### Project Summary

Tropical storms, hurricanes, and strong mid-latitude cyclones are governed by complex multi-scale processes in the sense of this SFB. Consider, e.g., a fully developed hurricane [Ema03, SM10]: The “eye” of the vortex with relatively quiet air (radial extent: ~10–50 km) is surrounded by an “eye wall” of similar size within which major precipitating updrafts generate strong latent heat release. This eye/eye wall structure is embedded in a dominantly circumferential .ow with two radial layers. The .rst layer covers the innermost ~50–150 km of the storm. In this layer, owing to very high rotational .ow speeds, the radial momentum balance is dominated by centripetal accelerations and radial pressure gradients while the Coriolis acceleration is of lesser importance (cyclostrophic balance). In the second layer with radial extension of ~150-500 km and lower flow velocities these three forces are instead of comparable magnitude (gradient wind balance). Even farther away (> 500 km) the horizontal .ow is no longer dominantly circumferential, and inertial accelerations are small relative to pressure gradients and Coriolis accelerations (geostrophic balance). This radial three-layer structure covers most of the troposphere vertically but rides on top of a turbulent boundary layer that mediates mass, heat, and moisture exchange with the ocean surface.

This large-scale view of the vortex structure already involves six substantially different scaling regimes (three radial times two vertical layers). Yet, this still covers only part of the pertinent cascade of scales. An essential part of the total .ow of heat and moisture in an atmospheric vortex is not mediated by these large-scale motions but rather by small-scale unstable convective updrafts and associated precipitation and evaporation. Here we think, e.g., of the intense turbulent updrafts in the eye wall, of spiraling cloud and rain bands that originate from unstable boundary layer outflow, or of the “vortical hot towers” that play a signi.cant role during vortex initiation.

The large-scale features of such vortices may be expected to be amenable to scale analysis and asymptotics, given some suitable averaged information from the small scales. But, whereas asymptotic methods can still provide a clue as to the characteristic scalings and local structure of the small-scale processes, it is unlikely that a closed theory can be derived solely through asymptotic analysis: the space-time distributions of the small-scale updrafts will involve a strong chaotic component because the updrafts originate from turbulent boundary layers and are driven by buoyancy instabilities. At the same time, the distributions also have a deterministic aspect due to .uid mechanical interactions between neighboring updrafts and the modulation of their local environment by the large-scale flow.

This project aims at a mathematically motivated approach to capturing process interactions across these partly deterministic and partly chaotic, or random, cascades of scales. The root model for this project is given by the compressible multiphase .uid .ow equations, yet its solutions are not accessible due to the vast scale range involved in the considered processes. The challenge here is to develop asymptotic analyses for the large scales of interest and surrogate models for the computationally inaccessible scales, and to couple these components in a mathematically transparent fashion. The obtained analytical models shall be validated against idealized numerical simulations and observational data.

### Project publications

Hittmeir, S. and Klein, R. and Müller, A. and Névir, P.
(2020)
*The Dynamic State Index with Moisture and Phase Changes.*
Mathematics of Climate and Weather Forecasting 2020
.
pp. 1-12.
(Submitted)

Hittmeir, Sabine and Klein, Rupert and Li, Jinkai and Titi, Edriss
(2020)
*Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes.*
Nonlinearity, 33
(7).
pp. 3206-3236.

Müller, A. and Névir, P. and Klein, R.
(2018)
*Scale Dependent Analytical Investigation of the Dynamic State Index Concerning the Quasi-Geostrophic Theory.*
Mathematics of Climate and Weather Forecasting, 4
(1).
pp. 1-22.
ISSN 2353-6438 (online)

Hittmeir, S. and Klein, R.
(2018)
*Asymptotics for moist deep convection I: Refined scalings and self-sustaining updrafts.*
Theoretical and Computational Fluid Dynamics, 32
(2).
pp. 137-164.
ISSN 0935-4964 (Print) 1432-2250 (Online)

Hittmeir, S. and Klein, R. and Li, J. and Titi, E.
(2017)
*Global well-posedness for passively transported nonlinear moisture dynamics with phase changes.*
Nonlinearity, 30
(10).
pp. 3676-3718.
ISSN 0951-7715

Dörffel, T. and Papke, A. and Klein, R. and Smolarkiewicz, P.
(2017)
*Intensification of tilted atmospheric vortices by asymmetric diabatic heating.*
SFB 1114 Preprint in arXiv:1708.07674
.
pp. 1-22.
(Unpublished)

Mazza, E. and Ulbrich, U. and Klein, R.
(2017)
*The Tropical Transition of the October 1996 Medicane in the Western Mediterranean Sea: A Warm Seclusion Event.*
Monthly Weather Review, 145
.
pp. 2575-2595.
ISSN Online: 1520-0493 Print: 0027-0644

### Theses

Papke, A.
(2017)
*Atmospheric vortex stability under vertical shear.*
PhD thesis, FU Berlin, FB Mathematik & Informatik.