1. Lecture Guidelines for GEOF110 Chapter 5 ( 2 hours) Chapter 6 (2 Hours) Ilker Fer Guiding for blackboard presentation. Following Pond & Pickard, Introductory Dynamical Oceanography

2. GEOF110 Guidelines / 3 2 Static Stability Atmosphere Ocean   Sketch: Incompressible fluid. Parcel 0 is in hydrostatic equilibrium. Atmosphere: Ocean: “Parcel” or “particle” : we mean a (infinitesimal) fluid sample of uniform T and composition in a continuous environment

3. GEOF110 Guidelines / 3 3 Level 1 z Level 2 z+  z PARCEL (P) ENVIRONMENT (W) Parcel at level 1 is displaced to level 2 with environment in situ properties given on the figure. At level 2, the restoring force on the parcel with volume  V is: F= buoyant upthrust – weight Archimedes: Buoyant upthrust = weight of displaced volume (=g  2  V)

4. GEOF110 Guidelines / 3 4

5. 5 E > 0 : STABLEparcel displaced vertically will tend to return to its position. Due to its inertia, overshoot equilibrium position  oscillate (with frequency N) E = 0 : NEUTRALvertically displaced parcel will remain there E < 0 : UNSTABLEoverturning

6. GEOF110 Guidelines / 3 6 In the upper 1 km of ocean E range between 100- 1000x10 -8 1/m, largest values at the pycnocline, in the upper a few 100 m. In deep ocean, deep trenches, E is close to 1x10 -8 1/m. i.e., in situ T increases about (0.1 – 0.2) K / km This is OK using eq. of state and an accurate relation for C. Use of in-situ density would look very stably stratified in neutral stability: compensates for compressibility These are comparable

7. GEOF110 Guidelines / 3 7 This is not very convenient, nor practical (e.g.  /  S taken while holding in situ values of T and P).

8. GEOF110 Guidelines / 3 8 NOTE: In a stable environment, an inviscid fluid parcel displaced a small distance  vertically will go through simple harmonic oscillations described by

9. GEOF110 Guidelines / 3 9 Double Diffusion Double-diffusion occurs when the density variations are caused by two different components with different rates of diffusion. In seawater heat and salt have different molecular rates of diffusion: K H / K S  100 z TS TS Salt Finger Double-Diffusive Convection Recall- diffusion and mixing is from high concentration of stuff to low concentration, i.e. downgradient. Double-Diffusion gives UPGRADIENT density flux.

10. GEOF110 Guidelines / 3 10 Equation of Motion (Ch. 6) Acceleration due to resultant force acting per unit mass Forces: Pressure Coriolis Gravity (gravitational acceleration, g f ) Other (e.g., tidal, friction) Will derive the terms, and then give the complete equation

11. GEOF110 Guidelines / 3 11 Pressure force on a surface element δA with outward normal vector n: i.e., directed towards the surface element. Derivation of the Pressure Term (Pressure Gradient Force) The net force δF p in the x-direction: Using δm =  δV = 1/  δV  Using all 3-D:

12. GEOF110 Guidelines / 3 12 Source: J.H.E.Weber

13. GEOF110 Guidelines / 3 13 Source: J.H.E.Weber

14. GEOF110 Guidelines / 3 14

15. GEOF110 Guidelines / 3 15 Eq. of motion relative to fixed axes: Eq. of motion relative to rotating Earth: For our use: Absolute frames is fixed  subscript f Relative frames is Earth  subscript e No translation of Earth  a o = 0 velocity relative to Earth  v  = 2  radians per sidereal day = 7.292x10 -5 radians/s Conversion from fixed to Earth frames: r = Distance to the centre of Earth

16. GEOF110 Guidelines / 3 16 Gravitation and Gravity m1m1 m2m2 r, directed along the line connecting masses. G : Gravitational constant Gravitation is the attractive force between two masses:  Equator  In our case, mass of Earth, m E. Gravitational force on body m, per unit mass: Centripetal acceleration : to move a body at distance r from the center of Earth to circulate about Earth’s axis with . Acceleration due to gravity: Note, g does not point to Earth’s center of mass due to centripetal acc.  Equatorial bulge.  Earth’s surface not spherical g=f(latitude) Max at poles Min. near eq. Variation about 0.5%  Assume g = 9.8 m/s 2

17. GEOF110 Guidelines / 3 17 Equation of motion Pressure Gravity Coriolis Other forces/m For convenience, move the coordinate to Earth’s surface. In Cartesian:

18. GEOF110 Guidelines / 3 18 Coordinate Systems Cartesian Coordinate System: The standard convention in geophysical fluid mechanics is x is to the east, y is to the north, and z is up. f-Plane is a Cartesian coordinate system in which the Coriolis parameter is assumed constant. It is useful for describing flow in regions small compared with the radius of the Earth (up to 100 km). β-plane is a Cartesian coordinate system in which the Coriolis force is assumed to vary linearly with latitude. It is useful for describing flow over areas as large as ocean basins. Spherical coordinates are used to describe flows that extend over large distances and in numerical calculations of basin and global scale flows.