Professor Ian Roulstone

We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd–Beltrami–Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

We introduce a new approach to Monge-Ampere geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampere geometry to the Poisson equation for the pressure that arises for incompressible Navier-Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via the (higher) Lagrangian submanifold it defines in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge-Ampere structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge-Ampere geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier-Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd-Beltrami-Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

We use a variational method to assimilate multiple data streams into the terrestrial ecosystem carbon cycle model DALECv2. Ecological and dynamical constraints have recently been introduced to constrain unresolved components of this otherwise ill-posed problem. Here we recast these constraints as a multivariate Gaussian distribution to incorporate them into the variational framework and we demonstrate their benefit through a linear analysis. Using an adjoint method we study a linear approximation of the inverse problem: firstly we perform a sensitivity analysis of the different outputs under consideration, and secondly we use the concept of resolution matrices to diagnose the nature of the ill-posedness and evaluate regularisation strategies.We then study the non linear problem with an application to real data. Finally, we propose a modification to the model: introducing a spin-up period provides us with a built-in formulation of some ecological constraints which facilitates the variational approach.

The behaviour of quadratic invariants of the velocity gradient tensor is explored when the time evolution is governed by semigeostrophic forms of the shallow water equations. The evolution equation of a certain Jacobian involving the geostrophic flow is formally similar to its counterpart under the primitive shallow water equations. The resultant deformation and the Frobenius norm do not behave in this symmetrical way. A product of the study is a straightforward derivation of the semigeostrophic potential vorticity conservation property. Results are extended to 3D baroclinic flow by using isentropic coordinates.

We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to Monge-Ampere structure, and Burgers’-type vortices are a canonical class of solutions associated with this structure. The mapping of such solutions, which are characterised by a linear dependence of the third component of the velocity on the coordinate defining the axis of rotation, to solutions of the incompressible equations in two dimensions is also shown to be an example of a symmetry reduction. The Monge-Ampere structure for incompressible flow in two dimensions is shown to be hyper-symplectic.

Changes in our climate and environment make it ever more important to understand the processes involved in Earth systems, such as the carbon cycle. There are many models that attempt to describe and predict the behaviour of carbon stocks and stores but, despite their complexity, significant uncertainties remain. We consider the qualitative behaviour of one of the simplest carbon cycle models, the Data Assimilation Linked Ecosystem Carbon (DALEC) model, which is a simple vegetation model of processes involved in the carbon cycle of forests, and consider in detail the dynamical structure of the model. Our analysis shows that the dynamics of both evergreen and deciduous forests in DALEC are dependent on a few key parameters and it is possible to find a limit point where there is stable sustainable behaviour on one side but unsustainable conditions on the other side. The fact that typical parameter values reside close to this limit point highlights the difficulty of predicting even the correct trend without sufficient data and has implications for the use of data assimilation methods.

We investigate a simplified form of variational data assimilation in a fully nonlinear framework with the aim of extracting dynamical development information from a sequence of observations over time. Information on the vertical wind profile, w(z), and profiles of temperature, T(z, t), and total water content, qt(z, t), as functions of height, z, and time, t, are converted to brightness temperatures at a single horizontal location by defining a two-dimensional (vertical and time) variational assimilation testbed. The profiles of T and qt are updated using a vertical advection scheme. A basic cloud scheme is used to obtain the fractional cloud amount and, when combined with the temperature field, this information is converted into a brightness temperature, using a simple radiative transfer scheme. It is shown that our model exhibits realistic behaviour with regard to the prediction of cloud, but the effects of nonlinearity become non-negligible in the variational data assimilation algorithm. A careful analysis of the application of the data assimilation scheme to this nonlinear problem is presented, the salient difficulties are highlighted, and suggestions for further developments are discussed.

© 2013 Royal Meteorological Society and Crown Copyright, the Met Office.Three data assimilation methods are compared for their ability to produce the best analysis: (i) 4DVar, four-dimensional variational data assimilation using linear and adjoint models with either a (perfect) 3D climatological background-error covariance or a 3D ensemble background-error covariance; (ii) EDA, an ensemble of 4DEnVars, which is a variational method using a 4D ensemble covariance; and (iii) the deterministic ensemble Kalman filter (DEnKF, also using a 4D ensemble covariance). The accuracy of the deterministic analysis from each method was measured for both perfect and imperfect toy model experiments. With a perfect model, 4DVar with the climatological covariance is easily beaten by the ensemble methods, due to the importance of flow-dependent background-error covariances. When model error is present, 4DVar is more competitive and its relative performance is improved by increasing the observation density. This is related to the model error representation in the background-error covariance. The dynamical time-consistency of the 4D ensemble background-error covariance is degraded by the localization, since the localization function and the nonlinear model do not commute. As a result, 4DVar with the ensemble covariance performs significantly better than the other ensemble methods when severe localization is required, i.e. for a small ensemble.

We describe a method for reconstructing spatially explicit maps of seasonal paleoclimate variables from site‐based reconstructions. Using a 3‐D‐Variational technique, the method finds the best statistically unbiased, and spatially continuous, estimate of the paleoclimate anomalies through combining the site‐based reconstructions and a prior estimate of the paleoclimate state. By assuming a set of correlations in the error of the prior, the resulting climate is smoothed both from month to month and from grid cell to grid cell. The amount of smoothing can be controlled through the choice of two length‐scale values. The method is applied to a set of reconstructions of the climate of the Last Glacial Maximum (ca. 21,000 years ago) for southern Europe derived from pollen data with a prior derived from results from the third phase of the Palaeoclimate Modelling Intercomparison Project. We demonstrate how to choose suitable values for the smoothing length scales for the data sets used in the reconstruction.

The formulation of four-dimensional variational data assimilation allows the incorporation of constraints into the cost function which need only be weakly satisfied. In this paper we investigate the value of imposing conservation properties as weak constraints. Using the example of the two-body problem of celestial mechanics we compare weak constraints based on conservation laws with a constraint on the background state. We show how the imposition of conservation-based weak constraints changes the nature of the gradient equation. Assimilation experiments demonstrate how this can add extra information to the assimilation process, even when the underlying numerical model is conserving.

We present a new global reconstruction of seasonal climates at the Last Glacial Maximum (LGM, 21 000 years BP) made using 3-D variational data assimilation with pollen-based site reconstructions of six climate variables and the ensemble average of the PMIP3—CMIP5 simulations as a prior (initial estimate of LGM climate). We assume that the correlation matrix of the uncertainties in the prior is both spatially and temporally Gaussian, in order to produce a climate reconstruction that is smoothed both from month to month and from grid cell to grid cell. The pollen-based reconstructions include mean annual temperature (MAT), mean temperature of the coldest month (MTCO), mean temperature of the warmest month (MTWA), growing season warmth as measured by growing degree days above a baseline of 5 ∘C (GDD5), mean annual precipitation (MAP), and a moisture index (MI), which is the ratio of MAP to mean annual potential evapotranspiration. Different variables are reconstructed at different sites, but our approach both preserves seasonal relationships and allows a more complete set of seasonal climate variables to be derived at each location. We further account for the ecophysiological effects of low atmospheric carbon dioxide concentration on vegetation in making reconstructions of MAP and MI. This adjustment results in the reconstruction of wetter climates than might otherwise be inferred from the vegetation composition. Finally, by comparing the uncertainty contribution to the final reconstruction, we provide confidence intervals on these reconstructions and delimit geographical regions for which the palaeodata provide no information to constrain the climate reconstructions. The new reconstructions will provide a benchmark created using clear and defined mathematical procedures that can be used for evaluation of the PMIP4–CMIP6 entry-card LGM simulations and are available at https://doi.org/10.17864/1947.244 (Cleator et al., 2020b).

© 2013 Royal Meteorological Society.Divergence and vorticity are well known to be geometrically invariant quantities in that their mathematical forms are independent of the orientation of the coordinate axes. Various other functions of the elements of the horizontal velocity gradient tensor are invariants in the same sense: examples are the resultant deformation and the determinant and Frobenius norm of the tensor. A brief account of these quadratic invariants is given, including expressions relating them to divergence and vorticity and to one another, and noting their occurrence in the divergence equation. Assuming shallow-water dynamics with background rotation, time-evolution equations for the resultant deformation and the other quadratic invariants are derived and compared. None rivals the vorticity and potential vorticity equations for compactness, but each may be written quite concisely in terms of familiar quantities. Corresponding time-evolution equations under quasi-geostrophic shallow-water dynamics are also derived, and lead to a simple prognostic equation for the ageostrophic vorticity.

Satellite infrared sounders are invaluable tools for making observations of the structure of the atmosphere. They provide much of the observational data used to initialize atmospheric models, especially in regions that do not have extensive surface-based observing systems, such as oceans. However, information is lacking in the presence of cloud, as the cloud layer is opaque to infrared radiation. This means that where information is most desired (such as in a developing storm) it is often in the shortest supply. In order to explore the mathematics of assimilating data from cloudy radiances, a study has been performed using an idealized single-column atmospheric model. The model simulates cloud development in an atmosphere with vertical motion, allowing the characteristics of a 2D-Var data assimilation system using a single simulated infrared satellite observation taken multiple times to be studied. The strongly nonlinear nature of cloud formation poses a challenge for variational methods. The adjoint method produces an accurate gradient for the cost function and minimization is achieved using preconditioned conjugate gradients. The conditioning is poor and varies strongly with the atmospheric variables and the cost function has multiple minima, but acceptable results are achieved. The assimilation system is provided with a prior forecast simulated by adding random correlated Gaussian error to the truth. Assimilating observations comparable to those available from current geostationary satellites allows vertical motion to be retrieved with an error of less than a centimetre per second in most conditions. Moreover, evaluating the second derivative of the cost function at the minimum provides an estimate of the uncertainty in the retrieval. This allows atmospheric states that do not provide sufficient information for retrieval of vertical motion to be detected (such as a cloudless atmosphere or a non-moving opaque cloud layer in the upper troposphere). Retrieval is most accurate with upwards motion.

We present a novel method to quantify the ecophysiological effects of changes in CO2 concentration during the reconstruction of climate changes from fossil pollen assemblages. The method does not depend on any particular vegetation model. Instead, it makes use of general equations from ecophysiology and hydrology that link moisture index (MI) to transpiration and the ratio of leaf-internal to ambient CO2 (χ). Statistically reconstructed MI values are corrected post facto for effects of CO2 concentration. The correction is based on the principle that e, the rate of water loss per unit carbon gain, should be inversely related to effective moisture availability as sensed by plants. The method involves solving a non-linear equation that relates e to MI, temperature and CO2 concentration via the Fu-Zhang relation between evapotranspiration and MI, Monteith’s empirical relationship between vapour pressure deficit and evapotranspiration, and recently developed theory that predicts the response of χ to vapour pressure deficit and temperature. The solution to this equation provides a correction term for MI. The numerical value of the correction depends on the reconstructed MI. It is slightly sensitive to temperature, but primarily sensitive to CO2 concentration. Under low LGM CO2 concentration the correction is always positive, implying that LGM climate was wetter than it would seem from vegetation composition. A statistical reconstruction of last glacial maximum (LGM, 21±1kyr BP) palaeoclimates, based on a new compilation of modern and LGM pollen assemblage data from Australia, is used to illustrate the method in practice. Applying the correction brings pollen-reconstructed LGM moisture availability in southeastern Australia better into line with palaeohydrological estimates of LGM climate.

Using Monge-Ampère geometry, we study the singular structure of a class of nonlinear Monge-Ampère equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Ampère geometry by examining the role of an induced metric on Lagrangian submanifolds of the cotangent bundle. In particular, we show that the signature of the metric serves as a classification of the Monge-Ampère equation, while singularities and elliptic-hyperbolic transitions are revealed by degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations.

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two spatial dimensions. Taking the divergence of the momentum equation leads, as usual, to a Poisson equation for the pressure: in this paper we study this equation using Monge-Amp`ere structures. In two dimensional flows where the laplacian of the pressure is positive, a K¨ahler geometry is described on the phase space of the fluid; in regions where the laplacian of the pressure is negative, a product structure is described. These structures can be related to the ellipticity and hyperbolicity (respectively) of a Monge-Amp`ere equation. We then show how this structure can be extended to a class of canonical vortex structures in three dimensions.

The volume examines and explains why such simplifications to Newton's second law produce accurate, useful models and, just as the meteorologist seeks patterns in the weather, mathematicians seek structure in the governing equations, such as ...