Sedimentation of buoyancy-driven flow
Julien Guillod
joint work with Anne-Laure Dalibard and Antoine Leblond
1. Introduction
Problem
Question
Stability of stratified flow and long-time behavior?
Motivation from oceanography
Lock exchange test case
Variables and domain
- Horizontal variable $x$ and vertical variable $z$
- Velocity field $\bu$ and density $\rho$
- Reynolds number $\mathrm{Re}$
- Periodic channel $\Omega=\mathbb{T}\times(0,1)$
Description of the model
- Boussinesq approximation $$\mathrm{Re}\bigl(\partial_{t}\bu+\bu\bcdot\bnabla\bu\bigr)-\Delta u+\bnabla p=-\rho\be_z$$
- Incompressible fluid $$\bnabla\bcdot\bu =0$$
- Pure transport of the density $$\partial_{t}\rho+\bu\bcdot\bnabla\rho=0$$
- Dirichlet boudnary conditions $$\bu|_{\partial\Omega}=\bzero$$
A priori estimates
- Potential energy $$E(\rho)=\int_{\Omega}z\rho$$
- From transport equation $$\frac{\mathrm{d}}{\mathrm{d}t}E(\rho)=\int_{\Omega}z\partial_{t}\rho=-\int_{\Omega}z\bnabla\bcdot(\rho\bu)=\int_{\Omega}\rho\bu\bcdot\be_{z}$$
- From velocity equation $$\frac{\mathrm{Re}}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\bu\|_{L^2}^{2}=-\|\bnabla\bu\|_{L^2}^2-\int_{\Omega}\rho\bu\bcdot\be_{z}$$
- Energy balance $$\frac{\mathrm{d}}{\mathrm{d}t}\biggl(\frac{\mathrm{Re}}{2}\|\bu\|_{L^2}^{2}+E(\rho)\biggr)=-\|\bnabla\bu\|_{L^2}^2$$
Steady states
Proposition
Steady states are precisely stratified density profiles, i.e
density profiles depending only on the vertical variable $z$.
Sketch of the proof
- From energy estimate $\|\bnabla\bu\|_{L^2}=0$, hence $\bu=\bzero$.
- Hence velocity equation reduces to $$\bnabla p=-\rho\be_{z}$$ so the pressure is independent of $x$, hence $\rho$ too.
Difficulty
Continuum of steady states, hence asymptotic stability is not expected to hold → slightly modified stratified profile.
Stokes-transport problem
- Neglecting the inertia of the fluid i.e. taking $\mathrm{Re}=0$: $$\tag{ST}\left\{\begin{aligned} \partial_{t}\rho+\bu\bcdot\bnabla\rho &=0\\ -\Delta\bu+\bnabla p &=-\rho\be_z\\ \bnabla\bcdot\bu &=0\\ \bu|_{\partial\Omega} &=\bzero\\ \rho|_{t=0} &= \rho_0 \end{aligned}\right.$$
- Valid for very viscous fluids
- Limit of infinitely many spherical solid particles sedimenting in a fluid (Höfer, Mecherbet, 2018)
- Simplification that keeps most of the interesting behavior
- Boussinessq ($\mathrm{Re}>0$) is a perturbation of Stokes ($\mathrm{Re}=0$) in the long-time limit (Park, 2024)
Global well-posedness
Theorem (Leblond, 2021, 2023)
Let $q\in(2,\infty]$. For any $\rho_{0}\in L^q$, (ST) has a unique solution
$$\rho\in L^{\infty}(\mathbb{R},L^{q}), \qquad \bu\in L^{\infty}(\mathbb{R},W^{1,\infty}).$$
Existence still holds for $q>1$.
Previous results
- For patches: Antontsev, Meirmanov & Yurinsky (2000)
- In $\mathbb{R}^3$: Höfer (2018), Mecherbet (2018), Höfer & Schubert (2021), Mecherbet & Sueur (2022), Inversi (2023)
Sketch of the proof
- Stokes bounds: $$\begin{align*} \|\bu\|_{W^{1,q}} &\lesssim \|\rho\|_{W^{-1,q}}\\ \|\bu\|_{W^{1,\infty}} \lesssim \|\bu\|_{W^{2,q}} &\lesssim \|\rho\|_{L^{q}} \end{align*}$$
- Method of characteristics, $\rho$ is the push-forward of $\rho_0$: $$\rho(t) = \rho_{0}\circ\mathbf{X}(-t)$$
- Stability estimate: $$\|\rho_1-\rho_2\|_{L^{\infty}_{T}W^{-1,q}} \lesssim \|\rho_{0}\|_{L^\infty}Te^{CKT}\|\bu_{1}-\bu_{2}\|_{L^{\infty}_{T}L^{q}}$$ where $$K=\|\bnabla\bu_1\|_{L^{\infty}_{T}L^{\infty}}$$
- Global solution by using the conservation of the $L^q$-norm
2. Stability of
continuous
density profiles
Stability of $\Theta(z)=1-z$
Theorem (Dalibard, Guillod & Leblond, 2024)
There exists $\varepsilon_0 >0$ small such that for any $\rho_0 \in H^6$
satisfying $\|\rho_0-\Theta\|_{H^6} \leq \epsilon_0$ and $\rho_0-\Theta\in H^2_0$,
the solution of (ST) satisfies:
$$\begin{align*}
\|\rho-\rho_\infty\|_{L^2(\Omega)} &\lesssim \frac{\varepsilon_0}{1+t},& \|\rho-\rho_\infty\|_{H^4(\Omega)} \lesssim \varepsilon_0,
\end{align*}$$
for some $\rho_\infty$ depending only on $z$.
Remarks
- The perturbation need to vanish near the boundary.
- Can be generalized to other profiles $\Theta(z)$ provided: $$\sup_{(0,1)}\partial_{z}\Theta<0$$
Identification of the limit
Corollary (Dalibard, Guillod & Leblond, 2024)
The long-time asymptotic $\rho_\infty$ is given by the decreasing vertical rearrangement of $\rho_0$:
$$\rho_\infty(z) = \rho_0^*(z) := \int_0^\infty \boldsymbol{1}_{0 \leq z \leq |\{\rho_0>\lambda\}|}\mathrm{d}\lambda.$$
Sketch of the proof
- $\rho_\infty$ depends only on $z$
- $\rho_\infty$ is strictly decreasing as a perturbation of $\Theta$
- levels sets of $\rho$ are preserved by the time evolution, so $\rho_\infty$ is a rearrangement of $\rho_0$
- unique decreasing vertical rearrangement
Strategy of proof
Rewriting
- Perturbation $\theta$: $$\rho = \Theta + \theta$$
- Decomposition into horizontal average and complement: $$\theta = \bar\theta + \theta' \qquad \bar\theta = \frac{1}{2\pi}\int_0^{2\pi}\theta \, \mathrm{d}x$$
- In these variables and using vorticity formulation: $$\left\{ \begin{aligned} \partial_t\theta' + (\bu\cdot\nabla\theta')' &= (1-\partial_z\bar\theta)u_2 & \theta'|_{t=0} &= \theta'_0 \\ \partial_t\bar\theta + \overline{\bu\cdot\nabla\theta'} &= 0 & \bar\theta|_{t=0} &= \bar\theta_0 \\ \Delta^2\psi &= \partial_x\theta' & \psi|_{\partial\Omega} &= 0 \\ \bu &= \nabla^\perp\psi & \partial_\bn\psi|_{\partial\Omega} &= 0 \\ \end{aligned} \right.$$
Strategy of proof
Dominant terms & Bootstrap
As $t\to\infty$, one expect that $\bar\theta\to\rho_\infty-\Theta$ and $\theta'\to0$,
thus the dominant part is $$\left\{ \begin{aligned} \partial_t\theta' &= (1-\partial_z\bar\theta)\partial_{x}\psi & \theta'|_{t=0} &= \theta'_0 \\ \Delta^2\psi &= \partial_x\theta' & \psi|_{\partial\Omega} &= \partial_\bn\psi|_{\partial\Omega} = 0 \end{aligned} \right.$$
thus the dominant part is $$\left\{ \begin{aligned} \partial_t\theta' &= (1-\partial_z\bar\theta)\partial_{x}\psi & \theta'|_{t=0} &= \theta'_0 \\ \Delta^2\psi &= \partial_x\theta' & \psi|_{\partial\Omega} &= \partial_\bn\psi|_{\partial\Omega} = 0 \end{aligned} \right.$$
Main steps
- Analysis when $\partial_z\bar\theta$ is small and given:
- Uniform bounds of $\|\Delta^{s/2}\theta'\|_{L^2}$ for $s=0,4$
- Decay of $\|\theta'\|_{L^2}$
- Bootstrap argument
Step 1. Uniform $H^s$ bounds
$$\partial_t\theta' = (1-\partial_z\bar\theta)\partial_{x}\psi \qquad \Delta^2\psi=\partial_x\theta'$$- In the torus $\mathbb{T}^2$ for any $s$: $$\begin{align} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \|\Delta^{s/2}\theta'\|_{L^2}^2 &= - \int\Delta^{s/2}((1-\partial_z\bar\theta)\psi) \Delta^{s/2+2}\psi \\ &= - \int\Delta^{s/2+1}((1-\partial_z\bar\theta)\psi) \Delta^{s/2+1}\psi \\ &\leq -(1-\bar C \|\partial_z\bar\theta\|_{H^{s+2}}) \|\Delta^{s/2+1}\psi\|_{L^2} \end{align}$$
- In $\Omega=\mathbb{T}\times(0,1)$ with Dirichlet boundary conditions:
- if $\partial_z\bar\theta=0$, works only for $s=0$ and $s=4$
because only the following traces are zero: $$\psi|_{\partial\Omega} \quad \partial_z\psi|_{\partial\Omega} \quad \Delta^2\psi|_{\partial\Omega} \quad \partial_z\Delta^2\psi|_{\partial\Omega}$$ - if $\partial_z\bar\theta\neq0$, some boundary terms remain for $s=4$
- if $\partial_z\bar\theta=0$, works only for $s=0$ and $s=4$
Step 2. Decay of $\|\theta'\|_{L^2}$
- Gagliardo–Nirenberg interpolation: $$\|\partial_{x}^{-1}\Delta^{2}\psi\|_{L^{2}}^2\lesssim \frac1{K}\|\Delta\psi\|_{L^{2}}^2+K^{2}\|\partial_{x}^{-3}\Delta^{4}\psi\|_{L^{2}}^2$$
- Energy inequality: $$\frac{\mathrm{d}}{\mathrm{d}t} \|\theta'\|_{L^{2}}^{2} \lesssim-\|\Delta\psi\|_{L^{2}}^{2} \lesssim K^{3}\|\partial_{x}^{-3}\Delta^{4}\psi\|_{L^{2}}^{2}-K\|\partial_{x}^{-1}\Delta^{2}\psi\|_{L^{2}}^{2}$$
- Taking $K \simeq (1+t)^{-1}$ and since $\Delta^2\psi = \partial_x\theta'$: $$\frac{\mathrm{d}}{\mathrm{d}t} \|\theta'\|_{L^{2}}^{2} + \frac{3}{1+t} \|\theta'\|_{L^{2}}^{2} \lesssim \frac{1}{(1+t)^3} \|\partial_{x}^{-2}\Delta^{2}\theta'\|_{L^{2}}^{2}$$
- Uniform $H^4$ bound: $$\|\Delta^{2}\partial_{x}^{-2}\theta'\|_{L^{2}}^{2} \lesssim \|\Delta^{2}\theta'\|_{L^{2}}^{2} \leq \|\theta'_0\|_{H^4}$$
- Gronwall: $$\|\theta'\|_{L^{2}} \lesssim \frac{\|\theta'_0\|_{H^4}}{1+t}.$$
Comparison with IPM
incompressible porous media
- IPM: Stokes replaced by Darcy $$\bu + \bnabla p = -\rho\be_z \qquad \bu\bcdot\bn=0$$
- No global well-posedness for IPM
- Similar stability result for IPM by Elgindi (2017) in $\mathbb{R}^2$ and $\mathbb{T}^2$
- Adapted by Castro, Córdoba & Lear (2019) to $\mathbb{T}\times(0,1)$
⤷ even traces $\partial_z^{2n}\rho_0|_{\Omega}$ need to vanish (preserved by the evolution)
⤷ almost no problem with integrations by part, like in $\mathbb{T}^2$
Nonlinear instability
Theorem (Leblond, 2023), adapted from Kiselev-Yao (2023)
Let $\Theta\in C^\infty$ be a stationary solution for (ST).
For any $\varepsilon>0$ and $0<\gamma<2$, there exists $\rho_0\in C^\infty$
satisfying $\|\rho_0-\Theta\|_{H^{2-\gamma}}\leq\varepsilon$ such that:
$$\limsup_{t\to\infty} t^{-s/4} \|\rho(t)-\Theta\|_{\dot{H}^s} = \infty$$
for all $s>0$.
Stability threashold
Stability in $H^4$ and unstability in $H^{2-}$, so Sobolev regularity threashold inbetween?
Asymptotic expansion
Theorem (Dalibard, Guillod & Leblond, 2024)
There exists $\varepsilon_0 >0$ small such that for any $\rho_0 \in H^{14}$ satisfying
$\|\rho_0-\Theta\|_{H^{14}} \leq \varepsilon_0$ and $\rho_0 - \Theta\in H^2_0$,
then the solution of (ST) satisfies:
\begin{align}
\rho = \rho_\infty + \rho_\BL + O(t^{-2}) \quad \text{in} \quad L^2 \quad \text{as} \quad t\to\infty,
\end{align}
where $\rho_\BL$ is the boundary layer part:
\begin{equation}
\rho_\BL = \frac{1}{t}\Theta_\mtop(x,t^{1/4}(1-z)) + \frac{1}{t}\Theta_\mbot(x,t^{1/4}z) + \text{l.o.t.}
\end{equation}
with $\Theta_\mtop$ and $\Theta_\mbot$ decaying exponentially in the second variable.
Remarks on the asymptotic expansion
- Optimal decay of $\|\rho-\rho_\infty\|_{L^2}$ like $t^{-9/8}$ since $$\|\rho_\BL\|_{L^2} \sim (1+t)^{-9/8}$$
- Boundary layer terms solves linear equations
- If the perturbation do not vanish near the boundary:
- lower order boundary layers present to lift these traces
- main order will be solution of a non-linear equation
- most probably way more technical
What is happening in general?
Proposition (Leblond, 2023)
For $\rho_0\in L^\infty$, let denote by $\rho_0^*$ the decreasing vertical rearrangement of $\rho_0$,
and by $\rho$ the solution of (ST).
- The potential energy $E(t)$ converges to some $E_\infty\geq E(\rho_0^*)$.
- The $\omega$-limit set $$\omega_{H^{-1}} := \bigcap_{t\geq0} \overline{\{\rho(\tau):\tau\geq t\}}^{H^{-1}}$$ is non-empty.
- Any $\rho_\infty\in\omega_{H^{-1}}$ is a stratified rearrangment of $\rho_0$ and $E(\rho_\infty) = E_\infty$.
- If $E_\infty=E(\rho_0^*)$, then $\omega_{H^{-1}} = \{\rho_0^*\}$ and $\rho(t)$ converges to $\rho_0^*$ in $H^{-1}$.
What can go wrong?
3. Stability of the interface problem
Stability of the constant interface
Theorem (Gancedo, Granero-Belinchón & Salguero, 2023)
There exists $\varepsilon_0>0$ small such that for any $h_0\in H^3(\mathbb{T})$ with $\|h_0\|_{H^3}\leq\varepsilon_0$,
the solution of (ST) in $\mathbb{T}\times\mathbb{R}$ with $\rho_0(x,z)=\mathbf{1}_{\{z<h_0(x)\}}$ is given by
$\rho(t,x,z)=\mathbf{1}_{\{z<h(t,x)\}}$ where:
$$\|h-\bar h_0\|_{L^2} \lesssim \frac{\varepsilon_0}{1+t^2}, \qquad \|h-\bar h_0\|_{H^3} \lesssim \varepsilon_0.$$
Numerical results
Influence of the boundary
$\|\zeta\|_{\dot{H}^2}$ ↘
$\|\zeta\|_{\dot{H}^2}$ ⤳
Numerical results
$\dot{H}^1$ and length not Lyapunov
$\|\zeta\|_{\dot{H}^1}$ ↘
$\|\zeta\|_{\dot{H}^1}$ ↗
length ⤳
Numerical results
Graph breaking
Numerical obervations
- Length of the interface can grow
- Graph nature can break