GFD オンラインセミナー (第 18 回)
              (RIMS 流体力学セミナーとの共同開催)

日時: 2024 年 12 月 5 日 (木) 16:30--18:00

場所: zoom / 京大数理研本館 204 号室

講演者: Prof. David G. Dritschel (Mathematical Institute, University of St Andrews)

講演タイトル: The onset of filamentation on vorticity interfaces

講演要旨:

As far back as the late 1800s, Lord Kelvin, who was one of the
greatest fluid dynamicists of all time, noted a curious property of
the dispersion relation governing linear waves propagating on a
vorticity interface.  Such an interface divides two regions of uniform
vorticity in a two-dimensional perfect (inviscid, incompressible)
fluid. Remarkably, small-amplitude waves on both a linear
interface (e.g. one lying on the $y$ axis in equilibrium)
"and" on the edge of a circular vortex patch oscillate at a
frequency equal to half of the vorticity jump across the interface,
$\Delta\omega/2$, when in a frame of reference moving with mean
velocity at the interface. Notably, the frequency does not depend on
the wavelength of the wave. As a result, in linear theory, any
disturbance to a vorticity interface --- which in general is made up
of many wavelengths --- oscillates at a fixed frequency, recovering
its initial form after a time equal to $4\pi/\Delta\omega$. Kelvin had
the insight to realise that, in the nonlinear equations, any
finite-amplitude disturbance will experience a "shear" (induced
by the equilibrium flow) which tends to slow down wave crests and
troughs relative to points near the equilibrium interface position.
He hypothesised that this shearing action would lead to progressive
steepening of the disturbance, and, eventually, wave breaking.

It took over a century before there were numerical methods capable of
accurately simulating the long-time behaviour of disturbances to
vorticity interfaces. The present author studied this problem with a
novel numerical method based on `contour dynamics', introduced a
decade earlier by Norm Zabusky and colleagues. In Dritschel
J. Fluid Mech. 194, 511--547, (1988), it was
confirmed that waves do break and form thin filaments extruding from
the interface, and moreover, they break repeatedly, once every period
of the linear oscillation.  This results in a huge growth in
complexity of the interface, which becomes covered in many filaments.
This eventuality has been termed `Kelvin's hair'!

The present author was unaware of Kelvin's hypothesis until he met
Alex Craik on a visit to the University of St Andrews in the early
1990s.  Professor Craik pulled down an original volume in the
collected works of Kelvin from his library at home, and found the
passage in which Kelvin hypothesised this scenario. Kelvin's insight
is all the more remarkable given that the wave steepening is quite
unlike what is found say in Burgers equation. Instead, the wave
`performs a dance' every linear period, changing its form
continuously: the wave crests at one instant of time later lie near
the mean interface position then later still on the other side of the
interface, etc.  Crests and troughs do not remain in position (as they
do in Burgers equation): the wave continuously distorts and then
distorts back to its original form (or nearly). Hence, it is less
obvious that shear will have the effect Kelvin hypothesised.

In Dritschel (1988), a weakly-nonlinear theory was developed to show
that there is net steepening of the wave every period, but this occurs
on a very long time scale inversely proportional to the square of the
wave slope. (This puts severe demands on any numerical simulation of
the process.) The present talk will discuss recent further
developments concerning the mathematical structure and properties of
the weakly nonlinear equation. In particular, we show that the
equation possesses a self-similar form describing the last stages in
the steeping of a disturbance --- the onset of
filamentation. Numerical evidence is provided which suggests that this
equation is an attractor for almost all wave forms, implying that
vorticity interfaces are generically prone to filamentation.


開催方法: 対面 + zoom のハイブリッド形式

トピック: 第１８回 GFD オンラインセミナー
時刻: 2024年12月5日 04:30 PM 大阪、札幌、東京

参加 Zoom ミーティング
https://kyoto-u-edu.zoom.us/j/94965679978?pwd=pFbCaMmavMMnlmEK3xwevCzxQMvfhD.1

ミーティング ID: 949 6567 9978
パスコード: 268689