INVESTIGATING THE OCEAN THROUGH COMPUTER MODELLING As mentioned earlier, the simple square oceans driven by simple wind

As the amount of water flowing into a space must equal the amount of water flowing out of it per unit time, the rate of flow, i.e. velocity, is also important in continuity considerations. For example, a broad, shallow current entering narrow straits will become faster as well as perhaps becoming deeper.

Figure 4.17 Schematic diagrams to illustrate one- and two-dimensional models for relatively simple, small-scale situations. Note that in these circumstances, the coordinate system shown in Figure 4.15 is applied differently, in that x and u are used for flow along the channel, and y and v are used for flow across the channel.

(a) A one-dimensional model for investigating flow through a channel. Such a model would require information about cross-sectional area (A1, A2 . . . . ) and the average current speed (~1, ~2, ~3 . . . . ) through each cross-section.

(b) A two-dimensional model of the channel can take account of cross-channel flow, vl, v2, v3 ...

between adjacent grid boxes, as well as along- channel flow, ul, u2, u3 . . . which can vary across the channel. The model takes account of flow into and out of each side of each grid box (with the exception of those corresponding to fixed

boundaries), but flow arrows are shown only for the nearest row of boxes. For clarity, flows v~, v2, v3 are showing at the front, but in reality they would be through the centres of the sides of the boxes. For a situation like an estuary, the grid boxes might have sides of -10-100 m.

The mathematical equation used to express the principle of continuity is:

du dv dw

+ ~ + ~ - 0

dx dy dz (4.3d)

which simply means that any change in the rate of flow in (say) the x-direction must be compensated for by a change in the rate of flow in the y- and/or ,z-direction(s). This continuity equation is used in conjunction with the equations of motion and provides extra constraints, enabling the equations to be solved for whatever dynamic situation is being investigated.

4.2.4 INVESTIGATING THE OCEAN THROUGH COMPUTER MODELLING

103

Not really. Models may be kept more simple by assuming that the sea- surface behaves as if a solid barrier is held against it (a so-called 'rigid lid'), but as you know, winds cause horizontal motion of surface water, so a 'rigid lid' is generally considered an unacceptable simplification. However, as most models are concerned with bulk movements resulting from ocean currents, the sea-surface is often assumed to be flat, because its ups and downs would average out over the duration of a model 'time-step'.

At a horizontal sea-bed, w must be zero. More generally (as the sea-bed might be sloping), the velocity

perpendicular

Models are driven or 'forced' by mathematically applying the effect of whatever is/are considered to be the principal driving force(s). The results are constrained by including in the model known, or assumed, values for one or more parameters - for example, in the simple case of flow through straits, it could be the total volume transport through the straits. If the aim of the exercise is to simulate the average situation - the 'mean flow' - the model is run until the flow pattern generated stops changing, i.e. has reached an equilibrium situation.

The early models of Sverdrup, Stommel and Munk, discussed earlier, were driven simply by the frictional influence of wind stress on the sea-surface.

Most modern three-dimensional models also include information about the transport of heat and salt, both into and out of the ocean, and within it. The importance of fluxes of heat and salt for the ocean circulation will be discussed in Chapter 6, but for now we should simply note that heat and salt are conservative properties, in that once a body of water has moved away from the sea-surface, its temperature, T, and salinity, S, can only be changed by mixing with another body of water with different T and S characteristics.

This means that if we can assign initial values of T and S to each grid box, and feed in information about addition and removal of heat and salt and/or freshwater at the sea-surface, we can predict fluxes of heat and salt through the faces of the subsurface grid boxes by using equations for the conservation of heat and salt, similar in type to the continuity equation (4.3d) for the conservation of mass. In this way, we can produce a pattern for the transport of heat and salt around our model ocean, enabling us not only to model a more realistic density distribution, but also to tackle problems of

environmental and climatic relevance.

Figure 4.18 Schematic diagram to illustrate three types of grids used in three-dimensional models. (a) Section showing the bathymetry of the region being modelled. In (b) to (d) the horizontal coordinate system is represented by fine vertical lines, while the vertical coordinate system is indicated by fine horizontal, sloping or curved lines. The most conventional system is (b), known as a 'level' model, because the vertical levels are fixed; in (c), the 'terrain-following' model, the heights of the boxes are a fixed proportion of the ocean depth; and in (d), known as an isopycnic model, the vertical coordinates are isopycnals. In each case, the equations relating to flow velocity are solved for points at the centres of the faces of the grid boxes (cf. Figure 4.17), while those relating to transport of heat and salt are solved for points in the centres of the grid boxes themselves.

How close a modelled situation comes to reality can only be judged by how closely the flow patterns, flow speeds, and T and S distributions generated by the model resemble those actually observed (although, of course a realistic flow pattern could be generated by a modelled situation in which the effects of two or more incorrect assumptions have cancelled out).

Paradoxically, insights into the processes really at work are sometimes provided by those parts of a modelled flow pattern that diverge most strongly from what is observed in reality. These may indicate that factors that were ignored (i.e. terms left out of the equations) because they were thought to be unimportant were, in fact, significant after all. Stommel's modelling of the North Atlantic gyre (Figure 4.12) provides a clear example of this: the fact that the model (2), which included no variation of the Coriolis parameter with latitude, also showed no western intensification of the flow, strongly suggested that variation with latitude was the significant factor (which was, of course, confirmed by model (3)).

Models like those of Stommel, Sverdrup and Munk, which omit unnecessary detail in order to reveal the fundamental processes at work, are known as process models. Another type of model includes as many factors as practicable, so as to be as realistic as possible. Such models are known as predictive models.

An example of a predictive model

Models are sometimes used to predict current patterns over a particular time period, under certain circumstances. For example, in 1988, oceanographic modellers at the Proudman Oceanographic Laboratory were asked by the British Olympic Sailing Team if they could predict racing conditions in the waters off Pusan in the Korea Strait (Figure 4.19(a)). Although current flow in the area is quite complex, the dominant features seem to be (1) strong tidal currents and (2) a mean flow to the north-east in the Tsushima Current, an offshoot of the Kuroshio (the equivalent of the Gulf Stream in the North Pacific).

Because time was short, it was decided to construct a simple two-dimensional model, in which current flow (the Tsushima Current plus tidal currents) would, of necessity, be depth-averaged. This was considered to be an acceptable simplification, but resulted in local wind-driven currents being under-represented, because the flow was spread over the whole depth instead of confined to the surface Ekman layer. It was decided to compensate for this effect at the time of racing by adding a surface current of 3% of the wind speed (from local weather forecasts), at a small angle to the right of the wind (cf. Section 3.1).

The aim was to predict the main features that were likely to occur in the race area off Pusan, particularly any effects of changes in the strength of the Tsushima Current. The model consisted of two parts. The first was a fine- scale model of the actual race area, with a resolution of 450 m (i.e. with grid boxes of 450 m • 450 m), sufficiently small to cope with the eddies

generated off headlands and the sharp divergences of surface currents observed in the area. This was 'nested' within a coarser-scale model of the Strait of Korea, with a grid size of about 10 km, which was used to provide tidal conditions and mean flow consistent with available observations. The Tsushima Current was represented by 'inputting' a north-easterly

o ~" --" =-c~ ~'o =_ o ~; _, -- = ~ ~~ ... ,-. ~ rlc~ 4 ---- ~. ~ .~'r

o~o- o_ ~.~

Figure 4.19 (b) (i) The coarse-scale model, results from which were used to force the fine-scale model. (ii) The fine-scale model of the area off Pusan. The black dots are locations of tide gauges which provided sea-level data against which the model could be checked or verified, i.e. run assuming conditions different from those specific ones used to calibrate the model. A, B, C and D were the centres of the different sail race courses.

107

Figure 4.19 (c) Current patterns off Pusan from the northern part of the fine-scale model, (i) 3 hours before and (ii) 3 hours after high water, assuming a Tsushima Current of 1 knot (-0.5 m s-l). The colours indicate the water depth; red is shallowest, blue deepest (___100 m).

In the South Atlantic Ocean, the shape of the Brazilian coastline diverts a large proportion of the water carried by the South Equatorial Current northwards so that it crosses the Equator as the Guyana Current and joins water from the North Equatorial Current (Figures 3.1 and 4.20(a) (overleaf)).

Much of this water enters the Caribbean Sea through the passages between the islands of the Lesser Antilles, and continues through the Yucatan Channel into the Gulf of Mexico; the remainder continues to flow north-westwards in the Antilles Current outside the chain of islands. Most of the flow entering the Gulf of Mexico takes a direct path from the Yucatan Channel to the Straits of Florida, although some takes part in the circulation within the Gulf itself (Figure 4.20(a)).

4.3.1 THE GULF STREAM SYSTEM

The Gulf Stream proper may be considered to extend from the Straits of Florida to the Grand Banks off Newfoundland. It does, however, have two fairly distinct sections, upstream and downstream of Cape Hatteras (at -~ 35 ~ N).

Between the Straits of Florida and Cape Hatteras, the current flows along the Blake Plateau, following the continental slope (Figure 4.20(b)), so that its depth is limited to about 800 m. In this region, the flow remains narrow and well defined. Its temperature and salinity characteristics show that it is supplemented by water from: (1) the Antilles Current (which includes deep water from the South Atlantic, kept out of the Caribbean Sea by the

shallowness of the passages between the Lesser Antilles); and (2) water that has recirculated in the Sargasso Sea.

Figure 4.20 (a) The Gulf Stream in relation to the surface circulation of the Atlantic. The Stream consists of water from the equatorial current system (much of which comes via the Caribbean/Gulf of Mexico) and water that has recirculated in the North Atlantic subtropical gyre. The broken lines represent cold currents.

(b) Map to show the sea-floor topography off the east coast of the United States, and geographical locations of places mentioned in the text.

109 As the Gulf Stream continues beyond Cape Hatteras, it leaves the continental slope and moves into considerably deeper water (4000-5000 m). While the current was following the continental slope, any fluctuations in its course had been limited and meanders had not exceeded about 55 km in amplitude.

Beyond Cape Hatteras there are no topographic constraints, the flow becomes more complex, and meanders with amplitudes in excess of 350 km are common. These meanders often give rise to the Gulf Stream 'rings' or eddies, mentioned in Section 3.5 (and discussed again shortly).

By the time it has reached the Grand Banks off Newfoundland (Figure 4.20(b)), the Gulf Stream has broadened considerably and become more diffuse. Beyond this area it is more correctly referred to as the North Atlantic Current (or, in older literature, the North Atlantic Drift). Much of the water in the North Atlantic Current turns south-eastwards to contribute to the Canary Current and circulate again in the subtropical gyre (Figures 3.1 and 4.20(a)): other flows become part of the subpolar gyre, or continue north-eastwards between Britain and Iceland.

Continuity and recent ideas about how the Gulf Stream is driven Because of the contributions from the recirculatory flow and the Antilles Cunent, the volume transport of the Gulf Stream increases as it flows northwards (cf. Question 4. l(b)). The average transport in the Florida Straits is about 30 x 10 ~' m 3 s-~: by the time the Gulf Stream leaves the shelf off Cape Hatteras this has been increased to (70-100) x 106m 3 s -l. The maximum transport of about 150 • 106 m 3 s -I is reached at about 65 ~ W, after which the transport begins to decrease again because of loss of water to the Azores Current and other branches of the recirculatory flow (Figure 4.20(a)). We should note here that volume transport values are often quoted in 'sverdrups' (after the distinguished oceanographer), where 1 sverdrup (Sv) = 106m 3 s -l.

Perhaps not surprisingly, these volume transports are much greater than the values that are obtained using Sverdrup's relationship to wind stress curl (Section 4.2.2), which cannot take account of any recirculatory flow, either at the surface or at depth. Another reason for the discrepancy is that, in winter, Gulf Stream/North Atlantic Current water sinks at subpolar latitudes, forming dense deep water which then flows equatorwards. Not only does the dense deep water contribute to the deep recirculatory flow but, for reasons of continuity (Section 4.2.3), the sinking of surface water 'draws' more Gulf Stream water polewards to take its place. In other words, the Gulf Stream is driven not only by the wind, but also by the deep thermohaline circulation.

(Do not worry if you do not fully understand this - we will be considering the thennohaline circulation and formation of deep water masses in Chapter 6.) 4.3.2 GEOSTROPHIC FLOW IN THE GULF STREAM

Figure 4.21(a) and (b) show the distributions of temperature and salinity for a section across the narrowest part of the Straits of Florida, based on measurements made in 1878 and 1914. Figure 4.21(c) shows the velocity of the geostrophic current, as calculated from these temperature and salinity data. Part (d) shows the distribution of current velocity based on direct measurements made by Pillsbury in the 1890s tSection 4.1.1 ): the correspondence between (c) and (d) is remarkable. This single example of agreement between observed current speeds and the values calculated using the geostrophic method (Section 3.3.3) did much to increase confidence in the use of geostrophic calculations. (It is worth mentioning that modem hydrographic sections across the Straits of Florida look very similar to those in Figure 4.21.)

Figure 4.21 The distribution of (a) temperature (~ and (b) salinity, plotted from measurements made in 1878 and 1914, for an east-west section in the narrowest part of the Straits of Florida, between Fowey Rocks, a short distance south of Miami, and Gun Cay, south of Bimini Islands (see Figure 4.20(b)). Note: Salinity values are effectively parts per thousand by weight, but for reasons explained later are usually given without units. (c) The velocity distribution as calculated from the temperature and salinity data, on the assumption that the current has decreased to zero by the depth of the O-contour; and (d), the velocity distribution on the basis of direct measurements by Pillsbury.

111 Study Figure 4.21, then answer Question 4.10, beating in mind that although the warmer water is also more saline, the ranges of temperature and salinity in the Straits of Florida are such that the density distribution is determined mainly by T rather than S. (If it were determined by salinity alone, the water column in the Straits of Florida would be unstable.)

Calculations of the sea-surface slope indicate that the sea-level is about 45 cm lower on the Florida side of the Straits than on the Bimini side.

Figure 4.22 shows the geostrophic velocity as calculated from T and S measurements made at hydrographic stations along an east-west line out from Cape Hatteras (cf. Figure 4.20(b)). The section illustrates how the current is no longer a relatively coherent flow, but instead typically consists of narrow vertical filaments separated by counter-currents - i.e. contrary flow (shaded blue).

Figure 4.22 Geostrophic current velocity (in m s -1) in the Gulf Stream off Cape Hatteras;

the section is east-west so the blue shaded areas (with negative velocity values) represent flow with a southerly component (probably south-westerly). The horizontal scale is smaller than that in Figure 4.21.

Towards the east. In the Gulf Stream, the flow is to the north (or strictly, north-east), and the Coriolis force acting to the right of the flow, i.e. towards the east, balances the horizontal pressure gradient acting to the left (west).

The density distribution is such that at the depth corresponding to the zero-velocity contour, the horizontal pressure gradient has become zero because the effects of the sea-surface slope and the density distribution have balanced out with depth. Below the zero-velocity contour, the horizontal pressure gradient reverses. Flow is to the south (or strictly south-west), and the horizontal pressure gradient force to the left of the flow (i.e. to the east) is balanced by the Coriolis force to the right of the flow (i.e. to the west).

The velocity section shown in Figure 4.22 is one of many that show a deep counter-current flowing south-westwards beneath the Gulf Stream. However, until the 1960s many oceanographers found the idea of a significant current close to the deep sea-bed, at depths of 3000-5000 m, hard to believe. The determination of distributions of temperature, salinity and velocity is fraught with difficulty, especially if the T, S and direct current measurements are widely spaced, so the sceptics could reasonably argue that other interpretations of the data, not involving counter-currents, were equally valid.

In 1965, Stommel developed a theory of the global thermohaline circulation that supported the idea of such equatorward deep currents. However, it was freely drifting floats that in the 1950s and 1960s first provided direct evidence of Gulf Stream counter-currents. We will consider these direct current measurements in Section 4.3.3.

First, however, look at Figure 4.23 which shows T and S sections across the Gulf Stream between Chesapeake Bay and Bermuda, i.e. downstream of Cape Hatteras. The horizontal scale is much smaller than those of Figures 4.21 and 4.22, and the oceanographic stations were too far apart to allow any filaments or counter-currents to be resolved.

The Gulf Stream is the region where isotherms and isohalines are close together and slope steeply down to the east, about 300 km offshore; in Figure 4.23(a) its 'warm core" (20-22 ~ may be clearly seen extending to depths of 200-300 m.

The zone of steeply sloping isotherms and isohalines indicates a boundary between two different water masses - cooler coastal water to landward and warmer Sargasso Sea water on the seaward side. As discussed in Section 3.5.2, fronts like this, with large lateral variations in density, have intense geostrophic currents flowing along them.

Note that the Sargasso Sea water is not only warmer but also more saline than the water on the coastal side of the Stream, which is influenced by cool freshwater input from land. As in the Straits of Florida (Figure 4.21), the salinity distribution on its o w n would result in an unstable situation. However, as is common in the oceans, the temperature distribution here has by far the greater effect on the density distribution and the slopes of the isopycnals.

113

Figure 4.23 (a) Temperature (~ and (b) salinity sections across the Gulf Stream between Chesapeake Bay and Bermuda, based on measurements made between 17 and 23 April, 1932. These cross-sections, like those in Figures 4.21 and 4.22, were plotted using Tand S measurements of water collected at widely spaced hydrographic stations (shown as crosses along the top), and at specific depths: the contours are interpolations based on the spot measurements. Note that as it was expected that there would be more variability in the western part of the section, the hydrographic stations were positioned closer together there.