Recent Developments in Numerical Earth System Modelling

In weather prediction and climate modelling, numerical models of the Earth System are used extensively. For both the atmosphere and ocean components such models consist of a fluid dynamics solver (dynamical core) coupled to physical parameterizations to represent processes that occur below the grid scale (physics). Over time these models have become capable of sophisticated simulations. Research and development is constantly being undertaken to improve the accuracy and efficiency of the dynamical core, the physics, and their coupling.

This session encompasses the development, testing and application of novel numerical techniques for Earth system models, including governing equations, horizontal and vertical discretizations, structure preserving methods, time stepping schemes, advection schemes, adaptive multi-scale models, physics-dynamics coupling, regional and global models, classical and stochastic physical parameterizations (that are not covered in other sessions).

solicited authors:
Co-organized by CL5.2
Convener: Werner Bauer | Co-conveners: Christian Kühnlein, Jemma Shipton, Hiroe Yamazaki
vPICO presentations
| Tue, 27 Apr, 09:00–10:30 (CEST)

vPICO presentations: Tue, 27 Apr

Chairpersons: Werner Bauer, Jemma Shipton
Maria De La Fuente, Sandra Arndt, Tim Minshul, and Héctor Marín-Moreno

Large quantities of methane (CH4) are stored in gas hydrates at shallow depths within marine sediments. These reservoirs are highly sensitive to ocean warming and if destabilized could lead to significant CH4 release and global environmental impacts. However, the existence of such a positive feedback loop has recently been questioned as efficient CH4 sinks within the sediment-ocean continuum likely mitigate the impact of gas hydrate-derived CH4 emissions on global climate. In particular, benthic anaerobic oxidation of methane (AOM) represents an important CH4 sink capable of completely consuming CH4 fluxes before they reach the seafloor. However, the efficiency of this benthic biofilter is controlled by a complex interplay of multiphase methane transport and microbial oxidation processes and is thus highly variable (0-100%). In addition, AOM potentially enhances benthic alkalinity fluxes with important, yet largely overlooked implications for ocean pH, saturation state and CO2 emissions. As a consequence, the full environmental impact of hydrate-derived CH4 release to the ocean-atmosphere system and its feedbacks on global biogeochemical cycles and climate still remain poorly quantified. To the best our knowledge, currently available modelling tools to assess the benthic CH4 sink and its environmental impact during hydrate dissociation do not account for the full complexity of the problem. Available codes generally do not explicitly resolve the dynamics of the microbial community and thus fail to represent transient changes in AOM biofilter efficiency and windows of opportunity for CH4 escape. They also highly simplify the representation of  multiphase CH4 transport processes and gas hydrate dynamics and rarely assess the influence of hydrate-derived CH4 fluxes on benthic-pelagic alkalinity and dissolved inorganic carbon fluxes. To overcome these limitations, we have developed a novel 1D thermo-hydro-biogeochemical hydrate model that improve the quantitative understanding of the benthic CH4 sink and benthic carbon cycle-climate feedbacks in response to methane hydrate dissociation caused by temperature and sea-level perturbations. Our mathematical model builds on previous thermo-hydraulic hydrate simulators, expanding them to include the dominant microbial processes affecting CH4 fluxes in a consistent and coupled mathematical formulation. The micro-biogeochemical reaction network accounts for the main redox reactions (i.e., aerobic degradation, organoclastic sulphate reduction (OSR), methanogenesis and aerobic-anaerobic oxidation of methane (AeOM-AOM)), carbonate dissolution/precipitation and equilibrium reactions that drive biogeochemical dynamics in marine hydrate-bearing sediments . In particular, the AOM rate is expressed as a bioenergetic rate law that explicitly accounts for biomass dynamics. Finally, the model allows tracking the carbon isotope signatures of all dissolved and solid carbon species. In this talk we will present the model structure for the multiphase-multicomponent hydrate system, describe the specific constitutive and reaction equations used in the formulation, discuss the numerical strategy implemented and illustrate the potential capabilities of the model.

How to cite: De La Fuente, M., Arndt, S., Minshul, T., and Marín-Moreno, H.: A novel 1D thermo-hydro-biogeochemical hydrate model to assess the full benthic environmental impact of methane gas hydrate dissociation, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-1093,, 2021.

New MESSy scavenging subroutine to treat aerosol particles gas-phase partitioning in convective clouds
Giorgio Taverna, Marc Barra, and Holger Tost
Roman Nuterman, Dion Häfner, and Markus Jochum

Until recently, our pure Python, primitive equation ocean model Veros 
has been about 1.5x slower than a corresponding Fortran implementation. 
But thanks to a thriving scientific and machine learning library 
ecosystem, tremendous speed-ups on GPU, and to a lesser degree CPU, are 
within reach. Leveraging Google's JAX library, we find that our Python 
model code can reach a 2-5 times higher energy efficiency on GPU 
compared to a traditional Fortran model.

Therefore, we propose a new generation of geophysical models: One that 
combines high-level abstractions and user friendliness on one hand, and 
that leverages modern developments in high-performance computing and 
machine learning research on the other hand.

We discuss what there is to gain from building models in high-level 
programming languages, what we have achieved in Veros, and where we see 
the modelling community heading in the future.

How to cite: Nuterman, R., Häfner, D., and Jochum, M.: Higher-level geophysical modelling, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-2127,, 2021.

Heena Patel, Konrad Simon, and Jörn Behrens

Urban canopies consist of buildings and trees that are aligned along a street in the horizontal direction. These canopies in cities and forests modulate the local climate considerably in a complex way. Canopies constitute very fine subgrid features that actually have a significant impact on other components of earth system models but their feedbacks on larger scales are by now represented in rather heuristic ways. The problem in simulating their impact is twofold: First, their local modeling is delicate and, secondly, the numerical modeling of the scale interaction between fine and large scales is complicated since the fine scale structure is global. We will mostly focus on the second aspect.


Multiscale finite element methods (MsFEM) in their classical form have been applied to various porous media problems but the situation in climate, and hence flow-dominated regimes is different from porous media applications. In order to study the effect of various parameters like the concentration of pollutants, or the dynamics of the background velocity and of the temperature in the atmospheric boundary layer, a semi-Lagrangian reconstruction based multiscale finite element framework (SLMsR) developed by [1, 2] for passive tracer transport modeled by an advection-diffusion equation with high-contrast oscillatory diffusion is applied.


These methods are composed of two parts: a local-in-time semi-Lagrangian offline phase that pre-computes basis functions and an online phase that uses these basis functions to compute the solution on a coarse Eulerian simulation mesh. The overhead of pre-computing the basis functions in each coarse block can further be reduced by parallelization. The online phase is approximately as fast as a low resolution standard FEM but using the modified basis that carries subgrid information still allows to reveal the fine scale features of a highly resolved solution and is therefore accurate. This approach is studied in order to reveal the feedback of processes in the canopy layer on different scales present in climate simulation models and in particular on the atmospheric boundary layer.


We will show the results of massively parallel simulations for passive tracer transport in an urban region using the new multiscale approach and compare them to classical approaches.

References :

[1] Simon, Konrad, and Jörn Behrens. "Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems.", Springer Journal of Scientific Computing (JOMP) (provisionally accepted), 2019

[2] Simon, Konrad, and Jörn Behrens. "Multiscale Finite Elements for Transient Advection-Diffusion Equations through Advection-Induced Coordinates.", Multiscale Modeling & Simulation 18.2 (2020): 543-571.

How to cite: Patel, H., Simon, K., and Behrens, J.: Massively Parallel Multiscale Simulations of the Feedback of Urban Canopies, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-2507,, 2021.

Martin Schreiber

Running simulations on high-performance computers faces new challenges due to e.g. the stagnating or even decreasing per-core speed. This poses new restrictions and therefore challenges on solving PDEs within a particular time frame in the strong scaling case. Here, disruptive mathematical reformulations, which e.g. exploit additional degrees of parallelism also along the time dimension, gained increasing interest over the last two decades.

This talk will cover various examples of our current research on (parallel-in-)time integration methods in the context of weather and climate simulations such as rational approximation of exponential integrators, multi-level time integration of spectral deferred correction (PFASST) as well as other methods.

These methods are realized and studied with numerics similar to the ones used by the European Centre for Medium-Range Weather Forecasts (ECMWF). Our results motivate further investigation for operational weather/climate systems in order to cope with the hardware imposed restrictions of future super computer architectures.

I gratefully acknowledge contributions and more from Jed Brown, Francois Hamon, Terry S. Haut, Richard Loft, Michael L. Minion, Pedro S. Peixoto, Nathanaël Schaeffer, Raphael Schilling

How to cite: Schreiber, M.: Next-Generation Time Integration targeting Weather and Climate Simulations, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-2734,, 2021.

James Kent

GungHo is the mixed finite-element dynamical core under development by the Met Office. A key component of the dynamical core is the transport scheme, which advects density, temperature, moisture, and the winds, throughout the atmosphere. Transport in GungHo is performed by finite-volume methods, to ensure conservation of certain quantaties. There are a range of different finite-volume schemes being considered for transport, including the Runge-Kutta/method-of-lines and COSMIC/Lin-Rood schemes. Additional horizontal/vertical splitting approaches are also under consideration, to improve the stability aspects of the model. Here we discuss these transport options and present results from the GungHo framework, featuring both prescribed velocity advection tests and full dry dynamical core tests. 

How to cite: Kent, J.: Transport Schemes in GungHo, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-2743,, 2021.

Stephanie Westerhuis and Oliver Fuhrer

Fog and low stratus pose a major challenge for numerical weather prediction (NWP) models. Despite high resolution in the horizontal (~1 km) and vertical (~20 m), operational NWP models often fail to accurately predict fog and low stratus. This is a major issue at airports which require visibility predictions, or for energy agencies estimating day-ahead input into the electrical grid from photovoltaic power.

Most studies dedicated to fog and low stratus forecasts have focused on the physical parameterisations or grid resolutions. We illustrate how horizontal advection at the cloud top of fog and low stratus in a grid with sloping vertical coordinates leads to spurious numerical diffusion and subsequent erroneous dissipation of the clouds. This cannot be prevented by employing a higher-order advection scheme. After all, the formulation of the terrain-following vertical coordinate plays a crucial role in regions which do not exhibit perfectly flat orography. We suggest a new vertical coordinate formulation which allows for a faster decay of the orographic signal with altitude and present its positive impact on fog and low stratus forecasts. Our experiments indicate that smoothing of the vertical coordinates at low altitudes is a crucial measure to prevent premature dissipation of fog and low stratus in high-resolution NWP models.

How to cite: Westerhuis, S. and Fuhrer, O.: A new vertical coordinate formulation to improve forecasts of fog and low stratus in high-resolution numerical weather prediction models, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-4429,, 2021.

Inherent dissipation of upwind-biased potential temperature advection and its feedback on model dynamics
Almut Gaßmann
Kevlahan Nicholas

This talk introduces WAVETRISK-OCEAN, an incompressible version of the atmosphere model WAVETRISK.  This new model is built on the same wavelet-based dynamically adaptive core as WAVETRISK, which itself uses DYNAMICO's mimetic vector-invariant multilayer shallow water formulation. Both codes use a Lagrangian vertical coordinate with conservative remapping.  The ocean variant solves the incompressible multilayer shallow water equations with a Ripa type thermodynamic treatment of horizontal density gradients.  Time integration uses barotropic-baroclinic mode splitting via an implicit free surface formulation, which is about 15 times faster than explicit time stepping.  The barotropic and baroclinic estimates of the free surface are reconciled at each time step using layer dilation. No slip boundary conditions at coastlines are approximated using volume penalization.  Results are presented for a standard set of ocean model test cases adapted to the sphere (seamount,  upwelling and baroclinic jet) as well as  turbulent wind-driven gyre flow in simplified geometries.  An innovative feature of WAVETRISK-OCEAN is that it could be coupled easily to the WAVETRISK atmosphere model, providing a simple integrated Earth system model using a consistent modelling framework.

How to cite: Nicholas, K.: WAVETRISK-OCEAN: an adaptive dynamical core for ocean modelling, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-6293,, 2021.

Rüdiger Brecht, Long Li, Werner Bauer, and Etienne Mémin

We introduce a new representation of the rotating shallow water equations based on a stochastic transport principle. The derivation relies on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved small-scale flow. The total energy of such a random model is demonstrated to be preserved along time for any realization. Thus, we propose to combine a structure-preserving discretization of the underlying deterministic model with the discrete stochastic terms. This way, our method can directly be used in existing dynamical cores of global numerical weather prediction and climate models. For an inviscid test case on the f-plane we use a homogenous noise and illustrate that the spatial part of the stochastic scheme preserves the total energy of the system. Finally, using an inhomogenous noise, we show  that the proposed random model better captures the structure of a large-scale flow than a comparable deterministic model for a barotropically unstable jet on the sphere.

How to cite: Brecht, R., Li, L., Bauer, W., and Mémin, E.: Rotating shallow water flow under location uncertainty with a structure-preserving discretization, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-7539,, 2021.

Werner Bauer and Colin Cotter

We introduce a higher order phase averaging method for nonlinear PDEs. Our method is suitable for highly oscillatory systems of nonlinear PDEs that generate slow motion through resonance between fast frequencies, such as is the case for rotating fluids with small but finite Rossby number. Phase averaging is a technique to filter fast motions from the dynamics whilst still accounting for their effect on the slow dynamics. In the small Rossby number limit of the phase averaged rotating shallow water equations, one recovers the quasi-geostrophic equations (as shown by Schochet, Majda and others). Peddle et al. 2017, Haut and Wingate 2014, have shown that phase averaging at finite Rossby number allows to take larger timesteps than would otherwise be possible. This was used as a coarse propagator (large timesteps at lower accuracy) for a Parareal method where corrections were made using a standard timestepping method with small timesteps.

In this contribution, we introduce an additional phase variable in the exponential time integrator that allows us to derive arbitrary order averaging methods that can be used as more accurate corrections to the basic phase averaged model, without needing small timesteps. We envisage their use as part of a time-parallel algorithm based on deferred corrections to the basic average. We illustrate the properties of this method on an ODE that describes the dynamics of a swinging spring, a model due to Peter Lynch. Although idealized, this model shows an interesting analogy to geophysical flows as it exhibits a high sensitivity of small scale oscillation on the large scale dynamics. On this example, we show convergence to the non-averaged (exact) solution with increasing approximation order also for finite averaging windows. At zeroth order, our method coincides with that in Peddle et al. 2017, Haut and Wingate 2014, but at higher order it is more accurate in the sense that it better approximates the faster oscillations around the slow manifold.

How to cite: Bauer, W. and Cotter, C.: Higher order phase averaging for big timesteps, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-7543,, 2021.

Jean-Pierre Croisille, Jean-Baptiste Bellet, and Matthieu Brachet

The Cubed Sphere is a grid commonly used in numerical simulation in climatology. In this talk we present recent progress
on the algebraic and geometrical properties of this highly symmetrical grid.
First, an analysis of the symmetry group of the Cubed Sphere will be presented: this group 
is identified as the group of the Cube, [1]. Furthermore, we show how to construct a discrete Spherical Harmonics (SH) basis associated to 
the Cubed Sphere. This basis displays a truncation scheme relating the zonal and longitudinal 
mode numbers reminiscent of the rhomboidal truncation on the Lon-Lat grid.
The new analysis allows to derive new quadrature rules of  interest for applications in any kind of spherical modelling. In addition,
we will comment on applications in mathematical climatology and meteorology, [2].

[1] J.-B. Bellet, Symmetry group of the equiangular Cubed Sphere, preprint, IECL, Univ. Lorraine, 2020, submitted

[2] J.-B. Bellet, M. Brachet and J.-P. Croisille, Spherical Harmonics on The Cubed Sphere, IECL, Univ. Lorraine, 2021, Preprint.

How to cite: Croisille, J.-P., Bellet, J.-B., and Brachet, M.: Interpolating data on the Cubed Sphere with Spherical Harmonics, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-7990,, 2021.

Hilary Weller, James Woodfield, and Christian Kuehnlein

Semi-Lagrangian advection schemes are accurate and efficient and retain accuracy and stability even for large Courant numbers but are not conservative. Flux-form semi-Lagrangian is conservative and in principle can be used to achieve large Courant numbers. However this is complicated and would be prohibitively expensive on grids that are not logically rectangular. 

Strong winds or updrafts can lead to localised violations of Courant number restrictions which can cause a model with explicit Eulerian advection to crash. Schemes are needed that remain stable in the presence of large Courant numbers. However accuracy in the presence of localised large Courant numbers may not be so crucial.

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 (once appropriate matrix solvers are available) and is conservative on any grid structure and can exploit improvements in solver efficiency and parallelisation. This talk will describe an implicit version of the MPDATA advection scheme and show results of linear advection test cases. To optimise accuracy and efficiency, implicit time stepping is only used locally where needed. This makes the matrix inversion problem local rather than global. With implicit time stepping MPDATA retains positivity, smooth solutions and accuracy in space and time.

How to cite: Weller, H., Woodfield, J., and Kuehnlein, C.: Long Time Steps for Advection: MPDATA with implicit time stepping, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-8815,, 2021.

Vladimir Gryanik, Christof Luepkes, Andrey Grachev, and Dmitry Sidorenko

Results of weather forecast, present-day climate simulations and future climate projections depend among other factors on the interaction between the atmosphere and the underlying sea-ice, the land and the ocean. In numerical weather prediction and climate models some of these interactions are accounted for by transport coefficients describing turbulent exchange of momentum, heat and moisture. Currently used transfer coefficients have, however, large uncertainties in flow regimes being typical for cold nights and seasons, but especially in the polar regions. Furthermore, their determination is numerically complex. It is obvious that progress could be achieved when the transfer coefficients would be given by simple mathematical formulae in frames of an economic computational scheme. Such a new universal, so-called non-iterative parametrization scheme is derived for a package of transfer coefficients.

The derivation is based on the Monin-Obukhov similarity theory, which is over the years well accepted in the scientific community. The newly derived non-iterative scheme provides a basis for a cheap systematic study of the impact of near-surface turbulence and of the related transports of momentum, heat and moisture in NWP and climate models. 

We show that often used transfer coefficients like those of Louis et al. (1982) or of Cheng and Brutsaert (2005) can be applied at large stability only with some caution, keeping in mind that at large stability they significantly overestimate the transfer coefficient compared with most comprehensive measurements. The latter are best reproduced by Gryanik et al. (2020) functions, which are part of the package. We show that the new scheme is flexible, thus, new stability functions can be added to the package, if required.


Gryanik, V.M., Lüpkes, C., Grachev, A., Sidorenko, D. (2020) New Modified and Extended Stability Functions for the Stable Boundary Layer based on SHEBA and Parametrizations of Bulk Transfer Coefficients for Climate Models, J. Atmos. Sci., 77, 2687-2716

How to cite: Gryanik, V., Luepkes, C., Grachev, A., and Sidorenko, D.: A Package of New Universal Non-Iterative Parametrizations for Stable Surface Layer Transfer Coefficients of Momentum, Heat and Moisture in Numerical Earth System Models, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-8850,, 2021.

Fabienne Schmid, Rupert Klein, Elena Gagarina, and Ulrich Achatz

This study introduces an efficient modeling framework for investigations of diabatic flows in the atmosphere. In particular, the spontaneous emission of inertia-gravity waves is addressed in idealized simulations of baroclinic-wave life cycles. Numerical simulations are perfomed using a finite-volume solver for the pseudo-incompressible equations on the f-plane with newly implemented semi-implicit time stepping scheme, adjusted to the staggered grid, which provides high stability and efficiency for long simulation runs with large domains. Furthermore, we have modified the entropy equation to include a heat source, allowing for a development of the vertically dependent reference atmosphere. Numerical experiments of several benchmarks are compared against an explicit third-order Runge-Kutta scheme as well as numerical models from the literature, verifying the accuracy and efficiency of the scheme. The proposed framework serves as a construction basis for an efficient simulation tool for the development and validation of a parameterization scheme for gravity-waves emitted from jets and fronts.

How to cite: Schmid, F., Klein, R., Gagarina, E., and Achatz, U.: A semi-implicit pseudo-incompressible flow solver for diabatic dynamics: Baroclinic-wave life cycles , EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-9204,, 2021.

Ryan Holmes, Jan Zika, Stephen Griffies, Andrew Hogg, Andrew Kiss, and Matthew England

Numerical mixing, the physically spurious diffusion of tracers due to the numerical discretization of advection, is known to contribute to biases in ocean circulation models. However, quantifying numerical mixing is non-trivial, with most studies utilizing specifically targeted experiments in idealized settings. Here, we present a precise method based on water-mass transformation for quantifying numerical mixing, including its spatial structure, that can be applied to any conserved variable in global general circulation ocean models. The method is applied to a suite of global MOM5 ocean-sea ice model simulations with differing grid spacings and sub-grid scale parameterizations. In all configurations numerical mixing drives across-isotherm heat transport of comparable magnitude to that associated with explicitly-parameterized mixing. Numerical mixing is prominent at warm temperatures in the tropical thermocline, where it is sensitive to the vertical diffusivity and resolution. At colder temperatures, numerical mixing is sensitive to the presence of explicit neutral diffusion, suggesting that much of the numerical mixing in these regions acts as a proxy for neutral diffusion when it is explicitly absent. Comparison of equivalent (with respect to vertical resolution and explicit mixing parameters) 1/4-degree and 1/10-degree horizontal resolution configurations shows only a modest enhancement in numerical mixing at the eddy-permitting 1/4-degree resolution. Our results provide a detailed view of numerical mixing in ocean models and pave the way for future improvements in numerical methods.

How to cite: Holmes, R., Zika, J., Griffies, S., Hogg, A., Kiss, A., and England, M.: The geography of numerical mixing in a suite of global ocean models, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-9319,, 2021.

Camilla Fiorini, Long Li, and Étienne Mémin

In this work we consider the surface quasi-geostrophic (SQG) system under location uncertainty (LU) and propose a Milstein-type scheme for these equations. The LU framework, first introduced in [1], is based on the decomposition of the Lagrangian velocity into two components: a large-scale smooth component and a small-scale stochastic one. This decomposition leads to a stochastic transport operator, and one can, in turn, derive the stochastic LU version of every classical fluid-dynamics system. 

    SQG is a simple 2D oceanic model with one partial differential equation, which models the stochastic transport of the buoyancy, and an operator which relies the velocity and the buoyancy.

    For this kinds of equations, the Euler-Maruyama scheme converges with weak order 1 and strong order 0.5. Our aim is to develop higher order schemes in time: the first step is to consider Milstein scheme, which improves the strong convergence to the order 1. To do this, it is necessary to simulate or estimate the Lévy area [2].

    We show with some numerical results how the Milstein scheme is able to capture some of the smaller structures of the dynamic even at a poor resolution. 


[1] E. Mémin. Fluid flow dynamics under location uncertainty. Geophysical & Astrophysical Fluid Dynamics, 108.2 (2014): 119-146. 

[2] J. Foster, T. Lyons and H. Oberhauser. An optimal polynomial approximation of Brownian motion. SIAM Journal on Numerical Analysis 58.3 (2020): 1393-1421.

How to cite: Fiorini, C., Li, L., and Mémin, É.: Higher order schemes in time for the surface quasi-geostrophic system under location uncertainty, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-10935,, 2021.

Peter Rutkevich, Georgy Golitsyn, and Anatoly Tur

Large-scale instability in incompressible fluid driven by the so called Anisotropic Kinetic Alpha (AKA) effect satisfying the incompressible Navier-Stokes equation with Coriolis force is considered. The external force is periodic; this allows applying an unusual for turbulence calculations mathematical method developed by Frisch et al [1]. The method provides the orders for nonlinear equations and obtaining large scale equations from the corresponding secular relations that appear at different orders of expansions. This method allows obtaining not only corrections to the basic solutions of the linear problem but also provides the large-scale solution of the nonlinear equations with the amplitude exceeding that of the basic solution. The fluid velocity is obtained by numerical integration of the large-scale equations. The solution without the Coriolis force leads to constant velocities at the steady-state, which agrees with the full solution of the Navier-Stokes equation reported previously. The time-invariant solution contains three families of solutions, however, only one of these families contains stable solutions. The final values of the steady-state fluid velocity are determined by the initial conditions. After account of the Coriolis force the solutions become periodic in time and the family of solutions collapses to a unique solution. On the other hand, even with the Coriolis force the fluid motion remains two-dimensional in space and depends on a single spatial variable. The latter fact limits the scope of the AKA method to applications with pronounced 2D nature. In application to 3D models the method must be used with caution.

[1] U. Frisch, Z.S. She and P. L. Sulem, “Large-Scale Flow Driven by the Anisotropic Kinetic Alpha Effect,” Physica D, Vol. 28, No. 3, 1987, pp. 382-392.

How to cite: Rutkevich, P., Golitsyn, G., and Tur, A.: Coriolis force influence on the AKA effect, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-13421,, 2021.

Takeshi Enomoto and Koji Ogasawara

Radial basis functions enable the use of unstructured quasi-uniform nodes on the sphere. Iteratively generated nodes such as the minimum energy nodes may not converge due to exponentially increasing local minima as the number of nodes grows. By contrast, deterministic nodes, such as those made with a spherical helix, are fast to generate and have no arbitrariness. It is noteworthy that the spherical helix nodes are more uniform on the sphere than the minimum energy nodes. Semi-Lagrangian and Eulerian models are constructed using radial basis functions and validated in a standard advection test of a cosine bell by the solid body rotation. With Gaussian radial basis functions, the semi-Lagrangian model found produces significantly smaller error than the Eulerian counterpart in addition to approximately three times longer time step for the same error. Moreover, the ripple-like noise away from the cosine bell found in the Eulerian model is significantly reduced in the semi-Lagrangian model. It is straightforward to parallelize the matrix–vector multiplication in the time integration. In addition, an iterative solver can be applied to calculate the inverse of the interpolation matrix, which can be made sparse.

How to cite: Enomoto, T. and Ogasawara, K.: Semi-Lagrangian advection models for quasi-uniform nodes on the sphere, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-13687,, 2021.

Long Li, Bruno Deremble, Noé Lahaye, and Etienne Mémin

In this work, a stochastic representation [Bauer2020a, Bauer2020b] based on a physical transport principle is proposed to account for mesoscale eddy effects on the the large-scale oceanic circulation. This stochastic framework [Mémin2014] arises from a decomposition of the Lagrangian velocity into a time-smooth component and a highly oscillating noise term. One important characteristic of this random model is that it conserves the energy of any transported tracer. Such an energy-preserving representation has been successfully implemented in a well established multi-layered quasi-geostrophic dynamical core ( The empirical spatial correlation of the small-scale noise is estimated from the eddy-resolving simulation data. In particular, a sub-grid correction drift has been introduced in the noise due to the bias ensuing from the coarse-grained procedure. This non intuitive term seems quite important in reproducing on a coarse mesh the meandering jet of the wind-driven double-gyre circulation. In addition, a new projection method has been proposed to constrain the noise living along the iso-surfaces of the vertical stratification. The resulting noise enables us to improve the intrinsic low-frequency variability of the large-scale current. From some statistical studies and energy transfers analysis, this improvement is well demonstrated.

  • [Bauer2020a] W. Bauer, P. Chandramouli, B. Chapron, L. Li, and E. Mémin. Deciphering the role of small-scale inhomogeneity on geophysical flow structuration: a stochastic approach. Journal of Physical Oceanography, 50(4):983-1003, 2020a.       
  • [Bauer2020b] W. Bauer, P. Chandramouli, L. Li, and E. Mémin. Stochastic representation of mesoscale eddy effects in coarse-resolution barotropic models. Ocean Modelling, 151:101646 (2020b).    
  • [Mémin2014] E. Mémin. Fluid flow dynamics under location uncertainty. Geophysical & Astrophysical Fluid Dynamics, 108(2):119-146, 2014.     

How to cite: Li, L., Deremble, B., Lahaye, N., and Mémin, E.: Stochastic modeling of the oceanic mesoscale eddies, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-15010,, 2021.