This is a continuation of a previous paper by Starius, where for the solution of the shallow water equations on the sphere, we consider Equator-Pole grid systems consisting of one latitude-longitude grid covering an annular band around the equator, and two orthogonal polar grids based on modified stereographic coordinates. Here, we generalise this by letting an equatorial band be covered with a reduced grid system, which can decrease the total number of grid points by at least 20%, with no substantial change in accuracy. Centred finite differences of high order are used for the spatial discretisation of the underlying differential equations and the explicit fourth order Runge-Kutta method for the integration in time. In the paper mentioned above, we demonstrate accuracy in the total mass, which is much higher than needed for NWP, for smooth solutions. Here we show that this holds also for non-smooth solutions, by considering the Cosine bell no. 1 and the mountain problem no. 5 in Williamson et al., the solutions of which have discontinuous second and first order derivatives, respectively.

In this continuation paper to Starius (

The methods of this paper give an accuracy in the total mass, which is considerably higher than needed for NWP. This was already shown for the smooth Rossby-Haurwitz wave example for the method in (S-E-P). Here it is also verified, with and without reductions, for non-smooth solutions by considering the Cosine bell no. 1 and the mountain problem no. 5 in Williamson et al. (

If

We now briefly comment on some other methods considered in the literature. Our main sources are the two overview papers Staniforth and Thuburn (

Cubed sphere grids (Sadourny,

Icosahedral grids (Williamson,

The more recently developed Yin-Yang grid, in Kageyama and Sato (

The great importance of the axis of the Earth for the dynamics of the weather is clear. However, the Yin-Yang grid does not have good rotational symmetry properties relative to this axis, not even near the poles, for details see, e.g. Section 1 in (S-E-P). At the end of Section 3.4, we compare the Yin-Yang grids and ours with respect to the total number of grid points, both with the same number on the equator.

The paper is organised as follows. In Section 2, we introduce the set of governing equations, some notation, and also some coordinate systems. This section is essentially a summary of Section 2 in (S-E-P).

Section 3 is devoted to reduced grid systems for annular bands around the equator. Two methods, I and II, to divide the band into segments and then to connect them by interpolation, are considered in Sections 3.1 and 3.2. Method I is quite general and easy to implement but requires more interpolation coefficients than method II. However, method II has limitations in its applicability, which is discussed in Section 3.2. In Section 3.3, two different measures of uniformity, used to generate suitable grid systems for the sphere, are compared. In Section 3.4, an E-P grid system is constructed with square net rectangles at mid-latitudes, which is often required for NWP models.

In Section 4, we investigate our E-P grid systems numerically, both with and without reductions. Centred finite differences of high order are used for the spatial discretisations and the explicit fourth order Runge-Kutta method for the time integration. First some notation for various errors is introduced and then, in Section 4.1, smoothing by addition of a ’hyper-diffusion’ term is considered. For a discretisation method of order 2

In (S-E-P) further numerical examples are given, and also investigations of grid imprinting and comparisons between formal and computational orders of discretisation errors are considered.

In this section, we introduce some notation and give a frame of reference for the paper. It is essentially a summary of

The transformation between Cartesian and spherical coordinates on the sphere is

Consider first the standard stereographic coordinates. The relation between these and spherical coordinates can be expressed as

Since the velocities in spherical coordinates,

Stereographic projection is only suitable for grid generation close to the poles. Here we will use a modification of (

The modified coordinates

In this Section, we consider the possibility to utilise a reduced grid system for an annular band around the equator, cf. Gates and Riegel (

We will later define latitudes _{i}_{i}_{i}_{i}

We now discretise the segments introduced above. Assume that all the latitudes _{i}_{i}_{i}

A small part

In this section, we will often use the fact that for a parallel with latitude

Here the number of segments

The transition from _{i}

Now _{i}

In

Reduced lat-lon grids with n = 360 and _{K}_{K}_{1}, that is _{i}

_{1}

_{2}

_{3}

_{K}

_{1}

_{2}

_{3}

_{K}

_{K}

_{K}

_{K}

A way to avoid scalability problems for the couplings between the reduced grid and the northern polar grid, say, is to let the former reach a high latitude. Since _{K}_{K}_{K}_{i}

We conclude this subsection with two examples. For initial _{3}. In a second example, with initial

This method is essentially applicable with a small number of segments and special choices of

The method is based on the following simple idea. In a consecutive manner we replace _{i}_{K}

We now determine _{i}

The above works only if

If we get a number

For high resolution, we indicate an improvement of the above modification of method II. If

We conclude this subsection by considering an example with ^{2}, then method II works perfectly well for ^{3}, we can use a modification as described above for the segments following

The measure of uniformity most commonly referred to in the literature is the ratio of maximum to minimum grid length, which we call

^{d}_{1}, that is ^{d}

The minimum of

For comparison we mention that the minimum of the maximal deviation factor for this segment is

This formula will be used in the next Section 3.4. ⊗

The ratio

where

We look at two cases, namely

In both cases

The sole reason why lat-lon coordinates are not suitable for the whole sphere is the convergence of meridians when we are moving towards a pole. Therefore, we find it inappropriate to use square net rectangles at the equator instead of rectangles with longer longitudinal than latitudinal sides. ⊗

The examples (i) and (ii) above convincingly show that the maximal deviation factor is a better tool for generation of suitable grid systems on the sphere, than the ratio of maximum to minimum grid length.

Grids for NWP are generally constructed to be uniform (having square net rectangles) at mid-latitudes,

We first use method I of

For our ’mid-latitude method’ we use notation with bars to distinguish the variables from those used for methods I and II. In order to avoid over-determined systems of equations we replace

We conclude this subsection with an example, in

Let _{i}

_{1}

_{2}

_{3}

_{4}

_{10}

_{1}

_{2}

_{3}

_{4}

_{10}

_{10}

For the Yin-Yang grid we let the Yin subgrid have square net rectangles at

Provided we accept less uniformity than in

For a full lat-lon grid, satisfying the mid-latitude condition and with

In this section, we investigate the E-P grid system numerically, both with and without reductions in the equatorial grid. Test examples no. 1 and 5 from Williamson et al. (

The spatial discretisation is obtained by using centred finite difference approximation of orders 6 or 4 on non-staggered grids, for which formulas can be found in

Explicit time integration seems to work quite well for the shallow water equations. However, for realistic NWP, 3d models are used and with vertical spacing much smaller than the horizontal ones, which would lead to very small time steps for explicit methods. This is why HEVI(horizontal explicit and vertical implicit) schemes are of interest in the atmospheric community.

In the tables below, 2

The geopotential

The global conservation properties for mass and energy will also be studied below. The integral formulas can be found, e.g. in Starius (2014) and their numerical computation is described in (S-E-P). The relative errors for the total mass after each day are stored in the vector

Smoothing is essential for the centred finite difference methods to work properly and will be achieved by adding a so called ‘hyper-diffusion’ term to each equation. Since ‘hyper-diffusion’ has no known physical meaning, we consider our smoothing a purely numerical stabilisation process.

Let _{h}

To the first, second, and third equation in (_{h}^{®} routine

In an earlier work we have used _{h}

We point out that the optimisations in this paper, are essential in order to ease comparisons between different methods.

In this example we consider a solid body rotation of the Cosine bell, according to test problem no. 1 from Williamson et al. (

The E-P grid system will be used for

Normalised errors for solid rotation of the Cosine bell, n = 360,

Despite the solution having discontinuous second order derivatives, the accuracy of

For the rest of this subsection we only consider the most important case

The following time dependent flow is a solution of the shallow water equations and was introduced in Läuter et al. (

We make some comments concerning

Normalised errors for time dependent flow(Läuter),

The third row of

In

Normalised errors for time dependent flow(Läuter),

The solutions (15) are very smooth and have very small variation, which explains the high accuracy for the geopotential, and for the total mass and energy cancellation of errors has further increased the accuracy, see

Test problem no. 5 in Williamson et al. (

Contour curves for the total height

It is fairly easy to experimentally determine a ’stability interval’
_{min}_{max}_{max}

In

Zonal flow over an isolated mountain,

In order to further investigate the reliability of our Equator-Pole grid systems, we have integrated the mountain problem for 5 years, using methods of order 6. In

The mountain problem,

The Equator-Pole grid system introduced in (S-E-P), for the solution of the shallow water equations on the sphere, consists of one equatorial and two polar grids. Here we generalise this by permitting the equatorial grid to be reduced, that is divided into segments. We give two methods of reduction, called method I and method II, in Sections 3.1 and 3.2, respectively. The reduction technique seems to work quite well, as can easiest be seen in

By using reductions the total number of grid points on the sphere can be decreased by 20% or more, and for high resolution also the computing time by a similar percentage. The change in accuracy for the height of the fluid is negligible, and for the total mass the accuracy is increasing with increased uniformity of the grid system, cf.

We have demonstrated high accuracy in the total mass for fairly simple problems, both with and without smooth solutions. For realistic NWP models the situation can be more complicated, e.g. water can contribute to the atmosphere’s mass and give rise to far less smooth problems than those considered here. We mention that Allen and Zerroukat (2015) have derived a two-dimensional generalisation of the shallow water equations, which includes moist.

We use high order centred finite difference approximations of first order spatial derivatives, in the shallow water equations. This requires only a fraction of the work per grid point compared with some other methods of the same order. This fraction decreases rapidly with the order of approximation. Further, one can very easily increase the order of approximation by increasing the order of approximation for first order derivatives. For methods of order 4, 6, 8 and 10 the corresponding spatial stencils contain only 9, 13, 17 and 21 grid points, respectively. The higher the order of the spatial approximation, up to a bound depending on the resolution, the better treatment of fronts in the solutions, see for instance

The Equator-Pole grids presented here only focus on uniformity, except in Section 3.4, and are mainly intended to be basic grid systems. To make them more flexible, for instance when taking jet streams and orography into account, they need to be modified, perhaps by using overlapping techniques.

I want to express my sincere gratitude to the anonymous reviewer for many valuable information about NWP, of which I was not aware.

No potential conflict of interest was reported by the authors.