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Understanding the Role of Friction in Ocean Dynamics and Circulations

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  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 layer • Horizontal variations in the surface winds produce vertical motions at the bottom of the Ekman layer, resulting in horizontal currents through ~1000m

  12. Dynamics Wind • Equations of Motion

  13. Ocean Dynamics • Basin Circulations

  14. Ocean Dynamics • Basin Circulations

  15. Ocean Dynamics • Basin Circulations Meatmos atmos Winds ocean Meocean

  16. Ocean Dynamics • Basin Circulations

  17. Ocean Dynamics • Basin Circulations

  18. Ocean Dynamics • Basin Circulations

  19. Ocean Dynamics • Basin Circulations

  20. Ocean Dynamics • Basin Circulations Contours of mass transport: “Sverdrups”=109kg/s or 106m3/s

  21. Ocean Dynamics Winds • Basin Circulations Winds Meocean we MSverdrup

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