Guidelines for GEOF110: Lecture 1. Relevant Mathematical Review (2 hours) Ilker Fer Guiding for blackboard presentation. Following Pond & Pickard, Introductory Dynamical Oceanography
GEOF110: Guidelines / Appendix 2 Co-ordinate system Right-handed, Cartesian Vectors with arrow on top x EAST y z UP NORTH (into paper) Scalars (e.g., T, S) : number and unit Vectors (e.g. velocity, force) : number, unit and direction
GEOF110: Guidelines / Appendix 3 x, u EAST y, v z, w UP NORTH (into paper) u EAST v NORTH
GEOF110: Guidelines / Appendix 4 Dot Product x y Length of a vector times the projection of b vector on a.
GEOF110: Guidelines / Appendix 5 Vector Product Vector product of a and b is perpendicular to both a and b vectors!! So- to get a unit vector simply divide the vector product by its magnitude: o o
GEOF110: Guidelines / Appendix 6 Taylor series expansion x means a small distance in x-direction Note: x is small; ( x) 2 is VERY small. First terms dominate. We can neglect order ( x) 2 or higher terms. In 2D:
GEOF110: Guidelines / Appendix 7 Differential and Derivative the differential of a function represents an infinitesimal change in its value The slope of the tangent line (red) is equal to the derivative of the function (black) at the marked point. The process of finding a derivative is called differentiation. It is the reverse process to integration.
GEOF110: Guidelines / Appendix 8 For infinitesimal x (dx) we successfully approximate the tangent. In finite differences, we deal with secants, but if x is small enough, approximation to the derivative is OK. x y x+ x y+ y tangent secant y=f(x)
GEOF110: Guidelines / Appendix 9 In 3-D:
GEOF110: Guidelines / Appendix 10
GEOF110: Guidelines / Appendix 11 Example u1u1 u2u2 u3u3 x y z A B C D The pipe has constant cross-section between A-B and C-D. There is a transition region with changing diameter between B-C. For a given volume flux, speed along the centerline will be u 3 >u 2 > u 1. If the volume flux is constant: 1. Steady state at all parts. Local acceleration is zero in all sections. 2. Cross-section does not change between A-B & C-D. Speed at the center will not change within these sections. 3. Between B-C flow will accelerate because the cross-section is gradually reduced.
GEOF110: Guidelines / Appendix 12 Kinematics
GEOF110: Guidelines / Appendix 13 =rate of change of V in the direction traverse to the flow =rate of change of direction in the downstream direction Shear and curvature is +ve, cyclonic, if in the same sense of Earth’s rotation looking down on the pole (CCW in NH). = shear + curvature = diffluence + stretching. / Confluence: Positive if streamlines are spreading apart downstream / Contraction: rate of change of speed in the downstream direction
GEOF110: Guidelines / Appendix 14 a)Shear flow- no curvature, diffluence, stretching or divergence b) Solid body rotation with cyclonic shear and curvature. No diffluence/stretching hence no divergence c) Radial, divergent flow. No curvature or shear, or vorticity d) Hyperbolic flow with i) diffluence and stretching ii) shear and curvature. Non-divergent because the two terms in i) exactly cancel. Irrotational (no vorticity) because two terms in ii) exactly cancel.
GEOF110: Guidelines / Appendix 15 Eulerian and Lagrangian Fluid Flow Eulerian: consider changes as they occur at a fixed point in the fluid. Lagrangian: consider changes which occur as you follow a fluid particle (i.e. along a trajectory). Identify a specific water parcel A with position x A,y A,z A at time t. Follow this parcel at all times giving the trajectory S A = S(x A,y A,z A,t)=f(t) only. The time rate of change along the trajectory is the Lagrangian velocity. When tracking of water parcels is difficult, sample at a control volume with fixed position, sample f(x,y,z,t). Eulerian velocity:
GEOF110: Guidelines / Appendix 16 (z) z z=-h z=-(h+ z) mg zz p(z=0)=p atm =0 g z=-H Hydrostatic Pressure Total p below a water parcel = p atm + p due to weight above parcel Typically, we assume p atm =0. Because water level adjusts to compansate for p atm changes
GEOF110: Guidelines / Appendix 17 (z) z z=-H z=0 h1h1 h2h2 h3h3 z Mixed layer Pycnocline No stratification Stepwise stratification Continuous stratification
GEOF110: Guidelines / Appendix 18 Slope Effects mg mgsin Friction P L =- gzP L =- g(z+ z) z+ z z A A’ Case 1: BAROTROPIC. Water with uniform . Whole body of water will move to the left, because P R > P L. Isobaric (eq. pressure) surfaces are parallel to each other. Note surface is an isobaric surface as well. Case 2: BAROCLINIC. Say above A-A’ decreased to the right so that pressure remained constant along AA’. Say below AA’ density is uniform. Upper layer moves to the left. Lower layer does not move. Isobaric surfaces are inclined to each other.
GEOF110: Guidelines / Appendix 19 Compressibility Incompressibility is = 0. Meaning 1/ d /dt = 0 or 1/V dV/dt=0
GEOF110: Guidelines / Appendix 20 Assumption of incompressibility in the previous slide is equivelant to the Bossinesq approx. :Density is nearly constant in the ocean, we can safely assume density is constant except when it is multiplied by g in calculations of pressure in the ocean. Boussinesq's assumption requires that: 1-Velocities in the ocean must be small compared to the speed of sound c. This ensures that velocity does not change the density. As velocity approaches the speed of sound, the velocity field can produces large changes of density such as shock waves. 2-The phase speed of waves or disturbances must be small compared with c. Sound speed in incompressible flows is infinite, and we must assume the fluid is compressible when discussing sound in the ocean. Thus the approximation is not true for sound. All other waves in the ocean have speeds small compared to sound. 3-The vertical scale of the motion must be small compared with c 2 /g. This ensures that as pressure increases with depth in the ocean, the increase in pressure produces only small changes in density. The approximations are true for oceanic flows, and they ensure that oceanic flows are incompressible. The Boussinesq Approximation