Elementary dispersion analysis of some mimetic discretizations on triangular C-grids

Spurious modes supported by triangular C-grids limit their application for modeling large-scale atmospheric and oceanic flows. Their behavior can be modified within a mimetic approach that generalizes the scalar product underlying the triangular C-grid discretization. The mimetic approach provides a discrete continuity equation which operates on an averaged combination of normal edge velocities instead of normal edge velocities proper. An elementary analysis of the wave dispersion of the new discretization for Poincaré, Rossby and Kelvin waves shows that, although spurious Poincaré modes are preserved, their frequency tends to zero in the limit of small wavenumbers, which removes the divergence noise in this limit. However, the frequencies of spurious and physical modes become close on shorter scales indicating that spurious modes can be excited unless high-frequency short-scale motions are effectively filtered in numerical codes. We argue that filtering by viscous dissipation is more efficient in the mimetic approach than in the standard C-grid discretization. Lumping of mass matrices appearing with the velocity time derivative in the mimetic discretization only slightly reduces the accuracy of the wave dispersion and can be used in practice. Thus, the mimetic approach cures some difficulties of the traditional triangular C-grid discretization but may still need appropriately tuned viscosity to filter small scales and high frequencies in solutions of full primitive equations when these are excited by nonlinear dynamics

Accuracy and stability analysis of horizontal discretizations used in unstructured grid ocean models

One important tool at our disposal to evaluate the robustness of Global Circulation Models (GCMs) is to understand the horizontal discretization of the dynamical core under a shallow water approximation. Here, we evaluate the accuracy and stability of different methods used in, or adequate for, unstructured ocean models considering shallow water models. Our results show that the schemes have different accuracy capabilities, with the A- (NICAM) and B-grid (FeSOM 2.0) schemes providing at least 1st order accuracy in most operators and time integrated variables, while the two C-grid (ICON and MPAS) schemes display more difficulty in adequately approximating the horizontal dynamics. Moreover, the theory of the inertia-gravity wave representation on regular grids can be extended for our unstructured based schemes, where from least to most accurate we have: A-, B, and C-grid, respectively. Considering only C-grid schemes, the MPAS scheme has shown a more accurate representation of inertia-gravity waves than ICON. In terms of stability, we see that both A- and C-grid MPAS scheme display the best stability properties, but the A-grid scheme relies on artificial diffusion, while the C-grid scheme doesn't. Alongside, the B-grid and C-grid ICON schemes are within the least stable. Finally, in an effort to understand the effects of potential instabilities in ICON, we note that the full 3D model without a filtering term does not destabilize as it is integrated in time. However, spurious oscillations are responsible for decreasing the kinetic energy of the oceanic currents. Furthermore, an additional decrease of the currents' turbulent kinetic energy is also observed, creating a spurious mixing, which also plays a role in the strength decrease of these oceanic currents

ICON-O: The Ocean Component of the ICON Earth System Model - Global simulation characteristics and local telescoping capability

Abstract We describe the ocean general circulation model ICON-O of the Max Planck Institute for Meteorology, which forms the ocean-sea ice component of the Earth system model ICON-ESM. ICON-O relies on innovative structure-preserving finite volume numerics. We demonstrate the fundamental ability of ICON-O to simulate key features of global ocean dynamics at both uniform and non-uniform resolution. Two experiments are analyzed and compared with observations, one with a nearly uniform and eddy-rich resolution of ?10?km and another with a telescoping configuration whose resolution varies smoothly from globally ?80?km to ?10?km in a focal region in the North Atlantic. Our results show first, that ICON-O on the nearly uniform grid simulates an ocean circulation that compares well with observations and second, that ICON-O in its telescope configuration is capable of reproducing the dynamics in the focal region over decadal time scales at a fraction of the computational cost of the uniform-grid simulation. The telescopic technique offers an alternative to the established regionalization approaches. It can be used either to resolve local circulation more accurately or to represent local scales that cannot be simulated globally while remaining within a global modeling framework

Low-Dissipation Simulation Methods and Models for Turbulent Subsonic Flow

The simulation of turbulent flows by means of computational fluid dynamics is highly challenging. The costs of an accurate direct numerical simulation (DNS) are usually too high, and engineers typically resort to cheaper coarse-grained models of the flow, such as large-eddy simulation (LES). To be suitable for the computation of turbulence, methods should not numerically dissipate the turbulent flow structures. Therefore, energy-conserving discretizations are investigated, which do not dissipate energy and are inherently stable because the discrete convective terms cannot spuriously generate kinetic energy. They have been known for incompressible flow, but the development of such methods for compressible flow is more recent. This paper will focus on the latter: LES and DNS for turbulent subsonic flow. A new theoretical framework for the analysis of energy conservation in compressible flow is proposed, in a mathematical notation of square-root variables, inner products, and differential operator symmetries. As a result, the discrete equations exactly conserve not only the primary variables (mass, momentum and energy), but also the convective terms preserve (secondary) discrete kinetic and internal energy. Numerical experiments confirm that simulations are stable without the addition of artificial dissipation. Next, minimum-dissipation eddy-viscosity models are reviewed, which try to minimize the dissipation needed for preventing sub-grid scales from polluting the numerical solution. A new model suitable for anisotropic grids is proposed: the anisotropic minimum-dissipation model. This model appropriately switches off for laminar and transitional flow, and is consistent with the exact sub-filter tensor on anisotropic grids. The methods and models are first assessed on several academic test cases: channel flow, homogeneous decaying turbulence and the temporal mixing layer. As a practical application, accurate simulations of the transitional flow over a delta wing have been performed

A split-explizit time-stepping scheme for ICON-Ocean

The development and implementation of advantageous time-stepping schemes in existing ocean models bears the potential to improve the model performance in terms of higher numerical accuracy as well lower numerical costs in terms of increased stability (larger possible time-steps). Stability and accuracy of time-stepping schemes should be considered in coupled space-time discretization. In that respect, the derivation and analysis of a new space-time discretization especially within the novel spatial framework of ICON-O (ocean component of the ICON earth system model) is of significant interest. In this thesis, adapting and implementing a split-explicit time-stepping scheme into ICON-O, we address both accuracy and stability: (a) We reduce the propagation error of barotropic signals by up to two orders of magnitude within mainly barotropic experiments. Furthermore, choosing a more advanced baroclinic time-stepping scheme results in increased accuracy of the baroclinic signal for relevant large Courant numbers. (b) The new space-time discretization shows increased numerical stability by a factor of up to 1.3 for the analysed experiments. In addition to the new split-explicit space-time discretization based on a Leap-Frog Adams-Moulthon-3 (LF-AM3) baroclinic step, we also adapt split-explicit time-stepping for the Adams-Bashfort-2 (AB2) scheme which is originally used in ICON-O together with a semi-implicit scheme. A major effort was to bring together these time- stepping schemes with the unique spatial framework of ICON-O, which is based on a C-type staggering of variables on a triangular grid. Following this spatial framework, we preserve a mass-matrix that filters out a spurious mode and furthermore fullfill discrete conservation of volume and tracers. In experiments with increasing complexity, we compare the two new split-explicit space-time discretizations with the original AB2 semi-implicit scheme. We show higher accuracy of the barotropic mode of the split-explicit schemes within various gravity wave experiments. In a lock-exchange experiment, we find for small Courant numbers that a coupling-error of both split-explicit time-stepping schemes results in smaller accuracy in velocity compared to the AB2 semi-implicit scheme. This coupling-error can be avoided with further improvements to the split-explicit algorithm. For desired large Courant numbers, the new LF-AM3 split-explicit space-time discretization is more accurate in the velocity, even for a time step that exceeds the stability limit of both AB2 schemes. Furthermore, the new LF-AM3 space-time discretization is more accurate for tracers independent of the Courant number. LF-AM3 shows slightly larger spurious mixing which we also find for smaller time steps with both AB2 schemes. We argue that this is caused by larger noise of the velocity on grid scale due to smaller numerical velocity diffusion. This results in gain of control over the total velocity diffusion when using ICON-O. Within the coupled space-time discretizations of ICON-O, the new LF-AM3 split-explicit discretization shows a stability limit that is 1.3 times larger compared to the AB2 semi-implicit and up to 1.5 times larger stability limit compared to the new AB2 split-explicit discretization for our experiments.Die Entwicklung und Implementierung von vorteilhaften Zeitschrittverfahren für vorhandene Ozeanmodelle birgt das Potenzial, deren Ergebnis hinsichtlich höherer numerischer Genauigkeit und geringerer numerischer Kosten bezogen auf erhöhte Stabilität (größere mögliche Zeitschritte) zu verbessern. Stabilität und Genauigkeit von Zeitschrittverfahren sollten im Kontext einer gekoppelten Raum-Zeit-Diskretisierung betrachtet werden. Diesbezüglich ist die Herleitung und Analyse eines neuen Zeitschrittverfahrens innerhalb der innovativen räumlichen Diskretisierung von ICON-O (Ozeankomponente des Erdsystemmodells ICON) von bedeutendem Interesse. In dieser Dissertation passen wir ein split-explizites Zeitschrittverfahren auf ICON-O an und implementieren dieses. Damit gehen wir die beiden Punkte Genauigkeit und Stabilität an: (a) Innerhalb überwiegend barotroper Experimente verringern wir den Fehler, der durch die Ausbreitung eines barotropen Signals entsteht, um bis zu zwei Größenordnungen. Zusätzlich wählen wir ein fortschrittliches Zeitschrittverfahren für den baroklinen Zeitschritt und verbessern damit das barokline Signal für die für uns relevanten, hohen Courant-Zahlen. (b) Die neue Raum-Zeit-Diskretisierung zeigt eine 1.3-fach erhöhte numerische Stabilität für die ausgewerteten Experimente. Zusätzlich zu dem neuem split-expliziten Zeitschrittverfahren, welches auf einem Leap-Frog Adams-Moulthon-3 (LF-AM3) baroklinem Zeitschritt basiert, entwickeln wir das split-explizite Zeitschrittverfahren für das in ICON-O ursprünglich mit einem semi-impliziten Zeitschritt verwendete Adams-Bashfort-2 (AB2) Verfahren. Eine der großen Leistungen dieser Arbeit war das Entwickeln dieser Zeitschrittverfahren innerhalb der besonderen r¨aumlichen Diskretisierung von ICON-O. Dieser folgend, erhalten wir die Massen-Matrix, welche eine numerische Mode aufhebt, und erfüllen diskrete Volumen- und Tracererhaltung. In Experimenten mit ansteigender Komplexität vergleichen wir die zwei neuen split-expliziten Raum-Zeit Diskretisierungen mit dem ursprünglichen semi-impliziten AB2 Verfahren. Wir zeigen die höhere Genauigkeit der barotropen Mode beider neuer split-expliziten Verfahren anhand mehrerer Experimente von Schwerewellen. In einem Lock-exchange Experiment zeigt sich, dass für kleine Courant-Zahlen beide split- expliziten Verfahren aufgrund eines Kopplungsfehlers geringere Genauigkeit in der Geschwindigkeit haben als das semi-implizite AB2 Verfahren. Dieser Kopplungsfehler kann durch weiterführende Verbesserungen des split-expliziten Verfahrens vermieden werden. Für die üblichen großen Courant-Zahlen ist die split-explizite LF-AM3 Raum-Zeit-Diskretisierung in der Geschwindigkeit genauer, sogar außerhalb des Stabilitätslimits beider AB2 Verfahren. Zusätzlich ist LF-AM3 für Tracer, unabhängig der Courant-Zahl, genauer. LF-AM3 zeigt ein etwas höheres numerisches Vermischen von Tracern, auch beobachtbar bei der Verwendung kleinerer Zeitschritte in den AB2 Verfahren. Wir begründen dies mit erhöhtem Rauschen in der Geschwindigkeit auf Gitterskala, bedingt durch eine geringere numerische Geschwindigkeitsdiffusion. Daraus ergibt sich für den Nutzer von ICON-O wiederum eine erhöhte Kontrolle über die gesamte Diffusion der Geschwindigkeit. Innerhalb der Raum-Zeit-Diskretisierung von ICON-O zeigt die neue split-explizite LF-AM3 Diskretisierung ein 1.3-fach höheres Stabilitätslimit als das semi-implizite AB2 Verfahren und bis zu 1.5-fache erhöhte Stabilität als die neue split-explizite AB2 Diskretisierung in den durchgeführten Experimenten