So All S Left Is Coriolis and Acceleration (We Re Neglecting the Field Acceleration Terms)

Oct 26

Inertial Motion

We have already shown both animations and drifter data showing inertial motion. Inertial motion is typically excited when the surface currents are impulsively excited by wind forcing. IF the wind were to remain steady—then the currents would eventually reach equilibrium and the time-dependent term would go to zero. But suppose the winds blew for a few hours then turned off. The current would be in motion—but once the winds turned off there would be no forcing. If we are far from boundaries we can neglect pressure gradients and friction (bottom is far away also)

So all’s left is Coriolis and acceleration (we’re neglecting the field acceleration terms).

(1)

(2)

Ok a couple of things we could do with this equation.

For example if you multiply the first equation by u and the second by v you get

(3&4)

Note that these are not the second derivative of velocity (which would be a rate of change of acceleration) but the time derivative of velocity squared

Which if you add these two equations together you get

(5)

Here we that , and thus we can rewrite

(6)

Equation 6 states that in inertial motion the kinetic energy remains constant . Even though the flow is accelerating (du/dt and dv/dt are not zero). What happens is (as you’ve seen from the drifter tracks and my animations) is that current speed remains constant but changes directions and a particle embedded in this would simply go around in a circle.

This can also be found by taking the time derivative of equation 1 and substituting in equation 2 for the right hand side to yield

(7)

Similarly taking the time derivative of equation 2 and substituting in equation 1 for the right hand yields

(8)

Equations 7 and 8 are wave equations—and have solutions that can use either the sin or cosine functions.

Let’s use the sin function

u(t)=Usin(wt) (9)

where U is the amplitude of the wave and w is the wave’s frequency

So substituting 9 into 7 we find

w2=f2 (10)

Which reveals that the frequency of the wave is the Coriolis frequency f.

Recall that f=2*pi/T where T is the local inertial period—or the time it takes a particle to complete a single loop.

So we can write

u(t)=Usin(ft)

By inspection of 1 and 2 we can see that if u varies with the sin(ft) than v (which is the derivative of u) must vary with –cos(ft).

Thus u = U sin (ft) and v=-Ucos(ft).

For those of you familiar with complex notation where u=u*iv this yields

u=Ueift

Where e ift = cos(ft)+i*sin(ft)

which traces out a circle over period T.

Around here T is 18.8 hours

At the pole it’s 12 hours

At 30 N/S its 24 hours

At the equation it’s infinity

Note that in the southern Hemisphere f is negative—so the rotation is in the opposite sense (counterclockwise).

In reality a storm that excites inertial oscillations has a finite length scale. This leads to spatial variability in the strength of inertial oscillations. This variability leads to horizontal flow divergences (and convergences) which then excites internal waves at or near the inertial frequency. The propagation of these waves away from the generation site will have a downward component and this can increase vertical shears in the main oceanic thermocline and reduce the Richardson number to a point where mixing may occur. Downward propagating near-inertial waves is an important mechanisms that transmits the surface wind forcing to the main oceanic thermocline. Furthemore, these inertial waves interact with the mean flow fields that can focus there energy in regions of negative vorticity. The details of the propagation of near-inertial waves an their interaction with the geostrophic shear is beyond the scope of this course)

Important concepts:

·  Inertial motion occurs is driven by impulsive winds

·  Inertial motion traces out an ellipse at period T (the local inertial period)

·  The Kinetic energy of inertial motion is constant

Ekman Transport

From Wikipedia

Vagn Walfrid Ekman

Vagn Walfrid Ekman (May 3, 1874 – March 9, 1954) was a Swedish oceanographer.

Born in Stockholm to Fredrik Laurentz Ekman, himself an oceanographer, he became committed to oceanography while studying physics at the University of Uppsala and, in particular, on hearing Vilhelm Bjerknes lecture on fluid dynamics.

During the expedition of the Fram, Fridtjof Nansen had observed that icebergs tend to drift not in the direction of the prevailing wind but at an angle of 20°-40°. Bjerknes invited Ekman, still a student, to investigate the problem and, in 1902, Ekman published his theory of the Ekman spiral which explains the phenomenon in terms of the balance between frictional effects in the ocean and the fictitious forces arising from planetary rotation.

On completing his doctorate in Uppsala in 1902, Ekman joined the International Laboratory for Oceanographic Research, Oslo where he worked for seven years, not only extending his theoretical work but also developing experimental techniques and instruments such as the Ekman current meter and Ekman water bottle.

From 1910 to 1939 he continued his theoretical and experimental work at the University of Lund, where he was professor of mechanics and mathematical physics. He was elected a member of the Royal Swedish Academy of Sciences in 1935.

A gifted amateur bass, pianist and composer, he continued working right up to his death in Gostad, near Stockaryd, Sweden.

The details of the Ekman spiral velocity profile as a function of depth are more of theoretical interest that practical importance. Current profiles closely resembling the theoretical result are seldom, if ever, observed.

The details of the spiral profile depend on the assumed eddy viscosity, and Av = constant is not a particularly good assumption. Recall that the size of turbulent eddies tends to scale with distance from the boundary so that Av is generally proportional to z which leads to the log-layer dependence. In Ekman dynamics, the log-layer structure is modified by Coriolis.

The fact that the surface current is to the right of the wind (f > 0) is a key result, but the magnitude of the angle will depend on the details.

In practical applications such as oil-spill tracking and search-and-rescue, empirical values for the angle of motion with respect to the wind direction are used based on experience and observation.

However, a robust and important result that is independent of these details is obtained if we integrate the equations over a depth large enough to encompass the whole Ekman layer (in practice, just a few times the Ekman scale depth).

Start with the Ekman equations expressed in terms of the stresses rather than eddy viscosity

u-momentum equation:

We need not actually integrate to –∞ because the Ekman currents decay exponentially fast.

In practice, it is sufficient to integrate from the surface to some depth z=-D, where D is a few times the Ekman depth, at which depth

Similarly, for the v-momentum equation:

These components of the Ekman transport describe depth integrated flow (in m2s-1) (velocity times depth) that is 900 to the right (left) of the wind stress in the northern (southern) hemisphere.

The details of the eddy viscosity profile have no influence on this result, and the calculation is very robust.

This Ekman balance between wind stress and Coriolis is established over several inertial periods, i.e. the balance is not established instantly when the wind starts blowing.

The ocean response to suddenly imposed winds is a set of inertial oscillations. The inertial oscillations decay over a period that is several times their natural oscillatory timescale f-1 leaving steady Ekman transports in their wake.

The vertical scale of the Ekman motion can be estimated with a scaling analysis.

fv is of magnitude (we also call this of order) o(fv)

Av d2u/dz2 is of order o(Av V/de2)

Since they are equal

fV= Avv/de2

from which we can determine the scale of depth to be

de~sqrt(Av/f)

Basically it is the depth that the vertical mixing can penetrate in an inertial period.

A more detailed approach would yield

de=p (2 Av/f)1/2

(Note that some text books does not include the p but this is only a scaling)

For Av=.01 Ekman depth is ~40 meters.

This is the depth of the wind driven layer.

It’s very small compared to the depth of the ocean.

Intuitively one would think that the stronger the wind the deeper should be the wind-driven layer. But the above equation doesn’t explicitly indicate this. Why?

Because the eddy viscosity is proportional to the wind speed! So as wind speed increases eddy viscosity increases and the Ekman depth increases.

Draw slaps of fluid along with free-body diagrams to show intuitive example of the

A subtle yet real important effect of the Ekman Transport on the ocean is that it produces pressure gradients. This occurs both at the basin scale due to the curl of the wind i.e westerlies in the temperate latitudes and easterlies in the tropics drives the subtropical gyre.

It also is responsible for coastal upwelling and Ekman Pumping, and the Bottom Ekman Layer—but we’ll discuss these in more detail later.

Important concepts:

·  Ekman currents are stronger at the surface, and decay approximately exponentially over a depth scale given by . Typical Ekman depths are of order 10 to 30 meters.

·  Regardless of the details of the eddy viscosity profile, the Ekman transports are:

which are directed perpendicular to the wind stress direction; right (left) in the northern (southern) hemisphere

·  Ekman transports are fully established after several inertial periods, i.e. 1 to 2 days

Ekman dynamics is a very practical way to estimate the oceanic response to winds on time scales of a few days.

Objects floating near the surface within the Ekman layer will be transported by Ekman currents, and their drift can be predicted with considerable skill using these simple equations.

However, Ekman transports have a far more significant impact on the entire upper ocean circulation (over much greater depths than the Ekman layer) through a rather subtle interaction with the oceanic pressure field.

Where Ekman transports converge and diverge they generate pressure gradients that are in turn balanced by the Coriolis force, and the resulting geostrophically balanced currents form the upper ocean pattern of gyres and western boundary currents.

Talk about wind stress curl and how Ekman motion can impact circulation well below the Ekman Layer Depth.

Show example of an eddy and the circulation associated with bottom Ekman Layers.

Here’s some notes from TAMU that are excellent.

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter09/chapter09_02.htm

Nansen's Qualitative Arguments
Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20°-40° to the right of the wind in the Arctic, by which he meant that the track of the iceberg was to the right of the wind looking downwind (See Figure 9.2) . He later worked out the balance of forces that must exist when wind tried to push icebergs downwind on a rotating Earth.

Figure 9.2 The balance of forces acting on an iceberg in a wind on a rotating Earth.

Nansen argued that three forces must be important:

1.  Wind Stress, W;

2.  Friction F (otherwise the iceberg would move as fast as the wind);

3.  Coriolis Force, C.

Nansen argued further that the forces must have the following attributes:

1.  Drag must be opposite the direction of the ice's velocity;

2.  Coriolis force must be perpendicular to the velocity;

3.  The forces must balance for steady flow.

W + F + C = 0

Ekman's Solution
Nansen asked Vilhelm Bjerknes to let one of Bjerknes students make a theoretical study of the influence of Earth's rotation on wind-driven currents. Walfrid Ekman was chosen, and he presented the results in his thesis at Uppsala. Ekman later expanded the study to include the influence of continents and differences of density of water (Ekman, 1905). The following follows Ekman's line of reasoning in that paper.

Ekman assumed a steady, homogeneous, horizontal flow with friction on a rotating Earth. Thus horizontal and temporal derivatives are zero:

/ (9.6)

Ekman further assumed a constant vertical eddy viscosity of the form (8:12):

/ (9.7)

where Txz, Tyz are the components of the wind stress in the x, y directions, and ρw is the density of sea water.

With these assumptions, and using (9.7) in (8.14), the x and y components of the momentum equation have the simple form:

/ (9.8a)
/ (9.8b)

where f is the Coriolis parameter. It is easy to verify that the equations (9.9) have solutions:

/ (9.9a)
/ (9.9b)

when the wind is blowing to the north (T = Tyz ). The constants are

/ (9.10)

and V0 is the velocity of the current at the sea surface.

Now let's look at the form of the solutions. At the sea surface z = 0, exp(z = 0) = 1, and

/ (9.11a)
/ (9.11b)

The current has a speed of V0 to the northeast. In general, the surface current is 45° to the right of the wind when looking downwind in the northern hemisphere. The current is 45° to the left of the wind in the southern hemisphere. Below the surface, the velocity decays exponentially with depth (Figure 9.3):

/ (9.12)
Figure 9.3. Ekman current generated by a 10m/s wind at 35°N.

Values for Ekman's Constants
To proceed further, we need values for any two of the free parameters: the velocity at the surface, V0; the coefficient of eddy viscosity, Az; or the wind stress T.

The wind stress is well known, and Ekman used the bulk formula (4.2):

/ (9.13)

where ρair is the density of air, CD is the drag coefficient, and U10 is the wind speed at 10m above the sea. Ekman turned to the literature to obtain values for V0 as a function of wind speed. He found:

/ (9.14)

With this information, he could then calculate the velocity as a function of depth knowing the wind speed U10 and wind direction.