl ak al = 0, for j = 1...N l=1

N

k=1 k=1

N N

system of nonlinear PDE’s reduced to a system of coupled ODE’s.

•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

Example: ocean gyres

Gulf stream: resulting from two gyres Subpolar Gyre

Subtropical GyreNot steady: •Temporal variability on many timescales•Results in low frequency signals in the climate system

“Western Intensification”

Temporal behaviour of gulf streamfrom observations from state-of-the-art models

Oscillation with 9-month timescale

Two distinct energy states(low frequency signal)

(After Schmeits, 2001)

• 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’

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)

Step 1

At R=82 this steady state becomes unstable. A linear stability analysisresults 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

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

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

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:

Two layer QG model

Instead of one layer, a second, active layer is introducedallows 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

•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:

•E = || full – proj||

|| full||E

=

•Some modes are active (clusters).•Which modes depends on R •Note weakly nonlinear beha- viour!!

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 knowledgeof 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.