BAROCLINIC AND
BAROTROPIC
INSTABILITY
22.04.2013
General circulation of the atmosphere
Represents the average air flow around the
globe
 It is created by unequal heating at earth’s
surface
 On global scale, earth is in radiative
equilibrium: energy in equals energy out
 General circulation’s function is to
transport heat pole ward

The atmosphere of the earth is being moved by the energy of
the sun. The atmosphere heated at the equator is then cooled
at the poles, forming a kind of thermal convection.
Looking at satellite pictures, the clouds around the equator do
seem to be moving in a convection-like fashion.
However, the cloud movement at
mid-latitudes doesn't look like a
convective current at all.
By looking at global movement of
clouds, we can see a prominent
east-west tendency in the wind,
i.e. easterly winds at low latitudes,
and westerly winds at middle
Latitudes.

The east and west winds
that circle the globe are
know as the trade winds
and the westerlies.
Hadley theorized that the
atmosphere moved in one
large thermal convection
current from the equator to
the poles.

The movement of the
atmosphere isn't quite that
simple.
Why? ….. Mostly due to the
rotation of the earth.



How does the rotation change the
thermal convection? Let's make a
simple experiment. We put water
into a donut-shaped container,
which will be the earth's
atmosphere. We'll heat the outer
edge of the container, and cool
the inner edge, so the inner edge
will be the poles, and the outer
edge will be the equator.
Because warm water is light, and
cold water is heavy, this difference
makes a pressure gradient
between the inner and outer
edges.
If the earth wasn't rotating, this
pressure gradient pushes the cold
water under the lighter warm
water, causing a convective
current.

However, in a rotating environment
things don't work in the same way. As
the pressure difference begins to move
water along the pressure gradient, the
Coriolis force changes the direction of
the flow. It ends up with the flow in
which the pressure gradient balances
with the Coriolis force resulting what is
called the geostrophic flow.

In this model of the atmosphere, if the
distribution of temperature is
concentric, then the resulting
geostrophic flow will also be concentric.
In this situation, the flow of water in the
upper layers is in the same direction as
the rotating table, like the westerlies
which we see in the earth's atmosphere.

In this photo, the flow of water is made
visible by liquid crystal capsules, and
temperature is represented by different
colors.



There is, however, something
wrong with this concentric flow of
water. A convection current is
meant to carry heat from hot
areas to cold areas, but with a
concentric flow of water, this
obviously doesn't happen. In other
words, a concentric flow of water
makes for an extremely inefficient
thermal convection.
If you speed up the rotation of the
model, this concentric flow
becomes unstable, and begins to
meander.
This is called baroclinic
instability. When water
meanders, it moves back and forth
from hot to cold areas, carrying
the heat as it goes. Compared to a
concentric flow, this meandering
flow transfers heat far more
efficiently.

If you look at the zonal
averages (the average along
the parallels) of the wind in the
atmosphere, you will find a
reverse circulation at midlatitude. This circulation is
called a Ferrel cell. The flow of
the atmosphere at the midlatitudes is characterized by
the baroclinic instability, which
causes high and low pressure
systems.

On the other hand, the
circulation of air at low
latitudes is close to the model
that Hadley proposed long ago,
so these areas are called
Hadley cells.
Development of Wind Patterns
The Nature of Wind

Direction of Movement
Pressure Gradient
 Coriolis Effect
 Friction

Cyclones and Anticyclones
 Wind Speed

Cyclones and Anticyclones
In centre air rising therefore low
pressure.
Air moves in an anticlockwise
direction.
This is when there is air in the
centre pushing down therefore,
high pressure. Wind clockwise,
sinking air and as sinks gets
warmer and therefore no rain.
Anticyclones
A high pressure center is where the pressure has been measured to be the
highest relative to its surroundings. That means, moving in any direction
away from the "High" will result in a decrease in pressure. A high
pressure center also represents the center of an anticyclone and is
indicated on a weather map by a blue "H".
Winds flow clockwise around a high pressure center in the northern
hemisphere, while in the southern hemisphere, winds flow
counterclockwise around a high.
Sinking air in the vicinity of a high pressure center suppresses the
upward motions needed to support the development of clouds and
precipitation. This is why fair weather is commonly associated with an
area of high pressure.
Cyclones
A low pressure center is where the pressure has been measured to be the
lowest relative to its surroundings. That means, moving in any horizontal
direction away from the "Low" will result in an increase in pressure.
Low pressure centers also represent the centers of cyclones.
A low pressure center is indicated on a weather map by a red "L" and winds flow
counterclockwise around a low in the northern hemisphere. The opposite is true in the
southern hemisphere, where winds flow clockwise around an area of low pressure.
Rising motion in the vicinity of a low pressure center favors the
development of clouds and precipitation, which is why cloudy weather
(and likely precipitation) are commonly associated with an area of low
pressure.
Cyclones and Anticyclones
Circulation Patterns
Hadley Cells
occur only in tropical latitudes
Cyclonic
System
Equator
Components of General
Circulation
Subtropical Highs
 Trade Winds
 Intertropical Convergence Zone
 The Westerlies
 Polar Highs
 Polar Easterlies
 Subpolar Lows

General
Circulation
Know:
•Polar Highs
•Westerlies
•Trade winds
•ITCZ
Inter-tropical
convergence zone
Trade Winds
25°N to 25°S
 Easterlies – named for the direction they
come from
 Most dominant winds

Westerlies
30° to 60° N & S
 Can be influenced by surface

Jet Stream
Polar to temperate latitudes
 Rossby wave influence

Global Atmospheric Circulation
Jet Streams

What is the purpose of Rossby Waves?
Rossby Waves are like rivers of air in the upper troposphere and they gradually
meander. The meander loops get bigger and bigger until their wavelength from
trough to trough could be as much as 8000 kms. When the Waves are well
developed and cover a wide range of latitude they are said to have a low zonal
index - which leads to the formation of ridges of blocking, high pressure
systems and dry stable conditions. When they are almost straight and cover a
narrow zone of latitude they are said to have a high zonal index - which leads to a
succession of low pressure systems and unsettled weather. The waves evolve
then they straighten up and then meanders form again in an endless cycle. The wave
evolution cycle lasts about 6 weeks. But what is the purpose of this?
What is the purpose of Rossby Waves?

The diagram shows that as a wave develops (in the mid-latitudes of the
northern hemisphere) cold Polar air is dragged southwards and surrounded by
warmer Tropical air. Similarly loops of warmer Tropical air are moving north
and being cut-off by cold Polar air. In this way heat transference is occurring cold air moving south and warming; warmer moving north and cooling. When
the loops become very pronounced, they detach the masses of cold, or warm,
air that become cyclones (depressions/low pressure) and anticyclones (high
pressure) areas that are responsible for day-to-day weather patterns at midlatitudes.
What is the relationship of Rossby Waves
to surface high and lows?
Surface high pressure areas and surface low pressures are thought
to correlate to particular parts of the wave trough section of the
Rossby Waves. The surface conditions in each location is the
opposite of what is happening in the Upper Air.
The Jet Stream and Rossby
Waves
Barotropic atmosphere
A barotropic atmosphere is
one in which the density
depends only in the
pressure, so that isobaric
surfaces are also surfaces
of constant density. For an
ideal gas, the isobaric
surfaces will be also
isothermal if the
atmosphere is barotropic.
For a barotropic atmosphere we have:
p    0
Because
p
RT
M
Mp
 pM  M
p    p  
p  p 
p  T

2
RT
 RT  RT
=0
p  T  0
  T  0
Two examples are the: homogenous atmosphere
isothermal atmosphere
p  0
T  0
Barotropic instability and Rossby waves


Consider the jet stream at midlatitudes blowing form west to east. If
some small disturbances (e.g. flow over mountains) causes the jet to
turn slightly northward, then the conservation of potential vorticity
causes the jet to meander north and south. This meander of the jet
stream is called a Rossby wave or planetary wave.
If we consider a zonal flow that has no relative vorticity, but it has
planetary vorticity related to the latitude of the flow, the conservation
of the potential vorticity can be written:
V

V

   fc 
 R  fc 

 initial  R
 final
We will focus on the curvature term V/R as the surrogate to the full relative
Vorticity.
If As
theitflow
goes
is southward
perturbed slightly
towardsatits
point
initial
2 to
latitude
turn to(point
the north,
4) it has less curvature
Abut
negative
shear
or curvature
R) must
form
into
the
(less
air will
relative
move
vorticity),
into higher
latitudes
still points
where
southeast.
the Coriolis
The(negative
jet will move
south
towards
flowstable
to
compensate
the
increased
planetary
shear, in
It initial
happens
thatuntil
ourthe
initial
(zonal)
flow
from point
1(point5).
has become
parameter
his
and
latitude
planetary
avorticity
region
where
are
greater.
planetary
vorticity
is less
orderistosaid
keep
potential
vorticity
constant.
(The jet
WAVY vorticity,
and
tothe
have
UNSTABLE
To preserve potential
it
develops
abecome
cyclonic
curvature
and heads
back
turns clockwise at point 3until it points southeast).
northeast.
What happens next?
The “wavy” flow is a Rossby wave, which requires a variation of Coriolis parameter
with latitude to create instability, which is called barotropic instability.
The parameter which gives the rate of change of Coriolis parameter with latitude is:

f c 2  

 cos
y Rearth
The path taken by the wave is given by:
Where y’ is the north-south displacement distance from the centre latitude
Y0 of the wave, x’ is the distance east from some arbitrary longitude, c is
the phase speed, A – north-south amplitude of the wave and λ is the
wavelength.
Typical values: λ = 6000km and A=1665km (at mid-latitudes)
These waves propagate relative to the mean zonal wind U0 at the phase
speed c0 of about:
  
c0     

 2 
2
Dispersion relation (because waves of different
wavelenghts propagate at different phase
speeds.
The negative sign indicates westward propagation relative to the mean flow.
A phase speed c relative to the ground is defined as:
c  U 0  c0
This gives the west-to-east movement of the wave
crest.
For typical values of c0 and zonal speed U0, the phase speed is positive.
In this case the mean wind pushes the waves toward the east rekative to
observers on the ground.
From this two equations we can say that waves of shorter
wavelenght (short waves) travel faster toward the east than
long waves.
Baroclinic atmosphere
An atmosphere in which
density depends on both
temperature and pressure is
called an baroclinic
atmosphere.
In a baroclinic atmosphere
the geostrophic wind
generally has vertical shear,
and this shear is related to
the horizontal temperature
gradient by the thermal
wind equation.
For a baroclinic atmosphere we have:
p    0
p  T  0
T    0
Or in terms of thermal wind:
u g
z


0: barotropic
 0: baroclinic
In a baroclinic atmosphere the geostrophic wind generally
has vertical shear which is related to the horizontal
temperature gradient by the thermal wind equation.
Baroclinicity – JET STREAM
Baroclinicity drives the west-to-east winds near the top of the troposphere,
via thermal wind relationship.
Air near the ground in warmer near
equator, colder at the poles, and there
is a frontal zone at mid-latitudes
where temperature decreases rapidly
toward the north. This north-south
gradient exists throughout the
troposphere.
The tropopause is lower near the
poles than near the equator. Because
of this the temperature begins
increasing with height at a lower
altitude near the poles than near the
equator. This causes a temperature
reversal in the stratosphere, where the
air is colder over the equator and
warmer over the poles.
If we apply the thermal wind equation to the temperature
filed we get the pressure filed.
In the troposphere, greater
thickness between pressure
surfaces in the warmer
equatorial air than in the
colder polar air causes the
isobars to become more
tilted at mid-latitudes as the
tropopause approaches.
Above the tropopause, tilt
decreases because the
north-south gradient is
revesed.
Regions with the greatest tilt have the greatest south-to-north
pressure gradient, which drives the fastest geostrophic
wind.
The maximum of westerly winds
is know as the Jet Stream and
occurs at the tropopause in the
mid-latitudes.
There are two extrema of southnorth temperature gradient, one in
the northern hemisphere and one in
the southern hemisphere. These are
the latitudes where we can find the
strongest jet velocities.
Although the temperature gradient in
the southern hemisphere has a sign
opposite to that in the north, the sign
of the Coriolis parameter also
changes. In this way, the Jet Stream
velocity is positive (west to east) in
both hemispheres.
North-south temperature gradient at sea
level and at 15 km altitude
Baroclinic instability
Decreasing temperature with respect to altitude can lead to the
development of convection, if the vertical temperature distribution
becomes statically unstable.
If the latitudinal distribution of temperature develops an equator to
pole temperature gradient which is too large, that indicates that the
tropics are too warm and the poles too cold. This equator to pole
imbalance in energy is fundamentally due to the excess net
radiational heating in tropical latitudes. Such an energy
distribution is unstable, and the name given to this unstable state
is baroclinic instability.
On the left is the typical wind regime under conditions of
baroclinic stability, when the imbalance of energy between
tropics and polar regions is not excessively large.
When the temperature gradient reaches an excessively large
value, the atmosphere becomes baroclinically unstable and
the wind currents respond by developing poleward energy
transporting modes of flow. This is accomplished by the
development of large meanders in the westerly flow, and
cut-off pressure centers, which provide pathways for warm
and cold air pools to move across latitudes, thus achieving
the required energy transports.
Baroclinic instability and planetary
waves
At midlatitudes the cold polar air slides under
the warmer tropical air. This causes the air to
be stable, as can be quantified by a Brunt
Vaisala frequency N.
The Brunt-Vaisala frequency is the angular frequency of internal waves driven by
negative density variations.
(Internal waves are buoyancy waves caused by variations in density with height).
In the baroclinic case both β and N have effects in an environment with a
north-south temperature gradient.
As an example we consider the simplest case and idealise the atmosphere as
being 2 two layer fluid, with a north-south sloping density interface.
Dark grey ribbon represent the jet-stream axis.
The white columns represent the vorticity of the jet.
Point 1 gives the initially zonal flow of the jet stream that has no relative vorticity,
but it does have planetary vorticity related to its latitude.
If the Jet Stream is perturbed northward by some outside influence like
a mountain, it rides up on the density surface.
The stratosphere is so statically stable that it acts like a lid on the tropopause,
resulting into a northward meandering of the air. The air will be squeezed
between the tropopause and the rising density interface, namely δZ shrinks.
The potential vorticity equation can be written as:
 f c  V / R 
 f c  V / R 




Z
Z

 initial 
 final
The column depth is less at point 3 than in point 2, hence the absolute vorticity at 3
must also be less that at 2, in order for the ratio of absolute vorticity to depth to
remain constant.
The planetary vorticity doesn’t help much, so the only way to conserve potential
vorticity is ofr the relative vorticity to decrease substantially. As it decreases
below its initial value of zero, the Jet Stream path curves anticyclonically at point
3.
The jet stream overshoots to the south and develops cyclonic relative vorticity and
turns back to the north. The resulting breakdown of the zonal flow into wavy flow
is called baroclinic
instability.
The waves look similar to those obtained for the barotropic instability, except
with shorter wavelenghts because now both βandδZ work togheter to cause the
oscillation.
Baroclinic vs. Barotropic
Barotropic
Baroclinic
=(p) only
=(p,T)
Implications:
1) isobaric and isothermal surfaces
coincide
2) no vertical wind shear
(thermal wind = 0)
3) no tilt of pressure systems with
height
Implications:
1) isobaric and isothermal surfaces
intersect
2) vertical wind shear
(thermal wind ≠ 0)
3) pressure systems tilt with height
Seasons:
Atmosphere is most baroclinic in winter.
Atmosphere is least baroclinic in summer.
Geographic:
Atmosphere is most baroclinic in midlatitudes
Atmosphere is least baroclinic in the Tropics
Bibliography


J. R. Holton: An Introduction to Dynamic Meteorology (2nd Ed.), Academic
Press
R. B. Stull: Meteorology for Scientists and Engineers (2nd Ed.), Brooks
Cole

J.P. Peixoto and A. H. Oort: Physics of Climate (2nd Ed.), American
Institute of Physics

Helenmary Hotz: Atmospheric Pressure and Wind (Ch5)
http://www.geog.umb.edu/HelenmaryHotz/Syllabus.htm


http://www.atmos.umd.edu/~owen/CHPI/IMAGES/barocln1.html
http://dennou-k.gaia.h.kyotou.ac.jp/library/gfd_exp/exp_e/doc/bc/guide01.htm