Unstable periodic orbits in models of the low

frequency atmospheric variability

Andrey Gritsun

Institute of Numerical Mathematics/RAS,Moscow (asgrit@mail.ru)

Motivation

1. Unstable periodic orbits (UPOs) are the part of the system attractor and define recurrent circulation regimes. UPOs may be important in understanding the system dynamics.

2. Many chaotic systems have infinite number of periodic orbits. For axiom A and Anosov systems UPOs are dense on the attractor. Any system characteristic can be approximated by the set of the UPOs.

Auerbach D., P. Cvitanovic, J.-P. Eckmann, G. Gunaratne, and I. Procaccia, 1987, Exploring chaotic motion through periodic orbits, Phys. Rev. Lett., 58, 2387-2389

Biham, O., Wenzel, W., 1898, Characterization of unstable periodic orbits in chaotic attractors and repellers, Phys. Rev.Lett., 63, 819–822,3..

R.Bowen, 1971, Periodic points and measures for axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154, p.377-397.

R.Bowen, 1972, Periodic orbits for hyperbolic flows, Amer. J.Math., 94, p.1-30.

Gallavotti G., 1998, Chaotic dynamics, fluctuations,nonequilibrium ensembles, Chaos, 8, N2, 384-392.

UPOs and Lorenz attractor(Galias, Tucker, 2007)

"Chaotic system with many degrees of freedom can be regarded as an Anosov system for the purpose of computing its Macroscopic properties“ (G.Gallavotti)

Atmospheric systems are chaotic and (likely) have nonzero Lyapunov exponents but (very likely) are not axiom A or Anosov systems.

Weather regimes and stationary points

In the vicinity of stationary point system moves slowly andwe may expect that the system PDF has maximum near it(weather regime).

1. “Lifetime” and predictability depends on stability characteristics.

2. System dynamics can be approximated by transitions between points.

Problems: dynamic is very complex, no clear regimes, no clear realtions between points and “regimes”.

Goals of the study

1. Develop effective numerical methods for finding UPOs for multidimensional chaotic atmospheric systems.

2. Find as many orbits as possible for simple (barotropic) atmospheric

system.

3. Try to approximate statistics of the system and its PDF with UPOs, use UPOs to identify variability of the system.

Couette flow: Kawahara, G., Kida, S., 2001, J. Fluid Mech., 449, 291–300;Kawahara, G., Kida. S., L. van Veen, 2006, Nonlin. Proc. in Geophysics, 5, 499-507; J.F. Gibson, J. Halcrow, and P.Cvitanović, 2008, J. Fluid Mech., 611, 107-130.

Barotropic ocean model: Kazantsev, E., 1998, Nonlin. Proc. in Geophysics, 5, 193-208; Kazantsev, E., 2001, Nonlin. Proc. in Geophysics, 8, 281-300.

Kuramoto-Sivashinsky system: Zoldi, S., Greenside, H., 1998, Phys. Rev.E, E57, R2511-2514

No results for atmospheric systems

Barotropic atmospheric system

The model is based on the barotropic vorticity equation on rotating sphere (=2D Navier-Stokes system +forcing + rotation + boundary friction + orography),

.

),(

2extf

HlJt

- Laplacian, - Jacobian, - Coriolis parameter, - streamfunction, - orography, - external forcing.

Galerkin T12 (spherical harmonics m<13), Phase space dim=78

J l H extf

.,),(

ut

Model orography H

.),( 2rrrrext HlJf

is streamfunction on 300mb surface from 1960-1990 NCEP/NCAR reanalysis dataset

)(tr

14.0k

turbulent viscosity boundary layer frictionorography normalization

3102

5106

External forcing f

Parameters

Model does a god job in reproducing real climate and variability:

Variance

(300mb NCEP data/model)

Average state

(300mb NCEP data/model)

Ktkk /)(

2/12 )/)))((( Ktkk

Model is chaotic

Attractor dimension = 13.56 positive Lyapunov exponents

Trajectory and PDF of T12 modelon (EOF1,EOF2) plane

Methodology for finding UPOs

Rewrite model equation in compact form

By the definition UPO with period T is the system trajectory satisfying

This equation has N+1 unknowns (initial point and period) that define UPO.

Damped Newton and inexact Quasi-Newton methods were used to solve the system (+ line search, multi shooting, tenzor correction etc….)

Initial guesses for u(0) and T were taken from the model trajectory {u(k), k=1..K} as local minimizers for

),()( 0utStu

)0())0(,()( uuTSTu

2|)()(| kuTku

Numerical realization

1. Run Newton procedure for ensemble of initial guesses on cluster computer system.

2. 2232 UPOs and 50 unstable stationary points found.

What is required?

1. Tangent model in full space (Newton)

2. Tangent and adjoint tangent models in Krylov space (inexact

methods).

3. 1 iteration for finding UPO with period T means 1 run of tangent

(adjoint tangent model) in full (Krylov) space for time T.

Convergence of Newton method for 250 initial guesses:

log10 of initial error**2 (green), log10 of final error**2 (black), log10 of 10**(-30) (yellow) (log scale)

14 UPOs (or stationary points) found

-Orbits have different periods and shape.-Some UPOs are outside the trajectory “attractor” -Shortest orbits do not approximate attractor-Orbits are “visually dense” on major part of the attractor

UPOs of the system (EOF1,EOF2 plane). Left: 6 selected orbits from previous picture; Center: 100 shortest orbits; Right: all orbits and stationary points. Black: system trajectory.

UPOs of the system on (EOF1,EOF2) plane

It is possible to find orbits that have very complicated structure,are highly unstable and have long periods.

Approximation of system statistics with UPO

))((1

lim)( iiT tT

pp

pppNP

lim)(

p

)exp(/1 Pp )1(

is the sum of the positive Lyapunov exponents of correspondent UPO

Average

Can be approximated as

Where is value of for p-periodic point with period NP.

Point weights must be calculated according to Axiom A theory as

calculated directly (y) vs using Axion A measure (x) (log-log scale).

ipp

ippi

PUPOOmes

)())((

)( iUPOO

1)( Pi )( iUPOOP

Let be a characteristic function of

)(Pi:)( iUPOO

1. Trajectory spends very few time in the vicinity of several least unstable UPOs.2. Axiom A formula underestimates time spent by system trajectory in the vicinity of very unstable orbits.

Measure of neighborhood of i-th UPO is

ii /

)exp(/1 Pp

is the average value for thesum of the positive Lyapunov exponentsin some neighborhood of correspondent UPO

)2(

ii / calculated directly (y) vs using Axion A measure (x)

calculated directly (y) vs using corrected measure (2) (x)

ii /

Approximation of the PDFs projections on (1-3) and (2-4) solution components

Left to right: real PDF; Ax.A weights (1); Ax.A weights (1) + unphys. orbits removed; weights (2) + unphys. orbits removed

Reproduction of the system average state and variance. Left: model; Center: calculated by UPOs with (1); Right: with (2).

Orbits form a skeleton for the system attractor and approximate statistical characteristics of the system with high accuracy. It is reasonable to expect that some of UPOs are connected with prominent patterns of model variability.

Periodic orbits and variability patterns

1. Apply direct (time) Fourier transform to every field component.

2. Apply π/2 time shift

3. Apply inverse Fourier transform

is a Hilbert transform for

4. Make a complex time series and calculate its covariance matrix.

5. Complex eigenvectors of are called Hilbert (or complex) EOFs (v.Storch, Zwiers,1999).

kkjj ibaXFX ][

kkkk iabiba

XiabFiab kkkk

~)(1

jX~

jX

*,~ ZZCXiXZ Hjjj

*ZZCH

Leading Hilbert EOF of the system defines dominant rotational component of the circulation.

Consider system trajectory jX TXXC

Hq

Leading Hilbert EOF for H500 field (NCEP data, Branstator, 1986) is a planetary wave, moving from east to west with characteristic period of around 25 days.

Leading Hilbert EOF for T21 barotropic system reflects basic features of HEOF from nature and has same period.

sin)Im(cos)Re( HH qq 4/3,2/,4/,0

Right: PDF for the states of T21 system when trajectory moves along HEOF plane (projected onto plane) ;

Left: projection of several least unstable UPOs of T21 model on HEOF plane.

))Im(),(Re( HH qq

))Im(),(Re( HH qq

Right: Leading Hilbert EOF of least unstable UPO (in 20-30days period range) of T21 barotropic system is almost identical to Leading Hilbert EOF (Left) of the model.

sin)Im(cos)Re( HH qq 4/3,2/,4/,0

Summary

1. Barotropic atmospheric model has (infinitely?) many periodic solutions.

2. UPOs approximate model statistical characteristics with sufficiently high accuracy. Axiom A assumptions however may not be exactly true (weight formula works only if proper UPOs are selected and with some correction).

3. UPOs define some important circulation regimes (25 day mode as an example).