Equations of Motion With Viscosity 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

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

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

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

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'

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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)

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

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

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 layer and 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

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

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

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

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.