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 $d_E$. The exact condition, derived from theoretical considerations of fluid dynamics, is $d_E^2 \lt\lt H^2$. For all practical applications the criterion $d_E \lt H$ is adequate to decide whether a given situation can be described by deep ocean dynamics or not. As stated in the previous chapter, the great advantage of deep ocean dynamics is that detailed knowledge of the current distribution in the Ekman layer is not essential to describe its role in the ocean circulation. This is no longer true in the shallow water situation, which requires an analysis of the vertical velocity profile. The first topic of this chapter is therefore a description of the Ekman layer structure.

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 $d_E$. At the surface the current is directed 45º 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 $d_E$. 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 45º 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 coefficient $\eta$ 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 viscosity $\nu = \eta \rho^{-1}$, where $\rho$ 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

\begin{equation} d = \sqrt{\frac{2 \nu}{f}} \label{eq:dmol} \end{equation}

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 centimeters 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 $A_v$, and the associated boundary layer thickness is then again given by

\begin{equation} d = \sqrt{\frac{2 A_v}{f}} \label{eq:drot} \end{equation}

This coefficient of turbulent viscosity $A_v$ 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 ν and varies from situation to situation. The coefficient of turbulent viscosity $A_v$ 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 $A_v$ 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 $A_v$. As it turns out, for the deep ocean ($D_E^2 << H^2$) the uncertainty in the magnitude of the Austausch coefficient is of not much consequence, because the dependence of $d_E$ and of the Ekman current velocity on $A_v$ cancel and the Ekman transport becomes independent of $A_v$. The Ekman layer transport can therefore be determined without any knowledge of the magnitude or time and space variability of $A_v$. This is one of the most important results of geophysical fluid dynamics and one of the reasons why deep ocean dynamics are so much simpler than the dynamics of the coastal ocean.

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 $A_v$. To this point, the discussion of Ekman layers assumed that $A_v$ is independent of depth. We now review the effect of depth-variability of $A_v$ 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 $A_v$ by a more realistic description is therefore to assume an exponential decrease of $A_v$ 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 $A_v$ 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 $A_v$ (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

\begin{equation} Ri = \dfrac{ \dfrac{g}{\rho} \dfrac{d \rho}{dz}}{ \left(\dfrac{du}{dz}\right)^2 } \end{equation}

Here, g is gravity ($g = 9.8\;m\;s^{-2}$), $\rho$ density ($\rho = 1025\;kg\;m^{-3}$ is a typical value for sea water) and u the velocity. The vertical density gradient $d \rho/dz$ 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 $A_v$ inversely proportional to Ri. Strong turbulence or small Ri then gives a large $A_v$, 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 $A_v$ 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 $A_v$.

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 $A_v$ 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 $A_v$ 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 $A_v$. It applies to an unstratified deep body of water with depth-independent mixing. These conditions are often satisfied 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 $A_v$ 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 $d_E^2 << H^2$ (deep ocean dynamics) or $d_E^2 >> H^2$ (shallow sea dynamics). The procedure of assessment is to derive an estimate of $d_E$ from eqn. 3.2 and check for the presence of a thermocline. The smaller of the two depths ($d_E$ 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 $1.25\; d_E$. As H decreases, the spiral changes shape. In the range $d_E >> H > 0.5 \;d_E$ 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 ($H < 0.2 \;d_E$) the flow is essentially controlled by wind stress and bottom friction and the Coriolis force becomes insignificant; the current aligns with the wind direction nearly completely and decreases nearly linearly with depth.

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 $H > 0.5\;d_E$). 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.