Chapter 8

1. Chapter 8 Equations of Motion With Viscosity Physical oceanography Instructor: Dr. Cheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 1 November 2003

2. Introduction • Friction • For most of the interior of the ocean and atmosphere: Fr 0  negligible • Boundary layer • A thin, viscous layer adjacent to the boundary • No-slip boundary condition  rapid change of velocity  outer boundary velocity • Fr is non-negligible within the boundary layer

3. The Influence of Viscosity • Molecular viscosity • Fig 8.1 • Collision between molecules and boundary • Transfer momentum • Inefficient  important only within few mms • The kinematic molecular viscosity n • n = 10-6m2/s for water at 200C • Definition • The ratio of the stress Tx tangential to the boundary of a fluid and the shear of the fluid at the boundary • Tzx = rnu/z • Extension to 3D

4. Turbulence • Reynolds number • the influence of a boundary transferred into the interior of the flow  turbulence • Reynolds experiment (1883) • Fig 8.2 • V  laminar  turbulent flow • Transition occurs at Re = VD/n  2000 • Same Re, same flow pattern (Fig 8.3) • Reynolds number Re • The ratio of the non-linear terms to the viscous terms of the momentum equation • Characteristic length L • Characteristic velocity U

5. Turbulence (cont.) • Turbulent stresses: the Reynolds stress • Prandtl and Karmen hypothesized that • Parcels of fluid in a turbulent flow played the same role in transferring momentum within the flow that molecules played in laminar flow • Separate the momentum equation into mean and turbulent components • u = U + u' ; v = V + v' ; w = W + w' ; p = P + p'

6. Turbulence (cont.) • Equations of motion with viscosity • X component

7. Calculation of Reynolds Stress • By Analogy with Molecular Viscosity • Fig 8.1 • Wind flow above the sea surface • Flow at the bottom boundary layer in the ocean  • Flow in the mixed layer at the sea surface • Steady state • /t = /x = /y  the turbulent frictional term: • Fx = (-1/r) Tz/z = (-/z) <u'w'> • In analogy with Tzx = rnu/z • -<u'w'> = Tzx = AzU/z Tzx/z = Az2U/z2 • Az : the eddy viscosity • Az cannot be obtained from theory. Instead, it must be calculated from data collected in wind tunnels or measured in the surface boundary layer at sea • If the molecular viscosity terms are comparatively smaller

8. Calculation of Reynolds Stress (cont.) • By Analogy with Molecular Viscosity (cont.) • Problems with the eddy-viscosity approach • Except in boundary layers a few meters thick, geophysical flows may be influenced by several characteristic scales • The height above the sea z • The Monin-Obukhov scale L discussed in §4.3, and • The typical velocity U divided by the Coriolis parameter U / f • The velocities u', w' are a property of the fluid, while Az is a property of the flow • Eddy viscosity terms are not symmetric, <u'v'> = <v'u'> but

9. Calculation of Reynolds Stress (cont.) • From a Statistical Theory of Turbulence • Relate <u'u'> to higher order correlations of the form <u'u'u'> • The closure problem in turbulence  determine the higher order terms  no general solution • Isotropic turbulence: turbulence with statistical properties that are independent of direction

10. Calculation of Reynolds Stress (cont.) • Summary: for most oceanic flows • Ax, Ay, and Az cannot be calculated accurately • Can be estimated from measurements • Measurements in the ocean  difficult • Measurements in the lab  cannot reach Re =1011 • The statistical theory of turbulence  gives useful insight into the role of turbulence in the ocean  an area of active research • Some values for n • nwater = 10-6 m2/s • ntar at 15°C = 106 m2/s • nglacier ice = 1010 m2/s • Ay = 104 m2/s

11. The Turbulent Boundary Layer Over a Flat Plate • The mixing-length theory • Empirical theory • G.I.Taylor (1886-1975), L. Prandtl (1875-1953), and T. von Karman (1818-1963) • Predicts well U(z) close to the boundary • U(z) = u*/k ln(z/z0) • Assuming: U z  Tzx = rnu/z  U = Tzxz/rn • Non-dimensionlizing: U/u* = u*z/n • Where u* 2 = Tzx/r is the friction velocity • By analogy with molecular viscosity: Tzx/r = AzU/z = u* 2 • Assuming: Az = kzu* • large eddies are more effective in mixing momentum than small eddies, and therefore Az ought to vary with distance from the wall • dU = u*/(kz)dz • For airflow over the sea, k = 0.4 and z0 = 0.0156u*2/g

12. Mixing in the Ocean • Mixing • Instability  mixing • Vertical mixing • Work against buoyancy  need more energy • Diapycnal mixing • Important  change the vertical structure • Equation:  Q/ t + W Q/ z =  / z(Az Q/ z) + S • Q : tracer, such as Salt and Temperature • Az: the vertical eddy diffusivity • W: mean vertical velocity • S: source term • Horizontal mixing • Larger

13. Average Vertical Mixing • Observation by Walter Munk (1966) • Simple observation  calculation of vertical mixing • Thermocline everywhere • Deeper part doesn‘t change for decades (Fig 8.4) steady state  balance between • Downward mixing of heat by turbulence • Upward transport of heat by a mean vertical current W • Steady state equation without source term: W T/ z =  / z(Az T/ z) • Solution: T(z)  T0exp(z/H) • H = Az/W: the scale length of the thermocline • T0 = T(z=0) • Simple observation T(z)  calculation of vertical mixing H • Vertical distribution of C14 Az (1.3  10-4m2/s) and W (1.2 cm/day)  H • W  the average vertical flux of water through the thermocline  25 ~ 30 Sv of water

14. Measured Vertical Mixing • Observation  require new techniques • Fine structure of turbulence • Including S and T with spatial resolution of cms • Detect tracer • SF6 ~ [g/Km3 sea waters] • Measured Az • Open ocean vertical eddy diffusivity: Az = 10-5m2/s • Rough bottom vertical eddy diffusivity: Az = 10-3m2/s • Indication: • Mixing occurs mostly at oceanic boundaries: along continental slopes, above seamounts and mid-ocean ridges, at fronts, and in the mixed layer at the sea surface

15. Measured Vertical Mixing (cont.) • Discrepancy • Deep convection  meridional overturning circulation • Deep convection  may mix properties rather than mass  smaller mass of upwelled water ( 8 Sv) • Mixing along the boundaries or in the source region • Example: mid Atlantic water  Gulf Stream  mid Atlantic

16. Measured Horizontal Mixing • Horizontal mixing = fn(Re) • A/g A/n ~ UL/n = Re • g: the molecular diffusivity of heat • Ax ~ UL • Measured Ax • Geostrophic Horizontal Eddy Diffusivity: Ax = 8  102 m2/s • Open-Ocean Horizontal Eddy Diffusivity: Ax = 1 – 3 m2/s

17. Stability • Three types of instability • Static stability  Dr(z) • Dynamic stability  velocity shear • Double-diffusion  S, T

18. Stability (cont.) • Static stability • z but r  unstable • Criterion • F = Vg(r2 - r')

19. Stability (cont.) • Static stability (cont.) • The stability of the water column E -a /(gdz) • More accurate form of E: E(S(z), t(z), z)

20. Stability (cont.) • Static stability (cont.) • The stability equation • In the upper kilometer of the ocean: (p/z)water >> (p/z)parcel E  (-1/r)dp/dz (-1/r)dst/dz • Below about a kilometer in the ocean, Dr(z)  0 E = E(S(z), t(z), z) • Stability is defined such that • E > 0 Stable • E = 0 Neutral Stability • E < 0 Unstable • In the upper kilometer of the ocean, z < 1,000m, E = (50-1000)×10-8/m • In deep trenches where z > 7,000m, E = 1×10-8/m

21. Stability (cont.) • Static stability (cont.) • Stability frequency N • The influence of stability is usually expressed by N • N2 -gE • N is often called the Brunt-Vaisala frequency or the stratification frequency • The vertical frequency excited by a vertical displacement of a fluid parcel • The maximum frequency of internal waves in the ocean • Fig 8.5: typical values of N

22. Stability (cont.) • Dynamic Stability and Richardson Number • If u = u(z) in a stable, stratified flow  current shear • if it is large enough  unstable • Example 1: wind blowing over the ocean • A step discontinuity in r  E    stable • If the wind is strong enough  waves  break  unstable • Example 2: the Kelvin-Helmholtz instability • Occurs when the density contrast in a sheared flow is much less than at the sea surface, such as in the thermocline or at the top of a stable, atmospheric boundary layer (Fig 8.7) • Richardson number • Definition • Ri > 0.25 Stable • Ri < 0.25 Velocity Shear Enhances Turbulence (+ High Re  instability)

23. Stability (cont.) • Double Diffusion and Salt Fingers • In some regions, r as z + no current  unstable?! • It occurs because the molecular diffusion of heat is about 100 times faster than the molecular diffusion of salt

24. Stability (cont.) • Four variations of T and S • Warm salty over colder less salty (Fig 8.8) • Salt fingering  r as z r(z) looks like stair steps • kT > kS •  a thin, cold, salty layer between the two initial layers •  the fluid sinks in fingers 1-5cm in diameter and 10 of centimeters long  fingers • Double diffusion (T and S) • It occurs in central waters of sub-tropical gyres, western tropical North Atlantic, and the North-east Atlantic beneath the out flow from the Mediterranean Sea • Recent report: Salt fingering mixed water 10 times faster than turbulence in some regions

25. Stability (cont.) • Four variations of T and S (cont.) • Colder less salty over warm salty • Diffusive convection • Double diffusion •  a thin, warm, less-salty layer at the base of the upper, colder, less-salty layerand a colder, salty layer forms at the top of the lower, warmer, salty layer •  sharpen the interface and reduce any small gradients of r • Less common than salt fingering • Mostly found at high latitudes • Diffusive convection also  r(z) looks like stair steps • Cold salty over warmer less salty • Always statically unstable • Warmer less salty over cold salty • Always stable and double diffusion diffuses the interface between the two layers

26. Important Concepts • Friction in the ocean is important only over distances of a few millimeters. For most flows, friction can be ignored. • The ocean is turbulent for all flows whose typical dimension exceeds a few centimeters, yet the theory for turbulent flow in the ocean is poorly understood. • The influence of turbulence is a function of the Reynolds number of the flow. Flows with the same geometry and Reynolds number have the same streamlines

27. Important Concepts (cont.) • Oceanographers assume that turbulence influences flows over distances greater than a few centimeters in the same way that molecular viscosity influences flow over much smaller distances. • The influence of turbulence leads to Reynolds stress terms in the momentum equation. • The influence of static stability in the ocean is expressed as a frequency, the stability frequency. The larger the frequency, the more stable the water column

28. Important Concepts (cont.) • The influence of shear stability is expressed through the Richardson number. The greater the velocity shear, and the weaker the static stability, the more likely the flow will become turbulent. • Molecular diffusion of heat is much faster than the diffusion of salt. This leads to a double-diffusion instability which modifies the density distribution in the water column in many regions of the ocean. • Instability in the ocean leads to mixing. Mixing across surfaces of constant density is much smaller than mixing along such surfaces

29. Important Concepts (cont.) • Horizontal eddy diffusivity in the ocean is much greater than vertical eddy diffusivity. • Measurements of eddy diffusivity indicate water is mixed vertically near oceanic boundaries such as above seamounts and mid-ocean ridges.