Wind blowing over
the sea surface exerts a stress on the ocean that imparts momentum. This drag
force at the surface slows the wind speed and forms a boundary layer in the
atmosphere. The precise details of how the atmosphere and ocean interact to
exchange momentum is complicated by the stratification or stability of the
atmospheric boundary layer, the presence of waves, wave breaking, and a host of
other processes.

For practical
applications in oceanography, it’s enough for us to use an empirical formula to
calculate wind stress from wind speed.

_{}

where

*U _{10}* = wind speed at 10 m above the sea surface

_{}= 1.22 kg m^{-3}

*C _{D}* = dimensionless drag coefficient

a typical value might be 0.0013

this gives _{} in units of N m^{-2},
or Pascals (Pa).

A popular formula
for a neutrally stable boundary layer is that of Large and Pond (1981), J.
Phys. Oceanog., 11, 324-336.

C_{D
}= 1.2 x 10^{-3 }for 4 < U_{10} < 11 m s^{-1}

C_{D}
= 10^{-3} (0.49 + 0.065 U_{10}) for 11 < U_{10} < 25 m s^{-1}

Example: U_{10} = 8 m s^{-1} _{} = 1.2 x 10^{-3}
* 1.22 * 64 = 0.0937 Pa

Typical oceanic
values of wind stress are around 0.1

Because wind stress
is a quadratic function of wind speed, gusty winds produce larger stresses than
would steady winds of the same average speed.

Stormy regions,
such as the Southern Ocean, have particularly high mean wind stress.

In practice, we
often have observations from instruments located as some height other than 10 m
above the sea surface. However, it is meteorological convention to report wind
speeds as equivalent 10 m values by using the log layer theory (Monin-Obukov theory) to adjust the observed wind speed to
that which would have been observed if the anemometer were at 10 m. This
adjustment can be a source of error.

At low wind speeds
there is considerable uncertainty about the correct parameterization of drag
coefficient, and this is an active research problem.

Direct estimates of
drag coefficients at low wind speed vary wildly, and there appear to be more
factors involved that simply the wind speed profile through the logarithmic
atmospheric boundary layer. These include:

·
whether the log layer assumption is valid at low wind speed

·
the presence of waves and swell

·
surfactants

·
meso-scale variability affecting
stability of boundary layer

We have now
introduced a complete description of the various forces that act on a fluid and
govern its motion.

_{}

which include the
forces:

·
pressure gradients

·
gravity

·
stresses (viscous and turbulent)

·
Coriolis

The sum of these
forces starts or keeps a fluid in motion by producing a net acceleration of the
fluid.

The acceleration is
comprised of a local time rate of change, but also changes following the fluid.
So even in steady flow, where nothing changes with time, fluid may gain or lose
momentum as it flows along.

This fluid motion
occurs subject to constraints on the conservation of mass, and since motion
transports temperature and salt which affect density and therefore pressure,
transport processes are directly linked to the dynamics of the flow itself.

Understanding fluid
motion requires an understanding of the balances of forces.

We already have
some intuition for how fluid behaves, and this stems from understanding simple
force balances:

·
water accelerates when you pour it (gravity : acceleration)

·
sticky fluids pour at a steady rate (gravity
: friction)

·
water pistol (pressure
: acceleration)

Geophysical fluid
dynamics is the study of fluid flow on scales large enough that the fact we are
on a rotating planet is of fundamental importance.

The effects of the Coriolis force are often counter-intuitive – but this is
just a matter of us developing a new intuition.

*Chapter 6 Knauss*

In discussing
friction we considered the Reynolds number

_{}

Re
= ratio of nonlinear momentum advection (or*
inertia*) terms to viscous terms.

The Rossby number we’ve met in previous lectures is

Ro
= ratio of *inertia** *terms (a.k.a. centrifugal acceleration) to the Coriolis
acceleration:

_{}

where U is a velocity scale

L a length

*f* is the Coriolis parameter 2W sin (latitude)

W = 2p/(86164
seconds) = 7.29 x 10^{-4} s^{-1}

typical *f* at latitude 45^{o} is 10^{-4} s^{-1}

Let’s consider the
dynamics that result for a very simple force balance for a flow with Ro ~ 1.

Shipwrecked: In a
wide expansive ocean with no coastlines and no gradients in anything in any
direction

·
no pressure gradient

·
no friction

·
no spatial gradients in velocity

·
the wind has stopped blowing, but the fluid is in motion

All that is left is
the balance between inertia and Coriolis

_{}

These equations can
be solved by differentiating one with respect to time, and substituting in the
other to give a single equation for *v *(or
*u*).

_{}

Applied
mathematicians, engineers and musicians will recognize this as a wave equation
with the simple solution

_{}

which is readily
verified by substitution into the equation. It follows that

_{}

The *magnitude* of the velocity is

_{}

which is *constant* in time.

The water is always
going the exact same *speed**. *This speed is the speed the water had attained when the wind (or
whatever started it going) stopped.

This fits with our
knowledge of

The peculiar thing
about this is that the direction keeps changing.

These velocity
components describe motion in a circle of radius

_{}

The direction of
motion is clockwise for *f>0*
(northern hemisphere) and counter-clockwise for *f<0* (southern hemisphere).

(We also describe
the direction as __anti-cyclonic__ regardless of the sign of *f* for reasons that will be become
apparent when we discuss the direction of rotation of cyclones and anticyclones
in the pressure field. __Anti-cyclonic__ implies *clockwise in the northern* hemisphere AND *counter-clockwise in the southern* hemisphere).

For example, water
in motion with a speed of U = 0.5 m s^{-1}, at latitude 42^{o}N
where *f* = 10^{-4} s^{-1},
describes a circle of radius

*r*
= 0.5/10^{-4} = 5000 m = 5 km.

The period of the
motion (once around the circle) is

_{}

which we call the
“inertial” period.

Notice that it
changes with latitude.

·
At 60^{o} latitude the inertial period is 13.8 hours.

·
At 45^{o} latitude the inertial period is 16.9 hours.

·
At 30^{o} latitude the inertial period is 23.9 hours.

Although we assumed
no spatial gradients in the flow, which would seem to make specifying a
characteristic length scale in the Rossby number
rather ill-posed, we see that a natural length scale (the radius of the
inertial circle) arises in the solution.

The Rossby number for this inertial oscillation phenomenon is:

*Ro
= _{}* = 1.

This says the inertia
forces are of the same size as Coriolis, which is
hardly surprising since this is precisely the simple balance we assumed at the
start.

These so-called
“inertial oscillations” are often observed in the ocean in situations such as
the response to the passage of an abrupt storm. They will be apparent in
current meter observations of *u(**t)* and *v(t)* at a fixed point, and also show in
the trajectories of drifting buoys.

Stewart: Figure 9.1 Inertial
currents in the North Pacific in October 1987 (days 275–300) measured by
holey-sock drifting buoy drogued at a depth of 15m.
Positions were observed 10–12 times per day by the

Further reading :

Stewart
chapter 9

Knauss chapter 6