Reports

Ideal (i.e. non-dissipative) fluids are characterized by a number of conservation laws, which are the defining features of the motion, such as

1. energy conservation;
2. mass conservation;
3. material conservation of generalized vorticity, e.g. potential vorticity,
4. model specific advected quantities, for instance, potential temperature in inviscid primitive equations. 

In many applications the dissipation can not be ignored, whereupon all of the above laws must be modified appropriately. However, in order to establish the ideas, it is useful to stick to the non-dissipative case initially. My work in TRR 181 is focused on developing “simplified” models that inherit the conservation laws from their parent system. This branches into two distinct sub-projects.

1. Lagrangian turbulence models.  Common approach to turbulence modeling is based upon averaging the equations of motion at each spatial location (e.g. Reynolds averaging). While very natural, this approach destroys material conservation laws. This can be remedied by using more elaborate procedures, known as Lagrangian averaging, based on averaging fluid parcels’ trajectories. Combining the recently introduced concept of geometric generalized Lagrangian mean (geometric GLM) with averaging of the variational principles, we developed a turbulence framework, which guarantees the inheritance of the above conservation laws by the model.

Using this framework, we derived a number of idealized turbulence models, namely for the primitive and Euler-Boussinesq equations, Euler’s equations of ideal fluid flow, and the multi-dimensional Burgers’ equations.  While working on these models, we found that our framework is highly adaptable to different physical contexts (compressible and incompressible flows, manifolds, various boundary conditions, anisotropy are all treatable within the framework) and leads to models with desirable mathematical properties (well-posedness, filtering of small scales). Another advantage is that the framework is not bound to a particular choice of scalar-averaging (e.g. time-averaging or statistical averaging). Recently we have learned how to combine the framework with stochastic turbulent closures, thereby making another step towards realistic turbulence models.

The bulk of work done up to now provides a proof of concept for our methodology. Several steps still need to be taken, in order to make the concept attractive to applied scientists. One is the inclusion of dissipation. Another avenue is further adapting the framework to a specific physical context (for instance, incorporating the concept of isopycnal averaging for the ocean models)  and developing corresponding parametrizations in the closure.

Three papers so far were written for this sub-project: one is published, one is submitted, and one is in preparation.

2. Variational approximations for rescaled fluid models. A powerful method of studying equations of mathematical physics is rescaling of equations of motion followed by their asymptotic analysis. However, when applied to Lagrangian system, this approach has an important drawback as it potentially destroys the variational structure of the problem and associated conservation laws.

How to construct asymptotic approximations to the rescaled equations that yield models inheriting conservation laws 1)-4) from the parent system then? The answer is fairly straightforward for an isotropic scaling and is based on first approximating the rescaled variational structure, then computing the equations of motion. However, for anisotropic scaling, the rescaled variational structure becomes more complex and not all its approximations have the desired property.

We completely resolved the question for inviscid homogeneous fluids, thereby developing a systematic procedure for constructing  conservation laws-preserving approximations to the rescaled systems. It turns out that many known models are trivial to derive using our framework. For instance, the inviscid primitive equations fall within the framework as an approximation to Euler-Boussinesq system. 

The problem that initiated this project comes from equatorial dynamics. There is much theoretical interest in the dynamics  of  planetary-scale  Kelvin  waves for the purpose of   atmospheric   and   oceanic   data   assimilation. This calls for a geophysical balance model, which retains equatorial Rossby waves in addition to Kelvin waves. The work on deriving the required balanced model is about to begin. Another direction of our research is to obtain models of front-formation in atmosphere and ocean due to strong meridional temperature gradients. Further applications are likely as the framework should prove useful whenever approximate models are sought in an anisotropic setting.

The theoretical paper for this sub-project will be submitted for publication in the nearest future. Two further papers on equatorial balance models and front-formation models are planned.

Hi, my name is Thomas, I am a PhD Student in subproject L2 “The interior energy pathway: internal wave emission by quasi-balanced flows” at the Max-Planck-Institute for Meteorology. This project tries to answer the question to what extend spontaneous wave generation contributes to the ocean’s route to dissipation.

The ocean’s route to dissipation is not yet fully understood. Both the large-scale currents and the meso-scale eddies in the ocean are essentially balanced. Moreover, those eddies tend to transfer energy upward towards larger scales. So the question arises how the energy is transported from the large scales to the small scales, where the energy can be dissipated. Spontaneously emitted waves can be refracted by the eddying flow and captured later. Wave capture is a possible way to transfer energy to smaller scales. I am contributing to this question by identifying spontaneously emitted waves in an OGCM.

To identify spontaneously emitted waves in a more or less realistic setting I use a high resolution global ocean model. The model runs in two configurations: One is a realistic setting forced by 6-hourly atmospheric fluxes obtained by reanalysis data and a second one with temporally constant forcing. By comparing the two simulations I found wavelike structures which are not generated by external forcing but by the eddying flows itself. The properties of these structures identify them as gravity waves which are likely to be generated by spontaneous emission. Further analysis may show to what extend these waves contribute to the ocean’s route to dissipation.

I am a PhD student in the subproject L2 at Jacobs University Bremen under supervision of Prof. Marcel Oliver.

Our role in the subproject can be briefly explained as follows: In large-scale ocean models, the ocean circulations are essentially balanced; however, this balance breaks down in small scales due to spontaneous generation of inertia-gravity waves by quasi-balanced circulations, and waves are maybe re-captured in later times. This spontaneous emission and wave capture is considered to contribute to the energy transfer from the essentially balanced large-scale circulation and mesoscale eddy fields down to smaller scales, which is a route to dissipation.

To analyse the role of inertia-gravity waves in interior dissipation, a reasonable approach is to diagnose the inertia-gravity waves by splitting the flow field into balance and imbalance components, which are the ocean circulation and the inertia-gravity waves, respectively. This balance-imbalance decomposition can be achieved by some diagnostic tools such as linear time filters, balance relations, and optimal potential vorticity balance. In this project, we want to provide a new numerical algorithm to separate spontaneously generated imbalanced flows from the vertical flows depending on a prior work called „optimal balance“.

We are currently working on application of the optimal balance algorithm to the shallow water model and the primitive equations will follow. The “optimal balance” algorithm is interesting to us not only for practical aspects but also mathematical features, so that we extensively worked on asymptotics-preserving schemes on a finite dimensional model in the algorithm. There are several other theoretically open questions, which are standing for the algorithm as its existence and uniqueness, to be considered.