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A standard test for AGCMs including their physical parametrizations: I: The proposal
Royal Meteorological Society discussion meeting on ‘New directions in mathematical modelling in numerical weather prediction’, 16th February, 2000.
Numerical weather prediction (NWP) is a branch of mathematical physics. This was the point-of-view adopted in the presentations and discussion at the meeting of the Society on 16 February. The aim of the meeting was to present a cursory review of new developments in subjects related to NWP which may have application in future generations of computer models. The meeting provided a forum for discussion on the latest advances in both the analysis and numerical integration of the Navier-Stokes equations, and applications to weather and climate forecasting. The topics covered by the speakers could be divided into roughly three areas: 1) Quantifying unsteady and singular behaviour in fluid dynamics; 2) New developments in numerical analysis; 3) Applications of (1) and (2) in numerical models. The discussion that ensued touched on a wide range of issues and so I shall restrict my review to two particular topics - the behaviour of vorticity in the 3D Euler equations and the concept of ‘geometric integrators’. These topics are of particular interest because it is important to represent the vertical motion as accurately as possible in numerical models, and geometric integrators offer the promise of numerical methods that respect the qualitative features of the flow that are important in meteorological contexts. These were, perhaps, the most technically challenging topics under discussion and therefore it is appropriate to try to summarise the salient points here. In the 2D Euler equations, it is well-known that the vorticity is a conserved quantity. That is, the directional derivative of the velocity in the direction of the vorticity, known as the vorticity stretching term, is zero. However, in 3D Euler this is not the case and if one defines a strain matrix, whose components are the skew-symmetric derivatives of the velocity with respect to position, then, if the vorticity aligns with an eigenvector of this matrix, the vorticity stretches (or compresses) exponentially depending on whether the corresponding eigenvalue is positive (or negative). John Gibbon (Imperial College, UK) posed the following questions: how fast is the growth? Is it fast
enough to produce a singularity in finite time? Is the direction in which the vorticity stretches or compresses important? Do any alignments occur? These are some of the outstanding classes of problems in modern applied mathematics. Gibbon reported on the progress to date with these issues. The Beale-KatoMajda theorem (1984) states that no quantity can blow up at a finite time without the integral (in time) of a norm of the vorticity also blowing up in finite time. The first point to make here is that it is a quantity based on the vorticity that controls the behaviour of the solutions. The vorticity field must, for the B-K-M theorem to apply, satisfy certain regularity, or smoothness, conditions (for example, being a square-integrable function is not sufficient). However, Constantin, Fefferman and Majda (1996) have managed to improve on the B-K-M result and reduce the regularity criterion, but at the expense of placing some technical constraints on the direction of the vorticity. A consequence of these results, from a modellers point-of-view, is that if one observes a singularity in a simulation then, for the growth and formation of the singularity to be genuine and not an artefact of the numerical scheme, the vorticity must grow like ‘(t – t0) to -α’, as t → t0, where α ≥ 1. Furthermore, these results show that solutions of the 3D Euler equations cannot develop singularities through the development of kinks or curvature singularities in the vortex lines. In order to address the questions concerning the directionality of vortex stretching, Gibbon defined a pressure hessian, that is, the matrix of second derivatives of the pressure with respect to position. A similar operator arises in the semi-geostrophic and non-linear balance equations. One can show that exponential growth in the vorticity is caused by near-alignment of the vorticity with a negative eigenvector of the pressure hessian. Furthermore, Gibbon demonstrated that the pressure hessian governs the direction of vorticity in fine scale motions. It remains to be seen what use we can make of these results in diagnosing the properties of solutions of numerical models. For example, the primitive equations actually used in NWP are an ‘averaged’ version of the Euler equations, and the average of a vortex singularity may be a quite misleading smooth distribution. Furthermore, the vorticity is not a very robust diagnostic of the flow as resolution is increased.
In meteorology and oceanography we come face-to-face with nonlinear dynamics. In applying traditional mathematical analysis to equations of motion, we want to derive precise qualitative results about a system, say: existence and uniqueness of solutions, dynamical features (e.g. asymptotic behaviour, sensitivity to parameters) and geometric structures (e.g. invariants, conservation laws, variational principles). Often, the only way of obtaining quantitative information is via numerical computation, in which we approximate the system via a discretisation. Naturally, the information we obtain depends on the discretisation, and consequently such methods cannot always be expected to respect certain qualitative features of the continuous system. These issues were central to Arieh Iserles’ (Cambridge) talk on ‘geometric integration’. Geometric integration is a new discipline at the intersection of numerical analysis and geometry, aiming to compute differential equations whilst respecting their known geometric and qualitative features. In geophysical fluid dynamics we are well-aware of the importance of conserved quantities like energy, potential temperature and potential vorticity. We also utilise certain transformations to simplify the process of solving a system of differential equations: the geostrophic momentum transformation being an important example in meteorology. Furthermore, there are features of the phase-space picture of the evolution of a system, such as Liouville's theorem stating that volume is preserved, that should be respected by numerical schemes too. The goal of geometric integration is to design schemes that will encompass some or all of these features, depending on the fluid-dynamical system under consideration. Roughly speaking, geometric integration utilises the fact that the qualitative features described above have group-theoretic descriptions and if one can devise integration schemes that solve the equations on the ‘group space’, you necessarily respect the qualitative features of the system. At the present time, the ideas have been successfully applied to the barotropic vorticity equation, and work is in progress studying the group-theoretic properties of the semi-geostrophic equations. Ian Roulstone, Meeting Organiser ([email protected])
Numerical weather prediction (NWP) is a branch of mathematical physics. This was the point-of-view adopted in the presentations and discussion at the meeting of the Society on 16 February. The aim of the meeting was to present a cursory review of new developments in subjects related to NWP which may have application in future generations of computer models. The meeting provided a forum for discussion on the latest advances in both the analysis and numerical integration of the Navier-Stokes equations, and applications to weather and climate forecasting. The topics covered by the speakers could be divided into roughly three areas: 1) Quantifying unsteady and singular behaviour in fluid dynamics; 2) New developments in numerical analysis; 3) Applications of (1) and (2) in numerical models. The discussion that ensued touched on a wide range of issues and so I shall restrict my review to two particular topics - the behaviour of vorticity in the 3D Euler equations and the concept of ‘geometric integrators’. These topics are of particular interest because it is important to represent the vertical motion as accurately as possible in numerical models, and geometric integrators offer the promise of numerical methods that respect the qualitative features of the flow that are important in meteorological contexts. These were, perhaps, the most technically challenging topics under discussion and therefore it is appropriate to try to summarise the salient points here. In the 2D Euler equations, it is well-known that the vorticity is a conserved quantity. That is, the directional derivative of the velocity in the direction of the vorticity, known as the vorticity stretching term, is zero. However, in 3D Euler this is not the case and if one defines a strain matrix, whose components are the skew-symmetric derivatives of the velocity with respect to position, then, if the vorticity aligns with an eigenvector of this matrix, the vorticity stretches (or compresses) exponentially depending on whether the corresponding eigenvalue is positive (or negative). John Gibbon (Imperial College, UK) posed the following questions: how fast is the growth? Is it fast
enough to produce a singularity in finite time? Is the direction in which the vorticity stretches or compresses important? Do any alignments occur? These are some of the outstanding classes of problems in modern applied mathematics. Gibbon reported on the progress to date with these issues. The Beale-KatoMajda theorem (1984) states that no quantity can blow up at a finite time without the integral (in time) of a norm of the vorticity also blowing up in finite time. The first point to make here is that it is a quantity based on the vorticity that controls the behaviour of the solutions. The vorticity field must, for the B-K-M theorem to apply, satisfy certain regularity, or smoothness, conditions (for example, being a square-integrable function is not sufficient). However, Constantin, Fefferman and Majda (1996) have managed to improve on the B-K-M result and reduce the regularity criterion, but at the expense of placing some technical constraints on the direction of the vorticity. A consequence of these results, from a modellers point-of-view, is that if one observes a singularity in a simulation then, for the growth and formation of the singularity to be genuine and not an artefact of the numerical scheme, the vorticity must grow like ‘(t – t0) to -α’, as t → t0, where α ≥ 1. Furthermore, these results show that solutions of the 3D Euler equations cannot develop singularities through the development of kinks or curvature singularities in the vortex lines. In order to address the questions concerning the directionality of vortex stretching, Gibbon defined a pressure hessian, that is, the matrix of second derivatives of the pressure with respect to position. A similar operator arises in the semi-geostrophic and non-linear balance equations. One can show that exponential growth in the vorticity is caused by near-alignment of the vorticity with a negative eigenvector of the pressure hessian. Furthermore, Gibbon demonstrated that the pressure hessian governs the direction of vorticity in fine scale motions. It remains to be seen what use we can make of these results in diagnosing the properties of solutions of numerical models. For example, the primitive equations actually used in NWP are an ‘averaged’ version of the Euler equations, and the average of a vortex singularity may be a quite misleading smooth distribution. Furthermore, the vorticity is not a very robust diagnostic of the flow as resolution is increased.
In meteorology and oceanography we come face-to-face with nonlinear dynamics. In applying traditional mathematical analysis to equations of motion, we want to derive precise qualitative results about a system, say: existence and uniqueness of solutions, dynamical features (e.g. asymptotic behaviour, sensitivity to parameters) and geometric structures (e.g. invariants, conservation laws, variational principles). Often, the only way of obtaining quantitative information is via numerical computation, in which we approximate the system via a discretisation. Naturally, the information we obtain depends on the discretisation, and consequently such methods cannot always be expected to respect certain qualitative features of the continuous system. These issues were central to Arieh Iserles’ (Cambridge) talk on ‘geometric integration’. Geometric integration is a new discipline at the intersection of numerical analysis and geometry, aiming to compute differential equations whilst respecting their known geometric and qualitative features. In geophysical fluid dynamics we are well-aware of the importance of conserved quantities like energy, potential temperature and potential vorticity. We also utilise certain transformations to simplify the process of solving a system of differential equations: the geostrophic momentum transformation being an important example in meteorology. Furthermore, there are features of the phase-space picture of the evolution of a system, such as Liouville's theorem stating that volume is preserved, that should be respected by numerical schemes too. The goal of geometric integration is to design schemes that will encompass some or all of these features, depending on the fluid-dynamical system under consideration. Roughly speaking, geometric integration utilises the fact that the qualitative features described above have group-theoretic descriptions and if one can devise integration schemes that solve the equations on the ‘group space’, you necessarily respect the qualitative features of the system. At the present time, the ideas have been successfully applied to the barotropic vorticity equation, and work is in progress studying the group-theoretic properties of the semi-geostrophic equations. Ian Roulstone, Meeting Organiser ([email protected])