Non-resonant non-linear interection in fluid equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 17H02860
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kyoto University
• Principal Investigator: Michio Yamada 京都大学, 数理解析研究所, 特任教授 (90166736)
• Co-Investigator(Kenkyū-buntansha): 竹広 真一 京都大学, 数理解析研究所, 准教授 (30274426) ; 米田 剛 一橋大学, 大学院経済学研究科, 教授 (30619086) ; 小布施 祈織 岡山大学, 環境生命科学学域, 准教授 (90633967) ; 横山 直人 同志社大学, 研究開発推進機構, 嘱託研究員 (80512730)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: 回転流体 / 帯状流 / ロスビー波

Outline of Final Research Achievements

Nonlinear and nonresonant interaction between linear waves is studied in nonlinear evolution equations of linear wave systems, including two-dimensional fluid dynamical equations on a rotating sphere where the Rossby waves are fundamental modes and the most noticeable phenomenon is the generation of zonal flows, which are steady Rossby waves uniform in east-west direction, as often observed in solar planets. Numerical simulation with very small viscosity strongly suggests that the generation of the zonal flows is possible without viscosity, although an interpretation of the zonal flow generation employs the energy transfer from Rossby waves to zonal flows through the viscous critical layers. The Rossby waves are divided into 4 groups according to zonal/nonzonal and whether resonant interaction is possible or not. Energy transport between these 4 groups shows that the non-resonant interaction transfers energy to the zonal modes which cannot receive energy by the resonant interaction.

Free Research Field

流体力学，応用数学

Academic Significance and Societal Importance of the Research Achievements

回転球面上の帯状流形成は惑星大気における典型的な流れパターンである．従来，帯状流形成はRossby波と帯状流の粘性臨界層を通しての相互作用によるという解釈が行われてきた．これに対し本研究の結果は，帯状流形成には非粘性非線形相互作用が重要であり，粘性は必ずしも必要ではないこと，また，共鳴相互作用は帯状流を形成できないため，非共鳴相互作用が帯状流へのエネルギー輸送を担っていること，を示している．即ち，帯状流形成においては共鳴相互作用に近い近共鳴相互作用が重要な役割を果たしていると考えられる．この帯状流へのエネルギー輸送では，波数空間が離散的であるため，非局所性が強く現れる．