Balanced Atmospheric Flows Tutorial
Flows or winds in the Earth's atmosphere are determined by five particular forces:
gravity, pressure gradient, coriolis, friction, and the centrifugal force. These forces
are all accounted for in one equation of motion; the momentum equation. Trying to
determine the direct effects of all these forces on the wind at any given moment
would prove to be a daunting task. Therefore, it is much easier to comprehend the
structure of the wind when certain forces are in balance with one another. Hence, a
balanced flow is a wind that involves a balance between two or more of the forces
described in the momentum equation.
There are some assumtions that are taken into consideration when applying
balanced flows. When these flows are analysed, they are assumed to be steady state
(time independant). Another assumption is that there is no vertical component to
the flow. These assumptions are made so that the balanced flows can be more
easily applied to certain applications such as weather maps. When all of these
assumptions are taken into account, there are five different balanced flows that
become apparent. These are Antitriptic, Cyclostrophic, Geostrophic, Gradient, and
Inertial flow.
A large portion of weather maps used today are in isobaric coordinates. Since the
momentum equation is typically seen in cartesian coordinates, it is necessary to
change it into isobaric coordinates. Furthermore, when applying the momentum
equation to the weather map, it is better to use a coordinate system for the
momentum equation that is defined along the flow field. This coordinate system is
called the natural coordinate system. The combination of these two systems allows
for the application of the momentum equation to the weather maps. For example,
changing the momentum equation into isobaric coordinates has the effect of
changing the pressure gradient term into a more easily applicable term with height
as the variable, rather than the pressure.
The momentum equation in isobaric natural coordinates is
where:
s is the coordinate defined as
n is the coordinate defined as
positive in the direction of the
flow field to the right of the n
axis
g is the acceleration of gravity
at the earth's surface
Ks is the friction term
f is the coriolis parameter
t is time
perpendicular and pointing
towards the left of the positive s
axis
z is geopotential height
rT is the radius of curvature
V is the wind speed
The six terms in the isobaric natural coordinate momentum equation are the
following:
is the tangential acceleration term
is the pressure gradient term in the s equation of motion
is the friction term
is the centrifugal acceleration term
is the pressure gradient term in the n equation of motion
is the coriolis term
Antitriptic flow is the only balanced flow in the atmosphere that is defined using
the s equation of motion, since there cannot be any tangential acceleration to the
flow. The following forces are in balance in antitriptic flow:
The pressure gradient:
The frictional force:
Antitripic flow is the least common of all the balanced flows in the atmosphere,
because of the balance of forces that are involved. By the definition of the forces
balanced, the flow must be purely cross-contour with frictional effects balancing
the cross-contour flow. In addition to that, there must be no coriolis, no pressure
gradient with respect to the n axis, no centrifugal force, and no tangential
acceleration. This means that the flow has to be either small in scale or near the
equator to neglect the effects of the coriolis. The curvature has to be equal to zero
as well. Since the likelihood of all these conditions being met is not very real, the
possibility of observing true antitriptic flow is minimal. The following graphic
illustrates the force balance that is required for antitriptic flow, showing how
unlikely it is:
Although the reality of true antitriptic flow is limited, the atmosphere does attempt
to simulate it. One atmospheric scenario that supports simulated antitriptic flow is a
valley wind. With mountains on either side of the valley, the effects of the coriolis
force turning the wind to the right are minimized. Therefore, the flow will flow
directly from high to low pressure in the valley, simulating antitriptic flow.
The Cyclostrophic Wind Equation
Cyclostrophic flow is a fairly common flow in the atmosphere. This balance
involves two forces in the n equation of motion, those being the following:
The centrifugal force:
The pressure gradient force:
With the balance between the pressure gradient force in the n equation of motion
and the centrifugal force, this constricts the possible types of flow to two types. The
flow can be either cyclonic or anti-cyclonic with a circular motion as a result of the
centrifugal force. However, the pressure gradient force always points inward,
making the center of circulation an area of low pressure. Since only two forces are
considered, there are certain assumptions the also have to be made. The flow must
be frictionless, always parallel to the height contours, and the scale of the flow is
either small in scale or near the equator, where the coriolis force is essentially zero.
The following picture illustrates cyclostrophic flow:
rT > 0
<0
rT < 0
>0
When we assume that a flow is cyclostrophic in nature, the coriolis force is defined
as being zero. Therefore, a method must exist to determine if the coriolis force can
be neglected. If the ratio of the centrifugal force to the coriolis force is large, then
the cyclostrophic assumption can be made. This ratio, called the Rossby number is
defined below:
where:
Ro is the Rossby number
There are some real world applications to cyclostrophic flow. Small scale
circulations, such as tornados, waterspouts, and dust devils are small enough so that
the coriolis force can be neglected. However, tornados typically have a cyclonic
circulation associated with them, because the rotation of the mesocyclone that
spawns the tornado is cyclonic. However, waterspouts and dust devils are not as
constrained to the weather systems that spawn them. Both of these circulations
have been observed to be both cyclonic and anti-cyclonic.
The Geostrophic Wind Equation
Geostrophic flow in the atmosphere is the most applied balanced flow among the
five that are defined. Specifically, the geostrophic assumption assumes that there is
a balance between the following forces in the n equation of motion:
The coriolis force:
The pressure gradient force:
A large portion of meteorological forecast computer models use the geostrophic
assumption when forecasting wind derived products in the upper atmosphere, such
as vorticity and divergence. The geostrophic assumption is good to use in areas
where the flow is straight and frictionless, such as in the upper atmosphere. This
assumption does not work when the amount of curvature is large in the upper
atmosphere, since curvature will allow the centrifugal force in the momentum
equation to have a more substantial magnitude. Although, the Gradient wind
equation does correct for curvature in the flow by incorporating all of the terms in
the n equation of motion. Although the geostrophic wind is not a perfect
assumption to make, it does save computational time and simplifies some of the
complicated dynamical equations.
The following picture shows how the forces balance out in geostrophic flow at
500mb in the northern hemisphere. Imagine the flow is moving from west to east,
thereby not changing latitude.
Notice that the flow is parallel to the lines of constant geopotential height, as they
should be by the definition of the geostrophic wind equation. To reiterate, since
there is no friction, there is no cross-contour flow, and therefore, no acceleration of
the flow.
The next picture will show geostrophic flow at 500mb that is moving from south to
north.
Notice that the flow decelerates with time as the flow moves from south to
north. This is because of the increase in the magnitude of the coriolis force
as air parcels move towards the north. At first, this deceleration of the flow
appears to violate the steady state condition of balanced flows.
Fortunately, this is not the case. If the total tangential acceleration term is
expanded, the answer becomes apparent:
where:
=0
=0
= --
=0
= ++
Although there is horizontal deceleration occurring, it is being counteracted by the
acceleration of the vertical component of the wind. This implies that, in fact, the
total tangential acceleration is zero, satisfying the steady state condition of balanced
flows.
The Gradient Wind Equation
where:
is the curvature term, being equal to
The gradient wind equation is a representation of the entire n equation of motion.
There is a balance between all of the forces present in the n equation, those being
the following:
The centrifugal force:
The pressure gradient force:
The coriolis
force:
The gradient wind is very much like the geostrophic wind, in that it is a frictionless
wind which allows for flow that is parallel to the height contours. The one
difference between the geostrophic wind and the gradient wind is that the gradient
wind includes the centrifugal force, thereby allowing curvature in the flow field. At
first glance, this may seem to be simply an addition to the geostrophic wind
equation, since the two flows are very similar to one another. For example, if the
curvature is equal to zero, then geostrophic flow is obviously the result. However,
the addition of the centrifugal force complicates the equation tremendously and
creates several different solutions to the gradient wind equation.
In order to understand the possibilities, it is necessary to study the gradient wind
equation more in depth. In the denominator of the gradient wind equation, there is a
positive root and a negative root. Mathematically, there are only four possible
solutions that can be computed. The table below illustrates these four possibilites:
Gradient Wind Possibilities
Sign of root
Positive Root
Negative Root
Sign of
Positive
Negative
Positive
Negative
Positive
Not Physical
Normal Low
Not Physical
Not Physical
Negative
Anomalous Low
Anomalous High
Not Physical
Normal High
This table shows that the four possible physical solutions to the flow are gradient
flows around normal high and low pressure systems, and anomalous high and low
pressure systems. Normal, meaning that the flow around these systems is anticyclonic for high pressure systems and cyclonic for low pressure systems.
Anomalous low pressure means that the flow is anti-cyclonic around the system.
Yet, with anomalous high pressure, the flow is still anti-cyclonic, but the wind
speed is much too high to represent a true high pressure system. The following
illustrates each of these four possible gradient flows:
Normal Low Pressure System
Normal High Pressure System
Anomalous Low Pressure System
Anomalous High Pressure System
In the cases of the normal high and normal low, the flow is baric, meaning that the
coriolis force and the pressure gradient force are in opposite directions of one
another. The same is true for the anomalous high pressure system. However, the
flow around the anomalous low pressure system is anti-baric and anti-cyclonic.
Because of these types of flow around the anomalous low pressure system, the
geostrophic velocity is in the opposite direction of the gradient wind velocity.
Hence, the anomalous low pressure system is not much of a reality.
When comparing gradient wind speeds with geostrophic wind speeds for the same
pressure gradient force, there are some differences. Gradient flow around a high
pressure system will be faster than the geostrophic flow if the pressure gradient
force is constant. The opposite is true when considering low pressure systems. In
this case, the gradient wind around the low pressure system is less than the
geostrophic wind if the pressure gradient
force is constant. The formula below
defines a ratio of geostrophic flow to
gradient flow:
where:
= Geostrophic wind speed
= Gradient wind speed
Thus, if the value of the ratio is greater than one, then the flow is cyclonic. If the
value is less than one, the flow is anti-cyclonic.
The Inertial Wind Equation
Inertial flow is not one of the more commonly seen flows in the atmosphere, yet it
does exist. The inertial wind is derived from the balance of the following forces in
the n equation of motion:
The centrifugal force:
The coriolis force:
Inertial flows are also known as inertial oscillations, since air parcels under the
influence of inertial balance follow circular paths in an anti-cyclonic manner. The
following graphic shows the illustration of the forces involved in inertial flow:
After examining the graphic and applying the inertial flow equation, it becomes
apparent that these oscillations are very regular and rotate with a certain period.
The period of these oscillations is:
where:
is the latitude and P is the period of the oscillation
As stated above, inertial flow is not one of the more commonly seen forces in the
atmosphere. The reason is that the pressure gradient force drives most flows in the
atmosphere. Since the pressure gradient force is assumed to be zero in inertial flow,
this line of reasoning is justified. The only situations where inertial flow may be
observed are in the center of large anti-cyclones or cyclones, where pressure
gradients are very weak.
Although atmospheric conditions may not be favorable at most times for inertial
flow, it is more prevalent in oceanic currents, where transient winds blowing across
the surface are more likely to drive the current, as opposed to internal pressure
gradients.
Greetings! My name is Ryan Turkington and I am currently a Senior at Plymouth
State College in Plymouth, New Hampshire. On May 17, 1997, I expect to graduate
with a Bachelor of Science degree in Meteorology, with a minor in Technical
Mathematics. I was born in Portland, Maine on August 4, 1974 and have lived in
Windham, Maine for my entire life (with the exception of being a temporary
resident of Plymouth on and off for the last four years). Currently, I am trying to
pursue a career in writing computer applications to better assist operational
meteorologists. Hopefully, I will find a job somewhere in New England that will
fulfill my interests.
My original motivation for pursuing a Meteorological career were my interests in
summer thunderstorms and 'noreasters that have frequented southern Maine
throughout the years. My interests started when I was four years old and has never
abated since. Once I started attending Plymouth State College, my interests in
Meteorology became more broad. I now enjoy Numerical Weather Prediction,
Mesoscale Meteorology, and Synoptic Meteorology. Other interests that I have are
writing computer applications, hiking, and spending time with my close friends.
The main reason that I wrote this tutorial was because I did not completely
understand the concept of balanced flows in the atmosphere. When I first took
Dynamic and Synoptic Meteorology, I was exposed to these concepts, yet I never
really absorbed the material. I thought that if I wrote a web tutorial to assist people
with this difficult concept, that they would better comprehend the material and that
I would come to understand them as well. Fortunately, I now comprehend the
concept of balanced flows. I only hope that this tutorial is beneficial for you as
well.