Stability of Couette flow for 2D tropical climate system with zero thermal dissipation

Abstract In this paper, we investigate the nonlinear stability of Couette flow for the two-dimensional tropical climate system with zero thermal dissipation in the domain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo>×</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:math> . To overcome the difficulty caused by the absence of the thermal dissipation, we fully exploit the structure of the system and the enhanced dissipation created by the non-self-adjoint operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>y</mml:mi> <mml:msub> <mml:mi>∂</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> </mml:mrow> </mml:math> to obtain the expected stability result.