# M4: Entropy Production in Turbulence Parameterizations

Principal investigators: Dr. Richard Blender (Universität Hamburg), Dr. Almut Gassmann (Institute for Atmospheric Sciences), Prof. Valerio Lucarini (Universität Hamburg)

## Objectives

We aim at a modification of sub-scale parameterisation schemes for heat, buoyancy, and momentum fluxes which are associated with energy dissipation and internal entropy production. We assess existing parameterisation schemes in view of the second law of thermodynamics and aim at their modification to conform with this law. Additionally, we wish to deal with the statistical mechanical fluctuations occurring on small scales, which can lead to temporary negative diffusion coefficients in sub-grid parameterisations. An outcome is the combination of dissipation and backscatter in a unified approach.

## Fluctuation analyses of energy fluxes

Analysis of inter-scale energy transfer rates reveals spatially organized patterns of up and down scale transports, impacting contemporary sub-grid closure approaches.

Lorenz energy cycle: Energy current densities are well approximated by generalized extreme value distributions. The fluctuation ratio reveals a linear behavior in a finite range.

## Turbulence parameterizations

Parametrization of turbulence at stable stratification in strong shear flows

Modeling of mesospheric inversion layers requires accounting for the kinetic energy loss when pushing isentropes down via a subscale buoyancy flux.

Non-Markovian subgrid closure: Reproducing statistics of single-time quantities requires reduced dynamics generated by nonlocal-in-time interactions.

## Phase 1

## Main conclusions drawn from M4

- Fluctuations of resolved scales reveal posititve and negative dissipation rates of aggregated coarse scales towards fine scales. A rigorously self-consistent dynamic Smagorinsky model is impossible.
- The 2nd law of thermodynamics enforces positive dissipation of resolved scales towards unresolved scales. The constitutive diffusion coefficient should be a combination of diffusive and antidiffusive parts and should require statistical information about the unresolved part. This conclusion is not contradictory to (1).

## Reports

## Coarse-graining, Entropy and the Unseen

Our goal is to unify two approaches currently taken to formulate closures for climate and weather simulations.

My name is Bastian and I’m currently a PhD student at the IAP, working in the subproject M4 “Entropy Production in turbulence parameterisations”. Our goal is to unify two approaches currently taken to formulate closures for climate and weather simulations.

The nature of weather and climate is such that their equations may not directly be solved mathematically. This forces us to use computer models. In these models we define the necessary equations on grids. Each grid-box represents one set of values associated with the volume that grid-box covers. This usually affords us horizontal resolutions between 10 and 100 km and vertical resolutions between several hundred down to a kilometer.

Not unlike the picture of a tree, which from far enough away seems convincing enough, but from close up lacks the details to show the little squirrel on that branch, we struggle with small scale contributions to the motions in our simulations. Namely such that would be small enough not to be resolved in our model, but large enough to have a significant effect on the model dynamics. We try to account for these using mathematical and physical models to incorporate the effects of what we cannot resolve on what we resolve, in order to get the dynamics right, thus correctly predicting the rain on your granny’s birthday party – or how quickly the polar caps melt. These models are called parameterizations.

My particular task is to retrieve the statistics and distribution of fluxes of energy between the resolved and unresolved part of the simulated atmosphere, in order to learn how to better model said fluxes in a

physically stringent way. This means to find formulations which do not only improve our model data, but are in line with the fundamental laws of energy conservation and the second law of thermodynamics. I do this by programming informed routines, which effectively slice our model data into another poor resolution model and high resolution reality. Computing the fluxes between these two regimes I hope to be able to extrapolate into what we don’t know.

So far I’ve had some very exciting findings, which indicate that we have good reason to apply new types of parameterizations, called backscatter parameterizations in conjunction with our old approaches. In addition to that there are hints to how to find formulations which do not violate our understanding of physics, which will then afford us a better understanding of the processes in our atmosphere and climate.

## Entropy production in turbulence parameterisations

The idea of M4 is the stochastic or counter-gradient parameterisation of momentum and heat fluxes in forced dissipative systems like the atmosphere and ocean.

Dear Reader,

Yesterday I dropped my beloved teacup. Viewing the broken fragments from the perspective of an enthusiastic tea addict: hope for occurrence of the backward process called self-repair. From the sober viewpoint of a physicist: second law of thermodynamics and entropy.

Macroscopic (isolated) systems evolve in a one-way direction of time, towards states with increased entropy seeming to be in contradiction to the underlying microscopic equations that are needed for their description. For instance, Newton’s law of motion for classical systems is symmetric under time reversal; no preference of a certain time direction; no preference of neither the forward nor backward process. Then why is there a break of time symmetry at macroscopic level (irreversibility)?

The second law of thermodynamics is valid in a statistical sense for large system sizes (average statement in thermodynamic limit) within the frame of equilibrium thermodynamics and can be generalised by the Fluctuation Theorem (FT). Resulting from statistical physics this theorem with its different versions connects microscopic and macroscopic behaviour for time reversal systems of arbitrary size arbitrary far driven out of equilibrium in form of an analytical expression of probability ratio of observing a trajectory of a system (in phase space) to its time reversed counterpart.

The essential quantity of the FT is the dissipation function, an entropy-like quantity in non-equilibrium related to internal entropy production under specific conditions. The latter quantity is especially important in regard to our project M4. Here, ‘our’ consists of the *Hamburg part*, Richard Blender, Valerio Lucarini (Reading) and me (since October last year), and of the *Rostock part* (IAP) composed of Almut Gaßmann and Bastian Sommerfeld.

The idea of M4 is the stochastic or counter-gradient parameterisation of momentum and heat fluxes in forced dissipative systems like the atmosphere and ocean. These sub-scale fluxes are related to energy dissipation and backscatter; as well related to positive and negative entropy production. Is it possible to put more physics in these turbulence parameterisation schemes with the usage of the FT? However, it requires the applicability of FT for non-time reversal systems (Navier–Stokes equations). As a representative example for turbulence toy models I use the class of shell models (as a first step) to get a basic notion for the incorporation of the FT with the final aim of modification and improvement of climate prediction models. I am looking forward to this challenge. Thank you for your attention.

## Publications

**Gassmann, A., & Blender, R.**(2019). Entropy Production in Turbulence Parameterizations. In*Energy Transfers in Atmosphere and Ocean*(pp. 225-244). Springer, Cham., doi: https://doi.org/10.1007/978-3-030-05704-6_7.**Blender, R., Gohlke, D.,**&**Lunkeit, F.**(2018). Fluctuation Analysis of the Atmospheric Energy Cycle.*Phys. Rev. E*,*98*(2), 023101, doi: 10.1103/PhysRevE.98.023101.Conti, G. &

**Badin, G.**(2017). Hyperbolic Covariant Coherent Structures in Two Dimensional Flows.*Fluids,*2017,*2*(4), 50., doi.org/10.3390/fluids2040050.**Blender, R.**&**Badin, G.**(2017).Construction of Hamiltonian and Nambu Forms for the Shallow Water Equations.*Fluids 2017*,*2 (2), doi:10.3390/fluids2020024*.**Blender, R.**&**Badin, G.**(2017). Viscous dissipation in 2D fluid dynamics as a symplectic process and its metriplectic representation.*Eur. Phys. J. Plus*,*132*(3), 137, doi: 10.1140/epjp/i2017-11440-x.