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Ilker Fer

Lecture Guidelines for GEOF110 Chapter 7 Until Re-averaging + movie = 2 h scaling/ hydrostatic equation = 2 h. Ilker Fer. Guiding for blackboard presentation. Following Pond & Pickard, Introductory Dynamical Oceanography. Relation

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Ilker Fer

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  1. Lecture Guidelines for GEOF110Chapter 7Until Re-averaging + movie = 2 hscaling/ hydrostatic equation = 2 h Ilker Fer Guiding for blackboard presentation. Following Pond & Pickard, Introductory Dynamical Oceanography

  2. Relation must hold otherwise a fluid parcel would accelerate in rotation to an infinite angular velocity as parcel becomes infinitesimal. For Newtonian incompressible fluid: zz yz z y xz x xy yx z zy zx zy y w zx u yx x xy The Viscous Stress • Viscous stress is friction force per unit area. In hydrostatics, the only stress on an element of fluid surface is a normal stress, called the pressure,acting perpendicular to the surface. In a viscous flow, friction acts on any surface– normal or tangent. The stress that acts along (parallel to) a surface is called shear stress. Only shearing motion or compression will give rise to viscous stress. xz means: shear stress acts in x-direction on a surface normal to z-direction. Newtonian fluid: flows like water—its stress vs rate of strain curve is linear and passes through the origin with slope = the viscosity. GEOF110 Guidelines / 4

  3. Normal Stresses Shear Stresses • is the (dynamic) viscosity. It is a property of the fluid. Units : kg/(ms). Kinematic viscosity :  = / with units m2/s. Water:  is about 1.2x10-6 m2/s Air:  is about 1.4x10-5 m2/s GEOF110 Guidelines / 4

  4. Just as the pressure gives rise to pressure force per unit volume of , the viscous stress gives rise to viscous force per unit volume. y z xz+xz xy+xy xx xy xx+ xx xz x The Viscous Force GEOF110 Guidelines / 4

  5. GEOF110 Guidelines / 4

  6. Reynold’s Experiment laminar Turbulent Turbulence arises from the non-linear terms in the momentum equation (u∂u/∂x, etc. in the advective acceleration). The importance of these terms is given by a non-dimensional number, the Reynolds Number Re, which is the ratio of the non-linear terms to the viscous terms: where, U is a typical velocity and L is a typical length describing the flow. Non-linear terms are important if Re > 10-1000. E.g., Gulf Stream U = 1 m/s and L = 100 km, so Re about 1011. E.g., coastal current U = 0.1 m/s and L = 10 km, so Re about 109. The ocean is turbulent. SHOW MOVIE by Stewart GEOF110 Guidelines / 4

  7. Typical scales of turbulence less than t1 Mean motion varies, slowly, at scale t2 We use a mean velocity component by time averaging over t1: Instantaneous u =mean u + turbulent u (Reynolds averaging): Averaging rules: [u=f(t); v=f(t); c=contant] In our equations, we will apply: GEOF110 Guidelines / 4

  8. Local acceleration: Coriolis: Pressure term: GEOF110 Guidelines / 4

  9. Friction term: Advective acceleration terms (non-linear terms): 3D  GEOF110 Guidelines / 4

  10. Equation for the Mean Flow Instantaneous: Mean Flow. Reynolds Equation: This came from the averaging of advective acceleration terms (non-linear terms). They represent the effect of turbulence on the mean motion. 3 more unknowns (u’,v’,w’)! Same number of equations. Big problem: Closure Parameterizations First, let’s tidy up this term using the continuity equation. GEOF110 Guidelines / 4

  11. Remember u-component of the turbulent part of the advective acceleration term: GEOF110 Guidelines / 4

  12. Example, mean Flow at x-dir: GEOF110 Guidelines / 4

  13. Eddy viscosity Process is similar: molecular viscosity, exchange of momentum due to molecules moving back-and-forth. Turbulent viscosity (eddy viscosity >> molecular) exchange of momentum due to eddies (which move larger parcels) in the fluid. Define: Eddy viscosity Ax, Ay, Az (all >> ) in x, y, z, directions. Eddy viscosity is a property of the flow - not the fluid. This is a simple parameterization where we assumed turbulent stresses are related to the mean gradients. Typically horizontal A components are similar, and larger than the vertical component: GEOF110 Guidelines / 4

  14. Drop over-bar for simplicity, Mean eq. of motion in x-dir (including for generality): Eddy Viscosity, Dimensions: [L2 T-1] Units: m2 s-1 Molecular + turbulent friction terms in x-dir: Since GEOF110 Guidelines / 4

  15. Summary: Governing equations of mean motion GEOF110 Guidelines / 4

  16. Scaling the equations of motion Introduce scales (order of magnitude) for large scale oceanographic features: Horizontal speed U = 0.10 m/s Horizontal length L = 1000 km = 106 m Vertical length H = 1000 m Time T = 106 s (about 11.5 days) Horizontal eddy viscosity Ax Ay = 105 m2/s Vertical eddy viscosity Az = 0.1 m2/s Other: at 45N (f =)2sin = 2cos  10-4 1/s; =1/ = 10-3 m3 /kg g = 10 m/s2 [Note, for H = 1000 m; z = -1000 m is p = 107 Pa.] Upper limits, range is factor 104 Example scaling of the continuity equation to estimate vertical velocity scale: GEOF110 Guidelines / 4

  17. Use scaling to get to the hydrostatic equation: Balance requires, to a high degree of accuracy: We assumed L >> H  good for ocean, but can be difficult to achieve in the atmosphere. GEOF110 Guidelines / 4

  18. Let’s scale the horizontal components of the eq. of motion: Same is true for y-component. First order balance is between the Coriolis term (fv) and the pressure gradient. Second important term, u/ t is 1% of fv and will be smaller for T>10 days. GEOF110 Guidelines / 4

  19. dy R d Say, d=/2 dy is ¼ of the circumference: dy= ¼2R dy=Rd Scaled equations to an order of accuracy of 1% become: In the interior ocean and away from the Equator Does this hold only for VERY large L?  valid also for less small horizontal scale, L. Use dy=Rd: GEOF110 Guidelines / 4

  20. GEOF110 Guidelines / 4

  21. h/2 r1 (r1+r2)/2 h h/2 r2 Dynamical Stability: Effect of Stratification • can have large Re, but no shear (e.g. uniform large U), and flow is not turbulent. • stratification can suppress turbulence (high Re not relevant) Mixing increases the potential energy ( i.e., raises the center of mass). When there is stratification, turbulence looses energy to do this job. GEOF110 Guidelines / 4

  22. Richardson number • low Ri : instability ; Kelvin-Helmholtz (K-H) billows ;shear dominates, turbulence is amplified • high Ri: stable; buoyancy forces dominate, turbulence is suppressed;Energy may be propagated in the form of internal gravity waves. They convey energy or momentum but no scalar flux and no vorticity. • Canonical critical value, Ri = 0.25. The relative importance of stratification and shear => Richardson number GEOF110 Guidelines / 4

  23. (Thorpe, 2007) Collapse of a Mixing Patch After one intertial period GEOF110 Guidelines / 4

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