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”

But can write:

€

dy

dt=y t − y t−1

Δt

So:

€

y t+1 − y tΔt

+y tτ

= a0υ t

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

What is noise?

Ex: Daily maximum temperature at SeaTac airport in 2002:

Anomalous temperature

• Now consider departure from normal (i.e., remove the annual cycle)

Histogram of anomalies

• Temperatures are most likely to be near normal, but there are a few days with extreme departures from normal.

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).

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’.

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

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

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

€

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

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

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