Scales of Motion, Reynolds averaging September 22

Recall the momentum equation for a fluid: F x is a frictional forcer per unit mass This includes all friction, including winds

Figure 8.1 in Stewart

Frictional Stresses Internal stress (frictional stress) Internal stress (frictional stress) Molecular friction stress related to velocity shear Molecular friction stress related to velocity shear Molecular Viscosity

del square Kinematic molecular viscosity

u y x constant stress, no acceleration in fluid u y x changing stress, acceleration of fluid “curvature of velocity field”

For an incompressible fluid, the frictional force per unit mass takes the form:

Turbulence Turbulence comes from the non-linear terms in the momentum equations Turbulence comes from the non-linear terms in the momentum equations localadvective, non-linear

Reynolds Number Reynolds Number R e, is the ratio of the non-linear terms to the viscous terms Reynolds Number R e, is the ratio of the non-linear terms to the viscous terms U: typical velocity scale u and ∂u ≈ U L: typical distance scale ∂x ≈ L

U L Pipe flow U = average velocity in center L = radius of pipe -Flow is not turbulent (non-linear terms are not important) if R e <1000 in pipes

Turbulence In more complicated flows, non-linear terms are important when R e ≈ 1 or larger In more complicated flows, non-linear terms are important when R e ≈ 1 or larger Flows become fully turbulent if R e >10 5 Flows become fully turbulent if R e >10 5 Gulf Stream: U~1 m/s L~100 km Gulf Stream: U~1 m/s L~100 km ν≈10 -6 m 2 s -1 R e ≈10 11 => turbulent ν≈10 -6 m 2 s -1 R e ≈10 11 => turbulent flow flow Non-linear effects are strong compared to molecular friction Non-linear effects are strong compared to molecular friction In open ocean, molecular friction is usually ignored! Turbulent (non-linear) terms are the important ones! In open ocean, molecular friction is usually ignored! Turbulent (non-linear) terms are the important ones! Turbulent fluctuations act like molecular friction – dissipate (redistribute) energy and other properties Turbulent fluctuations act like molecular friction – dissipate (redistribute) energy and other properties

Turbulent stresses mean over some period “eddy”

by definition

Non-linear terms in the momentum equation can be written as:

Using the previous equation, the continuity equation splits into two equations The x-component of the of the momentum equation becomes

At large Reynolds number,  is very small compared to other terms. The addition force per unit mass due to turbulence is: Reynolds Stresses

we assume turbulence acts like molecular viscocity: kinematic eddy viscosity

assume that A z is either constant or that it varies more slowly in the z direction than ∂ū/∂z. Similar for horizontal components but A z is much less than A x,y =A H

Figure 8.4 in Stewart The buoyancy force acting on the displaced parcel is: ‘

The acceleration of the displaced parcel is:

Stability Equation Stability is defined such that: E > 0 stable E = 0 neutral stability E < 0 unstable Influence of stability is expressed by a stability frequency N (also known as Brunt-Vaisala frequency):

Figure 8.6 in Stewart

Figure 8.7 in Stewart

Richardson Number The relative importance of static stability and dynamic instability is expressed by the Richardson Number: The relative importance of static stability and dynamic instability is expressed by the Richardson Number: R i > 0.25 Stable Ri < 0.25 Velocity shear enhances turbulence