Data Assimilation- A02. Deterministic ChaosShunji Kotsuki
Environmental Prediction Science Laboratory
Center for Environmental Remote Sensing (CEReS), Chiba University(shunji.kotsuki@chiba-u.jp)

DA Lectures A (Basic Course)‣ (1) Introduction and NWP‣ (2) Deterministic Chaos and Lorenz-96 model‣ (3) A toy model and Bayesian estimation‣ (4) Kalman Filter (KF)
‣ (5) 3D Variational Method (3DVAR)‣ (6) Ensemble Kalman Filter (PO method)‣ (7) Serial Ens. Square Root Filter (Serial EnSRF)‣ (8) Local Ens. Transform Kalman Filter (LETKF)‣ (9) Innovation Statistics‣ (10) Adaptive Inflations‣ (11) 4D Variational Method (4DVAR)Today’s Goal
‣ Lecture‣To understand the chaotic nature ofatmosphere‣ Training Course‣To implement Lorenz 96 (any language)‣To estimate error doubling time

Deterministic Chaos(Lecture)Deterministic Chaos and Predictability
Edward Lorenz

Lorenz 63 modelx  p( y  x)y   xz  rx  yz  xy  bz

p  10, r  28, b  8 / 3

Chaotic systems havelimits to predictability
even with the perfect model!!

Initial Conditions :: x=y=z=15.000, 15.001, 15.002, …, 15.009

Predictability differs
more predictableless predictable

Predictability depends on the initial state!

Ensemble Prediction (e.g. TC)

Observed(answer)
initial states－: deterministic－: ensemble© Japan Meteorological Agency

JAXA’s NWP: NEXRA
Kotsuki et al. (2019a, 2019b)

Press Released on Aug. 2020
Running Operationally on JAXA’s Supercomputing SystemEnsemble NWP by NEXRAFCST: 006hr

Thin: Ensemble Prediction
Bold: Analysis (reference)

6hr

Ensemble NWP by NEXRAFCST: 024hrThin: Ensemble Prediction
Bold: Analysis (reference)

1day

Ensemble NWP by NEXRAFCST: 072hr

Thin: Ensemble Prediction
Bold: Analysis (reference)

3dayEnsemble NWP by NEXRAFCST: 120hr

Thin: Ensemble Prediction
Bold: Analysis (reference)

5dayEnsemble NWP by NEXRAFCST: 168hr

Thin: Ensemble Prediction
Bold: Analysis (reference)

7day

Ensemble NWP by NEXRAFCST: 336hr

Thin: Ensemble Prediction
Bold: Analysis (reference)

14day

Ensemble NWP by NEXRAFCST: 504hr

Thin: Ensemble Prediction
Bold: Analysis (reference)

21day

Ensemble NWP by NEXRA
1-day forecasts

7-day forecasts
less predictable

more predictable・larger ensemble spread・smaller ensemble spread・ensemble predictions ≒ reference ・ensemble predictions ≠ reference・larger spread near extratropical cyclones

Training Course

DA Study w/ 40-variable Lorenz-96
Lorenz-96 model (Lorenz 1996)

Advection term

For j=1,…,N, Xj=Xj+N
Dissipation term Forcing termText Books① Training Description

pswd: ceres‣ https://kotsuki-lab.com/internal-pages/

Basic Task 1

Training Course Problem (1)The Lorenz-96 model (L96)
Journal of the Atmospheric Sciences, 55

The Lorenz-96 model (L96)Time Integration Schemes
from wikipedia

Euler method (EUL)

Runge-Kutta 2nd order (RK2)Runge-Kutta 4th order (RK4)DA Study w/ 40-variable Lorenz-96
Lorenz-96 model (Lorenz 1996)

Advection term

Initial Condition௜

For j=1,…,N, Xj=Xj+N
Dissipation term Forcing term

DA Study w/ 40-variable Lorenz-96
Lorenz-96 model (Lorenz 1996)

Advection term

For j=1,…,N, Xj=Xj+N
Dissipation term Forcing term

Lorenz et al. assumed 1 time unit = 5 days based on error doubling time.

Properties of system deduced without solving the equations.
1. When F > 0.895, steady solutions is unstable.
Properties of system deduced with solving the equations.
( J:40, dt : 0.05, with 4th order Runge-Kutta scheme)

1. When F < 4.0, the perturbations ultimately develop
into perfect wave number eight.
2. When F > 4.0, the a spatially irregular patternwith chaotic time variations appears.

Ultimate States (1000-days integration)
F=0.5

F=0.8

F=1.0

F=8.0

vars

Hovmöller Diagram

time steps

F=0.10F=0.50F=1.00HofmüllerF=3.00

F=5.00F=8.00Stable or Chaotic?
F=3.00

F=5.00Non-cyclic
 Unpredictable
F=8.00 (=chaos)cyclic predictable

steps

steps

Advection termsteps

Dissipation term Forcing term 2Advection term conserve the total energy defined as  X j / 2

Time Integration Schemes
day 15day 30day 45

Basic Task 2

Training Course Problem (2)To Calc. Error Doubling Time
1. Mersenne Twistter method: To generate pseudo-random number sampling

2. Box-Muller transformation: To generate pairs of normally distributed random numbers

3. Error propagation analysis (average RMSE)
t=0
Three year spin up

X ' j (t )  X j (t )  std N (0,1)
For std = 0.0010, 0.0100,0.1000

When std=1.000, some superposed run has diverged

Control run (CTL)1000
simulations

To Generate N(0,1) Numbers
1. Mersenne Twistter method: To generate pseudo-random number sampling

2. Box-Muller transformation: To generate pairs of normally distributed random numbers

Box-Muller’s method:

When X and Y obey uniform distribution (0,1),Z1 and Z2, defined by following equations,obey normal distribution N(0,1).

Z1  2 log X cos 2 YZ 2  2 log X sin 2 YError Propagation Analysis

std=0.050
Day 0: 0.0500
Day 1: 0.0654
Day 2: 0.1058
Day 3: 0.1539

error doubles inabout two days

std=0.500
Day 0: 0.5000
Day 1: 0.6466
Day 2: 1.0056
Day 3: 1.3954

error doubles inabout two days

Error Propagation Analysis

Lorenz(1996) Fig.2a std=0.001

Thank you for your attention!
Presented by Shunji Kotsuki(shunji.kotsuki@chiba-u.jp)
Further information is available athttps://kotsuki-lab.com/