Hoping is good! As I understand the conditions under which this Gibbs State could be achieved, is something a like a very distinct limiting case of renormalization of the (energy) scales below and above the Gibbs measure. That limiting case could be an asymptotic limit of two different flow statistics. Chorin in one of his papers, (On Turbulence Modeling, A. Chorin, 1994) discusses conditions under which this Gibbs State could hold. He discusses a limiting vortex temperature (after Onsager’s hydrodynamic treatment of turbulence, 1949) that could accept values from minus infinity to positive infinity. The temperature here is an equivalent analogue of temperature in the statistical mechanical sense. In 2 dimensions, the statistical theory finds strong numerical and observation supports, with Jupiter Great Red Spot being a prominent example of stable, time invariant, 2D turbulent flows for vortex equilibria, see P Marcus 1988.

]]>Ali wrote:

Do I make sense?

Yes. Saying we’re looking a fluid flowing around a spherical obstacle at high Reynolds number, with boundary conditions that say the flow is laminar and has fixed velocity far from the obstacle. Then we could try to model turbulence using a probability measure on the space of velocity vector fields obeying these boundary conditions. We would want to show there’s one that’s invariant under time evolution. And we could even hope it’s formally analogous to a Gibbs state, though it’s not a Gibbs state.

]]>Hi John, It is assumed that there is an external energy source to sustain the (hard) turbulent flow which is similar to maintaining an energy cascading process like the Kolmogorov spectrum. As I understand Chorin’s work, the assertion for the equilibrium mode is on a par with a kind of stable energy cascade, i.e., the external energy source ultimately driving the creation of smaller scale eddies until a universal dissipative structure is established. Interestingly he also provides a correlation between equilibrium and enstrophy (see his concluding thoughts at the end of section 5.1 of his Vorticity and Turbulence, where he makes an attempt to provide an effective and analogical representation of ‘entropy’ disguised under ‘enstrophy’. Do I make sense?

]]>The use of tools from statistical mechanics for problems in turbulence is a fascinating area of research, and is similar to certain ideas that have been discussed on this blog. Hopefully Tim Van Beek will discuss this subject in future posts in this series.

]]>That sounds interesting! Of course in real life turbulent flow gradually slows down and comes to a halt, so it’s not really in equilibrium. Is he saying there’s some sort of ‘quasi-equilibrium’?

]]>John,

I am reading Alexandre Chorin’s ‘Vorticity and Turbulence’ (1994) and noting his remarkable suggestion that in the very high Reynolds number (Re ) range, the inertial velocity field dissociates from the ‘time parameter’ and turns into an equilibrium statistical mechanics problem.

]]>Thanks. I should have remembered the name Ekman from the book I browsed some time ago,

* Vallis: “Atmospheric and Oceanic Fluid Dynamics”

]]>It depends on which Ekman layer you are referring to! For instance, the Ekman layer most commonly discussed in Oceanography 1 happens at the surface of the ocean, where the boundary condition is a prescribed surface (tangential) stress, parametrizing forcing (usually taken to be a constant wind). This phenomena has important implications for transport, since integration of the velocity profiles for all depths implies a non zero mass transport at to the direction of the wind, (where the sign depends on what hemisphere you are in).

An example of this is quite common during this time of year in California. Wind blows from the north, this leads to a transport of water to the right of the wind (f is greater than zero in the northern hemisphere) which causes surface waters to move offshore. Mass conservation then implies that water from depth moves up to replace the transported water, which tends to be much colder. Besides having biological implications, this makes surf sessions a bit chillier.

See for instance, here http://en.wikipedia.org/wiki/Ekman_layer

It is interesting to note that a phenomena so easy to mathematically model has been so difficult to observe. Indeed, Ekman presented his model in 1905, yet it was only very recently (somewhere in the ballpark of the 70s/80s) for the field techniques to reach a state where they could accurately measure the velocity profile in the Ekman layer. What people tend to observe is an “Ekman Spiral” that’s been flattened out vs what the model predicts.

Some possible explanations include the additional vortex force due to the interaction of the Stokes drift and the Coriolis force and the fact that we are assuming the eddy viscosity is constant with depth.

]]>That looks like a mathematical artefact that is of little importance for practical purposes: Just the right stuff to get mathematicians interested :-)

I’ll see if I can find a reference…

]]>Dumb question: Is the Ekman boundary layer a special kind of boundary layer solution but with the usual boundary condition that the fluid velocity at the boundary is zero? Or has it a different boundary condition?

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