Balanced flow

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In Atmospheric Science, Balanced Flow is an idealization of atmospheric motion when only selected forces acting on a parcel are balanced. Idealized, steady-state balanced flow is often an accurate approximation of the actual flow, and is useful in improving qualitative understanding of atmospheric motion.

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[edit] The Momentum Equations in Natural Coordinates

Let's consider a parcel of flow travelling along a trajectory on the horizontal plane and taken at a certain time t. The position of the parcel is defined by the distance s=s(t) it has travelled by then.

The trajectory at that position defines the tangent unit vector s (invariably pointing in the direction of growing s’s) and the normal unit vector n pointing to the local centre of curvature O (which can shift across either side of the trajectory according to the shape of it). (Such a frame of reference is shown by the red arrows in the figure.) The distance between the parcel location and the centre of curvature is the radius of curvature R at that position.

The velocity vector (V) is oriented like s and has intensity (speed) V = ds/dt. This speed, of course, is always a positive quantity, as the parcel moves along its own trajectory.

The acceleration vector of the parcel can be decomposed in the tangential acceleration (parallel to s; it changes the speed V) and in the centripetal acceleration (always towards the centre of curvature O and, therefore, along positive n; it changes the direction s of the forward displacement while the parcel moves on).

Also, the trajectory crosses a medium with spatially-varying atmospheric pressure p. (We can ignore the temporal changes of it.) The distribution of pressure is made visible through isobars that are curves of contiguous points where the pressure takes the same assigned value. (In the figure this is simplified by equally spaced straight lines.) At each location, the gradient vector of p (in symbols: grad p) points to the direction of maximum increase of p and is always normal to the isobar passing through there; also, this vector usefully visualises the effective direction which the pressure acts along. (In the figure this is drawn by the blue arrow.)

Friction always acts as a force opposing the forward motion, therefore invariably in the negative direction s with an effect to reduce the speed. For simplicity, we here assume that the frictional action adjusts to the parcel's speed proportionally through a constant coefficient K.

The horizontal momentum equations for a parcel in natural coordinates can be relatively simply expressed as follows:

\frac{DV}{Dt} = -\frac{\partial \phi}{\partial s} - K V

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

where

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.
  • - K V \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.
  • - f V \frac{}{} the coriolis acceleration.

In the following discussions, we consider steady state flow; that is, the stream lines do not change, and the tangential accelerations are also omitted (and so {DV}/{Dt} = 0 \frac{}{}). Omitting specific terms, we obtain one of the five following idealized flows: Antitriptic Flow, Cyclostrophic Flow, Geostrophic Flow, Gradient Flow, and Inertial Flow.

[edit] Antitriptic Flow

Antitriptic flow describes 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.

[edit] 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 ( - f V \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} = K V .

[edit] 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.[1]

[edit] Cyclostrophic Flow

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

[edit] Derivation

To obtain cyclostrophic balance, we neglect the frictional ( -K V \frac{}{}), and coriolis ( - f V \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}}.

[edit] 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 [2] make use of the cyclostrophic velocity equation to construct a theory for waterspouts. Winn, Hunyady, and Aulich [3] use the cyclostrophic approximation to compute the maximum tangential winds of a large tornado which passed near Allison, Texas on June 8, 1995.

[edit] 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.

[edit] Derivation

In the upper atmosphere, we can often neglect the effect of friction ( - K V \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} - f V .

Solving for V, we have

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

[edit] Application

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

[edit] 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.

[edit] Derivation

The assumptions made in deriving gradient flow are simple: we consider frictionless ( - K V = 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} - f V ,

for V yields:

V = -\frac{ f R }{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.

[edit] 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.

[edit] Inertial 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.

[edit] Derivation

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

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

which gives us

V = - f R \frac{}{}

[edit] 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.

[edit] See also

[edit] References

  1. ^ Schaefer Etling, J.; C. Doswell (1980). "The Theory and practical Application of Antitriptic Balance". Monthly Weather Review 108 (6): 746–456. doi:10.1175/1520-0493(1980)108<0746:TTAPAO>2.0.CO;2. 
  2. ^ Rennó, N.O.D.; H.B. Bluestein (2001). "A Simple Theory for Waterspouts". Journal of the Atmospheric Sciences 58 (8): 927–932. doi:10.1175/1520-0469(2001)058<0927:ASTFW>2.0.CO;2. 
  3. ^ Winn, W.P.; S.J. Hunyady G.D. Aulich (1999). "Pressure at the ground in a large tornado". Journal of Geophysical Research 104 (D18): 22,067–22,082. doi:10.1029/1999JD900387. 

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