Finite Elemente gleicher Ordnung von hydrostatischen Ströungsproblemen

Subject of this thesis is the issue of equal-order finite element discretization of hydrostatic flow problems. These flow problems typically arise in geophysical fluid dynamics on large scales and in flat domains. This small aspect ratio between the depth and the horizontal extents of the considered domain allows to efficiently reduce the complexity of the incompressible three dimensional Navier-Stokes equations, which form the basis of geophysical flows. In the resulting set of equations, the vertical momentum equation is replaced by the hydrostatic balance, which thus decouples the vertical pressure variations from the dynamic system, and the dynamically relevant pressure becomes two dimensional. Moreover, the vertical velocity component can be explicitely determined by the horizontal velocity components. Concomitant with this reduction is the replacement of the divergence constraint by a suitably modified version of it. As in the classical framework, it is known that these hydrostatic flow problems also show a saddle point structure, and there is a similar uncertainty concerning existence and uniqueness of solutions as is apparent for the classical case. Although the variational framework has been intensively treated, the issue of the discretization, in particular the finite element discretization of hydrostatic problems has hardly been considered yet. The present work dedicates to this topic. We indicate the tight relation between a finite element discretized hydrostatic flow problem and its two dimensional counterpart with respect to inf-sup stability. Moreover, we elaborate stabilization techniques in order to result to inf-sup stable schemes and to suitably treat the case of dominant advection. For each of these cases we can draw on classical stabilization schemes. For the isotropic hydrostatic Stokes problem we thus derive and examine residual-based as well as symmetric stabilization schemes. In the appropriate Oseen case we restrict to symmetric stabilization schemes. Beside the isotropic case, we also consider hydrostatic problems on vertical anisotropic meshes, i.e. although the mesh may be anisotropic, the surface mesh still shows isotropic structure. Therefore we derive an interpolation operator, which has suitable projection and stability properties in three dimensions. An appropriate operator for the two dimensional case for bilinear finite element spaces has been developed in Braack06. In this vertical anisotropic context we restrict to symmetric stabilizat on schemes for both problems, the hydrostatic Stokes and the hydrostatic Oseen problem. Further, we also examine the hydrostatic Stokes problem on meshes with anisotropy occurring also in the surface mesh. This may be necessary in regions with strong flows in one horizontal direction, e.g. in the Bering strait or along coastlines. In a following chapter we shortly discuss on the time discretization approach, particularly on the issue of pressure correction schemes. These schemes are discussed already in a couple of works for classical flow problems. But a proper analysis is still missing. Finally, after considering algorithmic aspects, which also includes the topic of parallelization, we numerically validate our theoretical results and numerically illustrate apparent physical phenomena occurring in ocean circulation regimes.Die vorliegende Arbeit widmet sich der Thematik der Diskretisierung von hydrostatischen Strömungsproblemen mittels Finiter Elemente gleicher Ordnung. Hydrostatische Strömungsprobleme treten typischerweise im Bereich der geophysikalischen Fluiddynamik auf grossen Skalen und in flachen Gebieten auf. Mathematische Grundlage bilden die inkompressiblen dreidimensionalen (3D) Navier-Stokes Gleichungen. Das kleine Aspektverhältnis zwischen der Gebietstiefe und der horizontalen Ausdehnung des Gebietes erlaubt es, die Komplexität der inkompressiblen 3D Navier-Stokes Gleichungen merkbar zu reduzieren. Anwendung der sogenannten hydrostatischen Approximation, welches das kleine Aspektverhältnis ausnutzt, führt dazu, dass die vertikale Gleichung der Impulserhaltung durch die hydrostatische Balance ersetzt wird. Dadurch wird der dynamisch relevante Druck zweidimensional (2D) und die vertikale Geschwindigkeit bestimmt sich direkt aus den horizontalen. Einhergehend mit dieser Reduktion ist eine Modifikation der Bedingung der Divergenzfreiheit. Das resultierende hydrostatische Strömungsproblem weist bekanntermaßen eine Sattelpunktstruktur auf, ähnlich dem klassischen Problem. Desweiteren herrscht auch im hydrostatischen Kontext eine ähnliche Unsicherheit bezüglich Existenz und Eindeutigkeit von Lösungen vor, wie sie auch in der klassischen Navier-Stokes-Thematik anzutreffen ist. Obwohl hydrostatische Probleme im variationellen Rahmen intensiv untersucht worden sind und werden, ist das Feld der Diskretisierung dieser Probleme, insbesondere die Finite-Elemente-Diskretisierung, größtenteils unbearbeitet. Die vorliegende Arbeit widmet sich dieser Thematik. Wir zeigen die enge Beziehung auf, die bezüglich der Inf-sup-Stabilität zwischen dem diskreten hydrostatischen Strömungsproblem und seinem 2D Pendant existiert. Desweiteren erarbeiten wir Stabilisierungsverfahren, um Inf-sup-Stabilität zu erlangen und den Fall der dominanten Advektion adäquat zu behandeln. Hierbei können wir auf klassische Stabilisierungsverfahren zurückgreifen. Neben dem isotropen Fall betrachten wir hydrostatische Probleme auf anisotropen Gittern. Für die Analyse entwickeln wir einen Interpolationsoperator, der passende Projektions- und Stabilitätseigenschaften in 3D besitzt. Ein entsprechender Operator für den 2D Fall für bilineare Finite Elemente wurde in Braack06 entwickelt. Für die Stabilisierung beschränken wir uns auf symmetrische Verfahren. Die Druckstabilisierung bleibt aufgrund der Dimension des Drucks auf vertikal anisotropen Gitter, d.h. obwohl Gitteranisotropie auftreten kann ist das Oberflächengitter isotrop, isotrop. Im Fall auftretender Gitteranisotropie auch im Horizontalen greifen wir auf anisotrope Druckstabilisierung zurück. Desweitern diskutieren wir kurz die Thematik der Zeitdiskretisierung. Insbesondere gehen wir auf Druckkorrektur-Verfahren ein. Diese Verfahren wurden bereits für klassische Strömungsprobleme diskutiert. Jedoch fehlt bislang eine Analyse dieser Thematik im hydrostatischen Kontext. Anschließend betrachten wir algorithmische Aspekte und gehen dabei auch auf die Thematik der Parallelisierung ein. Wir schließen die Arbeit mit einer numerischen Validierung der theoretischen Ergebnisse ab und illustrieren einige Phänomene der Ozeanzirkulation

Viscous-Plastic sea-ice solutions with Elastic-Viscous-Plastic sea-ice solvers

Most dynamic sea ice models for climate type simulations are based on the viscous-plastic (VP) rheology. The resulting stiff system of partial differential equations for ice velocity is either solved implicitly at great computational cost, or explicitly with added pseudo-elasticity (elastic- viscous-plastic, EVP). The more popular, because apparently faster EVP scheme has been found to create noisy solutions that do not converge to the VP rheology. A slight modification re- interprets EVP as a pseudotime VP solver and thus salvages the convergence to VP. In addition, the modification regularizes the EVP solutions so that they can be used in climate simulations at relatively low cost compared to efficient implicit methods. We present comparisons of two variants of the new EVP scheme with converged VP solution in Arctic. At coarse resolution (grid cell width of about 27km), the EVP solutions are very similar to the VP solutions. At higher resolution (4.5km), convergence of all schemes is more difficult to achieve and the solutions are obviously different

Impact of ocean and sea ice initialisation on seasonal prediction skill in the Arctic

There is a growing demand for skillful prediction systems in the Arctic. Using the Norwegian Climate Prediction Model (NorCPM) that combines the fully-coupled Norwegian Earth System Model and the Ensemble Kalman filter, we present a system that performs both, weakly-coupled data assimilation (wCDA) when assimilating ocean hydrogaphy (by updating the ocean alone) and strongly-coupled data assimilation (sCDA) when assimilating sea ice concentration (SIC) (by jointly updating the sea ice and ocean). We assess the seasonal prediction skill of this version of NorCPM, the first climate prediction system using sCDA, by performing retrospective predictions (hindcasts) for the period 1985 to 2010. To better understand origins of the prediction skill of Arctic sea ice, we compare this version with a version that solely performs wCDA of ocean hydrography. The reanalysis that assimilates just ocean data, exhibits a skillful hydrography in the upper Arctic ocean, and features an improved sea ice state, such as improved summer SIC in the Barents Sea, or reduced biases in sea ice thickness. Skillful prediction of SIE up to 10-12 lead months are only found during winter in regions of a relatively deep ocean mixed layer outside the Arctic basin. Additional DA of SIC data notably further corrects the initial sea ice state, confirming the applicability of the results of Kimmritz et al. (2018) in a historical setting. The resulting prediction skill of SIE is widely enhanced compared to predictions initialised through wCDA of only ocean data. Particularly high skill is found for July-initialised autumn SIE predictions.publishedVersio

Finite-Element Sea Ice Model (FESIM), version 2

Abstract. The Finite-Element Sea Ice Model (FESIM), used as a component of the Finite-Element Sea ice Ocean Model, is presented. Version 2 includes the elastic-viscous-plastic (EVP) and viscous-plastic (VP) solvers and employs a flux corrected transport algorithm to advect the ice and snow mean thicknesses and concentration. The EVP part also includes a modified approach proposed recently by Bouillon et al. (2013), which is characterized by an improved stability compared to the standard EVP approach. The model is formulated on unstructured triangular meshes. It assumes a collocated placement of ice velocities, mean thicknesses and concentration at mesh vertices, and relies on piecewise-linear (P1) continuous elements. Simple tests for the modified EVP and VP solvers are presented to show that they may produce very close results provided the number of iterations is sufficiently high

Seasonal predictions initialised by assimilating sea surface temperature observations with the EnKF

This study demonstrates that assimilating SST with an advanced data assimilation method yields prediction skill level with the best state-of-the-art systems. We employ the Norwegian Climate Prediction Model (NorCPM)—a fully-coupled forecasting system—to assimilate SST observations with the ensemble Kalman filter. Predictions of NorCPM are compared to predictions from the North American Multimodel Ensemble (NMME) project. The global prediction skill of NorCPM at 6- and 12-month lead times is higher than the averaged skill of the NMME. A new metric is introduced for ranking model skill. According to the metric, NorCPM is one of the most skilful systems among the NMME in predicting SST in most regions. Confronting the skill to a large historical ensemble without assimilation, shows that the skill is largely derived from the initialisation rather than from the external forcing. NorCPM achieves good skill in predicting El Niño–Southern Oscillation (ENSO) up to 12 months ahead and achieves skill over land via teleconnections. However, NorCPM has a more pronounced reduction in skill in May than the NMME systems. An analysis of ENSO dynamics indicates that the skill reduction is mainly caused by model deficiencies in representing the thermocline feedback in February and March. We also show that NorCPM has skill in predicting sea ice extent at the Arctic entrance adjacent to the north Atlantic; this skill is highly related to the initialisation of upper ocean heat content.publishedVersio

NorCPM1 and its contribution to CMIP6 DCPP

The Norwegian Climate Prediction Model version 1 (NorCPM1) is a new research tool for performing climate reanalyses and seasonal-to-decadal climate predictions. It combines the Norwegian Earth System Model version 1 (NorESM1) – which features interactive aerosol-cloud schemes and an isopycnic-coordinate ocean component with biogeochemistry – with anomaly assimilation of SST and T/S-profile observations using the Ensemble Kalman Filter (EnKF).publishedVersio

Wetter-Klima-Chemie-Kopplung am Beispiel der 1D-Flachwassergleichungen

Zur Verbesserung der Lösungen von Klima-Chemie-Modellen werden Wetter-Daten, auch als Beobachtungsdaten bezeichnet, verwendet. Eine Möglichkeit der Realisierung ist durch die Newtonsche Relaxation (oder auch dynamischer Antrieb) gegeben. In der Anwendung ird das Modell, bzw. werden die dazu benötigten Parameter empirisch und suboptimal vorgegeben. Ziel der vorliegenden Arbeit ist eine Auseinandersetzung mit der Thematik der Newtonschen Relaxation an einem vereinfachten Modell, den eindimensionalen Flachwassergleichungen. Dazu werden zuerst die eindimensionalen Flachwassergleichungen und ihre numerische Integration vorgestellt. Anschließend widmet sich die Arbeit der Problematik der Newtonschen Relaxation, in dem das Problem vorgestellt und Lösungswege mit den Mitteln der nichtlinearen Optimierung gezeigt werden. Abschließend erfolgt eine Herleitung eines Verfahrens, in dem die eindimensionalen Flachwassergleichungen mit initialen Unstetigkeiten mit der analytischen Lösung des vorgegebenen Problems dynamisch angetrieben werden

The adaptive EVP method for solving the sea ice momentum equation

Stability and convergence of the modified EVP implementation of the visco-plastic sea ice rheology by Bouillon et al., Ocean Modell., 2013, is analyzed on B- and C-grids. It is shown that the implementation on a B-grid is less restrictive with respect to stability requirements than on a C-grid. On C-grids convergence is sensitive to the discretization of the viscosities. We suggest to adaptively vary the parameters of pseudotime subcycling of the modified EVP scheme in time and space to satisfy local stability constraints. This new approach generally improves the convergence of the modified EVP scheme and hence its numerical efficiency. The performance of the new “adaptive EVP” approach is illustrated in a series of experiments with the sea ice component of the MIT general circulation model (MITgcm) that is formulated on a C-gri

On the convergence of the modified elastic–viscous–plastic method for solving the sea ice momentum equation

Most dynamic sea ice models for climate type simulations are based on the viscous–plastic (VP) rheology. The resulting stiff system of partial differential equations for ice velocity is either solved implicitly at great computational cost, or explicitly with added pseudo-elasticity (elastic–viscous–plastic, EVP). A recent modification of the EVP approach seeks to improve the convergence of the EVP method by re-interpreting it as a pseudotime VP solver. The question of convergence of this modified EVP method is revisited here and it is shown that convergence is reached provided the stability requirements are satisfied and the number of pseudotime iterations is sufficiently high. Only in this limit, the VP and the modified EVP solvers converge to the same solution. Related questions of the impact of mesh resolution and incomplete convergence are also addressed