Balanced flow

In atmospheric science, balanced flow is an idealisation of atmospheric motion. 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.

Balanced flow is often an accurate approximation of the actual flow, and is useful in improving the qualitative understanding and interpretation of atmospheric motion. In particular, the balanced-flow speeds can be used as estimates of the wind speed for particular arrangements of the atmospheric pressure on Earth’s surface.

The momentum equations are written primarily for the generic trajectory of a packet of flow travelling on a horizontal plane and taken at a certain elapsed time called t. The position of the packet is defined by the distance on the trajectory s=s(t) which it has travelled by time t. In reality, however, the trajectory is the outcome of the balance of forces upon the particle. In this section we assume to know it from the start for convenience of representation. When we consider the motion determined by the forces selected next, we will have clues of which type of trajectory fits the particular balance of forces.

The trajectory at a position s has one tangent unit vector s that invariably points in the direction of growing s’s, as well as one unit vector n, perpendicular to s, that points towards the local centre of curvature O. The centre of curvature is found on the 'inner side' of the bend, and can shift across either side of the trajectory according to the shape of it. The distance between the parcel position and the centre of curvature is the radius of curvature R at that position. 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). The frame of reference (s,n) is shown by the red arrows in the figure. 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.

The velocity vector (V) is oriented like s and has intensity (speed) V = ds/dt. This speed is always a positive quantity, since any parcel moves along its own trajectory and, for increasing times (dt>0), the trodden length increases as well (ds>0).

The acceleration vector of the parcel is decomposed in the tangential acceleration parallel to s and in the centripetal acceleration along positive n. The tangential acceleration only changes the speed V and is equal to DV/Dt, where big d's denote the material derivative. The centripetal acceleration always points towards the centre of curvature O and only changes the direction s of the forward displacement while the parcel moves on.

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