Monday, March 27


Introduction and Welcome


Peter Knippertz, The intricacies of identifying equatorial waves


Carsten Eden, Parameterized internal wave mixing in three ocean general circulation models


Coffee Break


Rosimar Rios-Berrios, Multi-scale interactions leading to tropical cyclogenesis in a convection-permitting aquaplanet simulation


Tobias Selz, The transition from practical to intrinsic predictability of midlatitude weather


Manita Chouksey, Balancing geophysical flows in different numerical settings


Short Break


Ted Shepherd, Combining physical reasoning with statistical practice in uncertainty quantification across scales

from 18:00

Posters, Food, and Drinks at the MIDS (Hohe Schule)

Tuesday, March 28


Michael Ghil, Data assimilation: from an eventful past to a bright future


Nedjeljka Žagar, Balance and four-dimensional data assimilation in the tropics


Coffee Break


Alberto Carrassi, Using machine learning in geophysical data assimilation (some of the issues and some ideas)


Mierk Schwabe, Machine learning to improve climate models and projections


Mariana Clare, Computer, how likely is it that I need my coat tomorrow? How bayesian neural networks can be used for probabilistic predictions of weather and climate


Eyke Hüllermeier, Representation and quantification of uncertainty in machine learning


Lunch Buffet


Sergey Danilov, Unstructured meshes and effective resolution


Coffee Break


Joanna Szmelter, Unstructured-mesh based non-oscillatory-forward-in-time integration for atmospheric flows


Sergiy Vasylkevych, TIGAR - a new global atmospheric model for the simulation of transient inertia-gravity and Rossby wave dynamics: perspectives and challenges


Colin Cotter, Structure preserving compatible finite element methods for geophysical fluid dynamics


Hans Burchard, Estuarine mixing and exchange flow in an isohaline framework

from 17:30

Tour at the German Medical History Museum

Wednesday, March 29


George Craig, Convergence of forecast probabilities with increasing ensemble size


Darryl Holm, Multiscale fluid interactions by Composition of Maps (CoM)


Coffee Break


Jin-Song von Storch, The integral effect and the intrinsic uncertainty (randomness) in dynamical systems


Alvaro de Diego, Data based computation of coherent sets in fluids


Frank Kwasniok, The structure of predictability in an intermediate-complexity atmospheric model: covariant Lyapunov vectors and finite-time Lyapunov exponents


David Dritschel, Lagrangian modelling of rotating stratified geophysical flows


Lunch Buffet


Hilary Weller, Unconventional ways of modelling the atmosphere: implicit advection and r-adaptivity


Coffee Break


Igor Shevchenko, An alternative approach to the ocean eddy parameterization problem


Ekaterina Bagaeva, Subgrid momentum closure: how to link kinetic and potential energy backscatter?


Moritz Pickl, Towards a process-level understanding of the impact of stochastic perturbations


Ivana Aleksovska, Post-processing ensemble forecasts using EMOS


Stephan Juricke, Simulating ocean eddies: intercomparison of novel mesoscale eddy parametrizations and diagnostics


Anton Kutsenko, Scale analysis on different basis systems may give different results for the energetic spectral slope

from 19:00

Conference Dinner

Thursday, March 30


Ulrich Achatz, Gravity waves and their influence on weather and climate: challenges and new approaches


Volkmar Wirth, Limitations of refractive index diagnostics for Rossby wave propagation


Coffee Break


Rupert Klein, Triple-deck boundary layers of synoptic motions and tropical storms


Closing Remarks


An Investigation of Large scale oceanic flows near shores

Khadeeja Afzal, Katholische Universität Eichstätt-Ingolstadt

We investigate the use and behavior of balance relations for studying the large-scale oceanic flows near shores. We study balance model for shallow water equations. Balance relations make use of the observation that large scale flows are dominated by potential vorticity and that all other flow quantities can be approximately derived from the potential vorticity by an inversion relation. This study aims to investigate theoretically the near-shore behavior of such models and by the direct numerical simulation. For the numerical investigation we will employ existing shallow water codes and will have access to full high resolution ocean models for comparative studies and model validation. The goal is to derive simple fully nonlinear models for the large scale flow in the vicinity of basin boundaries and find out their asymptotic behavior. This will help improve our conceptional understanding and provide benchmarks for the calibration of larger numerical ocean models.

This poster is joint work with Marcel Oliver.

Diagnosing spurious diapycnal mixing and its spatial distribution due to advection in Z-coordinate ocean models using discrete variance decay

Tridib Banerjee, Constructor University, Alfred Wegener Institute

Ocean models relying on geopotential (Z) vertical coordinates suffer from spurious diapycnal mixing created by advection due to the misalignment of isopycnal and grid-layer surfaces. Given the delicateness of diapycnal mixing in ocean models, several studies have been performed to determine its impact, mainly by means of global analyses. Here we present a local analysis of spurious diapycnal mixing based on tracer variance decay. We apply the discrete variance decay (DVD) method proposed by Klingbeil et al. (2014) to diagnose numerical mixing created by several third-order advection schemes used in FESOM (Finite volumE Sea ice Ocean Model). The analysis is applied for an idealized channel flow test setup with Z* vertical coordinates and a linear equation of state. This ensures numerical DVD to be entirely diapycnal enabling identification of its spatial distribution. Further modification of the DVD method is proposed which allows for splitting of total diapycnal mixing into individual contributions from advection and diffusion. The new modifications are then used to compare spurious diapycnal mixing due to advection and explicit horizontal diffusion with parameterized physical diapycnal mixing due to vertical diffusion.

Using deep learning to generate the ensemble spread from deterministic weather predictions

Rüdiger Brecht, Universität Hamburg

Spectral resolution of the ocean’s Lorenz energy cycle

Jan Niklas Dettmer, Universität Hamburg

Eddy kinetic energy (EKE) and the conversion terms of the Lorenz energy cycle are estimated from an eddy-resolving global ocean model and resolved spectrally per horizontal wavenumber. The baroclinic conversion term (BC) exhibits a dipolar structure, where it is a source for EKE at scales close to the first baroclinic Rossby radius and a sink for EKE at larger scales close to the Rhines scale. The geographical and vertical distributions of the BC term are explored. It is found that in the ocean interior negative BC is limited to regions poleward of approximately 30° north and south. It is suggested that the cause for this distribution is the transfer of eddy energy to Rossby waves and zonal jets equatorward of 30°. It removes eddy energy before it cascades up to the scale where negative BC takes place. Equatorward of 30° the existence of a closed energy loop is suggested. Positive BC produces EKE which cascades upscale where it is converted to available eddy potential energy (EAPE) by negative BC, which cascades downscale again. The sink of EKE partly balances the EKE produced by baroclinic instability. The energy loop traps a certain amount of energy. Finally, the baroclinic conversion term is explored further in idealized model setups. The goal of the idealized setups is to test the robustness of the diagnostic methods and gain physical understanding of the negative baroclinic conversion.

This poster is joint work with Carsten Eden.

Impacts of convective treatment on tropical rainfall variability in ICON-NWP simulations

Hyunju Jung, Karlsruhe Institute of Technology

Increased computing resources allow us to perform high-resolution convection-permitting numerical simulations on a global scale. One of the benefits using such simulations is that tropical rainfall and its coupling to large-scale circulation, such as convectively coupled wave, are better represented. This talk will assess this statement using the ICON-NWP model which is operationally used by the German Weather Service (DWD). Objectively identified equatorial waves will be shown in global simulations with various horizontal grid spacings (80-2.5km) and with different convective treatments (explicit vs. parameterized convection). To understand the impact of convective treatment in depth, idealized simulations are introduced with using a tropical aquachannel geometry and with four different combinations of deep and shallow convection. Differences among the idealized runs in the mean state, with a particular focus on rainfall, will be presented with a novel diagnostic tool to disentangle the processes important for the differences in a fully coupled and physically consistent way. Lastly, equatorial waves in the idealized runs will be shown and their potentials for forecast skill will be discussed.

This poster is joint work with Peter Knippertz, Yvonne Ruckstuhl, Robert Redl, Corinna Hoose, and Tijana Janjic.

Flow-dependent contribution of multivariate forecast uncertainties on skill and spread of convective-scale ensemble forecasts

Takumi Matsunobu, Meteorological Institute, LMU München

We investigate the spatial skill-spread relationship of ensemble forecasts of convection in summer 2021 and address the impact of considering model uncertainties from two physics parameterizations – microphysics and planetary boundary layer turbulence – in the presence of initial and lateral boundary condition uncertainties. To investigate their flow dependence the spatial skill-spread relationship and the impact of uncertainties are investigated conditionally to the strength of synoptic convective control. We find that the spatial skill-spread relationship is highly flow-dependent and the current operational ensemble forecasts are spatially underdispersive, especially during weak synoptic control, whereas a good agreement is found during strong synoptic control. A case study during weak synoptic control demonstrates that perturbations in the planetary boundary layer contribute to improving forecast skill and increasing spread at small scales while microphysical perturbations show a spread increase across scales. Overall, the combination of both perturbations seems to combine their individual impacts and thus benefits the spatial skill-spread relationship at most times and scales.

This poster is joint work with Christian Keil, Matjaž Puh, Christoph Gebhardt, and Chiara Marsigli.

Can a Block form as a “Traffic Jam” in the Jet Stream?

Christopher Polster, Johannes Gutenberg Universität Mainz

In 2018, Nakamura and Huang proposed a simplified model of blocking onset in which blocks form due to an obstruction of the zonal propagation of wave activity on the mid-latitude waveguide, analogous to how traffic jams emerge on a highway. The theory is derived from the budget of finite-amplitude local wave activity which can quantify blocking even during the non-linear stage accurately.

Using the quasi-geostrophic local wave activity framework, we investigate the development of a 2016 European winter block and evaluate the possible role of the ”traffic jam” mechanism in the flow transition. We determine the main processes contributing to the blocking onset by evaluating the terms of the wave activity budget with reanalysis data and quantify the impact of upstream precursor Rossby wave activity with an ensemble sensitivity analysis based on 200 ECMWF forecasts. The ensemble approach allows us further to create a case-specific “fundamental diagram”, characterizing the relationship between wave amplitude and wave activity flux that is central to the traffic jam mechanism.

Our findings from the NWP ensemble are consistent with the idealized traffic jam model of Nakamura and Huang, which we run in an analogue setup to provide a baseline for our expectations. Complementing previous composite studies, our results shows that a description of blocking onset in terms of the traffic jam mechanism can be meaningful for individual blocking events, though realistic blocking also exhibits features not accounted for in the 1D idealized theory. The combination of the local wave activity framework with an ensemble-based approach provides flow-dependent dynamical insight into the block development. We discuss implications for the predictability of blocking onset in medium-range weather forecasting.

This poster is joint work with Volkmar Wirth.

Implementation of the ultraviolet solar radiation and energetic particles in ICON-ART-LINOZ

Maryam Ramezani Ziarani, Mathematical Institute for Machine Learning and Data Science, MIDS, Katholische Universität Eichstätt-Ingolstadt

We have included variable solar forcing in the middle atmosphere from particle precipitation and solar UV radiation into ICON (ICOsahedral Nonhydrostatic) -ART (the extension for Aerosols and Reactive Trace gases) model system in the numerical weather prediction (NWP) configuration by further developments of the simplified ozone description (Linearized ozone scheme -LINOZ). The LINOZ scheme is computationally very cheap compared to a full middle atmosphere chemistry scheme, yet provides realistic ozone fields consistent with the stratospheric circulation and temperatures, and can therefore be used in climate models instead of prescribed ozone climatologies. To include the energetic particle precipitation indirect effect, we have implemented a “geomagnetic forcing” by downwelling of nitric oxides (NOy) produced by auroral and radiation belt electron precipitation in the upper mesosphere and lower thermosphere into the stratosphere during polar winter into ICON-ART. This was achieved by including an upper boundary condition of NOy, a parameterization using the geomagnetic Ap index, which was also recommended for chemistry-climate models for the CMIP6 experiments. With this extension, the model simulates realistic „tongues“ of NOy propagating downward from the model top in the upper mesosphere to the mid-stratosphere in every polar winter. We then expanded the simplified ozone description used in the model by applying LINOZ version 3, including a NOy-based tendency term. This NOy, coupled as an additional term in the linearized ozone chemistry, led to significant ozone losses in the polar upper stratosphere in both hemispheres. In a subsequent step, the ozone climatologies forming the basis of the LINOZ scheme were provided for solar maximum and solar minimum conditions separately and interpolated to the ICON using the F10.7 as a proxy for daily solar spectra (UV) variability to account for the solar UV forcing. This solar UV forcing in the model led to observed changes in ozone in the tropical stratosphere due to the photolysis of oxygen.

This poster is joint work with Miriam Sinnhuber and Thomas Reddmann.

Formulation of temporal mean flows with averaging

Juliane Rosemeier, University of Exeter, UK

Equations with a skew-hermitian linear operator, which has purely imaginary eigenvalues, and a bi-linear term are considered. Examples are the Rotating Shallow Water Equations and the Boussinesq Equations. Due to the skew-hermitian linearity fast oscillations arise in the solutions. A transformation can be applied to eliminate the skew-hermitian term. The resulting equation is denoted as modulation equation. The fast oscillations, which are damped but still in the equation, can further be mitigated by applying averaging. Inspired by the theory of fast singular limits in PDEs we investigate formulations where fast temporal oscillations are mitigated by applying a convolution with finite averaging windows. The formulations are used to define temporal mean flows. Using a finite averaging window has the advantage that the coarse behavior of the equations remains, that is oscillations with a period larger than the averaging window, and small scale features are not in the averaged equations anymore. This idea can also be used to design efficient numerical algorithms.

This poster is joint work with Beth Wingate and Terry Haut.

Influence of data assimilation on representation of tropical waves and prediction of precipitation

Yvonne Ruckstuhl

Forecasting precipitation in the tropics remains a challenge for numerical weather prediction today. The practical predictability realized with operational systems appears to be much further away from theoretical intrinsic predictability than at higher latitudes. Among the many reasons for this, is believed to be the strong coupling between tropical waves and precipitation, tying the representation of tropical waves in the initial conditions and the model to the quality of precipitation forecasts. In this work, we investigate the effect of tropical wave representation in the initial conditions to precipitation forecasts by performing twin experiments with the full ICON model in a tropical aqua channel with 13km horizontal resolution. The ensemble square root filter (EnSRF) with 40 ensemble members is employed as the DA algorithm. The observation coverage is varied, which affects the accuracy of the tropical waves representation in the initial conditions. Its influence on precipitation forecasts is quantified through traditional skill scores like the Fraction Skill Score (FSS). We find that a good representation of the Kelvin wave is most important, which is achieved by assimilating sufficient horizontal wind observations.

This poster is joint work with Hyunju Jung, Peter Knippertz, Tijana Janjic, and Robert Redl.

A toy model for quantification of unresolved scales error statistics

Florian Semrau, Katholische Universität Eichstätt-Ingolstadt

As observation error covariance matrix plays an important role in data assimilation (DA). Based on its characteristics, the information in observations is filtered to fit resolved scales of the numerical model. How to best specify observation error statistics to include error due to unresolved scales (Janjic et al. 2018) depends on the dynamics and physics of the problem at hand. For highly non-linear cloud microphysics processes, the specification of the unresolved scales error statistics is particularly hard during DA.

In this work, we derive a particle based stochastic model using the same assumptions as Wacker and Seifert (2001) and extend their definitions of number concentration and liquid water content to infinitesimally small length scales. The model is designed to allow for relatively easy generalization to more complex and realistic dynamics. Furthermore, our model explicitly distinguishes between stochastic and deterministic quantities and thus allows for direct calculation of the unresolved scales error covariance matrices. In addition, first DA results that consider approximate observation error correlations will be presented.

This poster is joint work with Tijana Janjic.

On the spontaneous generation of gravity waves

Marc Aurele Tiofack Kenfack, Katholische Universität Eichstätt-Ingolstadt

We investigate the spontaneous generation of gravity waves by balanced flows. It has been proven that this generation is linked to the complex poles of the balanced part of the flow. We generate a pseudo-random sequence of poles by using the Lorenz ’63 system, this last coupled to a simple finite-dimensional model that we consider. Our goal here is to construct a diagnostic tool for the emission of the unbalanced energy that takes the complex-time poles of the balanced flow as input.

Second, we implement the ”optimal balance” scheme, which is a non-linear algorithm for the separation of gravity waves (unbalanced part of the flow) and the Rossby waves (balanced part) in the TIGAR model. The main question that we address in this part is whether the equatorial waves namely the mixed Rossby-gravity and Kelvin waves do provide a ”fast” pathway in the energy transfer between Rossby and gravity waves.

This poster is joint work with Marcel Oliver.

Abstracts for talks

Gravity waves and their influence on weather and climate: challenges and new approaches

Ulrich Achatz, Goethe Universität Frankfurt

Even with present and foreseeable computational capabilities, the spatial resolution of atmospheric weather-forecast and climate models is and will remain insufficient to capture many essential processes. Next to clouds and turbulence, subgrid-scale waves and their parameterization are one of the challenges of the field. Here, especially buoyancy-driven gravity waves are in the focus. The talk will sketch the fundamental properties and atmospheric impacts of these waves. It will describe lead issues in their handling in models, and it will discuss recent developments towards their solution.

Post-processing ensemble forecasts using EMOS

Ivana Aleksovska, ECMWF

Numerical weather prediction models have systematic errors and biases, which can be reduced through statistical correction methods. The most intuitive approach to correcting weather forecasts is to establish a statistical relationship between the forecast and the corresponding observations. Once the relationship is established, it can be used to correct future forecasts. These approaches, also known as statistical adaptation, are used on a daily basis to improve the quality of operational forecasts. Many methods have been proposed in the literature to calibrate ensemble forecasts, and these can be divided into two main groups: parametric and non-parametric methods. One of the most widespread and used methods of the parametric group is the so-called Ensemble Model Output Statistics (EMOS). We show the performance of EMOS for post-processing temperature forecasts at 2m; specifically, the ensemble operational forecast at ECMWF versus SYNOP observations, that will serve as a benchmark for future studies

Ekaterina Bagaeva, Constructor University

Limited resolution of computational models of the ocean result in the need to parametrize processes smaller than the grid scale. In this work, we consider a range of parametrizations including deterministic and stochastic kinetic energy backscatter and potential benefits from its linkage with the potential energy backscatter. The implementations are tested on two intermediate complexity setups of the global ocean model FESOM2: a doubly-periodic channel and a double-gyre box model.

This talk is joint work with Stephan Juricke, Marcel Oliver, and Sergey Danilov.

Estuarine mixing and exchange flow in an isohaline framework

Hans Burchard, Leibniz Institute for Baltic Sea Research Warnemünde

For estuaries, which are transitions zones between rivers and the ocean and where the density is dominated by salinity, the isohaline framework offers new perspectives of analysing estuarine mixing and exchange flow.

Mixing. It has been recently shown that the long-term averaged mixing (reduction rate of salinity variance) in an estuarine volume bounded by a moving isohaline of salinity S amounts to M(S)=S^2Qr, where Qr is the river discharge. With this universal law of estuarine mixing, the mixing per salinity class amounts to m(S)=dM(S)/dS=2S*Qr. In a numerical model, this mixing is composed of parameterised (physical) mixing and numerical mixing due to the truncation error of advection schemes. To quantify this in an isohaline framework, the local mixing rates are binned into salinity classes for each water column giving the local mixing per salinity class, showing for each salinity class the horizontal composition of m. Long-term averaging and integration of total mixing (sum of physical and numerical mixing) over the isohaline surface gives the universal law.

Exchange flow. The diahaline velocity can be calculated by means of binning local layer thicknesses into salinity classes, taking the isohaline slope into consideration. Integration of over the entire isohaline surface and long-term averaging gives the diahaline volume transport which should be equal to Qr. The local composition shows a diahaline exchange flow with some regions of up-estuary (towards lower salinity) volume transport and more pronounced regions of down-estuary (towards higher salinity) volume transport. Integrating separately over all up estuary and down-estuary contributions gives the diahaline exchange flow.

Connection between mixing and exchange flow. Locally, mixing per salinity class and the diahaline velocity are connected through a fundamental relation: The diahaline velocity equals minus half of the S-gradient of the local mixing per salinity class. This relation is shown for a realistic estuarine application to the Baltic Sea. Applications of the analysis to large ocean scales are discussed, using isopycnals instead of isohalines.

This talk is joint work with Ulf Gräwe, Erika Henell, Knut Klingbeil, Xaver Lange, Xiangyu Li, Marvin Lorenz, and Nina Reese.

Using machine learning in geophysical data assimilation (some of the issues and some ideas)

Alberto Carrassi, Dept of Physics, University of Bologna, Italy

In recent years, data assimilation (DA), and more generally the climate science modelling enterprise have been influenced by the rapid advent of artificial intelligence, in particular machine learning (ML), opening the path to various form of ML-based methodology.

In this talk we will schematically show how ML can be included in the prediction and DA workflow in three different ways. First, in a so-called “non-intrusive” ML, we will show the use of supervised ML to estimate the local Lyapunov exponents (LLEs) based exclusively on the system’s state [1]. In this approach, ML is used as a supplementary tool, added to the physical model, to spot local instabilities. Our results prove ML is successful in retrieving the correct LLEs, although the skill is itself dependent on the degree of local homogeneity of the LLEs on the system’s attractor. Interestingly, the neutral mode, connected to the tangent to the flow is only moderately well predicted. In the second and third approach, ML is combined with DA in an integrated fashion to learn (i) a surrogate model from noisy and sparse data [2], or, (ii) to learn a data-driven parametrization of a physical’s model unresolved scales, again from noisy and sparse data [3]. The full surrogate model achieves great prediction skill up to 4 to 5 Lyapunov time, and its power spectra density as well as its Lyapunov spectra are almost identical to that of the original data. The parametrization of the unresolved scales has been studied using a coupled atmosphere-ocean model, whereby the truncated model misses 10 high frequencies atmospheric modes. We use strongly coupled DA [4] in the DA part of the combined DA-ML. The resulting hybrid physics-data driven model reduces the prediction error compared to the physical model by more than 50%. Importantly, the use of strongly coupled DA makes possible to exploit the data information from one model compartment (e.g., the ocean) to the other (e.g., the atmosphere), resulting in a surprising improvement in fully resolved yet poorly observed ocean. We then apply NN to a 1D sea-ice model in which ML is used to learn the parametrization of the melt-ponds.

Our results prove that DA is pivotal to extract much information from the sparse, noisy, data that ML would not be able to handle alone. DA not only provides a homogenous and complete pseudo-dataset (the analysis) for training, but also an estimate of its accuracy, which then enters the ML loss function formulation. Finally, the combined DA-ML approach is also implicitly physics-informed in that the analysis fields used as features in the ML incorporate the physical model signal in them and lead to physically consistent surrogate or parametrization models.

[1] Ayers D, J Amezcua, A Carrassi and V Ohija, 2022. Supervised machine learning to estimate instabilities in chaotic systems: estimation of local Lyapunov exponents, Under Review, Available at [2] Brajard J, A Carrassi, M Bocquet and L Bertino, 2020. Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model., J. Comp. Sci.,44,101171 [3] Brajard J, A Carrassi, M Bocquet and L Bertino. 2021. Combining data assimilation and machine learning to infer unresolved scale parametrisation., Phil. Trans A of the Roy Soc., 379 (2194). 20200086 [4] Tondeur M, A. Carrassi, S. Vannitsem and M. Bocquet: On temporal scale separation in coupled data assimilation with the ensemble Kalman filter. J. Stat. Phys., 44. 101171, 2020

This talk is joint work with M. Bocquet, J. Brajard, L. Bertino, D. Ayers, J. Amezcua, and S. Driscoll.

Balancing geophysical flows in different numerical settings

Manita Chouksey, Universität Bremen

Geophysical flow dynamics, such as atmospheric and oceanic dynamics, are heavily influenced by the slow balanced and fast unbalanced motions. These motions evolve at different temporal and spatial scales and exchange energy via nonlinear interactions, which makes is challenging to precisely separate and identify the balanced motions from the unbalanced ones. Here we compare two different methods to balance geophysical flows in shallow water model simulations with two different numerical model codes and two different initial conditions over a varied parameter range. Overall, the performance of both methods is comparable. Cross-balancing, i.e. balancing the model with one method and diagnosing the imbalance with the other, suggest that both methods find approximately the same balanced states. The resulting approximately balanced states are characterized by very small residual wave emission during the evolution of the flow field. Our results contradict previous claims of significant spontaneous wave emission from balanced flow, and show that the notion of balance in numerical models of geophysical flows is ultimately tied to the particular discretization, and an accompanying numerical model is essential.

This talk is joint work with Carsten Eden, Gökce Tuba Masur, and Marcel Oliver.

Computer, how likely is it that I need my coat tomorrow? How bayesian neural networks can be used for probabilistic predictions of weather and climate

Mariana Clare, ECMWF

The success of machine learning techniques over the years, and in particular neural networks, has opened up a new avenue of research for weather and climate predictions. However standard neural networks suffer as decision-making tools because they lack the ability to express uncertainty. In this talk I will present the relatively novel technique of Bayesian Neural Networks, which is a type of neural network that can make probabilistic predictions.

I will present two successful applications of Bayesian Neural Networks to weather and climate data, the first applied to ocean model data to understand uncertainty in dynamical oceans and the second applied to post-processing of numerical weather model data to add uncertainty information to deterministic model forecasts.

This talk is joint work with Maike Sonnewald, Redouane Lguensat, Thomas Haiden, and Zied Ben Bouallegue.

Structure preserving compatible finite element methods for geophysical fluid dynamics

Colin Cotter, Imperial College London

Compatible finite element methods provide an extension of the finite difference C-grid method that allows flexibility in the consistency order and in the wave dispersion properties. I will explain how compatible finite element methods can be used to construct energy preserving schemes for a hiearchy of models, even in the presence of upwinded transport. These energy preserving schemes can then be augmented with physical forcing and dissipation, leading to correct energy transfers between kinetic and potential energy.

This talk is joint work with Werner Bauer and Golo Wimmer.

Convergence of forecast probabilities with increasing ensemble size

George Craig, LMU München

Ensembles of numerical simulations are often used to make probabilistic predictions. Given the complexity of the atmosphere and Earth system, it is possible that very large ensembles, with thousands or millions of members, would be required to accurately sample the forecast distribution. Using large (up to 100,000 members) idealised convective-scale ensembles based on modified shallow water equations, we look at how sampling uncertainty decreases as ensemble sizes go far beyond those of currently operational forecasting systems. A bootstrapping technique is used on the distributions from the ensembles to create a convergence measure which quantifies how sampling uncertainty decreases with ensemble size. A universal power law in the sampling uncertainty is found across the model variables and all statistics tested, whereby when the ensemble is large enough, the sampling uncertainty decreases asymptotically, inversely proportional to the square root of ensemble size. The magnitude of uncertainty was found to differ depending on the synoptic situation, with a dependency on the shape of the ensemble distribution and the statistic of interest. Finally, it is discussed how to measure the convergence in a smaller ensemble of operational size and how to determine whether the asymptotic convergence can be applied.

This talk is joint work with Kirsten Tempest, Matjaž Puh, Christian Keil, and Jonas Brehmer.

Unstructured meshes and effective resolution

Sergey Danilov, Alfred Wegener Institute

Data based computation of coherent sets in fluids

Alvaro de Diego, TU München

In fluid flows one often finds subsets of material which mix little with their surroundings. Based on a geometrical characterization one can find such sets from the solutions of a partial differential equation involving the dynamic Laplacian. We present the theoretical background and a way to solve the resulting problem based on only tracer data.

This talk is joint work with Oliver Junge, Gary Froyland, and Peter Koltai.

Lagrangian modelling of rotating stratified geophysical flows

David Dritschel, University of St Andrews

A new Elliptical Parcel-In-Cell (EPIC) model is described and illustrated in several characteristic geophysical flow problems. The EPIC model enables one to represent mixing processes naturally without ad hoc stabilisers or hyperdiffusion. Simulations are presented for a novel Beltrami flow between parallel free-slip boundaries, a rotating Rayleigh-Taylor instability, and a rising moist bubble forming a cloud. A major advance is in the treatment of the free-slip boundary condition, where for the first time we permit non-zero boundary stress (equivalent to non-zero horizontal vorticity).

This talk is joint work with Matthias Frey and Steef Boeing.

Parameterized internal wave mixing in three ocean general circulation models

Carsten Eden, Universität Hamburg

The parameterization IDEMIX for vertical mixing by breaking internal gravity waves is presented and evaluated in three different non-eddy resolving ocean models: ICON-O, FESOM, and MITgcm. Three different products of wave forcing by tidal flow over topography, representing the current uncertainty, are applied and compared to reference simulations without IDEMIX, allowing the model-independent effects of the new closure to be assessed.

Common to all models is larger interior mixing work with stronger horizontal structure due to the inhomogeneous forcing functions in all simulations using IDEMIX, in better agreement to observations. Coherent model responses to the stronger mixing work are changes in the thermocline depth including IDEMIX related to stronger shallow overturning cells in the Indo-Pacific Ocean. Deeper mixed layer depths in the subpolar North Atlantic are related to an increase of the Atlantic overturning circulation which brings the model closer to observations, coming along with an increase in northward heat transport. The deep Indo-Pacific overturning circulation and the bottom cell of the Atlantic feature an incoherent model response, which may point towards the importance of excessive numerical mixing in the models.

This talk is joint work with Nils Brüggemann, Martin Losch, Patrick Scholz, Friederike Pollmann, Sergey Danilov, Oliver Gutjahr, Johann Jungclaus, Nikolay Koldunov, Peter Korn, and Dirk Olbers.

Data assimilation: from an eventful past to a bright future

Michael Ghil, ENS, UCLA

Vilhelm Bjerknes first described weather prediction as an initial-value problem in 1904. As John von Neumann and associates started using computers to implement this idea immediately after World War II, it quickly became apparent that the requisite initial data available then were incomplete. The appearance of weather satellites in the 1960s led further on to the concept of time-continuous assimilation of remote-sensing data. Nowadays, data assimilation (DA) is being applied across all the areas of the climate sciences and much beyond. This presentation traces the evolution of DA methodology from the successive corrections and polynomial interpolation of the beginnings through the development of sequential-estimation (“Kalman filtering”) and control-theoretical (“variational” or “adjoint”) methods to today’s machine-learning–aided methods. Key concepts, such as information transfer between variables and between regions, as well as parameter estimation and new areas of applications are emphasized. Cutting-edge developments covered in the presentation touch upon the application of concepts and tools from nonautonomous and random dynamical systems theory, as well as upon combining machine learning with DA and with knowledge-based models for weather and climate prediction.

References: Bach, E., and M. Ghil, 2022: A multi-model ensemble Kalman filter for data assimilation and forecasting, J. Advances Modeling Earth Systems, 15, e2022MS003123, doi:10.1029/2022MS003123 . Bengtsson, L., M. Ghil, and E. Källén (Eds.), 1981: Dynamic Meteorology: Data Assimilation Methods, Springer-Verlag, New York/Heidelberg/Berlin, doi:10.1007/978-1-4612-5970-1, reissued as an eBook by Springer in 2012, e-ISBN-13:978-1-4612-5970-1, 330 pp. Crisan, D., and M. Ghil, 2023: Asymptotic behavior of the forecast–assimilation process with unstable dynamics, Chaos, 33(2), doi: 10.1063/5.0105590. Ghil, M., and P. Malanotte-Rizzoli, 1991: Data assimilation in meteorology and oceanography, Adv. Geophys., 33, 141–266.

This talk is joint work with Eviatar Bach and Dan Crisan.

Multiscale fluid interactions by Composition of Maps (CoM)

Darryl Holm, Imperial College London

Why: SWOT data needs multiscale models of interacting swells, mesoscale and sub-mesoscale DoF. C◦M is natural for such multiscale fluid interactions.

How: Euler-Poincaré variational principles and their Lie-Poisson Hamiltonian forms provide multicomponent, multiphysics, multiscale fluid models.

What: Multiscale C◦M models can apply to either expected phenomena (e.g., WCI), or to projection onto modes (e.g., Littlewood-Paley modes).

Representation and quantification of uncertainty in machine learning

Eyke Hüllermeier, LMU München

Simulating ocean eddies: intercomparison of novel mesoscale eddy parametrizations and diagnostics

Stephan Juricke, Constructor University, Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Bremerhaven

An accurate representation of mesoscale eddy dynamics is crucial for simulating large-scale ocean currents as well as ocean and climate variability. Eddy-mean flow interactions affect strength, position and variations of ocean currents. Furthermore, eddies have a substantial impact on oceanic heat transport and the coupling between the atmosphere and ocean. However, at so-called eddy-permitting model resolutions around ¼°, eddy kinetic energy and variability is often fundamentally underestimated and an excessive dissipation of energy can be observed. To counter these substantial biases in the simulations, new eddy parametrizations need to be developed and implemented.

In this study, a set of oceanic eddy parameterizations is introduced and compared, including novel viscous and kinetic energy backscatter closures with different complexity. Various diagnostics that are suitable for applications on unstructured meshes are used to analyze their effect and guide further improvements.

The schemes are implemented in the unstructured grid, finite volume ocean model FESOM2 and tested in both global ocean and idealized channel simulations. We show that adjusted viscosity and kinetic energy backscatter parameterizations can substantially reduce some of the eddy related biases such as biases in mean currents, sea surface height variability and water mass properties. These novel eddy parametrizations outperform previously used viscous closures. A comparison to higher resolution simulations shows that they are also computationally less expensive while achieving similar results.

This talk is joint work with Sergey Danliov, Marcel Oliver, Ekaterina Bagaeva, and Anton Kutsenko.

Triple-deck boundary layers of synoptic motions and tropical storms

Rupert Klein, Freie Universität Berlin

Part one of this lecture will report on extensions of the asymptotic theory for strongly tilted atmospheric vortices in [1,2] by boundary layer effects. The boundary layer below such a vortex features a triple deck structure with a bottom friction layer and an intermediate ``convection controlling layer’’ (CCL) mediating the interaction between the bulk vortex above and the (ocean) surface below.

Strong flow convergence in the friction layer enforces the upward motion of moist, high-enthalpy air with positive convectively available potential energy (CAPE). Next up, the CCL hosts the lifting condensation level (LCL), the level of free convection (LFC), and the convective inhibition region (CIN), and tilt-induced asymmetries of the CCL flow determine where the boundary layer air will first breach the convection barrier, thereby controlling convection processes.

The triple-deck structure of the model equations couples the well-known equations for the friction boundary layer under an axisymmetric vortex, a version of the stationary Eliassen balanced vortex model - including its first Fourier mode asymmetric component - for the CCL, and Paeschke et al.’s tilted vortex model. We will present the full asymptotic model and show first numerical results of the boundary layer structure for the case of an upright axisymmetric vortex.

The shorter part II of the talk will summarize a spin-off theory that needed to be established before we could complete derivation of the vortex model: QG-DL-Ekman theory [3] provides a similar triple-deck model extending the classical QG theory for synoptic scale motions in the middle latitudes. The QG-DL-Ekman model is the natural asymptotic match in the radial direction to the vortex theory of part I. Moreover, it alleviates an important shortcoming of the classical QG-Ekman model which does not cover strong diabatic boundary layer processes, such as the development of a convective mixing layer in the morning of a sunny day.

[1] E. Päschke, P. Marschalik, A. Owinoh, R. Klein (2012), Motion and structure of atmospheric mesoscale baroclinic vortices: dry air and weak environmental shear, J. Fluid Mech., vol. 701, 137-170. [2] T. D"orffel, A. Papke, R. Klein, N. Ernst, P.K. Smolarkiewicz (2021), Dynamics of tilted atmospheric vortices under asymmetric diabatic heating, TCFD, vol. 35, 831-873. [3] R. Klein, L. Schielicke, S. Pfahl S., B. Khouider (2022), QG-DL-Ekman: Dynamics of a diabatic layer in the quasi-geostrophic framework, J. Atmos. Sci., vol. 79}, 887-905.

This talk is joint work with Tom Doerffel, Boualem Khouider, Stephan Pfahl, and Lisa Schielicke.

The intricacies of identifying equatorial waves

Peter Knippertz, Karlsruhe Institute of Technology

Equatorial waves (EWs) are synoptic- to planetary-scale propagating disturbances at low latitudes with periods from a few days to several weeks. Here, this term includes Kelvin waves, equatorial Rossby waves, mixed Rossby–gravity waves, and inertio-gravity waves, which are well described by linear wave theory, but it also other tropical disturbances such as easterly waves and the intraseasonal Madden–Julian Oscillation with more complex dynamics. EWs can couple with deep convection, leading to a substantial modulation of clouds and rainfall. EWs are amongst the dynamic features of the troposphere with the longest intrinsic predictability, and models are beginning to forecast them with an exploitable level of skill. Most of the methods developed to identify and objectively isolate EWs in observations and model fields rely on (or at least refer to) the adiabatic, frictionless linearized primitive equations on the sphere or the shallow-water system on the equatorial 𝛽-plane. Common ingredients to these methods are zonal wave-number–frequency filtering (Fourier or wavelet) and/or projections onto predefined empirical or theoretical dynamical patterns. This paper gives an overview of six different methods to isolate EWs and their structures, discusses the underlying assumptions, evaluates the applicability to different problems, and provides a systematic comparison based on a case study (February 20–May 20, 2009) and a climatological analysis (2001–2018). In addition, the influence of different input fields (e.g., winds, geopo- tential, outgoing long-wave radiation, rainfall) is investigated. Based on the results, we generally recommend employing a combination of wave-number–frequency filtering and spatial-projection methods (and of different input fields) to check for robustness of the identified signal. In cases of disagreement, one needs to carefully investigate which assumptions made for the individual methods are most probably not fulfilled. This will help in choosing an approach optimally suited to a given problem at hand and avoid misinterpretation of the results.

This talk is joint work with Maria Gehne, George N. Kiladis, Kazuyoshi Kikuchi, Athul Rasheeda Satheesh, Paul E. Roundy, Gui-Ying Yang, Nedjeljka Žagar, Juliana Dias, Andreas H. Fink, John Methven, Andreas Schlueter, Frank Sielmann, and Matthew C. Wheeler.

Scale analysis on different basis systems may give different results for the energetic spectral slope

Anton Kutsenko, Katholische Universität Eichstätt-Ingolstadt

Energy spectra can be computed in different basis systems. We discuss two popular ones: the classical Fourier trigonometric system and the Walch-Rademacher basis. While Kolmogorov’s slope -3 is the same for both systems, the slope, e.g., -5/3 differs. However, both systems provide a more or less similar qualitative analysis, which will be seen from the example of a channel simulation.

This talk is joint work with K. Bellinghausen, S. Danilov, S. Juricke, and M. Oliver.

The structure of predictability in an intermediate-complexity atmospheric model: covariant Lyapunov vectors and finite-time Lyapunov exponents

Frank Kwasniok, University of Exeter, UK

A comprehensive investigation of the predictability properties in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability is performed. The full spectrum of covariant Lyapunov vectors and associated finite-time Lyapunov exponents (FTLEs) is calculated. The statistical properties of the fluctuations of the FTLEs as well as the spatial localisation and entanglement properties of the covariant Lyapunov vectors are studied. We look at correlations between the FTLEs by means of a principal component analysis, identifying modes of collective excitation across the Lyapunov spectrum. We also investigate FTLEs conditional on underlying weather regimes. An advanced clustering algorithm is employed to decompose the state space into weather regimes associated with specific predictability properties as given by the FTLEs. Finally, the extreme value properties of the FTLEs are studied using generalised Pareto models for exceedances above a high and below a low threshold. Return levels as well as upper and lower bounds on the FTLEs are determined and extremely unstable or stable atmospheric states are identified.

Towards a process-level understanding of the impact of stochastic perturbations

Moritz Pickl, Karlsruhe Institute of Technology

Stochastic perturbations are a well-established technique to represent forecast uncertainty related to model errors and are used by numerous numerical weather prediction centers worldwide. Such stochastic perturbation schemes increase the ensemble dispersion and reduce systematic biases, and thus improve the performance of forecasts. A widely used approach is the stochastically perturbed parametrization tendencies (SPPT) scheme, which introduces uncertainty by randomly perturbing the net physics tendencies of the forecast model.

In this talk, a process-oriented perspective on SPPT is adopted by analyzing its impact on rapidly ascending air streams, so-called warm conveyor belts (WCBs), whose ascent is strongly enhanced by latent heat release from cloud-diabatic processes. As the magnitude of the perturbations of SPPT scales with the net physics tendencies, SPPT is expected to introduce large uncertainty in regions of WCB ascent. We show that although SPPT introduces zero-mean perturbations to the model, it systematically changes the occurrence frequency of WCBs and other diabatically enhanced weather systems, such as tropical convection. Based on our results, we formulate a hypothesis how Gaussian perturbations can result in a one-sided response. Finally, it is discussed to what extent the observed behavior has an impact on the model climate, focusing on processes that are linked to such rapidly ascending air streams.

This talk is joint work with Christian M. Grams and Simon T. K. Lang.

Multi-scale interactions leading to tropical cyclogenesis in a convection-permitting aquaplanet simulation

Rosimar Rios-Berrios, National Center for Atmospheric Research (NCAR)

Recent studies suggest that convectively coupled equatorial Kelvin waves can modulate the chances of tropical cyclogenesis. Some studies argue that the modulation happens primarily through kinematic processes (e.g., reduced vertical wind shear, enhanced lower-tropospheric vorticity), while other studies emphasize the role of moist convection and enhanced ascent associated with the waves. Here, we seek to clarify these discrepancies with an aquaplanet experiment with convection-permitting resolution in the tropics. This experiment captures the convective nature of both equatorial Kelvin waves and tropical cyclogenesis, while removing other known factors that influence the large-scale conditions of tropical cyclogenesis, such as interannual variability of the tropical atmosphere and oceans.

In this talk, I will first demonstrate that convection-permitting resolution is necessary to capture an accurate heating profile associated with Kelvin waves. I will then show that the aquaplanet experiment captures a modulation of tropical cyclogenesis by Kelvin waves. Process-based diagnostics further demonstrate that Kelvin waves aid cyclogenesis by promoting the amplification of easterly waves via enhanced ascent, enhanced cloud-radiative interactions, and enhanced surface heat fluxes. These novel results emphasize the role of thermodynamic processes that are critical for the mesoscale and convective-scale processes leading to tropical cyclogenesis.

Machine learning to improve climate models and projections

Mierk Schwabe, DLR Institut für Physik der Atmosphäre

Earth system models are fundamental to understanding and projecting climate change. The models have continued to improve over the years, but considerable biases and uncertainties in their projections remain. A large contribution to this uncertainty stems from differences in the representation of phenomena such as clouds and convection that occur at scales smaller than the resolved model grid. The long-standing deficiencies in cloud parameterizations have motivated developments of global high-resolution cloud-resolving models that can explicitly resolve clouds and convection. Short simulations from the computationally costly high-resolution models together with observations can serve as information to develop machine learning (ML)-based parameterizations that are then incorporated into Earth system models.

The ICOsahedral Non-hydrostatic (ICON) model is an open-access modelling framework, which is used on a variety of timescales and resolutions, ranging from numerical weather predictions to climate projections. Here we utilize existing regional and global cloud-resolving ICON simulations with data-driven techniques to train ML-based parametrizations. The newly developed parameterizations are coupled to the ICON Earth system model (ICON-ESM) via the Fortran-Keras Bridge, resulting in the ICON-ESM-ML hybrid model.

This talk is joint work with Arthur Grundner, Helge Heuer, Fernando Iglesias-Suarez, Tom Beucler, Pierre Gentine, Marco A. Giorgetta, and Veronika Eyring

The transition from practical to intrinsic predictability of midlatitude weather

Tobias Selz, Meteorologisches Institut, LMU München

Many studies indicate that the atmosphere possesses an intrinsic predictability limit and hence weather forecasts cannot be extended indefinitely, regardless of future improvements in technology. This study computes an estimate of the intrinsic limit and of the remaining improvement potential of midlatitude weather forecasts. For this purpose, current estimates of the initial condition uncertainty are reduced in several steps from 100% to 0.1% and propagated in time with a global numerical weather prediction model (ICON at 40 km resolution). The model is extended by a stochastic convection scheme to better represent error growth from unresolved motions. It is found that current forecasts could be extended by 4-5 days through perfecting the initial conditions. Close to the intrinsic limit the dominant error growth mechanism changes: With respect to physical processes, a transition occurs from rotationally-driven initial error growth to error growth dominated by latent heat release in convection and due to the divergent component of the flow. With respect to spatial scales, a transition from large-scale up-amplitude error growth to a very rapid initial error growth on small scales is found. These results confirm that planetary-scale predictability is intrinsically limited by rapid error growth due to latent heat release in clouds through an upscale-interaction process, while this interaction process is relatively unimportant on average for current levels of initial condition uncertainty.

This talk is joint work with Michael Riemer and George Craig.

Combining physical reasoning with statistical practice in uncertainty quantification across scales

Ted Shepherd, University of Reading

An alternative approach to the ocean eddy parameterization problem

Igor Shevchenko, Imperial College London

It is typical for low-resolution ocean simulations to miss not only small- but also large-scale patterns of the flow dynamics compared with their high-resolution analogues. It is usually attributed to the inability of coarse-grid models to properly reproduce the effects of the unresolved small-scale dynamics on the resolved large scales. In part, the reason for that is that coarse-grid models fail to at least keep the coarse-grid solution within the region of phase space occupied by the reference solution (the high-resolution solution projected onto the coarse grid).

In this talk we discuss two methods to solve this problem: (1) computation of the image point in the phase space restricted to the region of the reference flow dynamics, and (2) reconstruction of a dynamical system from the available reference solution data. The proposed methods show encouraging results for both low- and high-dimensional phase spaces.

One of the important and general conclusions that can be drawn from our results is that not only mesoscale eddy parameterisation is possible in principle but also it can be highly accurate (up to reproducing individual vortices). This conclusion provides great optimism for the ongoing parameterisation studies.

This talk is joint work with Pavel Berloff.

Unstructured-mesh based non-oscillatory-forward-in-time integration for atmospheric flows

Joanna Szmelter, Loughborough University

Effective simulations of all-scale atmospheric flows, e.g., cloud-resolving global weather, involve semi-implicit integrators of non-hydrostatic governing equations. A general unstructured/hybrid mesh Non-oscillatory-Forward-in-Time (NFT) framework is applied to generate a class of numerical schemes solving either the anelastic equations of motion for the small-scale models or the compressible Euler equations under gravity on a rotating sphere. The models use the finite volume discretisation in space with a collocated arrangement for all prognostic variables. The semi-implicit NFT integration builds on the second-order-accurate high-resolution Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) and non-symmetric Krylov-subspace elliptic solver. The schemes exploit MPDATA’s properties for Implicit Large Eddy Simulation (ILES). ILES appears to be particularly useful, as it enables the representation of high Reynolds number flows without a need for explicit subgrid scale models.

To illustrate the performance of the NFT-MPDATA models, the results will be shown of several diverse calculations of internal gravity waves associated with orographic flows, the numerical characterisation of stably stratified flows past a single and multiple spheres, and a global baroclinic instability epitomising evolution of weather systems.

TIGAR - a new global atmospheric model for the simulation of transient inertia-gravity and Rossby wave dynamics: perspectives and challenges

Sergiy Vasylkevych, Universität Hamburg

TIGAR is a new intermediate complexity global atmospheric model based on hydrostatic primitive equations. The mathematical underpinning of TIGAR is normal mode decomposition of atmospheric motions, which naturally leads to the expansion of dynamics in terms of physically identifiable structures (Hough harmonics) associated with Rossby, inertia-gravity (IG), Kelvin and MRG waves.

Numerically, high precision computation is achieved in TIGAR through the use of higher order integrating factor and exponential time-differencing schemes, which take advantage of the normal modes framework, leading to the major increase in computational efficiency and stability.

In the talk, I will present the result of standard validation tests of the TIGAR’s dynamical core, and discuss the opportunities and advantages offered by the framework as well as challenges and open problems associated with its development.

This talk is joint work with Nedjeljka Žagar.

The integral effect and the intrinsic uncertainty (randomness) in dynamical systems

Jin-Song von Storch, Max-Planck Institute for Meteorology

In the context of dynamical systems, the typical uncertainties are those about the initial state, or those encountered when formulating the governing equations (since the system considered is too complex). These uncertainties result from our inability, and are not intrinsic to dynamical systems. This leads to the general belief that dynamical systems are actually deterministic. Here I show that dynamical systems are not deterministic in an absolute sense. I do so in two steps.

First, I show that for a dynamical system described by evolution equations in form of dx/dt=f, there exists something that cannot be deciphered from the responsible dynamics f. This “something” is the ultra-low-frequency variability of an equilibrium solution x(t) reached under a constant external forcing. This comes about since the time derivative operator d/dt is a high-pass filter that suppresses variations of f at near-zero frequencies and makes variations of f to completely vanish at zero frequency. The situation is consistent with the fact that on the one hand, x(t) varies into infinite times (when left alone with the same constant external forcing), and on the other hand, f operates only on finite timescales, since otherwise no equilibrium solution can be reached. Consequently, ultra-low-frequency variability of an equilibrium solution cannot be determined by variations of f at the same frequencies.

Secondly, I show that these low-frequency variations are generated by the integral forcing obtained by integrating f over time. An integral forcing consists of a dissipating component and a fluctuating component. While f and the corresponding integral forcing are equivalent in determining the solution at a time, they produce difference types of variations. In the limiting situation, when integrated over sufficiently long time, the integration completely erases the information about time sequence. The dissipating component completely wipes out the memory of the initial condition, while the fluctuating component generates white variations. The ability of an integral forcing to generate white variations is referred to as the integral effect. Not all integral forcing produces an integral effect.

In the presence of integral effect, a dynamical system is intrinsically random. The total variance of an equilibrium solution of such a system can only be determined by integrating the system forward in time.

Unconventional ways of modelling the atmosphere: implicit advection and r-adaptivity

Hilary Weller, University of Reading

This talk will describe two unconventional ways of modelling the atmosphere: 1. Implicit time stepping to take long time steps for advection 2. Moving meshes over mountains for adaptivity.

Implicit time stepping for advection is not popular in atmospheric science because of the cost of the global matrix solution and the phase errors for large Courant numbers. However, implicit advection is simple to implement, conservative on any grid structure and can exploit improvements in solver efficiency and parallelisation. I will describe an implicit version of the MPDATA advection scheme and show that it is accurate for low Courant numbers, stable for very large Courant numbers and locally high Courant numbers (such as occur over the poles of a latitude-longitude grid) do not harm the accuracy.

I will also talk about adaptivity using moving meshes. When meshes adapt over mountains, the total volume of the fluid domain can change, making conservation impossible. I will introduce a fix that does not require calculating and keeping track of moving volumes but leads to exact conservation of volume and preservation of uniform fields.

This talk is joint work with Hiroe Yamazaki, James Woodfield, Christian Kühnlein, and Piotr Smolarkiewicz.

Limitations of refractive index diagnostics for Rossby wave propagation

Volkmar Wirth, Johannes Gutenberg University

The horizontal propagation of Rossby waves over planetary-scale distances has been studied for several decades. In recent years this topic was discussed in connection with the resonant amplification of Rossby waves, which requires a good zonal waveguide for its existence. If resonant amplification were a relevant mechanism in the Earth’s atmosphere, it would have important implications for the occurrence of extreme weather and the changes to be expected in a future climate.

The current contribution presents a critical appraisal of the often-used “refractive index” as a diagnostic for Rossby wave propagation and the existence of waveguides. It is shown that a straightforward application of this diagnostic may give rise to misleading results in relevant situations. In particular, the refractive index diagnostic may suggest the existence of a waveguide where most likely there is none. The reasons are, (1) that the underlying WKB assumption may not be satisfied, and (2) that finite-amplitude effects may produce artifacts. The presentation will discuss and illuminate both aspects with the help of barotropic model simulations. Furthermore, it is suggested that ideas from finite amplitude wave activity may help to circumvent some of the above issues.

This talk is joint work with Christopher Polster.

Balance and Four-Dimensional Data Assimilation in the Tropics

Nedjeljka Žagar, UHH

Many studies of the errors in weather predictions have focused on the midlatitude, quasi-geostrophic dynamics and often considered the error-free large-scale initial state. In contrast, the largest initial uncertainties in global numerical weather prediction (NWP) models and ensemble prediction systems are found in the tropics, and they have large scales. I will discuss scale-dependent properties and balance issues of analysis and short-term forecast uncertainties in the tropics using a perfect-model observing system assimilation experiments and contrast them with a state-of-the-art ensemble prediction system. The discussion will review the two decades of the application of equatorial linear wave theory for the data assimilation modelling and diagnosis of NWP in the tropics and the role of the tropics in global predictability.