The Equations of Motion 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: 24 October 2003

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

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

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

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

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

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  DT and DS 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.

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

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

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

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 Vi assume ri  ro Estimate the minimum flushing time Example Fig 7.2: Box model  qout = q/t dt + q/x dx + qin

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

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  10-5 radians/s Gravity fg = g Friction fr Dv/Dt = -p/r -2W  v + g + fr

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

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

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

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

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

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)

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

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

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)

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

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

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