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Chapter 9 Shallow water systems and the Ekman Layer

By Leon Andrews,2014-05-07 20:53
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Chapter 9 Shallow water systems and the Ekman Layer

Chapter 9: Shallow water systems and the Ekman Layer

In the late 1800’s biologist Fridtjof Nansen observed that icebergs drifting in arctic waters did not drift in the same direction as the local winds but rather in a direction

to the right of the wind. Examination of the balance between friction, Coriolis force and

the pressure force show that such a deflection is expected since the Coriolis force will

cause a rightward direction to the pressure induced wind force. Quantitative examination

of the balance between these forces was first proposed by oceanographer Vagn Ekman in

the early 1900s. The following analysis derives the equations that Ekman developed and

discusses their results.

1. Preliminaries - Turbulence and the effects of friction

When we derived the Navier-Stokes equations in Chapter 5, we assumed a constitutive relationship between the stress tensor the gradient of the velocity field. This

hypothesis is valid provided the flow is laminar; meaning that there are smooth variations

in the velocity field. The upper layer of the ocean, however, exhibits significant

turbulence and mixing and the constitutive relationship derived in Chapter 5 breaks down.

Instead, we must consider mean values of the flow field and assume that turbulent

stresses are proportional to the gradients of this mean flow field.

The most common approach to examining turbulent effects is to split a given

quantity into a mean flow, indicated by an over-bar, and a fluctuating contribution with a

mean value of zero. In other words, for scalar quantity, q, we can represent it in terms of

a mean and fluctuating contribution as:

q?q?q'

where

q'?0

By “mean value” we are referring to an integration over the time scale, T, of the problem

?uof interest. For example, to find the mean flow of , we average over the integrated ?t

flow field as follows:

Tuuu????Tu?1??0?dt? ?0tTtT??

By using this new notation of splitting up the equations of motion into mean and

fluctuating contributions and then taking the time average of these results, we can deduce the additional contributions that turbulent effects provide to the net frictional forces. One must be careful in this approach how we represent the product of averaged quantities. For example, for the advective component, ?v, breaking up the flow into two u?x

contributions and taking the average leads to:

?v?????????????????u?u?u'v?v'?uv?u'v?uv'?u'v' ?x?x?x?x?x?x

Mean values are considered constants in the process of integration and can be factored

out of the integration in the above equations. For example, in the second term

?v?v u'?u'?0?x?x

The term above is equal to zero because the mean of any fluctuation value is zero by

?vdefinition. The end result is the mean of is expressed as two terms: u?x?v?v?v'u?u?u'. ?x?x?x

Applying the above notation and averaging to the Navier-Stokes equation in a

rotating frame with small density fluctuations (Boussinesq approximation), we obtain,

Du?u'?u'?u'1?p2?u'?v'?w'?fv?????uDt?x?y?z?x?oDv?v'?v'?v'1?p2?u'?v'?w'?fu????v?Dt?x?y?z?y?o (1)

Dw?w'?w'?w'1?p?2?u'?v'?w'???g???wDt?x?y?z??z?oo?u?v?w???0?x?y?z

In the above equation, D??????u?v?w and, due to the Boussinesq Dt?t?x?y?z

approximation, the background density is independent of time. Both the conservative

body forces and the Coriolis parameter are also assumed to be independent of time as

compared to the turbulent time scales of our averages.

The continuity equation expressed in equation (1) is in its mean form. But the

original continuity equation also still holds,

?????????u?u'?v?v'?w?w'?0 ?x?y?z

Consequently we can see that continuity also holds for the fluctuating terms:

?u'?v'?w'???0 ?x?y?z

Use of the above continuity equation allows us to rearrange the non-linear terms in

equation (1). For example, the non-linear terms in the east-west equation of motion can ?????????u'?u'?u'?u'u'?u'v'?u'w'?u'?v'?w'??u'v'w'u'????????be written as: ???x?y?z?x?y?z?x?y?z?? ??????????0

?????????????u'u'u'v'u'w'??????x?y?z??Similar forms can be written for the non-linear terms in north-south and vertical

directions. The resulting equations of motion then take the form:

Du1p???????????2fvuu'u'u'v'u'w' ?????????Dtxxyz????o

Dv1p??????????2? (2) fuvv'u'v'v'v'w'?????????Dtyxyz????o

?Dw1p????2??????g?ww'u'w'v'w'w'????????Dtzxyz??????oo

We now apply a turbulent form of the constitutive hypothesis to the turbulent

non-linear components of the above equations of motion. The following analysis is from

the work of Kirwan, A. D. 19691. We can see from equations (2) that the eddy terms are second order tensors that may be related to the dissipation or frictional effects of the fluid

medium. We relate the above turbulent velocity components by introducing the

Reynold’s stress tensor

??u'u' (3) ijij

Recall from chapter 5, we derived the frictional term from the deviatoric term of the

stress tensor, ???p??d. The Reynold’s stress tensor is analogous to the deviatoric ijijijtensor in laminar flows. As such, we can establish the same constiuitive hypothesis as

1 Kirwan, A. D., Formulation of Constitutive Equations for Large-Scale Turbulent Mixing, Journal of

Geophysical Research, 74, 6953-6959, 1969

before and assume that the Reynold’s stress tensor is linearly proportional to the velocity

??Ke ijijklkl

Where (4)

????1uukl?? e??kl????2xxlk??

Notice there is an important additional point in the above hypothesis. In relating the

stresses to the velocity gradients or strain rates, we assume that they are related to the

mean velocity field elements as opposed to the turbulent contributions. The fourth order

tensor, K, can be simplified by utilizing ijkl

1. The symmetry of the Reynold’s stress tensor

2. Assuming incompressibility

3. Making a physical assumption about the distribution of the molecular medium.

In the laminar case, we assumed that molecular medium was isotropic. For the turbulent

case, we observe that the mixing scales (molecular motion) between the horizontal and

vertical are different. Thus we assume horizontal planar isotropy instead of uniformity

throughout the whole fluid medium. The resulting mathematics of how this modifies

K is beyond the scope of this chapter. Understanding the impact of the underlying ijkl

physical assumptions is all that is important for now. After assuming incompressibility

and planar isotropy, the resulting equations of motion become:

222????Du1puuu????????fv???????? hv222????Dtx??xyz???o????

222????Dv1pvvv????????fu???????? (5) hv222????Dty??xyz???o????

222????Dw1pwww?????????g???????? hv222????Dtz???xyz???oo?????u?v?w???0. ?x?y?z

Further, Reynold’s stress tensors are defined as

???u?uji????? ???ijij????xxij??

Before moving on to Ekmans quantitative analysis, let us examine how the 22Reynold’s stress tensor is applied to real world analysis. First looking at the Reynold’s mm5??10??10 and . In hvstress index notation itself we can see it that the first index indicates the direction of the ssforcing term. But note that the Reynold’s stress tensor is symmetric. In the lower 22mmthe upper ocean, typical coefficient values are ??100??0.01and . hvatmosphere, eddy coefficients have typical values of ss

^2Example: Let us now consider a mean flow field of the form, . A u?tanh(2?y)igraph of this field is shown in figure 1 below.

U(y)=tanh(2*pi*y)

1

y0

-1 0 x ^Figure 1 Flow profile of velocity field u?tanh(2?y)i

The associated Reynold’s stress for this mean flow is

?u2????????????21?tanh2?y and is shown in figure 2. Part of the forcing yxyy?y

associated with this flow is in the direction of the arrow and is observed to have a

spreading or diffusive effect on the flow by its lateral direction to the initial direction of

the flow field.

2 The following analysis follows from similar discussion in McWilliams, Fundamental of Fluid Dynamics,

Cambridge University Press, Cambridge UK, 2006, 83-85.

Associated Reynold's Stress Tensor to the flow u(y)=tanh(2*pi*y) 0.5

0.4

0.3

0.2

0.1

y0

-0.1

-0.2

-0.3

-0.4

-0.5-7-6-5-4-3-2-10x

2. Ekamn’s solution

Given the equation of Geostrophic balance derived in the chapter 7, we know that

the Coriolis force contribution of the Geostrophic flow will perfectly balance with the

pressure gradient provided the Coriolis vector is assumed constant.

1f?u???p g?o

This leaves the friction flow contribution of the Coriolis force in balance with the viscous

terms of the equation of motion. We will assume that the mixed layer is horizontally

homogeneous so that we neglect all terms for coefficients proportional to ?? and is hv

spatially constant. This leads us to the most basic form of the Ekman flow equation which assumes a steady state balance of the Coriolis force with the vertical viscous forces

in the open ocean.

2???uF???fv??Fv2???z?? (6) 2???vF???fu?Fv2???z??

^Now consider a constant wind stress, ?i, acting in the east-west or direction on the free

surface of the ocean, z=0.

?uF??????????? at z=0 (7) xxxyxzxzv?z

Notice in the above boundary condition that ????0 because vertical shear xxxydominates in the fluid medium (horizontal homogeneity).

We assume another boundary condition that the flow is negligible as we go to large

depths in the water to avoid any bottom friction contributions and to avoid any issues

with energy sources at large distances (The only energy source is due to the wind stress at

the surface given by equation (7)). A negligible flow field at large ocean depths is (8) u,v?0 as z???FFquantitatively expressed as Equations (6) (8) provide us all the information we need to solve for the Ekman or frictional contribution to the flow field.

To solve for the v component of the frictional flow field, take the second derivative of the north-south component of equation (6) with respect to z:

24?????uvvFF??? 24??f?z?z??

Now substitute the east-west component of equation (6) to obtain:

24?????v?vF????v?? F4????f?z????

A similar equation can be obtained for u: F

24?????u?vF????u?? F4????f?z????

3Both equations have a solution of the form:

????1?i?z?1?i?zu?Ae?Be F????1?i?z?1?i?zv?Ce?De F

where

f?? 2?v

We can see right away that the deep water boundary condition, equation (8),

requires B =D=0. Coefficients A and C are then related by the original coupled

equations by substituting in the general solutions:

22???v?2??1?i?z1?i?z????1?i?z2F????fu???fAe??Ce??Ce1?i? Fvvv22???z?z??

3 There is actually a more general solution with 2 additional terms in each velocity component. For

presentation purposes, we have only given the above reduced form to stick with main physical feature of

the problem. The alternative is to present the full general solution and impose another boundary condition;

likely an imposition to the velocity field at the surface which would seem too restrictive.

Therefore A?iC

Finally, using the free surface boundary condition, equation (7), we see that

du???????A1?i??? and vvdzz?0

??1?i??A?? 1?i2??????vv

With some algebraic manipulation and application of common trigonometric identities,

the general solutions of the equations of motion are:

????i?11????1??iz?zu?e??ecos??z???F???2f4??vv (9)

?????i11????1?i?z?zve?e?z??????sin??F??f?24??vv

For z?0

A graph of horizontal velocity with increasing depth is shown in Figure 1.

Scaled units of horizontal velocity with depth

0

?z?? 2?

v

z=0

0

u

Figure 1 Graph of solution of equation (9) with depth for a constant wind field in the

east-west direction. Values have been normalized such that ??uz?0?1. This type of

graph is also called a Hodograph.

Notice the following properties of the solution and from figure 1:

1. The velocity field at the surface makes a 45 degree angle to the right of the direction

of the wind stress at the surface in this steady state condition.

2. The exponential dependence shows that the flow decreases to 0 as we go deeper in the ?, the flow has reversed direction from southeast to northwest. z??ocean. ?3. At a distance This depth is arbitrarily taken as the effective depth of the Mixed layer and is called the

Ekman layer.

Unfortunately, it is very difficult to experimentally observe the above solutions in

the ocean. The primary causes of the inaccuracy of the above solution are due to the

assumption of a spatially independent vertical viscosity which was somewhat arbitrary ?v

for simplification purposes and the assumption of a constant stress field at the surface (we

know that winds are continuously changing with time). Further, no horizontal boundaries

were assumed so the results above are only applicable in the open ocean away from the

continents. Needless to say it is still quite on open area of research to relax some of the

above assumptions and still obtain a nice compact analytic solution. We can, however,

still appreciate the above results. Equation (9) and figure (1) show us a simple but

physically plausible coupling of the atmospheric conditions to the ocean. Further,

observations of the flow field in the mixed layer over a span of multiple days, confirm

approximation representations of the above solutions.

3: Transport and Upwelling

Often we are interested in examining the transport of a vertical fluid column

throughout the ocean basin. The appropriate quantity to consider is the horizontal mass

transport and is defined as

0

M??udz HH?h?

The mass transport is a measure of the quantity of a vertical fluid column that passes a

point per unit time. The depth value, h, is determined by the scaling of the mixed layer, or

alternatively where horizontal motions are assumed to vanish.

For the Ekamn solution discussed above, we can assume the ocean is

incompressible and thus examine a volumetric flow rate. Integration over the vertical

Ekman layer of equation (9) provides us with the components of the volume of fluid

transported within the mixed later:

00Volume transport in the east-west direction

?udz?udz?0 FF???????

00?Volume transport in the north-south direction ????vdzvdz FF??f?????

Recall that the applied stress was assumed in the east-west direction while the volume

transport is solely in the north-south direction. These results indicate that the transport of fluid volume is at right angles to the direction of the applied wind stress at the surface.

The net volume transport or mass transport to the right of the wind stress derived above

has significant implications on the ocean current patterns in the Atlantic and Pacific basin.

The average effects of the semi-permanent Bermuda high in the Atlantic Ocean leads to a long duration southerly wind flow along the east coast of the United States.

The water volume in the mixed layer is therefore transported to the east as calculated

above. Conservation of mass requires the above transported volume to be replaced by the

ocean water below the mixed layer. This leads to an upwelling or net vertical current

along the coastline. These upwelling effects have major implications on the ocean

climate as well as the biological conditions of the coastline environment. One can

imagine that nutrient rich conditions below the mixed layer provide the surrounding

marine life at the surface with an abundant source of food.

The arguments leading to upwelling along the coast are qualitatively sound but are not quantitatively relevant yet since we assumed no continental boundaries in section

2. In the next chapter, we will include these horizontal boundary conditions as well as

relax some of the above assumptions to obtain a more accurate representation of the

horizontal current along the coastline.

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