1 / 23

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

zubeda
Download Presentation

Ilker Fer

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related