1. 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

2. GEOF110 Guidelines / 4 2 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. x y z  xz  yz  zz  xz means: shear stress acts in x-direction on a surface normal to z-direction. x z  xy  zy u w y  zx  yx  zx Relation must hold otherwise a fluid parcel would accelerate in rotation to an infinite angular velocity as parcel becomes infinitesimal. For Newtonian incompressible fluid: Newtonian fluid: flows like water— its stress vs rate of strain curve is linear and passes through the origin with slope = the viscosity.

3. GEOF110 Guidelines / 4 3 Shear Stresses Normal Stresses  is the (dynamic) viscosity. It is a property of the fluid. Units : kg/(ms). Kinematic viscosity : =  /  with units m 2 /s. Water: is about 1.2x10 -6 m 2 /s Air: is about 1.4x10 -5 m 2 /s

4. 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. The Viscous Force  xx +  xx  xz +  xz  xx  xz  xy  xy +  xy x z y

5. GEOF110 Guidelines / 4 5

6. 6 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 10 11. E.g., coastal current U = 0.1 m/s and L = 10 km, so Re about 10 9. The ocean is turbulent. Reynold’s Experiment laminar Turbulent SHOW MOVIE by Stewart

7. GEOF110 Guidelines / 4 7 Typical scales of turbulence less than t 1 Mean motion varies, slowly, at scale t 2 We use a mean velocity component by time averaging over t 1 : Instantaneous u =mean u + turbulent u (Reynolds averaging): Averaging rules: [u=f(t); v=f(t); c=contant] In our equations, we will apply:

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

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

10. 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.

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

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

13. 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 A x, A y, A z (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:

14. GEOF110 Guidelines / 4 14 Eddy Viscosity, Dimensions: [L 2 T -1 ] Units: m 2 s -1 Since Molecular + turbulent friction terms in x-dir: Drop over-bar for simplicity, Mean eq. of motion in x-dir (including for generality):

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

16. GEOF110 Guidelines / 4 16 Scaling the equations of motion Introduce scales (order of magnitude) for large scale oceanographic features: Horizontal speedU = 0.10 m/s Horizontal lengthL = 1000 km = 10 6 m Vertical lengthH = 1000 m TimeT = 10 6 s (about 11.5 days) Horizontal eddy viscosity A x  A y = 10 5 m 2 /s Vertical eddy viscosity A z = 0.1 m 2 /s Other: at 45N (f =)2  sin  = 2  cos   10 -4 1/s;  =1/  = 10 -3 m 3 /kgg = 10 m/s 2 [Note, for H = 1000 m; z = -1000 m is p = 10 7 Pa.] Upper limits, range is factor 10 4 Example scaling of the continuity equation to estimate vertical velocity scale:

17. 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.

18. 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.

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

20. GEOF110 Guidelines / 4 20

21. GEOF110 Guidelines / 4 21 Dynamical Stability: Effect of Stratification Mixing increases the potential energy ( i.e., raises the center of mass). When there is stratification, turbulence looses energy to do this job. 11 22 h/2  1 +  2 )/2 h -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)

22. 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

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