Contour-Advective Semi-Lagrangian Research Articles

On contour crossings in contour-advective simulations – part 1 – algorithm for detection and quantification

This is the first of two papers devoted to the analysis of contour crossing errors that occur in contour-advective simulations of fluid motion. Here an algorithm is presented for quantifying the error due to contour crossings. The first step is to determine the relative proximity of all possible pairs of contours. A digital representation of each contour is produced to aid in the corresponding calculation. Simple analysis of functions is then used to find any crossings between contours deemed close to each other by the digital representation method. Next, the area in error of a pair of crossing contours is calculated by identifying the polygon or polygons that approximately bound the erroneous region. Finally, some preliminary results of analysis of contour crossings that occur in contour-advective semi-lagrangian (CASL) simulations of single layer quasigeostrophic turbulence are presented. It is shown that the error due to contour crossings is small in the simulations considered here.

On contour crossings in contour-advective simulations – part 2 – analysis of crossing errors and methods for their prevention

This is the second of two papers devoted to the analysis of contour crossing errors that occur in contour-advective simulations of fluid motion, where either vorticity or potential vorticity is represented by contours. We begin with a detailed discussion on some of the potential mechanisms for contour crossing. Past work has suggested that the formation of contour crossings is due to inadequate spatial resolution of contours [1]. The implementation of two schemes for preventing contour crossings within the framework of the Contour-Advective Semi-Lagrangian (CASL) algorithm is detailed here. We then present an analysis of contour crossing errors in simulations of quasigeostrophic turbulence on the f-plane and the quasigeostrophic motion of an initially circular vortex patch on the β-plane using the algorithm detailed in Part 1. We find that in general individual crossings occur at scales smaller than the inversion grid scale on which velocity is calculated, but at scales larger than that of the surgical scale that defines the smallest resolved features (vorticity) of a flow. If the resolution of a quasigeostrophic turbulence simulation on the f-plane is increased by doubling the number of grid points in each coordinate direction used in the calculation of the velocity field, then the total area in error due to contour crossings remains unchanged; a smaller number of crossings introducing larger scale area errors is replaced by a greater number of smaller local errors. Uniformly increasing the density of nodes along all contours and placement of nodes at points of close approach on contours are both effective methods for limiting contour crossings.

The combined Lagrangian advection method

We present and test a new hybrid numerical method for simulating layerwise-two-dimensional geophysical flows. The method radically extends the original Contour-Advective Semi-Lagrangian (CASL) algorithm [5] by combining three computational elements for the advection of general tracers (e.g. potential vorticity, water vapor, etc.): (1) a pseudo-spectral method for large scales, (2) Lagrangian contours for intermediate to small scales, and (3) Lagrangian particles for the representation of general forcing and dissipation. The pseudo-spectral method is both efficient and highly accurate at large scales, while contour advection is efficient and accurate at small scales, allowing one to simulate extremely fine-scale structure well below the basic grid scale used to represent the velocity field. The particles allow one to efficiently incorporate general forcing and dissipation.

Homostrophic vortex interaction under external strain, in a coupled QG-SQG model

The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic — Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius R at depth −2H interacting with a disk of uniform density anomaly, with radius R, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.

The Diabatic Contour-Advective Semi-Lagrangian Algorithms for the Spherical Shallow Water Equations

Abstract The diabatic contour-advective semi-Lagrangian (DCASL) algorithm is extended to the thermally forced shallow water equations on the sphere. DCASL rests on the partitioning of potential vorticity (PV) to adiabatic and diabatic parts solved, respectively, by contour advection and a grid-based conventional algorithm. The presence of PV in the source term for diabatic PV makes the shallow water equations distinct from the quasigeostrophic model previously studied. To address the more rapid generation of finescale structures in diabatic PV, two new features are added to DCASL: (i) the use of multiple sets of contours with successively finer contour intervals and (ii) the application of the underlying method of DCASL at a higher level to diabatic PV. That is, the diabatic PV is allowed to have both contour and grid parts. The added features make it possible to make the grid part of diabatic PV arbitrarily small and thus pave the way for a fully Lagrangian DCASL in the presence of forcing. The DCASL algorithms are constructed using a standard semi-Lagrangian (SL) algorithm to solve for the grid-based part of diabatic PV. The 25-day time evolution of an unstable midlatitude jet triggered by the action of thermal forcing is used as a test case to examine and compare the properties of the DCASL algorithms with a pure SL algorithm for PV. Diagnostic measures of vortical and unbalanced activity as well as of the relative strength of the grid and contour parts of the solution for PV indicate that the superiority of contour advection can be maintained even in the presence of strong, nonsmooth forcing.

The HyperCASL algorithm: A new approach to the numerical simulation of geophysical flows

We describe a major extension to the Contour-Advective Semi-Lagrangian (CASL) algorithm [D.G. Dritschel, M.H.P. Ambaum, A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields, Quart. J. Roy. Meteorol. Soc. 123 (1997) 1097–1130; D.G. Dritschel, M.H.P. Ambaum, The diabatic contour advective semi-Lagrangian algorithm, Mon. Weather Rev. 134 (9) (2006) 2503–2514]. The extension, called ‘HyperCASL’ (HCASL), uses Lagrangian advection of material potential vorticity contours like CASL, but a Vortex-In-Cell (VIC) method for the treatment of diabatic forcing or damping. In this way, HyperCASL is fully Lagrangian regarding advection. A grid is used as in CASL to deal with ‘inversion’ (computing the velocity field from the potential vorticity field). First, the novel aspects of the algorithm are described including several improvements to the underlying CASL algorithm. All numerical parameters are chosen so as to minimise the computational cost while improving conservation properties. Finally, a thorough inter-code comparison is conducted using a two-dimensional inviscid unforced turbulence test-case. This enables us to point out the advantages of this new algorithm in terms of resolution, computational cost and numerical diffusion compared to other existing methods, namely CASL, VIC and Pseudo-Spectral (PS) methods.

Assessing the Numerical Accuracy of Complex Spherical Shallow-Water Flows

Abstract The representation of nonlinear shallow-water flows poses severe challenges for numerical modeling. The use of contour advection with contour surgery for potential vorticity (PV) within the contour-advective semi-Lagrangian (CASL) algorithm makes it possible to handle near-discontinuous distributions of PV with an accuracy beyond what is accessible to conventional algorithms used in numerical weather and climate prediction. The emergence of complex distributions of the materially conserved quantity PV, in the absence of forcing and dissipation, results from large-scale shearing and deformation and is a common feature of high Reynolds number flows in the atmosphere and oceans away from boundary layers. The near-discontinuous PV in CASL sets a limit on the actual numerical accuracy of the Eulerian, grid-based part of CASL. For the spherical shallow-water equations, the limit is studied by comparing the accuracy of CASL algorithms with second-order-centered, fourth-order-compact, and sixth-order-supercompact finite differencing in latitude in conjunction with a spectral treatment in longitude. The comparison is carried out on an unstable midlatitude jet at order one Rossby number and low Froude number that evolves into complex vortical structures with sharp gradients of PV. Quantitative measures of global conservation of energy and angular momentum, and of imbalance as diagnosed using PV inversion by means of Bolin–Charney balance, indicate that fourth-order differencing attains the highest numerical accuracy achievable for such nonlinear, advectively dominated flows.

Revisiting the Rossby–Haurwitz wave test case with contour advection

This paper re-examines a basic test case used for spherical shallow-water numerical models, and underscores the need for accurate, high resolution models of atmospheric and ocean dynamics. The Rossby–Haurwitz test case, first proposed by Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow-water equations on the sphere, J. Comput. Phys. (1992) 221–224], has been examined using a wide variety of shallow-water models in previous papers. Here, two contour-advective semi-Lagrangian (CASL) models are considered, and results are compared with previous test results. We go further by modifying this test case in a simple way to initiate a rapid breakdown of the basic wave state. This breakdown is accompanied by the formation of sharp potential vorticity gradients (fronts), placing far greater demands on the numerics than the original test case does. We also go further by examining other dynamical fields besides the height and potential vorticity, to assess how well the models deal with gravity waves. Such waves are sensitive to the presence or not of sharp potential vorticity gradients, as well as to numerical parameter settings. In particular, large time steps (convenient for semi-Lagrangian schemes) can seriously affect gravity waves but can also have an adverse impact on the primary fields of height and velocity. These problems are exacerbated by a poor resolution of potential vorticity gradients.

Potential Vorticity and the Quasigeostrophic and Semigeostrophic MesoscaleVertical Velocity

Results of a variety of numerical simulations are presented and the accuracy of quasigeostrophic (QG) and semigeostrophic (SG) vertical velocity estimates of the total vertical velocity is analyzed. The authors examine the dependence of the results on the potential vorticity (PV) anomaly of the flow, its time evolution, and the amount of numerical diffusion. The SG ω equation is solved in a novel way in the original physical coordinates rather than in geostrophic coordinates. A three-dimensional numerical model is used that explicitly conserves the PV on isopycnal surfaces through a contour-advective semi-Lagrangian (CASL) algorithm. The numerical simulations consist initially of one or two horizontal cylinders of anomalous PV: a shear zone that induces two or three counterflowing jets. These jets destabilize and break into cyclones or anticyclones. This is accompanied by enhanced vertical motion, which exhibits a dominantly balanced quadrupole pattern in horizontal cross sections, depending on the ellipticity of the gyres, together with weak second-order inertia–gravity waves. For flows containing only negative PV anomalies the magnitude of both the QG and SG vertical velocities are smaller than the magnitude of the total vertical velocity, while the opposite occurs for flows containing only positive PV anomalies. The reason for this behavior is that the QG ω equation misses a term proportional to the Laplacian of the horizontal velocity. A new, more accurate, ω equation is proposed to recover the vertical velocity when both experimental density and horizontal velocity data are available. The SG solution is nearly always more accurate than the QG solution, particularly for the largest vertical velocity values and when the flow has single-signed PV anomalies. For flows containing both positive and negative PV anomalies, for example, mushroomlike eddies, the QG vertical velocity is a better approximation to the total vertical velocity than the SG solution. The reason for this anomalous behavior lies in one additional assumption concerning the conservation of volume that is usually adopted to derive the SG ω equation.

The CASL algorithm for quasi-geostrophic flow in a cylinder

The contour-advective semi-Lagrangian (CASL) algorithm is a hybrid Lagrangian–Eulerian numerical algorithm combining contour dynamics, for the advection of a materially conserved dynamical tracer, with spectral techniques, for the computation of the associated flow field. It benefits from the accuracy of contour dynamics, and the efficiency of spectral methods, and in applications to ‘quasi-geostrophic’ rotating, stratified flows, has no time-step restriction for numerical stability. It abandons the expensive aspect of contour dynamics – the computation of contour integrals to obtain the flow field – and the inaccurate (and costly) aspect of spectral methods – the advection of materially conserved tracers (requiring a time-step restriction for numerical stability and numerical diffusion). The two approaches are tied together by a fast method of converting tracer contours to gridded field values (Lagrangian→Eulerian) and the bi-linear interpolation of the flow field from the grid to points on the contours (Eulerian→Lagrangian). Here, we extend this method, originally developed for horizontally periodic flows, to flows within a cylindrical domain. Among other applications, this permits a comparison with experiments, which are often conducted in a cylindrical domain. We demonstrate the accuracy of the algorithm in a few test cases, then examine the differences between cylindrical and periodic domains for both simple vortex interactions and complex quasi-geostrophic turbulence.

On the representation of gravity waves in numerical models of the shallow‐water equations

AbstractGravity waves, or imbalanced motions, that develop during the evolution of vortical flows in numerical models of the shallow water (SW) equations are examined in detail. The focus here is on nearly‐balanced flows, with small but non‐zero gravity‐wave activity. For properly initialized flows, it is reasonable to expect small GW activity when Froude numbers Fr < 1 and Rossby numbers Ro ≲ 1.The guiding principle in the present study is that an accurate representation of potential vorticity (PV) is the pre‐requisite to a fair assessment of the generation of gravity waves. The contour‐advective semi‐Lagrangian (CASL) algorithm for the SW equations is applied, as it shows a remarkable improvement in the simulation of PV. However, it is shown that the standard CASL algorithm for SW leads to a noticeable numerical generation of gravity waves. The false generation of gravity waves can equivalently be thought of as the false, or numerical, breakdown of balance.In order to understand the maintenance of balance in the SW equations, a hierarchy of CASL algorithms is introduced. The main idea behind the new hierarchy is to implement PV inversion partially, balancing algorithms directly within the SW algorithm, while still permitting imbalanced motions. The results of the first three members of the hierarchy, CA0 (standard CASL), CA1, and CA2, are described and are compared with the results of two other SW algorithms, a pseudo‐spectral and a semi‐Lagrangian one. The main body of results is obtained for a highly ageostrophic regime of flow, with|Ro|max ∼ 1 and Frmax ∼ 0.5, where sub‐index 'max' denotes maximum over the domain. Other flow regimes in the relevant parts of the Ro‐Fr parameter space are also explored. It is found that, for a given resolution and Froude number, there is an optimal CASL algorithm, i.e. one which gives rise to the least numerical generation of gravity waves.

The Contour-Advective Semi-Lagrangian Algorithm for the Shallow Water Equations

A new method for integrating shallow water equations, the contour-advective semi-Lagrangian (CASL) algorithm, is presented. This is the first implementation of a contour method to a system of equations for which exact potential vorticity invertibility does not exist. The new CASL method fuses the recent contour-advection technique with the traditional pseudospectral (PS) method. The potential vorticity field, which typically develops steep gradients and evolves into thin filaments, is discretized by level sets separated by contours that are advected in a fully Lagrangian way. The height and divergence fields, which are intrinsically broader in scale, are treated in an Eulerian way: they are discretized on an fixed grid and time stepped with a PS scheme. In fact, the CASL method is similar to the widely used semi-Lagrangian (SL) method in that material conservation of potential vorticity along particle trajectories is used to determine the potential vorticity at each time step from the previous one. The crucial difference is that, whereas in the CASL method the potential vorticity is merely advected, in the SL method the potential vorticity needs to be interpolated at each time step. This interpolation results in numerical diffusion in the SL method. By directly comparing the CASL, SL, and PS methods, it is demonstrated that the implicit diffusion associated with potential vorticity interpolation in the SL method and the explicit diffusion required for numerical stability in the PS method seriously degrade the solution accuracy compared with the CASL method. Moreover, it is shown that the CASL method is much more efficient than the SL and PS methods since, for a given solution accuracy, a much coarser grid can be used and hence much faster computations can be performed.