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1. Nonlinear inviscid damping near monotonic shear flows

   https://doi.org/10.4310/ACTA.2023.v230.n2.a2  

   Ionescu, Alexandru D.; Jia, Hao (July 2023, Acta Mathematica)

   Tobias Ekholm (Ed.)

   We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $$\mathbb{T}\times[0,1]$$. More precisely, we consider shear flows $(b(y),0)$ given by a function $$b$$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $$u$$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $$u$$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved. 

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2. A Perturbative Solution for Nonlinear Stratified Upwelling over a Frictional Slope

   https://doi.org/10.1175/JPO-D-22-0191.1  

   Choi, Jang-Geun; Pringle, James; Lippmann, Thomas (October 2023, Journal of Physical Oceanography)

   Abstract A perturbative solution of simplified primitive equations for nonlinear weakly stratified upwelling over a frictional slope is found that resolves the vertical structure of velocity fields and can satisfy Ertel’s potential vorticity conservation in the stratified inviscid interior. The solution uses assumptions consistent with the model proposed by Lentz and Chapman, including a steady-state, constant cross-shore density gradient, no alongshore gradients, laterally inviscid, and consideration of cross-shore advection of alongshore momentum. The solution resolves the vertical structure of velocity fields (including subsurface maxima of compensational flow, not resolved by Lentz and Chapman) and can satisfy Ertel’s potential vorticity conservation in the stratified inviscid interior. The dynamics are similar to Lentz and Chapman; bottom stress balances alongshore wind stress in a homogeneous density ocean and is replaced by nonlinear cross-shore transport of alongshore momentum as the Burger number (S=αN/f, whereα,N, andfare the bottom slope, buoyancy frequency, Coriolis frequency, respectively) increases. When the solution uses the empirical relation between cross-shore and vertical density gradients proposed by Lentz and Chapman, vorticity conservation is not satisfied and the nonlinear momentum transport estimated by the solution linearly increases withS, asymptotically matching Lentz and Chapman forS< 1. When the solution conserves interior potential vorticity, the momentum transport is proportional toS²forS< 1 and is in better agreement with numerical simulations. 

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3. Structured input–output analysis of stably stratified plane Couette flow

   https://doi.org/10.1017/jfm.2022.648  

   Liu, Chang; Caulfield, Colm-cille P.; Gayme, Dennice F. (October 2022, Journal of Fluid Mechanics)

   We employ a recently introduced structured input–output analysis (SIOA) approach to analyse streamwise and spanwise wavelengths of flow structures in stably stratified plane Couette flow. In the low-Reynolds-number ( $Re$ ) low-bulk Richardson number ( $$Ri_b$$ ) spatially intermittent regime, we demonstrate that SIOA predicts high amplification associated with wavelengths corresponding to the characteristic oblique turbulent bands in this regime. SIOA also identifies quasi-horizontal flow structures resembling the turbulent–laminar layers commonly observed in the high- $Re$ high- $$Ri_b$$ intermittent regime. An SIOA across a range of $$Ri_b$$ and $Re$ values suggests that the classical Miles–Howard stability criterion ( $$Ri_b\leq 1/4$$ ) is associated with a change in the most amplified flow structures when the Prandtl number is close to one ( $$Pr\approx 1$$ ). However, for $$Pr\ll 1$$ , the most amplified flow structures are determined by the product $$PrRi_b$$ . For $$Pr\gg 1$$ , SIOA identifies another quasi-horizontal flow structure that we show is principally associated with density perturbations. We further demonstrate the dominance of this density-associated flow structure in the high $Pr$ limit by constructing analytical scaling arguments for the amplification in terms of $Re$ and $Pr$ under the assumptions of unstratified flow (with $$Ri_b=0$$ ) and streamwise invariance. 

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4. Stability threshold of nearly-Couette shear flows with Navier boundary conditions in 2D

   Bedrossian, J; He, S; Iyer, S; Wang, F (January 2025, Communications in Mathematical Physics)

   In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, $$\mathbb{T} \times [-1,1]$$, supplemented with Navier boundary conditions $$\omega|_{y = \pm 1} = 0$$. Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of $$\nu$$. On the other hand, the nonzero modes are assumed size $$O(\nu^{\frac12})$$ in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the \textit{inviscid damping}, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework. 

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5. Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha}$$ Velocity and Boundary

   https://doi.org/https://doi.org/10.1007/s00220-021-04067-1  

   Chen, Jiajie; Hou, Thomas Y (April 2021, Communications in mathematical physics)

   null (Ed.)

   Inspired by the numerical evidence of a potential 3D Euler singularity by Luo- Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $$C^{1,\alpha}$$ initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with $$C^{1,\alpha}$$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30,31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $$C^{1,\alpha}$$ initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with $$C^{1,\alpha}$$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. 

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