Balanced Flow - Balanced-flow Speeds Compared

Balanced-flow Speeds Compared

Each balanced-flow idealisation gives a different estimate for the wind speed in the same conditions. Here we focus on the schematisations valid in the upper atmosphere.

Firstly, imagine that a sample parcel of air flows 500 meters above the sea surface, so that frictional effects are already negligible. The density of (dry) air at 500 meter above the mean sea level is 1.167 kg/m3 according to its equation of state.

Secondly, let the pressure force driving the flow be measured by a rate of change taken as 1hPa/100 km (an average value). Recall that it is not the value of the pressure to be important, but the slope with which it changes across the trajectory. This slope applies equally well to the spacing of straight isobars (geostrophic flow) or of curved isobars (cyclostrophic and gradient flows).

Thirdly, let the parcel travel at a latitude of 45 degrees, either in the southern or northern hemisphere—so the Coriolis force is at play with a Coriolis parameter of 0.000115 Hz.

The balance-flow speeds also changes with the radius of curvature R of the trajectory/isobar. In case of circular isobars, like in schematic cyclones and anticyclones, the radius of curvature is also the distance from the pressure low and high respectively.

Taking two of such distances R as 100 km and 300 km, the speeds are (in m/s)

The chart shows how the different speeds change in the conditions chosen above and with increasing radius of curvature.

The geostrophic speed (pink line) does not depend on curvature at all, and it appears as a horizontal line. However, the cyclonic and anticyclonic gradient speeds approach it as the radius of curvature becomes indefinitely large—geostrophic balance is indeed the limiting case of gradient flow for vanishing centripetal acceleration (that is, for pressure and Coriolis force exactly balancing out).

The cyclostrophic speed (black line) increases from zero and its rate of growth with R is less than linear. In reality an unbounded speed growth is impossible because the conditions supporting the flow change at some distance. Also recall that the cyclostrophic conditions apply to small-scale processes, so extrapolation to higher radii is physically meaningless.

The inertial speed (green line), which is independent of the pressure gradient that we chose, increases linearly from zero and it soon becomes much larger than any other.

The gradient speed comes with two curves valid for the speeds around a pressure low (blue) and a pressure high (red). The wind speed in cyclonic circulation grows from zero as the radius increases and is always less than the goestrophic estimate.

In the anticyclonic-circulation example, there is no wind within the distance of 260 km (point R*) -- this is the area of no/low winds around a pressure high. At that distance the first anticyclonic wind has the same speed as the cyclostrophic winds (point Q), and half of that of the inertial wind (point P). Farther away from point R*, the anticyclonic wind slows down and approaches the geostrophic value with decreasingly larger speeds.

There is also another noteworthy point in the curve, labelled as S, where inertial, cyclostrophic and geostrophic speeds are equal. The radius at S is always a fourth of R*, that is 65 km here.

Some limitations of the schematisations become also apparent. For example, as the radius of curvature increases along a meridian, the corresponding change of latitude implies different values of the Coriolis parameter and, in turn, force. Conversely, the Coriolis force stays the same if the radius is along a parallel. So, in the case of circular flow, it is unlikely that the speed of the parcel does not change in time around the full circle, because the air parcel will feel the different intensity of the Coriolis force as it travels across different latitudes. Additionally, the pressure fields quite rarely take the shape of neat circular isobars that keep the same spacing all around the circle. Also, important differences of density occur in the horizontal plan as well, for example when warmer air joins the cyclonic circulation, thus creating a warm sector between a cold and a warm front.