![Page 1: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/1.jpg)
Hans Burchard
Leibniz Institute for Baltic Sea Research Warnemünde
From the Navier-Stokes equations via the Reynolds decomposition
to a working turbulence closure model for the shallow water equations:
The compromise between complexity and pragmatism.
![Page 2: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/2.jpg)
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?
![Page 3: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/3.jpg)
From stirring to mixing …
littl
e st
irri
ng s
tron
g st
irri
nglittle m
ixing strong mixing
![Page 4: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/4.jpg)
10cm
Tea mixing (analytical solution)
Put 50% of milk into tea.
Let m(z) be the milk fraction with m=1 at the bottom and m=0 at the surface.
With a constant mixing coefficient, the m-equation is this:
Conclusion: stirring leads to increased mixing.
Let us take the spoon and stir the milk-tea mix n-times such that we get a sinosodial milk-tea variation in the verticaland then see theresulting mixing after 1 min:
z
![Page 5: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/5.jpg)
Set of equations that describes turbulent mixing
6 equations for 6 unknowns (u1, u2, u3, p, , )
Navier-Stokes equations (for velocity vector u1, u2, u3):
Temperature equation:Equation of state:
tendency advection stress divergenceEarth rotation pressure gradient
gravitationalforce
Incompressibility condition:
stirring mixing
![Page 6: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/6.jpg)
Example for solution of Navier-Stokes equations (KH-instability)
Direct Numerical Simulation (DNS) by William D. Smyth, Oregon State University
![Page 7: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/7.jpg)
Reynolds decomposition
To reproduce system-wide mixing, the smallest dissipative scales must be resolved by numerical models (DNS).
This does not work in models for natural waters due to limited capacities of computers.
Therefore, the effects of turbulence needs to be partially (= Large Eddy Simulation, LES) or fully (Reynolds-averaged Navier-Stokes, RANS) parametersised.
Here, we go for the RANS method, which means that small-scale fluctuations are „averaged away“, i.e., it is only the expected value of the state variables considered and not the actual value.
![Page 8: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/8.jpg)
Reynolds decomposition (with synthetic tidal flow data)
Any turbulent flow can be decomposed into mean and fluctuating components:
![Page 9: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/9.jpg)
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).
Time averaging
Ensemble averaging
![Page 10: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/10.jpg)
Reynolds decompositionFor the ensemble average 4 basic rules apply:
Linearity
Differentiation
Double averaging
Product averaging
![Page 11: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/11.jpg)
The Reynolds equations
The Reynolds stress constitutes a new unknown which needs to be parameterised.
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:
![Page 12: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/12.jpg)
The eddy viscosity assumption
Reynolds stress andmean shear are assumed tobe proportional to each others:
eddy viscosity
![Page 13: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/13.jpg)
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.
![Page 14: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/14.jpg)
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)
![Page 15: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/15.jpg)
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
![Page 16: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/16.jpg)
Dynamic equation for the length scale (here: e eq.)
A dynamic equation for the dissipation rate of the TKE) is constructed:
with the adjustable empirical parameters c1, c2, c3, se.
With this, it can be calculated with simple
stability functions cm and cm‘.
All parameters can be calibrated to characteristic properties of the flow.
Example on next slide: how to calibrate c3.
![Page 17: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/17.jpg)
Layers with homogeneous stratification and shearFor stationary & homogeneous stratified shear flow,
Osborn (1980) proposed the following relation:
which is equivalent to (N is the buoyancy frequency),
a relation which is intensively used to derive the eddy diffusivity from micro-structure observations.
For stationary homogeneous shear layers, the k-e model reduces to
which can be combined to .
Thus, after having calibrated c1 and c2, c3 adjusts the effect of stratification on mixing.
Umlauf (2009), Burchard and Hetland (2010)
![Page 18: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/18.jpg)
Mixing = micro-structure variance decayExample: temperature mixing
Temperature equation:
Temperature variance equation:
Mixing
![Page 19: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/19.jpg)
Second-moment closures in a nut shell Instead of directly imposing the eddy viscosity assumption
With one could also derive a transport equation for
and the turbulent heat flux (second moments). These second-moment
equations would contain unknown third moments, for which also equations could
be derived, etc. The second-moments are closed by assuming local
equilibrium (stationarity, homogeneity) for the second moments. Together
with further emipirical closure assumptions, a closed linear system of equations
will then be found for the second moments. Interestingly, the result may be
formulations as follows: , where now cm and cm‘ are
functions of and with the shear squared, M2.
![Page 20: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/20.jpg)
Such two-equation second moment-closures are now the workhorses incoastal ocean modelling (and should be it in lake models) and have been consistently implemented in the one-dimensional
General Ocean Turbulence Model (GOTM)which has been released in 1999 by Hans Burchard and Karsten Boldingunder the Gnu Public Licence. Since then, it had been steadily developed and is now coupled to many ocean models.
![Page 21: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/21.jpg)
GOTM application: Kato-Phillips experiment
Stress-induced entrainment into linearly stratified fluid
Dm(t) Empirical G<0.2
G>0.2
Empirical:
![Page 22: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/22.jpg)
GOTM application: Baltic Sea surface dynamics
Reissmann et al., 2009
unstable
![Page 23: Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de](https://reader035.vdocuments.us/reader035/viewer/2022062501/56816950550346895de0f32d/html5/thumbnails/23.jpg)
Take home:Due to stirring, turbulence leads to an increase of effectivemixing and dissipation by several orders of magnitude.
For simulating natural systems, the Reynolds decomposition intomean (=expected) and fluctuating parts is necessary. Higher statistical moments are parameterised by means ofturbulence closure models. Algebraic second-moment closures provide a good compromisebetween efficiency and accuracy. Therefore such models areideal for lakes and coastal waters.
Question: Will we be able to construct a robost and more accurate closure model which resolves the second moments ( inclusion of budget equations for momentum and heat flux)?