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 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
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
Research Questions:
Can we
model
understand
predict
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 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
•usually NOT SELF-ADJOINT
dF
M dt + L(l) F + N(l,F) = F
Where •M : mass matrix, a linear operator.
In many problems M is singular
•L : linear operator
•N : nonlinear operator
• F : forcing vector
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 dt + J(l) f = 0
with the total jacobian J = L (l) + N (l,f,Feq)
with N linearized around Feq
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 M is singular, the eigenfunctions do not span the complete
function space!
Step 3: model reduction by Galerkin projection on eigenfunctions.
•Expand f in a FINITE number of eigenfunctions:
N
f = S rj aj(t)
j=1
•Insert F = Feq + f in the equations.
•Project on the adjoint eigenfunctions
for the amplitudes aj(t):
N
N
evolution equations
N
aj,t - k=1
Sbjk ak + S
S c a a = 0,
k=1 l=1 jkl k l
system of nonlinear PDE’s reduced to a
system of coupled ODE’s.
for j = 1...N
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
Example: ocean gyres
Gulf stream: resulting
from two gyres
Not steady:
•Temporal variability on
many timescales
•Results in low frequency
signals in the climate system
Subpolar Gyre
Subtropical Gyre
“Western Intensification”
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)
One layer QG model
• Geometry: square basin of 1000 by 1000 km.
• Forcing: symmetric, time-independent wind stress
Step ‘0’
• Equations:
+ appropriate b.c.
• Critical parameter is the Reynolds number R:
•High friction (low R): stationary
Route to chaos
•Low friction (high R): chaotic
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)
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
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
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:
•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
•Mode # 19 is essential: this mode is necessary to stabilize.
physical mechanism!
Low frequency
behaviour:
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:
•Different R
•Different frequency
N-S Reduced model
•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:
•Some modes are
active (clusters).
•Which modes
depends on R
•Note weakly
nonlinear behaviour!!
E=
|| ffull – fproj||
•E =
|| ffull||
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.