Semi-implicit semi-Lagrangian Research Articles

Error estimates for semi-Lagrangian finite difference methods applied to Burgers' equation in one dimension

We make an analytic study of the diffusive, dispersive and overall errors, which arise when using semi-implicit semi-Lagrangian (SISL) finite difference methods to approximate those travelling wave solutions of the one-dimensional Burgers' equation with small diffusion, which develop sharp fronts. For the case of a fixed uniform spatial mesh, with piecewise linear interpolation, a backward error analysis approach is used to construct a precise formal analytic description of the front profile of the numerical approximation to this solution. From this description it is possible to obtain precise estimates of the front width and the front speed in terms of the spatial and temporal step size and to express the overall solution error in terms of these. These formal estimates agree closely with numerical calculations, both qualitatively and quantitatively, and display a roughly periodic behaviour as the number Nx of mesh points increases, and the CFL number passes through integer values. In particular, they show that despite the otherwise poor resolution of the method, the front width is closely approximated when the CFL number is close to an integer, and the front speed is closely approximated when it is close to a half integer. The overall L2 error also shows super-convergence for certain values of Nx. This possibly motivates doing two calculations with different Nx when using the SISL method on such problems to separately minimise the diffusive and dispersive errors. Similar errors in the front width and speed are observed for a number of different interpolation schemes with and without flux limiters.

The moving mesh semi-Lagrangian MMSISL method

We introduce a novel location-based moving mesh algorithm MMSISL in which the arrival points in the Semi-Implicit Semi-Lagrangian (SISL) algorithm are located by using an equidistribution strategy. This algorithm gives a natural coupling between moving mesh methods and SISL methods. It involves little extra cost in implementation as it exploits the interpolation methods already embedded in the SISL algorithm. We apply this method to a number of partial differential equation problems in one-dimension, each of which have sharply defined features. We show that using MMSISL leads to a markedly improved performance over fixed mesh methods, with significantly reduced errors. We also show that unlike many adaptive schemes, no issues arise in the MMSISL algorithm from a CFL condition imposed restriction on the time step.

Long-Term Integration of a Global Non-Hydrostatic Atmospheric Model on an Aqua Planet

A global non-hydrostatic atmospheric model, i.e., GRAPES_YY (Global/Regional Assimilation and Prediction System on the Yin–Yang grid), with a semi-implicit semi-Lagrangian (SISL) dynamical core developed on the Yin–Yang grid was coupled with the physical parameterization package of the operational version of GRAPES. A 3.5-yr integration was carried out on an aqua planet to assess the numerical performance of this non-hydrostatic mo-del relative to other models. Specific aspects of precipitation and general circulation under two different sea surface temperature (SST) conditions (CONTROL and FLAT) were analyzed. The CONTROL SST peaked at the equator. The FLAT SST had its maximum gradient at about 20° latitude, giving a broad equatorial SST maximum in the tropics and flat profile approaching the equator. The tropical precipitation showed different propagation features in the CONTROL and FLAT simulations. The CONTROL showed tropical precipitation bands moving eastward with some envelopes of westward convective-scale disturbance. Less organized westward-propagating rainfall cells and bands were seen in the FLAT and the propagation of the tropical wave varied with the SST gradient. The Inter Tropical Convergence Zone (ITCZ), Hadley cell, and westerly jet core were weaker and more poleward as the SST profile flattened from the CONTROL to FLAT. The climatological structures simulated by GRAPES_YY, such as the distribution of precipitation and the large-scale circulation, fell within the bounds from other models. The stronger ITCZ precipitation, accompanied with stronger Hadley cells and convective heating in the CONTROL simulation, may be summed up as a result of stronger parameterized convection and the non-hydrostatic effects in GRAPES_YY. In addition, mechanism of the zonal mean circulation maintaining is analyzed for the different SST patterns referring the transient eddy flux.

An energy‐conserving restoration scheme for the shallow‐water equations

The numerical methods that solve the governing equations in an atmospheric dynamical core are designed to dissipate potential enstrophy and prevent the build‐up of kinetic energy at the grid scale. A side‐effect of this is the dissipation of total energy which should be conserved. Energy fixers are used in climate models to replace the dissipated energy by modifying the temperature in the thermodynamic equation, and stochastic backscatter schemes have also been developed for use in weather prediction models. Here, we present the first steps towards designing a deterministic energy‐conserving restoration scheme that considers the conversion of kinetic energy to heat, replacing kinetic energy lost due to model error, and the backscatter of kinetic energy.The energy‐conserving restoration scheme (ECRS) is presented in the context of the shallow‐water equations on the sphere. It is designed to be used with any existing shallow‐water equation scheme (called the preliminary scheme) which can adequately dissipate potential enstrophy, and in this article we use a semi‐implicit semi‐Lagrangian (SISL) scheme. For each prognostic variable, a spatial pattern is chosen; this is added to the preliminary scheme solution, and the amount added is calculated to ensure energy conservation. Results from short‐term test cases show that ECRS and SISL have very similar error norms. For long‐term simulations, ECRS conserves energy to a good approximation whereas SISL dissipates energy.

An alternative cell‐averaged departure point reconstruction for pointwise semi‐Lagrangian transport schemes

Convection‐permitting limited‐area models based on the same spectral semi‐implicit semi‐Lagrangian (SL) techniques which are used in the ECMWF global model, are run operationally in several countries of the ALADIN/HIRLAM consortium. Forecasters have reported a general tendency for these models to produce overestimated precipitation and unrealistic divergent winds at the edges of the cold outflows generated by the precipitation evaporation in the vicinity of convective clouds. These grid‐point storms have been associated with a spurious behaviour of the pointwise interpolation used in the SL scheme, where grid‐scale buoyant updraughts create strong small‐scale convergence near the surface. A modification of the interpolation weights in the SL transport scheme introduces the concept of cell‐averaging into the traditional pointwise SL scheme which improves the conservation property of the scheme and eliminates the spurious mode. The COMAD (COntinuous Mapping about Departure points) correction applied to the standard interpolation weights takes into account the deformation of the air parcels along each direction of interpolation in order to improve the continuity and the conservative property of the re‐mapping between the model grid points and the origin points of the backward trajectories. The method has been validated with the small planet configuration of the Integrated Forecast System at ECMWF and with the limited‐area version of the same dynamics used for the AROME (Météo‐France) and HARMONIE (HIRLAM) models. The pathological behaviour of grid‐scale buoyant flows permitted by these dynamical cores is corrected by the COMAD interpolations. The precipitation forecasts in the convection‐permitting models AROME/HARMONIE which show an overestimation of intense convective precipitation are systematically improved by the new weights.

Kinetic energy spectra characteristics of two convection‐permitting limited‐area models AROME and Meso‐NH

AbstractKinetic energy (KE) spectra are applied to evaluate two convection‐permitting models: the AROME numerical weather prediction operational model and the Meso‐NH research model, that share the same physics and differ only in the dynamics (semi‐Lagrangian semi‐implicit versus Eulerian explicit schemes).A first analysis of AROME spectra for winter and summer seasons shows that the model‐derived spectra match the observational spectra well, including the transition between k−3 and k−5/3 regimes. The vertical distribution of the spectra is coherent with previous observational and numerical studies and the diurnal cycle has a strong impact on the amount of KE in the mesoscale during summer.A comparative analysis of KE spectra for both models is then performed on a real case of individual convective cells that developed over plains, during the afternoon of 11 April 2007, characterized by a strong cold air outflow in the low levels.AROME spectra are characterized by a coarser effective resolution than Meso‐NH, even without explicit diffusion, revealing the impact of the implicit diffusion of the semi‐implicit semi‐Lagrangian scheme used in AROME. For large time steps, the damping increases and can be attributed preferentially to the SI part of the SISL (semi‐implicit semi‐Lagrangian) formulation. Adiabatic runs show that the transition to a shallow mesoscale regime is still apparent even if the mesoscale KE variance strongly depends on the presence of the physical processes.Effective resolution of Meso‐NH remains around 4–6Δx for horizontal grid spacings between 2.5 km and 250 m. The effects of subgrid mixing schemes are also investigated with Meso‐NH at 500 m horizontal grid spacing in the grey zone for turbulence, illustrating the difficulty in finding a good equilibrium between resolved and subgrid mixing at this scale.

An inherently mass‐conserving iterative semi‐implicit semi‐Lagrangian discretization of the non‐hydrostatic vertical‐slice equations

AbstractRecently an inherently mass‐conserving semi‐Lagrangian transport scheme has been successfully coupled to an iterative semi‐implicit scheme in a global shallow‐water‐equation (SWE) model. Here that methodology is extended and applied to an iterative semi‐implicit semi‐Lagrangian (SISL) compressible, non‐hydrostatic vertical‐slice model, in which the constant reference state of the SWE model is now replaced by a vertically varying reference state. An advantage of this approach is that it preserves the same basic structure as the interpolating, non‐mass‐conserving, iterative SISL model. The resulting mass‐conserving model is applied to a standard set of test problems for such models and compared with results from both the literature and the interpolating, iterative SISL version of the model. © Crown Copyright 2010. Reproduced with the permission of HMSO. Published by John Wiley & Sons, Ltd.

Non-hydrostatic semi-elastic hybrid-coordinate SISL extension of HIRLAM. Part II: numerical testing

The semi-implicit semi-Lagrangian (SISL), two-time-level, non-hydrostatic numerical scheme, based on the nonhydrostatic, semi-elastic pressure-coordinate equations, is tested in model experiments with flow over given orography (elliptical hill, mountain ridge, system of successive ridges) in a rectangular domain with emphasis on the numerical accuracy and non-hydrostatic effect presentation capability. Comparison demonstrates good (in strong primary wave generation) to satisfactory (in weak secondary wave reproduction in some cases) consistency of the numerical modelling results with known stationary linear test solutions. Numerical stability of the developed model is investigated with respect to the reference state choice, modelling dynamics of a stationary front. The horizontally area-mean reference temperature proves to be the optimal stabilitywarrant. The numerical scheme with explicit residual in the vertical forcing term becomes unstable for cross-frontal temperature differences exceeding 30 K. Stability is restored, if the vertical forcing is treated implicitly, which enables to use time steps, comparable with the hydrostatic SISL.

Non-hydrostatic semi-elastic hybrid-coordinate SISL extension of HIRLAM. Part I: numerical scheme

Two-time-level, semi-implicit, semi-Lagrangian (SISL) scheme is applied to the non-hydrostatic pressure coordinate equations, constituting a modified Miller—Pearce—White model, in hybrid-coordinate framework. Neutral background is subtracted in the initial continuous dynamics, yielding modified equations for geopotential, temperature and logarithmic surface pressure fluctuation. Implicit Lagrangian marching formulae for single time-step are derived. A disclosure scheme is presented, which results in an uncoupled diagnostic system, consisting of 3-D Poisson equation for omega velocity and 2-D Helmholtz equation for logarithmic pressure fluctuation. The model is discretized to create a non-hydrostatic extension to numerical weather prediction model HIRLAM. The discretization schemes, trajectory computation algorithms and interpolation routines, as well as the physical parametrization package are maintained from parent hydrostatic HIRLAM. For stability investigation, the derived SISL model is linearized with respect to the initial, thermally non-equilibrium resting state. Explicit residuals of the linear model prove to be sensitive to the relative departures of temperature and static stability from the reference state. Relayed on the stability study, the semi-implicit term in the vertical momentum equation is replaced to the implicit term, which results in stability increase of the model.

Linearly implicit time stepping methods for numerical weather prediction

The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer–Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations.

On the Accuracy of Semi-Lagrangian Numerical Simulation of Internal Gravity Wave Motion in the Atmosphere

We have investigated the accuracy ofthe semi-implicit semi-Lagrangian (SISL) method in simulating internal gravity wave (IGW) motion. We have focused on the relative accuracy of the hydrostatic, and nonhydrostatic IGW solutions. The analysis is based on a linearized model and a Global Circulation Model-Dynamic Core (GCM-DC) with a stretched grid.The nonhydrostatic version of the GCM-DC model produces the familiar IGW train disturbance anchored to an isolated hypothetical mountain. The wave has a distinct tilt away from the vertical direction, which is consistent with classical theory. For the hydrostatic version of the model, the axis of the resulting IGW train rests nearly perpendicular to the mountain top, thus again consistent with classical theory. Increasing the time step from 10 s; Courant number (Cn) = 0.5; to 60 s (Cn = 3.0), results in stable solutions for both the hydrostatic and nonhydrostatic versions of the model. The nonhydrostatic solution is in close agreement with the control run however, the hydrostatic solution exhibits large phase truncation errors.The solutions for the one-dimensional linearized SISL model confirm the GCM-DC results that the nonhydrostatic IGW train is less damped and shifted by the SISL scheme than the corresponding hydrostatic IGW motion. The linear solutions indicate very high accuracy of the physical mode of the solution, but it rapidly deteriorates when Cn exceeds unity. As Δt → 0 the amplitude of the computational mode tends to zero and its frequency to infinity. However, as Δt → ∞, the frequency of the computational SISL mode asymptotically approaches the value of the frequency of the corresponding SISL physical mode. Furthermore, the amplitude of the SISL computational mode is directly proportional to the size of the time step. Therefore, at large time steps, the amplification of the computational mode could offset some of the numerical damping of the physical mode by the SISL scheme.

A Study of the Spurious Orographic Resonance in Semi-Implicit Semi-Lagrangian Models

Abstract The semi-implicit semi-Lagrangian (hereafter SISL) numerical integration scheme has been widely adopted in atmospheric models as it permits, without loss of accuracy, the use of time steps much larger than those permitted by the Courant–Friedrichs–Lewy criterion for the corresponding Eulerian models. However, it is known that there is a problem with orographic forcing in SISL models, since spurious numerical resonance can occur in mountain regions for Courant numbers larger than unity. Several solutions to this problem have been proposed, but so far none has completely solved the problem. In recent work by Ritchie and Tanguay, two different treatments of the orographic forcing term, the semi-Lagrangian and the spatially averaged Eulerian (SAE), were analytically and numerically investigated. It was found that when the SAE method is used, the orographic resonance problem is less severe. The reason for this improvement is further investigated here and it is shown to be due to the modified growth ch...

Climate and climate change in western canada as simulated by the Canadian regional climate model

A þrst climate simulation performed with the novel Canadian Regional Climate Model (CRCM) is presented. The CRCM is based on fully elastic non‐hydrostatic þeld equations, which are solved with an efþcient semi‐implicit semi‐Lagrangian (SISL) marching algorithm, and on the parametrization package of subgrid‐scale physical effects of the second‐generation Canadian Global Climate Model (GCMII). Two 5‐year integrations of the CRCM nested with GCMII simulated data as lateral boundary conditions are made for conditions corresponding to current and doubled CO2 scenarios. For these simulations the CRCM used a grid size of 45 km on a polar‐stereographic projection, 20 scaled‐height levels and a time step of 15 min; the nesting GCMII has a spectral truncation of T32, 10 hybrid‐pressure levels and a time step of 20 min. These simulations serve to document: (1) the suitability of the SISL numerical scheme for regional climate modelling, (2) the use of GCMII physics at much higher resolution than in the nesting model, (3) the ability of the CRCM to add realistic regional‐scale climate information to global model simulations, and (4) the climate of the CRCM compared to that of GCMII under two greenhouse gases (GHG) scenarios.

A Global Nonhydrostatic Semi-Lagrangian Atmospheric Model with Orography

A global nonhydrostatic semi-implicit semi-Lagrangian (SISL) atmospheric model with orography has been developed. The height-based terrain-following σz coordinate of Gal-Chen and Somerville is used to incorporate the orography. A 3D vector form of the SISL formulation is proposed. It is based on the complete Navier–Stokes equations. The model is stable for large time steps of up to 1 h at horizontal/vertical resolution of 2.8125°/1200 m. Isolated bell-shaped mountain profiles and real orography are employed to evaluate the model performance. The sensitivity of the model with orography to the order of accuracy of the uncentering scheme, the reference temperature (T), and size of the time step are similar to that of the model without orography described in Semazzi et al. The authors find that for successful execution of the model, it is important that the orographic height Zs, the reference state mass variable (qs), and T satisfy the hydrostatic balance relationship in the terrain-following σz coordinate system. This formulation effectively controls the generation of unphysical orographic induced computational noise usually encountered in connection with semi-Lagrangian models. The global model used here is based on the complete dynamical Navier–Stokes equations, however, it is run at coarse resolutions, for which nonhydrostatic effects are negligible. In the future, when the computing resources permit, the model will be a valuable vehicle for investigating the role of multiple-scale interaction, including the effects of nonhydrostatic dynamics.

セミグランジュ・モデルにおけるトラジェクトリ・アンセンタリング法の研究

Application of trajectory uncentering is an effective remedy for ensuring noise-free solutions in semi-implicit semi-Lagrangian (SISL) models. In this study we have employed a one-dimensional SISL linear shallow-water model to investigate the impact of second-order trajectory uncentering on the meteorologically significant Rossby wave solutions. This study is an extension of the results reported in Semazzi and Dekker (1994) where first-order semi-Lagrangian trajectory uncentering was investigated. We show that for typical resolutions, in this case Δx=O(100 km) and Δt<1 hr, application of first-order uncentering results in large truncation errors for planetary-scale waves of the atmosphere. Our main result is that application of second-order uncentering drastically reduces these truncation errors.

A Global Nonhydrostatic Semi-Lagrangian Atmospheric Model without Orography

Abstract A global nonhydrostatic semi-implicit semi-Lagrangian (SISL) atmospheric model with orography has been developed. The height-based terrain-following σz coordinate of Gal-Chen and Somerville is used to incorporate the orography. A 3D vector form of the SISL formulation is proposed. It is based on the complete Navier–Stokes equations. The model is stable for large time steps of up to 1 h at horizontal/vertical resolution of 2.8125°/1200 m. Isolated bell-shaped mountain profiles and real orography are employed to evaluate the model performance. The sensitivity of the model with orography to the order of accuracy of the uncentering scheme, the reference temperature (T), and size of the time step are similar to that of the model without orography described in Semazzi et al. The authors find that for successful execution of the model, it is important that the orographic height Zs, the reference state mass variable (qs), and T satisfy the hydrostatic balance relationship in the terrain-following σz coor...

Optimal Accuracy in Semi-Lagrangian Models

Abstract A one-dimensional semi-implicit semi-Lagrangian (SISL) linear model and a nonlinear SISL global shallow-water model are employed to investigate the sensitivity of the solutions on (i) the order of interpolation applied at the departure points, (ii) trajectory uncentering adopted in recent studies to suppress computational gravitational noise, and (iii) the optimal truncation conditions. The linear model results show that the truncation errors associated with the semi-Lagrangian (SL) part of the SISL scheme dominate the error characteristics when the estimates of the departure point values are based on linear or quadratic polynomial Lagrangian interpolation. In these two cases, the rate of error growth is large and may significantly exceed the acceptable levels in operational numerical weather prediction. Application of cubic interpolation drastically reduces the overall truncation errors, and the accuracy is well within the acceptable range. In contrast with the schemes based on lower-order inter...