arXiv subjects

Peter Korn

Publications and source records attributed to Peter Korn.

10 recordsLinked to original sources

arXiv2026

Foundations of Global Ocean Climate Modelling at all Scales

We introduce a computational method that extends the range of current global ocean climate models to non-hydrostatic scales, and establishes a convection resolving ocean model. The main obstruction towards this goal at global scale -the calculation of the non-hydrostatic pressure- is resolved by computing the pressure locally in a manner that capitalizes on particular conditions of ocean dynamics, and the structure of ocean models. Our claims are substantiated 1) by a theoretical analysis, which shows that the proposed method is compatible with ocean physics, 2) by numerical experiments, which evidence by comparison that it represents faithfully and to high accuracy non-hydrostatic dynamics, and 3) by a performance analysis that establishes that our non-hydrostatic method exceeds the cost of a hydrostatic model by a fixed factor of about $1.2$ in operation count and $1.3$ in runtime at any resolution. We show how to use the additional information contained in non-hydrostatic scales can be used to calculate the diffusivity free from numerical contributions and the mixing efficiency, transforming both from prescribed values by the modeller to calculated values from the dynamics.

math.NA↗

arXiv2026

A no-go theorem and its resolution for the discrete compressible barotropic Navier--Stokes equations

The compressible barotropic Navier--Stokes equations in vector-invariant form preserve the vorticity structure of the system and underlie modern atmospheric and ocean dynamical cores, yet no PDE theory has been developed for the compressible discrete system in this form. On a Delaunay--Voronoi mesh we prove via discrete exterior calculus, that every density-independent mass matrix with integration-by-parts-consistent divergence carries a sharp $\OO(h^2)$ energy residual of indeterminate sign that no operator choice can eliminate. This no-go theorem covers A-, B-, C-, D-, and quasi-B-grid staggerings. The density-weighted mass matrix is the unique algebraic remedy: it restores exact total energy while preserving the vector-invariant momentum equation, Lamb antisymmetry, and the topological conservation laws, at the cost of an $\OO(h^{r_\star})$ Kelvin defect matching the convergence rate. The residual is the cause of the Hollingsworth instability that has shaped vector-invariant dynamical-core design; the density-weighted construction removes it structurally. For the density-weighted~(DW) scheme on closed oriented Riemannian manifolds in $d = 2, 3$ we establish global well-posedness for $\nu \ge 0$, convergence to smooth solutions uniformly in $\nu$, and asymptotic preservation in the low-Mach limit; the density-free residual diverges as $\OO(M^{-1})$. Via a discrete Arnold energy-Casimir construction, exact discrete conservation forces Lyapunov stability around three classes of equilibria, excluding Hollingsworth instability: unconditional stability around hydrostatic and constant-flow stratified states, and conditional stability around sheared baroclinic states under a discrete Charney--Stern criterion. The DW scheme admits genuine baroclinic instability only when the continuum equations themselves do.

math.AP↗

arXiv2026

Exact conservation as selection principle: discrete exterior calculus for the incompressible Navier-Stokes and Euler equations

We formulate a new discrete-exterior calculus based discretisation of the incompressible Euler and Navier-Stokes equations that preserves the geometric structure of the continuum, and establish a rigorous convergence and structure theory for a new discretisation. The discretisation operates on prismatic Delaunay-Voronoi meshes over closed Riemannian manifolds. The geometry of Euler and Navier-Stokes equations is maintained via a discrete Lie derivative that is built from an extrusion-based contraction for the nonlinear term in vector-invariant form. Conservation of energy and Kelvin circulation links the discrete scheme to the continuum: at the discrete level, energy conservation is a stability property, and in the vanishing-resolution limit it becomes both a constructive route into the conservative weak-solution theory of the continuum equations and a selection principle on the limits the scheme can reach. This correspondence appears in four regimes. \emph{Smooth solutions}: convergence at rate $\mathcal{O}(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|)$ in dimensions $d=2,3$, uniformly in viscosity $\nu \ge 0$; first order on general meshes, second order under centroid proximity and reconstruction symmetry. \emph{Leray-Hopf weak regime}: subsequential $L^2$ limits of the discrete Navier-Stokes system are weak solutions of the viscous equations. \emph{Inviscid measure-valued regime}: limits are conservative measure-valued Euler solutions, with concentration defect vanishing above the Onsager threshold $\alpha > 1/3$ provided the discrete solutions admit a uniform $C^{0,\alpha}$ bound; the scheme reaches the energy-conserving side of the Onsager landscape but not the dissipative side. \emph{Dissipative regime}: no subsequence converges to an energy-dissipating Euler solution at any H\"older regularity, an exclusion that follows from discrete energy conservation.

math.AP↗

arXiv2026

Structure-preserving stochastic parameterization of a barotropic coupled ocean-atmosphere model with Ornstein--Uhlenbeck noise

We present the first application of the stochastic advection by Lie transport (SALT) framework to an idealized coupled ocean-atmosphere system. SALT derives stochastic fluid equations from Hamilton's variational principle under a stochastic Lagrangian kinematic assumption, thereby preserving the geometric structure -- Kelvin circulation theorem, Lie-derivative advection operators, and local conservation laws -- of the underlying deterministic equations. The atmospheric component is rendered stochastic while the ocean remains deterministic, following Hasselmann's paradigm of a fast stochastic atmosphere driving a slow climate. The spatial correlation vectors encoding unresolved subgrid transport are estimated from high-resolution simulations via EOF analysis of Lagrangian trajectory differences. A central contribution is the replacement of the standard white-noise temporal model with Ornstein-Uhlenbeck (OU) processes, motivated by strong autocorrelation (decorrelation times of 50-150 time steps) in the dominant EOF modes; the OU process is the unique stationary Gaussian Markov process capable of capturing this temporal memory with a single parameter. Ensemble forecasts exhibit good spread-error agreement over 10-15 time units. Evaluated via the Continuous Ranked Probability Score -- a strictly proper scoring rule measuring discrepancy between the forecast measure and the true conditional distribution -- the stochastic ensemble consistently outperforms a size-matched deterministic ensemble, despite carrying higher RMSE. Well-posedness of the coupled stochastic system is identified as an important open problem.

physics.ao-ph↗

arXiv2024

Dynamic Deep Learning Based Super-Resolution For The Shallow Water Equations

Using the nonlinear shallow water equations as benchmark, we demonstrate that a simulation with the ICON-O ocean model with a 20km resolution that is frequently corrected by a U-net-type neural network can achieve discretization errors of a simulation with 10km resolution. The network, originally developed for image-based super-resolution in post-processing, is trained to compute the difference between solutions on both meshes and is used to correct the coarse mesh every 12h. Our setup is the Galewsky test case, modeling transition of a barotropic instability into turbulent flow. We show that the ML-corrected coarse resolution run correctly maintains a balance flow and captures the transition to turbulence in line with the higher resolution simulation. After 8 day of simulation, the $L_2$-error of the corrected run is similar to a simulation run on the finer mesh. While mass is conserved in the corrected runs, we observe some spurious generation of kinetic energy.

cs.LG↗

arXiv2023

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.

physics.flu-dyn↗

arXiv2023

Global Well-Posedness of the Primitive Equations of Large-Scale Ocean Dynamics with the Gent-McWilliams-Redi Eddy Parametrization Model

We prove global well-posedness of the ocean primitive equations coupled to advection-diffusion equations of the oceanic tracers temperature and salinity that are supplemented by the eddy parametrization model due to Gent-McWilliams and Redi. This parametrization forms a milestone in global ocean modelling and constitutes a central part of any general ocean circulation model computation. The eddy parametrization adds a secondary transport velocity to the tracer equation and renders the original Laplacian operators in the advection-diffusion equations nonlinear, with a diffusion matrix that depends via the equation of state in a nonlinear fashion on both tracers simultaneously. The eddy parametrization of Gent-McWilliams-Redi augments the complexity of the mathematical analysis of the whole system which we present here. We show first that weak solutions exist globally in time, provided the parametrization uses a regularized density. Then we prove by a detailed analysis of the eddy operators the global well-posedness. Our results apply also to the ``small-slope approximation'' that is commonly used in global ocean simulations.

math.AP↗

arXiv2021

Sea-ice dynamics on triangular grids

We present a stable discretization of sea-ice dynamics on triangular grids that can straightforwardly be coupled to an ocean model on a triangular grid with Arakawa C-type staggering. The approach is based on a nonconforming finite element framework, namely the Crouzeix-Raviart finite element. As the discretization of the viscous-plastic and elastic-viscous-plastic stress tensor with the Crouzeix-Raviart finite element produces oscillations in the velocity field, we introduce an edge-based stabilization. To show that the stabilized Crouzeix-Raviart approximation is qualitative consistent with the solution of the continuous sea-ice equations, we derive a $H^1$-estimate. In a numerical analysis we show that the stabilization is fundamental to achieve stable approximation of the sea-ice velocity field.

math.NA↗

arXiv2015

A study on boundary separation in an idealized ocean model

In numerical ocean models coast lines change the direction from one grid cell to its neighbor and the value for viscosity is set to be as small as possible. Therefore, model simulations are not converged with resolution and boundary separation points differ in essential properties from flow separation in continuous flow fields. In this paper, we investigate the quality of the representation of boundary separation points in global ocean models. To this end, we apply well established criteria for boundary separation within an idealized ocean model setup. We investigate an eddy-resolving as well as a steady test case with idealized and unstructured coast lines in a shallow water model that is based on a finite element discretization method. The results show that well established criteria for separation fail to detect boundary separation points due to an insufficient representation of ocean flows along free-slip boundaries. Along no-slip boundaries, most separation criteria provide adequate results. However, a very sophisticated criterion based on dynamical system theory reveals that the representation of boundary separation is limited for these flows as well. We conclude that the representation of boundary separation points in numerical ocean models is not satisfying. This will have an impact on the separation of boundary currents in global ocean models.

physics.flu-dyn↗

arXiv2015

The representation of boundary currents in a finite element shallow water model

We evaluate the influence of local resolution, eddy viscosity, coastline structure, and boundary conditions on the numerical representation of boundary currents in a finite element shallow-water model. The use of finite element discretization methods offers a higher flexibility compared to finite difference and finite volume methods, that are mainly used in previous publications. This is true for the geometry of the coast lines and for the realization of boundary conditions. For our investigations we simulate steady separation of western boundary currents from idealized and realistic coast lines. The use of grid refinement allows a detailed investigation of boundary separation at reasonable numerical cost.

physics.ao-ph↗