Momentum Equations in a Fluid (PD) Pressure difference (Co) Coriolis Force (Fr) Friction Total Force acting on a body = mass times its acceleration (W) Weight (PD) The forces on a cube of water Note the subscript “i”.

Navier Stokes Equation with Gravity and Coriolis Acceleration Advection Coriolis Pressure Grad Buoyancy Friction

Navier Stokes Equation with Gravity and Coriolis Component Form

Concept of Reynolds Number, Re Acceleration Advection Pressure Gradient Friction I II III IV Ignore Coriolis and Buoyancy and forcing

Example : a toothpick moving at 1mm/s Flow past a circular cylinder as a function of Reynolds number From Richardson (1961). Note: All flow at the same Reynolds number have the same streamlines. Flow past a 10cm diameter cylinder at 1cm/s looks the same as 10cm/s flow past a cylinder 1cm in diameter because in both cases Re = Example: a finger moving at 2cm/s Re<1 Re =174 Re = 20 Re = 5,000 Re = 14,480 Turbulent Cases Re = Laminar Cases Example: hand out of a car window moving at 60mph. Re = 1,000,000

Air Water Ocean Turbulence (3D, Microstructure) Wind Mixed Layer Turbulence Thermocline Turbulence Bottom Boundary Layer turbulence

Turbulent Frictional Effects: The Vertical Reynolds Stress Turbulence or in component form No Air Water

time Three Types of Averages Ensemble Time Space Ergodic Hypothesis: Replace ensemble average by either a space or time average Mean and Fluctuating Quantities

How does the turbulence affect the mean flow? 3D turbulence Mean Flow u’w’

Momentum Equations with Molecular Friction But Approach for Turbulence

Example Uniform unidirectional wind blowing over ocean surface Dimensional Analysis Boundary Layer Flow Gradient in “x” direction smaller than in “z” direction Example:Mean velocity unidirectional, no gradient in “y” direction

Now we average the momentum equation

Example: Tidal flow over a mound U H Laminar Flow Turbulent Flow

3 D Turbulence: Navier Stokes Equation (no gravity, no coriolis effect) Examples: tidal channel flow, pipe flow, river flow, bottom boundary layer) I. Acceleration II. Advection (non-linear) III. Dynamic Pressure IV. Viscous Dissipation

Surface Wind Stress  (Unstratified Boundary Layer Flow) Air Water wind What is the relationship between and ? Definition: Stress = force per unit area on a parallel surface

Definition Concept of Friction Velocity u* u* Characteristic velocity of the turbulent eddies Empirical Formula for Surface Wind Stress Drag Coefficient

Example. If the wind at height of 10m over the ocean surface is 10 m/sec, calculate the stress at the surface on the air side and on the water side. Estimate the turbulent velocity on the air side and the water side.  u*=? Since

Convention: When we deal with typical mean equations we drop the “mean” Notation! General Case of Vertical Turbulent Friction Note that we sometimes use 1,2,3 in place x, y, z as subscripts

Component form of Equations of Motion with Turbulent Vertical Friction Note: in many cases the mean vertical velocity is small and we can assume w = 0 which leads to the hydrostatic approximation and

Example : Steady State Channel flow with a constant surface slope,  (No wind) Role of Bottom Stress z = 0 z = D  Bottom Surface z Flow Direction Why? Bottom Stress Surface Stress Stress x

z = 0 z = D  Bottom Surface z Flow Direction Why? Bottom Stress x Typical Values

Turbulence Case: Eddy Viscosity Assumption Note. At a fixed boundary because of molecular friction. In general  =  (z). Relating Stress to Velocity Viscous (molecular) stress in boundary layer flow Low Reynolds Number Flow Note: Viscous Stress is proportional to shear. Mixing Length Theory: Modeling  l a characteristic length, a characteristic velocity of the turbulence

Back to constant surface slope example where we found that z = 0  z = D If we use the eddy viscosity assumption with constant k Example Values

Log Layer Note: in the previous example near the bottom,  independent of z Bottom Boundary Lay er z Note we have used the fact that

Typical Ocean Profile of temperature (T), density  20m- 100m 1km 4km Mixed Layer pycnocline thermocline T  But

Stratified Flow Vertical Equation: Hydrostatic condition No stratification Vertical Equation: Hydrostatic condition Stratification Horizontal Equation

Buoyancy Archimedes Principle Weight Buoyancy Force

 z z+  z Concept of Buoyancy frequency N

Gradient Richardson Number Turbulence in the Pycnocline Velocity Shear  Turbulence occurs when

Billow clouds showing a Kelvin-Helmholtz instability at the top of a stable atmospheric boundary layer. Photography copyright Brooks Martner, NOAA Environmental Technology Laboratory.

Depth(m) Distance (m) Turbulence Observed in an internal solitary wave resulting in Goodman and Wang (JMS, 2008)

Temperature (Heat)Equation with Molecular Diffusion Approach for Turbulence Eddy Diffusivity Model Case of Vertical Advection and Turbulent Flux Note: Heat Flux is given by

Advection Diffusion Equation drop bar notation

T(z) w Z=0 Z=D Surface (s) uu Example: Suppose the heat input is in water of depth 50m. The turbulent diffusivity is (a) For the case of no upwelling what is the heat transferred, H, at the surface, mid depth, and the bottom? What is the water temperature at the surface mid depth and the bottom? (b) Suppose there was an upwelling velocity of.1 mm/sec how would the results in part (a) change?

Z=0 Z=D Surface (s) (a) No Upwelling w=0

Z=0 Z=D Surface (s) (a)Upwelling w=.1 mm/sec wu u