Unsteady analytical solutions of the spherical shallow water equations

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Abstract

A new class of unsteady analytical solutions of the spherical shallow water equations (SSWE) is presented. Analytical solutions of the SSWE are fundamental for the validation of barotropic atmospheric models. To date, only steady-state analytical solutions are known from the literature. The unsteady analytical solutions of the SSWE are derived by applying the transformation method to the transition from a fixed cartesian to a rotating coordinate system. Fundamental examples of the new unsteady analytical solutions are presented for specific wind profiles. With the presented unsteady analytical solutions one can provide a measure of the numerical convergence in the case of a temporally evolving system. An application to the atmospheric model PLASMA shows the benefit of unsteady analytical solutions for the quantification of convergence properties.

Introduction

During the course of the development of dynamical cores in numerical weather prediction and climate modeling the development of a global barotropic model for atmospheric flows is a crucial step. For this reason, in the past barotropic models have been developed which are based on the spherical shallow water equations (SSWE). The SSWE comprise the essential physical phenomena that are included in the full set of primitive equations, e.g. large scale planetary waves and gravity waves. Furthermore, the SSWE present the major difficulties found in the horizontal discretization of the 3-dimensional primitive equations. In order to validate implemented numerical methods for the SSWE, reasonable test cases are required.

To date, the comprehensive test suite proposed by Williamson et al. [34] is the common basis for newly developed global shallow water models. Succeeding, additional tests have been proposed in the literature, e.g. [10]. In the following, an overview of the known test cases is given which can be divided into four categories:

Applying the common scale analysis, i.e. neglecting terms with small magnitude, simplified SSWE can be derived, like e.g. the linearized non-divergent barotropic vorticity equation or the geostrophic balance equation. Even though solutions of these systems are no analytical solutions of the full non-linear SSWE, integrations initialized by such initial states validate a SSWE model with respect to stability and noise generation.

For the first time Phillips [24] proposed Rossby–Haurwitz waves as initial fields for integrations of a SSWE model. Rossby–Haurwitz waves are solutions of the linearized non-divergent barotropic vorticity equation and move from west to east without change of their shape. Since Hoskins [15] showed that only Rossby–Haurwitz waves of zonal wave number 5 and less are stable, in the literature wave number 4 is applied to SSWE models, see e.g. [34], [17]. McDonald and Bates [21] proposed a geostrophic balanced initial state of zonal wave number 1. Due to the presence of divergence the wave rotates clockwise around the pole, see e.g. [11].

Steady-state analytical solutions of the SSWE mainly describe a purely zonal global flow with a balanced geopotential field. To our knowledge, solid body rotations on a non-rotating sphere have been described firstly by Dey [7] and on a rotating sphere by Umscheid and Sankar-Rao [32]. Taking into account that, except for the Coriolis term, the SSWE are invariant under a rotation of the spherical coordinates, an inclined solid body rotation with an inclination angle α = π/2 was given in [32] and for arbitrary α  [0,π] in [33]. Considering a more realistic zonal wind field, resembling a typical tropospheric jet, Browning et al. [3] deduced a steady-state analytical solution with compact support. Note that the balanced geopotential field is obtained by numerical integration. The mentioned solutions are inserted in [34, tests 2,3].

For this class of solutions, the basic idea is to prescribe the exact solution of a non-linear flow by adding appropriate forcing terms to the SSWE. For example, Rossby–Haurwitz waves [22], a linear combination of Hough-harmonics [6] or a translating low pressure center superimposed on a jet stream [3], [34] were applied as prescribed solutions.

Analytical solutions of simplified SSWE or the full non-linear SSWE, mentioned in (1.1)–(1.3), are not sufficient for the validation of SSWE models applied to more realistic flow situations. Instead, more elaborate initial states were used for SSWE models which describe real atmospheric phenomena, like e.g. the generation of planetary waves. For example, Takacs [29] introduced the initial state of a solid body rotation with a balanced geopotential corresponding to the global zonal geostrophic flow in [34, test 2], disturbed by an isolated mountain. Galewsky et al. [10] proposed a balanced zonal wind field with a given meridional wind profile similar to [34, test 3], disturbed by a given initial perturbation of the balanced geopotential field. In both cases, the perturbations to the barotropic zonal flow induce planetary waves. Williamson et al. [34, test 7] described integrations of several initial states which are derived from observational analyses of the 500 hPa height and wind field.

This overview on tests for the SSWE reveals that analytical solutions of the SSWE are rare and especially unsteady analytical solutions are not available in the literature.

The quantification of the model’s convergence properties is one of the main tasks of model validation. Reference solutions for a few non-analytical solutions were usually obtained with very high resolution spectral models, e.g. [17]. These numerical solutions suffer from inaccuracies and thus allow only a comparison of model results. Additionally, quantitative information about a model can be obtained by analyzing conservation properties, but this does not necessarily provide information about convergence properties. Finally, the distance of an analytical and a numerical solution yields an objective measure of the quality of the model result. For that reason, the knowledge about analytical solutions is crucial for convergence studies and thus model validation.

Here, the transformation method for the SSWE is introduced and applied to provide unsteady analytical solutions for the validation process of atmospheric models. This method is based on the transformation from a fixed cartesian to a rotating coordinate system. Fundamental examples for unsteady analytical solutions of the SSWE are derived which have not been available in the literature until now. The benefit of the new analytical solutions for the validation of numerical models is demonstrated. Due to the limited complexity of the known analytical solutions, the physical relevance of the presented examples is limited, too. However, Piani and Norton [25] showed that the simple solid body rotation is close to observed atmospheric data in the summer stratosphere.

To support a reliable formulation of the analytical solutions, two common formulations of the SSWE, namely in a cartesian coordinate system and in spherical coordinates are given in Section 2. In Section 3, we present the transformation method for the SSWE. By applying the transformation method, we derive fundamental examples for unsteady analytical solutions in cartesian and spherical coordinates in Section 4. Finally, we demonstrate the benefit of the new analytical solutions by analyzing the convergence properties of the barotropic model PLASMA (Parallel Large Scale Model of the Atmosphere) in Section 5.

Section snippets

Spherical shallow water equations

In spherical geometry, the description of functions depends on the choice of local coordinates. Because the SSWE are formulated as well in cartesian as in spherical coordinates in the literature, both formulations are described in the following. The examples of unsteady analytical solutions of the SSWE in both formulations are given in Section 4.

We introduce some notation for preparation. Let a = 6.371 × 106 m be the Earth radius, Ω = 7.292 × 10−5 s−1 the Earth’s angular velocity and I=(0,T)R a fixed

The transformation method

The transformation method for the SSWE consists of the transition from a fixed coordinate system to a rotating coordinate system. This method is well known from the literature, e.g. [13], [23], [8] for equations in R3. Due to the embedding of the sphere S in R3, the method leads to a transformation for equations on S. Practically, the transformation method adds an angular velocity to the original velocity field and generates the Coriolis term in the dynamical equations.

In a fixed coordinate

Analytical solutions

The transformation method in Section 3 allows us to derive a solution of the SSWE (2), (3), (4), if a solution of the SSWE without Coriolis force (8), (9), (10) is known. The results are unsteady solutions of the SSWE with a time period of 1 day. In the following, we derive three examples of unsteady solutions of the SSWE, an unsteady axially symmetric solution with an arbitrary wind profile and two special cases of it. The first is an unsteady solid body rotation and the second an unsteady

Convergence properties of the atmospheric model PLASMA

In this section, we quantify the convergence properties of the atmospheric model PLASMA (Parallel Large Scale Model of the Atmosphere) by means of the steady-state solution [34, case 2] and the unsteady analytical solutions from Section 4.2. Several features of the model PLASMA have been published recently, concerning the numerical method in [18], [19], concerning the grid generation in [1] and concerning the matrix solver in [9]. The reader may contact the authors by e-mail to get a FORTRAN 90

Summary

Models based on the SSWE are one way to realize a barotropic model of the atmosphere. For the validation process the quantification of convergence properties is an important task. Therefore, the necessary objective error measures are provided by analytical solutions of the SSWE. Whereas steady-state analytical solutions which consist essentially in axially symmetric solutions, are well known from the literature, unsteady analytical solutions of the SSWE have not been established until now.

We

Acknowledgements

This work is supported by the joint project PLASMA (Parallel LArge Scale Model of the Atmosphere), which is founded in the framework of DEKLIM (German climate research project) by the Federal Ministry of Education and Research of Germany (Grant No. 01LD0037). The authors thank the two anonymous reviewers for useful hints which improved the manuscript.

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