Balanced flow
The idealisation consists in considering the behaviour of one isolated parcel of air having constant density, its motion on a horizontal plane subject to selected forces acting on it and, finally, steady-state conditions.
The radius of curvature approaches an infinite length at the points where the trajectory becomes straight and the positive orientation of n is not determined in this particular case (discussed in geostrophic flows).
This frame is termed natural or intrinsic because the axes continuously adjust to the moving parcel, and so they are the most closely connected to its fate.
By reasoning on the balance of the remaining terms, we can understand The following yes/no table shows which contributions are considered in each idealisation.
The isobars of the ordinary weather charts summarise these pressure measurements, adjusted to the mean sea level for uniformity of presentation, at one particular time.
They also cannot describe the motion of the entire column from the contact surface with the Earth up to the outer atmosphere, because of the on-off handling of the friction forces.
Balanced-flow schematisations can be used to estimate the wind speed in air flows covering several degrees of latitude of Earth's surface.
The balanced-flow approach identifies typical trajectories and steady-state wind speeds derived from balance-giving pressure patterns.
In summary, the balanced-flow equations give out consistent steady-state wind speeds that can estimate the situation at a certain moment and a certain place.
These speeds cannot be confidently used to understand where the air is moving to in the long run, because the forcing naturally changes or the trajectories are skewed with respect to the pressure pattern.
The pressure gradient vector is only made by the component along the trajectory tangent s. The balance in the streamwise direction determines the antitriptic speed as
A positive speed is guaranteed by the fact that antitriptic flows move along the downward slope of the pressure field, so that mathematically
In the cross-stream momentum equation, the Coriolis force and normal pressure gradient are both negligible, leading to no net bending action.
Therefore, the antitriptic schematisation applies to flows that take place near the Earth's surface, in a region known as constant-stress layer.
Antitriptic flow can be used to describe some boundary-layer phenomena such as sea breezes, Ekman pumping, and the low level jet of the Great Plains.
entails that the trajectory must run along isobars, else the moving parcel would experience changes of pressure like in antitriptic flows.
Therefore, irrespective of the uncertainty in formally setting the unit vector n, the parcel always travels with the lower pressure at its left (right) in the northern (southern) hemisphere.
Because the Coriolis force is relevant, it normally fits processes with small Rossby number, typically having large length scales.
Although the balanced-flow equations do not allow for internal (air-to-air) friction, the flow directions in geostrophic streams and nearby rotating systems are also consistent with shear contact between those.
The etymology and the pressure charts shown suggest that geostrophic flows may describe atmospheric motion at rather large scales, although not necessarily so.
The balance-flow model gives no clue on the initial speed of an inertial circle, which needs to be triggered by some external perturbation.
whereby the cyclonic gradient speed V is smaller than the corresponding geostrophic, less accurate estimate, and naturally approaches it as the radius of curvature grows (as the inertial velocity goes to infinity).
This condition can be translated in the requirement that, given a high-pressure zone with a constant pressure slope at a certain latitude, there must be a circular region around the high without wind.
obtained by solving the above inequality for R. Outside this circle the speed decreases to the geostrophic value as the radius of curvature increases.
Firstly, imagine that a sample parcel of air flows 500 meters above the sea surface, so that frictional effects are already negligible.
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.
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.
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.
Additionally, the pressure fields quite rarely take the shape of neat circular isobars that keep the same spacing all around the circle.
