# M4: Entropy production in turbulence parameterisations

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

## Invited Guests

## Reports

## 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.**Blender, R., Gohlke, D., and Lunkeit, F.**(2018). Fluctuation Analysis of the Atmospheric Energy Cycle.*Physical Review E*,*98*(2), 023101.Conti, G., &

**Badin, G.**(2017). Hyperbolic Covariant Coherent Structures in Two Dimensional Flows.*Fluids*2017,*2*(4), 50. doi: 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.*The European Physical Journal Plus*,*132*(3), 137.