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# Workshop Programme

## for period 2 - 6 December 2013

### Partial Differential Equations and Geophysical Fluid Dynamics

2 - 6 December 2013

Timetable

 Monday 2 December 08:30-09:05 Registration 09:05-09:15 Welcome from Christie Marr (INI Deputy Director) 09:15-10:15 Gallagher, I (Université Paris 7 - Denis-Diderot) Trapping of Rossby waves in the equatorial betaplane model Sem 1 In this talk we shall report on joint works with Christophe Cheverry, Thierry Paul and Laure Saint-Raymond in which we study an equatorial shallow water system under the betaplane approximation. We prove that in some asymptotic regimes, Rossby waves are trapped around the equator while Poincaré waves disperse. This involves the use of microlocal techniques related to semiclassical analysis, such as Mourre estimates, and some ODE methods. 10:15-11:00 Gibbon, J (Imperial College London) Rescaled vorticity moments in the 3D Navier-Stokes equations Sem 1 Co-authors: D. D. Donzis (Texas A and M), A. Gupta (University of Rome Tor Vergata), R. M. Kerr (University of Warwick), R. Pandit (Indian Institute of Science Bangalore), D. Vincenzi (CNRS, Universite de Nice) The issue of intermittency in numerical solutions of the 3D Navier-Stokes equations is addressed using a new set of variables whose evolution has been calculated through three sets of numerical simulations. These variables are defined on a periodic box $[0,\,L]^{3}$ such that $D_{m}(t) = \left(\varpi_{0}^{-1}\Omega_{m}\right)^{\alpha_{m}}$ where $\alpha_{m}= 2m/(4m-3)$ \& the set of frequencies $\Omega_{m}$ for $1 \leq m \leq \infty$ are defined by $\Omega_{m}(t) = \left(L^{-3}\I |\mbox{\boldmath$\omega$}|^{2m}dV\right)^{1/2m}$\,; the fixed frequency $\varpi_{0} = \nu L^{-2}$. All three simulations unexpectedly show that the $D_{m}$ are ordered for $m = 1\,,...,\,9$ such that $D_{m+1} < D_{m}$. Moreover, the $D_{m}$ squeeze together such that $D_{m+1}/D_{m}\nearrow 1$ as $m$ increases. This regime is shown to connected to the depletion of nonlinearity. The first simulation is of very anisotropic decaying turbulence\,; the second pair is of decaying isotropic turbulence from random initial conditions \& of forced isotropic turbulence at constant Grashof number\,; the third $4096^{3}$ simulation is of very high Reynolds number forced, stationary, isotropic turbulence. 11:00-11:30 Morning Coffee 11:30-12:15 Methven, J (University of Reading) Recent observations of mesoscale structures in intense cyclones Sem 1 The impact of intense cyclones is often associated with coherent mesoscale structures embedded within them. Although the formation of cold fronts is to some extent described by semi-geostrophic theory, the dynamics of many of these features are not well understood. For example, whether they could be described by a form balance appropriate on the mesoscale, or in the presence of non-conservative processes. Some recent detailed observations of cyclones and fronts are presented and some of the unknown aspects of their dynamics highlighted. 12:30-13:30 Lunch at Wolfson Court 14:00-14:45 Bartello, P (McGill University) Between quasigeostrophic and stratified turbulence Sem 1 While it is well-established that the frequency disparity between vortical and wave motion is key to understanding the quasigeostrophic limit, i.e. strong rotation and stratification, the starting point for this contribution is that it has recently been established that there is no such frequency disparity in stratified turbulence without rotation. It remains to ask what happens in between these two limits, long held as the prevailing dynamics between deformation-scale eddies and the microscale where isotropy is recovered. To do this, ideas from numerical weather prediction were borrowed in order to explore numerically the nonhydrostatic Boussinesq equations starting from initial conditions that are close to our current fuzzy notions of balance for a variety of Rossby and Froude numbers. It is found that evolution is immediately away from this balance in the small scales, and from steep to much more shallow spectra. It will be argued that this conclusion is robust to unce rtainties in the definition of balance. 14:45-15:30 Petcu, M (Université de Poitiers) Exponential Decay of the Power Spectrum and Finite Dimensionality for Solutions of the Three Dimensional Primitive Equations Sem 1 The purpose of this talk is to estimate the number of modes, volumes and nodes, sufficient to describe well the solution of the three dimensional primitive equations; the physical meaning of these estimates is also discussed. We also study the exponential decay of the spatial power spectrum for the three dimensional primitive equations. 15:30-16:00 Afternoon Tea 16:00-16:45 Straub, D (McGill University) Energy cascades in the baroclinic ocean wind-driven double gyre problem Sem 1 Co-author: David Straub We consider the classic baroclinic quasigeostrophic wind-driven ocean double gyre problem over a range of deformation radii, wind stress amplitudes, and bottom friction coefficients with an aim of understanding transfer of energy across scales. In this beta-plane basin setting, we find significant differences from classic studies of gestrophic turbulence, which generally assume zonal periodicity. In a basin geometry, the beta term (related to a latitudinal dependence in the Corioils parameter) can play a key role. For example, it can be the dominant term allowing for energy transfer between the basin scale and the baroclinic mesoscale. We also find that barotropization of baroclinic mesoscale energy forces the barotropic mode at scales where the barotropic mode is most energetic. Related to this, the barotropic nonlinear inverse energy cascade does not extend between mesoscale injection and large scale dissipation wavenumbers, as is often assumed. Instead, it is part of a double cascade" of barotropic energy in which the nonlinear inverse cascade is nearly offset by a forward cascade associated with the beta term. This is particularly evident in weak bottom drag simulations, for which a time eddy-mean flow decomposition reveals the double cascade to beassociated with the eddy-only terms. 17:00-18:00 Welcome Drinks Reception
 Tuesday 3 December 09:15-10:15 Titi, E (UC, Irvine and Weizmann Institute of Science) Mathematical Study of Certain Geophysical Models: Global Regularity and Finite-time Blowup Results Sem 1 The basic problem faced in geophysical uid dynamics is that a mathematical description based only on fundamental physical principles, the so-called the \Primitive Equations", is often prohibitively expensive computationally, and hard to study analytically. In this talk I will discuss the main obstacles in proving the global regularity for the three-dimensional Navier-Stokes equations and their geophysical counterparts. However, taking advantage of certain geophysical balances and situations, such as geostrophic balance and the shallowness of the ocean and atmosphere, geophysicists derive more simpli ed and manageable models which are easier to study analytically. In particular, I will present the global well-posedness for the three-dimensional Benard convection problem in porous media, and the global regularity for a three-dimensional viscous planetary geostrophic models. Even though the primitive equations look as if they are more dicult to study analytically than the three-dimensional Navier-Stokes equations I will show, on the one hand, that the viscous primitive equations have a unique global (in time) regular solution for all initial data. On the other hand, I will show that in the non-viscous (inviscid) case there is a one-parameter family of initial data for which the corresponding smooth solutions develop nite-time singularities (blowup). 10:15-11:00 Temam, RM (Indiana University) Change of phase for the humid atmosphere Sem 1 In this lecture we will recall the atmospheric equations of water vapor with saturation. In their simplest form, these equations form a nonlinear system of partial differential equations with discontinuities. We will address the issue of the modelling of the system in the presence of singularities, and some questions on the existence, uniqueness and regularity of these solutions. 11:00-11:30 Morning Coffee 11:30-12:15 Chumakova, L (University of Edinburgh) Leaky lid: new dissipative modes in the troposphere Sem 1 Much of our understanding of tropospheric dynamics is based on the concept of discrete internal modes. Internal gravity waves, such as those associated with convective systems, propagate at definite speeds, typically associated with the first to third baroclinic vertical modes. These waves are the dynamical backbone of the tropospheric dynamics, even though their nature and speed can be altered significantly by nonlinearity, moist convection, mean wind shear, etc. These discrete modes are a signature of systems of finite extent, and are derived in a case when the atmosphere is bounded above by a rigid lid. In reality, the atmosphere does not have a definite top, and, some argue, should be modeled as semi-infinite, leading to a continuous spectrum. Are the discrete rigid lid modes then just a fallacy of overly simplified theoretical models? We present a correction to the rigid lid by using a boundary condition at the top of the troposphere, that allows for a fraction of waves to escape to the stratosphere. The new discrete leaky” modes decay with characteristic time-scales, which are in the ballpark of many atmospheric phenomena. We present both the non-rotating and rotating cases. 12:30-13:30 Lunch at Wolfson Court 14:00-14:45 Esler, JG (University College London) Adaptive stochastic trajectory modelling of transport in geophysical flows Sem 1 Motivated by the goal of improving and augmenting stochastic Lagrangian models of particle dispersion in turbulent geophysical flows, techniques from the theory of stochastic processes are applied to a model transport problem. The aim is to find an efficient and accurate method to calculate the total tracer transport between a source and a receptor when the flow between the two locations is weak, rendering direct stochastic Lagrangian simulation prohibitively expensive. Two methods are found to be useful. The first is Milstein's measure transformation method', which involves adding an artificial velocity to the trajectory equation, and simultaneously correcting for the weighting given to each particle under the new flow. Various difficulties associated with making an appropriate choice for the artificial velocity field are detailed and addressed. The second method is a variant of Grassberger's go-with-the-winners' branching process, which acts to remove particles unlikely to contribute to the net transport, and reproduces those that will contribute. A simple solution to the problem of defining a `winner' for flows in a high Peclet number chaotic advection regime is proposed. It is demonstrated that, used independently or together, the two methods can act to reduce the variance of estimators of the total transport by several orders of magnitude compared with direct simulation. 14:45-15:30 Vanneste, J (University of Edinburgh) Modelling the interactions of near-inertial waves and vortical motion in the ocean Sem 1 Co-author: Eric Danioux (University of Edinburgh) Wind forcing of the ocean generates a spectrum of inertia-gravity waves that is sharply peaked near the local inertial (or Coriolis) frequency. The corresponding near-inertial waves (NIWs) make a dominant contribution to the vertical velocity and vertical shear in the ocean; they therefore play an important role for mixing, biological productivity, pollutant dispersion and, arguably, the thermohaline circulation. An asymptotic model proposed by Young and Ben Jelloul describes the slow evolution of NIWs that results from weak dispersion and from their interactions with the quasi-two-dimensional vortical motion. We derive this YBJ model by applying a form of Whitham averaging to the variational formulation of the primitive equations for a rotating stratified fluid. This provides a direct route to the YBJ equation and elucidates its variational structure and conservation laws. We then consider the effect of turbulent vortical motion (modelled as a homogeneous random field) of a scale similar to that of the waves. Specifically, we derive a transport equation for NIWs that describes their scattering by the vortical motion and show how this scattering leads to an isotropization of the NIW field. Direct numerical simulations of the YBJ equations are used to test the predictions of the transport equation. Possible models of the two-way coupling between NIWs and vortical motion are also discussed. 15:30-16:00 Afternoon Tea 16:00-16:45 Tobias, S (University of Leeds) Direct Statistical Simulation of Jet Formation in Local and Global Geometries Sem 1 Co-author: Brad Marston (Brown University) We present Direct Statistical Simulation (DSS) of jet formation. We consider the simplest barotropic model both on a spherical surface and a local beta-plane. DSS involves the direct solution of the low-order statistics via an expansion in cumulants. In both cases we compare the results of our DSS with statistics obtained from long DNS simulations. We discuss in what circumstances truncating the cumulant expansion at second order (thereby including eddy – mean-flow interaction but neglecting eddy-eddy interactions for the fluctuating fields) gives a good description of the dynamics of the flow. We demonstrate that this depends on the degree of lack of statistical equilibrium in the flow (as measured by the Zonostrophy parameter). We discuss briefly how to proceed to higher order to include eddy-eddy interactions and the possibility of forward and inverse cascades.
 Wednesday 4 December 09:15-10:15 Holm, D (Imperial College London) Slice models Sem 1 Co-author: Colin Cotter (Imperial College) A variational framework is defined for vertical slice models with three dimensional velocity depending only on horizontal $x$ and vertical $z$. The models that result from this framework are Hamiltonian, and have a Kelvin-Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler--Boussinesq equations with a constant temperature gradient in the $y$-direction (the Eady--Boussinesq model), which is an idealised problem used to study the formation and subsequent evolution of weather fronts. We then introduce a new compressible extension of this model for testing compressible weather models running in a vertical slice configuration. (Joint work with CJ Cotter, Imperial College). 10:15-11:00 Zimmer, J (University of Bath) From simple particle models to PDE dynamics Sem 1 One often aims to describe systems out of equilibrium by the governing energy E and entropy S, as well as the corresponding evolution laws for E and S. How can we derive these ingredients of the macroscopic evolution from particle models? In recent years, a dynamic scale-bridging approach has been developed and applied to a number of problems; large deviation theory plays an important role. The talk will present some of these results, focussing on the derivation of the Wasserstein-entropy formulation of diffusion and the Vlasov-Fokker-Planck equation as a system driven by energy and entropy. Time permitting, an approach of deriving stochastic equations mimicking the fluctuations in underlying mesoscopic models will be sketched. This is joint work with Hong Duong, Rob Jack and Mark. A. Peletier. 11:00-11:30 Morning Coffee 11:30-12:15 Marston, B (Brown University) Direct Statistical Simulation of a Two-Layer Primitive Equation Model Sem 1 Co-authors: Wanming Qi (Brown University), Steve Tobias (University of Leeds) Low-order statistics of the large-scale circulation of planetary atmospheres may be directly accessed by solving the equations of motion for the equal-time statistics. We implement such Direct Statistical Simulation of a two-layer primitive equation model by systematic expansion in the cumulants. The first cumulant is the zonally averaged vorticity, divergence, and temperature as a function of latitude and level, and the second cumulant contains information about nonlocal teleconnections. At second order (CE2) the expansion retains the eddy – mean-flow interaction but neglects eddy-eddy interactions and is realizable. Eddy-eddy interactions appear at third (CE3) order, but care must be taken to maintain realizability with a non-negative probability distribution function. The cumulant expansion is conservative, order-by order, in the total angular momentum, total energy, and mean-squared potential temperature. An intermediate approximation, CE2.5, is related to the Edd y-Damped Quasi-Normal Markovian (EDQNM) approximation and maintains realizability at the expense of the introduction of a phenomenological timescale for eddy damping. First and second cumulants accumulated by time-integration of the two-layer primitive equations are compared with those obtained at the fixed points found at CE2, CE2.5, and CE3 levels of approximation, and against statistics obtained from reanalysis of the mid-level atmosphere of the Earth. CE2 reproduces qualitative features of the zonal mean general circulation such as the mid-latitude jets. CE2.5 and CE3 improve quantitative agreement in both the zonal means, and in the teleconnections. 12:30-13:30 Lunch at Wolfson Court 14:00-14:45 Wirosoetisno, D (Durham University) Navier-Stokes equations on a rotating sphere Sem 1 We showed that, as the rotation rate $1/\epsilon$ increases, the solution of the 2d Navier-Stokes equations on a rotating sphere becomes zonal, in the sense that the non-zonal component of the energy becomes bounded by $\epsilon$. This is obtained by estimating near-resonant interactions in the nonlinear term. As a consequence, the global attractor reduces to a single stable steady state when the rotation is fast enough (but still finite). 14:45-15:30 Wingate, B (University of Exeter) The influence of fast waves and fluctuations on the evolution of three slow solutions of the Boussinesq equations Sem 1 Co-authors: Jared P. Whitehead (Brigham Young University), Terry Haut (Los Alamos National Laboratory) We present results from a study of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We examine three known slow limits of the rotating and stratified Boussinesq equations: strongly stratified flow ($Fr \rightarrow 0, Ro \approx O(1)$), strongly rotating flow ($Ro \rightarrow 0, Fr \approx O(1)$) and Quasi-Geostrophy ($Ro \rightarrow 0, Fr \rightarrow 0, Fr/Ro = f/N$ finite). In order to understand how the flow approaches and interacts with the slow dynamics we decompose the full solution into a component that is projected onto the null space of the fast operator and everything else. We use this decomposition to find evolution equations for the flow (and corresponding energy) on and off the slow manifold. Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast waves act as a conduit, moving energy onto the slow manifold. This decomposition clarifies how the energy is exchanged when either the stratification or the rotation is weak. In the quasi-geostrophic limit the energetics are less clear, however it is observed that the energy off the slow manifold equilibrates to a quasi-steady value. At the end I will discuss how greater understanding of flow/fast dynamics could impact emerging numerical algorithms designed for future computer architectures. 15:30-16:00 Afternoon Tea 16:00-16:45 Zeitlin, V (Laboratoire de Météorologie Dynamique (LMD-ENS)) Resonant phenomena in the wave dynamics in the presence of boundaries in GFD Sem 1 Co-author: Grigory Reznik (In-t of Oceanography, Moscow) I will be discussing resonant interactions of coastal and shelf waves with free inertia-gravity waves and mean coastal currents in GFD. 19:30-22:00 Conference Dinner at Cambridge Union Society hosted by Cambridge Dining Company
 Thursday 5 December 09:15-10:15 Cullen, M (Met Office) Use of the semi-geostrophic model in understanding large-scale atmosphere and ocean flows Sem 1 The semi-geostrophic model is an accurate approximation to the Navier-Stokes equations when the Lagrangian Rossby number is small and the aspect ratio is less than f/N (where f is the Coriolis parameter and N the Brunt-Vaisala frequency). In practice this leads to a horizontal scale of around 1000km in the atmosphere and 100km in the ocean. The approximation is second order accurate in the limit epsilon tends to zero where the Rossby number is O(epsilon) and the Froude number O(sqrt(epsilon)). This approximation has a big advantage over the quasi-geostrophic approximation because it allow O(1) variations of the static stability, Coriolis parameter and orographic height. These features are essential in describing large-scale flows. The stability of the solutions of this system is consistent with the observed persistence of large-scale anomalies in the atmosphere and ocean. The failure of the approximation on smaller scales is associated with the lack of such structures on smal ler scales in the observed system. The talk will demonstrate how the semi-geostrophic model can be used to validate numerical methods, and how it can be extended to include a realistic model of the atmospheric boundary layer. 10:15-11:00 Feldman, M (University of Wisconsin-Madison) Lagrangian solutions for semigeostrophic system in physical space Sem 1 Co-author: Adrian Tudorascu In this introductory talk, we review results on existence of solutions to semigeostrophic system in physical and dual spaces. In particular, we show that Lagrangian solutions in physical space can be constructed for initial data satisfying a strict convexity condition. We also briefly discuss the recent joint work with A. Tudorascu, in which we relax the notion of Lagrangian solution to obtain existence for all convex initial data in physical space. 11:00-11:30 Morning Coffee 11:30-12:15 Tudorascu, A (West Virginia University) Renormalized relaxed Lagrangian solutions for SG in physical space Sem 1 A new, relaxed notion of Lagrangian solutions for SG in physical space will be introduced. The main motivation is the search for physical space solutions in the case of singular data in dual space. Existence and a weak stability result will be proved, along with an energy conservation result which comes as a consequence of the renormalization property. This presentation is based on joint work with M. Feldman (U. Wisconsin-Madison). 12:30-13:30 Lunch at Wolfson Court 14:00-14:45 Oliver, M (Jacobs University Bremen) Balance relations for rotating fluid flow Sem 1 We discuss different nonlinear elliptic balance relations for rotating shallow water flow and describe computational tests which compare their utility as an initialization or diagnostic tool in nearly geostrophic situations. 14:45-15:30 Vasylkevych, S (Jacobs University Bremen) Generalized Large Scale Semigeostrophic Equations: geometric structure and global well-posedness Sem 1 Co-authors: Marcel Oliver (Jacobs University), Mahmut Calik (Jacobs University) We derive and study a family of approximate Hamiltonian balance models (called GLSG) for rotating shallow water in the semigeostrophic limit with spatially varying Coriolis parameter and non-trivial bottom topology. The models can be formulated in terms of an advected potential vorticity with a nonlinear vorticity inversion relation and include L_1 and LSG models proposed by R. Salmon as special cases. We prove existence and uniqueness of global classical solutions to the GLSG equations for certain members of the family and study the PV invertibility as a function of the parameters. 15:30-16:00 Afternoon Tea 16:00-16:45 Pelloni, B (University of Reading) The semi-geostrophic system for large-scale atmospheric flows Sem 1 Co-author: Mike Cullen (Met Office) I could present results on the existence of solutions in 3D, free boundary setting as well as some preliminary results on the validity of the model as a reduction of the Euler system.
 Friday 6 December 09:00-09:45 Vallis, G (University of Exeter) The Range of Planetary Circulations Described by the Dry Primitive Equations Sem 1 Co-authors: Jonathan Mitchell (UCLA), Sam Potter (Princeton University) The dry primitive equations can, with appropriate forcing and dissipation, provide a reasonable simulation of the large-scale features of the Earth's atmosphere and ocean. In this talk I will describe the behaviour of these PDEs when they are taken out of the terrestrial parameter regime. In n particular, I will describe their behaviour when the thermal Rossby number, Ekman number and a radiative relaxation timescale are varied considerably, moving into a parameter regime more appropriate for Mars or Titan. I will pay particular attention to the formation of zonal jets, and in particular of equatorial superrotation, which is a feature of some other planets. It is well-known that zonal jets robustly arise in rotating atmospheres if there is a wavemaker at a particular latitude. Rossby waves are then generated that propagate away, and eastward momentum converges on the source region producing a zonal jet. The Earth's jet stream is, in part, formed this way. However, on slowly rotating atmospheres it seems unlikely that superrotation is produced by that mechanism. Rather, simulations indicate that, at small thermal Rossby number, a mechanism involving equatorial Kelvin waves is involved. 09:45-10:30 Roulstone, I (University of Surrey) Differential Geometry of the Semi-Geostrophic and Euler Equations Sem 1 The role of contact and symplectic geometry of the semi-geostophic (SG) equations, in describing their Legendrian and Hamiltonian properties, will be reviewed. Using the geometry of 2-forms in 4 dimensions and the geometry of 3-forms in 6 dimensions, we show that the incompressible Euler equations in 2 and 3 dimensions admit geometric structures akin to the those present in the SG equations. 10:30-11:00 Morning Coffee 11:00-11:45 Bokhove, O (University of Leeds) On Time Integration and the Use of Clebsch Variables in Shallow Water Equations Sem 1 Two topics will be covered in this lecture. The shallow water equations will be used as a test bed to introduce the ideas. (i) For forced variational systems such as the potential flow shallow water wave equations, variational and symplectic time integrators will be extended using a new finite element approach. Here, a standard variational finite element discretization will be applied in space. (ii) The shallow water equations formulated in terms of Clebsch variables will be discussed. The advantage of Clebsch variables is that they lead to canonical Hamilton's equations for shallow water dynamics, in the Eulerian framework. A disadvantage is that the the system, is less compactly expressed in comparison to the usual formulation in terms of the velocity and fluid depth. I will make a link between a symmetry in the Hamiltonian and the associated mass weighted potential vorticity conservation law, also within the Eulerian framework. This will be done in two dimensions (2D) and in a quasi-2D symmetric form. 11:45-12:30 Rubtsov, V (Université d'Angers) Monge-Ampère equations: geometry, invariants, and applications in 3D meteorological models Sem 1 I describe a symplectic geometric approach to Monge-Ampère equations, discuss some classical and modern geometric structures related to this class of non-linear equations, their invariants and their role in 3D meteorological models. My talk is based on joint works with B. Banos and I . Roulstone. 12:30-13:30 Lunch at Wolfson Court

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