Baroclinic Instability
H
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Cold
H
Advanced Synoptic
Warm
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M. D. Eastin
Baroclinic Instability
Baroclinic Instability
• Basic Idea
• Simple Models
• Classic Eady Framework
• Contributions from Barotropic Instability
• Examples of Observational Evidence
Advanced Synoptic
M. D. Eastin
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”
• 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
Advanced Synoptic
M. D. Eastin
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
Advanced Synoptic
M. D. Eastin
Baroclinic 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
Advanced Synoptic
M. D. Eastin
Baroclinic Instability
Review of Potential and Kinetic Energy:
“Available” Potential Energy
No kinetic energy
Unstable Situation
Advanced Synoptic
“Growth” of Wiley’s fall speed
due to extraction of potential energy
from the basic-state environment
(conversion of potential energy to kinetic energy)
M. D. Eastin
Baroclinic Instability
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”
Center of
Gravity
h
 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
Advanced Synoptic
h≈0
M. D. Eastin
Baroclinic Instability
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
Tropopause Light
P
Surface
Heavy
Equator
 If the atmosphere is given a small “push” (e.g. a weak
Pole
T
Light
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
Advanced Synoptic
Heavy
Equator
Pole
M. D. Eastin
Baroclinic Instability
The Basic Idea: “Simple Atmosphere”
• Several “events” occurred during this process
in our simple atmosphere that are commonly
observed in the real atmosphere:
Tropopause Light
P
• 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
Surface
Heavy
Equator
• 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.
Pole
T
Light
Heavy
Equator
Advanced Synoptic
Pole
M. D. Eastin
Baroclinic Instability
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
Advanced Synoptic
2
3
System continues to intensify
(increase its eddy kinetic energy)
until is can no longer generate
eddy potential energy
(becomes a closed occluded system)
M. D. Eastin
Baroclinic Instability
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)
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• 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
Cold
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Warm
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• Warm air rising poleward
• Cold air sinking equatorward
 Similar to warm/cold fronts
Advanced Synoptic
M. D. Eastin
Contributions from Barotropic Instability
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?
D  ug  vg

Dt 
2
2
2
Term A

u g
Rd




 T  u g vg
 
gp
y

Term B
See your text
(Section 2.7)
For full derivation
Term C
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…
Advanced Synoptic
M. D. Eastin
Contributions from Barotropic Instability
Barotropic Production of Eddy Kinetic Energy:
D  ug  vg

Dt 
2
2
Term A
2

u g
Rd







T

u
v

g g
gp
y

Term B
Term C
Term C: Perfectly Circular Eddy
Average of u’gv’g
over entire eddy
is zero
Circular systems
do NOT intensify
from barotropic
processes
(but they can from)
(baroclinic processes)
Advanced Synoptic
M. D. Eastin
Contributions from Barotropic Instability
Barotropic Production of Eddy Kinetic Energy:
D  ug  vg

Dt 
2
2
2

u g
Rd







T

u
v

g g
gp
y

Term A
Term B
Term C
Term C: Asymmetric Eddy with Negative Tilt
Average of u’gv’g
over entire eddy
is negative
“Negatively tilted” systems
CAN intensify
from barotropic
processes IF
located south
of the ͞u maximum
Advanced Synoptic
M. D. Eastin
Contributions from Barotropic Instability
Barotropic Production of Eddy Kinetic Energy:
t=0
t=0
Advanced Synoptic
t=0
t = +6 hrs
M. D. Eastin
Baroclinic/Barotropic Instability
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
Advanced Synoptic
Example of a Trough with an Upstream Jet Streak
+
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→ ascent
→ Psfc decreases
→ ΔP increases
→ EKE increases
→ enhances WAA / CAA
→ maintains any ongoing
baroclinic instability
process
M. D. Eastin
Baroclinic/Barotropic Instability
Observational Evidence for Possible Cyclogenesis?
Answer #2: When a diffluent trough
is above and west of a weak
surface low
Example of a Diffluent Trough
Diffluent upper-level troughs
induce deep-layer ascent
Ascent → Psfc decreases
→ ΔP increases
→ EKE increases
→ enhances WAA / CAA
→ maintains any ongoing
baroclinic instability
process
Advanced Synoptic
L
Note how the distance
between the 6 height
contours increases
downstream of the
trough axis
M. D. Eastin
Baroclinic/Barotropic Instability
Observational Evidence for Possible Cyclogenesis?
Example of a Negatively-tilted Trough
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
Advanced Synoptic
Y
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X
Note how the slope
of the trough axis
is negative in the
X-Y coordinate
system
M. D. Eastin
Baroclinic/Barotropic Instability
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
Advanced Synoptic
M. D. Eastin
References
Bluestein, 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.
Advanced Synoptic
M. D. Eastin