Turbulent properties: - vary chaotically in time around a mean value - exhibit a wide, continuous range of scale variations - cascade energy from large to small spatial scales “Big whorls have little whorls Which feed on their velocity; And little whorls have lesser whorls, And so on to viscosity.” (Richardson, ~1920)

- Use these properties of turbulent flows in the Navier Stokes equations - The only terms that have products of fluctuations are the advection terms - All other terms remain the same, e.g.,

0 Reynolds stresses are the Reynolds stresses arise from advective (non-linear or inertial) terms

Turbulent Kinetic Energy (TKE) An equation to describe TKE is obtained by multiplying the momentum equation for turbulent flow times the flow itself (scalar product) Total flow = Mean plus turbulent parts = Same for a scalar:

Turbulent Kinetic Energy (TKE) Equation Multiplying turbulent flow times u i and dropping the primes Total changes of TKETransport of TKEShear Production Buoyancy Production Viscous Dissipation fluctuating strain rate Transport of TKE. Has a flux divergence form and represents spatial transport of TKE. The first two terms are transport of turbulence by turbulence itself: pressure fluctuations (waves) and turbulent transport by eddies; the third term is viscous transport

interaction of Reynolds stresses with mean shear; represents gain of TKE represents gain or loss of TKE, depending on covariance of density and w fluctuations represents loss of TKE

In many ocean applications, the TKE balance is approximated as:

The largest scales of turbulent motion (energy containing scales) are set by geometry: - depth of channel - distance from boundary The rate of energy transfer to smaller scales can be estimated from scaling: u velocity of the eddies containing energy l is the length scale of those eddies u 2 kinetic energy of eddies l / u turnover time u 2 / (l / u ) rate of energy transfer = u 3 / l ~ At any intermediate scale l, But at the smallest scales L K, Kolmogorov length scale Typically, so that

Shear production from bottom stress z u bottom Vertical Shears (vertical gradients)

Shear production from wind stress z W u Vertical Shears (vertical gradients)

Shear production from internal stresses z u1u1 Vertical Shears (vertical gradients) u2u2 Flux of momentum from regions of fast flow to regions of slow flow

Parameterizations and representations of Shear Production Bottom stress: Near the bottom Law of the wall

Data from Ponce de Leon Inlet Florida Intracoastal Waterway Florida

Law of the wall may be widely applicable (Monismith’s Lectures)

Ralph Obtained from velocity profiles and best fitting them to the values of z 0 and u * (Monismith’s Lectures)

Shear Production from Reynolds’ stresses Mixing of momentum Mixing of property S Munk & Anderson (1948, J. Mar. Res., 7, 276) Pacanowski & Philander (1981, J. Phys. Oceanogr., 11, 1443)

With ADCP: and θ is the angle of ADCP’s transducers -- 20º Lohrmann et al. (1990, J. Oc. Atmos. Tech., 7, 19)

Souza et al. (2004, Geophys. Res. Lett., 31, L20309) (2002)

Day of the year (2002) Souza et al. (2004, Geophys. Res. Lett., 31, L20309)

S 1, T 1 S 2, T 2 S 2 > S 1 T 2 > T 1 Buoyancy Production from Cooling and Double Diffusion

Layering Experiment

From Kelley et al. (2002, The Diffusive Regime of Double-Diffusive Convection) Data from the Arctic

Layers in Seno Gala

Dissipation from strain in the flow (m 2 /s 3 ) (Jennifer MacKinnon’s webpage)

From: Rippeth et al. (2003, JPO, 1889) Production of TKE Dissipation of TKE

Example of Spectrum – Electromagnetic Spectrum

(Monismith’s Lectures)

Wave number K (m -1 ) S (m 3 s -2 ) Other ways to determine dissipation (indirectly) Kolmogorov’s K -5/3 law

(Monismith’s Lectures) P equilibrium range inertial dissipating range Kolmogorov’s K -5/3 law

(Monismith’s Lectures) Kolmogorov’s K -5/3 law -- one of the most important results of turbulence theory

Stratification kills turbulence In stratified flow, buoyancy tends to: i) inhibit range of scales in the subinertial range ii) “kill” the turbulence

(Monismith’s Lectures) U3U3

(responsible for dissipation of TKE) At intermediate scales --Inertial subrange – transfer of energy by inertial forces

(Monismith’s Lectures) Other ways to determine dissipation (indirectly)