1. 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

2. 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.)

3. 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.)

4. 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.)

5. 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

6. 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

7. 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.)

8. 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

9. 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.)

10. 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.)

11. 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 layerAt 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 ~1000mHorizontal variations in the surface winds produce vertical motions at the bottom of the Ekman layer, resulting in horizontal currents through ~1000m

12. Dynamics Equations of Motion Wind

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15. Ocean Dynamics Basin Circulations  ocean M e ocean  atmos M e atmos Winds

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20. Ocean Dynamics Basin Circulations Contours of mass transport: “Sverdrups”=10 9 kg/s or 10 6 m 3 /s

21. Ocean Dynamics Basin Circulations M Sverdrup M e ocean Winds wewe