1. 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

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 ( ), L. Prandtl ( ), and T. von Karman ( )
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 = u*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 = ( )×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 > Stable
Ri < 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 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

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.