San José State University Department of Economics |
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Thayer Watkins Silicon Valley & Tornado Alley U.S.A. |
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The Gradient Wind Approximation |
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The Gradient Wind Approximation is a useful construct but to understand it properly it is necessary to first review the conventions concerning such things as the radius of curvature of motion. In the diagram shown below the unit tangent vector is shown in red. It is a unit vector in the direction of the motion. The unit normal vector is defined to be a unit vector in the plane of the motion and oriented to the left of the motion. The unit normal is shown in blue in the diagram below. This convention is crucial in the definition of the sign of the radius of curvature. The radius of curvature is positive if the center of curvature of the motion is in the same direction as the unit normal.
The sign of the pressure gradient is also determined by the convention concerning the orientation of the unit normal vector. Below is shown the case for regular low and high pressure areas in the northern hemisphere. In both cases the unit normal vector points in the direction of lower pressures.
There are also what are called the anomalous cases. The designation of anomalous is clear in the case of the low pressure area in that the flow is clockwise instead of counterclockwise as is normal for a low pressure area in the northern hemisphere. The matter of whether or not there is justification for the use of the term anomalous for the high pressure area will have to be considered later.
The Southern Hemispheric cases are given below:
The momentum equations in natural (flow following) coordinates are:
where V is wind speed, ρ is density, p is pressure, s is distance along the stream-path, R is the signed radius of curvature, n is distance normal to the stream-path and f is the Coriolis parameter.
The second equation is just the quadratic equation
which has the solutions
Since f, R, ∂p/∂n can each be positive or negative and there are two choices for the sign of D^{1/2} there might seem to be 16 possible cases but because the solution V must be real and positive there are only eight cases, four for the northern hemisphere f>0 and four for the southern hemisphere f<0. In order for V to be real D must be positive. In order for D to be positive either -(R/ρ)(∂p/∂n) must be positive or (fR/2)^{2}>(R/ρ)(∂p/∂n). If -fR/2 is negative then only the positive root of D could be used and then only if -(R/ρ)(∂p/∂n) is positive.
The physically possible cases can be summarized in a table as shown below. In order to enhance the reability of the tables the following abreviations are used:
ABREVIATION | MEANING |
NH | Northern Hemisphere |
SH | Southern Hemisphere |
RL | Regular Flow Around Low Pressure Area |
RH | Regular Flow Around High Pressure Area |
AL | Anomalous Flow Around Low Pressure Area |
AH | Anomalous Flow Around High Pressure Area |
Case | Coriolis Parameter |
Signed Radius of Curvature | Pressure Gradient ∂p/∂n |
NH RL | positive | positive | negative |
NH AL | positive | negative | positive |
NH RH | positive | negative | positive |
NH AH | positive | negative | negative |
SH RL | negative | negative | positive |
SH AL | negative | positive | negative |
SH RH | negative | positive | positive |
SH AH | negative | positive | positive |
The signs of the parameters determine the signs of the separate elemnents in the quadratic formula as shown below.
Case | -fR/2 | -(R/ρ)(∂p/∂n) | root of D |
NH RL | negative | positive | positive |
NH AL | positive | positive | positive |
NH RH | positive | negative | negative |
NH AH | positive | negative | negative |
SH RL | negative | positive | positive |
SH AL | positive | positive | positive |
SH RH | positive | negative | negative |
SH AH | positive | negative | positive |
The signs of these parameters determine whether or not the positive or negative term (or both) in the quadratic formula is meaningful. There is an asymmetry between lows and highs in this matter. In the case of the regular low in either hemisphere, because the term -fR/2 is negative the positive root must be used. In the case of the anomalous low the -fR/2 is positive but because the -(R/ρ)(∂p/∂n) is positive the magnitude of the root is greater than the -fR/2 term so the positive root can be used but not the negative root. But in the case of the highs the -fR/2 is positive but the -(R/ρ)(∂p/∂n) term is negative so the magnitude of the root is less than that of the -fR/2 term so either value of the root may be used. The value of the gradient wind resulting from the use of the negative root is called the regular high case and the higher value that results from the use of the positive root is called the anomalous case. It is anomalous only because of the higher wind value. This is a different situation from the anomalous low case in which the direction of the wind is opposite to that of the regular case.
There cannot be a counterclockwise flow around a high pressure area in the norther hemisphere because that would imply R>0 and (∂p/∂n)>0 resulting in -fR/2<0 and -(R/ρ)(∂p/∂n)<0. This would mean the positive root would not be large enough to counter the -fR/2 to produce a positive gradient wind speed. The same thing applies to the possibility of a clockwise flow around a high pressure area in the southern hemisphere.
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