Coastal Ocean Dynamics Baltic Sea Research Warnemünde Third course: Turbulence and Mixing Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de

Development of a dense bottom current on a continental slope Burchard and Rennau (2008)

Why are we stirring our cup of coffee? Milk foam: light, because of foam and fat Coffee: relatively light, because hot Milk: less light, because colder than coffee Why the spoon? …OK, and why the coocky?

Why do you stirr the paint to mix it?

From stirring to mixing … strong stirring little stirring little mixing strong mixing

Consider the simple diffusion equation for velocity The molecular viscosity has a value of  = 10-6 m2/s. How long would it take that a storm mixes the water column to 30m from an ocean at rest?

Development of a velocity profile due to surface stress when only molecular viscosity is acting (no turbulence)

Temperature profile observations in the Northern North Sea storm event A storm event can mix the surface ocean down to 50m within 2 days. This is the effect of turbulent mixing (from Burchard, 2002).

Turbulent fluctuations are seen all over in natural waters Fluctuations are smooth on the centimetre scale due to the effect of viscosity. Small-scale velocity shear in Lago Maggiore (observation by Adolgf Stips, Ispra, Italy)

Eddy field observed by in-situ particle image velocimetry (PIV) Near-bottom obervations in shallow coastal water. Pictures are 1s apart from each others. Taken from Burchard et al. (2008)

Time series of velocity fluctuations in a tidal channel Acoustic Doppler Current Profiler observations (6 min) by University of Bangor (Wales)

Small-scale temperature fluctuations Small-scale temperature in Lago Maggiore (observation by Adolf Stips, Ispra, Italy)

Pipe flow experiment by Reynolds (1883) laminar turbulent transitional

Design of an experiment for flow around a cylinder Reynolds number:

Typical flow results for various Reynolds numbers laminar turbulent

Deterministic chaos (Lorenz 1963) A layer of oil heated from below shows Benard convection with chaotic fluctuations. These can be characterised by the following non-linear system of differential equatios:

Solution of the Lorenz equations For certain parameter combinations the solution shows chaotic fluctuations which appear to be random and are thus in principle unpredictable.

What is turbulence? Turbulence cannot be defined but described by the following properties: 1. Randomness: Turbulent fluctuations are unpredictable. 2. Diffusivity: Turbulence increases mixes. 3. Vorticity: Turbulent fluctuations are expressed as eddies. 4. Dissipation: Turbulence dissipates kinetic energy. 5. Non-linearity: Turbulence is a non-linear phenomenon (in contrast to waves).

Reynolds decomposition Any turbulent flow can be decomposed into mean and fluctuating components:

Reynolds decomposition There are many ways to define the mean flow, e.g. time averaging (upper panel) or ensemble averaging (lower panel). For the ensemble averaging, a high number N of macroscopically identical experiments is carried out and then the mean of those results is taken. The limit for N   is then the ensemble average (which is the physically correct one.

Reynolds decomposition For the ensemble average 4 basic rules apply: Linearity Differentiation Double averaging Product averaging

The Reynolds equations These rules can be applied to derive a balance equation for the ensemble averaged momentum. This is demonstrated here for a simplified (one-dimensional) momentum equation: The Reynolds stress constitutes a new unknown which needs to be parameterised.

The eddy viscosity assumption Reynolds stress and mean shear are assumed to be proportional to each others: eddy viscosity

The eddy viscosity assumption The eddy viscosity is typically orders of magnitude larger than the molecular viscosity. The eddy viscosity is however unknown as well and highly variable in time and space.

Parameterisation of the eddy viscosity Like in the theory of ideal gases, the eddy viscosity can be assumed to be proportional to a characteristic length scale l and a velocity scale v: In simple cases, the length scale l could be taken from geometric arguments (such as being proportional to the distance from the wall. The velocity scale v can be taken as proportional to the square root of the turbulent kinetic energy (TKE) which is defined as: such that (cl = const)

Dynamic equation for the TKE A dynamic equation for the turbulent kinetic energy (TKE) can be derived: with P: shear production B: buoyancy production e: viscous dissipation

Energy fluxes mean flow production of TKE energy spectrum turbulent flow turbulent flow direct dissipation dissipation spectrum large eddies small eddies dissipation of TKE molecular regime

Inertial subrange of a TKE spectrum Kolmogorov (1941), no viscous effects, no large scale effects: Wave number Spectral energy density Kolmogorov constant (1.4 … 1.8) Known as „Kolmogorov‘s minus 5/3 law“.

Questions Molecular viscosity and mixing rates are basically independent the dynamics. Why do we still stir our tea with milk with a spoon to get it mixed? How do stirring and mixing processes occur in the natural ocean? What is the significance of the Reynolds number? What is deterministic chaos and how is it related to turbulence? What is the Reynolds decomposition? What happens at the large, the intermediate and the small scales of turbulence?