Post on 05-Feb-2016
description
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.