stochastic climate model
DESCRIPTION
Stochastic climate model. a.k.a. a first-order autoregressive process, or AR(1) or “red noise”. y’ = some climate variable (PERTURBATION) t = characteristic timescale (“MEMORY”) n( t )= “noise” forcing. Simplest way of representing a system with memory and random - PowerPoint PPT PresentationTRANSCRIPT
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Stochastic climate model
€
dy'
dt+y '
τ= ν (t)
€
y’ = some climate variable (PERTURBATION) characteristic timescale
(“MEMORY”)t “noise” forcing
• Simplest way of representing a system with memory and random forcing
• a.k.a. a first-order autoregressive process, or AR(1) or “red noise”
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But can write:
€
dy
dt=y t − y t−1
Δt
So:
€
y t+1 − y tΔt
+y tτ
= a0υ t
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In discretized form (i.e., time stepping in increments of t):
€
y't = a1y 't−Δt +a0ν ta1 = 1- t/t = noise - a random event, (normalized)a0 = amplitude of noise.
memory randomforcing
currentstate
This is called a first order autoregressive process , or AR(1)Also known as ‘red noise’ process
The element of random noise makes it a stochastic process
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What is noise?
Ex: Daily maximum temperature at SeaTac airport in 2002:
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Anomalous temperature
• Now consider departure from normal (i.e., remove the annual cycle)
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Histogram of anomalies
• Temperatures are most likely to be near normal, but there are a few days with extreme departures from normal.
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No memory (uncorrelated)
= 5 yrs
= 1 yrs
= 25 yrs
= 0 yrs
Time (yrs) Long memory
Stochastic models with different characteristic timescales
• The greater the memory, the longer the timescale of the variability (i.e., length of interval above or below average).
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What does the spectrum of variability look like?How does the power (or energy) in the time series vary as a function of frequency (or period)?
Period in years (i.e. 1/frequency) note the log scale.
Pow
erPower spectrum
Time Series
= 0 yrs (no memory)
Time (yrs)
For no memory, energyIs the same at all periods (frequencies). Hence ‘white noise’.
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What does the spectrum of variability look like?How does the power (or energy) in the time series vary as a function of frequency (or period)?
Period in years (i.e. 1/frequency) note the log scale.
Pow
erPower spectrum
Time Series
= 1 yrs
Time (yrs)
Increased memoryincreases powerat longer periods:hence “red” noise
![Page 10: Stochastic climate model](https://reader036.vdocument.in/reader036/viewer/2022062309/5681482b550346895db54eb6/html5/thumbnails/10.jpg)
What does the spectrum of variability look like?How does the power (or energy) in the time series vary as a function of frequency (or period)?
Period in years (i.e. 1/frequency) note the log scale.
Pow
erPower spectrum
Time Series
= 5 yrs
Time (yrs)
Increased memoryincreases powerat longer periods:hence “red” noise
![Page 11: Stochastic climate model](https://reader036.vdocument.in/reader036/viewer/2022062309/5681482b550346895db54eb6/html5/thumbnails/11.jpg)
What does the spectrum of variability look like?How does the power (or energy) in the time series vary as a function of frequency (or period)?
Period in years (i.e. 1/frequency) note the log scale.
Pow
erPower spectrum
Time Series
= 25 yrs
Time (yrs)
Increased memoryincreases powerat longer periods:hence “red” noise
![Page 12: Stochastic climate model](https://reader036.vdocument.in/reader036/viewer/2022062309/5681482b550346895db54eb6/html5/thumbnails/12.jpg)
€
P( f ) =ao
2
1+ (2πfτ )2
Equation for power spectrum of a red noise process
P(f) = power per unit frequency, f
• Can also show (how?) that half the energy in the time seriesoccurs at periods which are 2 or longer.
See, e.g., Jenkins and Watts, 1968
By analogy: Pendulum time constant =
€
l
gl = length of stringg = gravity
Period of oscillation =
€
2πl
g
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Example from last time: Pacific Decadal Oscillation. Even though variability is decadal, time series consistent with a red noise processwith a timescale of ~1 yr.
Because any geophysical system at all will always have random noise, and some inertia (a tendency to remember previous states), red noiseshould always be the default expectation
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PDO index (top panel) compared to 2 random realizations of a an AR(1) process with a characteristic time scale of 1.2 years
-note the apparent cycles
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