GRADIENT WIND AND CYCLOSTROPHIC WIND

Looking back at the geostrophic approximation in s, n, z coordinate:

there are several properties of the geostrophic wind:

1. , so the geostrophic wind speed will not change with time.

2. , there is no pressure gradient along the direction of the geostrophic wind. This will force the geostrophic wind to go along the isobars on a constant z surface or along the height contours on a constant p surface.

3. , and therefore , so the geostrophic wind direction will not change with time.

4. , and therefore, (because the definition of the Natural coordinate implies that ), so the low pressure is to the left hand side of the geostrophic wind.

These properties of the geostrophic wind dictates that the geostrophic wind approximation can only be valid in a set of straight isobars or height contours. This is hardly the case on the weather map. A better approximation of the wind in a curved isobars of the height contours is the gradient wind.

Gradient wind

Definition: Tangential acceleration is zero.

As in the geostrophic wind, there will be no change of gradient wind speed with time; there is no pressure gradient along the direction of the gradient wind; and the gradient wind will go along the isobars on a constant z surface or along the height contours on a constant p surface. But the pressure gradient force and the Coriolis force are not balanced, leaving a net centripetal force to curve the wind along curved isobars and height contours. This is much closer to the reality of the pressure pattern and wind flow on the weather maps. The last equation can be used to solve for the gradient wind speed, as show later in this section.

Comparison of geostrophic wind speed and the gradient wind speed

Using the geostrophic wind equation

The gradient wind equation can be rewritten as

Rewrite this equation as

it can easily see that, for cyclonic flow (R>0),

.

For anticyclonic flow (R<0),

.

Since the gradient wind is a better approximation of the real wind, it means that the wind speed is usually subgeostrophic around a low and supergeostrophic around a high. This statement may give an erroneous impression that the wind speed around a low is weaker and around a high is stronger. In reality, the pressure gradient (and hence the geostrophic wind speed) around a low is much larger than the pressure gradient around a high.

Solution of the gradient wind speed

Rewrite the gradient wind equation as:

The gradient wind can be solved as:

For real solution, the quantity in the square root has to be larger or equal to zero. Since , the quantity in the square root is always larger or equal to zero for cyclonic flow (R>0). There is no limit of the allowable value of R. For anticyclonic flow (R<0)

or

This equations states that, for anticyclonic flow, the pressure gradient has to approach zero at the center of the high where |R| approaches zero. Another way to state it is that the curvature of the isobars or height contours in a high has to be small (|R| has to be large). Either way implies weak wind speed in the high pressure center.

Cyclostrophic wind

Definition: Tangential acceleration is zero and the Coriolis force is negligible.

The cyclostrophic wind can exist only around a low pressure and the centripetal force needed to change the wind direction is provided by the pressure gradient force. The cyclostrophic wind speed is computed as:

This is a good approximations for small scale rotations such as dust devils and, sometimes, tornadoes, in which the source of rotation is not produced by the Coriolis force. These rotations can be either clockwise or counter clockwise, as shown in the figures below.