or in a horizontal plane as

In the real atmosphere this pure rotation is found only in association of closed a circulation such as low and high pressure (geopotential). Another example of pure rotation would be the path a stationary air parcel within the atmosphere would make as it circumscribes the earth due to the earth's rotation. The contrast between these two rotations is quite apparent. The first involves the rotation of an air parcel about some point within the atmosphere. The amount of rotation depends on the physical factors forcing the rotation to occur.

The second is rotation about the center of the earth. Here the amount of rotation is dependent only upon the latitudinal position of the air parcel with respect to the earth. The amount of rotation can be precisely described as

The rotation that an air parcel makes can be referenced by the size of the rotation as being either a circulation or vorticity. Circulation is a large-scale measure of rotation indicative of such features as the Hadley cell or a low-pressure system. These features being of a size great enough to be 'seen' are referred to as macroscopic measurements of rotation.

On the other hand, vorticity is a measure of rotation which cannot be 'seen' (microscopic). Although quite small in dimension, the influence of vorticity can be noticed due to the macroscopic area that it uniformly effects. Furthermore, vorticity is the building block of circulation. Because of this the individual locations of vorticity describe pure rotation. However, the sum of the vorticity over an area (circulation) is not typically a description of pure rotation.

Gradient wind

Just as a wind type was defined using a pure translation field, a wind type can also be defined using a rotational field plus the translation field. If we start with the wind in a pure translation field (the geostrophic wind) and add the effects of curvature we can describe a wind known as the gradient wind. In this case we have taken the balance between the pressure gradient force and the Coriolis force and added the centrifugal force. This can be expressed as

In addition, an alteration of the directions in which the three forces are directed will result in some interesting flow conditions.

It should be noted that since the direction of n is positive to the left of the direction of displacement, that for a radius of curvature which extends to the right of the direction of motion the curvature is considered to be negative. Or simply, positive curvature curves to the left of the direction of motion while negative curvature curves to the right of the direction of motion. Thus, low pressures exhibit positive (cyclonic) curvature and high pressures exhibit negative (anticyclonic) curvature (the opposite is true in the Southern Hemisphere).

It should be obvious that with the inclusion of curvature to describe the gradient wind results in the gradient wind being a more accurate expression of the actual wind. As a result we can summarize that although the geostrophic wind permits use to simplify the atmosphere, in regions where curvature is great the geostrophic wind estimates will be in error. The amount of error will be directly proportional to the amount of curvature. Therefore, we should expect the geostrophic values to be in greatest error in the troughs (under estimates actual wind) and in the ridges (over estimates actual wind). And while the gradient wind is a more accurate estimate it is also more difficult and time consuming to estimate.

Furthermore, we can defined another wind type using the above gradient wind definition. Since we must have a balance of forces to maintain the kinematic structure of the atmosphere, there must be a minimum of two forces acting to have an atmosphere with motion. If we were to remove the pressure gradient force from the gradient wind relationship we would be left with a balance between the centrifugal and Coriolis forces. The resulting wind type is known as the inertial wind.

Vorticity

where for simplicity only the vertical component of each will be considered. The absolute vorticity has a nice property that when considered in three dimensions and integrated over the entire globe the result is the total angular momentum of the atmosphere. And since we have previously established that angular momentum is conserved for the earth-atmosphere system, then too the total absolute momentum is conserved. However, since we are considering only the vertical component of absolute vorticity, the question is whether we can consider the vertical component alone to be conservative. If we can then the expression for the vertical component of absolute vorticity becomes a tremendously powerful tool for both kinematic and dynamic analysis of the atmosphere.

An atmosphere, which is barotropic, must by its definition exhibit a conserved vertical component of absolute vorticity. This is easily explained in that there cannot be any horizontal components of vorticity, as these will generate isobaric-isosteric solenoids due to the vertical transport of heat. And since the synoptic scale atmosphere is approximately barotropic on the average, the assumption of a constant absolute vorticity is reasonable.

Although the planetary vorticity can be stated precisely with only the knowledge of the latitude, the relative vorticity is much more difficult. To estimate the relative vorticity the distribution of the atmospheric motions along the different horizontal components of the coordinate system in use must be known.

Through the use of natural coordinates this problem can be mitigated somewhat. First, by using the definition of rotation once more, then vorticity is simply the angular motion about a central reference. In natural coordinates where the displacement direction is always 'forward', the motion along this displacement path when curvature is present will describe the vorticity of the motion. This curvature vorticity is equivalent on a weather analysis chart to the turning of the wind along a streamline, which can be expressed as

where R_s denotes a radius of curvature of a streamline. It is sometimes important to quantify the degree to which the streamline (or trajectory) curves. The quantity, known as curvature, k, is related to the radius of curvature by

where is we again are using the radius of curvature of a streamline. A similar quantity for curvature can be obtained using the radius of curvature of a trajectory. However, the curvature of a streamline and that of a trajectory will generally be different values.

In addition to this curvature vorticity, it is not uncommon to see vorticity embedded within a translational field with non-uniform lateral wind speed gradient. This shear in the 'translation' field produces both a cyclonic and anticyclonic rotation in the flow. This shear vorticity appears whenever there is change in wind speed laterally across the flow and can be expressed as

The sum of these two component vorticities can be used to express the absolute vorticity

And since the absolute vorticity is approximately constant for the synoptic scale, any variation of a term on the right hand side of the above expression must be compensated by an adjustment in one or both of the remaining terms.