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回転球面上の電磁流体波動の理論的研究、及びその地球外核最上部と地磁気変動への応用

中島, 涼輔 NAKASHIMA, Ryosuke ナカシマ, リョウスケ 九州大学

2020.03.23

概要

トロイダル磁場が印加された回転球面上の水平2次元, 及び浅水系の電磁流体(MHD)線形波動について調べた。球面に張り付いた薄い流体層は, 電気伝導度が無限大の電導性流体, もしくは電気伝導度が有限の流体とする。このような系は, 地球液体核(外核)の最上部にあるとされる安定成層内の流体運動を表現しうる。この安定成層は, 近年の地震学的な研究により検出された可能性がある。また, 以前より地球物理学及び地球化学的な観点から, その存在が予想されてきた。このような層の存在は, 現在の地球磁場生成メカニズムや, 地球の熱・化学史を理解する上で重要であるが, ほとんど明らかになっていない。地球の外核内部をゆっくりと伝播する MHD 波動は, 地球磁場の永年変動を引き起こしうるという事実を利用すれば, 地震学とは異なる方法によって, このよくわからない安定成層に対して制約を与えることができるかもしれない。具体的には、回転成層流体における MHD 波動の包括的な理論研究と, 実際に起こっている地球磁場の変動の特徴を比較すれば良い。

線形波動を調べる上で, 2 種類の東西向き背景磁場分布を採用した。1つは, 強さが sinθ(θは余緯度)に比例する赤道対称な東西背景磁場, もう1つは, 強さが sinθcosθに比例する赤道反対称な東西背景磁場である。前者の場合については, 最近になって Márquez-Artavia et al.(2017)によって調べられ, 私たちはその結果を再現できることを確かめた。線形問題における固有モードの数値的決定を綿密に行ったところ, その研究には報告されていない新しい種類のモード(波動)が存在することも分かった。この新たなモードは, 地球外核内と同程度の磁場強度においても存在することができる。具体的には、Lehnert 数(Alfvén 波速度を, 自転角速度の 2 倍と球の半径で割った無次元数)が 0.5 以下となるような弱い磁場がかけられているときに存在が可能である。さらに, このモードは, 亜 Alfvén 波速度で西進し, また極域に振幅が局在化する。そのため, Márquez-Artavia et al.(2017)によって見つけられた, 強い磁場環境下(Lehnert 数が 0.5 以上)で存在できる, 極域に局在化する速い磁気 Rossby 波と似ている。これらの極域に局在化するいくつかの種類のモードを理解するために, 私たちは極域付近に張った円筒座標系での波動についても調べた。この簡単な系で求めた分散関係, 及び流れと磁場の構造は, 球面座標系で数値的に求められた極域に局在化するモードのふるまいをよく再現する。これは, 極付近に局在化するキンク不安定(Tayler 不安定)の解も例外ではない。この簡単な系での計算によって, 極に局在化する安定なモードは, 主に magnetostrophic バランス(Lorentz 力と Coriolis 力のつり合い)に支配されるということが分かった。また, Lorentz 力に含まれる曲率項によって, 一様磁場下では同一になる東進と西進のモードの位相速度に差ができる。東西波数が 1 で, 磁場が強いとき(Lehnert 数が 0.5 以上)に限り, Lorentz 力とCoriolis 力のつり合いが崩れ, そのつり合えなくなったLorentz 力がキンク不安定を引き起こす。

磁場強度が sinθcosθに比例する赤道反対称な東西背景磁場の場合は, Alfvén 波連続モードとその共鳴に由来する臨界緯度が現れることによって, 固有値問題がより困難になる。この複雑化は,散逸の効果がなく(電気伝導度が無限大), 東西背景磁場分布を sinθで割ったものがθに依存するときに限り起こる。私たちは, 有限の電気伝導度へ問題を拡張することで, Alfvén 波連続モードと臨界緯度付近の特異的なモードの構造を取り除くことが可能であることを, 数値的及び解析的に確かめた。水平2次元の問題においては, 解析的な研究によって, 特異的な構造を取り除く臨界緯度まわりの電気抵抗性境界層の厚みが Lundquist 数(電気伝導度の大きさを表す無次元数)の-1/3 乗に比例することが示される。加えて, “clamshell”不安定の一種と考えられる, 抵抗性不安定のモードも磁場が強い条件下で見つかった。この背景磁場分布においても, 極域に局在化する西進モードが, 再び地球に近い条件で見つかった。 もし地球外核最上部の安定成層が, Buffett(2014)によって推定された程度の成層強度であれば, このモードの伝播に起因する高緯度地域の地磁気変動が観測されるかもしれない。これらの波に着目した地球磁場モデルの時間変化についての解析を行えば, 過去になされた安定成層の性質の見積もりがもっともらしいかどうかを確かめることができるであろう。

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