1 Introduction

We consider an inviscid incompressible fluid in a periodic channel domain \( \Omega := \Omega _h \times (0,h) \subset \mathbb R^3 \), with horizontal periodic domain \( \Omega _h:= \mathbb T^2 = (0,1)^2 \) and vertical domain height \( h \in (0,\infty ) \). Denote by \( v \in \mathbb R^2 \) the horizontal velocity, \( w \in \mathbb R \) the vertical velocity, \( p \in \mathbb R \) the pressure, and \( \rho \in \mathbb R \) the density, respectively. Let the following be the typical characteristic physical scales for length, time, velocity, density, and pressure:

$$\begin{aligned} \begin{aligned}&L{} & {} \text {length scale} \\&U{} & {} \text {mean advective velocity} \\&T_e:= \dfrac{L}{U}{} & {} \text {eddy turnover time} \\&T_R:= f^{-1}{} & {} \text {rotation time} \\&\rho _b{} & {} \text {mean density} \\&{\overline{p}}{} & {} \text {mean pressure.} \end{aligned} \end{aligned}$$

Furthermore, set \( \overline{\rho }= \overline{\rho }(z) \) to be the background density stratification, which is assumed to be linear in the vertical coordinate, and decompose the density into the sum of stratification \( \overline{\rho }\) and deviation \( \rho _b\theta \), i.e.,

$$\begin{aligned} \rho = \rho _b \theta + \overline{\rho }\end{aligned}$$

The buoyancy (Brunt-Väisälä) frequency is defined as

$$\begin{aligned} N:= \biggl (- \dfrac{g \partial _z \overline{\rho }}{\rho _b} \biggr )^{1/2}, \end{aligned}$$

and the corresponding buoyancy time scale is

$$\begin{aligned} T_N:= N^{-1}. \end{aligned}$$

In this geophysical situation, one can introduce the following relevant non-dimensional numbers:

$$\begin{aligned}&\text {the Rossby number}{} & {} \textrm{Ro}:= \dfrac{U}{Lf} \\&\text {the Froude number}{} & {} \textrm{Fr}:= \dfrac{U}{LN} \\&\text {the Euler number}{} & {} \overline{P}:= \dfrac{{\overline{p}}}{\rho _bU^2} \\{} & {} {}&\Gamma := \dfrac{gL}{U^2}, \end{aligned}$$

see, e.g., [33]. With such notations, the dimensionless rotating Boussinesq equations are given by

$$\begin{aligned} \partial _t v + v \cdot \nabla _hv + w \partial _zv + \dfrac{1}{\textrm{Ro}} v^\perp + \overline{P} \nabla _hp = 0, \end{aligned}$$
(1.1a)
$$\begin{aligned} \partial _tw + v \cdot \nabla _hw + w \partial _zw + \overline{P}\partial _zp - \Gamma \theta = 0, \end{aligned}$$
(1.1b)
$$\begin{aligned} \partial _t\theta + v \cdot \nabla _h\theta + w \partial _z\theta + \dfrac{1}{\Gamma \cdot \textrm{Fr}^2} w = 0, \end{aligned}$$
(1.1c)
$$\begin{aligned} \textrm{div}_h\,v + \partial _zw = 0, \end{aligned}$$
(1.1d)

with

$$\begin{aligned} w \vert _{z=0,h} = 0&\qquad \text {i.e., the impermeable boundary condition,} \end{aligned}$$
(1.1e)

see, e.g., [33].

In this paper, we consider the quasi-geostrophic scale where

  • The Rossby number is small

    $$\begin{aligned} \textrm{Ro} = \varepsilon \ll 1; \end{aligned}$$

  • The flow is in geostropic balance, i.e., the rotation and the pressure forces are in balance,

    $$\begin{aligned} {\overline{P}} = \dfrac{1}{\textrm{Ro}}; \end{aligned}$$

  • The Froude number is small and equal to the Rossby number,

    $$\begin{aligned} \textrm{Fr} = \textrm{Ro}; \end{aligned}$$

  • The non-dimensional number \( \Gamma \) is in balance with the inverse of the Froude number

    $$\begin{aligned} \Gamma = \dfrac{1}{\textrm{Fr}}. \end{aligned}$$

Then the rotating Boussinesq equations (1.1) become

$$\begin{aligned} \partial _t v + v \cdot \nabla _hv + w \partial _zv + \dfrac{1}{\varepsilon } v^\perp + \dfrac{\nabla _hp}{\varepsilon } = 0, \end{aligned}$$
(1.2a)
$$\begin{aligned} \partial _tw + v \cdot \nabla _hw + w \partial _zw + \dfrac{\partial _zp}{\varepsilon } - \dfrac{\theta }{\varepsilon } = 0, \end{aligned}$$
(1.2b)
$$\begin{aligned} \partial _t\theta + v \cdot \nabla _h\theta + w \partial _z\theta + \dfrac{w}{\varepsilon } = 0, \end{aligned}$$
(1.2c)
$$\begin{aligned} \textrm{div}_h\,v + \partial _zw = 0, \end{aligned}$$
(1.2d)

with

$$\begin{aligned} w \vert _{z=0,h} = 0. \end{aligned}$$
(1.2e)

We refer the reader to [33, Sect. 7.4] for the detailed derivation of system (1.2). We remark that, the small Rossby number, i.e. \(\textrm{Ro}\ll 1 \), induces the fast Rossby waves, and the small Froude number, i.e. \( \textrm{Fr} \ll 1 \), induces the fast internal gravity waves. In our setting, i.e., system (1.2), both Rossby and gravity waves are fast and they are coupled. In particular, they have the same scale.

The goal of this work is to investigate the asymptotic limit of system (1.2) as \( \varepsilon \rightarrow 0^+ \) in the channel domain \( \Omega \)., i.e., the quasi-geostrophic approximation, taking into account the fast-slow waves interaction and their corresponding resonance terms.

Similar problem has been studied in the case of “well-prepared” initial data by Bourgeois and Beale in [10], where the convergence, as well as the convergence rate, of solutions to that of quasi-geostrophic equations ((2.27) and (2.29), below) is proved. In particular, the well-prepared initial data are chosen so that there are only slow waves in the dynamics and no contribution of the fast waves. That is, the initial data is close to the geostrophic balance (see (2.16)–(2.18), below). We remark that [10] assumes that \( \partial _zp^0\vert _{z=0,h} = 0 \) together with the balanced initial data. This guarantees that the system of equations satisfy some symmetry, and eventually can be extended periodically to a system into \( \mathbb T^3 \), i.e., there is no boundary effect as if one has a virtual boundary. The general convergence theory when \( \partial _zp^0\vert _{z=0,h} \ne 0 \) is still open. Here \( p^0 \) is the stream function associated with the potential vorticity as in (2.26). The existence of weak solutions for these quasi-geostrophic equations is established in [41, 44]

Taking into account the fast waves, but without physical boundary (i.e., in \( \mathbb T^3 \)), Embid and Majda studied the nonlinear resonances and established the asymptotic limit of system (1.2) in [16, 17, 34]. The limiting system is the quasi-geostrophic equation (2.27) with nonlinear resonances on the right-hand side, while the velocity and the temperature in the limiting quasi-geostrophic equations are given by (2.16) and (2.17), below, respectively.

In the case with vanishing viscosity, an Ekman boundary layer will arise in the channel domain, which leads to Ekman pumping. This is verified in [14], in the case with well-prepared initial data (i.e., slow waves only). To the best of the authors’ knowledge, the asymptotic limit taking into account both the fast waves and the Ekman pumping is open. The global well-posedness of solutions to the quasi-geostrophic system with Ekman pumping was established in [40].

In this paper, we introduce the notion of (full) generalized potential vorticity (i.e., \( \Phi \) and \( \Psi \) defined in (1.4) and (1.5), below, respectively), which allows us to separately describe the slow and the fast waves of the dynamics of system (1.2) in a channel domain without introducing any boundary layer. Moreover, the interaction between the slow and fast waves can be easily tracked and investigated. Therefore, we are able to establish the asymptotic limit as \( \varepsilon \rightarrow 0^+ \) in the channel for general initial data. In particular, we drop the requirement of well-prepared initial data or periodic spatial domain required in [10, 16], respectively. In addition, the fast waves correction to the slow dynamics is identified as a new resonance term.

We remark that in our context, the terms slow (fast) waves and slow (fast) dynamics, as well as well-prepared (ill-prepared or general) initial data and balanced (unbalanced) initial data are interchangeable, respectively. This terminology is widely used in the literature.

Before stating the main results in detail, we would like to put this work in the context of the study of asymptotic limit in the following subsection.

1.1 Asymptotic limit and boundary layer

We should stress that the following references are by no mean exhaustive.

The study of low Mach number limit of the compressible flows was pioneered by Klainerman and Majda in [27, 28], where the convergence with only slow waves (i.e., well-prepared initial data) was shown in domains without boundary. In \( \mathbb R^3 \), Ukai in [51] showed the dispersion of the fast acoustic waves and thus established the low Mach number limit with large acoustic waves. As pointed out in [15], such dispersion in \( \mathbb {R}^3 \) is characterized by the Strichartz estimate [26, 49]. In the case of \( \mathbb T^3 \), [30] showed the weak convergence of low Mach number limit for compressible flows by investigating the nonlinear resonances of fast acoustic waves. The general theory of fast singular limit was developed by Schochet in [47, 48] for hyperbolic systems, which was later extended to parabolic systems in [20]. We refer the reader to [1, 2, 12, 13, 18, 19, 36, 38] and the references therein for more studies of low Mach number limit in domains without boundary. When there is physical boundary in the underlying domain, the low Much number limit of viscous flows may give rise to a boundary layer. This is first studied in terms of eigenvalue-eigenfunction pairs in [23]. Recently in [37], by introducing uniform estimates in the co-normal Sobolev norm, together with some \( L^\infty \) estimates, the low Mach number limit of compressible viscous flows is established in smooth domain with Navier-slip boundary condition and general initial data. However, the corresponding low Mach number limit with no-slip boundary condition is still open.

Meanwhile, in the vanishing viscosity limit of the incompressible Navier–Stokes equations with no-slip boundary condition, the Prandtl boundary layer was introduced by Prandtl in 1904 [43] and became the paradigm of further mathematical studies. See, e.g., [52] for a derivation of the Prandtl equations. However it turned out to be the most singular. The boundary layer is due to the no-slip boundary condition for the Navier-Stokes and since this effect is not present at the level of the Euler equation, a discontinuity appears in the zero viscosity limit. Due to the nonlinearity of the problem such singularity may escape from the boundary layer and propagate in the fluid. This is one of the main source of turbulence, and as a consequence the Prandtl boundary layer is strongly unstable, and therefore may exist only for short time and under strict regularity hypothesis, see, e.g., [32, 45, 46]. A direct proof of such asymptotic limit, with the incompressible Euler equations as the limiting equations, without introducing the boundary layer correction can be found in [7, 39]. For general, smooth, but not analytic, initial data, the vanishing viscosity limit is still an open challenging problem. The pioneer work in this direction is by Kato [24]. See, also, [8, 9] and references therein for related results.

With fast rotation and vanishing viscosity (but no fast internal waves) in a domain with no-slip boundary condition, the Ekman boundary layer may arise, which is an important phenomenon in the atmospheric and oceanic study (see, for instance, [33, 42]). In [22] and [35], the asymptotic limit of fast rotation and vanishing viscosity with the Ekman boundary layer correction was established for flows with and without fast waves, respectively.

With only fast rotation in a domain without boundary (\( \mathbb T^3 \) or \( \mathbb R^3 \)), the asymptotic limit of the Euler or Navier–Stokes equations was studied in [4,5,6], where the limit dynamics is characterized by two dimensions three components (2D3C) flows, and the prolonging effect of fast rotation on the life-span of the solution was established. Such a regularizing effect of fast rotation was demonstrated in the case of a simple convection model in [3, 31]. See also [21, 29] for the study in the primitive equations, and [11] for some examples in the study of mathematical geophysics, including the aforementioned Ekman boundary layer.

As mentioned before, in this paper, we study the singular limit \( \varepsilon \rightarrow 0^+ \) of system (1.2) in the periodic channel domain \( \Omega = \mathbb T^2 \times (0,h) \). In particular, it will be established that the fast rotation induced by strong Coriolis force in (1.2a) suppresses the possible emergence of a boundary layer near the boundary.

1.2 Main results

The first main result of this paper is the following:

Theorem 1.1

(Uniform-in-\(\varepsilon \) estimate). Consider the initial data

$$\begin{aligned} (v_\textrm{in}, w_\textrm{in},\theta _\textrm{in}) \in H^3(\Omega ) \end{aligned}$$

of the solution \( (v,w,\theta )\) to system (1.2), satisfying the compatibility conditions \( \textrm{div}_h\,v_\textrm{in} + \partial _zw_\textrm{in} = 0 \) and \( w_\textrm{in}\vert _{z=0,h}=0 \). Then there exists \( T, C_\textrm{in} \in (0,\infty ) \), depending only on the initial data and independent of \( \varepsilon \), such that

(1.3)

Proof

The proof of this theorem is done in Sect. 3. \(\square \)

The local well-posedness theory of solutions in \( H^3(\Omega ) \) to system (1.2) for fixed \( \varepsilon \in (0,1) \) is classical and thus is omitted here. See, for instance, [25]. With continuity arguments, the uniform estimate (1.3) implies the uniform-in-\(\varepsilon \) local well-posedness with initial data as in the theorem.

To describe our second main result, define

$$\begin{aligned} \Phi (x,y,z,t)&: = \partial _z\theta + \textrm{curl}_h\,v, \end{aligned}$$
(1.4)
$$\begin{aligned} \Psi (x,y,z,t)&: = \nabla _h^\perp \theta + \nabla _hw - \partial _zv, \end{aligned}$$
(1.5)
$$\begin{aligned} H_0(x,y,t)&:= \theta \vert _{z=0}, \end{aligned}$$
(1.6)
$$\begin{aligned} H_h(x,y,t)&:= \theta \vert _{z=h}, \end{aligned}$$
(1.7)

and

$$\begin{aligned} Z(z,t)&:= \int _{\mathbb {T}^2} v(x,y,z,t) \,dxdy. \end{aligned}$$
(1.8)

Then our second main result of this paper is to investigate the limit system, as follows:

Theorem 1.2

(Convergence theory). Let \( T > 0 \) be as in Theorem 1.1, and let \( (\Phi , \Psi , H_0, H_h, Z) \) be defined as in (1.4)–(1.8). Then there exists a subsequence of \( \varepsilon \) that as \( \varepsilon \rightarrow 0^+ \), one has the following convergence in strong topology:

$$\begin{aligned} \Phi&\rightarrow \Phi _p{} & {} \text {in}{} & {} C([0,T];H^1(\Omega )), \end{aligned}$$
(1.9)
$$\begin{aligned} H_0, H_h&\rightarrow H_{p,0}, H_{p,h}{} & {} \text {in}{} & {} C([0,T];H^{3/2}(\mathbb T^2)), \end{aligned}$$
(1.10)
$$\begin{aligned} e^{\mp i \frac{t}{\varepsilon }}(\Psi \pm i \Psi ^\perp )&\rightarrow \psi _{p,\pm }{} & {} \text {in}{} & {} C([0,T];H^1(\Omega )), \end{aligned}$$
(1.11)

and

$$\begin{aligned} e^{\mp i \frac{t}{\varepsilon }}(Z \pm i Z^\perp )&\rightarrow z_{p,\pm }{} & {} \text {in}{} & {} C([0,T];H^2(\Omega )), \end{aligned}$$
(1.12)

and in suitable weak-\(*\) topology (see Sect. 4.1), the limit

$$\begin{aligned} (\Phi _p, H_{p,0}, H_{p,h}, \psi _{p,\pm }, z_{p\,pm}) \end{aligned}$$
(1.13)

satisfies system (4.46), below.

Proof

This is done in Sect. 4. In particular, the strong convergence can be found in (4.11), (4.12), (4.22), and (4.23), respectively. \(\square \)

Remark 1

In this paper, we have not explored the well-posedness, in particular, the uniqueness, of solutions to the limit system (4.46). For this reason, we only have the subsequence convergence in Theorem 1.2. However, if one manages to show the well-posedness of solutions to system (4.46), the convergence should be of the whole sequence of \( \varepsilon \rightarrow 0^+ \). Indeed, the well-posedness theory of the limit system (4.46) is non-trivial, due to the fact that it is no longer a symmetric hyperbolic system. The investigation of the limit system is left as a future study.

The rest of this paper is organized as follows. In Sect. 2, some preliminaries will be provided, including the notations and a boundary-to-domain extension (lifting) Lemma. The classical quasi-geostrophic approximation with only slow waves, i.e., well-prepared initial data, will be reviewed in Sect. 2.2. The key linear slow-fast waves structure will be discussed in Sect. 2.3. Section 3 is dedicated to the proof of Theorem 1.1. This paper will finish with the proof of Theorem 1.2 in Sect. 4.

2 Preliminaries

2.1 Notations and an extension Lemma

In this paper, we have been and will be using

$$\begin{aligned} \biggl ( \begin{array}{c} X_1 \\ X_2 \end{array} \biggr )^\perp = \biggl ( \begin{array}{c} - X_2 \\ X_1 \end{array} \biggr ) \end{aligned}$$
(2.1)

to denote the rotation of a two-dimensional vector. \( \textrm{div}_h\,\) and \( \textrm{curl}_h\,\) represent the horizontal divergence and curl operators, respectively. Then for any two-dimensional vector field \( X = (X_1, X_2)^\top \), one has

$$\begin{aligned} \textrm{div}_h\,X^\perp = - \textrm{curl}_h\,X \qquad \text {and} \qquad \textrm{curl}_h\,X^\perp = \textrm{div}_h\,X. \end{aligned}$$
(2.2)

For any functions A and B, the \(\mathcal X\) norms are written as

(2.3)

We will use \( \Delta _D^{-1} \) to represent the inverse Laplacian subject to the Dirichlet boundary condition at \( z = 0,h \) and the periodic boundary condition horizontally, i.e.,

$$\begin{aligned} \Delta \Delta _{D}^{-1} A = A \qquad \text {with} \quad (\Delta _D^{-1}A)\vert _{z=0,h} = 0. \end{aligned}$$
(2.4)

Therefore, the definition implies

$$\begin{aligned} \Delta \Delta _{D}^{-1} = \textrm{Id}. \end{aligned}$$
(2.5)

However, observe that

$$\begin{aligned} \Delta _D^{-1} \Delta \ne \textrm{Id}, \end{aligned}$$
(2.6)

which plays an important role in the proof of short time stability of analytic Prandtl boundary layer [32, 39].

Moreover, \( \Delta _h^{-1} \) is the inverse Laplacian in the horizontal variable with zero mean value. Therefore, one has that

$$\begin{aligned} \Delta _h^{-1} \Delta _hA = A - \int _{\mathbb T^2} A \,dxdy. \end{aligned}$$
(2.7)

We will need the following extension (lifting) Lemma:

Lemma 1

There exists a bi-linear extension operator

$$\begin{aligned} \mathrm E_b: \mathcal D'(\mathbb T^2) \times \mathcal D'(\mathbb T^2) \mapsto \mathcal D'(\Omega ), \end{aligned}$$
(2.8)

such that for any \( A, B \in H^{s-\frac{1}{2}}(\mathbb T^2) \), \( \mathrm E_b(A,B) \in H^s(\Omega ) \) satisfying

(2.9)

and

$$\begin{aligned} \mathrm E_b(A,B)\vert _{z=0} = A \qquad \text {and} \qquad \mathrm E_b(A,B)\vert _{z=h} = B. \end{aligned}$$
(2.10)

Moreover, the following property holds:

$$\begin{aligned} \partial _t\mathrm E_b(A,B) = \mathrm E_b(\partial _tA,\partial _tB). \end{aligned}$$
(2.11)

Proof

Let \( \chi _0:[0,h] \rightarrow [0,1] \) be a \( C^\infty ([0,h]) \) monotonic function such that

$$\begin{aligned} \chi _0(z) = {\left\{ \begin{array}{ll} 1 &{} \text {in} \quad z \in [0,h/4), \\ 0 &{} \text {in} \quad z \in (3h/4,h]. \end{array}\right. } \end{aligned}$$
(2.12)

Denote by, \( \textbf{x}_h = (x,y)^\top \in \mathbb T^2 \), for \( A, B \in \mathcal D'(\mathbb T^2) \),

$$\begin{aligned} A(x,y) = \sum _{\textbf{k} \in \mathbb Z^2}A_k e^{i 2\pi \textbf{k} \cdot \textbf{x}_h}, \qquad \text {and} \qquad B(x,y) = \sum _{\textbf{k} \in \mathbb Z^2}B_k e^{i 2\pi \textbf{k} \cdot \textbf{x}_h}. \end{aligned}$$
(2.13)

For \( z \in [0, h ]\), we define

$$\begin{aligned} \begin{aligned} \mathrm E_b(A,B)&= \sum _{\textbf{k} \in \mathbb Z^2} A_k e^{i 2\pi \textbf{k} \cdot \textbf{x}_h} e^{-|\textbf{k}|z} \chi _0(z) \\&\quad + \sum _{\textbf{k} \in \mathbb Z^2} B_k e^{i 2\pi \textbf{k} \cdot \textbf{x}_h} e^{-|\textbf{k}|(h-z)}( 1 - \chi _0(z)). \end{aligned} \end{aligned}$$
(2.14)

Then it is easy to verify that \( \mathrm E_b(A,B) \) satisfies the properties in the Lemma. This finishes the proof. \(\square \)

2.2 Classical quasi-geostrophic approximation and the potential vorticity formulation for inviscid flows

In this section, we review the formal quasi-geostrophic approximation with only slow waves of system (1.2), i.e., with well-prepared initial data. This is done by first introducing the formal asymptotic expansion ansatz

$$\begin{aligned} \psi (x,y,z,t): = \psi ^0(x,y,z,t) + \varepsilon \psi ^1(x,y,z,t) \end{aligned}$$
(2.15)

for \( \psi \in \lbrace v, w, p, \theta \rbrace \). Then, after substituting (2.15) in system (1.2) and matching the \( \mathcal {O}(\varepsilon ^{-1}) \) and \( \mathcal O(1) \) terms, one has

$$\begin{aligned}&(v^0)^\perp + \nabla _hp^0 = 0, \end{aligned}$$
(2.16)
$$\begin{aligned}&\partial _zp^0 - \theta ^0 = 0, \end{aligned}$$
(2.17)
$$\begin{aligned}&w^0 = 0, \end{aligned}$$
(2.18)
$$\begin{aligned}&\partial _tv^0 + v^0 \cdot \nabla _hv^0 + w^0 \partial _zv^0 + (v^1)^\perp + \nabla _hp^1 = 0, \end{aligned}$$
(2.19)
$$\begin{aligned}&\partial _tw^0 + v^0 \cdot \nabla _hw^0 + w^0 \partial _zw^0 + \partial _zp^1 - \theta ^1 = 0, \end{aligned}$$
(2.20)
$$\begin{aligned}&\partial _t\theta ^0 + v^0 \cdot \nabla _h\theta ^0 + w^0 \partial _z\theta ^0 + w^1 = 0, \end{aligned}$$
(2.21)
$$\begin{aligned}&\textrm{div}_h\,v^0 + \partial _zw^0 = 0, \end{aligned}$$
(2.22)

and

$$\begin{aligned} w^0\vert _{z=0,h} = 0. \end{aligned}$$
(2.23)

In addition, the \( \mathcal O(\varepsilon ) \) terms of (1.2d) and (1.2e) yield

$$\begin{aligned}&\textrm{div}_h\,v^1 + \partial _zw^1 = 0, \end{aligned}$$
(2.24)

and

$$\begin{aligned} w^1\vert _{z=0,h} = 0. \end{aligned}$$
(2.25)

Following [10, 16], we introduce the potential vorticity formulation. Indeed, from (2.16) and (2.17), it follows that

$$\begin{aligned} \Delta p^0 = (\Delta _h+ \partial _{zz}) p^0 = \textrm{curl}_h\,v^0 + \partial _z \theta ^0. \end{aligned}$$
(2.26)

In particular, the quantity on the right hand side of (2.26) is referred to as the potential vorticity in the literature, and \( p^0 \) is the corresponding steam function. In fact, this terminology is justified by observing that the potential vorticity is transported (see (2.27), below). After applying \( \textrm{curl}_h\,\) to (2.19), \( \partial _z\) to (2.21), and summing up the resulting equations, one arrives at Ertel’s conservation (transport) of the potential vorticity, i.e.,

$$\begin{aligned} \partial _t\Delta p^0 + v^0 \cdot \nabla _h\Delta p^0 = 0, \end{aligned}$$
(2.27)

where we have applied the fact, thanks to (2.16), (2.17), (2.18), and (2.22), that

$$\begin{aligned} \partial _zv^0 \cdot \nabla _h\theta ^0 = 0, \qquad w^0 = 0, \qquad \text {and} \qquad \textrm{div}_h\,v^0 = 0. \end{aligned}$$
(2.28)

In addition, thanks to (2.17), (2.21), and (2.25), one can show that

$$\begin{aligned} \partial _t(\partial _zp^0\vert _{z=0,h}) + v^0 \vert _{z=0,h} \cdot \nabla _h(\partial _zp^0\vert _{z=0,h}) = 0. \end{aligned}$$
(2.29)

The system formed by (2.16), (2.17), (2.27), and (2.29) is the well-known potential vorticity formulation of the classical quasi-geostrophic approximation. In particular, (2.29) describes the evolution of ‘boundary conditions’ for the stream function \( p^0 \), i.e., \( \partial _zp^0\vert _{z=0,h} \), which is used to invert the Laplacian in \( v^0 = \nabla _h^\perp p^0 = \nabla _h^\perp \Delta _N^{-1} (\Delta p^0) \), where \( \Delta _N^{-1} \) here is the inverse Laplacian with Neumann type boundary condition at \( z = 0,h \) and periodic boundary condition horizontally. Observe from (2.29) that if \( \partial _zp^0\vert _{z=0,h} = 0 \) initially, it remains zero. This is one of the underlying observation behind the well-prepared initial data in [10]. In addition, observe that \( \Delta _N^{-1} \) is unique up to a constant, which, without loss of generality, can be taken to be zero, justifying the notation of inverse.

2.3 The slow–fast waves structure: linear analysis

Our goal in this section is to investigate the linear slow-fast waves structure of system (1.2). This will guide us to obtain uniform-in-\(\varepsilon \) estimates as well as nonlinear waves interaction analysis in the next sections. Without loss of generality, we write \( (v_l, w_l, \theta _l) \) and \( p_l \), i.e., the linear variables, and the linear system associated with system (1.2) as follows:

$$\begin{aligned}&\partial _tv_l + \dfrac{1}{\varepsilon } v_l^\perp + \dfrac{\nabla _hp_l}{\varepsilon }= 0, \end{aligned}$$
(2.30a)
$$\begin{aligned}&\partial _tw_l + \dfrac{\partial _zp_l}{\varepsilon }- \dfrac{\theta _l}{\varepsilon }= 0, \end{aligned}$$
(2.30b)
$$\begin{aligned}&\partial _t\theta _l + \dfrac{w_l}{\varepsilon }= 0, \end{aligned}$$
(2.30c)
$$\begin{aligned}&\textrm{div}_h\,v_l + \partial _zw_l = 0, \end{aligned}$$
(2.30d)

with

$$\begin{aligned} w_l\vert _{z=0,h} = 0 \qquad \text {i.e., impermeable boundary condition}, \end{aligned}$$
(2.30e)

and periodic boundary condition horizontally.

The linear version of Ertel’s conservation (transport) of the potential vorticity \((\partial _z\theta _l + \textrm{curl}_h\,v_l)\) and the corresponding stream function \( p_l \) read, thanks to (2.30a), (2.30d), and (2.30e),

$$\begin{aligned} \Delta _h p_l + \partial _{zz} p_l = \partial _z\theta _l + \textrm{curl}_h\,v_l, \quad \partial _t(\Delta _h p_l + \partial _{zz} p_l) = \partial _t(\partial _z\theta _l + \textrm{curl}_h\,v_l) = 0.\nonumber \\ \end{aligned}$$
(2.31a)

Meanwhile, taking the trace of (2.30c) to the channel boundary yields

$$\begin{aligned} \partial _t\theta _l\vert _{z=0,h} = 0. \end{aligned}$$
(2.31b)

On the other hand, one can verify that

$$\begin{aligned} \partial _t(\nabla _h^\perp \theta _l + \nabla _hw_l - \partial _zv_l) + \dfrac{1}{\varepsilon }(\nabla _h^\perp \theta _l + \nabla _hw_l - \partial _zv_l)^\perp = 0. \end{aligned}$$
(2.31c)

Last but not least, integrating (2.30a) in the horizontal variables yields

$$\begin{aligned} \partial _t\int _{\mathbb T^2} v_l(x,y,z)\,dxdy + \dfrac{1}{\varepsilon }\biggl (\int _{\mathbb T^2} v_l(x,y,z)\,dxdy\biggr )^\perp = 0. \end{aligned}$$
(2.31d)

Moreover, observe that (2.30b) and (2.30c) imply

$$\begin{aligned} \partial _t(\partial _zp_l\vert _{z=0,h}) = 0. \end{aligned}$$
(2.32)

Equations (2.31a) and (2.31c) form the linear full generalized potential vorticity equations. A few remarks about this linear structure are in order:

  • While system (2.30) is stable with respect to the \( L^2 \) norm, i.e., one can get uniform-in-\(\varepsilon \) \( L^2 \) estimate by taking the \( L^2 \)-inner product of (2.30a), (2.30b), and (2.30c) with respect to \( v_l \), \( w_l \), and \( \theta _l \), the same can not be said about the \( H^s \) estimate for \( s \ge 1 \). This is due to the absence of boundary condition for the higher order derivatives of \( p_l \) and \( w_l \). For this reason, only in the case of periodic spatial domains (e.g., [16]), or in the case with well-prepared initial data and \( \partial _z p_l\vert _{z=0,h} = 0 \) (e.g., [10]; see (2.32)), one can verify the uniform \(H^s\) estimates and the asymptotic limit as \( \varepsilon \rightarrow 0^+ \);

  • On the other hand, (2.31a), (2.31c), and (2.31d) completely eliminate \( p_l \), and in particular, the underlying quantities in this system are stable with respect to any spatial derivatives. Therefore, one can get uniform-in-\(\varepsilon \) \( H^s \) estimates without any restriction for these quantities;

  • To be more precise, the estimates of the horizontal derivatives can be achived from (2.30). Then from (2.31a), (2.31c), and (2.30d), one can derive the estimates of \( \partial _z\theta _l \), \( \partial _zv_l \), and \( \partial _zw_l \), respectively, in terms of the horizontal derivatives. Bootstrap arguments will lead to \( H^s \) estimates;

  • One can regard (2.31a) and (2.31b) as the equations of the slow waves (dynamics), and (2.31c) and (2.31d) as the equations of the fast waves (dynamics). That is, one is able to separate the slow and fast state variables;

  • From (2.30c) and (2.31c), one can conclude that as \( \varepsilon \rightarrow 0 \), \( w_l, \nabla _h^\perp \theta _l - \partial _zv_l \rightharpoonup 0 \), weakly in the sense of distribution. This is consistent with (2.16), (2.17,) and (2.18).

Now we shall write down the slow-fast waves of linear system (2.30). Denote by

$$\begin{aligned} \Phi _l(x,y,z,t)&:= \partial _z\theta _l + \textrm{curl}_h\,v_l \qquad (\text {the potential vorticity}), \end{aligned}$$
(2.33)
$$\begin{aligned} \Psi _l(x,y,z,t)&:= \nabla _h^\perp \theta _l + \nabla _hw_l - \partial _zv_l, \end{aligned}$$
(2.34)
$$\begin{aligned} H_{l,0}(x,y,t)&:= \theta _l\vert _{z=0}, \end{aligned}$$
(2.35)
$$\begin{aligned} H_{l,h}(x,y,t)&:= \theta _l\vert _{z=h}, \end{aligned}$$
(2.36)

and

$$\begin{aligned} Z_l(z,t)&:= \int _{\mathbb T^2} v_l(x,y,z) \,dxdy. \end{aligned}$$
(2.37)

Correspondingly, let \( \Phi _\textrm{in} \), \(\Psi _\textrm{in}\), \(H_{0,\textrm{in}}\), \(H_{h,\textrm{in}}\), and \(Z_\textrm{in}\) be the initial data at \( t = 0 \) for \( \Phi _l \), \( \Psi _l \), \( H_{l,0} \), \( H_{l,h} \), and \( Z_l \), respectively. In particular, \( \Phi _l \) and \( \Psi _l \) form the generalized potential vorticity, and are the main ingredient of, and to be explored later in, this work. Then it follows from system (2.31), that

$$\begin{aligned} \begin{aligned} \text {linear slow variables:}&\quad \Phi _l(t) \equiv \Phi _\textrm{in}, \quad H_{l,0}(t) \equiv H_{0,\textrm{in}}, \quad H_{l,h}(t) \equiv H_{h,\textrm{in}}, \\ \text {linear fast variables:}&\quad \Psi _l(t) = e^{it/\varepsilon } \dfrac{\Psi _\textrm{in} + i \Psi _\textrm{in}^\perp }{2} + e^{-it/\varepsilon } \dfrac{\Psi _\textrm{in} - i \Psi _\textrm{in}^\perp }{2},\\ \text {and}&\quad Z_l(t) = e^{it/\varepsilon } \dfrac{Z_\textrm{in} + i Z_\textrm{in}^\perp }{2} + e^{-it/\varepsilon } \dfrac{Z_\textrm{in} - i Z_\textrm{in}^\perp }{2}. \end{aligned} \end{aligned}$$
(2.38)

We claim that \( (\Phi _l, \Psi _l, H_{l,0}, H_{l,h}, Z_l) \) as in (2.38) provide complete information on the solutions of system (2.30). This can be seen by writing \( (v_l, w_l, \theta _l) \) in terms of \( (\Phi _l, \Psi _l, H_{l,0}, H_{l,h}, Z_l) \). First, taking \( \textrm{div}_h\,\) and \( \textrm{curl}_h\,\) to (2.34) yields that, respectively, thanks to (2.30d) and (2.33),

$$\begin{aligned} \Delta _hw_l + \partial _{zz} w_l = \textrm{div}_h\,\Psi _l \end{aligned}$$
(2.39)

and

$$\begin{aligned} \Delta _h\theta _l + \partial _{zz} \theta _l = \partial _{z} \Phi _l + \textrm{curl}_h\,\Psi _l \qquad \text {or, equivalently} \nonumber \\ \Delta ( \theta _l - \mathrm E_{b}(H_{l,0},H_{l,h})) = \textrm{curl}_h\,\Psi _l + \partial _z\Phi _l - \Delta \mathrm E_{b}(H_{l,0},H_{l,h}). \end{aligned}$$
(2.40)

Note that, thanks to (2.10), (2.30e), (2.35), and (2.36),

$$\begin{aligned} w_l\vert _{z=0,h} = 0 \qquad \text {and} \qquad (\theta _l - \mathrm E_{b}(H_{l,0},H_{l,h}))\vert _{z=0,h} = 0. \end{aligned}$$

Therefore, let \( \Delta ^{-1}_D \) be the three-dimensional inverse Laplacian with Dirichlet boundary condition on \( \lbrace z = 0,h \rbrace \) and periodic boundary condition in the horizontal directions. From (2.39) and (2.40), one has

$$\begin{aligned} w_l&= \Delta _D^{-1} \textrm{div}_h\,\Psi _l \end{aligned}$$
(2.41)

and

$$\begin{aligned} \theta _l&= \mathrm E_{b}(H_{l,0},H_{l,h}) + \Delta _D^{-1}(\textrm{curl}_h\,\Psi _l + \partial _z\Phi _l - \Delta \mathrm E_{b}(H_{l,0},H_{l,h})). \end{aligned}$$
(2.42)

To calculate \( v_l \), let \( \Delta _h^{-1} \) be the two-dimensional inverse Laplace with zero horizontal mean value. Then, thanks to (2.30d) and (2.33), one has

$$\begin{aligned} \textrm{div}_h\,v_l = - \partial _zw_l \qquad \text {and} \qquad \textrm{curl}_h\,v_l = \Phi _l - \partial _z\theta _l, \end{aligned}$$
(2.43)

and, therefore, it follows that

$$\begin{aligned} v_l = Z_l + \nabla _h\Delta _h^{-1} \textrm{div}_h\,v_l + \nabla _h^\perp \Delta _h^{-1}\textrm{curl}_h\,v_l, \end{aligned}$$

or, after substituting (2.43), (2.41), and (2.42) in the above expression, one has

$$\begin{aligned} \begin{aligned} v_l&= Z_l - \nabla _h\Delta _h^{-1} \partial _z(\Delta _D^{-1}\textrm{div}_h\,\Psi _l)\\&\quad + \nabla _h^\perp \Delta _h^{-1} [\Phi _l - \partial _z\mathrm E_{b}(H_{l,0},H_{l,h}) \\&\quad - \partial _z\Delta _D^{-1}(\textrm{curl}_h\,\Psi _l + \partial _z\Phi _l - \Delta \mathrm E_{b}(H_{l,0},H_{l,h}))]. \end{aligned} \end{aligned}$$
(2.44)

We remind the reader that \( (\Phi _l, \Psi _l, H_{l,0},H_{l,h}, Z_l) \) are as in (2.38), with \( (\Psi _l, Z_l) \) being fast state variables and \( (\Phi _l, H_{l,0},H_{l,h}) \) slow state variables. Therefore, one can decompose \( v_l,w_l, \theta _l \) in terms of slow and fast waves in an unambiguous fashion.

3 Uniform-in-\( \varepsilon \) Estimates of the Euler Equations with Fast Rossby and Gravity Waves

In this and the following sections, we will proceed to the nonlinear analysis. In particular, we focus in this section on the uniform-in-\(\varepsilon \) estimates for system (1.2) in this section. Inspired by the discussion in Sect. 2.3, recall that we have defined \( \Phi , \Psi , H_0, H_h, Z \) as in (1.4)–(1.8), i.e.,

$$\begin{aligned} \Phi (x,y,z,t)&: = \partial _z\theta + \textrm{curl}_h\,v, \end{aligned}$$
(1.4')
$$\begin{aligned} \Psi (x,y,z,t)&: = \nabla _h^\perp \theta + \nabla _hw - \partial _zv, \end{aligned}$$
(1.5')
$$\begin{aligned} H_0(x,y,t)&:= \theta \vert _{z=0}, \end{aligned}$$
(1.6')
$$\begin{aligned} H_h(x,y,t)&:= \theta \vert _{z=h}, \end{aligned}$$
(1.7')

and

$$\begin{aligned} Z(z,t)&:= \int _{\mathbb {T}^2} v(x,y,z,t) \,dxdy. \end{aligned}$$
(1.8')

Recall that \( \Phi \) and \( \Psi \) form the generalized potential vorticity. From (1.2a), (1.2b), (1.2c), and (1.2d), one can write down the following equations

$$\begin{aligned}&\partial _t\textrm{curl}_h\,v + v \cdot \nabla _h\textrm{curl}_h\,v + w \partial _z\textrm{curl}_h\,v \nonumber \\&\quad + \textrm{curl}_h\,v \cdot \textrm{div}_h\,v + \partial _zv \cdot \nabla _h^\perp w - \dfrac{\partial _zw}{\varepsilon } = 0, \end{aligned}$$
(3.1)
$$\begin{aligned}&\partial _t\partial _zv + v \cdot \partial _zv + w \partial _z\partial _zv + \dfrac{\partial _zv^\perp }{\varepsilon } + \dfrac{\nabla _h\partial _zp}{\varepsilon } \nonumber \\&\quad + \partial _zv \cdot \nabla _hv + \partial _zw \partial _zv = 0, \end{aligned}$$
(3.2)
$$\begin{aligned}&\partial _t\nabla _hw + v \cdot \nabla _h\nabla _hw + w \partial _z\nabla _hw + \dfrac{\nabla _h\partial _zp}{\varepsilon } - \dfrac{\nabla _h\theta }{\varepsilon } \nonumber \\&\quad + (\nabla _hv)^\top \nabla _hw + \partial _zw \nabla _hw = 0, \end{aligned}$$
(3.3)
$$\begin{aligned}&\partial _t\nabla _h\theta + v \cdot \nabla _h\nabla _h\theta + w \partial _z\nabla _h\theta + \dfrac{\nabla _hw}{\varepsilon } \nonumber \\&\quad + (\nabla _hv)^\top \nabla _h\theta + \partial _z\theta \nabla _hw = 0, \end{aligned}$$
(3.4)
$$\begin{aligned}&\partial _t\partial _z\theta + v \cdot \nabla _h\partial _z\theta + w \partial _z\partial _z\theta + \dfrac{\partial _zw}{\varepsilon } \nonumber \\&\quad + \partial _zv \cdot \nabla _h\theta + \partial _zw \partial _z\theta = 0. \end{aligned}$$
(3.5)

Consequently, one has, from system (1.2), that

$$\begin{aligned}&\partial _t\Phi + v \cdot \nabla _h\Phi + w \partial _z\Phi + N_1 = 0, \end{aligned}$$
(3.6a)
$$\begin{aligned}&\partial _t\Psi + v \cdot \nabla _h\Psi + w \partial _z\Psi + \dfrac{1}{\varepsilon } \Psi ^\perp + N_2 = 0, \end{aligned}$$
(3.6b)
$$\begin{aligned}&\partial _tH_0 + v\vert _{z=0} \cdot \nabla _hH_0 = 0, \end{aligned}$$
(3.6c)
$$\begin{aligned}&\partial _tH_h + v\vert _{z=h} \cdot \nabla _hH_h = 0, \end{aligned}$$
(3.6d)
$$\begin{aligned}&\partial _tZ + \dfrac{1}{\varepsilon } Z^\perp + N_3 = 0, \end{aligned}$$
(3.6e)

where

$$\begin{aligned} N_1&:= \textrm{curl}_h\,v \cdot \textrm{div}_h\,v + \partial _zv \cdot \nabla _h^\perp w + \partial _zv \cdot \nabla _h\theta + \partial _zw \partial _z\theta , \end{aligned}$$
(3.6f)
$$\begin{aligned} N_2&:= ((\nabla _hv)^\top \nabla _h\theta )^\perp + \partial _z\theta \cdot \nabla _h^\perp w + (\nabla _hv)^\top \nabla _hw + \partial _zw \nabla _hw \nonumber \\&\quad - \partial _zv \cdot \nabla _hv - \partial _zw \partial _zv, \end{aligned}$$
(3.6g)
$$\begin{aligned} N_3&:= \int _{\mathbb T^2} \partial _z(w v) \,dxdy. \end{aligned}$$
(3.6h)

We continue with the uniform-in-\(\varepsilon \) estimates in the following steps: 1. establish estimates for the horizontal derivatives; then 2. establish estimates for the vertical derivatives; finally, 3. close the estimates.

3.1 Estimates for the horizontal derivatives

Let \( \partial _h\in \lbrace \partial _x, \partial _y \rbrace \) and \( \alpha \in \lbrace 0,1,2,3 \rbrace \). Applying \( \partial _h^\alpha \) to system (1.2) leads to

$$\begin{aligned}&\partial _t\partial _h^\alpha v + (v \cdot \nabla _h+ w \partial _z) \partial _h^\alpha v + \dfrac{1}{\varepsilon } \partial _h^\alpha v^\perp + \dfrac{\nabla _h\partial _h^\alpha p}{\varepsilon } \nonumber \\&\quad + \partial _h^\alpha (v\cdot \nabla _hv + w \partial _zv) - (v\cdot \nabla _h+ w \partial _z) \partial _h^\alpha v = 0, \end{aligned}$$
(3.7)
$$\begin{aligned}&\partial _t\partial _h^\alpha w + (v \cdot \nabla _h+ w \partial _z) \partial _h^\alpha w + \dfrac{\partial _z\partial _h^\alpha p}{\varepsilon } - \dfrac{\partial _h^\alpha \theta }{\varepsilon } \nonumber \\&\quad + \partial _h^\alpha (v\cdot \nabla _hw + w \partial _zw) - (v \cdot \nabla _h+ w \partial _z) \partial _h^\alpha w = 0, \end{aligned}$$
(3.8)
$$\begin{aligned}&\partial _t\partial _h^\alpha \theta + (v \cdot \nabla _h+ w \partial _z) \partial _h^\alpha \theta + \dfrac{\partial _h^\alpha w}{\varepsilon } \nonumber \\&\quad + \partial _h^\alpha (v\cdot \nabla _h\theta + w \partial _z\theta ) - (v \cdot \nabla _h+ w \partial _z) \partial _h^\alpha \theta = 0, \end{aligned}$$
(3.9)
$$\begin{aligned}&\textrm{div}_h\,\partial _h^\alpha v + \partial _z\partial _h^\alpha w = 0, \qquad \partial _hw\vert _{z=0,h} = 0. \end{aligned}$$
(3.10)

Taking the \( L^2 \)-inner product of (3.7)–(3.9) with \( 2\partial _h^\alpha v, 2\partial _h^\alpha w, 2\partial _h^\alpha \theta \), respectively, applying integration by parts, and summing up the resultants lead to

(3.11)

for some generic constant \( C \in (0,\infty ) \), where in the last inequality we have applied the Hölder inequality, the Gagliardo–Nirenberg inequality, and the Sobolev embedding inequality.

3.2 Estimates for the vertical derivatives

As before, let \( \partial \in \lbrace \partial _x, \partial _y, \partial _z \rbrace \) and \( \beta \in \lbrace 0,1,2 \rbrace \). Applying \( \partial ^\beta \) to equations (3.6a) and (3.6b) leads to

$$\begin{aligned}&\partial _t\partial ^\beta \Phi + (v \cdot \nabla _h+ w \partial _z) \partial ^\beta \Phi + \partial ^\beta N_1\nonumber \\&\quad + \partial ^\beta (v \cdot \nabla _h\Phi + w \partial _z\Phi ) - (v \cdot \nabla _h+ w \partial _z) \partial ^\beta \Phi = 0, \end{aligned}$$
(3.12)
$$\begin{aligned}&\partial _t\partial ^\beta \Psi + (v\cdot \nabla _h+ w \partial _z) \partial ^\beta \Psi + \dfrac{1}{\varepsilon } \partial ^\beta \Psi ^\perp + \partial ^\beta N_2 \nonumber \\&\quad + \partial ^\beta (v \cdot \nabla _h\Psi + w \partial _z\Psi ) - (v \cdot \nabla _h+ w \partial _z)\partial ^\beta \Psi = 0. \end{aligned}$$
(3.13)

Taking the \( L^2 \)-inner product of (3.12) and (3.13) with \( 2\partial ^\beta \Phi \) and \( 2\partial ^\beta \Psi \), respectively, applying integration by parts, and summing up the resultants lead to

(3.14)

for some absolute constant \( C \in (0,\infty ) \), where in the last inequality we have applied the Hölder inequality, the Gagliardo–Nirenberg inequality, and the Sobolev embedding inequality.

3.3 Closing the estimates

Define the total “energy” functional by

(3.15)

We observe that

(3.16)

for some generic constant \( C \in (0,\infty ) \). Indeed, the right-hand side inequality in (3.16) follows directly from the definition of \( \Phi \) and \( \Psi \) in (1.4) and (1.5). To show the left-hand side inequality, notice that

$$\begin{aligned}&\partial _z v = - \Psi + \nabla _h^\perp \theta + \nabla _hw, \qquad \partial _z\theta = \Phi - \textrm{curl}_h\,v, \end{aligned}$$

and

$$\begin{aligned} \partial _zw = - \textrm{div}_h\,v. \end{aligned}$$

Thus,

Similarly, following a bootstrap argument on the derivatives implies the left-hand side part of (3.16).

Consequently, (3.11) and (3.14) yield

$$\begin{aligned} \dfrac{d}{dt} \mathfrak E \le C \mathfrak E^{3/2}, \end{aligned}$$
(3.17)

for some generic constant \( C \in (0,\infty ) \). In particular, from (3.17) and (3.16), one concludes that there exists \( T \in (0,\infty ) \), depending only on the initial data and independent of \( \varepsilon \), such that

(3.18)

for the same constant C as in (3.16). This finishes the proof of Theorem 1.1.

4 Convergence Theory

4.1 Convergence theory: part 1, compactness

What is left is to establish the convergence of the solutions to system (1.2) as \( \varepsilon \rightarrow 0^+ \), which we will do in two steps. In this subsection, we will conclude the weak and strong compactness, thanks to the uniform estimate (3.18). In the next subsection, we will deal with the convergence of the nonlinearities.

In the rest of this paper, we denote by \( T \in (0,\infty ) \) the uniform-in-\(\varepsilon \) existence time established in Sect. 3 at (3.18). \( C_\textrm{in}\in (0,\infty ) \) will denote a constant that is independent of \( \varepsilon \), different from line to line, depending only on the initial data. With such notations, thanks to (3.18), by virtue of the definitions of \( \Phi \), \( \Psi \), \( H_0 \), \( H_h \), and Z in (1.4)–(1.8), respectively, we have

(4.1)

Similarly, from (3.6f)–(3.6h), it follows that

(4.2)

From (3.6a)–(3.6e), one has, thanks to (3.18), (4.1), and (4.2), that

(4.3)

Consequently, by virtue of the Aubin compactness theorem [50, Theorem 2.1], there exist

$$\begin{aligned} \begin{aligned} \Phi _p, \Psi _p \in L^\infty (0,T; H^2(\Omega )), \qquad H_{p,0}, H_{p,h} \in L^\infty (0,T; H^{5/2}(\mathbb T^2)), \\ \text {and} \qquad Z_p, v_p, w_p, \theta _p \in L^\infty (0,T;H^3(\Omega )), \end{aligned} \end{aligned}$$
(4.4)

with

$$\begin{aligned} \partial _t\Phi _p \in L^\infty (0,T;H^1(\Omega ), \qquad \partial _tH_{p,0}, \partial _tH_{p,h} \in L^\infty (0,T;H^{3/2}(\mathbb T^2), \end{aligned}$$
(4.5)

such that there exists a subsequence of \( \varepsilon \) that as \( \varepsilon \rightarrow 0^+ \),

$$\begin{aligned} \Phi , \Psi&{\mathop {\rightharpoonup }\limits ^{*}} \Phi _p, \Psi _p&\text {weak-}*\text { in}&\qquad L^\infty (0,T;H^2(\Omega )), \end{aligned}$$
(4.6)
$$\begin{aligned} H_0, H_h&{\mathop {\rightharpoonup }\limits ^{*}} H_{p,0}, H_{p,h}&\text {weak-}*\text { in}&\qquad L^\infty (0,T;H^{5/2}(\mathbb T^2)), \end{aligned}$$
(4.7)
$$\begin{aligned} Z, v, w, \theta&{\mathop {\rightharpoonup }\limits ^{*}} Z_p, v_p, w_p, \theta _p&\text {weak-}*\text { in}&\qquad L^\infty (0,T;H^3(\Omega )), \end{aligned}$$
(4.8)
$$\begin{aligned} \partial _t\Phi&{\mathop {\rightharpoonup }\limits ^{*}} \partial _t\Phi _p&\text {weak-}*\text { in}&\qquad L^\infty (0,T;H^1(\Omega )) \end{aligned}$$
(4.9)
$$\begin{aligned} \partial _tH_0, \partial _tH_h&{\mathop {\rightharpoonup }\limits ^{*}} \partial _tH_{p,0}, \partial _tH_{p,h}&\text {weak-}*\text { in}&\qquad L^\infty (0,T;H^{3/2}(\mathbb T^2)) \end{aligned}$$
(4.10)

and

$$\begin{aligned} \Phi&\rightarrow \Phi _p&\text {in}&\qquad C([0,T];H^1(\Omega )), \end{aligned}$$
(4.11)
$$\begin{aligned} H_0, H_h&\rightarrow H_{p,0}, H_{p,h}&\text {in}&\qquad C([0,T];H^{3/2}(\mathbb T^2)) \end{aligned}$$
(4.12)

Furthermore, from (1.2c), (3.6b) and (3.6e), after sending \( \varepsilon \rightarrow 0^+ \), one can verify that \( w_p = \Psi _p = Z_p \equiv 0 \). In fact, after taking the inner product of corresponding equations with \( \varepsilon \) and a test function in \( \mathcal D'((0,T)\times \Omega )\) and passing the limit \( \varepsilon \rightarrow 0^+ \), it is easy to verify that \( w_p = \Psi _p = Z_p \equiv 0 \) in the sense of distribution. Then it follows from the regularity of \( w_p, \Psi _p \), and \( Z_p \) that they are equal to zero. Following similar arguments from the definition, it is easy to show that,

$$\begin{aligned} \begin{aligned} w_p = 0, \quad \Phi _p = \partial _z\theta _p + \textrm{curl}_h\,v_p, \quad \nabla _h^\perp \theta _p + \nabla _hw_p - \partial _zv_p = 0, \\ \textrm{div}_h\,v_p + \partial _zw_p = 0, \quad H_{p,0} = \theta _p \vert _{z=0}, \quad H_{p,h} = \theta _p\vert _{z=h}, \\ \quad \text {and} \quad \int _{\mathbb {T}^2} v_p(x,y,z) \,dxdy = 0, \end{aligned} \end{aligned}$$
(4.13)

or, equivalently, repeating similar calculation as in (2.39)–(2.44), one has

$$\begin{aligned} \begin{aligned}&w_p = 0,{} & {} \qquad \theta _p = \mathrm E_b(H_{p,0}, H_{p,h}) + \Delta _D^{-1} ( \partial _z\Phi _p - \Delta \mathrm E_b(H_{p,0}, H_{p,h})), \\&\text {and}{} & {} \qquad v_p = \nabla _h^\perp \Delta _h^{-1} [\Phi _p - \partial _z\mathrm E_b(H_{p,0}, H_{p,h})\\{} & {} {}&\qquad \qquad \qquad - \partial _z\Delta _D^{-1}(\partial _z\Phi _p - \Delta \mathrm E_b(H_{p,0}, H_{p,h}))]. \end{aligned} \end{aligned}$$
(4.13')

Remark 2

We can perform the following calculation to rewrite \( \theta _p \). Let \( P:= \Delta _h^{-1} [\Phi _p - \partial _z\mathrm E_b(H_{p,0}, H_{p,h}) - \partial _z\Delta _D^{-1}(\partial _z\Phi _p - \Delta \mathrm E_b(H_{p,0}, H_{p,h}))] \). Then direct calculation shows that

$$\begin{aligned} \partial _zP&= \Delta _h^{-1} [\partial _z\Phi _p - \partial _{zz} \mathrm E_b(H_{p,0}, H_{p,h})\\&\quad - (\Delta - \Delta _h) \Delta _D^{-1}(\partial _z\Phi _p - \Delta \mathrm E_b(H_{p,0}, H_{p,h}))] \\&= \underbrace{\mathrm E_b(H_{p,0}, H_{p,h}) + \Delta _D^{-1} ( \partial _z\Phi _p - \Delta \mathrm E_b(H_{p,0}, H_{p,h}))}_{=\theta _p}\\&\quad - \underbrace{\int _{\mathbb T^2}\bigl [\mathrm E_b(H_{p,0}, H_{p,h}) + \Delta _D^{-1} ( \partial _z\Phi _p - \Delta \mathrm E_b(H_{p,0}, H_{p,h})) \bigr ]\,dxdy}_{=:Q(z)}, \end{aligned}$$

where we have applied (2.5) and (2.7). Together with (4.13’), we have

$$\begin{aligned} \theta _p = \partial _z(P + \int _0^z Q(z')\,dz') \qquad \text {and} \qquad v_p = \nabla _h^\perp (P + \int _0^z Q(z')\,dz'). \end{aligned}$$

This is consistent with the classical theory of the quasi-geostrophic approximation. See, for instance, [10, 16].

Next, to handle the fast waves, i.e., \( \Psi \) and Z, following Schochet’s theory [48], from (3.6b) and (3.6e), one has

$$\begin{aligned}&\partial _t[e^{\mp i\frac{t}{\varepsilon }}(\Psi \pm i \Psi ^\perp )] = - v\cdot \nabla _h[e^{\mp i\frac{t}{\varepsilon }}(\Psi \pm i \Psi ^\perp )] \nonumber \\&\quad - w \partial _z[e^{\mp i\frac{t}{\varepsilon }}(\Psi \pm i \Psi ^\perp )] - e^{\mp i\frac{t}{\varepsilon }}(N_2 \pm i N_2^\perp ), \end{aligned}$$
(4.14)
$$\begin{aligned}&\text {and} \qquad \partial _t[e^{\mp i\frac{t}{\varepsilon }}(Z \pm i Z^\perp )] = - e^{\mp i\frac{t}{\varepsilon }}(N_3 \pm i N_3^\perp ). \end{aligned}$$
(4.15)

From (4.14) and (4.15), thanks to (3.18), (4.1), and (4.2), it follows that

(4.16)

Therefore, by the Aubin compactness theorem [50, Theorem 2.1], there exist

$$\begin{aligned} \begin{aligned} \psi _{p,\pm } \in L^\infty (0,T;H^2(\Omega )), \qquad \qquad z_{p,\pm } \in L^\infty (0,T;H^3(\Omega )),\\ \partial _t\psi _{p,\pm } \in L^\infty (0,T;H^1(\Omega )), \qquad \text {and}\qquad \partial _tz_{p,\pm } \in L^\infty (0,T;H^2(\Omega )), \end{aligned} \end{aligned}$$
(4.17)

such that there exists a subsequence of \( \varepsilon \) that as \( \varepsilon \rightarrow 0^+ \),

$$\begin{aligned} \Psi _{\pm }&{\mathop {\rightharpoonup }\limits ^{*}} \psi _{p,\pm }&\text {weak-}*\text {in}&\qquad L^\infty (0,T;H^2(\Omega )), \end{aligned}$$
(4.18)
$$\begin{aligned} Z_{\pm }&{\mathop {\rightharpoonup }\limits ^{*}} z_{p,\pm }&\text {weak-}*\text { in}&\qquad L^\infty (0,T;H^3(\Omega )), \end{aligned}$$
(4.19)
$$\begin{aligned} \partial _t\Psi _{\pm }&{\mathop {\rightharpoonup }\limits ^{*}} \partial _t\psi _{p,\pm }&\text {weak-}*\text { in}&\qquad L^\infty (0,T;H^1(\Omega )), \end{aligned}$$
(4.20)
$$\begin{aligned}&\quad \partial _tZ_{\pm }&{\mathop {\rightharpoonup }\limits ^{*}} \partial _tz_{p,\pm }&\text {weak-}*\text {in}&\qquad L^\infty (0,T;H^{2}(\Omega )), \end{aligned}$$
(4.21)

and

$$\begin{aligned} \Psi _{\pm }&\rightarrow \psi _{p,\pm }&\text {in}&\qquad C([0,T];H^1(\Omega )), \end{aligned}$$
(4.22)
$$\begin{aligned} Z_{\pm }&\rightarrow z_{p,\pm }&\text {in}&\qquad C([0,T];H^2(\Omega )), \end{aligned}$$
(4.23)

where

$$\begin{aligned} \Psi _{\pm }:= e^{\mp i\frac{t}{\varepsilon }}(\Psi (t) \pm i \Psi ^\perp (t)) \quad \text {and}\quad Z_{\pm }:= e^{\mp i\frac{t}{\varepsilon }}(Z(t) \pm i Z^\perp (t)). \end{aligned}$$
(4.24)

In particular, directly one can verify that

$$\begin{aligned} \begin{aligned} 2 \Psi - (e^{i\frac{t}{\varepsilon }}\psi _{p,+} + e^{-i\frac{t}{\varepsilon }}\psi _{p,-}) = e^{i\frac{t}{\varepsilon }} (\Psi _+ - \psi _{p,+}) + e^{-i\frac{t}{\varepsilon }}(\Psi _- - \psi _{p,-}) \\ \rightarrow 0 \qquad \text {in}\qquad L^\infty (0,T;H^1(\Omega )), \quad \text {as} ~ \varepsilon \rightarrow 0^+, \end{aligned} \end{aligned}$$
(4.25)

and

$$\begin{aligned} \begin{aligned} 2 Z - (e^{i\frac{t}{\varepsilon }} z_{p,+} + e^{-i\frac{t}{\varepsilon }}z_{p,-}) = e^{i\frac{t}{\varepsilon }} (Z_+ - z_{p,+}) + e^{-i\frac{t}{\varepsilon }}(Z_- - z_{p,-}) \\ \rightarrow 0 \qquad \text {in}\qquad L^\infty (0,T;H^2(\Omega )), \quad \text {as} ~ \varepsilon \rightarrow 0^+. \end{aligned} \end{aligned}$$
(4.26)

To conclude this section, we write the fast-slow-error decomposition of \( v,w,\theta \). Let

$$\begin{aligned} \begin{aligned} W_\pm := \dfrac{1}{2} \Delta _D^{-1}\textrm{div}_h\,\psi _{p,\pm }, \qquad \Theta _\pm := \dfrac{1}{2} \Delta _D^{-1}\textrm{curl}_h\,\psi _{p,\pm }, \qquad \text {and}\\ V_\pm := \dfrac{1}{2} \bigl (z_{p,\pm } - \nabla _h\Delta _h^{-1}\partial _z\Delta _D^{-1} \textrm{div}_h\,\psi _{p,\pm } - \nabla _h^\perp \Delta _h^{-1}\partial _z\Delta _D^{-1} \textrm{curl}_h\,\psi _{p,\pm } \bigr ). \end{aligned} \end{aligned}$$
(4.27)

Thanks to (4.54.17), one has that

$$\begin{aligned} \begin{aligned} W_\pm , \Theta _\pm , V_\pm \in L^\infty (0,T;H^3(\Omega )) \\ \text {and} \qquad \partial _tW_\pm , \partial _t\Theta _\pm , \partial _tV_\pm \in L^\infty (0,T;H^2(\Omega )). \end{aligned} \end{aligned}$$
(4.28)

Repeating the exact calculation as in (2.39)–(2.44) leads to

$$\begin{aligned} \begin{aligned} w&= \Delta _D^{-1} \textrm{div}_h\,\Psi = \underbrace{e^{i\frac{t}{\varepsilon }}W_+}_{=:w_{\textrm{fast},+}} + \underbrace{e^{-i\frac{t}{\varepsilon }}W_-}_{=: w_{\textrm{fast},-}} + w_\textrm{err}, \end{aligned} \end{aligned}$$
(4.29)
$$\begin{aligned} \begin{aligned} \theta&= \mathrm E_b(H_0,H_h) + \Delta _D^{-1}(\textrm{curl}_h\,\Psi + \partial _z\Phi - \Delta E_b(H_0,H_h)) \\&= \underbrace{\mathrm E_b(H_0,H_h) + \Delta _D^{-1}(\partial _z\Phi - \Delta E_b(H_0,H_h))}_{=:\theta _{\textrm{slow}}} + \underbrace{e^{i\frac{t}{\varepsilon }}\Theta _+}_{=:\theta _{\textrm{fast},+}} + \underbrace{e^{-i\frac{t}{\varepsilon }}\Theta _-}_{=:\theta _{\textrm{fast},-}} \\ {}&\quad + \theta _\textrm{err}, \end{aligned} \end{aligned}$$
(4.30)

and

$$\begin{aligned} \begin{aligned} v&= Z - \nabla _h\Delta _h^{-1} \partial _z(\Delta _D^{-1}\textrm{div}_h\,\Psi ) \\&\quad + \nabla _h^\perp \Delta _h^{-1} [\Phi - \partial _z\mathrm E_b(H_0,H_h) \\&\quad - \partial _z\Delta _D^{-1}(\textrm{curl}_h\,\Psi + \partial _z\Phi - \Delta \mathrm E_b(H_0,H_h))] \\&= \underbrace{\begin{array}{c} \nabla _h^\perp \Delta _h^{-1} [\Phi - \partial _z\mathrm E_b(H_0,H_h) \\ \qquad - \partial _z\Delta _D^{-1}(\partial _z\Phi - \Delta \mathrm E_b(H_0,H_h))] \end{array}}_{=:v_\textrm{slow}} + \underbrace{e^{i\frac{t}{\varepsilon }}V_+}_{=:v_{\textrm{fast},+}} + \underbrace{e^{-i\frac{t}{\varepsilon }}V_-}_{=:v_{\textrm{fast},-}} + v_\textrm{err}, \end{aligned} \end{aligned}$$
(4.31)

where, thanks to (4.1), (4.54.17), (4.25), and (4.26), the error terms satisfy

(4.32)

In addition, thanks to (2.9), (4.3), (4.6), (4.7), (4.11), (4.12) and (4.13’), we have

(4.33)

Moreover, there exists a subsequence of \( \varepsilon \) that as \( \varepsilon \rightarrow 0^+ \), we also have

$$\begin{aligned} \begin{aligned} v_\textrm{slow}, \theta _\textrm{slow} \qquad \rightarrow \qquad v_p, \theta _p \qquad \text {in} \qquad C(0,T;H^{2}(\Omega )), \\ \text {and} \qquad v_\textrm{slow}, \theta _\textrm{slow} \qquad {\mathop {\rightharpoonup }\limits ^{*}} \qquad v_p, \theta _p \qquad \text {weak-* in} \qquad L^\infty (0,T;H^3(\Omega )),\\ \text {as} ~ \varepsilon \rightarrow 0^+. \end{aligned}\nonumber \\ \end{aligned}$$
(4.34)

4.2 Convergence theory: part 2, convergence of the nonlinearities

In this section, we finish the convergence theory by investigating the convergence of the nonlinearities.

4.2.1 Convergence of the slow waves (3.6a), (3.6c), and (3.6d)

First, we investigate \( N_1 \), defined in (3.6f). Notice that \( N_1 \) is quadratic. substituting (4.29)–(4.31), we write

Then thanks to (3.18), (4.28), (4.32), and (4.34), we have, as \( \varepsilon \rightarrow 0^+ \),

$$\begin{aligned} N_{1,\textrm{slow}}&\rightarrow \textrm{curl}_h\,v_p \cdot \textrm{div}_h\,v_p + \partial _zv_p \cdot \nabla _h\theta _p = 0 \quad \text {in} ~ C([0,T];H^1(\Omega )), \end{aligned}$$
(4.35)
$$\begin{aligned} N_{1,\text {fast},1}, N_{1,\text {fast},2}&\rightharpoonup 0 \qquad \text {weakly in} \quad L^p(0,T;H^{1}(\Omega )) \quad \forall p\in (1,\infty ), \end{aligned}$$
(4.36)
$$\begin{aligned} N_{1,\textrm{err}}&\rightarrow 0 \qquad \qquad \quad ~ \text {in} \quad L^\infty (0,T;H^1(\Omega )), \end{aligned}$$
(4.37)

and

$$\begin{aligned} \begin{aligned} N_{1,\textrm{res}}&\rightarrow \textrm{curl}_h\,V_{\pm }\cdot \textrm{div}_h\,V_{\mp } + \partial _zV_{\pm } \cdot \nabla _h^\perp W_{\mp } + \partial _zV_{\pm }\cdot \nabla _h\Theta _{\mp } + \partial _zW_{\pm } \partial _z\Theta _{\mp }\\ \text {in}&\qquad L^\infty (0,T;H^2(\Omega )). \end{aligned} \end{aligned}$$
(4.38)

Consequently, as \( \varepsilon \rightarrow 0^+ \), in the sense of distribution, the limit of equation (3.6a) is

$$\begin{aligned} \begin{aligned} \partial _t\Phi _p + v_p \cdot \nabla _h\Phi _p + w_p \partial _z\Phi _p + \textrm{curl}_h\,V_{\pm }\cdot \textrm{div}_h\,V_{\mp } \\ + \partial _zV_{\pm } \cdot \nabla _h^\perp W_{\mp } + \partial _zV_{\pm }\cdot \nabla _h\Theta _{\mp } + \partial _zW_{\pm } \partial _z\Theta _{\mp } = 0. \end{aligned} \end{aligned}$$
(4.39)

Here we have omitted the convergence of the advection terms, which is left to the reader.

The limit equations of (3.6c) and (3.6d) follow similarly. The proof is left to the reader and we only state the result as follows:

$$\begin{aligned} \partial _tH_{p,0} + v_p\vert _{z=0} \cdot \nabla _hH_{p,0} = 0, \end{aligned}$$
(4.40)
$$\begin{aligned} \partial _tH_{p,h} + v_p\vert _{z=h} \cdot \nabla _hH_{p,h} = 0. \end{aligned}$$
(4.41)

We remind the reader that \( v_p, w_p, \theta _p \) (\(V_\pm , W_\pm , \Theta _\pm \), respectively) are determined by \( \Phi _p, H_{p,0}, H_{p,h} \) (\(\psi _{p,\pm },z_{p,\pm })\), respectively), as in (4.13’) ((4.27), respectively). Therefore, the equations for \( \Phi _p \), \( H_{p,0}\), and \( H_{p,h} \), i.e., (4.39), (4.40), and (4.41), can be considered as the equations of \( v_p, w_p, \theta _p \), with source terms given by the resonances involving \( V_\pm \), \( W_\pm \), and \( \Theta _\pm \) (equivalently \( \psi _{p,\pm } \) and \( z_{p,\pm } \)). To close the system, we will investigate the limit equations of (4.14) and (4.15) in the following.

4.2.2 Convergence of the fast waves (4.14) and (4.15)

Using the notation of (4.24), (4.14) and (4.15) can be written as

$$\begin{aligned}&\partial _t\Psi _\pm + v\cdot \nabla _h\Psi _\pm + w \partial _z\Psi _\pm + e^{\mp i\frac{t}{\varepsilon }}(N_2 \pm i N_2^\perp ) = 0 , \end{aligned}$$
(4.14')
$$\begin{aligned} \partial _tZ_\pm + e^{\mp i \frac{t}{\varepsilon }}(N_3 \pm i N_3^\perp )=0. \end{aligned}$$
(4.15')

Thanks to (4.8) and (4.94.18)–(4.23), we only need to investigate the limit of \( e^{\mp i \frac{t}{\varepsilon }}N_2 \) and \( e^{\mp i \frac{t}{\varepsilon }}N_3 \).

Repeating the same arguments as for \( N_1 \), above, one can show that

After substituting (4.29)–(4.31) and sending \( \varepsilon \rightarrow 0^+ \), it follows that

$$\begin{aligned} \begin{aligned}&e^{\mp i \frac{t}{\varepsilon }}N_{2} \rightharpoonup \bigl ( (\nabla _hV_\pm )^\top \nabla _h\theta _p + (\nabla _hv_p)^\top \nabla _h\Theta _\pm \bigr )^\perp \\&\qquad + \partial _z\theta _p \cdot \nabla _h^\perp W_\pm + (\nabla _hv_p)^\top \nabla _hW_\pm \\&\qquad - \partial _zV_\pm \cdot \nabla _hv_p - \partial _zv_p \cdot \nabla _hV_\pm - \partial _zW_\pm \partial _zv_p =: N_{\psi }\\&\qquad \text {in}\quad L^p(0,T;H^1(\Omega ))\quad \forall p \in (1,\infty ). \end{aligned} \end{aligned}$$
(4.42)

Therefore, the limit of (4.14) as \( \varepsilon \rightarrow 0^+ \) is

$$\begin{aligned} \partial _t\psi _{p,\pm } + v_p \cdot \nabla _h\psi _{p,\pm } + w_p \partial _z\psi _{p,\pm } + (N_\psi \pm i N_\psi ^\perp ) = 0. \end{aligned}$$
(4.43)

Last but not least, one has that

and thus

$$\begin{aligned} \begin{aligned} e^{\mp i \frac{t}{\varepsilon }} N_3 \rightharpoonup \int _{\mathbb {T}^2} \partial _z( W_\pm v_p) \,dxdy =: N_z \\ \qquad \text {in}\quad L^p(0,T;H^1(\Omega ))\quad \forall p \in (1,\infty ). \end{aligned} \end{aligned}$$
(4.44)

Consequently, as \( \varepsilon \rightarrow 0^+ \), the limit of (4.15) is

$$\begin{aligned} \partial _tz_{p,\pm } + (N_z \pm i N_z^\perp ) = 0. \end{aligned}$$
(4.45)

4.2.3 Conclusion

The limit system for the slow limit variables \( \Phi _p, H_p \) and the fast limit variables \( \psi _{p,\pm }, z_{p,\pm } \) is then, from (4.39), (4.40), (4.41), (4.43), and (4.45),

$$\begin{aligned}&\partial _t\Phi _p + v_p \cdot \nabla _h\Phi _p + N_\Phi = 0, \end{aligned}$$
(4.46a)
$$\begin{aligned}&\partial _tH_{p,0} + v_p\vert _{z=0} \cdot \nabla _hH_{p,0} = 0, \end{aligned}$$
(4.46b)
$$\begin{aligned}&\partial _tH_{p,h} + v_p\vert _{z=h} \cdot \nabla _hH_{p,h} = 0, \end{aligned}$$
(4.46c)
$$\begin{aligned}&\partial _t\psi _{p,\pm } + v_p \cdot \nabla _h\psi _{p,\pm } + (N_\psi \pm i N_\psi ^\perp ) = 0, \end{aligned}$$
(4.46d)
$$\begin{aligned}&\partial _tz_{p,\pm } + (N_z \pm i N_z^\perp ) = 0, \end{aligned}$$
(4.46e)

where

$$\begin{aligned} N_\Phi := \textrm{curl}_h\,V_{\pm }\cdot \textrm{div}_h\,V_{\mp } + \partial _zV_{\pm } \cdot \nabla _h^\perp W_{\mp } + \partial _zV_{\pm }\cdot \nabla _h\Theta _{\mp } + \partial _zW_{\pm } \partial _z\Theta _{\mp }, \end{aligned}$$
(4.46f)

and \( N_\psi \) and \(N_z\) are defined in (4.42) and (4.44), above, respectively. We remind the reader that \( v_p, \theta _p, W_\pm , \theta _\pm \), and \( V_\pm \) are functions of \( \Phi _p, H_{p,0}, H_{p,h}, \psi _{p,\pm } \), and \( z_{p,\pm } \) as in (4.13’) and (4.27). This finishes the proof of Theorem 1.2.

Remark 3

In the absence of fast waves, all the fast wave variables in system (), i.e., \( N_\Psi , \psi _{p,\pm }, z_{p,\pm }, N_\psi , N_z \) vanish. Therefore system () reduces to the classical quasi-geostrophic approximation as studied in [10]. Meanwhile, in the case of periodic domains, with additional symmetry, the traces on the upper and bottom boundaries, i.e., \( H_{p,0}, H_{p,h}\) vanish. Hence system () reduces to a special case in [16, 17]. Notably, in [16] as well as [17], only the special case that avoids the resonance in the slow dynamics is explicitly written down. Recalling that in [16, 17], the authors treat the fast waves case subject to the periodic boundary condition (i.e., without solid boundaries). In comparison, in our paper, we treat the case with solid boundaries and identify an explicit form of such resonance terms.