Transcript Document 7902207

Paper
G. Gyarmati, I. Szunyogh, and D.J. Patil, 2003:
Local predictability in a simple model of
atmospheric balance,
Nonlinear Processes in Geophysics 10:183-196
http://www.copernicus.org/EGU/npg/10/183.htm
Introduction
Atmospheric models are chaotic: small differences in the
initial conditions can lead to big differences in the
forecasts.
linear
analysis
non-linear
PDF
control
truth
ball of small errors
L
tangent linear operator
ensemble member
An ensemble forecast aims at predicting the PDF of the
atmospheric state in the state space. We want to capture the
fastest growing error.
Two measures of local predictability
Leading Singular Value
L(x(t),)vi (x(t), )=i (x(t),)ui (x(t),)
v1 : most unstable direction
vn : most stable direction
|| L(x(t),  )l(x(t   )) ||
|| l(x(t   )) ||
l (x(t), )  u1 (x,)
un
t
1
l(x(t))
l(x(t-))
L (t-,t-)
t-
(x(t),  ) 
v1 L(t-,t)
vn
t-
Local Lyapunov Number
ball of
small
errors
u1
L(t-,t)
t-
t

Motivations
• Szunyogh , Toth and Kalnay (1997) compared
SVs and Lyapunov vectors for a low resolution
(T10) version of the NCEP MRF.
• They found that singular vectors had a quickly
decaying (stable) unbalanced component.
Geostrophic adjustment.
• Lyapunov vectors were nearly balanced.
• Conjecture: From dynamical systems point of
view, the SVs are initially not on the attractor, but
they quickly converge to it. On the other hand, the
LVs are always located on the attractor.
Motivations II.
• Lorenz (1986) derived the L5 model to
prove the existence of the slow manifold.
• L5 is the highest truncated version of the
shallow water equations.
• Existence of the slow manifold for the L5
model had not been proved for the nondissipative case for 10 years.
• Bokhove and Sheperd (1996) finally
showed that the slow manifold can exist in
the non-dissipative case.
Main Goal
• To demonstrate, that the characteristic difference
between the SVs and LVs in terms of balance is
due to the existence of multiple time scales (the
existence of an attractor, i.e dissipative forces, is
not a necessary condition.)
• We choose a simple chaotic model that maintains
two distinct time scales, but which is nondissipative (i.e has no attractor).
The model
• Camassa (1995): the L5 is a nonlinearly coupled
system of a nonlinear pendulum and a linear
oscillator (spring). There are only 4 independent
dynamical variables.
• Lynch (1996, 2002): A slight modification of the
coupling term in L5 leads to the swinging spring
model. It has two time scales, but no attractor.
Model equations
1 1 2
1 2
2
H   ( p   )  p (1 1/ 2 )2  cos (1 1/ 2 )
2
2
 pendulum
g l0


 spring
k m
Non-dimensional, rescaled form
Ý 

units: m, l0, 1/pendulum
p (1  )
1/ 2
2
pÝ   (1 1/ 2  )sin 
Ý   p

1

pÝ  1/ 2 p (1 1/ 2 )3  1  1/ 2 cos 
Balance equation:
2
0  1/ 2 p (1 1/ 2 )3  1  1/
cos 

1/2
Two time scales
=0.32
=0.025
5
time
=0.25
time
=0.4
time

Symmetries in the state space
• The symplectic structure of the elastic
pendulum equations implies the following
symmetries of the singular values and
Lyapunov numbers:
 i (x(t),  )   1
ni1 (x(t),  )
• Conservation of energy implies that there
must be at least one neutral direction.
n=4
1   1
4
and
 2   1
3 1
Numerical Solutions
• The system can not be solved analytically.
• We need to solve with a numerical method.
• A high order symplectic integrator is needed,
because
• we want to preserve the symmetry of the singular values
• structure preserving schemes are know to be more efficient in
preserving energy, than energy preserving schemes in
conserving the structure
• For a non-separable Hamiltonian system structure
preserving integrators are symmetric composition
schemes.
• We created a family of symplectic integrators for
the elastic pendulum based on McLachlan’s
(1995) generalized theory.
Numerical experiments
• Four parameter setup: H=1.8, =0.025, 0.25,
0.325, 0.4 based on Lynch (2002).
• 10 trajectories for each setup were calculated with
symplectic integrator No.12. (Most efficient
scheme to achieve global RMS energy error
smaller than 10-10)
• Deriving the tangent linear model was also needed
for the calculation of the SVs and LVs.
Visualization
Technique
Poincare section
trajectory
=0, p <0
2
3
1.5
2
1
1
0.5
0
0
-0.5
-1
-1
-2
-1.5
-3 -3 -2 -1 0 1 2 3
-2

slow plane
=0, p <0
3
1.5
1.2
1.09
1.065
1.04
1
0.99
0.97
0.94
=0.025
-1
0

fast plane
=0.25
=0.325
=0.4
1
2
Summary of the results
for neraly balanced motions
LV: The least and most predictable states are in
the neigborhood of hyperbolic fix point. The
direction of the motion determines whether the
state is extremely well or poorly predictable.
max
LV
SV: in the vicinity of the hyperdolic fixpoint
the predictibility is low regardles the direction
of the motion.
stable
motions
unstable
motions
SV
max
SV: the least predictable states are located in
the vicinity of the elliptic fix point.
SV can be strongly unbalanced.
LV is mainly balanced.
No attractor
9.
=0, p <0
Local Lyapunov number
 = 0.25
leading singular value
H = 1.8

< LV, v4 >
Unbalanced part of
the displacement
1.5
0.9
0.6
0.3
0.2
0
-0.1
-0.3
-0.5
-0.8
-1.6
unbalanced part of LV
< u4 , v 4 >
unbalanced SV
3
1.5
1.2
1.09
1.065
1.04
1
0.99
0.97
0.94
0.5
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.7
0.9
0.6
0.3
0.2
0
-0.1
-0.3
-0.5
-0.8
-1.6
Meteorological Example
Strong instabilities and stabilities can be present in the atmosphere at the
same time. This can lead to a difference between SV and LV.
Example: jet exit zone
unstable
baroclinic wave
COLD
Basic flow:
baroclinic
unstable
jet
barotropic
instability/
stability
u
WARM
horizontal
axis of
the wave
barotrpic „stability”
leads to the decay of
the baroclinic wave
barotropic
instability leads
to the
amplification
of the baroclinic
wave.
isolines: negative
colors:baroclinic energy conversion (J*day/kg)barotropic energy conversion
baroclinic
-400 - -600
-400 - -600
barotropi
c
-400 - -1600
baroclinic
Mike Oz
Bred vectors and Lyapunov vectors
The local Lyapunov number and the growth
rate of the bred vector are significantly
correlated.
Lyap.number <1
corr=0.78
Sample=5x105
Lyap. number >1
corr=0.95
Sample=5x105
local Lyapunov number
Conclusions
• When the atmospheric state is nearly balanced, the SVs
can have a strongly unbalanced component, while the LVs
are also nearly balanced. This seems to be fundamental
property of systems with more time scales, which exists
independently of the dissipative or non-dissipative nature
of the system.
• There are cases when the Lyapunov number indicates
extremely high predictability, while the leading singular
value indicates extremely low predictability. Barotropic
instability is a process that can lead to similar behavior in
the atmosphere. (We are planning to carry out more
research in this area.)