Why do we have storms in atmosphere?

Mid-atmosphere (500 hPa) DJF temperature map

What are the features of the mean state on which storms grow?

Zonal thermal winds, and zonal mean cross sections

The DJF Atlantic Jet

10 m/s

10 m/s20 m/s

20 m/s30 m/s

30 m/s

0 m/s

0 m/s

40 m/s

Sector zonal mean cross section

Thermal wind map (250 hPa – 850 hPa)

Are atmospheric jets stable?Will perturbations to the jet immediately decay? Does the jet support instabilities?

Analogy: Ball on top of a hill

Answer #1: Perturbations to the jet seem to grow in magnitude;The jet is unstable.

Answer #2: The jets are the time mean of the atmospheric flow;Something most kill the instability and return the flow to the time

mean.

Velocity Shear

The velocity shear between the two layers leads to an instability

Consider two different mean stateswith different velocity shears

5 m/s

- 5 m/s - 10 m/s

10 m/s

The systems have the same linear momentumBut the system with more shear has more kinetic energy

x

y

Can the perturbations extract energy from the flow?

• Key concept: If the perturbations can decrease the velocity shear in the flow, the flow will have less energy

• This energy will go to the perturbations

• Therefore, in order for the perturbations to grow in shear flow, they must transport momentum against the shear of the flow

Lets put this in mathematical terms

dUdy > 0

x

y

MEAN STATE Which perturbation stream functionwill grow?

A

B

C

Which way do perturbations transport momentum

When v’ is positive, u’ is zeroWhen u’ is negative, u’ is zero

u’v’ = 0

When v’ is positive, u’ is negativeWhen v’ is negative, u’ is positive

u’v’ < 0

u’v’ > 0

When v’ is positive, u’ is positiveWhen v’ is negative, u’ is negative

No momentum transport

Negative momentum transportIn the y direction

Positive momentum transportIn the y direction

Barotropic Conversion

Mean State Shear

Perturbation Stream Function

Energy flow from mean state to the perturbation if :

u’v’ dUdy

< 0

And the initial perturbationwill grow

• Simple analogy: If we perturb a ball, lying on top

topography from its resting position, will it continue to move away or will it return to the original position

Stable Neutrally stable

Ideal Flow and most unstable mode

Barotropic Normal Modes

'

''

'ddU v

t dydx

0D

Dt

Linearize about a basic state with zonally invariant zonal velocity

( , ) ( ( ), 0)U V U y

The equation becomes

In the spatial domain, we require that the solution at each grid point grows linearly

We assume a sinusoidal structure in x and discrete structure in y ' 2 /( ) i mx D

nG y e

1 1 11 1 1 1

1

1

n Nn n n

n k N n nN N N

N k N N N

G M M M G G

G M M M G GtG M M M G G

Is this the form of instability which is most pronounced in the atmosphere

10 m/s

10 m/s20 m/s

20 m/s30 m/s

30 m/s

0 m/s

0 m/s

40 m/s

Sector zonal mean cross section

Thermal wind map (250 hPa – 850 hPa)

There is much more vertical sheer than horizontal sheer in the jetThis is a different form of instability because it is in hydrostatic and geostophic balance

Mid-atmosphere (500 hPa) DJF temperature map

Can we extract energy from this?

How do Storms Grow? Perturbations (Storms) extract energy from the

mean state via two different mechanisms

Meridional Temperature Gradient(Baroclinic)

Velocity Shear(Barotropic)

High Energy

Low Energy

High Energy

Low Energy

Wind VectorsIsotherms

Hot

Cold

Not too hot

Not too cold

Atmospheric available potential energy- energy decrease by flattening isotherms

Temperature Potential Temperature

Equator Pole

Equator Pole

Equator Pole

Equator Pole

Height

Height

Height

Height

cold

coldhot

hot

More Energy

Less Energy

Simple model of baroclinic instability

x

y

z

Top layer

Bottom layer

MEAN STATE

•2 layer model

•Uniform meridional temperature gradient

•No zonal variations

warm

cold

FASTEST GROWING PERTURBATION

Contours = perturbation stream functionColors = tendency in stream function

•Peturbations slant against the vertical shear

•This causes the energy to be converted to the eddies

• The maximum growth of perturbations is: .31 f N-1 Eady Growth Rate

• Using values for the mid-latitude atmosphere, the perturbations will double in magnitude every 2 days, ocean = 20 days

• The horizontal spatial scale of the perturbations is 4000 km in the atmosphere, 400 km in the ocean

Results of the simple model

dUdz

Tilt of baroclinic modes

What happens to eddies as they are advected by the mean flow

T=0, u’v’ dU =0 dY

u’v’ dU >0 dY

T=1

Layered eddy advection

UpperLevel mean

flow

LowerLevel mean

flow

T= 0 days T= 2 days T= 4 days

v’t’ < 0

u’v’ = 0

v’t’ = 0

u’v’ > 0

dTdydUdy

dTdydUdy

Atmospheric Energy Reservoirs

Mean StatePotential EnergyEquator to Pole TemperatureGradient

Mean StateKinetic Energy

Large Scale Circulation

Transient Eddy Potential Energy

Transient thermal

anomalies

Transient EddyKinetic Energy

Transient Circulations

10X >

-v’t’ dTdy

w’α’ N2

u’v’ dUdy

BaroclinicGrowth

BarotropicDecay

Lifecycles in simple models

Chang and Orlanski , 1993

Baroclinic Instability explains the Locations of storm tracks

Storm activity

High pass height variance at 250 hPa

75 m2

75 m2

105 m2

Baroclinicity

Eady Growth Rate

.6 /day

.6 /day

Hoskins, Valdes, 1989

Different Ways of defining a storm-trackHigh Pass Eulerian variancesHigh pass filter a field and take the variance

z’2 (250 hPa) v’2 (300 hPa) slp’2

v’t’ (850 hPa)u’v’ (250 hPa)

Chang, 2002

Alternative approach, Lagrangian tracking

Dotted red lines = tracksColored boxes = size

Defining a storm track, continuedTrack features (Lagrangian) and keep track of where they pass and

how strong their central value is

Sea level pressure features- track density.

The picture in 1888

Hinman, 1888

What seeds storms?

Cyclogenesis density at 850 hPa.

Courtesy of Sandra Penny

Seasonality of storminess

Pacific StorminessV’2 (300 hPa)

Atlantic StorminessV’2 (300 hPa)

Pacific BaroclinicityU(500 hPa) – U(925 hPa)

Atlantic storm activity follows the baroclinicity, but Pacific does notChang, 2002

Mid-winter suppression of the Pacific storm track

Courtesy Camille Li

Ideas on Midwinter Suppression

DJF

SON

Weaker seeding in winter?From Sandra, again

Last Glacial Maximum Storm Tracks

V’T’ throughout the troposphere (colors) and upper level jet (contours) during the winter

LGM has less storm activity

LGM - MODERN

MODERN

LGM

LGM baroclinicity

LGM

Modern

LGM - MODERN

LGM has more baroclinicity

Do faster growing storms meanbigger storms?

Composite Storm size (m)

Composite storm growth

Rate (per day)

LGM

MODERN

Do faster growing storms meanbigger storms?

LGM

MODERNProbabilitydistributionfunction

of initial size,growth rate, and growth period for LGM and Modern

Predicted and observed storm sizedistribution for LGM and modern

LGM

MODERN

Predicted frominitial size

and growthPDF

Observed inmodel

Bottom line, storm size not proportional to growth rate/instability in mean state

Baroclinic instability in the ocean