Seminars (unless otherwise stated) will take place on **Tuesdays at 3.00pm in 20 Bedford Way Room 822 **- see the map for further details. There will be tea afterwards in **Maths Room 606** (6th Floor, 25 Gordon Street) - see map for further details. If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof Ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).

## 14 January 2020 - CANCELLED

## 21 January 2020

### Speaker: Mikhail Cherdantsev, Cardiff University.

##### Title: Stochastic homogenisation of high-contrast MEDIA

**Abstract: **

We study the homogenisation problem for a divergence type operator $A_\e$ with high-contrast random coefficients. In particular, we are interested in the behaviour of their spectra. We assume that on one of the components (referred to as ``stiff''), of the composite described by the operator, the coefficients are "of order one", and on the complementary ``soft" component, consisting of randomly distributed inclusions of size $\e$, the coefficients are of order $\e^2$, $\e<<1$.

Our interest in high-contrast homogenisation problems is motivated by the band-gap structure of their spectra. From a mathematically rigorous perspective, these were first analysed in [Zhikov 2000, 2004] for the case of periodic coefficients. It was shown that for both the bounded domain and the whole space settings the spectra of the operators converge in the sense of Hausdorff to the spectrum of a certain homogenised two-scale operator $A_hom$. In our previous work [Ch,Ch,V 2008] we have considered the case of a bounded domain for which we proved a version of the resolvent convergence of $A_\e$ to a homogenised operator $A_hom$. Due to the bounded domain setting, entailing relevant compactness statements, this implied, in particular, the Hausdorff convergence of the spectra. To a certain extent, our findings were similar to those of [Zhikov 2000] in the periodic case.

The present work is concerned with the whole space setting. Unlike the periodic high-contrast operators, whose spectra are described by similar multiscale arguments for a bounded domain and for the whole space, in the stochastic case the situation is different. We show that in the whole space setting the spectrum of $A_hom$ is, in general, a proper subset of the limit set of the spectra of $A_\e$, which we rigorously describe. In fact, the case when the spectrum of $A_\e$ is occupies the positive half of the real line is not uncommon. This additional spectrum, which does not appear in the bounded-domain case, is attributed to the stochastic nature of the problem. At the same time, it is deemed to be a mathematical artefact of our imperfect model for a random distribution of inclusions, rather than a physically relevant object (at least in the context of applications we have in mind) - we thus refer to this part of the spectrum as ``irrelevant". We devise a procedure which allows an ``asymptotic'' (i.e. with a certain error) classification of the ``relevant'' and ``irrelevant'' parts of the spectrum of $A_\e$. We prove the Hausdorff convergence of the ``relevant'' spectrum of $A_\e$ to the spectrum of $A_hom$.

## 28 January 2020

### Speaker: Professor Herbert Huppert, Cambridge University

##### Title: Stokes drift in an ocean above a coral layer: theory, field confirmation and biological implications

**Abstract: **

The talk will start with a simple desk-top experiment of waves on an effectively infinitely deep ocean. We will briefly demonstrate and describe: the velocity of the waves (the phase velocity); the velocity of the energy in the waves (the group velocity); and the velocity of junk particles on the surface (the Stokes drift velocity).

## 04 February 2020

### Speaker: Matthew Crowe (UCL)

##### Title: Theoretical models of turbulent mixing in ocean fronts

**Abstract: **

Fronts, or regions with large horizontal density gradients, are common and important features of the upper ocean. On sufficiently large scales, the hydrostatic pressure gradient associated with the horizontal density gradient is nearly balanced by a (geostrophic) along-front flow. However, fronts in the ocean and atmosphere co-exist with small-scale turbulence which disrupts this balance. I will present a simple asymptotic model in which turbulent mixing is represented using a turbulent viscosity. Leading order solutions show that mixing acts to drive a cross-front flow and maintain a vertical density stratification with the correlation between these two effects leading to shear dispersion. Using the leading order solutions, a nonlinear diffusion equation for the background density can be derived and used to study the long term evolution of the fronts. I will present the case of an unforced front in which shear dispersion leads to self-similar spreading of the frontal region and the case of a forced front in which surface forcing may act to oppose the spreading. Finally I will consider the effects of turbulent mixing on frontal baroclinic instability and show that mixing can act to suppress instability growth and modify the direction of the fastest growing wave-vectors.

## 11 February 2020

##### See the Departmental Colloquium webpage

## 18 February 2020

**READING WEEK - NO SEMINAR**

## 25 February 2020 - SEMINAR CANCELLED

## 03 March 2020 - SEMINAR CANCELLED

## 10 March 2020 - SEMINAR CANCELLED

## 17 March 2020 - SEMINAR CANCELLED