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).
u
 fv
t
v
  fu
t
(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
u
1 u 2

 fuv
t
2 t
v
1 v 2
v

  fuv
t
2 t 2
u
(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
u
u
v 1 u 2 1 v 2
 1

v  ( 2 
)  ( (u 2  v 2 ))  (ke)  0
2
t
t 2 t
2 v
t 2
t
(5)
1
Here we that u
u 1 u 2
, and thus we can rewrite

x 2 t
1  u 2 v 2   (ke)



0
2  t
t 
t
(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
 2u
 f 2 u
2
t
(7)
Similarly taking the time derivative of equation 2 and substituting in equation 1 for the
right hand yields
2v
 f 2v
2
t
(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(t)
(9)
where U is the amplitude of the wave and is the wave’s frequency
So substituting 9 into 7 we find
=f
(10)
Which reveals that the frequency of the wave is the Coriolis frequency f.
2
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
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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.
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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:
 x
 fv 
z
0
 f  v dz 

 x
x
 z dz  
0
 
0

 x
 fVE   x
VE  
x
f
in m 2 s 1
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 exp(  D /  e )  0
Similarly, for the v-momentum equation:
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fu 
 y
z
UE 
y
f
in m 2 s 1
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= (2 Av/f)1/2
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(Note that some text books does not include the 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 winddriven 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
2 Av
e 
f . Typical Ekman depths are
exponentially over a depth scale given by
of order 10 to 30 meters.

Regardless of the details of the eddy viscosity profile, the Ekman transports are:
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VE  
UE 
x
f
y
f
in m 2 s 1
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.
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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):
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(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)
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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.
Ekman Layer Depth
The thickness of the Ekman layer is arbitrary because the Ekman currents decrease
exponentially with depth. Ekman proposed that the thickness be the depth DE at
which the current velocity is opposite the velocity at the surface, which occurs at a
depth DE = π/a, and the Ekman layer depth is:
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(9.15)
Using (9.13) in (9.10) , dividing by U10, and using (9.14) and (9.15) gives:
(9.16)
in SI units; wind in meters per second gives depth in meters. The constant in (9.16)
is based on ρw = 1027 kg/m3, ρair = 1.25 kg/m3, and Ekman's value of CD = 2.6×10-3
for the drag coefficient.
Using (9.16) with typical winds, the depth of the Ekman layer varies from about 45m
to 300m (Table 9.3), and the velocity of the surface current varies from 2.5% to
1.1% of the wind speed depending on latitude.
Table 9.3 Typical Ekman Depths
U10(m/s)
Latitude
15°
45°
5
40m
30m
10
90m
50m
20
180m
110m
The Ekman Number: Coriolis and Frictional Forces The depth of the Ekman
layer is closely related to the depth at which frictional force is equal to the Coriolis
force in the momentum equation (9.9) . The Coriolis force is fu, and the frictional
force is Az ∂2U/∂z2. The ratio of the forces, which is non dimensional, is called the
Ekman Number Ez:
(9.17)
where we have approximated the terms using typical velocities u, and typical depths
d. The subscript z is needed because the ocean is stratified and mixing in the vertical
is much less than mixing in the horizontal. Note that as depth increases, friction
becomes small, and eventually, only the Coriolis force remains.
Solving (9.17) for d gives
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(9.18)
which agrees with the functional form (9.15) proposed by Ekman. Equating (9.18)
and (9.15) requires Ez = 1/(2π2) = 0.05 at the Ekman depth. Thus Ekman chose a
depth at which frictional forces are much smaller than the Coriolis force.
Bottom Ekman Layer
The Ekman layer at the bottom of the ocean and the atmosphere differs from the
layer at the ocean surface. The solution for a bottom layer below a fluid with velocity
U in the x-direction is:
u = U [1 - exp(-az) cos az
(9.19a)
v = U exp(-az) sin az
(9.19b)
The velocity goes to zero at the boundary, u = v = 0 at z = 0. The direction of the
flow close to the boundary is 45° to the left of the flow U outside the boundary layer
in the northern hemisphere; and the direction of the flow rotates with distance above
the boundary (Figure 9.4). The direction of rotation is anti-cyclonic with distance
above the bottom.
Figure 9.4 Ekman layer for
the lowest kilometer in the
atmosphere (solid line),
together with wind velocity
measured
by
Dobson
(1914) - - - . The numbers
give height above the
surface in meters. The
boundary layer at the
bottom of the ocean has a
similar
shape.
From
Houghton (1977).
Winds above the planetary boundary layer are perpendicular to the pressure gradient
in the atmosphere and parallel to lines of constant surface pressure. Winds at the
surface are 45° to the left of the winds aloft, and surface currents are 45° to the
right of the wind at the surface. Therefore we expect currents at the sea surface to
be nearly in the direction of winds above the planetary boundary layer and parallel to
lines of constant pressure. Observations of surface drifters in the Pacific tend to
confirm the hypothesis (Figure 9.5).
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Figure 9.5 Trajectories of
surface drifters in April
1978 together with surface
pressure
in
the
atmosphere averaged for
the month. Note that
drifters tend to follow lines
of
constant
pressure
except in the Kuroshio
where ocean currents are
fast
compared
with
velocities in the Ekman
layer in the ocean. From
McNally, et al. (1983).
Examining Ekman's Assumptions
Before considering the validity of Ekman's theory for describing flow in the surface
boundary layer of the ocean, let's first examine the validity of Ekman's assumptions.
He assumed:
1.
2.
3.
4.
No boundaries. This is valid away from coasts.
Deep water. This is valid if depth >> 200m.
f-plane. This is valid.
Steady state. This is valid if wind blows for longer than a pendulum day. Note
however that Ekman also calculated a time-dependent solution, as did
Hasselmann (1970).
5. Az is a function of U 210 only. It is assumed to be independent of depth. This is
not a good assumption. The mixed layer may be thinner than the Ekman
depth, and Az will change rapidly at the bottom of the mixed layer because
mixing is a function of stability. Mixing across a stable layer is much less than
mixing through a layer of a neutral stability. More realistic profiles for the
coefficient of eddy viscosity as a function of depth change the shape of the
calculated velocity profile. We reconsider this problem below.
6. Homogeneous density. This is probably good, except as it effects stability.
Observations of Flow Near the Sea Surface
Does the flow close to the sea surface agree with Ekman's theory? Measurements of
currents made during several, very careful experiments indicate that Ekman's theory
is remarkably good. The theory accurately describes the flow averaged over many
days. The measurements also point out the limitations of the theory.
Weller and Plueddmann (1996) measured currents from 2 m to 132 m using 14
vector-measuring current meters deployed from the Floating Instrument Platform
FLIP in February and March 1990 500 km west of point Conception, California. This
was the last of a remarkable series of experiments coordinated by Weller using
instruments on FLIP.
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Davis, DeSzoeke, and Niiler (1981) measured currents from 2m to 175m using 19
vector-measuring current meters deployed from a mooring for 19 days in August and
September 1977 at 50°N, 145°W in the northeast Pacific.
Ralph and Niiler (2000) tracked 1503 drifters drogued to 15m depth in the Pacific
from March 1987 to December 1994. Wind velocity was obtained every 6 hours from
the European Centre for Medium-Range Weather Forecasts ECMWF.
The results of the experiments indicate that:
1. Inertial currents are the largest component of the flow.
2. The flow is nearly independent of depth within the mixed layer for periods
near the inertial period. Thus the mixed layer moves like a slab at the inertial
period. Current shear is concentrated at the top of the thermocline.
3. The flow averaged over many inertial periods is almost exactly that calculated
from Ekman's theory. The shear of the Ekman currents extends through the
averaged mixed layer and into the thermocline. Ralph and Niiler found:
(9.20)
(9.21)
4. The Ekman-layer depth DE is almost exactly that proposed by Ekman (9.16),
but the surface current V0 is half his value (9.14).
5. The transport is 90° to the right of the wind in the northern hemisphere. The
transport direction agrees well with Ekman's theory.
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