Institute of Numerical Mathematics RAS The sensitivity of climate

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					Unstable periodic orbits in models of the low
     frequency atmospheric variability

                 Andrey Gritsun

 Institute of Numerical Mathematics/RAS,Moscow
 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)
Atmospheric systems are chaotic and (likely) have nonzero
Lyapunov exponents but (very likely) are not axiom A or
Anosov systems.

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

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

1. “Lifetime” and predictability depends on stability

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   (u, )   ,   .
  +forcing + rotation + boundary friction + orography),

       J ( ,   l  H )   
      f ext .

     - Laplacian, J - Jacobian, l - Coriolis parameter,
     - streamfunction, H - orography, f ext - external forcing.
Galerkin T12 (spherical harmonics m<13), Phase space dim=78
 Model orography H

 External forcing f
                      fext  J ( r ,  r  l  H )   r  2 r .
  r (t ) is streamfunction on
300mb surface from 1960-1990
NCEP/NCAR reanalysis dataset

                       turbulent viscosity       6 105
                       boundary layer friction   2 103
                       orography normalization k  0.14
Model does a god job in reproducing real climate and variability:

                                       Average state
                                           k (tk ) / K
                                       (300mb NCEP data/model)

                                         ( k ( (tk )  ))2 / K )1/ 2
                                       (300mb NCEP data/model)
Model is chaotic

Attractor dimension = 13.5
6 positive Lyapunov exponents

                                Trajectory and PDF of T12 model
                                on (EOF1,EOF2) plane
              Methodology for finding UPOs
Rewrite model equation in compact form
 u ( t )  S ( t , u0 )
By the definition UPO with period T is the system trajectory satisfying
 u (T )  S (T , u (0))  u (0)
This equation has N+1 unknowns (initial point and period) that define

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 | u(k  T )  u(k ) |2
What is required?
1. Tangent model in full space (Newton)
2. Tangent and adjoint tangent models in Krylov space (inexact
3. 1 iteration for finding UPO with period T means 1 run of tangent
   (adjoint tangent model) in full (Krylov) space for time T.

Numerical realization
1. Run Newton procedure for ensemble of initial guesses on cluster
computer system.
2. 2232 UPOs and 50 unstable stationary points found.
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
    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.

-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 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
Average       ( )  lim T   i  ( (ti ))
                                                p  p p
Can be approximated as         ( )  lim NP
                                                  p p
Where       p is value of         for p-periodic point with period NP.

Point weights must be calculated according to Axiom A theory as
 p  1/ exp( P)                                                    (1)

     
          is the sum of the positive Lyapunov exponents of correspondent UPO
                                  Let i (P ) be a characteristic function
                                  of O (UPOi ) :
                                           i ( P )  1 P  O (UPOi )
                                  Measure of   neighborhood
                                  of i-th UPO O (UPOi ) is
                                                          p  p i ( P )
                                   mes(O (UPOi ))                          i
                                                              p p

i /  i calculated directly (y) vs
using Axion A measure (x)
(log-log scale).                       1. Trajectory spends very few time in
                                       the vicinity of several least unstable
                                       2. Axiom A formula underestimates
                                       time spent by system trajectory in the
                                       vicinity of very unstable orbits.
                       p  1/ exp( P)                                    ( 2)
             is the average value for thesum of the positive Lyapunov exponents
         in some neighborhood of correspondent UPO

i /  i calculated directly (y) vs       i /  i calculated directly (y) vs
using Axion A measure (x)                 using corrected measure (2) (x)
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
  Consider system trajectory   Xj     C  XX T 
1. Apply direct (time) Fourier transform   X j  F [ X j ]  ak  ibk
   to every field component.
                                            ak  ibk  bk  ia k
2. Apply π/2 time shift

                                                             1       ~
3. Apply inverse Fourier transform
                                           bk  iak  F (bk  iak )  X
   ~                                 Xj
   Xj   is a Hilbert transform for
4. Make a complex time series and          Z j  X j  iX j , CH  ZZ * 
   calculate its covariance matrix.

5. Complex eigenvectors qH of CH  ZZ               
                                                         are called Hilbert
  (or complex) EOFs (v.Storch, Zwiers,1999).

    Leading Hilbert EOF of the system defines dominant
    rotational component of the circulation.
  Re(qH ) cos   Im( qH ) sin             0, / 4, / 2,3 / 4
Leading Hilbert EOF for H500 field    Leading Hilbert EOF for T21 barotropic
(NCEP data, Branstator, 1986) is a    system reflects basic features of HEOF
planetary wave, moving from east to   from nature and has same period.
west with characteristic period of
around 25 days.
Right: PDF for the states of T21 system when trajectory moves
along HEOF plane (projected onto (Re( qH ), Im( qH )) plane) ;

Left: projection of several least unstable UPOs of T21 model on
HEOF (Re( qH ), Im( qH )) plane.
Re(qH ) cos   Im( qH ) sin        0, / 4, / 2,3 / 4
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.
1. Barotropic atmospheric model has (infinitely?) many periodic

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).

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