# 地球流体セミナー (GFD seminar)

毎年夏に定期的に開催しています. 「各参加者(聴衆)個々人による(自己満足的)理解の追求」を合言葉に, 話題提供者による地球流体力学に関連した個々の問題に対してじっくりと時間をかけて議論することを目的としています.

地球流体セミナー運営グループ (gfdsemi-admin at gfd-dennou.org) が中心になって開いています.

## 次回セミナー, 関連研究集会予定

### 夏の GFD セミナー

- TBA

### FDEPS (Fluid Dynamics in Earth and Planetary Sciences)

### 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, 場所:

2024 年 12 月 5 日 (木) 16:30 - 18:00

Zoom あるいは京大数理研 204 号室

話題提供者とタイトル

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

「The onset of filamentation on vorticity interfaces」

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.

参加方法:

対面 + zoom にて開催します. zoom 参加希望者は gfdsemi-admin (at) gfd-dennou.org までその旨ご連絡ください.

## GFD セミナー過去の資料

これまでに行われた地球流体セミナー (GFD seminar) の講演資料はこちら. 姉妹編ワークショップ「地球惑星科学における流体力学」(FDEPS)の資料も並べてあります.

## 裏方

- 地球流体セミナー運営グループ
- 相木 秀則 (名大・宇地研)
- 伊賀 啓太 (東大・大気海洋研)
- 石岡 圭一 (京大・理)
- 石渡 正樹 (北大・理)
- 岩山 隆寛 (福大・理)
- 樫村 博基 (神戸大・理)
- 酒井 敏 (京大・総人)
- 佐々木 洋平 (北海道情報大)
- 杉本 憲彦 (慶大・法)
- 杉山 耕一朗 (松江高専)
- 高木 征弘 (京産大・理)
- 高橋 芳幸 (神戸大・理)
- 竹広 真一 (京大・数理研)
- 中島 健介 (九大・理)
- 西澤 誠也 (理化学研究所)
- はしもとじょーじ (岡山大・自然科学)
- 林 祥介 (神戸大・理)
- 村上 真也 (宇宙航空研究開発機構)
- 柳澤 孝寿 (海洋研究開発機構)
- 山田 道夫 (京大・数理研)
- 吉田 茂生 (九大・理)
- 余田 成男 (京大・理)

- 提供 : 「mosir プロジェクト」
- 問い合わせ先 : 地球流体セミナー運営グループ
- gfdsemi-admin at gfd-dennou.org