S. Lovejoy McGill, Montreal

The weather and climate as problems

in physics: scale invariance and

multifractals

McGillApril 13, 2012

The Weather and ClimateEmergent Laws and Multifractal Cascades

SHAUN LOVEJOY and DANIEL SCHERTZER

(in press, Oct. 2012)

Required reading for this course

The Weather

3

Meteorologists

The Emergence of physical laws

Low level (fundamental)

high level(simpler if applicable)

Quantum mechanics Classical MechanicsLarge scales(usually)

Statistical mechanicsContinuum mechanics, Fluid mechanics thermodynamics

Large numbers of particles

General Relativity Special RelativityLow energy mass density

Special Relativity

Velocities << speed of light

Classical (Galilean) Relativity

Example: The emergence of Thermodynamics from

Newton’s laws

F=ma

F=ma

F=ma

First law: conservation of energySecond law: increase in entropy

Newton’s laws: Thermodynamics:

ex.: Boyle’s law: (pressure) x (volume) = constant

Low level, (difficult to handle for many

particles)

High level (Valid when many particles

are present)

Large number ofparticles

Pioneers of turbulence

7

Richardson1881 - 1953

Kolmogorov1903 – 1987

Corrsin1920 – 1986

8

Obukhov1918 – 1989

Ralph Bolgiano, Jr. 1922 — 2002

The emergence of turbulence dynamics (Classical)

Laws of turbulenceClassical:

Richardson, Kolmogorov, Corrsin, Obukhov, Bolgiano

Fluid mechanics

Strong stirring (nonlinearity)

Low level (fundamental)

High level

Vortices in strongly turbulent fluid(M. Wiczek, numerical simulation, 2010)“Spaghetti”

Emergent laws reduce seeming complexity to

simplicity at another level

Mandelbrot1924-2010

Complex?

The Mandelbrot set

(“self-similar”, scale invariant, fractal)

Blowing up gives the same type of shapes

Or simple?

Generating the Mandelbrot set

-Take a number.

-Multiply it by itself.

-Add a constant.

-Repeat.

(I forgot to mention: take a COMPLEX number)

Complex?

20 steps: 1:1 20 steps: 5:1

20,000 steps: 1:1 20,000 steps: 5:1

200 steps: 1:1 200 steps: 5:1

Drunkard’s walk

Δx

Or simple?

(distance) x (distance) = number of bars visited

(Brownian motion)

From initial barAverage number of bars visited (or displacements made)

Complex?… or simple?

20481536102451201100

1200

1300

1400

1500

1600

1700

t

v(t)

1 second of wind data (roof of Rutherford

building, McGill)

Infra Red satellite effective temperatures, January 16, 2008

The Atmosphere

mm/s

Brute force…

Atmosphere: Laws of Fluid mechanics(low level)

windEarth angular velocity

pressure

Gravitational potential

Specific heat Heating ratetemperature

Specific volume=1/ρ

Gas constant

density

Governing atmospheric laws

Friction

Brute force numerical solution of the equations (2)…

Discretization of the equations

Brute force numerical solution of the equations (3)…

Earth system modelling

Or simplicity?

Atmosphere: Emergent laws(high level)

Fluctuations ≈ (turbulent flux) x (scale)H

Size:Anisotropic Space-time Scale function

Fluctuation/conservation exponent

Cascading Turbulent flux

Differences, tendencies, wavelet coefficients

Fluctuation = change in time and/or spaceScale = sizeTurbulent flux = strength of stirring

Power law

These laws are scale invariant

Which Richardson?The father of

Numerical Weather

Prediction…

The father of numerical weather prediction

1922

The weather prediction factory

(artist: Francois Schuiten)

Richardson’s numerical

grid for integrating

Each column was divided into 5 vertical cells and defined 7 quantities: pressure, temperature, density, water content, 3 velocity components

“It took me the best part of six weeks to draw up the computing forms and to work out the new distribution in two vertical columns for the first time. My office was a heap of hay in cold rest billet. With practice the work of an average computer might go perhaps ten times faster. If the time-step were 3 hours, then 32 individuals could just compute two-points so as to keep up with the weather.”

-Richardson 1922

... or the grandfather of cascades?

Weather prediction by Numerical Process 1922, p.66

Scale by scale simplicity: cascades

CASCADE LEVELS

0 --

1 --

2 -- . . .

n --

xy

ε

0l

l0 / λ1

2

n

l0 / λ

l0 / λ

multiplication by 4independent random(multiplicative)increments

multiplication by 16independent random(multiplicative)increments

“Does the wind have a velocity?”

0.2 0.4 0.6 0.8 1

-2

-1

1

2

3

4

W(t)

t

“Although at first sight strange, the question grows upon acquaintance…” - Richardson 1926

Richardson suggested that the trajectory of a particle could be like a Wierstrass function (1872)

20481536102451201100

1200

1300

1400

1500

1600

1700

t

v(t)

1 second of wind data (roof of Rutherford

building, McGill)

mm/s

Scale invarianceand fractals

0.2 0.4 0.6 0.8 1

-2-1

1234

0.2 0.4 0.6 0.8

-2

-1

1

2

3

4 λΗ

1

λ

1

Wierstrass function showing scale invariance under anisotropic “blowup” (H=1/3 in this example, λ=3)

blowup

λ=3

Cantor set

• Let us start with:

and let us iterate:

x3A small part is same as the whole if “blown up” by a factor 3 (“scale invariance”, “self-similarity”)

Sierpinski Triangle

1 2

Koch snowflakeLet us start with:

and let us iterate:

Sierpinski Pyramid

• First iteration:

10 th iteration:

Menger Sponge

• motif:

iterations:

Hom

ogen

eous Interm

ittent

Parent eddy

Daughter eddies

Grand-daughter eddies

CASCADES

β-modelFractal set

“active”

“calm”

Cascades and Multifractals

Aircraft temperature transect (12km altitude)

500 1000 1500 2000 x (km)

- 53

- 52

- 51

- 50

- 49Temperature (oC)

Turbulent flux (gradient of the above)

1 σ

500 1000 1500 2000 x5

101520 ε

500 1000 1500 2000 x48

12 ε

500 1000 1500 2000 x2468 ε

500 1000 1500 2000 x12345 ε

500 1000 1500 2000

500 1000 1500 2000 x0.51.5

3.0 ε

500 1000 1500 2000 x0.51.01.5 ε

x0.40.81.2 ε

Temperature turbulent flux ε

at 280m resolution

High to low Resolution:degrading by factors of 4

km

ε0 ε1

Cascades and Multifractals

(“α model”)

Simulations: adding small scale details (low resolution to high)

Cascades

DRESSEDDRESSEDENERGYENERGY

FLUX FLUX DENSITYDENSITYFIELDSFIELDS

BAREBAREENERGYENERGY

FLUX FLUX DENSITYDENSITYFIELDSFIELDS

Dressed densityaveraged over

16x16

y εx y Π

x

1.1.

1.1.

1.1.

1.1.

1.

Cascade • level 1

bare density

1.

Cascade • level 3

bare density

Cascade • level 4

bare density

Cascade • level 5

bare density

Cascade • level 6

bare density

Cascade • level 7

bare density

Dressed densityaveraged over

8x8

Dressed densityaveraged over

4x4

Dressed densityaveraged over

2x2

Multifractal

0.5 1.0 1.5 2.0

Log10λ0.1

0.2

0.3

0.4 Log10

Mq

0.5 1.0 1.5 2.0

0.1

0.2

0.3

0.4

0.5 1.0 1.5 2.0

0.1

0.2

0.3

0.4

0.5 1.0 1.5 2.0

0.1

0.2

0.3

0.4

EW wind NS wind

Temphs humidity

Log10

Mq

Log10λ

Log10

Mq Log10

Mq

Mq ≈ λK q( )

q=2

q=1.5

q=2

q=1.5

q=2

q=1.5

q=2

q=1.5

20000km20000km

20000km 20000km

200km

200km

200km

200km

Cascades work!!

Cloud liquid water (top)

Cloud liquid water (side)

Cloud top, infra red

Cloud top visible

Cloud bottom visible

Cascade modeling: clouds and radiative transfer

Cascade Simulations

The Climate

48

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What is the climate?

“Climate is conventionally defined as the long-term statistics of the weather…”. -Committee on Radiative Forcing Effects on Climate, 2005 US National Academy of Science

"Climate is what you expect, weather is what you get.”

-Farmers Almanac

3 Three regimes:

three types of variability: not two!

51

100 200 300 400 500 600 700

- 2

2

4

6

8

10

1 Century,Vostok, 20-92kyr BP

20 days,75oN,100 oW, 1967 - 2008

1 hour,Lander, 10 Feb.-12 March, 2005

0

�T/

t

�

Climate(10-30 yrs to 50,000 yrs)

Weather(up to 10 days)

Low frequencyweather(10 days to 10-30 yrs)

TemperatureFluctuations Growing

Fluctuations Growing

Fluctuations Decreasing

100 200 300 400

-8

-6

-4

-2

2ΔT (K)

Age (kyr)

Vostok core (Antarctica ≈75oS) ≈ 130 yr resolution (average)

Holocene

≈40yr resolution

≈600yr resolutionLast four Glacials

Today

53

20 40 60 80

-

-

-

-

-

-

44

42

40

38

36

34

Age (kyr BP)

Holo

cene

Last glacial

“Dansgaard events”

� 8K

�18O

Paleotemperatures: GRIP (GReenland Ice core Project), summit location (≈75oN), High (5.2 yr ) resolution section

NOW

Moburg

Ljundquist

Huang

Esper

Crowley

1600 1700 1800 1900

-

-

-

-

0.8

0.6

0.4

0.2

0.2

BriffaMann

Jones

T(K)Date

1980

Eight “Reconstructions” of the Northern hemisphere

temperatures(30 year average applied)

Multiproxy Temperatures

post 2003

1900 1920 1940 1960 1980 2000

-

-

0.4

0.2

0.2

0.4

0.6T (K)

Date

Global temperature anomaly mean and “spread” from

three different instrumental estimates

The climate?

Do Global Climate models predict...

56

...or low frequency weather?

1 year 100 years10 years

- 0.5 0.0 0.5 1.0 1.5 2.0 2.5

-

-

-

-

0.9

0.7

0.5

0.3

Atmosphere (3km)

Mean surface

IPSL

Log10

Δt (yrs)

0.2

0.5

1

0.1

- 0.1

0.1

0.34

GCM

Data

averagetemperaturechange

OC

OC

OC

OC

OC

2

-0.5

0.5

Global Surface

>2003 multiproxies, 1500-1980

Vostok(Antarctic)

105yrs

10-2

Log10Δt (yrs)

±2K

±3K5oC

0.1oC

20CR global scale

0.4

weather Low frequency weather climate

Low frequency climate

1oC

GlobalTemperature

averagetemperaturechange

1 year10000 years100 years10 days

GCM’s

Implications for global warming

• By comparing model and natural variability, we found that GCM’s seem to be missing a long-time mechanism of internal variability such as land-ice.

• Anthropogenic contributions to 20th warming and 21st C warming scenarios may thus be either over - or under estimated.

59

2. Emergent Atmospheric laws are power lawsFluctuations are scaling, their exponents are scale invariant

Conclusions1. Low level laws: complex (Fluid mechanics)High level laws simplicity (emergent turbulent laws)

3. There are three different regimes:Weather to ≈ 10 days, Low frequency weather to ≈ 10-30 yrs,Climate to ≈ 50- 100kyrs.

4. Without special forcing GCM’s produce low frequency weathernot climate type variability