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.,

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 ui and dropping the primes Total changes of TKE Transport of TKE Shear 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:

- distance from boundary 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 u2 kinetic energy of eddies l / u turnover time u2 / (l / u ) rate of energy transfer = u3 / l ~ At any intermediate scale l, But at the smallest scales LK, Kolmogorov length scale Typically, so that

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

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

Shear production from internal stresses z Vertical Shears (vertical gradients) u1 u2 Flux of momentum from regions of fast flow to regions of slow flow

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

Florida Intracoastal Waterway Data from Ponce de Leon Inlet Florida Intracoastal Waterway Florida

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

(Monismith’s Lectures) Ralph Obtained from velocity profiles and best fitting them to the values of z0 and u*

Shear Production from Reynolds’ stresses Mixing of property S Mixing of momentum 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)

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

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

Buoyancy Production from Cooling and Double Diffusion S1, T1 S2, T2 S2 > S1 T2 > T1

Layering Experiment http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html

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

Layers in Seno Gala

Dissipation from strain in the flow (m2/s3) (Jennifer MacKinnon’s webpage)

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

Example of Spectrum – Electromagnetic Spectrum http://praxis.pha.jhu.edu/science/emspec.html

(Monismith’s Lectures)

Other ways to determine dissipation (indirectly) Kolmogorov’s K-5/3 law S (m3 s-2) Wave number K (m-1)

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

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

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)

(Monismith’s Lectures)

(Monismith’s Lectures)

At intermediate scales --Inertial subrange – (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) (Monismith’s Lectures)