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Implicit time-stepping for advection is applied locally in space and time where Courant numbers are large, but standard explicit time-stepping is used for the remaining solution which is typically the majority. This adaptively implicit advection scheme facilitates efficient and robust integrations with long time-steps while having negligible impact on the overall accuracy, and achieving monotonicity and local conservation on general meshes. A novel and important aspect for the efficiency of the approach is that only one linear solver iteration is needed for each advection solve. The implementation in this paper uses a second-order Runge-Kutta implicit/explicit time-stepping in combination with a second/third-order finite volume spatial discretisation. We demonstrate the adaptively implicit advection in the context of deformational flow advection on the sphere and a fully compressible model for atmospheric flows. Tracers are advected over the poles of latitude-longitude grids with very large Courant numbers and through hexagonal and cubed-sphere meshes with the same algorithm. Buoyant flow simulations with strong local updrafts also benefit from adaptively implicit advection. Stably stratified flow simulations require a stable combination of implicit treatment of gravity and acoustic waves as well as advection in order to achieve long stable time-steps., Comment: 40 pages, 18 figures

Semi-implicit time-stepping schemes for atmosphere and ocean models require elliptic solvers that work efficiently on modern supercomputers. This paper reports our study of the potential computational savings when using mixed precision arithmetic in the elliptic solvers. The essential components of a representative elliptic solver are run at precision levels as low as half (16 bits), and accompanied with a detailed evaluation of the impact of reduced precision on the solver convergence and the solution quality. A detailed inquiry into reduced precision requires a model configuration that is meaningful but cheaper to run and easier to evaluate than full atmosphere/ocean models. This study is therefore conducted in the context of a novel semi-implicit shallow-water model on the sphere, purposely designed to mimic numerical intricacies of modern all-scale weather and climate (W&C) models with the numerical stability independent on celerity of all wave motions. The governing algorithm of the shallow-water model is based on the non-oscillatory MPDATA methods for geophysical flows, whereas the resulting elliptic problem employs a strongly preconditioned non-symmetric Krylov-subspace solver GCR, proven in advanced atmospheric applications. The classical longitude/latitude grid is deliberately chosen to retain the stiffness of global W&C models posed in thin spherical shells as well as to better understand the performance of reduced-precision arithmetic in the vicinity of grid singularities. Precision reduction is done on a software level, using an emulator. The reduced-precision experiments are conducted for established dynamical-core test-cases, like the Rossby-Haurwitz wave number 4 and a zonal orographic flow. The study shows that selected key components of the elliptic solver, most prominently the preconditioning, can be performed at the level of half precision., Comment: 38 pages, 12 figures; to be submitted to the Journal of Computational Physics (JCP)

A numerical study of stably stratified flows past spheres at Reynolds numbers $Re=200$ and $Re=300$ is reported. In these flow regimes, a neutrally stratified laminar flow induces distinctly different near-wake features. However, the flow behaviour changes significantly as the stratification increases and suppresses the scale of vertical displacements of fluid parcels. Computations for a range of Froude numbers $Fr\in [0.1,\infty]$ show that as Froude number decreases, the flow patterns for both Reynolds numbers become similar. The representative simulations of the lee-wave instability at $Fr=0.625$ and the two-dimensional vortex shedding at $Fr=0.25$ regimes are illustrated for flows past single and tandem spheres, thereby providing further insight into the dynamics of stratified flows past bluff bodies. In particular, the reported study examines the relative influence of viscosity and stratification on the dividing streamline elevation, wake structure and flow separation. The solutions of the Navier-Stokes equations in the incompressible Boussinesq limit are obtained on unstructured meshes suitable for simulations involving multiple bodies. Computations are accomplished using the finite volume, non-oscillatory forward-in-time (NFT) Multidimensional Positive Definite Transport Algorithm (MPDATA) based solver. The impact and validity of the numerical approximations, especially for the cases exhibiting strong stratification, are also discussed. Qualitative and quantitative comparisons with available laboratory experiments and prior numerical studies confirm the validity of the numerical approach.

The work extends the computational model EULAG-MHD to include Hall magnetohydrodynamics (HMHD)---important to explore physical systems undergoing fast magnetic reconnection at the order of the ion inertial length scale. Examples include solar transients along with reconnections in magnetosphere, magnetotail and laboratory plasmas. The paper documents the results of two distinct sets of implicit large-eddy simulations in the presence and absence of the Hall forcing term, initiated with an unidirectional sinusoidal magnetic field. The HMHD simulation while benchmarking the code also emphasizes the complexity of three dimensional (3D) evolution over its two dimensional (2D) counterpart. The magnetic reconnections onset significantly earlier in HMHD. Importantly, the magnetic field generated by the Hall term breaks any inherent symmetry, ultimately making the evolution 3D. The resulting 3D reconnections develop magnetic flux ropes and magnetic flux tubes. Projected on the reconnection plane, the ropes and tubes appear as magnetic islands, which later break into secondary islands, and finally coalesce to generate an X-type neutral point. These findings are in agreement with the theory and contemporary simulations of HMHD, and thus verify our extension of the EULAG-MHD model. The second set explores the influence of the Hall forcing on generation and ascend of a magnetic flux rope from sheared magnetic arcades---a novel scenario instructive in understanding the coronal transients. The rope evolves through intermediate complex structures, ultimately breaking locally because of reconnections. Interestingly, the breakage occurs earlier in the presence of the Hall term, signifying faster dynamics leading to magnetic topology favorable for reconnections., Comment: Accepted for publication in ApJ

It is tested whether machine learning methods can be used for preconditioning to increase the performance of the linear solver -- the backbone of the semi-implicit, grid-point model approach for weather and climate models. Embedding the machine-learning method within the framework of a linear solver circumvents potential robustness issues that machine learning approaches are often criticized for, as the linear solver ensures that a sufficient, pre-set level of accuracy is reached. The approach does not require prior availability of a conventional preconditioner and is highly flexible regarding complexity and machine learning design choices. Several machine learning methods are used to learn the optimal preconditioner for a shallow-water model with semi-implicit timestepping that is conceptually similar to more complex atmosphere models. The machine-learning preconditioner is competitive with a conventional preconditioner and provides good results even if it is used outside of the dynamical range of the training dataset., Comment: To be submitted to GRL, 15 pages, 3 figures

Various models of solar subsurface stratification are tested in the global EULAG-MHD solver to simulate diverse regimes of near-surface convective transport. Sub- and superadiabacity are altered at the surface of the model ($ r > 0.95~R_{\odot}$) to either suppress or enhance convective flow speeds in an effort to investigate the impact of the near-surface layer on global dynamics. A major consequence of increasing surface convection rates appears to be a significant alteration of the distribution of angular momentum, especially below the tachocline where the rotational frequency predominantly increases at higher latitudes. These hydrodynamic changes correspond to large shifts in the development of the current helicity in this stable layer ($r<0.72R_{\odot}$), significantly altering its impact on the generation of poloidal and toroidal fields at the tachocline and below, acting as a major contributor towards transitions in the dynamo cycle. The enhanced near-surface flow speed manifests in a global shift of the toroidal field ($B_{\phi}$) in the butterfly diagram - from a North-South symmetric pattern to a staggered anti-symmetric emergence.

Various models of solar sub-surface stratification are tested in the global EULAG-MHD solver to simulate diverse regimes of near-surface convective transport. Sub- and superadiabacity are altered at the surface of the model ($ r > 0.95~R_{\odot}$) to either suppress or enhance convective flow speeds in an effort to investigate the impact of the near-surface layer on global dynamics. A major consequence of increasing surface convection rates appears to be a significant alteration of the distribution of angular momentum, especially below the tachocline where the rotational frequency predominantly increases at higher latitudes. These hydrodynamic changes correspond to large shifts in the development of the current helicity in this stable layer ($r<0.72R_{\odot}$), significantly altering its impact on the generation of poloidal and toroidal fields at the tachocline and below, acting as a major contributor towards transitions in the dynamo cycle. The enhanced near-surface flow speed manifests in a global shift of the toroidal field ($B_{\phi}$) in the butterfly diagram -- from a North-South symmetric pattern to a staggered anti-symmetric emergence., Comment: The wrong article was accidentally submitted, it will be resubmitted in its full form along with a link to the published article in ApJ, so that it can be more easily linked to NASA ADS

This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA). The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include: homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; shallow-water system compared with analytical solution (originally derived for a 2D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensioanl array containers, and is released as free/libre and open-source software., Comment: Among the changes are: new chapter 5.4, updated presentation in chapter 6.3 and editorial corrections