## Applied Projects

## Nils Andersson: Gravitational wave sources

With several hypersensitive kilometer-long interferometric gravitational wave detectors due to come on line within the next year, we are standing on the doorstep of a revolution in astronomy. Once Einstein's elusive gravitational waves are caught, we can hope to learn much about the very extremes of physics. This project provides the student with an insight into this fascinating area of applied general relativity. Following an introduction to the nature of gravitational waves, the main focus will be on the various sources that are likely to produce detectable waves, such as colliding black holes or rapidly spinning neutron stars. The aim of the project is to produce an up-to-date survey of the many exciting possibilities. Although it is not an absolute requirement for this project, the student would benefit from the first half unit MATH 3006 Relativity, Black Holes and Cosmology , which includes a brief consideration of gravitational waves. [Level 3]

Background reading:

R. A. d'Inverno, Introducing Einstein's Relativity , Clarendon Press

B. F. Schutz, A First Course in General Relativity , CUP

K. S. Thorne, in Three Hundred Years of Gravitation (ed.S.Hawking, W.Israel), CUP

## Nils Andersson: Gravitational wave asterology

Recent evidence supports the notion that most stars pulsate. Such stellar oscillations share many properties with waves in the Earth's oceans and atmosphere. This project is aimed at providing the student with an introduction to the stellar pulsation, with particular focus on those modes of oscillation that may prove relevant for gravitational-wave astronomy. Starting from the standard Euler equations from hydrodynamics, we will study the nature of waves in spherical bodies, the possible relevance of various pieces of physics (equation of state of matter, viscosity) etc. One possible extension of this work would make contact with current research on oscillations in rapidly rotating (and therefore no longer spherical) stars. Although it is not an absolute requirement for this project, the student would benefit from the first half unit MATH 3006 Relativity, Black Holes and Cosmology , which includes a brief consideration of gravitational waves. [Level 3]

Background reading:

R. A. d'Inverno, Introducing Einstein's Relativity , Clarendon Press

B. F. Schutz, A First Course in General Relativity , CUP

J. P. Cox, Theory of Stellar Pulsation , Princeton

## Leor Barack: The causal structure of black holes (not available 2013-14)

The aim of this project is to expose you to some of the most fascinating consequences of Einstein's General Relativity. You will first familiarise yourself with the idea of *conformal diagrams*---one of the most elegant and useful methods of relativistic theory. With the help of these, you will define notions such as those of "null infinity", black hole and white hole, event horizon and Cauchy horizon; and will then be able to explore the causal structure of spacetime outside and inside different types of black holes. Looking first at a non-rotating black hole, you will learn about the maximal extension of the Schwarzschild geometry, and explore the fate of an observer falling through the event horizon. You will then investigate into the much richer structure of charged and rotating black holes, strolling through a wonderland of ring singularities, closed timelike curves, wormholes and multiverses. [Level 3 or 4]

Prerequisite course: General Relativity.

Background reading:

Gravitation (Misner, Thorne and Wheeler)

An Introduction to Einstein's General Relativity (Hartle)

A Relativist's Toolkit (Poisson)

## Leor Barack: Black hole orbital mechanics (not available 2013-14)

The project will explore the behavior of objects moving in the vicinity of a black hole, looking at orbits of test-mass satellites as well as the trajectories of light particles. Using the mathematical tools of General Relativity, you will try to describe and understand curious phenomena such as the gravitational deflection of light; periastron precession; the existence of an innermost stable orbit; the bizarre "zoom--whirl" behavior; and the possibility of light rays moving in a closed circular orbit. The project will begin with a review of Keplerian celestial mechanics, and, through analogy, proceed to investigate general-relativistic orbits around a non-rotating (Schwarzschild) black hole. [Level 3 or 4]

Prerequisite course: General Relativity.

Background reading: Gravitation (Misner, Thorne and Wheeler)

An Introduction to Einstein's General Relativity (Hartle)

A Relativist's Toolkit (Poisson)

## Giampaolo D'Alessandro: Using conformal maps to solve potential problems

**Aim:** Analytical and (optionally) numerical study of how to use conformal maps to solve partial differential equations for two dimensional potentials. [Level 3 or 4]**Description:** A conformal map is an analytic function with non-zero derivative. This rather abstract definition hides the fact that these functions can be used in one of the most intriguing applications of complex analysis, namely solving potential problems. A potential, in this respect, is a function of two variables, defined in a region of the plane, that satisfies Laplace's equation. Examples of potentials are the electrostatic potential and the fluid velocity potential.

It is normally quite easy to solve Laplace's equation in rectangular or circular domains. It becomes much more complicated to do so when the domain does not have such a simple shape. It is in this case that conformal maps come to the rescue: we can solve the problem in a simple region and transform it, using the map, in the solution for the same problem in a much more complicated region.

In this project the student will learn the main theory of conformal maps and apply it to some simple problems. If time allows we will also explore how conformal maps are implemented numerically.

## Giampaolo D’Alessandro: Solving partial differential equations numerically without using a mesh

**Aim:** It is possible to solve partial differential equations numerically by scattering a set of points in the domain of definition and solving a linear system? In this project you will learn how to do this.

**Description: **In the numerical methods module you learn the method of finite differences to solve partial differential equations. This method is simple to implement on a rectangular domain. Suppose, though, that you need to solve Laplace’s equation in a three dimensional “blob”, an object that has no “nice” geometrical shape. In this case, it is rather hard to define a mesh and, hence, apply a finite difference method.

Many other methods have been developed to solve partial differential equations in arbitrary geometries. In this project we will study Radial Basis Function methods: these consist in scattering a random set of points (called collocation points) in the domain of definition of the solution (the “blob”) and imposing that the function satisfies the equation at these points. It is a very simple method to implement, but requires us to do some mathematics first: in particular, we need to define appropriately what it means to “satisfy the equation at the collocation points”. We will study the mathematics behind these methods and also, if time allows, code them in Matlab. In a one semester project we will consider only one type of Radial Basis Function methods. In the two semester project will compare different methods and/or, depending on the interests of the students, apply them to solve some specific mathematical model. Prerequisites: MATH3018 - Numerical methods. [Level 3 or 4]

Further reading:

M. D. Buhmann, Radial Basis Functions : Theory and Implementations, Cambridge University

Press, West Nyack, NY, USA, 2003.

## Carsten Gundlach: Making small black holes

A massive object will collapse under its own weight, and turn into a black hole, if it is sufficiently dense. Interesting things happen at the black hole threshold: when the object is only just dense enough, or only just fails to be dense enough. The black hole threshold shows many mathematical similarities to a critical phase transition, for example in a fluid: a small change in temperature and pressure turns a liquid into a gas, or vice versa. Understanding these phenomena requires a little general relativity and a little knowledge of dynamical systems, and brings together two widely separated areas of physics: gravitational collapse and statistical mechanics. Some of the basic mechanisms can also be explored using a nonlinear wave equation as a toy model. A one semester project [Level 3] could be centered on reading, and a two semester project [Level 4] on writing a numerical code to simulate gravitational collapse.

(MATH3006 Relativity, Black Holes and Cosmology is helpful but not required.)

## Carsten Gundlach: Detecting gravitational waves

General relativity predicts that whenever masses are accelerated, they produce gravitational waves: distortions in space itself that travel at the speed of light. In principle, these waves can be detected because they deform objects they pass through. In practice, the waves are very weak: even violent astrophysical events such as the collision of two stars, once the waves reach us, produce a deformation of objects of only one part in 10^{24} or so. Nevertheless, if current mathematical models of the sources are correct, and if the instruments work as planned, it seems likely that gravitational waves will be detected in the next few years by instruments already under construction. These instruments use laser interferometers to measure tiny changes (a fraction of the size of a nucleus) in the length of a pair of arms 4 kilometers long. The project could focus on the theory of why gravitational waves can be measured at all, or on how specific detectors work, and on the weird and interesting sources of experimental noise one has to consider when trying to measure something to a precision of one part in 10^{24} [Level 3 or Level 4]

(MATH3006 Relativity, Black Holes and Cosmology is helpful but not required.)

## Ian Hawke: Dam breaking and shocks

Some physical phenomena involve large changes on very short scales, which are often modelled using discontinuous functions. Examples include tsunamis and shock waves in gas and fluid dynamics. This project will look at the structure and evolution of simple systems containing shocks, particularly the dam break problem for the shallow water equations.**Pre-requisites:** A good knowledge of PDEs. At level 3 some use of Maple will be expected. At level 4 some more detailed knowledge of numerical methods will be required.**Background reading:**

E. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer)

R. Leveque: Finite Volume Methods for Hyperbolic Problems (CUP)

## Wynn Ho: Neutron star cooling

Neutron stars are created in the collapse and supernova explosion of massive stars, and they begin their lives very hot but cool rapidly. The processes that govern this cooling depend on the detailed but uncertain properties of matter at the extremely high densities that exist inside neutron stars. By comparing astronomical observations to predictions from theoretical models, we can learn about not just the neutron star interior but also fundamental physics. The aim of this project is to understand the energy and heat flow equations that govern neutron star cooling and calculate (including computational work) some simple evolutions.

Background reading: Page, Geppert, and Weber, "The cooling of compact stars", Nuclear Physics A, vol 777, pg 497-530 (2006)

## Wynn Ho: Modelling stellar and planetary spectra

Measurements of the brightness as a function of photon energy (or spectrum) of stars and planets can reveal important information about their physical properties, for example, the temperature and composition of the star or planet. The aim of this project is to study the radiative transfer equation which determines the spectrum of stars and planets and to use this equation to calculate (including computational work) spectra for simple situations.

Background reading: Rybicki and Lightman, Radiative Processes in Astrophysics, Wiley

## Chris Howls: Lenses with Negative Refractive Index (Not available 2013-14)

The art of making lenses involves geometry and Snell's law sin(*i*)/sin(*r*)=*n* where *i* is the angle of the incident ray to the surface normal, *r* is the angle of the refractive ray and *n* is the refractive index of the lens. In naturally occurring material *n* >1 (*n* = 1 for a vacuum).

Lens design effectively involves fixing the lens shape in order to focus the rays, etc. Recent developments in photonic materials have led to the possibility of man-made substances with negative refractive index (*n* < -1), which would have remarkable focussing properties.

The purpose of this project is to investigate how the geometrical optics of lens design would be affected by using materials with negative refractive index, using geometry and ray-tracing methods. This project requires no prior knowledge of physics. [Level 3]

## Chris Howls: Conflict modelling (Not available 2013-2014)

You may already be familiar with predator-prey models of populations. Similar coupled differential equation models have been applied to attrition rates in military combats, most notably by Lanchester. Different types of warfare require different consideration. For example, conventional warfare on an island with regiment-sized deployments should be treated differently from a model of an asymmetrical guerilla war where the opportunities for each side to kill one-other are much smaller. Effects such as reinforcements and resupply can also be included. Such models have been used retrospectively to exam the causes of defeat of large powers in certain conflicts in the post World War 2 era.

This project will consider a variety of warfare scenarios and apply Lanchester-type models to them. One aim is to carry out a time-dependent model of a recent guerilla conflict based on available public data. If time allowed, a stochastic element might also be considered. Skills involved: interest in mathematical modelling, ability to solve differential equations, phase plane analysis, use of software packages, ability to research data, ability to search out papers, ability to write reports.

Background reading:

Lanchester FW, 1956,Mathematics in warfare, in The world of mathematics vol 4, Newman JR, ed. (Simon and Shuster: New York)

Tung, KK, 2007 Topics in Mathematical Modelling, (Princeton University Press).

## Ian Jones: Testing General Relativity using binary pulsars

Ever since its conception in the early 20th Century, experimenters have attempted to put general relativity to the test. The most impressive tests so far have involved making use of laboratories supplied by Nature herself - the so-called binary pulsar systems. These are distant neutron star-neutron star binaries, where (at least) one star is seem as a pulsar, i.e. a rotating lighthouse illuminating the Earth with radio waves once per rotation. This rotation is remarkably steady. This in effect means that Nature has supplied us with a very accurate clock moving in a relativistic spacetime. By analysing the received time of arrival of these pulses, astronomers have carried out remarkably precise tests of many predictions of General Relativity, including energy loss to gravitational waves, which won Hulse & Taylor the 1993 Nobel Prize. The aim of this project is to understand the theory behind these tests and what they tell us about relativistic gravity. The semester 2 course MATH3006 Relativity, Black Holes and Cosmology is very useful but not essential for this project. [Level 3/4]

Background reading:

R. A. d'Inverno, Introducing Einstein's Relativity , Clarendon Press

B. F. Schutz, A First Course in General Relativity , CUP

A. Lyne and F. Graham-Smith, Pulsar Astronomy, CUP

## Giles Richardson: Modelling of Magnetically targetted drug delivery

In conventional (systemic) drug delivery the drug is administered by intravenous injection; it then travels to the heart from where it is pumped to all regions of the body. Where the drug is aimed at a small target region this method is extremely inefficient and leads to much larger doses (often of toxic drugs) being used than necessary. In order to overcome this problem a number of targeted drug delivery methods have been developed. One of these, magnetically targeted drug delivery, involves binding a drug to micron-sized biocompatible magnetic particles, injecting these into the blood stream and using a high gradient magnetic field to pull them out of suspension in the target region. Once on the vessel wall the drug can either be released directly into the blood stream or a biological technique can be used to ensure uptake of the particles into the tissue.

The aim of this project is to formulate a simple model for the transport of drug carrying magnetic particles in the blood stream, use it to track the distribution of particles within the bloodstream and to assess where the particles are pulled out of suspension. [Level 3 or level 4]

## Giles Richardson: Organic Solar Cells

In semiconductors electric current is transported both by negatively charged excited electrons lying in the conduction band of the material and by the positively charged ‘holes’ that they leave behind in the valence band of the material. In an organic solar cell two organic semiconducting materials, with very different properties, are joined together to form a junction. These materials are chosen so that one shows a strong affinity for holes and the other a strong affinity for excited electrons and are termed hole and electron carriers, respectively. The junction between the two materials is used to separate hole-electon pairs created by incident sunlight so that positively charged holes end up in the hole carrier while negatively charged electrons end up in the electron carrier. This charge separation between the two halves of the solar cell can be used to drive an electric current around a circuit and is thus a means of converting solar power to electric power. The project will focus on the modelling of the charge transport in the device and the generation of charged pairs via solar radiation. Charge transport within semiconductors can be modelled by straight-forward advection diffusion equations for the motion of the electrons, and the holes, that account for the diffusive motion of these particles and their motion due to the forces they experience from the electric field in the device. In turn the electric field in the device is coupled to the densities of the charged particles. [Level 3 or 4].

## Janne Ruostekoski: Numerical integration of nonlinear propagation equation

Nonlinear propagation equations play an important role in numerous applications and frequently require numerical solutions. In this project you will learn how to integrate numerically the Gross-Pitaevskii equation (also known as the nonlinear Schroedinger equation),

using the Crank-Nicholson algorithm. The Gross-Pitaevskii equation is an important PDE that describes the dynamics of coherent matter waves, superfluidity and light propagation in nonlinear optics. [Level 3 or Level 4]

Prerequisites : Differential equations. Some knowledge of a numerical software (e.g., Matlab or Maple) or a programming language (e.g. Fortran).

## Ruben Sanchez-Garcia: Community detection in networks

Community structure is one of the most relevant features of graphs representing real-world systems (networks). It consists on identifying groups of vertices (clusters or communities) highly connected among themselves while poorly connected to other vertices.

This project consists firstly on a description of the basic terminology on network theory and community detection, and an overview of the most popular methods, with an emphasis on spectral and modularity-based methods.

Then the student will choose one of the community detection methods to be studied in depth and implemented as a computer algorithm. The algorithm will then be validated on a small set of real-world network examples.

[Level 3 or 4]

Reference: Fortunato, Santo. Community detection in graphs. Physics Reports (2010).

Pre-requisites: None, but programming experience (e.g. Matlab in MATH3018) desirable.

## Kostas Skenderis: Spacetimes of constant curvature

Spacetimes of constant curvature are solutions of Einstein's equations with a cosmological constant. The solution with positive cosmological constant, the de Sitter spacetime, was found by de Sitter in 1917 and describes a Universe undergoing a rapid, exponential expansion. The dynamics of such a University is dominated by the cosmological constant which corresponds to dark energy in our Universe or the inflaton field in the very early Universe. When the cosmological constant is negative, the solution, the Anti-de Sitter solution, describes a negatively curved Universe. This solution has played a prominent role in many recent developments in theoretical physics. The aim of this project is to understand the main properties of these spacetimes. (Level 3 or 4)

Prerequisite course: General Relativity.

Background reading:

Gravitation (Misner, Thorne and Wheeler)

An Introduction to Einstein's General Relativity (Hartle)

A First Course in General Relativity ( Schutz)

## Tim Sluckin: Monte Carlo simulation of liquid crystals

This project involves the investigation of the properties of liquid crystals by using a computational molecular model, and weighting the individual states according to specific rules. It is the analogue in mechanics of what is known as Markov Chain Monte Carlo methods in statistics. The project will suit those who like computational rather than analytic methods in solving physical problems. This project will concentrate on models of liquid crystals in porous media, a new device area. [Level 3 or Level 4]

Prerequisites : Some background in and much enthusiasm for computational scientific methods.

## Tim Sluckin: The optics of liquid crystals

The optical properties of liquid crystals, i.e. the way light is changed as it propagates through a liquid crystal, are the reasons why liquid crystals are used in all sorts of displays, from large TV screens to mobile phone displays.

In this project you will learn how to describe the configuration of a liquid crystal and its interaction with light. The project is very open ended and it is expected that its general direction will be decided jointly by the student and the supervisor. A student more interested in analytical results may wish to apply suitable asymptotic methods to analyse various models of liquid crystal optical phenomena. On the other hand, the project is also suitable for more numerically oriented students, who would be apply PDE or optimisation methods to study the configuration of liquid crystals in geometries that do not lend themselves easily to analytical studies. [Level 3 or Level 4]

Prerequisites : Differential equations. Some knowledge of Maple and of a programming language is an advantage, but is not essential.

Background to liquid crystal projects : There is a large group of liquid crystal scientists, both experimental and theoretical, in the university, grouped together in the Southampton Liquid Crystal Institute. This group contains members in Physics, Chemistry, E-Science as well as in Mathematics, who will be in a position to give advice and help.

## Tim Sluckin: Modelling in theoretical biology

This project will involve ecological models of multitrophic systems (e.g. prey, predator, superpredator). The project involves modelling the ecological interactions in several different ways, and then solving the resulting equations, probably using an equation-solver. By changing relevant parameters, the student will be able to explore regions of equilibrium and limit cycle behaviour, showing that ecological systems do not always have a steady state. The sensitivity of ecological systems to disruption by man or other external disruption can also be investigated. [Level 3 or Level 4]

Background to the project : There is an active collaboration between Mathematics and Biology, with a number of Biologists interested in problems in this area, and the student will be able to attend the weekly Mathematical Biology seminar.

Prerequisites : Differential equations. Some knowledge of Maple and of a programming language is an advantage, but is not essential.

## Tim Sluckin: Linguistic phylogeny

Languages are similar either because they share a common origin, or because of borrowing. This project will use publicly available phylogeny software and word lists or other information to recreate linguistic family trees. The project will also involve a literature search of current work in this area. [Level 3 or Level 4]

Prerequisites : An interest in languages, competence in some foreign languages, and background in statistical methods will be an advantage. This is a computational project, and you will have to either write software or run rather complex packages.

## Marika Taylor: Perturbation and stability methods

Many problems of physical interest can be dealt with using asymptotic limits, in which a parameter or coordinate in the problem takes large or small values. Such perturbative, asymptotic, methods can be used to gain insight into the underlying structure and in many cases give a sufficiently accurate solution of the problem at hand. Given a solution, the next question is that of stability, which is very important in most physical problems: if we make small changes to the initial state, will the behaviour at late times be similar or very different? In this project a number of perturbation and stability methods will be explored and then applied to a range of physical problems. For example, the so-called matched asymptotic expansion method for solving differential equations is applicable to many fields, ranging from fluid problems involving regions of rapid variation right through to astrophysical and quantum mechanics problems. The physical applications explored in this project will be decided jointly with the student, following their interests. [Level 3/4]

Prerequisites: Confidence in analysis, and some prior interest and knowledge about fluid dynamics and electromagnetism, such as is covered in MATH 2044 and MATH 3071.

Background reading: E.J. Hinch, "Perturbation Methods", Cambridge University Press (1991)

## James Vickers: Stellar evolution

This project will consider possible mathematical models for the structure of various types of star. In particular it will consider problems relating to the equilibrium and stability of stars -- including White Dwarfs, Neutron Stars and Supermassive stars -- and to an understanding of how stars evolve. Most of the models will use the Newtonian theory of gravitation, but there will also be an opportunity to study the relativistic stages of the evolution of cosmic objects, including a study of collapse to a Black Hole. [Level 4]

S. Chandrasekhar, An Introduction to the Study of Stellar Structure , Dover.

## James Vickers: Gravitational waves in General Relativity

This project follows on from the first-semester unit MATH3006 Relativity, Black Holes and Cosmology , which includes a brief consideration of gravitational waves. The project will develop this further, first looking at the books of Schutz and Griffiths. Various extensions are possible, depending on the interests developed by the student. [Level 3]

R. A. d'Inverno, Introducing Einstein's Relativity , Clarendon Press B. F. Schutz, A First Course in General Relativity , CUP J. B. Griffiths, Colliding Plane Waves in General Relativity , Clarendon Press.

## James Vickers: Dynamical systems and cosmology

Even if one makes assumptions about the symmetries of our universe it can be very difficult to find exact solutions of Einstein's equations with those symmetries. However in the study of certain types of cosmology it is possible to reduce the Einstein's equations to a non-linear system of ordinary differential equations. Although it is usually still not possible to solve the resulting equations exactly, one can use techniques from the study of dynamical systems to understand the qualitative behaviour of the solutions. This project will look at how one can make such a reduction to a finite dimensional dynamical system and use these methods to study models of both stable and chaotic cosmologies. This is a Level 3 project for a student who has taken MATH3006 Relativity, Black Holes and Cosmology.

Kostas Zygalakis: Numerical solution of Stochastic Differential Equations

Many physical phenomena cannot be described realistically in deterministic terms and we need to allow for the effect of randomness. This randomness might be inherent in the system (quantum mechanics), coming from incomplete information (weather prediction), or coming from dimensional reduction, i.e. describing a complicated deterministic system with many degrees of freedom, using a lower dimensional one (a heavy particle colliding with lighter ones). In all the aforementioned cases the correct mathematical framework for describing our variables is stochastic.

The aim of this project (which can be offered both as a third or fourth year project) is to expose the student to the ideas of stochastic modelling and in particular modelling using stochastic differential equations (SDEs). The project will then focus on some different numerical methods for solving SDEs. [Levels 3 or 4]

Pre-requisites: Some understanding of basic probability and numerical analysis and some familiarity with a numerical software (most preferably Matlab)

Background reading:

[1] Peter E. Kloeden and Eckhard Platen. Numerical solution of stochastic differential equations. Springer, 1992.

[2] Peter E. Kloeden, Eckhard Platen and Henri Schurz. Numerical solution of SDE through computer experiments. Springer, 1994.

[3] Desmond J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM J. Numer. Anal., 43(3): 525–546, 2001

## Kostas Zygalakis: Data Assimilation

Data assimilation is the problem of estimating the state variables of a dynamical system, given observations of the output variables. It is a challenging and fundamental problem area, of importance in a wide range of applications (weather prediction, control theory to name a few). If one considers the class of linear systems, then it is possible to estimate in an exact (probabilistic way) the state variable of the linear dynamical system in question. However, the situation is more complicated when one deals with a non-linear dynamical system since then performing such an exact estimation becomes much more difficult.

The aim of this project is to introduce the student to some of the fundamental concepts of data assimilation. In addition, the student will become familiar with some basic numerical algorithms used in data assimilation and get to apply them in some interesting but simple dynamical systems. [Levels 3 or 4]

Pre-requisites: Some understanding of basic probability, especially Bayes rule and some familiarity with a numerical software (most preferably Matlab)

Background reading:

[1] Tzyh-Jong Tarn; Rasis, Y.; , Observers for nonlinear stochastic systems, Automatic Control, IEEE Transactions on , vol.21, no.4, pp. 441- 448, Aug 1976

[2] Duan, L. and Farmer, C. L. and Moroz, I. Sequential Inverse Problems Bayesian Principles and the Logistic Map Example. 8th International Conference of Numerical Analysis and Applied Mathematics, AIP Proceedings (2010)

[3] E. Kalnay. Atmospheric modeling, data assimilation, and predictability.

Cambridge Univ Pr, 2003.

[4] K. Law and A.M. Stuart: Evaluating Data Assimilation Algorithms. Mon. Wea. Rev. doi:10.1175/MWR-D-11-00257.1, in press