This link contains the slides that I used on a recent discussion with my colleagues at the Marine Biology Laboratory of NYUAD led by prof. John Burt.
They document some preliminary results obtained by my student Chenhao Xu. They compare the temperature at the bottom of the sea simulated through a water column model with those observed in-situ, in an attempt to elucidate which are the physical factors that cause (or avoid) excessive bottom temperatures during the summer.
It turns out that the main factor keeping the water cool (or, at least, not unbearably hot) during summer is the sustained presence of strong winds. Those enhance the evaporative cooling, which, at the temperatures of the Summer in the Gulf can lead to peaks in excess of 400 W/m\(^2\) in the latent heat flux.
Our work is progressing by fine-tuning the model and switching to the new ERA5 reanalysis for the meteorological parameters that force the water column model.
A post-doctoral position is available at NYU Abu Dhabi, under my supervision, to work on Lagrangian numerical methods for ocean modelling.
The focus will be mostly on biogeochemical problems, using the techniques set forth in this paper. One of the main goals is to produce a state-of-the-art, parallel particle library that can work in sync with one or more of the major ocean models (e.g. ROMS).
Those interested should apply through this web page. Please remember that the application will not be considered as completed (and therefore will not be taken in consideration) until at least two recommendation letters are received.
From February 5th to February 7th there has been a workshop on PDEs and fluid mechanics at the “Centro Ricerche Matematiche Ennio De Giorgi” of the Scuola Normale Superiore in Pisa, where I was very honored to be invited to give a talk.
The slides of my talk are linked here: CRM_2018. The subject is very dear to me: fingering convection, a peculiar form of convection that occurs with two buoyancy-changing scalars having different diffusivities and competing effects on density (think temperature and salinity in sea water).
Numerical models of the ocean (and of the atmosphere) are always underresolved. In the ocean tracers such as temperature, salinity, dissolved oxigen, iron and other nutrients have been measured to have fluctuations down to tiny scales, which in some instances may be submillimetric. Even a regional model that only encompasses a limited portion of the world’s oceans (say, just the Gulf) can only resolve, in the best of cases, scales of the order of one kilometer (and often much less).
Just about any existing ocean model works by representing the relevant physical quantities on a fixed grid, spanning latitude, longitude and depth. This is called an Eulerian representation. What happens when some structure (say, fluctuations in temperature) becomes so small as to be unresolvable on a model’s grid? Details change from model to model, but, by and large, all model smear out the fluctuations, so that the values recorded at the grid nodes are representative not of the pointwise value of that quantity, but of its average over a region of size comparable with the size of a computational mesh.
That’s the same thing that happens when you look at a newspaper’s page from a distance too large to discern the individual characters that compose the printed words. In that case you don’t see black letters printed over white paper: you see gray areas corresponding to the region of the page covered by text.
Now imagine that there were bugs living by eating concentrated ink: on a real newspaper page those bugs would thrive, at least until they hadn’t eaten every printed word. But if you attempted to model that process with an unresolved model, it would not quite work in the same way: because the simulated bugs would only be offered a smeared-out version of reality, they might find that the ink is not concentrated enough to survive. And thus the model may wrongly predict that the bugs would get extinct without wiping out all the ink from the page.
Biases of this sort are commonplace in ocean models that attempt to go beyond physics, and describe the chemical and biological processes that occur in seawater. What can be done to fix that problem?
At first sight, the situation appears hopeless, because there’s no way that a computer will have enough number crunching strength to reach the necessary resolution in the foreseeable future. But not all is lost. Lack of resolution is bad in itself, but it becomes truly evil when it smears out the fluctuations that would otherwise be present. Is it then possible to perform simulations that, albeit underresolved, still retain a realistic range of fluctuations at the smallest resolved scales? If the answer were “yes” then we could avoid the worst of the problems that comes with underresolution.
Well, it turns out that the answer is “yes”. But the price to pay is to change completely the simulation framework. Rather than representing quantities on a fixed grid, we must think in terms of moving water masses, going with the flow, each carrying a certain amount of temperature, salinity, oxygen, etc. This is called a Lagrangian description of the flow, and is immune from the smearing effects that are inevitable with the Eulerian approach.
My last paper, just accepted for publication on Journal of Computational Physics, works out the gory mathematical details and describes a general framework for simulation biogeochemical processes without incurring in nasty biases due to the smearing-out effect.
As an example of the performance of the method, here are two simulations. The one on the left is obtained with a high-resolution code (\(4096^2\) grid nodes). On the right, the same simulation is performed with the Lagrangian approach, using just \(128^2\) particles.
Even though the Lagrangian simulation is underresolved, and (by design) can’t capture all the delicate intricacies of the orange filaments visible in the high-resolution simulation, the range of fluctuations, and the overall shape of the structures is the same in both simulations. This is what really matters for ocean biogeochemistry: we want to get the statistics right! And with a Lagrangian approach we will…
The final exam for the Fall 2017 Numerical Methods class consisted in performing a project. Each student had to study some additional theory and write a code to numerically solve a mathematical problem.
The depth, breadth and quality of their work is outstanding! Here’s a brief summary of what they did.
Titas Geryba: Nonlinear differential algorithm to compute all the zeros of a generic polynomial. A recent paper by prof. Francesco Calogero shows that the zeros of a polynomial may be found by solving a certain system of nonlinear differential equations involving the polynomial’s coefficients. For polynomials of arbitrary order the solutions can’t be found with paper and pen, therefore numerical methods must be used. The following figure shows the solutions in the complex plane of these ODEs for the polynomial \(P(z)=(z-20)(z+20)(z-10)\). Starting from arbitrary initial conditions, the solutions at time t=1 approximate the zeros. These approximations may be taken as new initial conditions, and the solution process may be iterated a few times in order to achieve a very high accuracy.
Ziyuan (Jennifer) Huang: Simulations of a bouncing rod.
A rod of elastic material bouncing vertically over a rigid floor is simulated as a chain of point masses connected by linear, dissipationless springs. Each time that the lowest point mass hits the floor its velocity reverses. The simulation illustrates the transfer of potential gravitational energy into energy of the vibrational modes.
Frederik Jensen: Liquidity Reserve Control.
The need for a currency broker to maintain her reserves liquid is vital to business. If the broker runs low on any reserve, possible consequences range from increased commission rates, a cap on withdrawal, or discontinuement of supply until the reserve has been replenished. This projects aims at building a revenue maximising solution for strongly competitive brokers by casting the liquidity issue as a control problem and adopting a reinforcement learning approach. In particular, the scope of the problem considered extends to brokers that deal with cryptocurrencies as well.
Myera Rashid: Pattern formation in the FitzHugh-Nagumo equations.
The foundations for a mathematical understanding of pattern formation in biological systems were laid down by Alan Turing (yes, that Turing) in a seminal paper. An example of pattern-forming equations is obtained by adding diffusion terms to the FitzHugh-Nagumo equations. The movie shows the early phases of pattern formation, starting from random initial conditions, on a doubly-periodic domain with 50×50 grid nodes. An explicit FTCS scheme is used to numerically solve the equations.
Jin Shang: A simple model of traffic flow.
An important topic in civil engineering is modelling the flow of cars along roads. The earliest model, due to Lighthill and Whitham captures a wide variety of traffic flow phenomena in a surprisingly simple conservation law. The following figure shows a rarefaction wave of a long queue of cars some time after a street light has become green.
Boyan Stoychev: Solving PDEs via pseudospectral methods.
Pseudospectral methods are a class of powerful numerical methods for solving PDEs in simple domains. In this project a simple Galerkin method is used to numerically solve Burger’s equation and Schrödinger’s equation with a variety of potentials. The following movie shows the modulus of the wavefunction of a Gaussian wavepacket (blue) impinging on a very tall potential barrier (black). In classical mechanics a particle with the same energy of the wavepacket could not pass the barrier. In quantum mechanics, if the barrier is sufficiently thin, there is a non–negligible probability to find the particle on the other side of the barrier, signaled by the splitting of the wavefunction in two (roughly Gaussian) parts.
Silviu Udrescu: Three bodies restricted problem and symplectic integrators.
Some ODEs have a special structure, known as symplectic. The most notable application that gives rise to such ODEs is Hamiltonian classical mechanics. Standard numerical integrators do not respect the symplectic structure of ODEs, giving rise to subtle errors (diagnosable by a lack of conservation of energy) that pile up in long integrations, until the result becomes unusable. Symplectic integrators are numerical methods that approximate a symplectic ODE with a symplectic map, thereby imposing conservation properties as close as possible to those of he original equations. The following animation shows the chaotic orbit of a tiny planet in a binary stellar system (it’s called the restricted three-body problem) integrated through a sixth-order, symplectic Runge-Kutta method.
Alexandra Urbanikova: Numerical analysis of bifurcations.
Upon changing the numerical value of a parameter in an ODE, the number and stability of equilibria may change. When this occurs, it is said that the system has undergone a bifurcation. A numerical continuation code is able, as the parameter changes, to follow the position of the equilibria and detect bifurcations by monitoring stability changes. The following figure shows an example of numerical continuation for the celebrated Lorenz system.
Sunyi Wang: Simulations of shallow wave equations in 1D.
Shallow wave equations describe thin layers of fluid with a free surface. They have strong similarities with compressible flows. In the following figure it is shown a nonlinear wave that evolved from an initially sinusoidal bump.
Lin Zhu: Numerical solution of 2D Navier-Stokes equations.
Incompressible fluid motions require the solution of an elliptic problem to be numerically computed. In 2D, when one uses the streamfunction-vorticity formulation (which is convenient to avoid dealing with pressure), to this difficulty one must add that simple centered finite differences methods tend to be unstable. An elegant fix is to use Arakawa’s discretization of the Jacobian. The following animation shows the evolution of vorticity starting from a symmetric initial condition.
