Cyclostrophic, antitriptic, and inertial winds

Forecasters analysing weather charts, especially upper level charts, generally assume that the wind is in geostrophic or balance (Note 12.C) or gradient wind balance (Note 12.D). Either assumption is, at best, approximately valid in mid and high latitudes. At the surface, away from mountains and coastlines, the wind is significantly weaker than geostrophic. For instance, at a station in Denmark during 1980-84, the wind measured at 10 m was, on average, 37% of the geostrophic wind, and it turned 23º to the left (1). The differences are larger at night. The wind at 10 metres over the Canadian Great Plains tends to blow in a direction about 30º anticlockwise of the gradient wind at 1,000 m, and be only half as strong, at 5 pm local time (00 UTC), and about 60º anticlockwise at just 10% of the speed at 5 am (2).

At upper levels, away from jet streaks and deep Rossby shortwaves, the actual wind normally departs no more than 10% from the geostrophic wind.

The geostrophic wind assumption results from the estimation of the magnitudes of all terms in the full equations of motion. The horizontal momentum conservation equation looks like this:

For large-scale motion the pressure gradient force and the Coriolis force are an order of magnitude larger than the other terms, hence the geostrophic balance. Other balances are conceivable though (Table 1). All of them yield a steady flow, i.e. locally the wind speed nor the wind direction change. In other words, the local acceleration is zero. The advection of horizontal momentum can be thought of, in coordinates following the flow, as a centrifugal force.

Table 1. Various force balances and types of horizontal flow.

The winds swirling rapidly in the wall of a tropical cyclone are affected by a powerful centrifugal force, opposing the steep gradient of pressure inwards. By comparison, the Coriolis force of the relevant low latitudes is negligible, beyond determining the direction of rotation early in the development of the cyclone. The balance of centrifugal and pressure-gradient forces is known as ‘cyclostrophic’ (Note 13.F). Smaller vortices, such as tornadoes and dust devils, are cyclostrophically balanced at any latitude. The reason is that the centrifugal force is proportional to the flow's curvature and the square of the wind speed, whereas the Coriolis force is linearly proportional to the wind speed.

A coastal sea breeze (Note 14.C) reaches its typical strength around noon and continues to blow at about the same speed for several hours. Under a steady onshore pressure gradient force (which is due to the temperature gradient, Note 12.E) the sea breeze should continue to gain strength. The main force opposing this pressure gradient force is surface friction. This balance is known as 'antitriptic'. Density currents, such as shallow outflows of cooler air from a mature thunderstorm, are largely antitriptically balanced.

Imagine a current of air trapped along a mountain range. Suddenly the mountain range gives way to an open plain and the airflow is left 'on its own'. In the absence of a large-scale pressure gradient, it will start to circle, and this slow rotation is known as 'inertial' motion. The Coriolis force initiates the rotation, which is then opposed by the centrifugal force. The circulation is anticyclonic, i.e. counterclockwise in the southern hemisphere. The inertial period (the time needed to complete one circle) equals 2p /f, where f is the Coriolis parameter. At 30° latitude the period is exactly 1 day, and at 60° it is about 14 hours. Inertial oscillations occur both in the atmosphere and in the oceans.

1. WASA 1998. Changing waves and storms in the northeast Atlantic? Bull. Amer. Meteor. Soc., 79, 741-60.

(2) Hare, F.K. and J.E. Hay 1976. The climates of Canada & Alaska. World Survey of Climatology (Elsevier), 11, 49-192.