Relationship Between the Geostrophic Wind and the Real Wind: A Context

 Can the "real" wind or ocean current be viewed as geostrophic?

A. What Does The Equation of Motion Tell Us?

The frictionless equation of motion in concept form says:

Total Acceleration Experienced by (Air/Water) Parcel DUE TO the Pressure Gradient Acceleration and Coriolis Acceleration

 

 

The upper air charts we've been looking at are constant pressure charts. For example, the 500 mb chart is a chart upon which all pressures are 500 mb. Clearly, you can not express a pressure gradient on a map upon which all the pressures are constant. But we can perform a mathematical trick (you won't in this class, but in more advanced classes) to transform the horizontal pressure gradient acceleration to its equivalent to be used on a constant pressure map. The pressure gradient acceleration is kind of "ugly" because it has density in it (see the horizontal pressure gradient acceleration in natural coordinates shown below)*. It turns out that making the mathematical transformation eliminates density (which can't be measured directly anyway) into a simpler expression that only has the gradients (differences) in heights (determined by radiosonde) of the given pressure level along the horizontal coordinate axes. The version below is the Equation of Motion in natural coordinates (without the friction acceleration)

 

* The horizontal pressure gradient acceleration can be expressed in terms of a height gradient (on an upper level chart) or a pressure gradient, as here PGA

 

 

Equation of Motion
(1)*

 

In the absence of friction, in the real atmosphere or ocean, as will be demonstrated in class, a fluid parcel initially at rest experiencing these accelerations would begin to move, but eventually its velocity would no longer change because the accelerations balance.

In that restrictive circumstance, the fluid parcel no longer accelerates and tthe left hand side of the expression is zero. The equation can be solved for the "special case" wind velocity, the "geostrophic wind" velocity, which will be entirely due to the balance of the pressure gradient normal to the flow and Coriolis. Making these substitutions into (1) and solving for V gives:

Geostrophic Wind Equation (2)

and the subscript "g" for the wind tells us that this is an approximation related to the geostrophic assumption, explained above. Thus, in all cases in which the total acceleration is small compared to the Coriolis acceleration, for a given pressure gradient, the wind will be nearly geostrophic.

Equation (2) simply states that when the balance between pressure gradient acceleration and Coriolis acceleration occurs, say, at 500 mb, since the air parcel will no longer be accelerated, the wind speed should be simply determined by gravity, the Coriolis parameter (both of which are constants at a given latitude) and the variation of heights NORMAL (along the n-axis) to the flow.

Geostrophic Wind -- The wind that flows parallel to height contours or isobars resulting from an exact balance between the pressure gradient acceleration and the Coriolis acceleration.

Let's see to what extent we can use the "geostrophic" wind idea to explain the real wind we see at the various scales we've already defined.

B. Is the Geostrophic Wind a Real Wind?

For the geostrophic flow concept to work, the wind must not be changing speed (is unaccelerated or the acceleration is almost zero). We can measure the tendency for wind to be accelerated at the various scales of circulation we discussed. Meanwhile, of course, the Coriolis acceleration is only related to the speed of the object and its latitude. Thus, the flow (both in the ocean and atmosphere) approaches geostrophic the smaller the actual acceleration is relative to the Coriolis acceleration.

Obviously, if the total acceleration is small compared to the Coriolis acceleration, the ratio of the former to the latter will yield a very small number. This ratio is called the Rossby Number, and can be used to decide to what extent the "geostrophic wind" corresponds to the real wind, as a function of scale. Ratios of 0.1 or less indicates that the total accleration is one to two orders of magnitude smaller than the Coriolis acceleration and, thus, the total acceleration can be dropped on an order of magnitude basis.

Values approaching 1 or more indicate that the total acceleration experienced by an air parcel is very large, and cannot be dropped out of the equation. In those cases, using the idea of the geostrophic wind to explain what you see on a weather map will produce confusion. In those cases, it will appear that air is moving at right angles to the isobars from higher values of pressure to lower values of pressure.

In Middle Latitudes

Scale of Circulation
(Distance Scale; no Friction)

Ratio of Total Acceleration to Coriolis Acceleration

Real Wind Generally Explained by Geostrophic Wind?

10,000 km

0.01

Yes

1,000 km

0.1

Yes

100 km

1.0

Not really

10 km

3.0

No

1 km

4.0

No

Another way to look at this is that real wind is always made up of a portion that is geostrophic and a portion that is not geostrophic (called AGEOSTROPHIC). Here's a simple algebraic expression that summarizes this concept. In this case, the symbol, V, indicates the real (two dimensional) horizontal wind in natural coordinates.

V = Vgeo + Vageo

 

(3)

It turns out that even at the synoptic and macroscales, there are situations in which the total acceleration (ageostrophic wind) is significant. These situations generally occur in certain portions, or levels, of the troposphere. Also, there is a portion of the troposphere, or level, in which the total acceleration (ageostrophic wind) is small. There the wind will appear to be flowing parallel to the isobars (or height contours) with a magnitude exactly determined by the pressure gradient. The table below shows the extent to which a student may visualize the wind as geostrophic at various levels of the atmosphere, because the real wind can always be viewed as an addition of the geostrophic wind concept and the extent to which the wind is not geostrophi. The far right hand column contains information we will use in a future lecture after the notion of Divergence/Convergence is discussed in class. Finally, there is a portion of the atmosphere in which the notion of the Rossby Number makes no sense because there is no Coriolis Effect, namely, at the Equator. In that case, all the wind is ageostrophic, which means NONE of the wind is moving parallel to the isobars, and the original concept that air motion is from high to low pressure directly across isobars works.

Level of the Troposphere

Vgeo

Vageo

Real Wind "Looks" Like Geostrophic Wind?

Actual Accelerations

Divergence

Upper (300-200 mb level)

Large

Large

Somewhat

Large

"Large"

Middle (600 to 450 mb)

Large

None to small

Yes

None to small

None to small

Lower (Sfc to 925 mb)

Large

Large

Somewhat

Large

"Large "

Thus, students who embrace the concept of the geostrophic approximation "love" the 500 mb chart, because there the wind appears to flow parallel to the contours and in direct proportion to the pressure (or height) gradient. That's the level at which the geostrophic approximation corresponds most closely to the real wind.

This is not the case on 200 and 300 mb charts. It's true that even the largest ageostrophic motions there are relatively small compared to the geostrophic motion. So, at first glance it will appear that the wind is in geostrophic balance. A closer look will reveal areas of cross contour flow, and, also, areas in which the wind speeds do not match what one would expect from solution of the geostrophic wind relation.

Finally, at the ground, there is also substantial cross contour flow. This "disruption" of the geostrophic wind balance occurs chiefly because of friction. That's why the surface wind will most closely resemble the geostrophic wind in regions in which frictional effects are minimal (e.g., over oceans, over flat featureless plains).