Basic forces within the atmosphere and ocean

There are five basic forces underlying the circulation of the atmosphere and the ocean: (i) gravity; (ii) pressure gradients generated by density or mass differences; (iii) drag, or the momentum gain or loss across the air/sea, air/land or land/sea interface; (iv) the Coriolis force due to the intrinsic rotation of the Earth; and (v) tidal motion due to the astronomical influence of the Moon and the Sun. Gravity and the pressure gradient force provide the basic balance in the vertical but horizontal motions in the atmosphere and ocean are driven differently. In the atmosphere the pressure gradient force and the Coriolis force strongly dominate, with surface drag acting to modify their balance, and tidal forces being almost irrelevant (discussion of drag in the atmosphere will be left until §2.8). All five forces can be involved in horizontal motion in the ocean.

Local balance is provided by momentum transfer from the wind and the Coriolis force (the Ekman spiral). On larger scales density differences and the interaction of wind, sea and coastal boundaries produce pressure gradients, and also height differences. Tidal motions occur with regular periodicities to provide a strong mixing influence, and are often the predominant driving force in coastal zones (see §2.7). Motion resulting from all these influences is then affected by the Coriolis force. Detailed discussion of these motions in the sea will be considered in §§2.9-2.11, while some particular features of cloud physics and the marine atmospheric circulation will be considered in §2.13, although the major links between clouds and the ocean will be considered in Chapters 3 and 4. A basic description of the key common processes in the ocean will, however, be given here.

2.6.1 Hydrostatic balance

The atmosphere and ocean are generally close to being in hydrostatic balance.ln the extreme this means that no vertical motion occurs because the force of gravity acting downwards is exactly balanced by the pressure gradient force acting upwards, away from the high pressures of the lower atmosphere or deep ocean generated by the weight of overlying air or water respectively. This balance is expressed by the hydrostatic equation:

where dP is the change in pressure, P, over a very small, differential, height change dz, p is the density of the appropriate fluid, g is the gravitational acceleration and z is the vertical coordinate, which is positive upward. The ocean is essentially incompressible; a change of 10 m in depth changes the pressure by about 105 Nm-2, or the equivalent of one atmosphere. The density of the compressible atmosphere changes exponentially with height; near sea level a change of 10 m in height changes the pressure by about 1 millibar (102 Nm-2).

The hydrostatic approximation is not always obeyed in the atmosphere or ocean. During localized vigorous convection, such as in a cumulus cloud, deep water formation or fast flows down steep slopes, the balance is broken. However, even in situations where there is an element of large-scale vertical motion, such as in atmospheric frontal systems or the oceanic thermohaline circulation, hydrostatic balance is maintained locally. It is sometimes useful for such processes to be represented by a coordinate system whose vertical coordinate changes height slightly with position, following a density surface. In the atmosphere such a system is known as sigma coordinates while in the ocean these are isopycnals.

2.6.2 The Coriolis force

The motion induced by the pressure gradient force is affected by a number of other processes to produce the full, complex, three-dimensional, nature of the circulation of the ocean and atmosphere. Foremost amongst these is the Coriolis force, a consequence of motion occurring on the surface of a rotating body, the Earth. The Coriolis force is often called a pseudo-force because if

Fig. 2.15. Illustration of the Coriolis effect on a flat disc, rotating counter-clockwise. (a) The path drawn by the pen over the surface of the disc (solid) as it is moved directly to the right as the disc is set spinning counter-clockwise (dashed). (b) The path drawn by the pen as seen from the rotating disc. [Adapted from Fig. 4 of Persson (2000). Reproduced with permission from the Royal Meteorological Society.]

Fig. 2.15. Illustration of the Coriolis effect on a flat disc, rotating counter-clockwise. (a) The path drawn by the pen over the surface of the disc (solid) as it is moved directly to the right as the disc is set spinning counter-clockwise (dashed). (b) The path drawn by the pen as seen from the rotating disc. [Adapted from Fig. 4 of Persson (2000). Reproduced with permission from the Royal Meteorological Society.]

we were observing the Earth from space the effects ascribed to the Coriolis force could be completely explained by rotational dynamics. However, as the climate system is observed from the rotating Earth itself the introduction of such a pseudo-force makes it easier to describe the effects of the rotation.

To understand how the rotation of the Earth affects the motion of a moving body on its surface consider the following experiment. Take a disc and a water-soluble felt pen. Place the pen at a point half way between the edge and the centre of the stationary disc lightly. Then rotate the disc anticlockwise, while moving the pen directly to the right (Fig. 2.15(a)). When you take your pen off the disc you will see that the path traced was a curve, that is, the pen acted as if it was 'deflected' to the right of its path. This occurred because the disc was spinning underneath the moving pen (Fig. 2.15(b)). This deflection is due to the Coriolis force experienced by the pen acting on objects moving relative to the disc's rotation.

Thus, for a disc spinning anticlockwise a body moving on it will experience an apparent force to the right - the Coriolis Force, named after the French nineteenth century scientist who first correctly described the phenomenon. An anticlockwise spin is experienced by objects moving on the Northern Hemisphere of the Earth; the Southern Hemisphere spins clockwise, seen from space below the South Pole, and so moving objects in this hemisphere experience an apparent force to the left.

A moment's thought will show the possibility of your pen ending where it started if the rotation of the disc continues for long enough. If an object is set in motion, and there are only very weak forces then affecting it other than the Coriolis force, it will continue to be deflected to the right (in the Northern Hemisphere) until it arrives back at its starting point. The Coriolis force is acting as a centripetal force so that mu

where m is the mass of the object, r is the radius of the motion, f is the Coriolis parameter quantifying the effect of the Coriolis force (f = 2m sin 0, where m is the rotation rate of the Earth and 0 is the latitude at which the object is moving; see (2.10)) and u is the object's tangential velocity. This is known as an inertial oscillation. The angular velocity of the circular motion is thus f so that the period is 2n/f. The period therefore depends on latitude, but is 12-24 hours in r

Fig. 2.16. Illustration of the conservation of angular momentum on a spinning Earth. As the ship steams north its distance from the rotation axis decreases causing it to move eastwards, thus gaining an eastward speed v, adding to the underlying rotation speed and so spinning faster (see equation (2.9)).

Fig. 2.16. Illustration of the conservation of angular momentum on a spinning Earth. As the ship steams north its distance from the rotation axis decreases causing it to move eastwards, thus gaining an eastward speed v, adding to the underlying rotation speed and so spinning faster (see equation (2.9)).

mid-latitudes. Such oscillations tend to occur in the ocean after sudden changes in the wind.

The Earth's Coriolis force is, however, acting on a sphere rather than a flat disc. We can see the origin of the Coriolis force on the sphere using some simple rotational dynamics.

Consider a ship steaming in the Northern Hemisphere, as shown in Fig. 2.16. As the ship steams north the distance from the ship to the Earth's axis of rotation decreases. The angular momentum, I, of the ship is decreased also, as

where r is the radius from the axis of rotation and v is the rotational velocity (the local speed of the Earth in this case). In the absence of other forces angular momentum is conserved by a moving body. This can be illustrated by sitting in a revolving chair and spinning yourself around. If you hug your arms to your body, you decrease the radius of your centre of mass from the axis of rotation and consequently increase your speed, so that I remains constant. The northward-moving ship is in exactly the same situation: it needs to accelerate. It does this by turning to the east, so that its rotational speed becomes greater than that of the Earth, which is also rotating towards the east. Naturally, the opposite happens if the ship moves south; now it needs to decelerate and so turns to the west. In the Southern Hemisphere the opposite must happen.

The Coriolis force is also observed to act on objects moving east or west. The reason for this is not as easily seen as for the case of north-south, or meridional, motion, but again relies on rotational dynamics. When an object rotates it is kept in motion by a centripetal force, as shown in Fig. 2.17. This is directed at right angles to, and towards, the rotation axis. The object, however, has a velocity tangential to the radius, and so the force acts to pull the object towards the axis. As the object moves it is still subject to the centripetal force, but from a different direction. Continual adjustment to the direction of travel of the object maintains it in a circular path. The magnitude of this force can be shown to be mv2/r or mw2r, where m is the mass of the object and rn is the rotation rate.

To an observer sitting on the rotating object (or on a stationary ship at sea) there appears to be no motion. For this observer, therefore, there must be another pseudo-force balancing the centripetal force. This is known as the centrifugal force, which acts in the direction of the radius, but away from the axis, as shown in Fig. 2.17. We have all experienced this force on a merry-go-round. On the surface of the Earth this force can be expressed as the result of two right-angled components: one tangential to the surface, and a vertical component. When the object is at rest this tangential force, mw2r sin 0 - 0 is the latitude - does not

Fig. 2.17. Illustration of the apparent centrifugal force needed by an observer on the rotating Earth to balance the centripetal force acting towards the rotation axis, and thus reconcile the absence of motion of a stationary object on the Earth's surface. This centrifugal force can be split into two components: one acting parallel to the local Earth's surface and another acting outwards from the centre of the Earth, in a local vertical direction.

induce any motion (as the Earth itself has distorted as a response to a balance between gravity and centripetal force, making the equatorial radius greater than the polar by 21 km). However, if the ship is moving to the east at speed u, it is effectively rotating faster than the Earth. Its centrifugal force is then u\ mm2 mu m( m +— =--1---+ 2m mu r ) r r

The first term on the right of (2.10) expresses the basic centrifugal balance of the Earth, while the second is much smaller in magnitude than the third. The horizontal component of this third term, 2musin 0, where 2m sin 0 is known as the Coriolis parameter, f gives the Coriolis force on a unit mass under zonal motion. At the equator there is no force, but away from the equator an object moving eastwards is pushed towards the equator, while one moving west is pushed poleward.

2.6.3 Geostrophy

The basic horizontal driving force in both the ocean and atmosphere is the pressure gradient force. The principal balance observed is with the Coriolis force; this is called geostrophy. However a horizontal difference in pressure is established a flow will be generated from high to low pressure. The Coriolis force acts to deflect this to the right in the Northern Hemisphere, or left in the Southern, until the pressure gradient and Coriolis forces are equal and opposite. Steady flow proceeds, at right angles to the driving pressure gradient (Fig. 2.18). If the pressure gradient is known then this steady velocity can be calculated from the balance equations:

p d y where (u,v) is the horizontal velocity in the (x,y), or (east,north), direction and, for example, d P/dx is the change in pressure per unit distance in the x direction (a partial derivative).

In the atmosphere it is simple to measure pressure gradients, particularly at the surface, as pressure is one of the commonly observed quantities. In the ocean,

Fig. 2.18. Illustration of the geostrophic balance in the atmosphere. An initial pressure gradient pushes flow northwards. Once flow begins it is deflected to the right by the Coriolis force. Eventually, the Coriolis force is large enough to balance the pressure gradient force and the resulting steady flow is at right angles to both forces.

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Balanced forces however, pressure is often used as an approximate proxy for depth (because the pressure is mostly made up purely from the weight of overlying water) and the small horizontal gradients in pressure present have to be inferred from horizontal and vertical changes in density. This requires measurement of the variation of both temperature and salinity with depth. Even with such data it is rarely possible to obtain an absolute calculation of geostrophic velocity because there are slight gradients of the sea surface which need to be known, but are barely measurable even by the most advanced satellite instruments. Geostrophic velocities in the ocean are therefore usually calculated assuming some level of no motion, typically deep in the water column. This can introduce considerable error.

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