Geostrophic balance

 

We now consider the effect of the Coriolis force on atmospheric flow. Above heights of about 1 km, the frictional effects of the surface can be neglected. If the flow is driven by pressure differences (high to low pressure) but is steady and non-accelerating, which also means that the flow is in the straight line, a balance should exist between the pressure gradient force and the Coriolis force. This is called geostrophic balance, and the wind in this state is called the geostrophic wind .

 

We can see how this would come about by considering the hypothetical situation in which a parcel of air starts from rest in a “frozen” atmosphere with an applied pressure field.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here, P is the pressure gradient force  acting perpendicular to the isobars and towards lower pressure. At the origin (point O) the parcel starts from rest and moves towards lower pressure under the influence of the pressure gradient. As soon as the parcel acquires a velocity, a Coriolis force acts to deviate it to the right. This continues until a state of balance exists when the pressure gradient force P and the Coriolis force C are equal in magnitude and directly opposed one another. Inspection of the diagram shows that this can only be the case when the wind is directed along the isobars. 

 

Buys-Ballot’s law relates the direction of the geostrophic wind to the pressure pattern: “with your back to the wind, low pressure is to your left”. In the southern hemisphere, the rule must be reversed, and low pressure is on the right.

 

The balance between the pressure gradient and the Coriolis force can be expressed quantitatively as

 

 

and, rearranging

 

Recall that the distance n is measured perpendicular to the isobars and in the direction of decreasing pressure such that  is, by definition, a positive quantity.

The above expression can be used to calculate  based on the spacing of the isobars on a surface weather map.

 

 

Constant pressure surfaces

 

Upper air weather charts present the pressure field as the height of a constant pressure surface rather than as the pressure at a fixed height, such as mean sea level. However, there are equivalent ways of presenting the same information: high pressure areas relate to high heights, and low pressure areas relate to low heights of a constant pressure surface. For example, consider a cross-section showing the slope of the 500 mb surface.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

There are several advantages to display upper air charts in this manner, one of which we will see shortly.

 

Using the hydrostatic equation, which expressed the rate of decrease of pressure with height, we can show that

 

 

 

 

and geostrophic balance can be written in a form appropriate to upper air charts as

 

 

 

and      

 

 

Which has the advantage that it does not include air density, a quantity that is not routinely measured. In performing a calculation, dz/dn is approximated by  where  is the height interval between adjacent contour lines (60m, say), and  is the geographical distance between contour lines, and measured perpendicularly between the lines.

 

 

 

 

 

 

 

 

 

 

 

 

 

Conveniently, latitude lines can be used as a map scale, knowing that 1o latitude = 111km. Care must be taken, however, since the scale changes across the map because of the particular kind of projection used.

 

Just as the geostrophic wind at the surface blows such that low pressures are on the left, the geostrophic wind blows on upper air maps such that low height is on the left. The wind is greatest where the contours are closely packed, and weakest when contour lines are far apart.

 

 

The effect of curvature in the flow

 

Any brief examination of surface or upper air weather maps will reveal that atmosphere flows are rarely in a straight line but, rather, they contain substantial curvature around highs, low, ridges and troughs

 

Any object or mass of atmosphere that is constrained to a curved path is undergoing an acceleration towards the center of that rotation. This acceleration is called centripetal acceleration and is the consequence of a force, or resultant of two or more forces, that is called the centripetal force. (Note: a centrifugal force is an apparent force that tends to fling an object outwards from a constrained curved path. In the treatment of rotation, here, we will only consider the real forces acting inwards to produce a centripetal acceleration).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The force F towards the center, the centripetal force, maintains an acceleration that keeps the object in a curved path. This acceleration must be a function only of the radius of curvature r and either the linear velocity v or the angular velocity .

 

Mathematically, we can show that

 

Centripetal acceleration =

 

Note: it is always a good plan to check the dimensions or units to verify the correctness of expression such as these. Alternatively, if you forget the form of the expression, you could use dimensional arguments to arrive at the correct way to put the variables together;  and  are the only combinations that have the proper units of acceleration.

 

 

Cyclostrophic flow

 

In some atmospheric circulations, notably hurricanes, tornadoes, water spouts, and dust devils, a pressure gradient acts as the centripetal force towards the center of rotation.

 

 

 

 

 

 

 

 

 

 

 

 

Equating the force (per unit mass) to the centripetal acceleration, we obtain an equation describing the motion

 

         

 

This type of flow is called cyclostrophic and is not uncommon in the atmosphere as indicated by the examples above. The Coriolis force can be neglected because, for very large wind speeds, the centripetal acceleration, being proportional to the square of the speed while the Coriolis force is proportional to the first power, is so much larger. The fact that hurricanes form at low latitudes (between 5 and 20 degrees, north or south latitude) also contributes to the neglect of the Coriolis force.

 

 

The gradient wind

 

In general, we must consider that both the pressure gradient and the Coriolis force are important and that an imbalance between the two results in a centripetal acceleration. A calculation of the wind speed based on both forces and on the centripetal acceleration is called the gradient wind. Consider the following pressure pattern and resulting flow field:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For cyclonic flow, around a low or trough, there is a net force inwards to produce the centripetal acceleration. We see that the Coriolis force must be smaller than the pressure gradient force. Therefore the wind (the gradient wind) is less than the geostrophic value () because C is proportional to v.

 

On the other hand, around a high or ridge the Coriolis force must exceed the pressure gradient. This is equivalent to saying that the wind speed must exceed the geostrophic value (). Thus, flow is subgeostrophic around a low or trough, and supergeostrophic around a high or ridge.

 

The gradient wind  can be expressed quantitatively:

 

         

 

where r, the radius of curvature, is taken to be positive for cyclonic flow, when P > C.

 

The pressure gradient can be expressed in terms of the geostrophic wind:

 

                       

while the Coriolis force

         

 

Solving the quadratic: (actually, divide through by  and find solution to quadratic in 1/)

 

         

 

when r > 0 (cyclonic)       subgeostrophic

when r < 0 (anticyclonic)   supergeostrophic

 

 

Does this mean that the wind speed is greater in the vicinity of a high pressure than near a low? Generally no, because usually the pressure gradient near a low is much greater. In fact, there is a theoretical upper limit to the pressure gradient near a high, while there is no such limit around a low. It is relatively easy to show this upper limit by examining an equation formulated for the gradient wind (how to prove it?).

 

 

The effect of friction

 

Near the earth’s surface, and extending to a height of about 1 km, frictional drag retards atmospheric motion. Surface winds are virtually always less than the geostrophic value based on the pressure gradient. Exceptions would be when the air is funneled through channels or when cold air drains down valleys.

 

In addition, the direction of the wind is altered so that it crosses the isobars from high to low pressure. The following diagram will explain this:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The effect of friction (force F) is to slow the wind, which in turn reduces the Coriolis force. The pressure gradient is then able to force the wind towards lower pressure. A balance can exist again when the wind is turned inwards and the combination of friction and Coriolis balance the pressure gradient, as shown in the final diagram.

 

This cross-isobaric angle is typically 20 to 30o but is quite dependent on the roughness of the surface (greater angles with rougher surfaces) and on the stability of the lower layers of the atmosphere (greater angles when the air is stable such as at night and surface layer air is decoupled from the air aloft).

 

Surface wind patterns typically show air spiraling out from the center of a high pressure and into a low pressure.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Such patterns of convergence and divergence can only be maintained if the air is ascending over the low and subsiding over the high. Otherwise, the low would fill and disappear, and the loss of mass from the high would cause it to subside. Later, we will see that upper air flows create these vertical motions to support lows and highs at the surface.