Equations of Motion in Physical Oceanography

1. Chapter 7 Some Mathematics: The Equations of Motion Physical oceanography Instructor: Dr. Cheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 24October 2003

2. Introduction • Response of a fluid to • Internal force • External force •  basic equations of ocean dynamics • Chapter 8: viscosity • Chapter 12: vorticity • Table 7.1 • Conservation laws  basic equations

3. Dominant Forces for Ocean Dynamics • Gravity Fg • WwaterP(x)  P • Revolution and rotation  DFg  tides, tidal current, tidal mixing • Buoyancy FB • DT  Dr  FB (vertical direction)  upward or sink • Wind Fw • Wind blows  momentum transfer  turbulence  ML • Wind blows  P(x)  P  waves

4. Dominant Forces for Ocean Dynamics (cont.) • Pseudo-forces •  motion in curvilinear or rotating coordinate systems • a body moving at constant velocity seems to change direction when viewed from a rotating coordinate system  the Coriolis force • Coriolis Force • The dominant pseudo-force influencing currents • Other forces: Table 7.2 • Atmospheric pressure • Seismic

5. Coordinate System • Coordinate System  find location • Cartesian Coordinate System • Most commonly use • Simpler  spherical coordinates • Convention: • x is to the east, y is to the north, and z is up. • f-plane • Fcor = const (a Cartesian coordinate system) • Describing flow in small regions

6. Coordinate System (cont.) • b-plane • Fcor  latitude (a Cartesian coordinate system) • Describing flow over areas as large as ocean basins • Spherical coordinates • (r, q, f) • Describe flows that extend over large distances and in numerical calculations of basin and global scale flows

7. Types of Flow in the Ocean • Flow due to currents • General Circulation • The permanent, time-averaged circulation • Meridional Overturning Circulation • The sinking and spreading of cold water • Also known as the Thermohaline Circulation • the vertical movements of ocean water masses  Dr DTandDS • The circulation in meridional plane driven by mixing • Wind-Driven Circulation • The circulation in the upper kilometer  wind • Gyres • Wind-driven cyclonic or anti-cyclonic currents with dimensions nearly that of ocean basins.

8. Types of Flow in the Ocean (cont.) • Flow due to currents (cont.) • Boundary Currents • Currents owing parallel to coasts • Western boundary currents  fast, narrow jets • e.g. the Gulf Stream and Kuroshio • Eastern boundary currents  weak • e.g. the California Current • Squirts or Jets • Long narrow currents • with dimensions of a few hundred kilometers • Nearly  west coasts • Mesoscale Eddies • Turbulent or spinning flows on scales of a few hundred kilometers

9. Types of Flow in the Ocean (cont.) • Oscillatory flows due to waves • Planetary Waves • The rotation of the Earth  restoring force • Including Rossby, Kelvin, Equatorial, and Yanai waves • Surface Waves (gravity waves) • The waves that eventually break on the beach • The large Dr between air and water  restoring force • Internal Waves • Subsea wave ~ surface waves • r = r (D)  restoring force • Tsunamis • Surface waves with periods near 15 minutes generated by earthquakes

10. Types of Flow in the Ocean (cont.) • Oscillatory flows due to waves (cont.) • Tidal Currents •  tidal potential • Shelf Waves • Periods  a few minutes • Confined to shallow regions near shore • The amplitude of the waves drops off exponentially with distance from shore

11. Conservation of Mass and Salt • Dm = 0 & DS = 0  net fresh water loss  minimum flushing time • Net fresh water loss = R + P – E • QL  bulk formula  large amount of ship measurements (T, q, …)  impossible • Dm = 0  Vi + R + P = Vo + E • DS = 0  ri Vi Si = ro Vo So • Measure Viassume ri ro • Estimate the minimum flushing time • Example • Fig 7.2: Box model  qout = q/t dt + q/x dx + qin

12. The Total Derivative (D/Dt) • D/Dt = ∂/dt + u·( ) • A simple example of acceleration of flow in a small box of fluid • qout = q/t dt + q/x dx + qin • Dq/Dt = q/t + u q/x • 3D case: D/Dt = /t + u/x + v/y + w/z • The simple transformation of coordinates from one following a particle to one fixed in space converts a simple linear derivative into a non-linear partial derivative

13. Conservation of Momentum: Navier-Stokes equation • Newton’s 2nd law • F = D(mv)/Dt • Dv/Dt = F/m = fm = fp+ fc+ fg + fr • Pressure gradient fp = -p/r • Coriolis force fc = -2W v • W = 7.292  10-5radians/s • Gravity fg = g • Friction fr • Dv/Dt = -p/r -2W v + g + fr • W = 7.292  10-5 radians/s

14. Conservation of Momentum: Navier-Stokes equation (cont.) • Pressure term • ax = -(1/r) (p/x) • dFx = pdy dz-(p + dp) dy dz = -dpdydz Source:http://oceanworld.tamu.edu/resources/ocng_textbook/chapter07/chapter07_06.htm

15. Conservation of Momentum: Navier-Stokes equation (cont.) • Gravity term • g = gf - W (W R) Source:http://oceanworld.tamu.edu/resources/ocng_textbook/chapter07/chapter07_06.htm

16. Conservation of Momentum: Navier-Stokes equation (cont.) • The Coriolis term

17. Conservation of Momentum: Navier-Stokes equation (cont.) • Momentum Equation in Cartesian Coordinates

18. Conservation of mass: the continuity equation • For compressible fluid Source:http://oceanworld.tamu.edu/resources/ocng_textbook/chapter07/chapter07_06.htm

19. Conservation of mass: the continuity equation (cont.) • Oceanic flows are incompressible • Boussinesq's assumption • v << c (sound speed) • When v  c, Dv Dr • Phase speed of waves << c • c   in incompressible flows • Vertical scale of the motion << c2/g • The increase in pressure produces only small changes in density • r  const, except the pressure term (rg)

20. Conservation of mass: the continuity equation (cont.) • For incompressible flow • The coefficient of compressibility b • b = 0 for incompressible flows

21. Solutions to the Equations of Motion • Solvable in principle • Four equations • 3 momentum equations • 1 continuity equation • Four unknowns • 3 velocity components: u, v, w • 1 pressure p • Boundary conditions • No slip condition: v//(boundary) = 0 • No penetration condition: v(boundary) = 0

22. Solutions to the Equations of Motion (cont.) • Difficult to solve in practice • Exact solution • No exact solutions for the equations with friction • Very few exact solutions for the equations without friction • Analytic solution • For much simplified forms of the equations of motion • Numerical solution • Solutions for oceanic flows with realistic coasts and bathymetric features must be obtained from numerical solutions (Chapter 15)

23. Important concepts • Gravity, buoyancy, and wind are the dominant forces acting on the ocean • Earth's rotation produces a pseudo force, the Coriolis force • Conservation laws applied to flow in the ocean lead to equations of motion; conservation of salt, volume and other quantities can lead to deep insights into oceanic flow

24. Important concepts (cont.) • The transformation from equations of motion applied to fluid parcels to equations applied at a fixed point in space greatly complicates the equations of motion. The linear, first-order, ordinary differential equations describing Newtonian dynamics of a mass accelerated by a force become nonlinear, partial differential equations of fluid mechanics. • Flow in the ocean can be assumed to be incompressible except when de-scribing sound. Density can be assumed to be constant except when density is multiplied by gravity g. The assumption is called the Boussinesq approximation

25. Important concepts (cont.) • Conservation of mass leads to the continuity equation, which has an especially simple form for an incompressible fluid.