At this colloquium, we are happy to welcome:

Hannah Bergner (Uni Mainz)

Ice clouds as nonlinear oscillator

Clouds usually cover about two thirds of the Earth’s surface and play a key role in the water and energy cycle of the Earth. They impact the energy budget by interacting with incoming solar radiation and outgoing thermal radiation. For pure ice clouds, the net impact of the different radiative effects is still unknown, and there is no generally accepted theory of clouds in terms of a closed system of partial differential equations or similar.

In this talk, I will present a simple but physically consistent ice cloud model which is a 3D nonlinear ODE system (depending on several parameters). The model constitutes a nonlinear oscillator with two Hopf bifurcations in the relevant parameter regime. Apart from the equilibrium points and bifurcations, limits cycles and scaling behaviours of the system for varying parameters can be determined. Finally, the model shows very good agreement with measurement data, indicating that the main physics is captured and such a simple model might be a helpful tool for investigating ice clouds.

This is joint work with Peter Spichtinger.

Ojesh Koul (KU Eichstätt-Ingolstadt)

Fast-Slow splitting of geophysical flows with non uniform Coriolis parameter

We investigate the method of optimal balance for the decomposition of flows into slowly evolving, balanced part and fast, unbalanced part. This decomposition finds application in areas such as improving weather predictions, gravity wave parametrizations. The method works due to the balanced component evolving on an almost adiabatic invariant manifold under slowly varying perturbations. A ramp function is used to slowly deform the fully non linear system into a linear system where the balanced part and unbalanced part of the system are clearly delineated. While in the linear state, the unbalanced part of the flow is projected out. This requires explicit knowledge of the projection operator, which is available when the Coriolis parameter is uniform. This is not the case when a non-uniform Coriolis parameter is involved. In this study we extend the method by using the ramping process to eliminate the Coriolis force as well, deforming to a linear, non-rotating system where the balanced component is stationary and can be obtained by either a projection operator or by averaging out the fast varying unbalanced component. We show, through numerical experiments, that the balanced component is an adiabatic invariant of the ramping process and that the leakage into unbalanced components decreases rapidly with decreasing Rossby number and increasing ramping period. We supplement these results with analytical estimates of leakage from the balanced component into the unbalanced component showing the invariance of eigenstates of the linear system when the Coriolis terms are slowly ramped to zero. In this study we have examined the validity of the method on a shallow water model. We aim to investigate a full 3D stratified model in the future and diagnose the generation of internal waves.

This is a joint work with Silvano Rosenau, Marcel Oliver, Manita Chouksey and Carsten Eden.

Richard Höfer (Uni Regensburg)

From Spheres to Rods: Deriving Suspension Models

We consider inertialess rigid non-Brownian particles in a Stokes flow with the goal to rigorously derive effective models in suitable limits of many small particles for vanishing small particle volume fraction. The problem is significantly easier for identical spherical particles than for non-spherical particles. 

For spherical particles, I will present results on the derivation of a coupled transport-Stokes system for sedimentation models of buoyant particles, i.e. particles with much higher mass density than the fluid. To leading order, the buoyant particles create an additional force in the Stokes equation. Moreover, they effectively increase the viscosity of the fluid by a term proportional to the (small) particle volume fraction.For non-spherical particles, the (nonperturbative) treatment of buoyant particles is widely open. For neutrally buoyant particles, so called Doi-type models have been proposed, which couple a Stokes equation with increased viscosity to a transport equation for the particle density in position and orientation space.  While Doi models accurately describe the effective evolution of the spatial particle density to the first order in the particle volume fraction, this accuracy fails regarding the evolution of the particle orientations. We rigorously attribute this failure to the singular interaction of the particles via a −3-homogeneous kernel. In the situation that the particles are initially distributed according to a stationary ergodic point process, we identify the limit of this singular interaction term.

The talk is based on joint work with David Gérard-Varet, Amina Mecherbet and Richard Schubert.