Ocean Dynamics
Previous Lectures
So far we have discussed the equations of motion
ignoring the role of friction
In order to understand ocean circulations we also
need to consider the effects of friction since it is
friction imparted by winds which is the primary
driver for these circulations
Today, we will see the effect friction has on the
dynamic of the ocean
Dynamics
Equations of Motion
For the “frictional layer”, vertical gradients in the velocity (“shear”)
is produced by frictional force applied to the ocean by the
atmosphere (and vice versa), which in turn produces a strong
momentum flux in the vertical (called “shear stress”)
For individual molecules, the stress is proportional to:
• The vertical gradient in velocity
• The “kinematic viscosity” of water, which is a molecular property
of a fluid that measures the internal resistance to deformation
• The density of fluid
(Eqn.)
For large-scale motions, or “eddies”, we estimate stress based
upon the gradient in current:
(Eqn.)
Dynamics
Equations of Motion
For the momentum equations, the frictional force is related to the
vertical convergence/divergence of momentum flux associated
with this stress:
(Eqn.)
We can now plug this into our momentum equations from before
and come up with:
(Eqn.)
Ocean Dynamics
Basin Circulations
To examine the role of friction, first assume that the system is in
steady state (I.e. no change with time):
(Eqn.)
Lets also assume the currents are comprised of a wind-driven
component and a pressure-driven (I.e. geostrophic) component:
(Eqn.)
Then the equations for the wind-driven components only can be
written as:
(Eqn.)
Ocean Dynamics
Basin Circulations
For the wind-driven equations, we can take the second-derivative
of the first equation and plugging it into the second, which gives:
(Eqn.)
Solving this equation eventually gives the currents as a function of
depth:
(Eqn.)
Using, this equation we can plot the currents at various fractions of
the Ekman height and produce a 3-dimension image of the Ekman
spiral
Ocean Dynamics
Basin Circulations
In addition to looking at the current at a given level, we can
also look at the “mass flux” of water in the Ekman layer,
which just represents a vertical integral of the currents with
depth:
(Eqn.)
For the ocean, the mass flux can be written as:
(Eqn.)
Hence, the net mass flux associated with the Ekman
velocities is directed 90-degrees to the right of the surface
wind direction
Ocean Dynamics
Basin Circulations
Net mass transport in the ocean and atmosphere
In the geographic coordinate frame, the mass transport of
the ocean can be written as:
(Eqn.)
Ocean Dynamics
Basin Circulations
Now we want to consider variations in winds and their
relation to convergence or divergence of mass in various
regions
If we assume that water is incompressible, then we can use
the equation of “mass continuity” to write:
(Eqn.)
Integrating with depth and assuming that the vertical
velocity at the surface is zero, gives the vertical velocity at
the bottom of the Ekman layer:
(Eqn.)
Hence, horizontal variations in the winds can produce
vertical motions into and out of the Ekman layer
Ocean Dynamics
Basin Circulations
In the mid-latitudes, the strong meridional gradient in between
subtropical easterlies and mid-latitude westerlies
From before we know that the wind-stress is related to the Ekman
mass transport by:
(Eqn.)
Taking the derivatives of each side and plugging into the equation
for vertical velocities then gives:
(Eqn.)
Ocean Dynamics
Basin Circulations
If we consider the mass transport to be the
integrated transport through the entire column,
then the continuity equation is
(Eqn.)
Then we find that the mass transport below the
Ekman layer is:
(Eqn.)
Ocean Dynamics
Basin Circulations
Hence, the interior circulation in the oceans is a
consequence of secondary circulations forced by Ekman
pumping associated with wind forcing at the surface
Ocean currents associated with wind forcing
• At the surface, ocean currents tend to be parallel to the
geostrophic winds above the atmospheric boundary layer
• Horizontal variations in the surface winds produce vertical
motions at the bottom of the Ekman layer, resulting in
horizontal currents through ~1000m
Dynamics
Equations of Motion
Wind
Ocean Dynamics
Basin Circulations
Ocean Dynamics
Basin Circulations
Ocean Dynamics
Basin Circulations
Meatmos
atmos
Winds
ocean
Meocean
Ocean Dynamics
Basin Circulations
Ocean Dynamics
Basin Circulations
Ocean Dynamics
Basin Circulations
Ocean Dynamics
Basin Circulations
Ocean Dynamics
Basin Circulations
Contours of mass transport:
“Sverdrups”=109kg/s or 106m3/s
Ocean Dynamics
Basin Circulations
Winds
Winds
Meocean
we
MSverdrup