PGF = -1/r Dp/Dx
where r is the air density,
Dp is the difference in poressure, and
Dx is the distance over which the pressure difference occurs.
This formula says that the pressure gradient force is:
Lines of equal pressure at a given elevation are shown by isobars. Close spacing indicates high pressure gradients and high winds, whereas wide spacing: low pressure gradient.
Objects on a rotating body feel a pull away from the axis of rotation. This is because the objects have inertia, or a tendency to continue to travel in a straight line with a constant velocity. On a rotating body, this means that objects will 'want' to travel off at a tangent, resulting in an acceleration or 'pull' away from the axis of rotation. This is a familiar effect, well-known to children on roundabouts, and is known as the centrifugal force, C.
C = U2/R
where U = the rotational velocity (m sec-1), and R is the radius (metres).
For a rotating disc, C increases with distance from the centre. On the Earth, C is at a maximum at the equator (where U and R are greatest) and zero at the poles (where U and R are zero). Because the Earth is spherical (or nearly so), the centrifugal force is directed at right angles away from the Earth's axis, and so acts straight upwards at the equator, and slants increasingly towards the local horizontal with increasing latitude. At any given latitude, C can be divided into a vertical component, C(v), pointing straight up, and a horizontal component, C(h) pointing along the ground towards the equator. We have seen that the vertical component is 100% of the total at the equator. The horizontal component accounts for 100% of the total at the poles, but C is zero there. Thus, on the Earth, the horizontal centrifugal force is at a maximum at 45o North and South.
Q: Why does this force not pull us towards the equator, analogous to the force we feel on a roundabout?
A: Because the Earth is not quite a sphere. The Earth bulges slightly at the equator, and is flattened at the poles, so it formas an oblate spheroid. Because of this, on most of the earth's surface, the straight down direction does not point to the Earth's centre, as it would on a perfect sphere. (The Earth's centre is straight down at the equator and at the poles) Because of this, the gravitational force G is tilted slightly poleward of straight down over most of the Earth. Because of this tilt, we can divide G into two components: a vertical component, pointing straight down, and a horizontal component, G(h), pointing along the ground towards the pole. For a stationary point on the Earth, the horizontal component of G exactly balances the centrifugal force C. There is thus no net force, and no tendency for objects to be flung off into space. (Indeed, the early Earth adopted its oblate shape so that the forces would be in balance)
Now, for a body moving over the surface of the Earth, this force balance breaks down. The body has a velocity which differs from that of the Earth below, and thus experiences a different value of C. Because of this, the body will experience a net force, and will thus be deflected from its original course. This is to the right in the northern hemisphere, and to the left in the southern hemisphere. Thus, the Coriolis 'force' is not an independent force at all, but arises from an imbalance between two 'real' forces, the centrifugal force C and the gravitational force G. The magnitude of the force imbalance - the Coriolis effect - is given by:
2W sin f V (metres sec-2)
where W is the angular velocity of the earth's spin (The Earth rotates
through 360o day-1, or 15o hour-1.
In this equation, W is expressed in radians per second: 2P
radians = 360°, so W = 7.29 x 10-5)
f is the latitude, and
V is the horizontal velocity of the moving body (metres per second)
The Coriolis effect is thus directly proportional to:
(a) wind speed: i.e. air moving at 10 m sec-1 is subject
to half the deflective force as air moving at 20 m sec-1.
(b) the sine of the latitude (sin 0o = 0; sin 90o = 1). Thus the Coriolis effect increases from zero at the Equator, and is largest at the poles.
The component 2W sin f is known as the Coriolis parameter f, which varies with latitude as follows:
0° 10° 20° 43° 90°
f (x10-4) 0 0.25 0.5 1.0 1.46
Winds blowing due to the pressure gradient force will be deflected by
the Coriolis effect until the deflection acts in the opposite direction
to the pressure gradient force: only then will the deflection cease. Thus,
for equilibrium, the pressure gradient force and the Coriolis effect will
be in balance, and winds will therefore tend to parallel to the isobars.
Such winds are known as geostrophic winds (in the northern hemisphere,
the geostrophic ("Earth-turning") wind will blow along the isobars with
the high pressure to right, and low pressure to left - the opposite is
true in the southern hemisphere. Near the equator, the Coriolis force is
negligible, and so winds will blow at right angles to isobars.
The local centrifugal force is given by:
where v is the wind velocity, and
r is the radius of the curve around which the wind is blowing.
The centrifugal force is only significant where windspeeds are high,
and r is small, i.e. where isobars are tightly curved and pressure gradients
are at very large.
Winds blowing at constant speed around curved isobars are termed gradient winds.
z = 2w
w (omega) is the angular velocity of the spinning system (radians per second).
Note that this is similar to the formula for the Coriolis parameter. There is a good reason for this: air which is stationary with respect to the Earth still has vorticity because it is turning with the spinning Earth. In this case, its vorticity is 2W sin f. Systems which are rotating with respect to the Earth thus have two components of vorticity:
(1) relative vorticity, due
to the rotation of the system with respect to the Earth, and
(2) planetary vorticity, which is the latitudinal value of the Coriolis parameter.
The total vorticity (absolute vorticity) of a system tends to remain constant (ignoring energy losses due to heat exchange or friction). This is due to the conservation of angular momentum. Momentum is a measure of the inertia of a body, or its tendency to remain in its current state of motion, and is defined as:
mass x velocity
(that is, heavy, fast moving objects - such as a lorry - possess greater momentum than light, slowly moving objects - such as a lecturer on a bicycle). When an object is moving in an arc, momentum is expressed as angular momentum, defined as:
= r m Vt
where r is the distance from the
axis of rotation (metres)
m is the mass of the body (kg), and
Vt is the tangential velocity (metres per second),
Angular momentum is conserved (ignoring losses to friction and so on). Thus, for a spinning system, any changes in the radius of the system must result in changes in its velocity (assuming the mass stays constant). A familiar example of this is an ice skater in a spin: the skater begins a spin with arms outstretched, then draws them in towards her body, causing her to spin faster. To exit the spin, she extends her arms, thus reducing her angular velocity and allowing her to move on in a controlled manner. For air masses, a decrease in the radius of the system occurs during convergence, such as when uplift of air draws air inwards. Conversely, an increase in the radius occurs during divergence, such as when subsidence of air causes air to spread outwards. Thus, a converging and uplifting air mass (such as a cyclone or tornado) will gain positive or cyclonic relative vorticity in order to conserve angular momentum as air is drawn inwards (rotating anticlockwise in the northern hemisphere), whereas a spreading and subsiding air mass (such as an anticyclonic high pressure cell) will tend to acquire negative or anticyclonic vorticity (clockwise in the northern hemisphere).
Planetary vorticity can also be converted into relative vorticity if
an air mass changes latitude. Because the Coriolis parameter f increases
with latitude, the planetary vorticity will increase for a poleward moving
air mass. To conserve the absolute vorticity, the relative vorticity must
decrease, becoming less cyclonic (i.e. anticyclonic). Conversely, for equatorward-moving
air masses, the planetary vorticity decreases, so the relative vorticity
must become more cyclonic.
Another possible way of bolstering up the north-south opposite spin idea is to assume that the bath-plug whirlpool results from the acceleration of the planetary vorticity of the bathwater as the water converges. This would make the water rotate anti-clockwise in the northern hemisphere, and clockwise in the southern (i.e. increasing positive vorticity due to convergence). However, I calculate that for this effect to be significant on the timescales involved, you would need a plughole in the centre of a circular tub at least 14 metres across.
The real explanation for the bath-plug whirlpool is that it results from the acceleration of water currents running obliquely to the direct line to the plughole. Convergence (and the conservation of angular momentum) can easily produce the observed velocities. This explanation indicates that the whirlpool has its origins in the shape of your tub, and has nothing whatever to do with Coriolis.
For an entertaining debunking of the plughole myth, see Alistair Fraser's Bad Coriolis page (Part of the Bad Meteorology site, which looks at lots of incorrect explanations of weather phenomena) http://www.ems.psu.edu/~fraser/Bad/BadCoriolis.html
All of the principles explored in this lecture (plus some from earlier lectures) can be illustrated using the example of tropical cyclones. Tropical cyclones (also known as hurricanes in the Caribbean; and typhoons in the west Pacific) are intense circular storms, spiralling around a low pressure centre. They are defined as having maximum sustained surface wind speeds over 33 m sec-1, and in many storms they exceed 50 m sec-1. The central pressure is commonly below 950 mb, and can be below 900 mb. Cyclones are typically 650 km in diameter, and Pacific examples may be much larger. Around 80 occur each year, causing on average 20,000 fatalities and immense damage to property and posing serious hazards to shipping, due to the effects of winds, high seas, flooding from heavy rainfall and storm surges.
The largest observed Tropical cyclone was in the Caribbean: Hurricane Gilbert, generated on 9th Sept 1988 east of Barbados. The central pressure was 888 mb, and maximum winds near the core were 55 m sec-1 (125 mph). Over 500 mm rain fell on upland Jamaica in 9 hours. Gilbert was 3 times larger than the average Caribbean storm, with a maximum diameter of 3500 km, disrupting the ITCZ over 1 sixth of the earth’s circumference, and drawing in air from a vast region from Florida to the Galapagos Islands.
Such intense spinning storms require a special combination of circumstances for their formation:
At 27.5° C, convection suddenly becomes much more efficient, and warmer-than-27.5°C areas are often identifiable in satellite images by the strong development of rainstorms penetrating high into the atmosphere. Intense convection is a prerequisite for cyclone development, creating low pressure, convergence zones which draw in air from the surrounding area. Cyclones occur when intense convection combines with upper level divergence to produce uplift and low-level covergence. This uplift is not simply due to the buoyancy of heated air. Most of the energy for the storm comes from latent heat released by the condensation of water vapour in clouds. The need for an abundant moisture supply explains why tropical cyclones form over warm oceans, and dissipate rapidly over land. Uplift occurs in large numbers of cumulus cells arranged in spirals, which begin to converge and rotate around a central core area. Uplift is further ebcouraged by the release of latent heat as clouds form from the uplifting warm, moist air. This latent heating warms the core, typically through the action of 100-200 huge cumulonimbus towers, known as hot towers. The rapid uplift of air in this region intensifies low pressure at the centre of the system, and high pressure at high levels of the atmosphere, which encourages upper-air outflow. In turn, this sustains low-level inflow, convective uplift, and latent heat release in a positive feedback mechanism which feeds the developing storm. Cyclones are, in effect, vast mechanisms for releasing and transporting the energy stored in hot, humid air.
A distinctive feature of tropical cyclones is the presence of a distinct eye, a central quiet amphitheatre a few tens of km across surrounded by the furious, rotating eye wall of towering cloud. We can explain the presence and characteristics of the eye using the principles we have covered in this and earlier lectures. First, the presence of the eye itself. We have seen that the pressure-gradient force, the Coriolis effect, and surface friction combine to give surface winds that spiral in towards the centre of a storm system. We have also seen that the conservation of angular momentum requires that if the radius of a rotating system decreases, its velocity must increase. Thus, for winds spiralling in towards a storm centre, the velocities will thus tend to rise continuously. However, as the velocity of a rotating wind system increases, so the centrifugal force - acting outward from the centre of rotation - also increases. There comes a point where the winds are fast enough for the outward-acting centrifugal force to exactly balance the net inward-acting forces. At this point, the winds cease to spiral inwards, but rotate around a circular path instead. This defines the eye wall, and the windless zone within is the quiet eye. The centrifugal force also limits the wind velocities in cyclones. Calculated geostrophic flows may be as high as 500 m sec-1, but actual winds are typically 75 m sec-1. Thus, centrifugal force is important in limiting the destructive effect of cyclones.
A feature of the eye is that it is warmer (by up to 5o) than the surrounding parts of the storm, providing its alternative name of the warm core. The eye is also an area of subsidence, in sharp contrast to the vigorous uplift experienced in the violent, outer parts of the system. The warmth and subsidence are related. Condensation of water droplets from vapour in rising cumulonimbus towers releases latent heat, so the rising air cools at the saturated adiabatic lapse rate (Lecture 4). When the water droplets form precipitation, the water falls back to Earth but much of the heat released during condensation remains behind. Most of the warmed air spreads outward at the top of the storm, but some sinks into the centre, and during its descent, it warms at the dry adiabatic lapse rate. This means that the air arrives back at sea level warmer than when it began its ascent. This increase in temperature is not a 'free lunch': it merely results from energy transfers from one form to another. In the prelude to the storm, solar energy was used to evaporate water. This energy was transferred as latent heat during condensation, adding to the potential temperature of the air. Adiabatic descent completes the cycle, with the solar energy converted to a sensible temperature increase.
A significant Coriolis parameter is required to generate the spin necessary for cyclone formation. The spin can be achieved purely by the convergence of the air mass, generating positive relative vorticity from the planetary vorticity of a larger area. McIlveen (1992, p. 428) gives a formula for working out the wind velocity U of a cyclone produced by the convergence of a larger circular air mass:
U = (f R02)/2R
where f is the Coriolis parameter, R0 is the initial radius of the air mass, and R is the new, smaller radius. Taking U as 50 m sec-1, R as 30 km (typical for the ring of maximum winds in a tropical cyclone), and the value of f for 20o N or S, R0 would be 145 km. For 5o, the value would increase to 500 km. Thus, large areas of warm ocean are required to 'feed' cyclone development. In practice, the areas would be even larger because wind velocities are also eroded by friction at the sea surface. Nevertheless, the estimates indicate that tropical cyclones represent the condensed energy (vorticity, thermal and latent heat) of considerable areas of ocean. They also demonstrate why cyclones only form more than 5o poleward of the equator.
The two requirements of convection and spin thus determine the geographical distribution of cyclones and their annual cycles of occurrence. Cyclones form in late summer and autumn in the tropical oceans when the stored energy in the oceans is at a maximum. Cyclones occur in the northern hemisphere (Atlantic, Pacific, northern Indian Ocean) in June-November and in the southern hemisphere (South Pacific and southern Indian Ocean) in January-March.
Tracks of hurricanes generally move towards the west, then veer polewards and eastwards, but are also steered by mid-tropospheric winds. Some may reach the mid-latitudes and form vigorous frontal cyclones: in such cases, energy is carried far from the tropics. Cyclones therefore play an important role in transporting energy from the tropics towards the mid latitudes.
Cyclones and climate change
It is interesting to speculate on the effect of even a minor overall warming of the tropical oceans. There are large regions of water with temperature close to 27.5°C (either slightly warmer or slightly cooler), so even a small general temperature change could produce large changes in the area in which convection takes place. For example, the area of the tropical Pacific that has mean SST greater than 27°C is 20% larger than that greater than 27.5°C. Thus, a mere one-half degree uniform increase could produce a significant change in the amount of tropical convection, and the frequency of cyclones.
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