Walker circulation
Encyclopedia
The Walker circulation, also known as the Walker cell, is a conceptual model of the air flow in the tropics
in the lower atmosphere (troposphere
). According to this model parcels of air follow a closed circulation in the zonal
and vertical directions. This circulation, which is roughly consistent with observations, is caused by differences in heat distribution between ocean and land. It was discovered by Gilbert Walker
. In addition to motions in the zonal and vertical direction the tropical atmosphere also has considerable motion in the meridional direction as part of, for example, the Hadley Circulation.
Walker then selected a number of "centers of action", which included areas such as the Indian Peninsula. The centers were in the hearts of regions with either permanent or seasonal high and low pressures. He also added points for regions where rainfall, wind or temperature was an important control.
He examined the relationships of the summer and winter values of pressure and rainfall, first focusing on summer and winter values, and later extending his work to the spring and autumn.
He concludes that variations in temperature are generally governed by variations in pressure and rainfall. It had previously been suggested that sunspots could be the cause of the temperature variations, but Walker argued against this conclusion by showing monthly correlations of sunspots with temperature, winds, cloud cover, and rain that were inconsistent.
Walker made it a point to publish all of his correlation findings, both of relationships found to be important as well as relationships that were found to be unimportant. He did this for the purpose of dissuading researchers from focusing on correlations that did not exist.
, the autocorrelation function at lag 
is defined as:
The value of the autocorrelation function at lag
is the power of 
, or its variance if the mean value of 
is zero:
Moreover,
is the mean value for random processes.
The autocorrelation function may be used to detect deterministic components masked in a random background because autocorrelation functions of deterministic data (like sine wave) persist over all time displacements, while autocorrelation functions of stochastic processes tend to zero for large time displacement (for 0-mean time series).
is mathematical modeling of a time series based on the assumption that each value of the series depends only on a weighted sum of the previous values of the same series plus "noise".
If
is the 
-th value of the time series, the AR model of order 
is given by:
where
is the noise. The order, 
, can be considered as an index of the lag within the time series of which data will be considered for the analysis. The larger the lag, the larger the system of equations to be solved.
The AR coefficients can be estimated from the autocorrelation sequence by solving the Yule-Walker equations.
The generalized matrix version of the AR(p) model is given by the equation
Gidon Eshel provides a useful breakdown of the Yule-Walker equations that discusses their relation to between the least squares approach for fitting an AR(p) model.
It is based on parameters
where 
. There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which is itself obtained from the covariances). This is done using the Yule-Walker equations:
where
, yielding 
equations. 
is the autocovariance function of 
, 
is the standard deviation
of the input noise process, and
is the Kronecker delta function.
Because the last part of the equation is non-zero only if
, the equation is usually solved by representing it as a matrix for 
, thus getting equation
solving all
. For 
have
which allows us to solve
.
The above equations (the Yule-Walker equations) provide one route to estimating the parameters of an AR(p) model, by replacing the theoretical covariances with estimated values. One way of specifying the estimated covariances is equivalent to a calculation using least squares regression of values
on the 
previous values of the same series.
Changes in the Walker Circulation with time occur in conjunction with changes in surface temperature. Some of these changes are forced externally, such as the seasonal shift of the Sun into the Northern Hemisphere in summer. Other changes appear to be the result of coupled ocean-atmosphere feedback in which, for example, easterly winds cause the sea surface temperature to fall in the east, enhancing the zonal heat contrast and hence intensifying easterly winds across the basin. These anomalous easterlies induce more equatorial upwelling and raise the thermocline in the east, amplifying the initial cooling by the southerlies. This coupled ocean-atmosphere feedback was originally proposed by Bjerknes. From an oceanographic point of view, the equatorial cold tongue is caused by easterly winds. Were the earth climate symmetric about the equator, cross-equatorial wind would vanish, and the cold tongue would be much weaker and have a very different zonal structure than is observed today.
The Walker cell is indirectly related to upwelling
off the coasts of Peru
and Ecuador
. This brings nutrient
-rich cold water to the surface, increasing fishing stocks.
The Walker circulation is caused by the pressure gradient force
that results from a high pressure system
over the eastern Pacific ocean, and a low pressure system over Indonesia
. When the Walker circulation weakens or reverses, an El Niño results, causing the ocean surface to be warmer than average, as upwelling of cold water occurs less or not at all. An especially strong Walker circulation causes a La Niña
, resulting in cooler ocean temperatures due to increased upwelling.
A scientific study published in May 2006 in the journal Nature
indicates that the Walker circulation has been slowing since the mid-19th Century. The authors argue that global warming
is a likely causative factor in the weakening of the wind pattern. However, a new study from The Twentieth Century Reanalysis Project shows that the Walker circulation has not been slowing (or increasing) from 1871-2008.
Tropics
The tropics is a region of the Earth surrounding the Equator. It is limited in latitude by the Tropic of Cancer in the northern hemisphere at approximately N and the Tropic of Capricorn in the southern hemisphere at S; these latitudes correspond to the axial tilt of the Earth...
in the lower atmosphere (troposphere
Troposphere
The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 80% of the atmosphere's mass and 99% of its water vapor and aerosols....
). According to this model parcels of air follow a closed circulation in the zonal
Zonal
Zonal can refer to:* Zonal and meridional, directions on a globe* Zonal and poloidal, directions in a toroidal magnetically confined plasma* Zonal polynomial, a symmetric multivariate polynomial...
and vertical directions. This circulation, which is roughly consistent with observations, is caused by differences in heat distribution between ocean and land. It was discovered by Gilbert Walker
Gilbert Walker
Sir Gilbert Thomas Walker, CSI, FRS, was a British physicist and statistician of the 20th century. He is best known for his groundbreaking description of the Southern Oscillation, a major phenomenon of global climate, and for greatly advancing the study of climate in general.He was born in...
. In addition to motions in the zonal and vertical direction the tropical atmosphere also has considerable motion in the meridional direction as part of, for example, the Hadley Circulation.
Walker's methodology
Walker determined that the time scale of a year (used by many studying the atmosphere) was unsuitable because geospatial relationships could be entirely different depending on the season. Thus, Walker broke his temporal analysis into December–February, March–May, June–August, and September–November.Walker then selected a number of "centers of action", which included areas such as the Indian Peninsula. The centers were in the hearts of regions with either permanent or seasonal high and low pressures. He also added points for regions where rainfall, wind or temperature was an important control.
He examined the relationships of the summer and winter values of pressure and rainfall, first focusing on summer and winter values, and later extending his work to the spring and autumn.
He concludes that variations in temperature are generally governed by variations in pressure and rainfall. It had previously been suggested that sunspots could be the cause of the temperature variations, but Walker argued against this conclusion by showing monthly correlations of sunspots with temperature, winds, cloud cover, and rain that were inconsistent.
Walker made it a point to publish all of his correlation findings, both of relationships found to be important as well as relationships that were found to be unimportant. He did this for the purpose of dissuading researchers from focusing on correlations that did not exist.
Mathematical basis
The statistical model involved in the analysis of atmospheric data that led to the discovery of the Walker circulation is called an autoregressive (AR) process.Autocorrelation function
As background, first consider the autocorrelation function. An autocorrelation function in a measure of the dependence of time series values at one time on the values at another time. Given the time series, the autocorrelation function at lag is defined as:The value of the autocorrelation function at lag
is the power of , or its variance if the mean value of is zero:Moreover,
is the mean value for random processes.The autocorrelation function may be used to detect deterministic components masked in a random background because autocorrelation functions of deterministic data (like sine wave) persist over all time displacements, while autocorrelation functions of stochastic processes tend to zero for large time displacement (for 0-mean time series).
Autoregressive model
Next, consider the autoregressive model proposed by Walker. Autoregressive ModelingAutoregressive model
In statistics and signal processing, an autoregressive model is a type of random process which is often used to model and predict various types of natural phenomena...
is mathematical modeling of a time series based on the assumption that each value of the series depends only on a weighted sum of the previous values of the same series plus "noise".
If
is the -th value of the time series, the AR model of order is given by:where
is the noise. The order, , can be considered as an index of the lag within the time series of which data will be considered for the analysis. The larger the lag, the larger the system of equations to be solved.The AR coefficients can be estimated from the autocorrelation sequence by solving the Yule-Walker equations.
The generalized matrix version of the AR(p) model is given by the equation
Gidon Eshel provides a useful breakdown of the Yule-Walker equations that discusses their relation to between the least squares approach for fitting an AR(p) model.
Yule-Walker equations
The AR(p) model is given by the equationIt is based on parameters
where . There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which is itself obtained from the covariances). This is done using the Yule-Walker equations:where
, yielding equations. is the autocovariance function of , is the standard deviationStandard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of the input noise process, and
is the Kronecker delta function.Because the last part of the equation is non-zero only if
, the equation is usually solved by representing it as a matrix for , thus getting equationsolving all
. For havewhich allows us to solve
.The above equations (the Yule-Walker equations) provide one route to estimating the parameters of an AR(p) model, by replacing the theoretical covariances with estimated values. One way of specifying the estimated covariances is equivalent to a calculation using least squares regression of values
on the previous values of the same series.Oceanic effects
The Walker Circulations of the tropical Indian, Pacific, and Atlantic basins result in westerly surface winds in Northern Summer in the first basin and easterly winds in the second and third basins. As a result the temperature structure of the three oceans display dramatic asymmetries. The equatorial Pacific and Atlantic both have cool surface temperatures in Northern Summer in the east, while cooler surface temperatures prevail only in the western Indian Ocean. And these changes in surface temperature reflect changes in the depth of the thermocline.Changes in the Walker Circulation with time occur in conjunction with changes in surface temperature. Some of these changes are forced externally, such as the seasonal shift of the Sun into the Northern Hemisphere in summer. Other changes appear to be the result of coupled ocean-atmosphere feedback in which, for example, easterly winds cause the sea surface temperature to fall in the east, enhancing the zonal heat contrast and hence intensifying easterly winds across the basin. These anomalous easterlies induce more equatorial upwelling and raise the thermocline in the east, amplifying the initial cooling by the southerlies. This coupled ocean-atmosphere feedback was originally proposed by Bjerknes. From an oceanographic point of view, the equatorial cold tongue is caused by easterly winds. Were the earth climate symmetric about the equator, cross-equatorial wind would vanish, and the cold tongue would be much weaker and have a very different zonal structure than is observed today.
The Walker cell is indirectly related to upwelling
Upwelling
Upwelling is an oceanographic phenomenon that involves wind-driven motion of dense, cooler, and usually nutrient-rich water towards the ocean surface, replacing the warmer, usually nutrient-depleted surface water. The increased availability in upwelling regions results in high levels of primary...
off the coasts of Peru
Peru
Peru , officially the Republic of Peru , is a country in western South America. It is bordered on the north by Ecuador and Colombia, on the east by Brazil, on the southeast by Bolivia, on the south by Chile, and on the west by the Pacific Ocean....
and Ecuador
Ecuador
Ecuador , officially the Republic of Ecuador is a representative democratic republic in South America, bordered by Colombia on the north, Peru on the east and south, and by the Pacific Ocean to the west. It is one of only two countries in South America, along with Chile, that do not have a border...
. This brings nutrient
Nutrient
A nutrient is a chemical that an organism needs to live and grow or a substance used in an organism's metabolism which must be taken in from its environment. They are used to build and repair tissues, regulate body processes and are converted to and used as energy...
-rich cold water to the surface, increasing fishing stocks.
El Niño
Pressure gradient force
The pressure gradient force is not actually a 'force' but the acceleration of air due to pressure difference . It is usually responsible for accelerating a parcel of air from a high atmospheric pressure region to a low pressure region, resulting in wind...
that results from a high pressure system
High pressure area
A high-pressure area is a region where the atmospheric pressure at the surface of the planet is greater than its surrounding environment. Winds within high-pressure areas flow outward due to the higher density air near their center and friction with land...
over the eastern Pacific ocean, and a low pressure system over Indonesia
Indonesia
Indonesia , officially the Republic of Indonesia , is a country in Southeast Asia and Oceania. Indonesia is an archipelago comprising approximately 13,000 islands. It has 33 provinces with over 238 million people, and is the world's fourth most populous country. Indonesia is a republic, with an...
. When the Walker circulation weakens or reverses, an El Niño results, causing the ocean surface to be warmer than average, as upwelling of cold water occurs less or not at all. An especially strong Walker circulation causes a La Niña
La Niña
La Niña is a coupled ocean-atmosphere phenomenon that is the counterpart of El Niño as part of the broader El Niño-Southern Oscillation climate pattern. During a period of La Niña, the sea surface temperature across the equatorial Eastern Central Pacific Ocean will be lower than normal by 3–5 °C...
, resulting in cooler ocean temperatures due to increased upwelling.
A scientific study published in May 2006 in the journal Nature
Nature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...
indicates that the Walker circulation has been slowing since the mid-19th Century. The authors argue that global warming
Global warming
Global warming refers to the rising average temperature of Earth's atmosphere and oceans and its projected continuation. In the last 100 years, Earth's average surface temperature increased by about with about two thirds of the increase occurring over just the last three decades...
is a likely causative factor in the weakening of the wind pattern. However, a new study from The Twentieth Century Reanalysis Project shows that the Walker circulation has not been slowing (or increasing) from 1871-2008.
See also
- Atmospheric circulationAtmospheric circulationAtmospheric circulation is the large-scale movement of air, and the means by which thermal energy is distributed on the surface of the Earth....
- Earth's atmosphereEarth's atmosphereThe atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...