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CATEGORIES:Fluid Mechanics (DAMTP)
SUMMARY:2nd year PhD student talks - James Cummins\, Tom D
aggitt\, William Oxley\, Alistair Hales\, Jenny Di
ngwall\, Matt Davison\, Elvinas Ribinskas\, James
Mason\, Joseph Webber
DTSTART;TZID=Europe/London:20220527T144500
DTEND;TZID=Europe/London:20220527T170100
UID:TALK173495AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/173495
DESCRIPTION:* 1445 - 1458 James Mason\n* 1458 - 1511 Tom Daggi
tt\n* 1511 - 1524 William Oxley\n* 1524 - 1537 Ali
stair Hales\n* 1537 - 1550 Jenny Dingwall\n\n* 155
0 - 1610 coffee\n\n* 1610 - 1623 Matt Davison\n* 1
623 - 1636 Elvinas Ribinskas\n* 1636 - 1649 James
Cummins \n* 1649 - 1701 Joseph Webber\n\n---------
---------------------------------\n\n*James Mason*
\n\nMacroscopic behaviour in a two-species exclusi
on process via the method of matched asymptotics\n
\nWe consider a two-species simple exclusion proce
ss on a periodic lattice. We use the method of mat
ched asymptotics to derive evolution equations for
the two population densities in the dilute regime
\, namely a cross-diffusion system of partial diff
erential equations for the two species densities.
First\, our result captures non-trivial interactio
n terms neglected in the mean-field approach\, inc
luding a non-diagonal mobility matrix with explici
t density dependence. Second\, it generalises the
rigorous hydrodynamic limit of Quastel [Commun. Pu
re Appl. Math. 45(6)\, 623--679 (1992)]\, valid fo
r species with equal jump rates and given in terms
of a non-explicit self-diffusion coefficient\, to
the case of unequal rates in the dilute regime. I
n the equal-rates case\, by combining matched asym
ptotic approximations in the low- and high-density
limits\, we obtain a cubic polynomial approximati
on of the self-diffusion coefficient that is numer
ically accurate for all densities. This cubic appr
oximation agrees extremely well with numerical sim
ulations. It also coincides with the Taylor expan
sion up to the second order in the density of the
self-diffusion coefficient obtained using a rigoro
us recursive method. \n\n\n*Tom Daggitt*\n\nVariat
ions in Observations of Geosynchronous Magnetopaus
e and Last Closed Drift Shell Crossings with Magne
tic Local Time\n \nWe present an analysis of event
s in which the number of electrons trapped in Eart
h's outer radiation belts drops rapidly due to inw
ard movement of the outer edge of Earth's magnetic
field. These observations are compared to models
of the largest trapped electron orbits derived fro
m models of Earth's magnetic field and particle tr
acing models. These models of the largest trapped
orbits agree well with the losses seen over the ti
mescale of hours\, but fail to reproduce more rapi
d decreases in the number of electrons measured on
the timescale of minutes. We show that different
satellites in geostationary orbit observe differen
t trends in the trapped electron population on tim
escales of less than a day during geomagnetic stor
ms due to their separation in longitude. These dif
ferences demonstrate that data from at least three
satellites in geostationary orbit\, ideally more\
, may be required for accurate\, high time resolut
ion forecasting and reconstruction of Earth's radi
ation belts during geomagnetic storms.\n\n*William
Oxley*\n\nAmplitude Thresholds for Zombie Vortex
Instability\n \nRotating\, stratified and sheared
flows\, where the shear is in a direction perpendi
cular to the (stable) vertical stratification\, ar
e found in a wide range of settings that include t
he atmosphere and oceans\, as well as protoplaneta
ry disks. The addition of stable stratification to
rotating shear flows is of interest as it can hav
e a destabilising effect. A number of studies of t
hese types of flows\, through numerical experiment
s\, show vortices appearing in lattice like struct
ures\, after spreading across the domain due to an
initial localised perturbation. This finite ampli
tude instability was first identified by Marcus et
. al. 2013\, who gave it the name ‘Zombie Vortex I
nstability’ (ZVI) due to its formation mechanism.
The individual vortices result from the excitation
of baroclinic critical layers\, and each vortex t
hen acts as a source for a new initial perturbatio
n\, which allows the process to repeat. Baroclinic
critical layers are similar to their classical na
mesake\, and take the form of sharp changes in per
turbation variables around locations which are sin
gular in the linear inviscid theory. In contrast t
o classical critical layers\, which form where the
perturbation phase speed matches the speed of the
mean flow\, baroclinic critical layers form at th
e locations where the phase speed of a perturbatio
n (relative to the background shear) matches the c
haracteristic gravity wave speed. This instability
has been suggested by Marcus et. al. (2013) as a
candidate to destabilize dead zones in protoplanet
ary discs\, although that suggestion has been ques
tions by other authors (e.g. Lesur & Latter 2016).
Building on previous analytical and numerical stu
dies\, I have been investigating ZVI numerically t
o try and gain more insight into the formation mec
hanism\, and in particular what are the conditions
we must impose on the initial perturbation in ord
er to produce ZVI. One of the key challenges to ov
ercome is how to actually identify ZVI\, and how t
o focus on one individual occurrence of the replic
ation process. To do this\, a carefully chosen ini
tial condition is used and the problem is simulate
d using various different parameters.\n\n*Alistair
Hales*\n\nReduction of Leading-Edge Noise Using T
ailored Turbulence Anisotropy\n\nIn this talk I gi
ve an outline of a new mathematical model to appro
ximate the leading edge noise from the scattering
of anisotropic flow off a rigid aerofoil. Thin aer
ofoil theory is used to model an aerofoil as a sem
i-infinite plate and the scattering of incoming tu
rbulence is solved via the application of the Wien
er-Hopf technique. This theoretical is integrated
over a wavenumber-frequency spectrum to account fo
r general incoming turbulence which is obtained us
ing the method of Gaussian decomposition\, account
ing for and modelling anisotropy in the incoming t
urbulence.\n\n*Jenny Dingwall*\n\nModelling the ac
cumulation of buoyant particles under wind-driven
and convective turbulence using large-eddy simulat
ions\n \nBuoyant material such as microplastics te
nd to accumulate near the ocean surface in regions
with convergent surface currents where they can b
e harmful to marine life. We investigate the accum
ulation of buoyant tracers and Lagrangian surface
particles by small-scale turbulence in the ocean m
ixed layer under combined wind and convective forc
ing using large-eddy simulations. Surface cooling
drives convection\, and under this regime persiste
nt convective vortices form which trap buoyant mat
erial\, leading to large concentrations. For suffi
ciently weak winds\, convective vortices survive b
ut become less effective at clustering material as
the wind stress increases. Under strong wind forc
ing\, convective vortices are no longer visible\,
but some particle clustering occurs in downwelling
regions associated with longitudinal wind rolls.\
n\n*Matt Davison*\n\nReaction-Diffusion Dynamics o
f the Tropical Atmosphere\n \nThe Madden-Julian Os
cillation (MJO) is a precipitating disturbance tha
t propagates eastward across the Indian and Pacifi
c oceans every 30-60 days. There is no single acce
pted theoretical explanation for the MJO and in cl
imate models the MJO is often poorly represented.
It has been suggested that there is a relation bet
ween the MJO and the phenomenon of convective aggr
egation observed in 'convection-resolving' numeric
al simulations. I will describe an extension of a
previous theoretical model of convective aggregati
on\, based solely on reaction-diffusion\, to inclu
de large scale dynamics. The new model consists of
the shallow water equations and a moisture equati
on coupled through the effects of moisture on radi
ation and latent heat release. When the latter are
chosen such that there are two stable states alon
gside the initial unstable radiative-convective eq
uilibrium\, convective aggregation is observed. Si
nce this model has dynamics we can include rotatio
n\, and on an equatorial beta-plane the dynamical
model exhibits coherent\, moist\, eastward propaga
ting disturbances\, and therefore appears to repre
sent the mechanism behind the MJO.\n\n*Elvinas Rib
inskas*\n\nReduced modelling of ice sheet response
to climate perturbations\n\nIce sheet melting ass
ociated to climate change poses the risk of signif
icantly increasing the sea level. Modelling how ic
e sheets respond to climate perturbations allows u
s predict this change. Reduced models in particula
r can be used to also infer how the underlying phy
sical processes influence ice sheet response to ex
ternal forcings. I will present a reduced model of
a land-terminating ice sheet that is forced by a
prescribed ablation-accumulation law. The model co
nsists of a simplified ice flow law\, basic meltwa
ter network and a basal boundary condition that co
uples ice flow to the basal meltwater pressure. Th
e steady-state solutions of the model are found to
be non-unique. Also\, we learn that longitudinal
stresses have to be included into the model for it
to represent a realistic ice sheet. In the end\,
I will present preliminary results of the improved
model and possible directions for future work.\n\
n*James Cummins*\n\nUnconventional computing & the
n-queens problem\n\nTraditional computers can not
keep up with the increasing speed and power expec
ted of them. Unconventional computing architecture
s overcome this by using gain-dissipative systems
to improve computational performance. These platfo
rms may comprise of lasers\, superconducting qubit
s\, polariton condensates\, or photon condensates.
In this talk we explore how such systems can solv
e one of the oldest problems in mathematics: the n
-queens problem.\n\n*Joseph Webber*\n\nDynamics of
super-absorbent hydrogels\n\nHydrogels are soft m
aterials formed from a hydrophilic polymer scaffol
d surrounded by adsorbed water molecules\, and may
comprise over 99% water by volume in their fully-
swollen state. As such\, when allowed to swell and
dry\, their volumes may change to an extreme degr
ee\, rendering linear poroelastic models invalid o
wing to the large magnitudes of the strains involv
ed. Models proposed in the literature to describe
this extreme swelling and drying often rely on a c
omplex molecular-scale understanding of polymer-wa
ter interactions and formulate gel dynamics in ter
ms of a free energy density. In this talk\, I will
summarise a new formulation for the dynamics of h
ydrogels which treats them as instantaneously inco
mpressible linear-elastic materials\, whilst allow
ing for nonlinearities in the isotropic strains co
rresponding to swelling and drying. Key features o
f this model include a complete description of any
gel using only three material parameters\, a nonl
inear diffusion equation governing swelling and dr
ying\, and an associated equation to describe the
displacement field\, from which the shape of the h
ydrogel can be deduced. Applications of this model
to a number of different problems will then be di
scussed briefly.
LOCATION:MR2\, Centre for Mathematical Sciences\, Wilberfor
ce Road\, Cambridge
CONTACT:Prof. Jerome Neufeld
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