arXiv subjects

Hiroe Yamazaki

Publications and source records attributed to Hiroe Yamazaki.

4 recordsLinked to original sources

arXiv2025

A vertical slice frontogenesis test case for compressible nonhydrostatic dynamical cores of atmospheric models

A new test case is presented for evaluating the compressible dynamical cores of the atmospheric models. The test case is based on a compressible vertical slice model that can be obtained by simple modification of a standard three dimensional compressible dynamical core. On the one hand, an advantage of the test case is that is quasi-2D, so it can be run quickly on a standard workstation, enabling rapid experimentation with numerical schemes and discretisation choices. On the other hand, the test case exhibits frontogenesis, a challenging regime for numerical discretisations which usually only arises in 3D model configurations for the compressible case. Numerical results of the test case using an implicit time-stepping method with a compatible finite element discretisation are presented as a reference solution. An example comparison between advective and vector-invariant forms for the advective nonlinearity in the velocity equation demonstrates one possible use of the scheme. The comparison shows a Hollingsworth-like instability when the vector invariant form is used.

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arXiv2023

Time parallel integration and phase averaging for the nonlinear shallow water equations on the sphere

We describe a proof-of-concept development and application of a phase averaging technique to the nonlinear rotating shallow water equations on the sphere, discretised using compatible finite element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables) so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: 1) we have developed a stable matrix exponential specific to finite elements and 2) we have developed a parallel finite averaging proceedure. Following Peddle et al (2019), we consider finite width phase averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi-implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work.

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arXiv2021

Conservation with moving meshes over orography

Adaptive meshes have the potential to improve the accuracy and efficiency of atmospheric modelling by increasing resolution where it is most needed. Mesh re-distribution, or r-adaptivity, adapts by moving the mesh without changing the connectivity. This avoids some of the challenges with h-adaptivity (adding and removing points): the solution does not need to be mapped between meshes, which can be expensive and introduces errors, and there are no load balancing problems on parallel computers. A long standing problem with both forms of adaptivity has been changes in volume of the domain as resolution changes at an uneven boundary. We propose a solution to exact local conservation and maintenance of uniform fields while the mesh changes volume as it moves over orography. This is solved by introducing a volume adjustment parameter which tracks the true cell volumes without using expensive conservative mapping. A finite volume solution of the advection equation over orography on moving meshes is described and results are presented demonstrating improved accuracy for cost using moving meshes. Exact local conservation and maintenance of uniform fields is demonstrated and the corrected mesh volume is preserved. We use optimal transport to generate meshes which are guaranteed not to tangle and are equidistributed with respect to a monitor function. This leads to a Monge-Ampère equation which is solved with a Newton solver. The superiority of the Newton solver over other techniques is demonstrated in the appendix. However the Newton solver is only efficient if it is applied to the left hand side of the Monge-Ampère equation with fixed point iterations for the right hand side.

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arXiv2016

Vertical slice modelling of nonlinear Eady waves using a compatible finite element method

A vertical slice model is developed for the Euler-Boussinesq equations with a constant temperature gradient in the direction normal to the slice (the Eady-Boussinesq model). The model is a solution of the full three-dimensional equations with no variation normal to the slice, which is an idealized problem used to study the formation and subsequent evolution of weather fronts. A compatible finite element method is used to discretise the governing equations. To extend the Charney-Phillips grid staggering in the compatible finite element framework, we use the same node locations for buoyancy as the vertical part of velocity and apply a transport scheme for a partially continuous finite element space. For the time discretisation, we solve the semi-implicit equations together with an explicit strong-stability-preserving Runge-Kutta scheme to all of the advection terms. The model reproduces several quasi-periodic lifecycles of fronts despite the presence of strong discontinuities. An asymptotic limit analysis based on the semi-geostrophic theory shows that the model solutions are converging to a solution in cross-front geostrophic balance. The results are consistent with the previous results using finite difference methods, indicating that the compatible finite element method is performing as well as finite difference methods for this test problem. We observe dissipation of kinetic energy of the cross-front velocity in the model due to the lack of resolution at the fronts, even though the energy loss is not likely to account for the large gap on the strength of the fronts between the model result and the semi-geostrophic limit solution.

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