Paper

1
Paper

• G. Gyarmati, I. Szunyogh, and D.J. Patil, 2003
  Local predictability in a simple model of
  atmospheric balance,
• Nonlinear Processes in Geophysics 10183-196
• http//www.copernicus.org/EGU/npg/10/183.htm

2
Introduction
Atmospheric models are chaotic small differences
in the initial conditions can lead to big
differences in the forecasts.
non-linear
linear
PDF
analysis
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.
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Two measures of local predictability
u1
Leading Singular Value
v1
vn
L(x(t),?)vi (x(t), ?)?i (x(t),?)ui (x(t),?)
un
v1 most unstable direction vn most stable
direction
l(x(t))
l(x(t-?))
Local Lyapunov Number
l (x(t), ?) ? u1 (x,?)
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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.

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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 non-dissipative case
  for 10 years.
• Bokhove and Sheperd (1996) finally showed that
  the slow manifold can exist in the
  non-dissipative case.

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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
  non-dissipative (i.e has no attractor).

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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.

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Model equations
Non-dimensional, rescaled form
units m, l0, 1/?pendulum
?1/2?
Balance equation
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Two time scales
?0.325
displacement of the spring
?0.025
time
?0.25
?0.4
time
time
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Symmetries in the state space

• The symplectic structure of the elastic pendulum
  equations implies the following symmetries of the
  singular values and Lyapunov numbers
• Conservation of energy implies that there must be
  at least one neutral direction.

and
n4
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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 McLachlans (1995)
  generalized theory.

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Numerical experiments

• Four parameter setup H1.8, ?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.

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VisualizationTechnique
?0, p ?lt0
?0, p ?lt0
3 1.5 1.2 1.09 1.065 1.04 1 0.99 0.97 0.94
3 2 1 0 -1 -2 -3
p?
p?
?0.025
-3 -2 -1 0 1 2 3
-2 -1 0 1 2
?
?
Poincare section
fast plane
slow plane
trajectory
?0.25
?0.325
?0.4
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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.
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.
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?0, p ?lt0
Local Lyapunov number
leading singular value
? 0.25 H 1.8
p?
?
lt LV, v4 gt
lt u4 , v4 gt
Unbalanced part of the displacement
unbalanced part of LV
unbalanced SV
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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
barotropic instability leads to the
amplification of the baroclinic wave.
u
WARM
horizontal axis of the wave
barotrpic stability leads to the decay of the
baroclinic wave
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isolines negative barotropic energy conversion
colorsbaroclinic energy conversion (Jday/kg)
baroclinic
-400 - -600
-400 - -1600
barotropic
-400 - -600
baroclinic
Mike Oz
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Bred vectors and Lyapunov vectors
The local Lyapunov number and the growth rate of
the bred vector are significantly correlated.
Lyap.number lt1
corr0.78
Sample5x105
growth rate of bred vector
Lyap. number gt1
corr0.95 Sample5x105
local Lyapunov number
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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.)