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Balanced flow
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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 in natural coordinates

Trajectories

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.

Kinematics

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.

Forces

In the balanced-flow idealization we consider a three-way balance of forces that are:

  • Pressure force. This is the action on the parcel arising from the spatial differences of atmospheric pressure p around it. (Temporal changes are of no interest here.) The spatial change of pressure is visualised through isobars, that are contours joining the locations where the pressure has a same value. In the figure this is simplistically shown by equally spaced straight lines. The pressure force acting on the parcel is minus the gradient vector of p (in symbols: grad p) – drawn in the figure as a blue arrow. At all points, the pressure gradient points to the direction of maximum increase of p and is always normal to the isobar at that point. Since the flow packet feels a push from the higher to the lower pressures, the effective pressure vector force is contrary to the pressure gradient, whence the minus sign before the gradient vector.
  • Friction. This is a force always opposing the forward motion, whereby the vector invariably acts in the negative direction s with an effect to reduce the speed. The friction at play in the balanced-flow models is the one exerted by the roughness of the Earth's surface on the air moving higher above. For simplicity, we here assume that the frictional force (per unit mass) adjusts to the parcel's speed proportionally through a constant coefficient of friction K. In more realistic conditions, the dependence of friction on the speed is non-linear except for slow laminar flows.
  • Coriolis force. This action, due to the Earth's rotation, tends to displace any body travelling in the northern (southern) hemisphere towards its right (left). Its intensity per unit mass is proportional to the speed V and increases in magnitude from the equator (where it is zero) towards the poles proportionally to the local Coriolis frequency f (a positive number north of the equator and negative south). Therefore, the Coriolis vector invariably points sideways, that is along the n axis. Its sign in the balance equation may change, since the positive orientation of n flips between right and left of the trajectory based solely on its curvature, while the Coriolis vector points to either side based on the packet's position on the Earth. The exact expression of the Coriolis force is a bit more complex than the product of the Coriolis parameter and parcel's velocity. However, this approximation is consistent with having neglected the curvature of the Earth's surface.

In the fictitious situation drawn in the figure, the pressure force pushes the parcel forward along the trajectory and inward with respect to the bend; the Coriolis force pushes inwards (outwards) of the bend in the northern (southern) hemisphere; and friction pulls (necessarily) rearwards.

Governing equations

For the dynamical equilibrium of the parcel, either component of acceleration times the parcel's mass is equal to the components of the external forces acting in the same direction. As the equations of equilibrium for the parcel are written in natural coordinates, the component equations for the horizontal momentum per unit mass are expressed as follows:

in the forward and sideway directions respectively, where ρ is the density of air.

The terms can be broken down as follows:

  • is the temporal rate of speed change to the parcel (tangential acceleration);
  • is the component of the pressure force per unit volume along the trajectory;
  • is the deceleration due to friction;
  • is the centripetal acceleration;
  • is the component of the pressure force per unit volume perpendicular to the trajectory;
  • is the Coriolis force per unit mass (the sign ambiguity depends on the mutual orientation of the force vector and n).

Steady-state assumption

In the following discussions, we consider steady-state flow. The speed cannot thus change with time, and the component forces producing tangential acceleration need to sum up to zero. In other words, active and resistive forces must balance out in the forward direction in order that . Importantly, no assumption is made yet on whether the right-hand side forces are of either significant or negligible magnitude there. Moreover, trajectories and streamlines coincide in steady-state conditions, and the pairs of adjectives tangential/normal and streamwise/cross-stream become interchangeable. An atmospheric flow in which the tangential acceleration is not negligible is called allisobaric.

The velocity direction can still change in space along the trajectory that, excluding inertial flows, is set by the pressure pattern.

General framework

The schematisations

Omitting specific terms in the tangential and normal balance equations, we obtain one of the five following idealized flows: antitriptic, geostrophic, cyclostrophic, inertial, and gradient flows. By reasoning on the balance of the remaining terms, we can understand

  • what arrangement of the pressure field supports such flows;
  • along which trajectory the parcel of air travels; and
  • with which speed it does so.

The following yes/no table shows which contributions are considered in each idealisation. The Ekman layer's schematisation is also mentioned for completeness, and is treated separately since it involves the internal friction of air rather than that between air and ground.

Antitriptic flow Geostrophic flow Cyclostrophic flow Inertial flow Gradient flow Ekman flow
curvature N N Y Y Y N
friction Y N N N N Y
pressure Y Y Y N Y Y
Coriolis N Y N Y Y Y

The limitations

Vertical differences of air properties

The equations were said to apply to parcels of air moving on horizontal planes. Indeed, when one considers a column of atmosphere, it is seldom the case that the air density is the same all the height up, since temperature and moisture content, hence density, do change with height. Every parcel within such a column moves according to the air properties at its own height.

Homogeneous sheets of air may slide one over the other, so long as stable stratification of lighter air on top of heavier air leads to well-separated layers. If some air happens to be heavier/lighter than that in the surroundings, though, vertical motions do occur and modify the horizontal motion in turn. In nature downdrafts and updrafts can sometimes be more rapid and intense than the motion parallel to the ground. The balanced-flow equations do not contain either a force representing the sinking/buoyancy action or the vertical component of velocity.

Consider also that the pressure is normally known through instruments (barometers) near the ground/sea level. 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. Such values represent the weight of the air column overhead without indicating the details of the changes of the air's specific weight overhead. Also, by Bernoulli's theorem, the measured pressure is not exactly the weight of the air column, should significant vertical motion of air occur. Thus, the pressure force acting on individual parcels of air at different heights is not really known through the measured values. When using information from a surface-pressure chart in balanced-flow formulations, the forces are best viewed as applied to the entire air column.

One difference of air speed in every air column invariably occurs, however, near the ground/sea, also if the air density is the same anywhere and no vertical motion occurs. There, the roughness of the contact surface slows down the air motion above, and this retarding effect peters out with height. See, for example, planetary boundary layer. Frictional antitriptic flow applies near the ground, while the other schematisations apply far enough from the ground not to feel its "braking" effect (free-air flow). This is a reason to keep the two groups conceptually separated. The transition from low-quote to high-quote schematisations is bridged by Ekman-like schematisations where air-to-air friction, Coriolis and pressure forces are in balance.

In summary, the balanced-flow speeds apply well to air columns that can be regarded as homogeneous (constant density, no vertical motion) or, at most, stably stratified (non-constant density, yet no vertical motion). An uncertainty in the estimate arises if we are not able to verify these conditions to occur. 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.

Horizontal differences of air properties

Even if air columns are homogeneous with height, the density of each column can change from location to location, firstly since air masses have different temperatures and moisture content depending on their origin; and then since air masses modify their properties as they flow over Earth's surface. For example, in extra-tropical cyclones the air circulating around a pressure low typically comes with a sector of warmer temperature wedged within colder air. The gradient-flow model of cyclonic circulation does not allow for these features.

Balanced-flow schematisations can be used to estimate the wind speed in air flows covering several degrees of latitude of Earth's surface. However, in this case assuming a constant Coriolis parameter is unrealistic, and the balanced-flow speed can be applied locally. See Rossby waves as an example of when changes of latitude are dynamically effective.

Unsteadiness

The balanced-flow approach identifies typical trajectories and steady-state wind speeds derived from balance-giving pressure patterns. In reality, pressure patterns and the motion of air masses are tied together, since accumulation (or density increase) of air mass somewhere increases the pressure on the ground and vice versa. Any new pressure gradient will cause a new displacement of air, and thus a continuous rearrangement. As weather itself demonstrates, steady-state conditions are exceptional.

Since friction, pressure gradient and Coriolis forces do not necessarily balance out, air masses actually accelerate and decelerate, so the actual speed depends on its past values too. As seen next, the neat arrangement of pressure fields and flow trajectories, either parallel or at a right angle, in balanced-flow follows from the assumption of steady flow.

The steady-state balanced-flow equations do not explain how the flow was set in motion in the first place. Also, if pressure patterns change quickly enough, balanced-flow speeds cannot help track the air parcels over long distances, simply because the forces that the parcel feels have changed while it is displaced. The particle will end up somewhere else compared to the case that it had followed the original pressure pattern.

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.

Antitriptic flow

Antitriptic flow describes a steady-state flow in a spatially varying pressure field when

  • the entire pressure gradient exactly balances friction alone; and:
  • all actions promoting curvature are neglected.

The name comes from the Greek words 'anti' (against, counter-) and 'triptein' (to rub) – meaning that this kind of flow proceeds by countering friction.

Formulation

In the streamwise momentum equation, friction balances the pressure gradient component without being negligible (so that K≠0). 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 . Provided the product KV is constant and ρ stays the same, p turns out to vary linearly with s and the trajectory is such that the parcel feels equal pressure drops while it covers equal distances. (This changes, of course, when using a non-linear model of friction or a coefficient of friction that varies in space to allow for different surface roughness.)

In the cross-stream momentum equation, the Coriolis force and normal pressure gradient are both negligible, leading to no net bending action. As the centrifugal term vanishes while the speed is non-zero, the radius of curvature goes to infinity, and the trajectory must be a straight line. In addition, the trajectory is perpendicular to the isobars since . Since this condition occurs when the n direction is that of an isobar, s is perpendicular to the isobars. Thus, antitriptic isobars need to be equispaced circles or straight lines.

Application

Antitriptic flow is probably the least used of the five balanced-flow idealizations, because the conditions are quite strict. However, it is the only one for which the friction underneath is regarded as a primary contribution. Therefore, the antitriptic schematisation applies to flows that take place near the Earth's surface, in a region known as constant-stress layer.

In reality, the flow in the constant-stress layer has a component parallel to the isobars too, since it is often driven by the faster flow overhead. This occurs owing to the so-called free-air flow at high quotes, which tends to be parallel to the isobars, and to the Ekman flow at intermediate quotes, which causes a reduction of the free-air speed and a turning of direction while approaching the surface.

Because the Coriolis effects are neglected, antitriptic flow occurs either near the equator (irrespective of the motion's length scale) or elsewhere whenever the Ekman number of the flow is large (normally for small-scale processes), as opposed to geostrophic flows.

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.[1]

Geostrophic flow

Nearly parallel isobars supporting quasi-geostrophic conditions
Geostrophic flow (westerly)
A westerly stream flow of global scale spans, approximately along parallels, from Labrador (Canada) across the North Atlantic Ocean as fas as inner Russia
Geostrophic flow (easterly)
An easterly westerly stream flow of global scale spans, approximately along parallels, from Russia over Europe as fas as the mid latitude Atlantic Ocean
Geostrophic flow (northerly)
A northerly air stream flows from Arctic to mid latitudes south of the 40th parallel
Geostrophic flow (north-westerly)
A northwesterly stream sets in between two large-scale counter-rotating curved flows (cyclone and anticyclone ). Close isobars indicate high speeds

Geostrophic flow describes a steady-state flow in a spatially varying pressure field when

  • frictional effects are neglected; and:
  • the entire pressure gradient exactly balances the Coriolis force alone (resulting in no curvature).

This condition is called geostrophic equilibrium or geostrophic balance (also known as geostrophy). The name 'geostrophic' stems from the Greek words 'ge' (Earth) and 'strephein' (to turn). This etymology does not suggest turning of trajectories, rather a rotation around the Earth.

Formulation

In the streamwise momentum equation, negligible friction is expressed by K=0 and, for steady-state balance, negligible streamwise pressure force follows.

The speed cannot be determined by this balance. However,








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