Rules are along Claire DC's pic comp post,

 

I'll ask a question and then whoever gets it right gets to ask there question so on and so forth!!!

**Normal Bushy Quiz Rules Apply**

 

Question:

 

What is the capital city of Outer Mongolia!

Ulanbator innit?

 

Not been though.

congrats flakey!

 

Your question!

 

Uhm....

 

'whats the basic principle of physics which allows a supertanker to float,not sink to a watery grave?'

 

Sad question,can't think of anythin else.

gravity and volume

Uhm....

 

'whats the basic principle of physics which allows a supertanker to float,not sink to a watery grave?'

 

Sad question,can't think of anythin else.

 

Is it not the area covered, im a bit tipsy so cant think of the word, but the area of the base against the density/resistance of the water or somethin?!?! Im good at science too, i should do this when not had a drink!!

The starting point for an understanding of the dynamics of the coastal ocean is the dynamics of the deep ocean. This chapter therefore presents, in a very compressed form, the essentials of the dynamics of deep ocean basins. It can be regarded an abbreviated version of chapters 2 - 4 of Regional Oceanography: an Introduction (Tomczak and Godfrey, 1994), where a fuller treatment, on the same elementary level used in this text, can be found.

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 r is expressed in kg/m3 and represented by st = r - 1000. As is common oceanographic practice, st 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 preceeding 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 priciples, 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 w sinf, where w is the angular velocity of the earth equal to 2p / Td, with Td = 86,300 s the length of a day, and f 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.

 

 

 

geostrophic balance

In most of the deep ocean, currents can be regarded as frictionless and maintained by a balance between the horizontal pressure gradient and the Coriolis force. Figure 2.1 shows the principle. Currents established by a balance between the horizontal pressure gradient and the Coriolis force are called geostrophic currents, and the method to derive them by determining the pressure field from observations is called the geostrophic method.

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.

 

 

 

Rule 1: In geostrophic flow, water moves along isobars, with the higher pressure on its left in the southern hemisphere and to its right in the northern hemisphere.

 

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

 

 

 

Rule 2: In a hydrographic section across a current, looking in the direction of flow, the isopycnals slope upward to the right of the current in the southern hemisphere, downward to the right in the northern hemisphere.

 

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.

 

 

 

Ekman layer transports

As mentioned before, the generation of currents by wind requires frictional transfer of momentum from the atmosphere to the ocean. The effect of wind stress on oceanic movement is best analyzed by excluding for the moment the effect of the pressure field on the balance of forces and assuming that the ocean is unstratified, ie of uniform density, and its surface horizontal. The balance of forces is then between friction and the Coriolis force. Since wind blowing over water is always associated with turbulent mixing, the condition of uniform density is usually satisfied in the wind affected surface layer, and the balance between friction and the Coriolis force prevails. This layer is often called the Ekman layer, after Vagn Walfrid Ekman who made the first quantitative study of the dynamics of wind driven currents. Ekman's major finding is summarized in

 

 

Rule 3: The wind driven transport of water in the surface layer of the sea (the Ekman layer) is directed perpendicular to the direction of the wind, to the left of the wind in the southern hemisphere and to the right of the wind in the northern hemisphere.

 

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.

 

 

 

the Sverdrup balance

If we now are to describe the circulation in the real ocean we have to allow for the presence of all three major forces - pressure gradient, Coriolis force and friction - and combine them in a single balance. This was first achieved by the oceanographer Hans Ulrik Sverdrup, and the balance of forces found in most of the ocean is therefore known as the Sverdrup balance. Briefly stated it says that friction is only important in the Ekman layer at the surface, below which all movement is friction free. Movement below the Ekman layer is induced by transfer of water from the Ekman layer to the depths below as a result of flow convergence or divergence in the Ekman layer. In the case of flow convergence, water is forced downwards to the depths below; this process is known as Ekman pumping. In the case of flow divergence, water is pumped upwards into the Ekman layer; this process is known as Ekman suction (or negative Ekman pumping).

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 kilometres 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 permanent and seasonal thermocline

To complete this very brief discussion of deep ocean dynamics we have to introduce and explain the terms permanent and seasonal thermocline. A thermocline is a layer of rapid temperature change from the warm waters of the upper ocean to the colder waters below. It is thus identified by a local maximum of the vertical temperature gradient. The seasonal thermocline is found in all but polar climates during spring, summer and autumn underneath the surface mixed layer (Ekman layer). Its development during that period is best understood by beginning with the winter situation, when the temperate and subtropical ocean shows uniform temperature over its top 200 m or more. Increasing solar radiation during spring produces a warming of the surface mixed layer, where the heat received from the atmosphere is uniformly distributed by mixing. As a result a zone of rapid transition from warm to cold water, the seasonal thermocline, develops at the base of the mixed layer. Its depth is determined by the depth to which the wind mixes the surface layer. Frequent spring storms initially set it at several tens of metres. Winds tend to moderate during summer resulting in a shallowing of the seasonal thermocline, which then often lies above a remnant of the maximum temperature gradient developed during spring (Fig. 2.3). Throughout spring and summer the determining factor for the depth of the thermocline is wind mixing. This situation changes in autumn when the surface layer cools and its density increases, causing instability of the water column. Convection sets in, drawing in cold water from underneath the mixed layer until the stratification is stable again. The resulting deepening of the mixed layer is usually quite rapid, causing the depth of the seasonal thermocline to increase by a hundred netres or more within a matter of a few weeks. In winter the heat stored from the summer is exhausted and the seasonal thermocline disappears (Fig. 2.3).

The transition from the upper ocean, which experiences seasonal heat exchange with the atmosphere, to the deeper layers is knwon 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.

 

 

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© 1996 M. Tomczak

contact address: matthias.tomczak@flinders.edu.au

 

Ekman layer dynamics for shallow seas with stratification

 

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The simple relationship between the direction of the wind and the direction of the Ekman layer transport in the deep ocean is valid as long as the total water depth H is larger than the depth of the Ekman layer dE. The exact condition, derived from theoretical considerations of fluid dynamics, is dE2

Figure 3.1 shows how current speed and direction change with depth in an Ekman layer generated by a wind blowing over a deep ocean. Current speed is largest at the surface and decreases rapidly with depth. Current direction also changes with depth, and we see the remarkable result that at some depth the current actually opposes the surface current; however, at that depth the current is so small that it can be considered negligible. This depth is therefore taken as the bottom of the Ekman layer or Ekman layer thickness dE. At the surface the current is directed 45° to the right (left) of the wind in the northern (southern) hemisphere. Somewhere further down in the water column the current flows at right angle to the wind, while below that depth it flows at various angles against the wind direction. The total transport in the Ekman layer is the combined effect of water movement in the Ekman layer, ie the integral from the surface to the depth dE. This explains why in the deep ocean the Ekman layer transport is directed at right angle to the wind direction: The transport contributions in the direction of the wind found in the upper Ekman layer are cancelled by the contributions in opposite directions found in the lower Ekman layer. Only the transport components perpendicular to the wind direction contribute to the final integral.

 

The rather intricate structure of the Ekman layer is the result of a balance between friction and the Coriolis force. Friction transfers momentum from the atmosphere to the ocean. In a non-rotating frame of reference this would result in water movement in the direction of the wind. Rotation gives rise to an apparent force (the Coriolis force) which acts perpendicularly to the direction of movement. The combined action of friction and the Coriolis force produces a surface current directed at 45° from the wind direction and further deflection from the wind direction down the water column.

 

The details of the Ekman layer structure depend on several assumptions which are not always easy to verify. The most important assumption, and the one associated with the greatest uncertainties, concerns the process of momentum transfer from the sea surface to greater depths. In the absence of turbulence, momentum would be transferred by friction between the water molecules. Frictional effects can be quantified through a molecular friction coefficientl, which is a property of the medium and a measure of the viscosity of the fluid; it can be determined in the laboratory and has the units kg m-1 s-1. A quantity often used is the kinematic molecular viscosityn = lr-1, where r is the water density with units kg m-3. If momentum is transferred by molecular friction, the frictional boundary layer thickness, ie the distance over which the velocity is under the influence of the drag force of the wind, can be shown to be given by

 

 

 

 

 

 

 

where f is the Coriolis parameter or Coriolis frequency (a typical value for mid-latitudes is 10-4 s-1). The kinematic molecular viscosity of water is of the order of 10-6 m2 s-1, so the frictional boundary layer is typically about 0.1 m thick. Such molecular boundary layers are easily produced in laboratory tanks and sometimes seen when a light breeze blows over a tranquil pond. Floating leaves or other suspended matter will then indicate swift water movement right at the surface, progressively slower movement in the next few centimetres and no movement below. This is, however, not the everyday situation in the coastal ocean, where the frictional boundary layer (the Ekman layer) is tens of metres thick. The conclusion must be that molecular friction cannot be responsible for the transfer of the wind's energy to the water. Transfer of momentum in the ocean is achieved by turbulence.

 

Unlike viscosity, turbulence is not a property of the medium but of the flow; its intensity and structure depend on the current shear (both horizontal and vertical), the stratification, the wave field, the roughness of the ocean floor and other factors. The major mechanism which contributes to oceanic turbulence are eddies of different size, from the smallest swirls a few metres across to the large geostrophic eddies with diameters of 200 km or more. Wind waves contribute to the turbulence at the sea surface, and other processes contribute to turbulent motion on the centimeter scale. The water parcels moved by the eddies are several orders of magnitude larger than the water molecules. By exchanging their properties with their surroundings they are much more effective in transporting momentum downward from the sea surface than molecular diffusion.

 

To describe the effect of turbulent momentum transfer in exact detail requires the knowledge of the details of the eddy field, under most circumstances an impossible task. Fluid dynamicists have convinced themselves that for nearly all situations its effect can again be described through a viscosity coefficient Av, and the associated boundary layer thickness is then again given by

 

 

 

 

 

 

 

This coefficient of turbulent viscosity Av has again units of m2 s-1 but is no longer a material constant; it is several orders of magnitude larger than the kinematic molecular viscosity n and varies from situation to situation. The coefficient of turbulent viscosity Av is often called the turbulent friction or mixing coefficient. Since eddies are the main mechanisms how oceanic turbulence transfers momentum it is also known as the eddy coefficient. Often it is referred to as the Austausch coefficient (Austausch = German for exchange indicating exchange of momentum through eddies). Typical values for Av are in the vicinity of 0.1 m2 s-1 but can vary by an order of magnitude or more to either side, giving a range of 15 - 150 m for the Ekman layer thickness.

 

One of the most important quantities in the theory of the oceanic circulation is the Ekman layer transport. It might appear that calculating the transport is a rather unreliable operation since it is based on an integral of velocity over the Ekman layer, which are both functions of Av. As it turns out, for the deep ocean (dE2

 

Before we proceed to discuss the modifications of the Ekman layer in shallow seas it is probably helpful to look at some observations of Ekman layer currents. Figure 3.2 shows a photograph of an experiment in which a vertical streak of dye was brought into the upper ocean (The ship used in the experiment is visible in the photo). The water was reasonably clear and the dye could be seen nearly through the entire water column. After some time the shape of the dye streak had changed to the configuration shown. If water movement were the same at all depths the dye streak would appear from the air as a single blob. The fact that it turned into a patch of elongated shape with a distinct curvature indicates a decrease of water movement with depth with a systematic change in direction, in agreement with the Ekman spiral concept.

 

We can find further evidence for the existence of Ekman spirals if we turn our attention to the ocean floor. A current flowing over a rough bottom experiences a drag in a very similar way as a quiescent ocean experiences drag from a wind blowing over its surface. Whether the effect of the drag is to move the water along (the wind) or to hold it back (the bottom) does not make much difference; we could just as well imagine that the water is at rest and the bottom moving in the opposite direction. There exists therefore an Ekman layer above the bottom which serves to bring the current down from whatever its strength is above the bottom Ekman layer to nothing at the sea floor. Figure 3.3 shows an example of such a situation. The observations were taken in 70 m water depth; the surface Ekman layer was only 30 m thick and not covered by the observations. The bottom Ekman layer is seen to be about 25 m thick. Between the two Ekman layers is the region of frictionless geostrophic flow (seen in the data at 25 m and 35 m above the bottom).

 

 

 

Form of the Austausch coefficient

In contrast to the Ekman layer transport, which is independent of the Austausch coefficient, details of the velocity profile in the Ekman layer are affected by the details of Av. To this point, the discussion of Ekman layers assumed that Av is independent of depth. We now review the effect of depth-variability of Av on the velocity profile and possible reasons why the coefficient might vary with depth.

Since the surface Ekman layer is a result of wind action it is reasonable to assume that the turbulence elements responsible for the transfer of momentum are mainly the wind waves. Particle movement in wind waves in deep water is on orbital paths in a vertical plane. The diameters of the orbital paths decrease exponentially with depth; hence it can be argued that the intensity of the turbulence and thus the Austausch coefficient also decrease exponentially with depth. The depth over which this decrease occurs is a function of the dominant wave period (since the exponential decrease of the particle path diameters is a function of wave period), which in turn is some function of wind speed. One way of replacing the simple assumption of constant Av by a more realistic description is therefore to assume an exponential decrease of Av with depth and make the decrease dependent on wind speed.

 

Implementation of this idea is not trivial, and we shall not pursue the details further. We only note that the effect of an exponential decrease of Av is to concentrate most of the mixing in the upper wave zone, thereby reducing the Ekman layer depth. Our "first guess" estimate of 50 - 150 m for the Ekman layer thickness is therefore an upper bound for what we can expect. An example of observations supporting the notion of a depth-dependence of Av (though not strictly exponential in this case) is shown in Figure 3.4.

 

Waves are not always the most important turbulence- generating mechanism. Current shear tends to produce eddies. Currents in the sea nearly always display much stronger shear in the vertical than in the horizontal (on the 1 - 100 m length scale, current speed and direction change much faster vertically than horizontally), so the formation of small overturning eddies is more common than the formation of swirls with a vertical axis of rotation (on the same scale). Acting against the formation of overturning eddies is the stratification, since it is more difficult to move water up or down in the water column against a strong density gradient. It is possible to quantify the tendency for the formation of turbulence by comparing a measure for the stratification with a measure of the vertical current shear. The Richardson number Ri is a non- dimensional number which achieves this. It is defined as

 

 

 

 

 

 

 

Here, g is gravity (g = 9.8 m s-2), r density (r = 1025 kg m-3 is a typical value for sea water) and u the velocity. The vertical density gradient dr/dz measures the stratification, the vertical change of velocity du/dz gives the current shear. The larger Ri, the larger the relative role of stratification and the less likely the presence of active turbulence. Inversely, the smaller Ri, the larger the relative role of current shear and the more likely the presence of turbulence. Observations show that turbulence sets in if Ri falls below a critical value; most researchers give this value as Ri = 1/4, others suggest that it is slightly smaller.

 

The Richardson number can be used to derive a depth-dependence for the Austausch coefficient which somehow reflects the different levels of turbulence at different depth. A commonly used approach is to make Av inversely proportional to Ri. Strong turbulence or small Ri then gives a large Av, which makes sense. Figure 3.5 shows a typical summer situation on a shelf with weak tidal mixing. The heat received at the surface is mixed downward by wave action. Winds in summer are usually light, so the resulting warm mixed layer is relatively shallow and separated from the colder water underneath by a strong thermocline. The stability of the water column is largest in the thermocline, where the Richardson number shows a maximum and the Austausch coefficient a minimum. The extremely low values of Av mean that it is difficult to expand the Ekman layer beyond the thermocline depth even under strong wind conditions. The Ekman layer will not deepen in autumn, despite the seasonal increase in mean wind speed, until cooling at the surface initiates convective mixing, pushing the thermocline downward and with it the minimum of Av.

 

Another example for a shallow surface Ekman layer as a result of a shallow thermocline is indicated in Figure 3.3. The fact that current speed and direction do not change between 25 m and 35 m from the bottom in 74 m water depth suggests an Ekman layer of less than 39 m extent, significantly less than what is normally observed. The shallowness of the thermocline is the result of coastal upwelling, which lifts the thermocline towards the surface, reducing the Ekman layer thickness.

 

The principle that the magnitude and variation with depth of the Austausch coefficient depends on the character of the turbulence applies at the ocean floor as well. In contrast to the free surface, the sea floor imposes a rigid boundary to all movement. As a result, the turbulent eddies close to the floor are restricted in size; their diameter cannot exceed the distance from their centre to the boundary. Larger eddies can occur further away from the sea floor. If this is expressed in terms of turbulent viscosity it is found that Av is zero at the bottom and grows linearly with distance, until it reaches the value typical for turbulence in the ocean interior. This results in a velocity profile where the current increases logarithmically in strength with distance from the sea floor and does not change direction. In the atmosphere, this logarithmic boundary layer determines the wind profile in the first tens of metres from the ground upward. In the ocean it rarely exceeds a few metres in thickness. It occurs as a modification of the Ekman layer over the first few metres and is of no consequence to the direction and magnitude of the total Ekman layer transport discussed below. Figure 3.6 compares the logarithmic layer with the Ekman layer for an oceanic situation. The importance of the logarithmic boundary layer is in its role for the movement of sand and fine sediment in the coastal zone. Coastal engineers concerned with beach stabilization and shallow water circulation spend significant effort on understanding the logarithmic boundary layer in detail.

 

It is easy to see that in many situations it is impossible to make a definite statement about the depth-dependence of Av since the information necessary to estimate the size of turbulence elements or to calculate the Richardson number is not always available. Nevertheless, some general conclusions about the vertical extent of the Ekman layer emerge from our considerations, and it is worth summarizing them here.

 

The starting point for an estimate of Ekman layer depth is eqn. 3.2 with constant Av. It applies to an unstratified deep body of water with depth-independent mixing. These conditions are often satisifed at the sea floor where the stratification is weak and mixing is not associated with waves. The estimate from eqn. 3.2 would include the logarithmic layer as a sub-layer next to the ocean floor. At the surface, mixing is nearly always associated with wind waves. The resulting exponential decrease of Av with depth reduces the estimate of Ekman layer depth. The presence of a shallow thermocline can reduce the Ekman layer thickness further, since turbulence does not penetrate the thermocline.

 

In coastal oceanography it is often important to ascertain whether observations were made under conditions dE2> H2 (shallow sea dynamics). The procedure of assessment is to derive an estimate of dE from eqn. 3.2 and check for the presence of a thermocline. The smaller of the two depths (dE and the thermocline depth) is then used for comparison with the water depth to decide whether deep or shallow dynamics apply to the situation. The importance of a correct assessment becomes evident when we now turn to the question of Ekman layer transport in shallow water.

 

Ekman layer transports

Figure 3.7 shows how current speed and direction change as the water depth H becomes increasingly smaller. The Ekman spiral observed in the deep ocean (seen in Figure 3.1) is found as long as H is larger than 1.25 dE. As H decreases, the spiral changes shape. In the range dE > H > 0.5 dE the current maintains its speed but does not turn against the wind direction at depth. Current speed is reduced when H decreases further, and the current aligns more and more with the direction of the wind. In very shallow water (H

In all situations the Ekman layer transport is independent of the details of the Austausch coefficient, but in shallow water it is no longer directed perpendicular to the wind direction (as it is for H > 0.5 dE). The shallower the water, the more does the Ekman layer transport - which under these conditions becomes the transport of the total water column - point in the direction of the wind. This is of particular importance for the understanding of wind-driven upwelling in shallow seas. Chapter 6 will develop these ideas further.

 

 

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© 1996 M. Tomczak

contact address: matthias.tomczak@flinders.edu.au

 

Sea level set-up and buoyancy-driven flows

 

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This chapter deals with ocean circulation processes that occur in the deep ocean but are usually not of enough significance to warrant much attention. The presence of a coast magnifies these processes, giving them much more importance in coastal oceanography. They relate to the ocean's response to atmospheric forcing and can be divided into two groups.

 

The atmosphere forces the ocean in two ways, through tangential stress (wind), and through modification of the ocean's buoyancy (heating, cooling, evaporation, precipitation). The response of the coastal ocean to the first type of forcing is sea level set-up; the response to the second type of forcing is buoyancy-driven flow.

 

Sea level set-up

Wind-generated surface currents reflect the structure of the wind field on a variety of scales. On the oceanic scale the global system of Trade Winds, Westerlies and Polar Easterlies creates the large oceanic gyres that dominate the circulation of entire ocean basins. On the synoptic weather scale, atmospheric low pressure systems create storms, which introduce significant variability into the oceanic current field on time scales of days to weeks. Variations of the wind field in space generate convergences and divergences of the oceanic flow field on all scales.

In principle, a region of surface flow convergence means an accumulation of water in the convergence region. However, in the open ocean such accumulation is kept small, because a rise of the sea level from accumulation of mass immediately causes a depression of the thermocline, followed by vertical water movement from the upper ocean to greater depth. The main effect of a surface convergence is thus vertical exchange of water between the upper ocean and the underlying regions. This phenomenon is commonly known as Ekman pumping. Variations of sea level in the open ocean are therefore modest, of the order of tens of centimetres, and rarely reach 1 m in height.

 

In the coastal ocean the situation is quite different. To begin with, currents in the open ocean rarely oppose each other on scales of tens of kilometres, and convergences (or divergences) are usually the result of slow changes in current speed and direction. The presence of a coast inhibits horizontal water flow and produces vastly stronger convergence (and divergence) effects than are ever encountered in the open ocean. Another factor is the restricted water depth on the shelf. The depth of the coastal zone usually does not exceed 200 m or so, which poses a severe restriction on vertical water movement on a large scale. As a result, water can pile up against the coast to great height, a phenomenon known as sea level set-up or surge. Sea level set-up can pose a severe threat to coastal land and can lead to large scale flooding and loss of life in low lying coastal regions.

 

The major reason for sea level set-up are atmospheric storm systems. Stronger storms are of course more dangerous than moderate storms; but the severity of the resulting storm surges depends on the coastal topography just as much as on the absolute wind strength. (Theoretical arguments show that when water is piled up against a coast the sea surface slope dz/dx is proportional to the wind stress t and inversely proportional to the water depth h; g is gravity and r the water density; compare the sketch on the right.) Some parts of the world's shelf regions are more susceptible to storm surges than others. Figure 4.1 shows an example of a storm surge in the North Sea. The storm system responsible for a surge in the North Sea is usually generated in the region of the Westerlies over the North Atlantic Ocean. It moves with the westerly wind in a generally north eastward direction, passing north of Scotland and continuing along the Norwegian coast. The storm produces a weak surface elevation in the open ocean, which enters the North Sea from the north and builds up to significant height as it crosses the continental slope and enters shallower water. Under the influence of the Coriolis force, which in the northern hemisphere acts to the right of the movement, the surge leans against the British coast and increases in height as it moves south towards the Dutch and German coast.

 

The severity of a storm surge depends to a large extent on its timing relative to the tidal cycle. Many storm systems pass quickly and produce a surge of not much more than 12 hours in duration. If the peak of the surge occurs at low tide the effect will be minimal. If, on the other hand, the peak of the surge coincides with high tide, the water level can reach quite unusual height. The most dangerous storm surges are produced by long lasting storm systems that coincide with spring tides. There have been surges during which the water level did not fall after high tide but kept rising despite of ebb tide, bringing the following high tide close to the height of the dykes along the coastline. The constant pounding from the sea under the lashing of the storm is a severe test for any dyke system. Much land can be flooded when a dyke breaks, and nothing can control the force of the water as it comes rushing in. The following table list some major flood events produced by storm surges.

 

 

Some major historical storm surges

Date Shelf region Estimated maximum

surge height Estimate of lives lost

November 1218 Zuider Zee (Dutch North Sea) unknown 100,000

October 1737 India and Bangladesh 12 m 300,000

1864 Bangladesh unknown 100,000

October 1876 Bangladesh 15 m 100,000

1897 Bangladesh unknown 175,000

September 1900 Galveston, Texas (Gulf of Mexico) 4.5 m 6000

Jan/Febr 1953 Southern North Sea 3.0 m 2000

March 1962 Atlantic coast, USA 2.0 m 32

November 1970 Bangladesh 9.0 m 500,000

 

The table indicates that the two ocean regions most exposed to severe storm surges are the North Sea and the Bay of Bengal, particularly the coastline of Bangladesh. Whether a storm surge develops into a national disaster is determined by several factors. The disproportionate number of casualties in Bangladesh is to a large degree the result of the country's underdevelopment. Holland is one of the most densely populated countries of the world. Most of the country is below sea level and relies on dykes keeping the sea out. It experiences severe storm surges every year but is very rarely flooded because an infrastructure built over centuries and maintained by a rich nation secures the coastline. In comparison, the coastal defences of Bangladesh are of ancient design and the country lacks the resources to improve and maintain them.

 

Another factor that turns Bangladesh's storm surges into national calamities is its closeness to the equator. Atmospheric low pressure systems, which bring the storms experienced in temperate climate, can develop into cyclones in the tropics. Tropical cyclones (known as hurricanes in America) are extreme events of nature; wind and rain are very much more powerful in cyclones than in the storm systems experienced in the North Sea. Cyclones can therefore produce intense surges, with the sea level sometimes rising rapidly by several metres in a matter of hours. Figure 4.2 shows examples of sea level set-up from tropical cyclones experienced in various towns around Australia. Although none of the cyclones caused flooding with loss of life, severe flooding usually occurs with all cyclones because they are associated with record rainfall. The flooding events that occurred with storm surges in Bangladesh are to a large extent also the result of overflowing rivers following heavy rain.

 

Cyclones form over the sea and can create large surges when they come close to land. They often do not cross into land immediately but continue on a path parallel to the coast. This can amplify the surge if the propagation of the cyclone is in resonance with the propagation of coastal Kelvin waves. The dynamics of these waves is discussed in chapter 8, where it is shown that along the east coast of ocean basins Kelvin waves can only travel poleward, while on the west coast they can only travel towards the equator. Cyclones are features of the tropics; they form in the vicinity of the equator and move away towards the from it. Resonance between cyclone propagation and Kelvin wave propagation can therefore only occur on the east coast of ocean basins (the west coast of continents). Figure 4.3 verifies this for the Australian continent. It shows that sea level set-up from cyclones is generally stronger on the west coast than on the east coast.

 

Most cyclone tracks over the Bay of Bengal move poleward parallel to the Indian coast. Many reach landfall in Bangladesh after travelling in close proximity to the Indian coastline. This favours resonant build-up of extreme surges. Figure 4.4 shows examples of cyclone tracks over the Bay of Bengal and associated surge heights. Managing the storm surges in the Bay of Bengal remains one of humanity's greatest challenges.

 

Buoyancy-driven flow

The coastal zone is generally a region of increased temperature and salinity variability in the ocean, and density differences between the open ocean and the shelf are a common occurrence. Variations of temperature are enhanced through increased tidal mixing, shallow water depth and advection of warm or cold air from land. This leads to enhanced variability of the near-shore current field.

While the effect of temperature on the density field of the coastal ocean manifests itself as an enhancement of processes that occur in all ocean regions to some degree, the salinity contribution can be much more substantial. The coastal zone is not only exposed to the same freshwater balance (precipitation - evaporation) as the deep ocean; it also receives the freshwater that collects from rain over the land. This river run-off can have a substantial impact on the oceanic density field on the shelf and create its own circulation.

 

Figure 4.5 is a sketch of a situation where the freshwater input from land is the result of outflow from numerous rivers along the coast and can be described as a continuous freshwater source for the shelf region. Because a freshwater source lowers the density of the sea water on the shelf, the isopycnals slope downwards as they approach the shelf from the deep ocean. The resulting thermocline slope produces a geostrophic current parallel to the coast, directed equatorward on the west coasts and poleward on the east coasts of the ocean.

 

It is seen that river run-off from the continents produces a cyclonic circulation around the ocean basins. This circulation is generally restricted to the shelf and superimposed on the wind driven general circulation and therefore not always easily recognised in observations. On the west coasts of the ocean it is overwhelmed by the western boundary currents. In the subtropics the circulation of the deep ocean is anti-cyclonic, so the subtropical western boundary currents oppose the direction of the buoyancy driven flow, but buoyancy driven flow can occur in the form of a narrow countercurrent on the shelf. Currents on the east coasts of the ocean are generally weaker than the western boundary currents, and buoyancy driven flow can occur as countercurrents.

 

A prominent example of buoyancy driven flow is found along the Pacific coastline of Canada and Alaska. These coasts are in high latitudes, and the circulation in the adjacent ocean is cyclonic as part of the subpolar gyre of the North Pacific Ocean. Numerous rivers enter the ocean from the Canadian and Alaskan coast, to the effect that the continent acts as a nearly continuous line source of freshwater. The resulting buoyancy driven flow enhances the general cyclonic circulation of the region by increasing the current speed on the shelf. The strongest currents are usually found within 25 km off the coast. Figure 4.6 shows how the Alaska Coastal Current is intensified through river run-off along its path. The freshwater output from rivers amounts to only 4% of the current's transport but the current reacts with a lag of about one month, doubling its transport if the freshwater input is doubled. The variations of the Alaska Coastal Current are thus not simply the result of additional freshwater carried by the current but a response to the freshwater-induced changes in the density field.

 

 

 

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© 2000 M. Tomczak

contact address: matthias.tomczak@flinders.edu.au

 

Tides in shallow seas and estuaries

 

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Although the tide-generating forces are all-pervasive and no water particle can evade its influence, they are of no consequence to the mean large-scale oceanic circulation. The reason is that the tides are a form of low amplitude periodic motion and do not contribute to the balance of forces for the steady state. A more formal way of expressing this is that tidal motion and the mean water movement of the steady state circulation can be studied independently, and their joint effect on the movement of water particles and on the distribution of water properties can be found by adding the results from the two independent studies. (In the framework of mathematics this is known as a linear system. Linear systems have the useful property that the sum of any two solutions, such as the solution to the mean steady state equations and the solution to the tidal equations, is again a solution to the system. This allows us to build the full description of the water's movement through a succession of solutions to simple problems.) Our review of deep ocean dynamics in Chapter 2 therefore did not include a review of the tides.

The situation is different when it comes to the balance of forces in shallow seas, even if we consider only the steady state. In many shallow seas tidal movement, though still periodic, is no longer weak, and can result in mean water movement known as the residual flow. More importantly, tidal currents cause mixing strong enough to determine the stratification of some shallow seas. A description of shallow water dynamics therefore has to include the effect of tidal movement.

 

Estuarine dynamics depend entirely on the tides. The fact that in estuaries even the mean circulation cannot be understood without consideration of a strictly periodic phenomenon makes the study of estuarine dynamics intrinsically more difficult than the study of the dynamics of the deep ocean. (Mathematically, estuaries represent a non-linear system. Solutions to non-linear systems cannot be found by adding solutions of simpler sub-systems.) The balance of forces in estuaries inevitably includes a representation of tidal effects. This is not always evident from the equations used for the study of estuarine dynamics, which rarely if ever include the tide-generating forces in their original periodic form. Most models of estuaries include the effect of tidal mixing by choosing appropriate formulations for the terms which represent the effects of friction and diffusion. The presence of tidal motion is not obvious from the resulting equations. As we shall see in Part 2, it can also take on different forms depending on the type of estuary. For a correct assessment of the circulation in an estuary it is important to understand how the tides act in the estuary and how they can be represented in the balance of forces.

 

This chapter gives a very brief review of the tide-generating forces and their impact on the world ocean. Emphasis is on those aspects which are of consequence for the coastal ocean and estuaries. Many important aspects of the tides (for example details about constituents, inequalities etc.) are not included; these are left to dedicated texts on tides. The approach here is similar to the procedure adopted with earlier topics: We begin with a review of the situation in the deep ocean and proceed to a discussion of the modifications produced by shallow water.

 

 

 

The tide-generating forces

Tides are the result of gravitational attraction between stellar bodies. A discussion of the tide-generating forces therefore begins with a look at the forces which act on the earth as it travels through space.

Newton's Law of Gravitation states that the force of gravity between two stellar bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Since the distances from our earth to nearly all other large objects in space are so enormous, the only stellar bodies of tidal significance are the moon and the sun. The effect of the two can be studied separately, and most properties of the tides as they are observed can be understood by assuming that earth, moon and sun are all perfect spheres of homogeneous mass composition. This allows us to apply Kepler's First Law which describes the movement of stellar bodies around each other and states that the earth and the moon, or the sun and the earth, follow elliptical paths around one of the two focal points of an ellipse. Since the sun's mass is about 332,000 times larger than the mass of the earth, the focal point of the sun-earth system is located inside the sun. Likewise, the focal point of the earth-moon system is located inside the earth, since the moon's mass is only 1.2% of the mass of the earth.

 

The movement of the earth on its elliptical path is illustrated in Figure 5.1. To understand what generates the tides it is convenient to forget for the moment that the earth rotates around its axis once every day and consider its movement with respect to sun or moon without its daily rotation. The earth revolves around the focal point without changing its orientation in space. This type of movement is known as revolution without rotation. We can see an example of revolution with rotation when looking at the moon: It does not maintain its orientation in space but always faces the earth with the same side; the moon's "backside" was not known to us until we could send spacecraft up to look at the moon from behind. The moon's movement around the earth is therefore similar to the movement of a stone twirled around at the end of a string. In contrast, to model the earth's rotation at the end of a stick would require a mechanism which keeps the orientation of the earth's axis fixed in space.

 

Tides are the result of the balance between the force of gravity and the centrifugal force; they cannot exist under revolution with rotation. Figure 5.1 shows the balance of forces under revolution without rotation. The gravitational force varies with distance from the attracting body; it is larger at points on the earth's surface closer to the sun (or moon) and smaller at points on the opposite side. The centrifugal force, on the other hand, is determined by the angular velocity of the earth's movement, which under revolution without rotation is the same for all points inside and on the surface of the earth. The two forces balance each other exactly at the earth's centre and when integrated over the mass of the earth. On the earth's surface the balance is not exact, and the remaining force varies in strength and direction. It is directed outward, acting in the opposite direction to gravity (ie in the vertical, indicated by the open arrows in the Figure), at the point directly under the sun (or moon) and at the point directly opposite. This produces a minuscule variation of gravity, not enough to be noticeable without extremely sensitive instruments and certainly not enough to produce oceanic tides of the observed magnitude. More importantly, over most of the earth's surface the remaining force acts horizontally, ie in a direction which in an ocean at rest does not experience any other forcing. It is this horizontal component (indicated by the full brown arrows) which is responsible for tides in the ocean and is therefore known as the tide-generating force.

 

The tide-generating force is periodic around the earth and produces a series of convergences and divergences (see Figure 5.1). If we now allow for the daily rotation around the earth's axis, we find that the sequence of divergences and convergences of the tide-generating force sweeps around the earth once every day. As a result, water is moved and accumulates in one region and is drained away in another region, and these bulges and depressions travel across the ocean. In other words, oceanic tides are waves of very long wavelength driven by currents which are produced by the horizontally acting tide-generating force. To put it even more succinctly, the rise and fall of the sea level which we generally associate with the word "tides" is a result of horizontal water movement; it is not the primary response to the tide-generating force.

 

 

 

The equilibrium tide, the dynamic theory of tides

It is tempting to think that on an earth without continents the tidal wave follows the distribution of the tide-generating force exactly, so that the arrows of Figure 5.1 can be interpreted directly as a depiction of the tidal currents: Water accumulates under the sun or moon and at the opposite point on the earth's surface, while tidal currents are largest along a line half-way between these points; the entire pattern is tied to the sun's or moon's position and follows its apparent path around the earth. This theoretical idea is known as the equilibrium tide. It was first proposed by Bernoulli in 1740 who used it to explain many tidal features such as periodicity, inequalities between successive high waters and low waters, and the occurrence of spring tides near full and new moon. The equilibrium tide theory was a great step forward in explaining the tides, being based on a correct understanding of the tide-generating force. Before Bernoulli's theory the tides were explained as being generated by the ocean breathing in and out and other unrealistic ideas.

Nevertheless, it is clear that the tides cannot react to the tide-generating force in the way assumed by the equilibrium tide. To move the tidal wave around the earth within one day would require the movement of enormous amounts of water with the speed of modern aircraft and is physically simply not possible. This was recognized by Laplace in 1775 when he developed the dynamic theory of tides. Laplace accepted Bernoulli's idea of the tides as long waves but pointed out that every finite volume of fluid has its own preferred wave frequencies. If some force tries to excite periodic motion, the reaction of the fluid will be much stronger if the forcing occurs at one of these resonance frequencies than if it occurs at other frequencies. As an example, imagine you hold a bowl or a small tank filled with water and move it gently back and forth. At frequencies other than a resonance frequency the water in the vessel will only follow the movement of the vessel. If the vessel movement occurs at a resonance frequency the water level will undergo strong oscillation, possibly causing some water to spill over. It is not difficult to identify the resonance frequencies of the vessel by slowly varying the frequency of the forcing. Figure 5.2 shows how the response to the forcing varies as you come close to resonance. The amplitude of the response grows rapidly as the resonance frequency is approached, while the phase indicates a change from direct to inverse response.

 

The resonance frequencies depend on the dimensions of the vessel; the larger (wider or deeper) the vessel, the longer the resonance periods. It is possible to determine the resonance frequencies for a water body of a given size from its dimensions. Alternatively, it is possible to determine the size a water body must have to resonate at given tidal frequencies. Laplace calculated the depth the ocean must have to be at resonance with the major periods of the lunar and solar tide, assuming again an earth without continents. He found that the ocean is at resonance with the semidiurnal lunar tide if its depth equals 1965 m and again at 7937 m; resonance with the semidiurnal solar tide occurs at 2248 m and at 8894 m. At other depths the ocean would not be at resonance but would show an inverse response for all depths between the two resonance depths, while a shallower or deeper ocean would show a direct response. (Thus, an ocean of 2000 m depth would have a direct semidiurnal solar tide but an inverse semidiurnal lunar tide.)

 

Later refinements of the dynamic theory repeated Laplace's calculations for ocean basins limited by coastlines of various shapes. This modifies the resonance frequencies of the basins, but the principle remains: The closeness of the frequency of the tide-generating force to one of the resonance frequencies determines the amplitude of the tidal wave which is generated in the basin. It also determines the phase, ie the occurrence of low and high water relative to the passage of the moon and sun. Newton's dynamic theory was the first theory capable of explaining why tidal amplitudes and phases vary widely throughout the world ocean.

 

In propagating waves (such as wind waves seen to travel across the sea surface) all points on the sea surface undergo periodic uplift and sinking and experience horizontal movement. The experiment with the bowl or small tank demonstrates that tides of ocean basins are standing waves. Particles moving in a standing wave do not all experience the same type of motion. Some particles - in our bowl or tank experiment the particles near the walls of the vessel - experience only periodic uplift and sinking. Halfway between the ends of the vessel the particles experience only horizontal motion. These particles are located on a line known as a nodal line or node (Fig. 5.3a).

 

Not many ocean basins contain tidal nodes. The reason is that on a rotating earth the Coriolis force deflects particle movement from a straight path, causing water to accumulate on the right in the northern hemisphere and on the left in the southern hemisphere. As a result, the standing plain wave observed in the non-rotating tank is changed into a system in which the wave moves along the vessel wal

Damn, I didn't even answer the question did I?

The starting point for an understanding of the dynamics of the coastal ocean is the dynamics of the deep ocean. This chapter therefore presents, in a very compressed form, the essentials of the dynamics of deep ocean basins. It can be regarded an abbreviated version of chapters 2 - 4 of Regional Oceanography: an Introduction (Tomczak and Godfrey, 1994), where a fuller treatment, on the same elementary level used in this text, can be found.

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 r is expressed in kg/m3 and represented by st = r - 1000. As is common oceanographic practice, st 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 preceeding 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 priciples, 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 w sinf, where w is the angular velocity of the earth equal to 2p / Td, with Td = 86,300 s the length of a day, and f 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.

 

 

 

geostrophic balance

In most of the deep ocean, currents can be regarded as frictionless and maintained by a balance between the horizontal pressure gradient and the Coriolis force. Figure 2.1 shows the principle. Currents established by a balance between the horizontal pressure gradient and the Coriolis force are called geostrophic currents, and the method to derive them by determining the pressure field from observations is called the geostrophic method.

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.

 

 

 

Rule 1: In geostrophic flow, water moves along isobars, with the higher pressure on its left in the southern hemisphere and to its right in the northern hemisphere.

 

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

 

 

 

Rule 2: In a hydrographic section across a current, looking in the direction of flow, the isopycnals slope upward to the right of the current in the southern hemisphere, downward to the right in the northern hemisphere.

 

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.

 

 

 

Ekman layer transports

As mentioned before, the generation of currents by wind requires frictional transfer of momentum from the atmosphere to the ocean. The effect of wind stress on oceanic movement is best analyzed by excluding for the moment the effect of the pressure field on the balance of forces and assuming that the ocean is unstratified, ie of uniform density, and its surface horizontal. The balance of forces is then between friction and the Coriolis force. Since wind blowing over water is always associated with turbulent mixing, the condition of uniform density is usually satisfied in the wind affected surface layer, and the balance between friction and the Coriolis force prevails. This layer is often called the Ekman layer, after Vagn Walfrid Ekman who made the first quantitative study of the dynamics of wind driven currents. Ekman's major finding is summarized in

 

 

Rule 3: The wind driven transport of water in the surface layer of the sea (the Ekman layer) is directed perpendicular to the direction of the wind, to the left of the wind in the southern hemisphere and to the right of the wind in the northern hemisphere.

 

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.

 

 

 

the Sverdrup balance

If we now are to describe the circulation in the real ocean we have to allow for the presence of all three major forces - pressure gradient, Coriolis force and friction - and combine them in a single balance. This was first achieved by the oceanographer Hans Ulrik Sverdrup, and the balance of forces found in most of the ocean is therefore known as the Sverdrup balance. Briefly stated it says that friction is only important in the Ekman layer at the surface, below which all movement is friction free. Movement below the Ekman layer is induced by transfer of water from the Ekman layer to the depths below as a result of flow convergence or divergence in the Ekman layer. In the case of flow convergence, water is forced downwards to the depths below; this process is known as Ekman pumping. In the case of flow divergence, water is pumped upwards into the Ekman layer; this process is known as Ekman suction (or negative Ekman pumping).

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 kilometres 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 permanent and seasonal thermocline

To complete this very brief discussion of deep ocean dynamics we have to introduce and explain the terms permanent and seasonal thermocline. A thermocline is a layer of rapid temperature change from the warm waters of the upper ocean to the colder waters below. It is thus identified by a local maximum of the vertical temperature gradient. The seasonal thermocline is found in all but polar climates during spring, summer and autumn underneath the surface mixed layer (Ekman layer). Its development during that period is best understood by beginning with the winter situation, when the temperate and subtropical ocean shows uniform temperature over its top 200 m or more. Increasing solar radiation during spring produces a warming of the surface mixed layer, where the heat received from the atmosphere is uniformly distributed by mixing. As a result a zone of rapid transition from warm to cold water, the seasonal thermocline, develops at the base of the mixed layer. Its depth is determined by the depth to which the wind mixes the surface layer. Frequent spring storms initially set it at several tens of metres. Winds tend to moderate during summer resulting in a shallowing of the seasonal thermocline, which then often lies above a remnant of the maximum temperature gradient developed during spring (Fig. 2.3). Throughout spring and summer the determining factor for the depth of the thermocline is wind mixing. This situation changes in autumn when the surface layer cools and its density increases, causing instability of the water column. Convection sets in, drawing in cold water from underneath the mixed layer until the stratification is stable again. The resulting deepening of the mixed layer is usually quite rapid, causing the depth of the seasonal thermocline to increase by a hundred netres or more within a matter of a few weeks. In winter the heat stored from the summer is exhausted and the seasonal thermocline disappears (Fig. 2.3).

The transition from the upper ocean, which experiences seasonal heat exchange with the atmosphere, to the deeper layers is knwon 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.

 

 

--------------------------------------------------------------------------------

© 1996 M. Tomczak

contact address: matthias.tomczak@flinders.edu.au

 

Ekman layer dynamics for shallow seas with stratification

 

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The simple relationship between the direction of the wind and the direction of the Ekman layer transport in the deep ocean is valid as long as the total water depth H is larger than the depth of the Ekman layer dE. The exact condition, derived from theoretical considerations of fluid dynamics, is dE2

Figure 3.1 shows how current speed and direction change with depth in an Ekman layer generated by a wind blowing over a deep ocean. Current speed is largest at the surface and decreases rapidly with depth. Current direction also changes with depth, and we see the remarkable result that at some depth the current actually opposes the surface current; however, at that depth the current is so small that it can be considered negligible. This depth is therefore taken as the bottom of the Ekman layer or Ekman layer thickness dE. At the surface the current is directed 45° to the right (left) of the wind in the northern (southern) hemisphere. Somewhere further down in the water column the current flows at right angle to the wind, while below that depth it flows at various angles against the wind direction. The total transport in the Ekman layer is the combined effect of water movement in the Ekman layer, ie the integral from the surface to the depth dE. This explains why in the deep ocean the Ekman layer transport is directed at right angle to the wind direction: The transport contributions in the direction of the wind found in the upper Ekman layer are cancelled by the contributions in opposite directions found in the lower Ekman layer. Only the transport components perpendicular to the wind direction contribute to the final integral.

 

The rather intricate structure of the Ekman layer is the result of a balance between friction and the Coriolis force. Friction transfers momentum from the atmosphere to the ocean. In a non-rotating frame of reference this would result in water movement in the direction of the wind. Rotation gives rise to an apparent force (the Coriolis force) which acts perpendicularly to the direction of movement. The combined action of friction and the Coriolis force produces a surface current directed at 45° from the wind direction and further deflection from the wind direction down the water column.

 

The details of the Ekman layer structure depend on several assumptions which are not always easy to verify. The most important assumption, and the one associated with the greatest uncertainties, concerns the process of momentum transfer from the sea surface to greater depths. In the absence of turbulence, momentum would be transferred by friction between the water molecules. Frictional effects can be quantified through a molecular friction coefficientl, which is a property of the medium and a measure of the viscosity of the fluid; it can be determined in the laboratory and has the units kg m-1 s-1. A quantity often used is the kinematic molecular viscosityn = lr-1, where r is the water density with units kg m-3. If momentum is transferred by molecular friction, the frictional boundary layer thickness, ie the distance over which the velocity is under the influence of the drag force of the wind, can be shown to be given by

 

 

 

 

 

 

 

where f is the Coriolis parameter or Coriolis frequency (a typical value for mid-latitudes is 10-4 s-1). The kinematic molecular viscosity of water is of the order of 10-6 m2 s-1, so the frictional boundary layer is typically about 0.1 m thick. Such molecular boundary layers are easily produced in laboratory tanks and sometimes seen when a light breeze blows over a tranquil pond. Floating leaves or other suspended matter will then indicate swift water movement right at the surface, progressively slower movement in the next few centimetres and no movement below. This is, however, not the everyday situation in the coastal ocean, where the frictional boundary layer (the Ekman layer) is tens of metres thick. The conclusion must be that molecular friction cannot be responsible for the transfer of the wind's energy to the water. Transfer of momentum in the ocean is achieved by turbulence.

 

Unlike viscosity, turbulence is not a property of the medium but of the flow; its intensity and structure depend on the current shear (both horizontal and vertical), the stratification, the wave field, the roughness of the ocean floor and other factors. The major mechanism which contributes to oceanic turbulence are eddies of different size, from the smallest swirls a few metres across to the large geostrophic eddies with diameters of 200 km or more. Wind waves contribute to the turbulence at the sea surface, and other processes contribute to turbulent motion on the centimeter scale. The water parcels moved by the eddies are several orders of magnitude larger than the water molecules. By exchanging their properties with their surroundings they are much more effective in transporting momentum downward from the sea surface than molecular diffusion.

 

To describe the effect of turbulent momentum transfer in exact detail requires the knowledge of the details of the eddy field, under most circumstances an impossible task. Fluid dynamicists have convinced themselves that for nearly all situations its effect can again be described through a viscosity coefficient Av, and the associated boundary layer thickness is then again given by

 

 

 

 

 

 

 

This coefficient of turbulent viscosity Av has again units of m2 s-1 but is no longer a material constant; it is several orders of magnitude larger than the kinematic molecular viscosity n and varies from situation to situation. The coefficient of turbulent viscosity Av is often called the turbulent friction or mixing coefficient. Since eddies are the main mechanisms how oceanic turbulence transfers momentum it is also known as the eddy coefficient. Often it is referred to as the Austausch coefficient (Austausch = German for exchange indicating exchange of momentum through eddies). Typical values for Av are in the vicinity of 0.1 m2 s-1 but can vary by an order of magnitude or more to either side, giving a range of 15 - 150 m for the Ekman layer thickness.

 

One of the most important quantities in the theory of the oceanic circulation is the Ekman layer transport. It might appear that calculating the transport is a rather unreliable operation since it is based on an integral of velocity over the Ekman layer, which are both functions of Av. As it turns out, for the deep ocean (dE2

 

Before we proceed to discuss the modifications of the Ekman layer in shallow seas it is probably helpful to look at some observations of Ekman layer currents. Figure 3.2 shows a photograph of an experiment in which a vertical streak of dye was brought into the upper ocean (The ship used in the experiment is visible in the photo). The water was reasonably clear and the dye could be seen nearly through the entire water column. After some time the shape of the dye streak had changed to the configuration shown. If water movement were the same at all depths the dye streak would appear from the air as a single blob. The fact that it turned into a patch of elongated shape with a distinct curvature indicates a decrease of water movement with depth with a systematic change in direction, in agreement with the Ekman spiral concept.

 

We can find further evidence for the existence of Ekman spirals if we turn our attention to the ocean floor. A current flowing over a rough bottom experiences a drag in a very similar way as a quiescent ocean experiences drag from a wind blowing over its surface. Whether the effect of the drag is to move the water along (the wind) or to hold it back (the bottom) does not make much difference; we could just as well imagine that the water is at rest and the bottom moving in the opposite direction. There exists therefore an Ekman layer above the bottom which serves to bring the current down from whatever its strength is above the bottom Ekman layer to nothing at the sea floor. Figure 3.3 shows an example of such a situation. The observations were taken in 70 m water depth; the surface Ekman layer was only 30 m thick and not covered by the observations. The bottom Ekman layer is seen to be about 25 m thick. Between the two Ekman layers is the region of frictionless geostrophic flow (seen in the data at 25 m and 35 m above the bottom).

 

 

 

Form of the Austausch coefficient

In contrast to the Ekman layer transport, which is independent of the Austausch coefficient, details of the velocity profile in the Ekman layer are affected by the details of Av. To this point, the discussion of Ekman layers assumed that Av is independent of depth. We now review the effect of depth-variability of Av on the velocity profile and possible reasons why the coefficient might vary with depth.

Since the surface Ekman layer is a result of wind action it is reasonable to assume that the turbulence elements responsible for the transfer of momentum are mainly the wind waves. Particle movement in wind waves in deep water is on orbital paths in a vertical plane. The diameters of the orbital paths decrease exponentially with depth; hence it can be argued that the intensity of the turbulence and thus the Austausch coefficient also decrease exponentially with depth. The depth over which this decrease occurs is a function of the dominant wave period (since the exponential decrease of the particle path diameters is a function of wave period), which in turn is some function of wind speed. One way of replacing the simple assumption of constant Av by a more realistic description is therefore to assume an exponential decrease of Av with depth and make the decrease dependent on wind speed.

 

Implementation of this idea is not trivial, and we shall not pursue the details further. We only note that the effect of an exponential decrease of Av is to concentrate most of the mixing in the upper wave zone, thereby reducing the Ekman layer depth. Our "first guess" estimate of 50 - 150 m for the Ekman layer thickness is therefore an upper bound for what we can expect. An example of observations supporting the notion of a depth-dependence of Av (though not strictly exponential in this case) is shown in Figure 3.4.

 

Waves are not always the most important turbulence- generating mechanism. Current shear tends to produce eddies. Currents in the sea nearly always display much stronger shear in the vertical than in the horizontal (on the 1 - 100 m length scale, current speed and direction change much faster vertically than horizontally), so the formation of small overturning eddies is more common than the formation of swirls with a vertical axis of rotation (on the same scale). Acting against the formation of overturning eddies is the stratification, since it is more difficult to move water up or down in the water column against a strong density gradient. It is possible to quantify the tendency for the formation of turbulence by comparing a measure for the stratification with a measure of the vertical current shear. The Richardson number Ri is a non- dimensional number which achieves this. It is defined as

 

 

 

 

 

 

 

Here, g is gravity (g = 9.8 m s-2), r density (r = 1025 kg m-3 is a typical value for sea water) and u the velocity. The vertical density gradient dr/dz measures the stratification, the vertical change of velocity du/dz gives the current shear. The larger Ri, the larger the relative role of stratification and the less likely the presence of active turbulence. Inversely, the smaller Ri, the larger the relative role of current shear and the more likely the presence of turbulence. Observations show that turbulence sets in if Ri falls below a critical value; most researchers give this value as Ri = 1/4, others suggest that it is slightly smaller.

 

The Richardson number can be used to derive a depth-dependence for the Austausch coefficient which somehow reflects the different levels of turbulence at different depth. A commonly used approach is to make Av inversely proportional to Ri. Strong turbulence or small Ri then gives a large Av, which makes sense. Figure 3.5 shows a typical summer situation on a shelf with weak tidal mixing. The heat received at the surface is mixed downward by wave action. Winds in summer are usually light, so the resulting warm mixed layer is relatively shallow and separated from the colder water underneath by a strong thermocline. The stability of the water column is largest in the thermocline, where the Richardson number shows a maximum and the Austausch coefficient a minimum. The extremely low values of Av mean that it is difficult to expand the Ekman layer beyond the thermocline depth even under strong wind conditions. The Ekman layer will not deepen in autumn, despite the seasonal increase in mean wind speed, until cooling at the surface initiates convective mixing, pushing the thermocline downward and with it the minimum of Av.

 

Another example for a shallow surface Ekman layer as a result of a shallow thermocline is indicated in Figure 3.3. The fact that current speed and direction do not change between 25 m and 35 m from the bottom in 74 m water depth suggests an Ekman layer of less than 39 m extent, significantly less than what is normally observed. The shallowness of the thermocline is the result of coastal upwelling, which lifts the thermocline towards the surface, reducing the Ekman layer thickness.

 

The principle that the magnitude and variation with depth of the Austausch coefficient depends on the character of the turbulence applies at the ocean floor as well. In contrast to the free surface, the sea floor imposes a rigid boundary to all movement. As a result, the turbulent eddies close to the floor are restricted in size; their diameter cannot exceed the distance from their centre to the boundary. Larger eddies can occur further away from the sea floor. If this is expressed in terms of turbulent viscosity it is found that Av is zero at the bottom and grows linearly with distance, until it reaches the value typical for turbulence in the ocean interior. This results in a velocity profile where the current increases logarithmically in strength with distance from the sea floor and does not change direction. In the atmosphere, this logarithmic boundary layer determines the wind profile in the first tens of metres from the ground upward. In the ocean it rarely exceeds a few metres in thickness. It occurs as a modification of the Ekman layer over the first few metres and is of no consequence to the direction and magnitude of the total Ekman layer transport discussed below. Figure 3.6 compares the logarithmic layer with the Ekman layer for an oceanic situation. The importance of the logarithmic boundary layer is in its role for the movement of sand and fine sediment in the coastal zone. Coastal engineers concerned with beach stabilization and shallow water circulation spend significant effort on understanding the logarithmic boundary layer in detail.

 

It is easy to see that in many situations it is impossible to make a definite statement about the depth-dependence of Av since the information necessary to estimate the size of turbulence elements or to calculate the Richardson number is not always available. Nevertheless, some general conclusions about the vertical extent of the Ekman layer emerge from our considerations, and it is worth summarizing them here.

 

The starting point for an estimate of Ekman layer depth is eqn. 3.2 with constant Av. It applies to an unstratified deep body of water with depth-independent mixing. These conditions are often satisifed at the sea floor where the stratification is weak and mixing is not associated with waves. The estimate from eqn. 3.2 would include the logarithmic layer as a sub-layer next to the ocean floor. At the surface, mixing is nearly always associated with wind waves. The resulting exponential decrease of Av with depth reduces the estimate of Ekman layer depth. The presence of a shallow thermocline can reduce the Ekman layer thickness further, since turbulence does not penetrate the thermocline.

 

In coastal oceanography it is often important to ascertain whether observations were made under conditions dE2> H2 (shallow sea dynamics). The procedure of assessment is to derive an estimate of dE from eqn. 3.2 and check for the presence of a thermocline. The smaller of the two depths (dE and the thermocline depth) is then used for comparison with the water depth to decide whether deep or shallow dynamics apply to the situation. The importance of a correct assessment becomes evident when we now turn to the question of Ekman layer transport in shallow water.

 

Ekman layer transports

Figure 3.7 shows how current speed and direction change as the water depth H becomes increasingly smaller. The Ekman spiral observed in the deep ocean (seen in Figure 3.1) is found as long as H is larger than 1.25 dE. As H decreases, the spiral changes shape. In the range dE > H > 0.5 dE the current maintains its speed but does not turn against the wind direction at depth. Current speed is reduced when H decreases further, and the current aligns more and more with the direction of the wind. In very shallow water (H

In all situations the Ekman layer transport is independent of the details of the Austausch coefficient, but in shallow water it is no longer directed perpendicular to the wind direction (as it is for H > 0.5 dE). The shallower the water, the more does the Ekman layer transport - which under these conditions becomes the transport of the total water column - point in the direction of the wind. This is of particular importance for the understanding of wind-driven upwelling in shallow seas. Chapter 6 will develop these ideas further.

 

 

--------------------------------------------------------------------------------

© 1996 M. Tomczak

contact address: matthias.tomczak@flinders.edu.au

 

Sea level set-up and buoyancy-driven flows

 

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This chapter deals with ocean circulation processes that occur in the deep ocean but are usually not of enough significance to warrant much attention. The presence of a coast magnifies these processes, giving them much more importance in coastal oceanography. They relate to the ocean's response to atmospheric forcing and can be divided into two groups.

 

The atmosphere forces the ocean in two ways, through tangential stress (wind), and through modification of the ocean's buoyancy (heating, cooling, evaporation, precipitation). The response of the coastal ocean to the first type of forcing is sea level set-up; the response to the second type of forcing is buoyancy-driven flow.

 

Sea level set-up

Wind-generated surface currents reflect the structure of the wind field on a variety of scales. On the oceanic scale the global system of Trade Winds, Westerlies and Polar Easterlies creates the large oceanic gyres that dominate the circulation of entire ocean basins. On the synoptic weather scale, atmospheric low pressure systems create storms, which introduce significant variability into the oceanic current field on time scales of days to weeks. Variations of the wind field in space generate convergences and divergences of the oceanic flow field on all scales.

In principle, a region of surface flow convergence means an accumulation of water in the convergence region. However, in the open ocean such accumulation is kept small, because a rise of the sea level from accumulation of mass immediately causes a depression of the thermocline, followed by vertical water movement from the upper ocean to greater depth. The main effect of a surface convergence is thus vertical exchange of water between the upper ocean and the underlying regions. This phenomenon is commonly known as Ekman pumping. Variations of sea level in the open ocean are therefore modest, of the order of tens of centimetres, and rarely reach 1 m in height.

 

In the coastal ocean the situation is quite different. To begin with, currents in the open ocean rarely oppose each other on scales of tens of kilometres, and convergences (or divergences) are usually the result of slow changes in current speed and direction. The presence of a coast inhibits horizontal water flow and produces vastly stronger convergence (and divergence) effects than are ever encountered in the open ocean. Another factor is the restricted water depth on the shelf. The depth of the coastal zone usually does not exceed 200 m or so, which poses a severe restriction on vertical water movement on a large scale. As a result, water can pile up against the coast to great height, a phenomenon known as sea level set-up or surge. Sea level set-up can pose a severe threat to coastal land and can lead to large scale flooding and loss of life in low lying coastal regions.

 

The major reason for sea level set-up are atmospheric storm systems. Stronger storms are of course more dangerous than moderate storms; but the severity of the resulting storm surges depends on the coastal topography just as much as on the absolute wind strength. (Theoretical arguments show that when water is piled up against a coast the sea surface slope dz/dx is proportional to the wind stress t and inversely proportional to the water depth h; g is gravity and r the water density; compare the sketch on the right.) Some parts of the world's shelf regions are more susceptible to storm surges than others. Figure 4.1 shows an example of a storm surge in the North Sea. The storm system responsible for a surge in the North Sea is usually generated in the region of the Westerlies over the North Atlantic Ocean. It moves with the westerly wind in a generally north eastward direction, passing north of Scotland and continuing along the Norwegian coast. The storm produces a weak surface elevation in the open ocean, which enters the North Sea from the north and builds up to significant height as it crosses the continental slope and enters shallower water. Under the influence of the Coriolis force, which in the northern hemisphere acts to the right of the movement, the surge leans against the British coast and increases in height as it moves south towards the Dutch and German coast.

 

The severity of a storm surge depends to a large extent on its timing relative to the tidal cycle. Many storm systems pass quickly and produce a surge of not much more than 12 hours in duration. If the peak of the surge occurs at low tide the effect will be minimal. If, on the other hand, the peak of the surge coincides with high tide, the water level can reach quite unusual height. The most dangerous storm surges are produced by long lasting storm systems that coincide with spring tides. There have been surges during which the water level did not fall after high tide but kept rising despite of ebb tide, bringing the following high tide close to the height of the dykes along the coastline. The constant pounding from the sea under the lashing of the storm is a severe test for any dyke system. Much land can be flooded when a dyke breaks, and nothing can control the force of the water as it comes rushing in. The following table list some major flood events produced by storm surges.

 

 

Some major historical storm surges

Date Shelf region Estimated maximum

surge height Estimate of lives lost

November 1218 Zuider Zee (Dutch North Sea) unknown 100,000

October 1737 India and Bangladesh 12 m 300,000

1864 Bangladesh unknown 100,000

October 1876 Bangladesh 15 m 100,000

1897 Bangladesh unknown 175,000

September 1900 Galveston, Texas (Gulf of Mexico) 4.5 m 6000

Jan/Febr 1953 Southern North Sea 3.0 m 2000

March 1962 Atlantic coast, USA 2.0 m 32

November 1970 Bangladesh 9.0 m 500,000

 

The table indicates that the two ocean regions most exposed to severe storm surges are the North Sea and the Bay of Bengal, particularly the coastline of Bangladesh. Whether a storm surge develops into a national disaster is determined by several factors. The disproportionate number of casualties in Bangladesh is to a large degree the result of the country's underdevelopment. Holland is one of the most densely populated countries of the world. Most of the country is below sea level and relies on dykes keeping the sea out. It experiences severe storm surges every year but is very rarely flooded because an infrastructure built over centuries and maintained by a rich nation secures the coastline. In comparison, the coastal defences of Bangladesh are of ancient design and the country lacks the resources to improve and maintain them.

 

Another factor that turns Bangladesh's storm surges into national calamities is its closeness to the equator. Atmospheric low pressure systems, which bring the storms experienced in temperate climate, can develop into cyclones in the tropics. Tropical cyclones (known as hurricanes in America) are extreme events of nature; wind and rain are very much more powerful in cyclones than in the storm systems experienced in the North Sea. Cyclones can therefore produce intense surges, with the sea level sometimes rising rapidly by several metres in a matter of hours. Figure 4.2 shows examples of sea level set-up from tropical cyclones experienced in various towns around Australia. Although none of the cyclones caused flooding with loss of life, severe flooding usually occurs with all cyclones because they are associated with record rainfall. The flooding events that occurred with storm surges in Bangladesh are to a large extent also the result of overflowing rivers following heavy rain.

 

Cyclones form over the sea and can create large surges when they come close to land. They often do not cross into land immediately but continue on a path parallel to the coast. This can amplify the surge if the propagation of the cyclone is in resonance with the propagation of coastal Kelvin waves. The dynamics of these waves is discussed in chapter 8, where it is shown that along the east coast of ocean basins Kelvin waves can only travel poleward, while on the west coast they can only travel towards the equator. Cyclones are features of the tropics; they form in the vicinity of the equator and move away towards the from it. Resonance between cyclone propagation and Kelvin wave propagation can therefore only occur on the east coast of ocean basins (the west coast of continents). Figure 4.3 verifies this for the Australian continent. It shows that sea level set-up from cyclones is generally stronger on the west coast than on the east coast.

 

Most cyclone tracks over the Bay of Bengal move poleward parallel to the Indian coast. Many reach landfall in Bangladesh after travelling in close proximity to the Indian coastline. This favours resonant build-up of extreme surges. Figure 4.4 shows examples of cyclone tracks over the Bay of Bengal and associated surge heights. Managing the storm surges in the Bay of Bengal remains one of humanity's greatest challenges.

 

Buoyancy-driven flow

The coastal zone is generally a region of increased temperature and salinity variability in the ocean, and density differences between the open ocean and the shelf are a common occurrence. Variations of temperature are enhanced through increased tidal mixing, shallow water depth and advection of warm or cold air from land. This leads to enhanced variability of the near-shore current field.

While the effect of temperature on the density field of the coastal ocean manifests itself as an enhancement of processes that occur in all ocean regions to some degree, the salinity contribution can be much more substantial. The coastal zone is not only exposed to the same freshwater balance (precipitation - evaporation) as the deep ocean; it also receives the freshwater that collects from rain over the land. This river run-off can have a substantial impact on the oceanic density field on the shelf and create its own circulation.

 

Figure 4.5 is a sketch of a situation where the freshwater input from land is the result of outflow from numerous rivers along the coast and can be described as a continuous freshwater source for the shelf region. Because a freshwater source lowers the density of the sea water on the shelf, the isopycnals slope downwards as they approach the shelf from the deep ocean. The resulting thermocline slope produces a geostrophic current parallel to the coast, directed equatorward on the west coasts and poleward on the east coasts of the ocean.

 

It is seen that river run-off from the continents produces a cyclonic circulation around the ocean basins. This circulation is generally restricted to the shelf and superimposed on the wind driven general circulation and therefore not always easily recognised in observations. On the west coasts of the ocean it is overwhelmed by the western boundary currents. In the subtropics the circulation of the deep ocean is anti-cyclonic, so the subtropical western boundary currents oppose the direction of the buoyancy driven flow, but buoyancy driven flow can occur in the form of a narrow countercurrent on the shelf. Currents on the east coasts of the ocean are generally weaker than the western boundary currents, and buoyancy driven flow can occur as countercurrents.

 

A prominent example of buoyancy driven flow is found along the Pacific coastline of Canada and Alaska. These coasts are in high latitudes, and the circulation in the adjacent ocean is cyclonic as part of the subpolar gyre of the North Pacific Ocean. Numerous rivers enter the ocean from the Canadian and Alaskan coast, to the effect that the continent acts as a nearly continuous line source of freshwater. The resulting buoyancy driven flow enhances the general cyclonic circulation of the region by increasing the current speed on the shelf. The strongest currents are usually found within 25 km off the coast. Figure 4.6 shows how the Alaska Coastal Current is intensified through river run-off along its path. The freshwater output from rivers amounts to only 4% of the current's transport but the current reacts with a lag of about one month, doubling its transport if the freshwater input is doubled. The variations of the Alaska Coastal Current are thus not simply the result of additional freshwater carried by the current but a response to the freshwater-induced changes in the density field.

 

 

 

--------------------------------------------------------------------------------

© 2000 M. Tomczak

contact address: matthias.tomczak@flinders.edu.au">matthias.tomczak@flinders.edu.au

 

Ekman layer dynamics for shallow seas with stratification

 

--------------------------------------------------------------------------------

The simple relationship between the direction of the wind and the direction of the Ekman layer transport in the deep ocean is valid as long as the total water depth H is larger than the depth of the Ekman layer dE. The exact condition, derived from theoretical considerations of fluid dynamics, is dE2

Figure 3.1 shows how current speed and direction change with depth in an Ekman layer generated by a wind blowing over a deep ocean. Current speed is largest at the surface and decreases rapidly with depth. Current direction also changes with depth, and we see the remarkable result that at some depth the current actually opposes the surface current; however, at that depth the current is so small that it can be considered negligible. This depth is therefore taken as the bottom of the Ekman layer or Ekman layer thickness dE. At the surface the current is directed 45° to the right (left) of the wind in the northern (southern) hemisphere. Somewhere further down in the water column the current flows at right angle to the wind, while below that depth it flows at various angles against the wind direction. The total transport in the Ekman layer is the combined effect of water movement in the Ekman layer, ie the integral from the surface to the depth dE. This explains why in the deep ocean the Ekman layer transport is directed at right angle to the wind direction: The transport contributions in the direction of the wind found in the upper Ekman layer are cancelled by the contributions in opposite directions found in the lower Ekman layer. Only the transport components perpendicular to the wind direction contribute to the final integral.

 

The rather intricate structure of the Ekman layer is the result of a balance between friction and the Coriolis force. Friction transfers momentum from the atmosphere to the ocean. In a non-rotating frame of reference this would result in water movement in the direction of the wind. Rotation gives rise to an apparent force (the Coriolis force) which acts perpendicularly to the direction of movement. The combined action of friction and the Coriolis force produces a surface current directed at 45° from the wind direction and further deflection from the wind direction down the water column.

 

The details of the Ekman layer structure depend on several assumptions which are not always easy to verify. The most important assumption, and the one associated with the greatest uncertainties, concerns the process of momentum transfer from the sea surface to greater depths. In the absence of turbulence, momentum would be transferred by friction between the water molecules. Frictional effects can be quantified through a molecular friction coefficientl, which is a property of the medium and a measure of the viscosity of the fluid; it can be determined in the laboratory and has the units kg m-1 s-1. A quantity often used is the kinematic molecular viscosityn = lr-1, where r is the water density with units kg m-3. If momentum is transferred by molecular friction, the frictional boundary layer thickness, ie the distance over which the velocity is under the influence of the drag force of the wind, can be shown to be given by

 

 

 

 

 

 

 

where f is the Coriolis parameter or Coriolis frequency (a typical value for mid-latitudes is 10-4 s-1). The kinematic molecular viscosity of water is of the order of 10-6 m2 s-1, so the frictional boundary layer is typically about 0.1 m thick. Such molecular boundary layers are easily produced in laboratory tanks and sometimes seen when a light breeze blows over a tranquil pond. Floating leaves or other suspended matter will then indicate swift water movement right at the surface, progressively slower movement in the next few centimetres and no movement below. This is, however, not the everyday situation in the coastal ocean, where the frictional boundary layer (the Ekman layer) is tens of metres thick. The conclusion must be that molecular friction cannot be responsible for the transfer of the wind's energy to the water. Transfer of momentum in the ocean is achieved by turbulence.

 

Unlike viscosity, turbulence is not a property of the medium but of the flow; its intensity and structure depend on the current shear (both horizontal and vertical), the stratification, the wave field, the roughness of the ocean floor and other factors. The major mechanism which contributes to oceanic turbulence are eddies of different size, from the smallest swirls a few metres across to the large geostrophic eddies with diameters of 200 km or more. Wind waves contribute to the turbulence at the sea surface, and other processes contribute to turbulent motion on the centimeter scale. The water parcels moved by the eddies are several orders of magnitude larger than the water molecules. By exchanging their properties with their surroundings they are much more effective in transporting momentum downward from the sea surface than molecular diffusion.

 

To describe the effect of turbulent momentum transfer in exact detail requires the knowledge of the details of the eddy field, under most circumstances an impossible task. Fluid dynamicists have convinced themselves that for nearly all situations its effect can again be described through a viscosity coefficient Av, and the associated boundary layer thickness is then again given by

 

 

 

 

 

 

 

This coefficient of turbulent viscosity Av has again units of m2 s-1 but is no longer a material constant; it is several orders of magnitude larger than the kinematic molecular viscosity n and varies from situation to situation. The coefficient of turbulent viscosity Av is often called the turbulent friction or mixing coefficient. Since eddies are the main mechanisms how oceanic turbulence transfers momentum it is also known as the eddy coefficient. Often it is referred to as the Austausch coefficient (Austausch = German for exchange indicating exchange of momentum through eddies). Typical values for Av are in the vicinity of 0.1 m2 s-1 but can vary by an order of magnitude or more to either side, giving a range of 15 - 150 m for the Ekman layer thickness.

 

One of the most important quantities in the theory of the oceanic circulation is the Ekman layer transport. It might appear that calculating the transport is a rather unreliable operation since it is based on an integral of velocity over the Ekman layer, which are both functions of Av. As it turns out, for the deep ocean (dE2

 

Before we proceed to discuss the modifications of the Ekman layer in shallow seas it is probably helpful to look at some observations of Ekman layer currents. Figure 3.2 shows a photograph of an experiment in which a vertical streak of dye was brought into the upper ocean (The ship used in the experiment is visible in the photo). The water was reasonably clear and the dye could be seen nearly through the entire water column. After some time the shape of the dye streak had changed to the configuration shown. If water movement were the same at all depths the dye streak would appear from the air as a single blob. The fact that it turned into a patch of elongated shape with a distinct curvature indicates a decrease of water movement with depth with a systematic change in direction, in agreement with the Ekman spiral concept.

 

We can find further evidence for the existence of Ekman spirals if we turn our attention to the ocean floor. A current flowing over a rough bottom experiences a drag in a very similar way as a quiescent ocean experiences drag from a wind blowing over its surface. Whether the effect of the drag is to move the water along (the wind) or to hold it back (the bottom) does not make much difference; we could just as well imagine that the water is at rest and the bottom moving in the opposite direction. There exists therefore an Ekman layer above the bottom which serves to bring the current down from whatever its strength is above the bottom Ekman layer to nothing at the sea floor. Figure 3.3 shows an example of such a situation. The observations were taken in 70 m water depth; the surface Ekman layer was only 30 m thick and not covered by the observations. The bottom Ekman layer is seen to be about 25 m thick. Between the two Ekman layers is the region of frictionless geostrophic flow (seen in the data at 25 m and 35 m above the bottom).

 

 

 

Form of the Austausch coefficient

In contrast to the Ekman layer transport, which is independent of the Austausch coefficient, details of the velocity profile in the Ekman layer are affected by the details of Av. To this point, the discussion of Ekman layers assumed that Av is independent of depth. We now review the effect of depth-variability of Av on the velocity profile and possible reasons why the coefficient might vary with depth.

Since the surface Ekman layer is a result of wind action it is reasonable to assume that the turbulence elements responsible for the transfer of momentum are mainly the wind waves. Particle movement in wind waves in deep water is on orbital paths in a vertical plane. The diameters of the orbital paths decrease exponentially with depth; hence it can be argued that the intensity of the turbulence and thus the Austausch coefficient also decrease exponentially with depth. The depth over which this decrease occurs is a function of the dominant wave period (since the exponential decrease of the particle path diameters is a function of wave period), which in turn is some function of wind speed. One way of replacing the simple assumption of constant Av by a more realistic description is therefore to assume an exponential decrease of Av with depth and make the decrease dependent on wind speed.

 

Implementation of this idea is not trivial, and we shall not pursue the details further. We only note that the effect of an exponential decrease of Av is to concentrate most of the mixing in the upper wave zone, thereby reducing the Ekman layer depth. Our "first guess" estimate of 50 - 150 m for the Ekman layer thickness is therefore an upper bound for what we can expect. An example of observations supporting the notion of a depth-dependence of Av (though not strictly exponential in this case) is shown in Figure 3.4.

 

Waves are not always the most important turbulence- generating mechanism. Current shear tends to produce eddies. Currents in the sea nearly always display much stronger shear in the vertical than in the horizontal (on the 1 - 100 m length scale, current speed and direction change much faster vertically than horizontally), so the formation of small overturning eddies is more common than the formation of swirls with a vertical axis of rotation (on the same scale). Acting against the formation of overturning eddies is the stratification, since it is more difficult to move water up or down in the water column against a strong density gradient. It is possible to quantify the tendency for the formation of turbulence by comparing a measure for the stratification with a measure of the vertical current shear. The Richardson number Ri is a non- dimensional number which achieves this. It is defined as

 

 

 

 

 

 

 

Here, g is gravity (g = 9.8 m s-2), r density (r = 1025 kg m-3 is a typical value for sea water) and u the velocity. The vertical density gradient dr/dz measures the stratification, the vertical change of velocity du/dz gives the current shear. The larger Ri, the larger the relative role of stratification and the less likely the presence of active turbulence. Inversely, the smaller Ri, the larger the relative role of current shear and the more likely the presence of turbulence. Observations show that turbulence sets in if Ri falls below a critical value; most researchers give this value as Ri = 1/4, others suggest that it is slightly smaller.

 

The Richardson number can be used to derive a depth-dependence for the Austausch coefficient which somehow reflects the different levels of turbulence at different depth. A commonly used approach is to make Av inversely proportional to Ri. Strong turbulence or small Ri then gives a large Av, which makes sense. Figure 3.5 shows a typical summer situation on a shelf with weak tidal mixing. The heat received at the surface is mixed

Fukin hell!

I wish Id just asked what the capital of new zealand is now.

 

If anyone's arsed the answer is that the weight of the tanker is in equal proportion to the amount of water it has displaced,thus allowing it to float.

Or some old bollox like that.

 

 

Fukin hell!

I wish Id just asked what the capital of new zealand is now.

 

If anyone's arsed the answer is that the weight of the tanker is in equal proportion to the amount of water it has displaced,thus allowing it to float.

Or some old bollox like that.

 

 

The one word answer would have simply been displacement...

Whose question's next then?