baroclinic instability

Advanced Synoptic M. D. Eastin

Baroclinic Instability

LH

LH

Trough

Ridge

Northerlies

Southerlies

WarmCold



• Basic Idea• Simple Models• Classic Eady Framework• Contributions from Barotropic Instability• Examples of Observational Evidence



Definition

Spontaneous growth of a small-scale perturbations within a basic-state environment Energy source for the growth is drawn from the basic-state environment

Different Types of “Instability”

• Convective instability → Convective clouds grow as parcels tap into the background CAPE

• Kelvin-Helmholtz instability → Wave-like clouds grow (and “break”) as parcels tap into the background vertical shear

Concept of Instability


Definition

Spontaneous growth of a small-scale perturbations within a basic-state environment Energy source for the growth is drawn from the basic-state environment

Different Types of “Instability”

• Barotropic instability → Disturbances grow by extracting kinetic energy from the background flow

→ suction-vortices in tornadoes → meso-vortices in hurricanes → short-waves in jet stream

• Baroclinic instability → Disturbances grow by extracting potential energy from the background flow

→ Synoptic-scale waves

Concept of Instability


Questions:

QG theory and polar-front theory have taught us that the development and intensification

of a surface cyclone requires the interaction of the fledgling surface cyclone or stationary front with a pre-existing upper-level wave.

What mechanism develops the upper-level waves?

What determines the size, structure, and intensity of the upper-level waves?

What basic-state conditions are required for the waves to develop?

Our Approach:

• Your text (Chapter 7) provides a very well-written and thorough explanation of baroclinic instability via the classic theoretical framework first presented by Eady (1949)• This will be (has been) presented in detail in Advanced Dynamics

• Here, we will address the relevant results from a practical perspective



Review of Potential and Kinetic Energy:


“Available” Potential EnergyNo kinetic energyUnstable Situation

“Growth” of Wiley’s fall speed due to extraction of potential energy

from the basic-state environment(conversion of potential energy to kinetic energy)


The Basic Idea: “Coin Model”

• Consider a coin resting on its edge (an “unstable” situation)

• Its center of gravity (or mass) is located some distance (h) above the surface• As long as h > 0, the coin has some “available potential energy”

If the coin is given a small push to one side, it will fall over and come to rest on its side (a “stable” situation) The instability was “released” and “removed”

• Its center of gravity was lowered and thus its potential energy was decreased• The coin’s motion represents kinetic energy that was converted from the available potential energy


Center ofGravity

h

h ≈ 0


The Basic Idea: “Simple Atmosphere”

• Consider a stratified four-layer atmosphere with the most dense air near the surface at the pole and the least dense near the tropopause above the equator (an “unstable” situation)

• Each layer has a center of gravity ( ) located some distance above the surface• Each layer has some available potential energy• The entire atmosphere also has a center of gravity ( ) and some available potential energy

If the atmosphere is given a small “push” (e.g. a weak cyclone) then the layers will move until they have adjusted their centers of gravity to the configuration that provides lowest possible center of gravity for the atmosphere (the most “stable” situation) The baroclinic instability was released and removed

• Each layer’s motion represents a portion of the total atmospheric kinetic energy that was converted from the atmosphere’s available potential energy


T

P

Equator Pole

Surface

Tropopause Light

Heavy

Equator Pole

Light

Heavy


The Basic Idea: “Simple Atmosphere”

• Several “events” occurred during this process in our simple atmosphere that are commonly observed in the real atmosphere:

• Kinetic energy (or wind) was generated similar to the increase in winds as a weak low pressure system intensifies

• Warm (less dense) air was lifted over cold (more dense) air in a manner very similar to fronts

• There is a poleward transport of warm air and an equatorward transport of cold air similar to the typical temperature advection pattern around a low pressure system.


T

P

Equator Pole

Surface

Tropopause Light

Heavy

Equator Pole

Light

Heavy


Eady Framework: Energy Conversion Processes

• Basic-state environment consists of a strong north-south temperature gradient with an upper-level zonal jet stream (atmosphere in thermal wind balance)• Basic-state environment contains both available potential energy and kinetic energy

• Instability is initiated by (1) perturbation flow inducing weak localized WAA and CAA across the thermal gradient → warm and cold air parcels (or eddy potential energy)• Eddy kinetic energy is then generated (2) as warm parcels rise and cold parcels sink• Acceleration of initial parcels away from their origin creates (via mass continuity) more WAA and CAA [or creates more eddy potential energy (3)]


1 2

3

System continues to intensify(increase its eddy kinetic energy)

until is can no longer generateeddy potential energy

(becomes a closed occluded system)


Eady Framework: Idealized Situation

• Maximum growth rate occurs for waves

with wavelengths of 3000-6000 km

Synoptic scale

• Maximum growth rate occurs for waves

tilting west with height (21º phase shift)

• Greater tilt → no intensification• Less tilt → no intensification

We look for westward tilt Stacked systems are mature

• Eddy kinetic energy develops from an upward heat flux

• Warm air rising poleward• Cold air sinking equatorward

Similar to warm/cold fronts


LH

LH

Trough

Ridge

Northerlies

Southerlies

WarmCold


Production of Eddy Kinetic Energy

• Our analysis of baroclinic instability showed that the synoptic waves/cyclones are essentially

systems of eddy kinetic energy → What else can create eddy kinetic energy?

Term A: Eddy Kinetic Energy (EKE) Production

Term B: Baroclinic Generation → Upward heat flux produces EKE → Warm air rising / cold air sinking

Term C: Barotropic Generation → Function of location relative to zonal mean jet stream → Function of mean momentum flux → Let’s examine a few scenarios…

Contributions from Barotropic Instability

y

uvuT

gp

Rvu

Dt

D ggg

dgg

2

22

Term A Term B Term C

See your text(Section 2.7)

For full derivation


Barotropic Production of Eddy Kinetic Energy:

Term C: Perfectly Circular Eddy


y

uvuT

gp

Rvu

Dt

D ggg

dgg

2

22


Average of u’gv’g over entire eddy

is zero

Circular systemsdo NOT intensifyfrom barotropic

processes

(but they can from) (baroclinic processes)



Term C: Asymmetric Eddy with Negative Tilt


y

uvuT

gp

Rvu

Dt

D ggg

dgg

2

22


Average of u’gv’g over entire eddy

is negative

“Negatively tilted” systemsCAN intensify

from barotropicprocesses IF located south

of the Ku maximum




t = 0 t = 0

t = 0 t = +6 hrs


Observational Evidence for Possible Cyclogenesis?

Answer #1: When a jet core is upstream of a trough axis that is above and west of a weak surface low

Upstream jet streaks have large positive vorticity on their poleward flank with PVA downstream near the trough axis and over the weak surface low

PVA → ascent

→ Psfc decreases → ΔP increases

→ EKE increases→ enhances WAA / CAA→ maintains any ongoing baroclinic instability process

Baroclinic/Barotropic Instability

Example of a Trough with an Upstream Jet Streak

L+



Answer #2: When a diffluent trough is above and west of a weak surface low

Diffluent upper-level troughs induce deep-layer ascent

Ascent → Psfc decreases → ΔP increases

→ EKE increases→ enhances WAA / CAA→ maintains any ongoing baroclinic instability process


Note how the distancebetween the 6 heightcontours increasesdownstream of the

trough axis

Example of a Diffluent Trough

L



Answer #3: When a negatively-tilted trough is above and west of a weak surface low and south of the zonal mean jet core

Negative tilts permit barotropic processes to generate a net increase in eddy kinetic energy

↑ EKE → enhances WAA / CAA→ maintains any ongoing baroclinic instability process


Example of a Negatively-tilted Trough

L Note how the slopeof the trough axisis negative in theX-Y coordinate

systemX

Y


Observational Analysis Tips:

Not all negatively-tilted troughs intensify Not all diffluent troughs intensify Not all troughs with upstream jet cores intensify

Must evaluate the vertical tilt of the system

Westward – may intensify (21º optimal) • Stacked – should not intensify much• Eastward – should not intensify

Must evaluate the latitude of the zonal mean jet core

Negatively-tilted systems south of the jet core may intensify Positively-tilted systems north of the jet core may intensify

• Negatively-tilted systems north of the jet core should not intensify• Positively-tilted systems south of the jet core should not intensify

Should evaluate all forcing terms in modified QG Omega equation Should evaluate all forcing terms in modified QG Height Tendency equation



ReferencesBluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume II: Observations and Theory of Weather

Systems. Oxford University Press, New York, 594 pp.

Bretherton, F. P., 1966: Critical layer instability in baroclinic flows, Quart. J. Roy. Meteor. Soc., 92, 325-334.

Charney, J. G., 1947: the dynamics of long waves in a baroclinic westerly current. J. Meteor., 6, 56-60.

Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, 33-52.

Lackmann, G., 2011: Mid-latitude Synoptic Meteorology – Dynamics, Analysis and Forecasting, AMS, 343 pp.

Orlanski, I., 1968; Instability of frontal waves. J. Atmos. Sci., 25, 178-200.

baroclinic instability

Documents

extraction of potential

basicstate conditions

eastinthe basic idea

eastinreview of potential

advanced dynamics

upperlevel waves

coin model

fledgling surface cyclone