Movement in fluids is produced by the action of forces. Where this movement is in a steady state it presents an equilibrium between forces. Dynamical oceanography is to a large part a study of balances of forces that can bring about steady state circulations. A starting point for this chapter is therefore a review of the forces that can be found acting in the ocean and their possible balances. Three forces are sufficient to describe and understand most ocean currents. They are the interior pressure field, friction, and the Coriolis force, which will now be reviewed very briefly.
The structure of the interior pressure field can be described by its horizontal and vertical gradients. The vertical gradient is the result of the pressure increase with depth that exists in the fluid regardless of its state of motion; it is of no relevance to the study of fluid motion. Horizontal pressure gradients, on the other hand, cannot be sustained without movement; fluid particles experience a force directed from regions of high pressure towards regions of low pressure and try to move in the direction of the (negative) pressure gradient. Pressure at a point in the ocean is determined by the weight of the water above it, which is determined in turn by the height of the water column and its density. Horizontal pressure gradients can therefore be the result of differences in the height of the water above the horizon in question, in other words variations of sea level in space, and they can be the result of differences in density.
The density of seawater is a function of temperature and salinity. At temperatures above about 5°C it decreases with increasing temperature and increases with increasing salinity. At temperatures of about 1°C and below, the temperature dependence becomes inverted; density then decreases with decreasing temperature (but still increases with increasing salinity). In the temperature range 1 - 5°C temperature has little effect on density, which is then controlled nearly exclusively by salinity (in the usual way). The details of the relationship between seawater density, temperature and salinity, known as the International Equation of State of Seawater, need not concern us here; readers may look them up in Millero and Poisson (1981) or, for example, Pond and Pickard (1983). In the present context it is important to note that we can give a full description of the interior pressure field if we know the distribution of temperature and salinity in space and time. This allows us to calculate horizontal pressure gradients and draw conclusions about the associated water movement.
A few words about units in oceanography are necessary at this point. This text follows the recommendations of Unesco (1981) and expresses temperature in degrees Celsius (degree C) and pressure in kiloPascal (kPa, 10 kPa = 1 dbar, 0.1 kPa = 1 mbar; for most applications, pressure is proportional to depth, with 10 kPa equivalent to 1 m). Salinity is evaluated on the Practical Salinity Scale (see Pond and Pickard (1983) or Unesco (1981) for details) and therefore carries no units. Density ρ is expressed in kg/m3 and represented by σt = ρ - 1000. As is common oceanographic practice, σt carries no units (although strictly speaking it should be expressed in kg/m3 as well).
Returning to the discussion of forces, it is clear from the preceding discussion that we can determine the oceanic pressure field by measuring temperature and salinity as functions of space and using the International Equation of State to evaluate the density field. The horizontal pressure gradients are then obtained by determining the weight of the water above the horizons of interest, ie by integrating density from the surface down.
The second important force in the ocean, the Coriolis force, is an apparent force, ie it is only apparent to an observer on the rotating earth. Basic physical principles tell us that in the absence of any forces, moving objects follow a straight path at constant speed. Observation shows that objects moving over long distances on the surface of the earth experience a deflection from a straight path. This deflection is the consequence of conservation of angular momentum, another basic principle of physics. Since the earth rotates, all points on its surface have their own angular momentum proportional to the distance from the axis of rotation. An object moving poleward gradually comes closer to the earth's axis; in order to maintain its angular momentum and make up for the loss of rotational speed it must move eastward. To an earthly observer (ie one standing on the earth's surface and sharing its rotation) this appears as a deflection from the original poleward path. An observer from space does not share this illusion but sees the object move on a straight path with the earth rotating beneath it. When viewed from a fixed point in space the object observes both principles, movement on a straight path at constant speed and conservation of angular momentum. Viewed from a fixed point on earth it appears under the influence of a force that causes it to deflect from a straight path. This apparent force, which results from the fact that we express all oceanic and atmospheric movement in coordinates that rotate with the earth, is called the Coriolis force. It is always directed normal to the direction of the movement and proportional in magnitude to the speed of the moving body. It acts to the right of the direction of movement in the northern hemisphere and to the left of the direction of movement in the southern hemisphere.
Quantitatively, the Coriolis force is expressed as the product of velocity and a factor known as the Coriolis parameter f = 2 ω sinφ, where ω is the angular velocity of the earth equal to 2π / Td, with Td = 86,300 s the length of a day, and φ is the latitude. f has the dimension s-1; it is therefore also known as the Coriolis frequency.
The most important role of the third important force, friction, is the transfer of momentum from the atmosphere to the ocean. Without it, winds would glide over the surface of the ocean without the build-up of waves and the transport of water in wind-driven currents. Friction can also be important were strong currents run along the sea bed, a situation not usually found in the deep ocean but often encountered in shallow seas and always in estuaries.
Since the oceanic pressure field can be calculated by integration of the density field, which in turn can be derived from observations of temperature and salinity, the principle of geostrophy enables us to derive the oceanic current field from observations of temperature and salinity.
The essence of geostrophic flow can be formulated in a few simple but important rules. These rules express the results of theoretical analysis in a form easy to remember and to apply to field data. A complete derivation of the rules is beyond these notes; interested students should consult texts on geophysical fluid dynamics. Tomczak and Godfrey (1994) give a detailed but still elementary discussion. The rules are as follows.
It can be shown that the oceanic thermocline is depressed in regions of high pressure and raised in regions of low pressure. This allows expression of Rule 1 in terms of hydrographic properties, which leads to
In most deep ocean situations salinity does not change enough to influence the density field to the same extent as does temperature, and the word isopycnals in Rule 2 can be replaced by the word isotherms. This is particularly useful since temperature is the quantity most easily measured in the field and the direction of the geostrophic current can then be deduced from the shape of the thermocline. Caution has to be exercised in the coastal ocean, where salinity variations resulting from land runoff can affect the density distribution at least as much if not more than temperature.
The character of oceanic turbulence responsible for the transfer of momentum between atmosphere and ocean is still not entirely understood. It is therefore important to note that the theoretical result formulated in Rule 3 is independent of the details of the turbulence. This is one of the most important findings in oceanography, since it allows very definite statements about the wind driven oceanic circulation without knowledge of the physics of turbulence. We shall see in the following chapter that this is no longer true in shallow water situations, which makes the dynamics of shallow water regions more difficult to understand.
Note that Rule 3 establishes a relationship between the wind direction and the direction of the Ekman layer transport, not the current in the Ekman layer. The transport is the integral of the current velocity over the layer; it indicates the net water movement effected by the layer. The current direction can and does vary across the layer; but when the effect of the current at all levels within the layer is taken into account, net movement is perpendicular to the wind direction. This will be discussed in much more detail in the following chapter.
How are convergences or divergences of the Ekman transport generated? They cannot be the product of convergences or divergences of the wind field. Figure 2.2 illustrates why this is so.
Theory shows that the key quantity responsible for Ekman layer pumping is the curl of the wind stress. The curl of a field of vectors measures the tendency of the vector field to induce rotation. It has three components (curlx , curly and curlz), each measuring the°of rotation around one of the three axes (two horizontal and one vertical). In oceanography only the third component which relates to rotation around the vertical is of interest, and the expression "curl of the wind field" or "wind stress curl" always refers to that component only. The relationship between Ekman layer pumping and rotation in the wind field can be visualized from the examples given in Figure 2.2.
The principle of Ekman pumping becomes important in a discussion of upwelling which will be developed in detail in chapter 6. In the present context it is sufficient to remember that convergences and divergences of the Ekman layer transport are associated with the presence of wind stress curl. Since divergences are simply negative convergences, we take it as understood from now on that the term "convergence" includes both.
In a steady state, the volume of water contained in any given region of the ocean cannot change. Another way of saying this is that the same amount of water that enters the region has to leave it again. In other words, the circulation through the region is free of convergence (otherwise the sea level would rise without bounds). How can this be achieved in the presence of wind stress curl? The Sverdrup balance shows that the convergence of the Ekman layer transport is compensated by a divergence of the geostrophic flow in the frictionless layer underneath. Instead of accumulating along the convergence line in the Ekman layer (seen, for instance, in the example of Fig. 2.2 on the right), water is moved downward by Ekman pumping and flows away from the region in the depths below the Ekman layer. As a result, variations in sea level produced by winds are extremely small in the open ocean and do not exceed one meter. However, the Sverdrup balance operates only on scales of hundreds of kilometers and fails in shallow water or in the vicinity of coastlines. This means that the wind can produce large departures of the sea surface from its normal position in the coastal ocean, an effect known as sea level set-up or, in extreme situations, a storm surge.
The transition from the upper ocean, which experiences seasonal heat exchange with the atmosphere, to the deeper layers is known as the permanent or oceanic thermocline. In this layer, which usually spans the depth range from below the seasonal thermocline to about 800 m depth, temperature changes gradually to the very low temperatures found in the abyssal ocean (Fig. 2.4). Because of its depth the permanent thermocline cannot exist in the coastal ocean. Its existence is, nevertheless, important for coastal oceanography since it usually occupies the depth range adjacent to the shelf and its properties determine the effect which exchange processes between the coastal ocean and the open sea will have on water properties on the shelf. This is all the more important since other hydrographic properties such as oxygen and nutrients also show large vertical gradients in the permanent thermocline, and the depth to which the coastal ocean can interact with the deep sea is of large consequence for the exchange of nutrients and other properties. This aspect will be discussed in detail again in Chapter 6.