Damping effects in boundary layers for rotating fluids with small viscosity

Abstract

The goal of this paper is to study the system of rotating fluids between two infinite parallel plates with Dirichlet boundary conditions and with small viscosity which vanishes when the Rossby number goes to zero. We want to improve the convergence result of [18] and show the global in time convergence of the weak solution of the system of rotating fluids toward the solution of a two-dimensional damped Euler system with three components, using the decay in time of the $H^s$-norm ($s>2$) of the limiting solution.

Appendix A: Construction of the Ekman layers

The construction of Ekman boundary layers is classical and can be found in [9, 18] for instance. In this paper, we will only give very brief recall of this construction. The typical approach consists in looking for the appropriate solutions of system (1.1) in the following form

$$\begin{aligned} \begin{aligned}&u^\varepsilon = u^{I,0} + u^{B,0} + {\tilde{u}}^{B,0} + \varepsilon \left( u^{I,1} + u^{B,1} + {\tilde{u}}^{B,1}\right) + \varepsilon ^2 \left( u^{I,2} + u^{B,2} + {\tilde{u}}^{B,2}\right) + \cdots \\&p^\varepsilon = \frac{1}{\varepsilon }\left( p^{I,-1} + p^{B,-1} + {\tilde{p}}^{B,-1}\right) + p^{I,0} + p^{B,0} + {\tilde{p}}^{B,0} + \varepsilon \left( p^{I,1} + p^{B,1} + {\tilde{p}}^{B,1}\right) + \cdots . \end{aligned} \end{aligned}$$

(A.1)

The “interior part” functions with the index “I” depend on $\left( t,x_1,x_2,x_3\right)$ and the “boundary layer part” with the index “B” consists in smooth functions of the form

$$\begin{aligned}&u^{B,j} = u^{B,j}\left( t,x_1,x_2,\frac{x_3}{\varepsilon }\right) ,&p^{B,j} = p^{B,j}\left( t,x_1,x_2,\frac{x_3}{\varepsilon }\right) ,\\&{\tilde{u}}^{B,j} = {\tilde{u}}^{B,j}\left( t,x_1,x_2,\frac{1-x_3}{\varepsilon }\right) ,&{\tilde{p}}^{B,j} = {\tilde{p}}^{B,j}\left( t,x_1,x_2,\frac{1-x_3}{\varepsilon }\right) , \end{aligned}$$

and which rapidly (exponentially) decrease when the third space variable goes to infinity. The justification of the size of the boundary layers being $\varepsilon$ and the discussion about other sizes of the layers can be found in [8, 9, 18, 26].

1.1 In the interior part of the domain

Away from the boundary, all the boundary terms in (A.1) rapidly decrease to zero. Thus, at the leading order $\varepsilon ^{-1}$, the divergence-free conditions and the fast rotation yield

$$\begin{aligned} u^{I,0}_2 = \partial _1 p^{I,-1}, \; u^{I,0}_1 = -\partial _2 p^{I,-1}, \; \partial _3 p^{I,-1} = 0, \end{aligned}$$

which leads to

$$\begin{aligned} \left\{ \begin{aligned}&\partial _3 u^{I,0} = 0, \; \partial _3 p^{I,-1} = 0\\&\partial _1 u^{I,0}_1 + \partial _2 u^{I,0}_2 = 0. \end{aligned} \right. \end{aligned}$$

(A.2)

The limiting velocity $u^{I,0}$ is then a two-dimensional divergence-free vector field with three components, which means that the fluid has tendency to move in columns when the rotation is fast, as predicts the Taylor–Proudman theorem.

Now, looking at the zeroth order ($\varepsilon ^0$), one obtains the governing equation of the fluid at the limit as $\varepsilon \rightarrow 0$

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t u^{I,0}_1 + u^{I,0}_h\cdot \nabla _h u^{I,0}_1 - u^{I,1}_2 + \partial _1 p^{I,0} = 0\\&\partial _t u^{I,0}_2 + u^{I,0}_h\cdot \nabla _h u^{I,0}_2 + u^{I,1}_1 + \partial _2 p^{I,0} = 0\\&\partial _t u^{I,0}_3 + u^{I,0}_h\cdot \nabla _h u^{I,0}_3 + \partial _3 p^{I,0} = 0. \end{aligned} \right. \end{aligned}$$

(A.3)

Here, we remark that the fast rotation implies the action of the first-order term, i.e., the added vector field $\left( -u^{I,1}_2, u^{I,1}_1, 0\right)$, on the limiting behavior of the fluid. In the next paragraph, we will show that the interaction with the boundary allows to explicitly calculate this vector field, which turns out to be a dissipative term (the Ekman pumping). We also remark that $p^{I,0}$ is independent of $x_3$, so under the hypothesis of Theorems 1.1 and 1.2 , we deduce from the third equation of system (A.3) that $u^{I,0}_3 \equiv 0$.

1.2 In the Ekman boundary layers

We will focus on the boundary layer near $\left\{ x_3 = 0\right\}$. The other layer near $\left\{ x_3 =1\right\}$ can be obtained in a similar way. Performing the change of variable $y = \dfrac{x_3}{\varepsilon }$, we deduce the following “divergence-free properties” of $u^{B,j}$, for any $j \ge 0$,

$$\begin{aligned} \left\{ \begin{aligned}&\partial _y u^{B,0}_3 = 0\\&\partial _1 u^{B,j}_1 + \partial _2 u^{B,j}_2 + \partial _y u^{B,j+1}_3 = 0, \quad \forall \, j\ge 0. \end{aligned} \right. \end{aligned}$$

(A.4)

We remark that the first equation of (A.4) implies that $u^{B,0}_3 \equiv 0$ because it goes to zero as $y \rightarrow +\infty$.

The Dirichlet boundary condition $u^\varepsilon (t,x_1,x_2,0) = 0$ is rewritten as follows

$$\begin{aligned} u^{B,j}(t,x_1,x_2,0) = -u^{I,j}(t,x_1,x_2,0), \quad \forall \, j\ge 0, \end{aligned}$$

(A.5)

which implies $u^{I,0}_3 \equiv 0$, since $u^{I,0}_3$ is independent of $x_3$ and $u^{I,0}(t,x_1,x_2,0) = -u^{B,0}(t,x_1,x_2,0) = 0$.

Now, putting the Ansatz into the first equation of (1.1) and looking at the leading order $\varepsilon ^{-2}$, we simply get

$$\begin{aligned} \partial _y p^{B,-1} = 0, \end{aligned}$$

and so, $p^{B,-1} = 0$ since $p^{B,-1} \rightarrow 0$ as $y\rightarrow +\infty$. This is a classical principle of fluid mechanics which claims that the pressure does not vary in a boundary layer. At the next order $\varepsilon ^{-1}$, combining the boundary condition (A.5) and the fact that $u^{I,0} = u^{I,0}(t,x_1,x_2)$ is independent of $x_3$, we recover the following system

$$\begin{aligned} \left\{ \begin{aligned}&\beta \partial _y^2 u^{B,0}_1 + u^{B,0}_2 = 0\\&\beta \partial _y^2 u^{B,0}_2 - u^{B,0}_1 = 0\\&u^{B,0}_1\vert _{y=0} = -u^{I,0}_1, \; u^{B,0}_2\vert _{y=0} = -u^{I,0}_2\\&\lim _{y\rightarrow +\infty } u^{B,0}_1 = \lim _{y\rightarrow +\infty } u^{B,0}_2 = 0. \end{aligned} \right. \end{aligned}$$

(A.6)

This system can be explicitly solved and that gives

$$\begin{aligned} \left\{ \begin{aligned}&u^{B,0}_1 = -\, {\mathrm{e}}^{-\frac{y}{\sqrt{2\beta }}} \left( u^{I,0}_1 \cos \frac{y}{\sqrt{2\beta }} + u^{I,0}_2 \sin \frac{y}{\sqrt{2\beta }}\right) = -\, {\mathrm{e}}^{-\frac{x_3}{\varepsilon \sqrt{2\beta }}} \left( u^{I,0}_1 \cos \frac{x_3}{\varepsilon \sqrt{2\beta }} + u^{I,0}_2 \sin \frac{x_3}{\varepsilon \sqrt{2\beta }}\right) \\&u^{B,0}_2 = -\, {\mathrm{e}}^{-\frac{y}{\sqrt{2\beta }}} \left( u^{I,0}_2 \cos \frac{y}{\sqrt{2\beta }} - u^{I,0}_1 \sin \frac{y}{\sqrt{2\beta }}\right) = -\, {\mathrm{e}}^{-\frac{x_3}{\varepsilon \sqrt{2\beta }}} \left( u^{I,0}_2 \cos \frac{x_3}{\varepsilon \sqrt{2\beta }} - u^{I,0}_1 \sin \frac{x_3}{\varepsilon \sqrt{2\beta }}\right) . \end{aligned} \right. \end{aligned}$$

(A.7)

Using (A.4), (A.7) and the fact that $\partial _1 u^{I,0}_1 + \partial _2 u^{I,0}_2 = 0$, we have

$$\begin{aligned} \partial _y u^{B,1}_3 = - \partial _1 u^{B,0}_1 - \partial _2 u^{B,0}_2 = \exp \left( -\frac{y}{\sqrt{2\beta }}\right) \sin \left( \frac{y}{\sqrt{2\beta }}\right) \text{ curl }\,u^{I,0}_h, \end{aligned}$$

where $\text{ curl }\,u^{I,0}_h = \partial _1 u^{I,0}_2 - \partial _2 u^{I,0}_1$. Integrating the above equation with respect to y and recalling that $u^{B,1}_3 \rightarrow 0$ as $y\rightarrow +\infty$, we obtain

$$\begin{aligned} u^{B,1}_3 = -\sqrt{\beta } {\mathrm{e}}^{-\frac{y}{\sqrt{2\beta }}} \sin \left( \frac{y}{\sqrt{2\beta }} + \frac{\pi }{4}\right) \text{ curl }\,u^{I,0}_h = -\sqrt{\beta } {\mathrm{e}}^{-\frac{x_3}{\varepsilon \sqrt{2\beta }}} \sin \left( \frac{x_3}{\varepsilon \sqrt{2\beta }} + \frac{\pi }{4}\right) \text{ curl }\,u^{I,0}_h. \end{aligned}$$

(A.8)

Remark A.1

We remark that unlike the irrotational case where the boundary layers are described by the Prandtl equations, in this case, the main boundary layer term $\left( u^{B,0}_1,u^{B,0}_2,u^{B,1}_3\right)$ can be explicitly computed from the limiting velocity in the interior domain, which allows to directly treat these layers without using the Prandtl equations. We also remark that in the case where the rotation axis is not $e_3$, the main boundary layer term $\left( u^{B,0}_1,u^{B,0}_2,u^{B,1}_3\right)$ can be described by an evolutionary Prandtl-like system and will be much more difficult to study. We refer for example to [22] for more details.

The same arguments near the boundary $\left\{ x_3 = 1\right\}$ lead to ${\tilde{p}}^{B,-1} = 0$ and

$$\begin{aligned} \left\{ \begin{aligned}&{\tilde{u}}^{B,0}_1 = -\, {\mathrm{e}}^{-\frac{1-x_3}{\varepsilon \sqrt{2\beta }}} \left( u^{I,0}_1 \cos \frac{1-x_3}{\varepsilon \sqrt{2\beta }} + u^{I,0}_2 \sin \frac{1-x_3}{\varepsilon \sqrt{2\beta }}\right) \\&{\tilde{u}}^{B,0}_2 = -\, {\mathrm{e}}^{-\frac{1-x_3}{\varepsilon \sqrt{2\beta }}} \left( u^{I,0}_2 \cos \frac{1-x_3}{\varepsilon \sqrt{2\beta }} - u^{I,0}_1 \sin \frac{1-x_3}{\varepsilon \sqrt{2\beta }}\right) \\&{\tilde{u}}^{B,1}_3 = \sqrt{\beta } {\mathrm{e}}^{-\frac{1-x_3}{\varepsilon \sqrt{2\beta }}} \sin \left( \frac{1-x_3}{\varepsilon \sqrt{2\beta }} + \frac{\pi }{4}\right) \text{ curl }\,u^{I,0}_h. \end{aligned} \right. \end{aligned}$$

(A.9)

We come back to (A.3). Taking $\text{ curl }\,u^{I,0}_h = \partial _1 u^{I,0}_2 - \partial _2 u^{I,0}_1$, we have

$$\begin{aligned} \partial _t \text{ curl }\,u^{I,0}_h + u^{I,0}_h \cdot \nabla _h \text{ curl }\,u^{I,0}_h = - \partial _1 u^{I,1}_1 - \partial _2 u^{I,1}_2 = \partial _3 u^{I,1}_3. \end{aligned}$$

(A.10)

Since $u^{I,0}$ is independent of $x_3$, by integrating (A.10) with respect to $x_3$ over [0, 1], one gets

$$\begin{aligned} \partial _t \text{ curl }\,u^{I,0}_h + u^{I,0}_h \cdot \nabla _h \text{ curl }\,u^{I,0}_h = {u^{I,1}_3}_{\vert _{x_3=1}} - {u^{I,1}_3}_{\vert _{x_3=0}} = {u^{B,1}_3}_{\vert _{x_3=0}} - {\tilde{u}}^{B,1}_3{}_{\vert _{x_3=1}} = -\sqrt{2\beta } \text{ curl }\,u^{I,0}_h. \end{aligned}$$

Using the Biot–Savart law, we can finally rewrite system (A.3) in the same form as (1.3)

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t u^{I,0}_1 + u^{I,0}_h\cdot \nabla _h u^{I,0}_1 + \sqrt{2\beta }\, u^{I,0}_1 + \partial _1 q = 0\\&\partial _t u^{I,0}_2 + u^{I,0}_h\cdot \nabla _h u^{I,0}_2 + \sqrt{2\beta }\, u^{I,0}_2 + \partial _2 q = 0\\&\partial _3 u^{I,0} \equiv 0, \; u^{I,0}_3 \equiv 0,\\&\partial _1 u^{I,0}_1 + \partial _2 u^{I,0}_2 = 0. \end{aligned} \right. \end{aligned}$$

(A.11)