Definitions

# 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, in 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.

## The Momentum Equations in Natural Coordinates

Let's consider a packet of flow travelling along its trajectory on the horizontal plane and taken at a certain time t. The position of the packet is defined by the distance on the trajectory s=s(t) that it has travelled by then. In reality, as we will see, the trajectory is the outcome of the balance of forces upon the particle, and we assume to know it from the start for convenience of representation. When we consider the motion determined by selected forces, we will have clues of which type of trajectory fits the particular balance of forces.

The trajectory at one position has a tangent unit vector s that invariably points in the direction of growing s’s, as well as a normal unit vector n that points to 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 following 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 approaches an infinite length at the points where the trajectory becomes straight and, in this particular case, the positive orientation of n is not determined (we will see how to cope with this). 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 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, of course, is always a positive quantity, since the parcel moves along its own trajectory and, for increasing times, the trodden length increases as well.

The acceleration vector of the parcel can be 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.

### Actions

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. In the current idealization those vector forces are:

• Pressure force. This is the action arising from the difference of pressure around the parcel body, when the trajectory crosses a medium with spatially-varying atmospheric pressure p. (Temporal changes are of no interest here.) The spatial change of pressure can be 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) - in the figure this vector is drawn 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 force has an orientation contrary to the properly defined pressure gradient, whence the minus sign attached to the force 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 action (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 very slow, creeping 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 in proportion with the local Coriolis parameter 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 changes 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, as the curvature of the Earth's surface has been neglected in the first instance, this is a consistent approximation.

In the fictitious situation drawn in the figure, you may check that the pressure force pushes the parcel forward along the trajectory and inward with respect to the bend; that the Coriolis force pushes inward (outward) in the northern (southern) hemisphere; and that the friction pulls in the rearward direction.

### Governing equations

Writing down the equations of equilibrium for the parcel in natural coordinates, the component equations for the horizontal momentum (per unit mass) can be relatively simply expressed as follows:

$frac\left\{DV\right\}\left\{Dt\right\} = -frac\left\{1\right\}\left\{rho\right\}frac\left\{partial p\right\}\left\{partial s\right\} - K V$

$frac\left\{V^2\right\}\left\{R\right\} = - frac\left\{1\right\}\left\{rho\right\}frac\left\{partial p\right\}\left\{partial n\right\} pm f V$,

where ρ is the density of air.

The terms can be broken down as follows:

• $\left\{DV\right\}/\left\{Dt\right\}$ is the temporal rate of speed change to the parcel (tangential acceleration);
• $-\left\{partial p\right\}/\left\{partial s\right\}$ is the component of the pressure force along the trajectory;
• $-K V$ is the deceleration due to friction;
• $\left\{V^2\right\}/\left\{R\right\}$ is the centripetal acceleration;
• $-\left\{partial p\right\}/\left\{partial n\right\}$ is the component of the pressure force perpendicular to the trajectory;
• $pm f V$ is the Coriolis force per unit mass.

Omitting specific terms in the balance of the tangential and normal equations, we obtain one of the five following idealized flows: Antitriptic Flow, Geostrophic Flow, Cyclostrophic Flow, Inertial Flow, and Gradient Flow. By reasoning on the remaining terms, we can understand what arrangement of the pressure field can support 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.

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

In the following discussions, we consider steady-state flow. The speed cannot 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 $DV/Dt=0$. No assumption is made yet on whether the right-hand side forces are of 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.

## 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, the component (sole) pressure force and friction balance one another without being negligible (so that K≠0). This balance determines the antitriptic speed:

$V = - frac\left\{1\right\}\left\{Krho\right\}frac\left\{partial p\right\}\left\{partial s\right\}$

A positive speed is guaranteed by the fact that antitriptic flows move along the downward slope of the pressure field, so that $\left\{partial p\right\}/\left\{partial s\right\} <0$. Also, so long as the product KV is constant and ρ stays the same, p turns out to vary linearly with s. Therefore, the trajectory is such that the parcel feels equal pressure drops as it covers equal distances. (This would change accordingly, of course, when using a non-linear model of friction or a coefficient of friction that varies in space.)

In the cross-stream momentum equation, the Coriolis and component pressure forces are both negligible, leading to no net bending action. As the centrifugal term $\left\{V^2\right\}/\left\{R\right\}$ vanishes while the speed is non-zero, the radius of curvature goes to infinity, and the trajectory is a straight line. In addition, since $partial p/partial n=0$, the trajectory is perpendicular to the isobars - this is because the n direction needs to be that of an isobar for the gradient to vanish as required. To keep all trajectories straight and in steady state, finally, isobars should be equispaced straight lines or circles.

### Application

Antitriptic flow is probably the least used of our five idealizations, because the conditions are quite strict. However, it is the only one for which friction is regarded as a primary contribution. Therefore, the antitriptic schematisation applies to flows that take place near the Earth's surface (boundary-layer processes). Because the Coriolis effects are neglected, antitriptic flow occurs either near the equator (irrespective of the motion's lengthscale) 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.

## Geostrophic Flow

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).

The name 'geostrophic' stems from the Greek words 'ge' (Earth) and 'strephein' (to turn).

### 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, $partial p/partial s=0$ entails that the trajectory must run along isobars, else the moving parcel would experience changes of pressure like in antitriptic flows.

In the cross-stream momentum equation, non-negligible Coriolis force is balanced by the pressure force, in a way that the parcel does not experience any bending action. Since the trajectory does not bend, there is no way to determine the positive orientation of n for lack of a centre of curvature. However, while the signs of the vector components become uncertain in this case, the pressure force must exactly counterbalance the Coriolis force anyway: so the parcel of air needs to travel with the Coriolis force contrary to the decreasing sideways slope of pressure. Therefore, irrespective of the conventional orientation of n, the parcel always travels with the lower pressure at its left (right) in the northern (southern) hemisphere.

The geostrophic speed is

$V = frac\left\{1\right\}\left\{rho\right\} left| frac\left\{1\right\}\left\{f\right\} frac \left\{partial p\right\}\left\{partial n\right\} right|$.

The expression of the geostrophic speed resembles that of antitriptic speed: here the speed is determined by the magnitude of the pressure gradient across (instead of along) the trajectory that develops along (instead of across) an isobar.

As trajectories were seen to be isobars, no bending is only possible if the isobars are straight lines in the first instance. So long as ρ stays the same and the Coriolis parameter is approximately constant in the region where the flow develops, isobars need to keep their spacing at one point equal all along to support such idealised conditions.

### Application

Modelers, theoreticians, and operational forecasters frequently make use of geostrophic/quasi-geostrophic approximation. Because friction is unimportant, the geostrophic balance fits flows that take place high enough above the Earth's surface. Because the Coriolis force is relevant, it normally fits processes with small Rossby number, typically having large lengthscales. Geostrophic conditions are also realised for flows having small Ekman number, as opposed to antitriptic conditions. However, since the centripetal acceleration vanishes, geostrophic isobars are straight lines, which seldom occurs in the upper atmosphere. The above etymology is in fact slightly misleading in that there is no turning of trajectories - rather rotation "around the Earth".

See the article on Geostrophic wind for a description of applications.

## Cyclostrophic Flow

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

• the frictional and Coriolis actions are neglected; and:
• the centripetal acceleration is entirely sustained by the pressure gradient.

Trajectories do bend. The name 'cyclostrophic' stems from the Greek words 'kyklos' (circle) and 'strephein' (to turn).

### Formulation

Like in geostrophic balance, the flow is frictionless and, for steady-state motion, the trajectories follow the isobars.

In the cross-stream momentum equation, only the Coriolis force is discarded, so that the centripetal acceleration is just the cross-stream pressure force per unit mass

$frac\left\{V^2\right\}\left\{R\right\} = -frac\left\{1\right\}\left\{rho\right\}frac\left\{partial p\right\}\left\{partial n\right\}$.

This implies that the trajectory is subject to a bending action, and that the cyclostrophic speed is

$V = sqrt\left\{ -frac\left\{R\right\}\left\{rho\right\} frac\left\{partial p\right\}\left\{partial n\right\}\right\}$.

So, the cyclostrophic speed is determined by the magnitude of the pressure gradient across the trajectory and by the radius of curvature of the isobar. The flow is faster, the farther away from its centre of curvature, albeit less than linearly.

Another implication of the cross-stream momentum equation is that a cyclostrophic flow can only develop next to a low-pressure area. This is implied in the requirement that the quantity under the square root is positive. Recall that the cyclostrophic trajectory was found to be an isobar. Only if the pressure increases from the centre of curvature outwards, the pressure derivative is negative and the cyclostrophic speed is well defined - the pressure in the centre of curvature must thus be a low. The above mathematics gives no clue whether the cyclostrophic rotation ends up to be clockwise or anticlockwise, meaning that the eventual arrangement is a consequence of effects not allowed for in the idealisation (for example, the Coriolis force).

### Application

The cyclostrophic schematisation is realistic when Coriolis and frictional forces are both negligible, that is for flows having large Rossby number and small Ekman number. Coriolis effects are ordinarily negligible in lower latitudes or on smaller scales. Cyclostrophic balance can be achieved in systems such as tornadoes, dust devils and waterspouts. Cyclostrophic speed can also be seen as one of the components of the speed in gradient balance, as shown next.

Among the studies using the cyclostrophic schematisation, Rennó and Buestein use the cyclostrophic speed equation to construct a theory for waterspouts; and Winn, Hunyady, and Aulich use the cyclostrophic approximation to compute the maximum tangential winds of a large tornado which passed near Allison, Texas on 8 June 1995.

## Inertial Flow

Unlike all other flows, inertial balance implies a uniform pressure field. In this idealisation:

• the flow is frictionless;
• no pressure gradient (and force) is present at all.

The only remaining action is the Coriolis force, which imparts curvature to the trajectory.

### Formulation

As before, frictionless flow in steady-state conditions implies that $partial p / partial s =0$. However, isobars are not defined in this case and we cannot draw any anticipation about the trajectory from the arrangement of the pressure field.

In the cross-stream momentum equation, after omitting the pressure force, the centripetal acceleration is the Coriolis force per unit mass. The sign ambiguity disappears, because the bending is solely determined by the Coriolis force that sets unchallenged the side of curvature - so this force has always a positive sign. The inertial rotation will be clockwise (anticlockwise) in the northern (southern) hemisphere. The momentum equation

$frac\left\{V^2\right\}\left\{R\right\} = left| f right| V$,

gives us the inertial speed

$V = left| f right| R$.

The inertial speed's equation only helps determine either the speed or the radius of curvature (a result of the pressure field in the other flows) once the other is given. The trajectory resulting from this motion is also known as inertial circle.

### Application

Since atmospheric motion is due largely to pressure differences, inertial flow is not very applicable in atmospheric dynamics. However, the inertial speed appears as a contribution to the solution of the gradient speed (see next). Moreover, inertial flows are observed in the ocean streams, where flows are less driven by pressure differences than in air because of higher density -- inertial balance can occur at a certain depth such that the friction transmitted from above by the surface winds vanishes.

Gradient flow is an extension of geostrophic flow as it accounts for curvature too, making this a more accurate approximation for the flow in the upper atmosphere. However, mathematically gradient flow is slightly more complex, and geostrophic flow may be fairly accurate, so the gradient approximation is not as frequently mentioned.

Gradient flow is also an extension of the cyclostrophic balance, as it allows for the effect of the Coriolis force, making it suitable for flows with any Rossby number.

Finally, it is an extension of inertial balance, as it allows for a pressure force to drive the flow.

### Formulation

Like in all but the antitriptic balance, frictional and pressure forces are neglected in the streamwise momentum equation, so that it follows from $partial p / partial s = 0$ that the flow is parallel to the isobars.

Solving the full cross-stream momentum equation as a quadratic equation for V yields

$V = pm frac\left\{ f R \right\}\left\{2\right\} pm sqrt\left\{ frac\left\{f^2 R^2\right\}\left\{4\right\} - frac\left\{R\right\}\left\{rho\right\}frac\left\{partial p\right\}\left\{partial n\right\} \right\}$.

The first sign ambiguity follows from the mutual orientation of the Coriolis force and unit vector n, whereas the second follows from the square root. Not all solutions of the gradient wind speed yield physically plausible results. The summed-up right-hand side needs be positive because of the definition of speed, and the quantity under square root needs to be non negative. The important cases of cyclonic and anticyclonic circulations are discussed next.

#### Pressure lows and cyclones

For regular cyclones (air circulation around pressure lows), the pressure force is inward (positive term) and the Coriolis force outward (negative term) irrespective of the hemisphere:

$frac\left\{V^2\right\}\left\{R\right\} = frac\left\{1\right\}\left\{rho\right\}left|frac\left\{partial p\right\}\left\{partial n\right\}right| - left| f right| V$.

Dividing both sides by |f|V, one recognizes that

$frac\left\{ V_\left\{geostrophic\right\} \right\}\left\{ V \right\} = 1 + frac\left\{ V \right\}\left\{ V_\left\{inertial\right\} \right\} > 1$,

whereby the cyclonic gradient speed V is smaller than the geostrophic, less accurate estimate, and naturally approaches it as the radius of curvature grows. In cyclones, therefore, curvature slows down the flow compared to the no-curvature value of geostrophic speed.

The positive root of the equation for V is

$V = -frac\left\{ V_\left\{inertial\right\} \right\}\left\{2\right\} + sqrt\left\{ frac\left\{V_\left\{inertial\right\}^2\right\}\left\{4\right\} + V_\left\{cyclostrophic\right\}^2 \right\}$

that is always well defined as the quantity under the square root is positive.

#### Pressure highs and anticyclones

In anticyclones (air circulation around pressure highs), the Coriolis force is always inward (and positive), and the pressure force outward (and negative) irrespective of the hemisphere:

$frac\left\{V^2\right\}\left\{R\right\} = -frac\left\{1\right\}\left\{rho\right\}left|frac\left\{partial p\right\}\left\{partial n\right\}right| + left| f right| V$.

Dividing both sides by |f|V, we obtain

$frac\left\{ V_\left\{geostrophic\right\} \right\}\left\{ V \right\} = 1 - frac\left\{ V \right\}\left\{ V_\left\{inertial\right\} \right\} < 1$,

whereby the anticyclonic gradient speed V is larger than the geostrophic value while approaching it as the radius of curvature becomes larger and larger. In anticyclones, therefore, the curvature of isobars speeds up the airflow compared to the (geostrophic) no-curvature value.

There are two positive roots for V, but the only one consistent with the geostrophic approximation is

$V = frac\left\{ V_\left\{inertial\right\} \right\}\left\{2\right\} - sqrt\left\{ frac\left\{ V_\left\{inertial\right\}^2 \right\}\left\{4\right\} - V_\left\{cyclostrophic\right\}^2 \right\}$,

that requires that $V_\left\{inertial\right\} ge 2 V_\left\{cyclostrophic\right\}$ to be meaningful. 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. On its circumference the air blows at half the corresponding inertial speed (at the cyclostrophic speed), and the radius is

$R^* = frac\left\{4\right\}\left\{rho f^2\right\} left| frac\left\{partial p\right\}\left\{partial n\right\} right|$,

obtained by solving the above inequality for R. Outside this circle the speed decreases to the geostrophic value as the radius of curvature increases.

### Application

Gradient Flow is useful in studying atmospheric flow rotating around high and low pressures centers with small Rossby numbers. This is the case where the radius of curvature of the flow about the pressure centers is small, and geostrophic flow no longer applies with a useful degree of accuracy.

## 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

R=100 km 7.45 9.25 11.50 N/A 5.15
R=300 km 7.45 16.00 34.50 10.90 6.30

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 but 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. In particular, here the isobars are thought as being equally spaced all over, while in reality the flow may slow down as soon as the spacing widens.

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 this example, there is no anticyclonic 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 it 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.