# Balanced flow

I

## Geostrophic flow

Nearly-parallel isobars supporting quasi-geostrophic conditions

A westerly stream flow of global scale spans, approximately along parallels, from Labrador (Canada) across the North Atlantic Ocean as fas as inner Russia
An easterly westerly stream flow of global scale spans, approximately along parallels, from Russia over Europe as fas as the mid latitude Atlantic Ocean
An northerly air stream flows from Arctic to mid latitudes south of the 40th parallel
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).

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, $\partial p/\partial s=0$ entails that the trajectory must run along isobars, else the moving parcel would experience changes of pressure like in antitriptic flows. No bending is thus only possible if the isobars are straight lines in the first instance. So, geostrophic flows take the appearance of a stream channelled along such isobars.

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, the positive orientation of n cannot be determined for lack of a centre of curvature. The signs of the normal vector components become uncertain in this case. However, 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 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.

The geostrophic speed is

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

The expression of 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.

### Application

Modelers, theoreticians, and operational forecasters frequently make use of geostrophic/quasi-geostrophic approximation. Because friction is unimportant, the geostrophic balance fits flows 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.

It is frequent that the geostrophic conditions develop between a well-defined pair of pressure high and low; or that a major geostrophic stream is flanked by several higher- and lower-pressure regions at either side of it (see images). 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 speed of a geostrophic stream is larger (smaller) than that in the curved flow around a pressure low (high) with the same pressure gradient: this feature is explained by the more general gradient-flow schematisation. This helps use the geostrophic speed as a back-of-the-envelope estimate of more complex arrangements—see also the balanced-flow speeds compared below.

The etymology and the pressure charts shown suggest that geostrophic flows may describe atmospheric motion at rather large scales, although not necessarily so.

## 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

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

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

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

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 square root 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 relationship, namely the rotation of the parent cell.

### 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 contribution of the gradient balance-speed, as shown next.

Among the studies using the cyclostrophic schematisation, Rennó and Bluestein [2] use the cyclostrophic speed equation to construct a theory for waterspouts; and Winn, Hunyady, and Aulich [3] 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, in this case isobars are not defined in the first place. 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

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

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 once the other is given. The trajectory resulting from this motion is also known as inertial circle. The balance-flow model gives no clue on the initial speed of an inertial circle, which needs to be triggered by some external perturbation.

### 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 depths such that the friction transmitted by the surface winds downwards vanishes.

A nearly-uniform pressure field covers Central Europe and Russia with pressure differences smaller than 8 mbar over several tens of degrees of latitude and longitude. (For the conditions over the Atlantic Ocean see geostrophic and gradient flow) © British Crown Copyright 2009, The Met Office

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

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

Not all solutions of the gradient wind speed yield physically plausible results: the right-hand side as a whole needs be positive because of the definition of speed; and the quantity under square root needs to be non negative. 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.

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. The cross-trajectory momentum equation is

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

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

$\frac{ V_{geostrophic} }{ V_{cyclone} } = 1 + \frac{ V_{cyclone} }{ V_{inertial} } > 1$,

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). In cyclones, therefore, curvature slows down the flow compared to the no-curvature value of geostrophic speed. See also the balanced-flow speeds compared below.

The positive root of the cyclone equation is

$V_{cyclone} = -\frac{ V_{inertial} }{2} + \sqrt{ \frac{V_{inertial}^2}{4} + V_{cyclostrophic}^2 }$.

This speed is always well defined as the quantity under the square root is always 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. The cross-trajectory momentum equation is

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

Dividing both sides by |f|V, we obtain

$\frac{ V_{geostrophic} }{ V_{anticyclone} } = 1 - \frac{ V_{anticyclone} }{ V_{inertial} } < 1$,

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

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

$V_{anticyclone} = \frac{ V_{inertial} }{2} - \sqrt{ \frac{ V_{inertial}^2 }{4} - V_{cyclostrophic}^2 }$

that requires that $V_{inertial} \ge 2 V_{cyclostrophic}$ 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

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

obtained by solving the above inequality for R. Outside this circle the speed decreases to the geostrophic value as the radius of curvature increases. The width of this radius grows with the intensity of the pressure gradient.

### 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.

Surface pressure charts supporting gradient-wind conditions

Low pressure W of Ireland and cyclonic conditions.
High pressure over the British Isles and anticyclonic conditions.

## 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)

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 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.

## References

1. ^ Schaefer Etling, J.; C. Doswell (1980). "The Theory and Practical Application of Antitriptic Balance". Monthly Weather Review 108 (6): 746–456. Bibcode 1980MWRv..108..746S. doi:10.1175/1520-0493(1980)108<0746:TTAPAO>2.0.CO;2. ISSN 1520-0493.
2. ^ Rennó, N.O.D.; H.B. Bluestein (2001). "A Simple Theory for Waterspouts". Journal of the Atmospheric Sciences 58 (8): 927–932. Bibcode 2001JAtS...58..927R. doi:10.1175/1520-0469(2001)058<0927:ASTFW>2.0.CO;2. ISSN 1520-0469.
3. ^ Winn, W.P.; S.J. Hunyady G.D. Aulich (1999). "Pressure at the ground in a large tornado". Journal of Geophysical Research 104 (D18): 22,067–22,082. Bibcode 1999JGR...10422067W. doi:10.1029/1999JD900387.