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 middleLatitudes.

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 mid-latitude. This circulation is called a Ferrel cell. The flow of the atmosphere at the mid-latitudes 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 MovementPressure GradientCoriolis EffectFriction

Cyclones and AnticyclonesWind Speed

Cyclones and Anticyclones

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.

In centre air rising therefore low pressure.Air moves in an anticlockwise direction.

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 Cellsoccur only in tropical latitudes

Equator

CyclonicSystem

Components of General Circulation

Subtropical HighsTrade Winds Intertropical Convergence ZoneThe WesterliesPolar HighsPolar EasterliesSubpolar Lows

General Circulation

Know:

• Polar Highs

• Westerlies

• Trade winds

• ITCZInter-tropical convergence zone

Trade Winds25°N to 25°SEasterlies – named for the direction they

come fromMost dominant winds

Westerlies30° to 60° N & SCan be influenced by surface

Jet StreamPolar to temperate latitudesRossby 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 mid-latitudes.

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.

What is the relationship of Rossby Waves to surface high and lows?

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.

0 p

M

RTp

For a barotropic atmosphere we have:

Because

TpRT

Mppp

RT

M

RT

pMpp

2

= 0

0

0

T

Tp

Two examples are the: homogenous atmosphere 0p

isothermal atmosphere 0T

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:

finalc

initialc f

R

Vf

R

V

We will focus on the curvature term V/R as the surrogate to the full relative Vorticity.

If the flow is perturbed slightly at point 2 to turn to the north, the air will move into higher latitudes where the Coriolisparameter and planetary vorticity are greater.

A negative shear or curvature (negative R) must form into the flow to compensate the increased planetary shear, in order to keep the potential vorticity constant. (The jet turns clockwise at point 3until it points southeast).

As it goes southward towards its initial latitude (point 4) it has less curvature (less relative vorticity), but still points southeast. The jet will move south towards his initial latitude until a region where planetary vorticity is less (point5). To preserve potential vorticity, it develops a cyclonic curvature and heads back northeast.

What happens next?

It happens that our initial stable (zonal) flow from point 1 has become WAVY and is said to have become UNSTABLE

The “wavy” flow is a Rossby wave, which requires a variation of Coriolis parameterwith latitude to create instability, which is called barotropic instability.

The parameter which gives the rate of change of Coriolis parameter with latitude is:

cos

earth

c

Ry

f 2

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:

2

0 2

c

The negative sign indicates westward propagation relative to the mean flow.

A phase speed c relative to the ground is defined as:

00 cUc 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 toobservers on the ground.

Dispersion relation (because waves of different wavelenghts propagate at different phase speeds.

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:

0

0

0

T

Tp

p

barotropic

baroclinic

g

z

u :

:

0

0

Or in terms of thermal wind:

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 STREAMBaroclinicity 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 south-north 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

In the baroclinic case both β and N have effects in an environment with a north-south temperature gradient.

The Brunt-Vaisala frequency is the angular frequency of internal waves driven by negative density variations.

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.

(Internal waves are buoyancy waves caused by variations in density with height).

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.

The white columns represent the vorticity of the jet.

Dark grey ribbon represent the jet-stream axis.

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:

final

c

initial

c

Z

RVf

Z

RVf

//

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 Baroclinicr=r(p) only

Implications:1) isobaric and isothermal surfaces coincide2) no vertical wind shear(thermal wind = 0)3) no tilt of pressure systems with height

r=r(p,T)

Implications:1) isobaric and isothermal surfaces intersect2) 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 midlatitudesAtmosphere 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