Balanced flow

In Atmospheric Science, Balanced Flow is an idealization of atmospheric motion in which flow is considered steady-state. This occurs when the forces acting on a parcel in the direction normal to it's trajectory are balanced. Balanced flow is often an accurate approximation, and is useful in improving qualitative understanding.

The Momentum Equations in Natural Coordinates

We make two important assumptions to obtain balanced flow:

• Flow is steady state
• Vertical motions are negligible

With these assumptions in mind, the momentum equations for a parcel in natural coordinates can be relatively simply expressed as follows:

${\frac {DV}{Dt}}=-{\frac {\partial \phi }{\partial s}}-KV$

$0={\frac {V^{2}}{R}}-{\frac {\partial \phi }{\partial n}}-fV$,

where

• V is the speed of the parcel,
• t is the time,
• Φ is the geopotential,
• s is the distance along the parcel's trajectory,
• K is the coefficient of friction,
• R is the radius of curvature,
• n is the distance normal to the parcel's trajectory,
• f is the coriolis parameter, and
• D/Dt is the material derivative.

The terms can be broken down as follows:

• ${DV}/{Dt}{\frac {}{}}$ is the parcel's acceleration along its trajectory.

• $-{\partial \phi }/{\partial s}{\frac {}{}}$ is the component of the pressure gradient acceleration along the particle's trajectory.

• $-KV{\frac {}{}}$ is the acceleration due to friction.

• $-{V^{2}}/{R}{\frac {}{}}$ is the centrifugal acceleration.

• $-{\partial \phi }/{\partial n}{\frac {}{}}$ the component of the pressure gradient acceleration normal to the parcel's trajectory.

• $-fV{\frac {}{}}$ the coriolis acceleration.

By omitting specific terms, we obtain one of the five following idealized flows: Antitriptic Flow, Cyclostrophic Flow, Geostrophic Flow, Gradient Flow, and Inertial Flow.

Antitriptic Flow

Antitriptic flow describes steady state, non-accelerating flow in a straight line from high to low pressure. Antitriptic flow is probably the least used of our five idealizations, because the conditions are quite strict, as we will see.

Derivation

To obtain the equation for antitriptic flow we assume that antitriptic flow has no curvature; that is, we let the radius of curvature go to infinity, and thus the centrifugal term (${V^{2}}/{R}{\frac {}{}}$) goes to zero. We also assume that there is no pressure gradient force normal to the trajactory, or we omit ($-{\partial \phi }/{\partial n}$). We also neglect the coriolis force, eliminating ($-fV{\frac {}{}}$). Finally, we assume that the pressure gradient force normal to the parcel's trajectory and the frictional force are in perfect balance. Thus the equation for antitriptic flow is

$-{\frac {\partial \phi }{\partial s}}=KV$.

Application

Antitriptic Flow can be used to describe some boundary-layer phenomenon such as sea breezes, Ekman Pumping, and the low level jet of the Great Plains.

Cyclostrophic Flow

Some small-scale rotational flow patterns can be described as cyclostrophic. Cyclostrophic balance can be achieved in systems such as as tornatoes, dust devils and waterspouts.

Derivation

To obtain cyclostrophic balance, we neglect the frictional ($-KV{\frac {}{}}$), and coriolis ($-fV{\frac {}{}}$), and tangential pressure gradient ($-{\partial \phi }/{\partial s}{\frac {}{}}$) terms. Our equations of motion then reduce to

$0=-{\frac {V^{2}}{R}}-{\frac {\partial \phi }{\partial n}}$,

which implies

$V={\sqrt {-R{\frac {\partial \phi }{\partial n}}}}$.

Application

Since cyclostrophic flow ignores the coriolis effect, it is restricted to use in lower latitudes or on smaller scales. Cyclostrophic flow is frequently used in the study of small scale, intense vortices, such as dust devils, tornadoes, and waterspouts. Rennó and Buestein make use of the cyclostrophic velocity equation to construct a theory for waterspouts. Winn, Hunyady, and Aulich use the cyclostrophic approximation to compute the maximum tangential winds of a large tornado which passed near Allison, Texas on June 8, 1995.

Geostrophic Flow

Main article: Geostrophic wind

Geostrophic flow describes straight-line flow parallel to geopotential height contours. This occurs frequently in earth's upper atmosphere, and modelers, theoreticians, and operational forecasters frequently make use of geostrophic and quasi-geostrophic theory.

Derivation

In the upper atmosphere, we can often neglect the effect of friction ($-KV{\frac {}{}}$) and of curvature ($-V^{2}/R{\frac {}{}}$). Assuming steady state flow parallel to geopotential height contours ($-\partial \phi /\partial s=0$), we are left with

$0=-{\frac {\partial \phi }{\partial n}}-fV$.

Solving for V, we have

$V=-{\frac {1}{f}}{\frac {\partial \phi }{\partial n}}$.

Application

See the article on Geostrophic wind for a description of applications.

Gradient Flow

Geostrophic flow is generally a fair assumption in the upper atmosphere. However, it is purely a straight-line flow. However, looking at a 500mb geopotential height map, one will notice that the geopotential height lines are rarely, if ever, straight. Gradient flow accounts for curvature in height-parallel flow, and is generally a more accurate approximation than geostrophic flow. However, mathematically gradient flow is more complex, and geostrophic flow is farily accurate, so the gradient approximation is not as frequently mentioned.

Derivation

The assumptions made in deriving gradient flow are simple: we consider frictionless ($-KV=0{\frac {}{}}$) flow parallel to geopotential height contours ($-\partial \phi /\partial s=0$). Solving the remaining momentum equation,

$0=-{\frac {V^{2}}{R}}-{\frac {\partial \phi }{\partial n}}-fV$,

for V yields:

$V=-{\frac {fR}{2}}\pm \left({\frac {f^{2}R^{2}}{4}}-R{\frac {\partial \phi }{\partial n}}\right)^{1/2}$

Not all solutions of the gradient wind equation yield physically plausible results. For regular cyclonic and anticyclonic rotation, it can be shown that the geostrophic wind equation underestimates the actual wind in cyclonic rotation, and overestimates the actual wind in anticyclonic rotation.

Application

Gradient wind is useful in studying atmospheric flow around high and low pressures centers, particularly where the radius of curvature of the flow about the pressure centers is small, and geostrophic flow no longer applies with a useful degree of accuracy.

Ineritial Flow

Although rarely observed in the atmosphere, we can consider rotational flow that is sustained with no pressure gradient present. This type of flow is called inertial flow.

Derivation

In omitting the pressure gradient acceleration ($-\partial \phi /\partial n{\frac {}{}}$ and $-\partial \phi /\partial s{\frac {}{}}$) and frictional ($-KV{\frac {}{}}$) terms, we are left with

$-{\frac {V^{2}}{R}}-fV=0$,

which gives us

$V=-fR{\frac {}{}}$

Application

Since atmospheric motion is due largely to pressure differences, inertial flow is not very applicable in atmospheric dynamics. However, inertial flows are often observed in the ocean, where flows are driven less by pressure differences than by surface winds.

Further Reading

• Holton, James R.: An Introduction to Dynamic Meteorology, 2004. ISBN 0-12-354015-1

External Links

• Plymouth State Weather Center Balanced Flows Tutorial