Ocean DynamicsPrevious 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

DynamicsEquations 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.)

DynamicsEquations 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 DynamicsBasin 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 DynamicsBasin 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 DynamicsBasin 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 DynamicsBasin 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 DynamicsBasin 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 DynamicsBasin 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 DynamicsBasin 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 DynamicsBasin Circulations

Hence, the interior circulation in the oceans is a consequence of secondary circulations forced by Ekman pumping associated with wind forcing at the surfaceOcean 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 moti

ons at the bottom of the Ekman layer, resulting in horizontal currents through ~1000m

DynamicsEquations of Motion

Wind

Ocean DynamicsBasin Circulations

Ocean DynamicsBasin Circulations

Ocean DynamicsBasin Circulations

ocean

Meocean

atmos

Meatmos

Winds

Ocean DynamicsBasin Circulations

Ocean DynamicsBasin Circulations

Ocean DynamicsBasin Circulations

Ocean DynamicsBasin Circulations

Ocean DynamicsBasin Circulations

Contours of mass transport:

“Sverdrups”=109kg/s or 106m3/s

Ocean DynamicsBasin Circulations

MSverdrup

Meocean

Winds

Winds

we