2.1 Transformation of the linear operator into the vertical mode basis

As is commonly done in geophysical fluid dynamics (see Gill (Reference Gill1982) or Vallis (Reference Vallis2006), for example), we decompose into a vertical mode basis which we use in much of the rest of the paper.

The appropriate transformation (see, for example, Ball Reference Ball1964) can be found to be

(2.16a ) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{m} & = & \displaystyle L_{m}H_{1}\boldsymbol{u}_{1}+H_{2}\boldsymbol{u}_{2},\end{eqnarray}$$

(2.16b ) $$\begin{eqnarray}\displaystyle p_{m} & = & \displaystyle \frac{1}{c_{m}}(L_{m}H_{1}p_{1}+H_{2}p_{2}),\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle c_{m}^{2}=g\frac{H_{1}+H_{2}}{2}+mg\sqrt{\left(\frac{H_{1}-H_{2}}{2}\right)^{2}+H_{1}H_{2}r_{\unicode[STIX]{x1D70C}}}, & \displaystyle\end{eqnarray}$$

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle L_{m}H_{1}=\frac{H_{1}-H_{2}}{2}+m\sqrt{\left(\frac{H_{1}-H_{2}}{2}\right)^{2}+H_{1}H_{2}r_{\unicode[STIX]{x1D70C}}}. & \displaystyle\end{eqnarray}$$

Here, $m=\pm$ ; $m$ defines the vertical mode; in this form, the system is split into modes $m=+,-$ instead of the two layers $i=1,2$ as previously.

It should be noticed that the transformation operates on the pressure in each layer, not the perturbation heights $\unicode[STIX]{x1D702}_{i}$ . More simply, using (2.3) and (2.6), we can write the direct transform from $\unicode[STIX]{x1D702}_{i}$ ,

(2.19) $$\begin{eqnarray}\displaystyle p_{m}=gL_{m}c_{m}\unicode[STIX]{x1D702}_{1}+g\unicode[STIX]{x1D702}_{2}. & & \displaystyle\end{eqnarray}$$

This transforms the linear operator into the following in Fourier space:

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D647}=\left[\begin{array}{@{}cccccc@{}}0 & -f & c_{+}k\text{i} & 0 & 0 & 0\\ f & 0 & c_{+}l\text{i} & 0 & 0 & 0\\ c_{+}k\text{i} & c_{+}l\text{i} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -f & c_{-}k\text{i}\\ 0 & 0 & 0 & f & 0 & c_{-}l\text{i}\\ 0 & 0 & 0 & c_{-}k\text{i} & c_{-}l\text{i} & 0\\ \end{array}\right]. & & \displaystyle\end{eqnarray}$$

It can be seen from (2.20) that the two modes are entirely decoupled in the linear part; this means that all coupling will now appear in the nonlinear interactions. Throughout the rest of this paper, the modes are interchangeably referred to as both internal/baroclinic ( $m=-$ ) and external/barotropic ( $m=+$ ). We have adopted this convention in reference to the rigid lid limit (discussed in § 2.4; see also Salmon (Reference Salmon1998), for example).

The frequencies and modal functions correspond to the eigenvalues and eigenvectors of the matrix. The frequencies are

(2.21a-f ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714} & = & \displaystyle -\sqrt{c_{+}^{2}|\boldsymbol{k}|^{2}+f^{2}},\quad 0,\quad \sqrt{c_{+}^{2}|\boldsymbol{k}|^{2}+f^{2}},\nonumber\\ \displaystyle & & \displaystyle -\sqrt{c_{-}^{2}|\boldsymbol{k}|^{2}+f^{2}},\quad 0,\quad \sqrt{c_{-}^{2}|\boldsymbol{k}|^{2}+f^{2}},\end{eqnarray}$$

and in this basis the corresponding (orthonormal) modal functions are respectively

(2.22a-f ) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\left[\begin{array}{@{}c@{}}\displaystyle \frac{(-k\unicode[STIX]{x1D714}+\text{i}f\,l)}{\sqrt{2}|k|\unicode[STIX]{x1D714}}\\ \displaystyle \frac{(-\text{i}f\,k-l\unicode[STIX]{x1D714})}{\sqrt{2}|k|\unicode[STIX]{x1D714}}\\ \displaystyle \frac{|k|c_{+}}{\sqrt{2}\unicode[STIX]{x1D714}}\\ 0\\ 0\\ 0\end{array}\right],\quad \left[\begin{array}{@{}c@{}}\displaystyle -\frac{\text{i}lc_{+}}{\unicode[STIX]{x1D714}}\\ \displaystyle \frac{\text{i}kc_{+}}{\unicode[STIX]{x1D714}}\\ \displaystyle \frac{f}{\unicode[STIX]{x1D714}}\\ 0\\ 0\\ 0\end{array}\right],\quad \left[\begin{array}{@{}c@{}}\displaystyle \frac{(k\unicode[STIX]{x1D714}+\text{i}f\,l)}{\sqrt{2}|k|\unicode[STIX]{x1D714}}\\ \displaystyle \frac{(-\text{i}f\,k+l\unicode[STIX]{x1D714})}{\sqrt{2}|k|\unicode[STIX]{x1D714}}\\ \displaystyle \frac{|k|c_{+}}{\sqrt{2}\unicode[STIX]{x1D714}}\\ 0\\ 0\\ 0\end{array}\right],\\ \left[\begin{array}{@{}c@{}}0\\ 0\\ 0\\ \displaystyle \frac{(-k\unicode[STIX]{x1D714}+\text{i}f\,l)}{\sqrt{2}|k|\unicode[STIX]{x1D714}}\\ \displaystyle \frac{(-\text{i}f\,k-l\unicode[STIX]{x1D714})}{\sqrt{2}|k|\unicode[STIX]{x1D714}}\\ \displaystyle \frac{|k|c_{-}}{\sqrt{2}\unicode[STIX]{x1D714}}\end{array}\right],\quad \left[\begin{array}{@{}c@{}}0\\ 0\\ 0\\ \displaystyle -\frac{\text{i}lc_{-}}{\unicode[STIX]{x1D714}}\\ \displaystyle \frac{\text{i}kc_{-}}{\unicode[STIX]{x1D714}}\\ \displaystyle \frac{f}{\unicode[STIX]{x1D714}}\end{array}\right],\quad \left[\begin{array}{@{}c@{}}0\\ 0\\ 0\\ \displaystyle \frac{(k\unicode[STIX]{x1D714}+\text{i}f\,l)}{\sqrt{2}|k|\unicode[STIX]{x1D714}}\\ \displaystyle \frac{(-\text{i}f\,k+l\unicode[STIX]{x1D714})}{\sqrt{2}|k|\unicode[STIX]{x1D714}}\\ \displaystyle \frac{|k|c_{-}}{\sqrt{2}\unicode[STIX]{x1D714}}\end{array}\right].\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Henceforth, these modes will be referred to by $\boldsymbol{r}_{\boldsymbol{k}}^{\unicode[STIX]{x1D6FC}m}$ .

Each vertical mode is equivalent to those in the one-layer system (see equation (2.51) in Embid & Majda (Reference Embid and Majda1996)). Properties then transfer from their analysis: the 0 modes are in geostrophic balance, and the fast modes have 0 linear potential vorticity (defined in § 2.2). In the remainder of this work, we refer to fast modes (inertial gravity waves) where $\unicode[STIX]{x1D714}\neq 0$ and slow modes (potential vorticity modes) where $\unicode[STIX]{x1D714}=0$ .

The nonlinear parts for the equations of $\boldsymbol{u}_{m}$ and $p_{m}$ are transformed similarly, giving

(2.23) $$\begin{eqnarray}\displaystyle N_{u}^{m} & = & \displaystyle \mathop{\sum }_{m_{1},m_{2}}A_{m_{1}m_{2}}^{m}(\boldsymbol{u}_{m_{1}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}_{m_{2}},\end{eqnarray}$$

(2.24) $$\begin{eqnarray}\displaystyle N_{p}^{m} & = & \displaystyle \mathop{\sum }_{m_{1},m_{2}}\frac{c_{m}}{c_{m_{1}}}A_{m_{1}m_{2}}^{m}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{m_{1}}\boldsymbol{u}_{m_{2}}),\end{eqnarray}$$

where

(2.25) $$\begin{eqnarray}\displaystyle A_{m_{1}m_{2}}^{m}=\frac{m_{1}m_{2}}{(L_{+}-L_{-})^{2}}\left[\frac{L_{m}}{H_{1}}+\frac{L_{-m_{1}}L_{-m_{2}}}{H_{2}}\right]. & & \displaystyle\end{eqnarray}$$

Here, there is coupling between all of the different combinations of vertical modes as $A$ is, in general, non-zero.

2.2 Conservation of potential vorticity

As remarked previously, all coupling between layers in the equations is due to the pressure terms. We define the potential vorticity (hereafter referred to as PV) similarly to that of the one-layer equations (see Vallis Reference Vallis2006). The pressure term is redundant in the calculation of the PV, and this term is the only part that couples the equations. It follows that there can be no coupling between the PV of each layer; the potential vorticity is conserved per layer,

(2.26a,b ) $$\begin{eqnarray}\displaystyle \frac{\text{D}_{i}q_{i}}{\text{D}t}=0,\quad q_{i}=\frac{\unicode[STIX]{x1D701}_{i}+f}{h_{i}}, & & \displaystyle\end{eqnarray}$$

where $\text{D}_{i}/\text{D}t$ is the Lagrangian derivative for the flow in the $i$ th layer and $\unicode[STIX]{x1D701}_{i}=(\unicode[STIX]{x1D735}\times \boldsymbol{u}_{i})_{z}$ is the relative vorticity. For the linear part of the problem, we have

(2.27a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}Q_{i}}{\unicode[STIX]{x2202}t}=0,\quad Q_{i}=\unicode[STIX]{x1D701}_{i}-\frac{f\unicode[STIX]{x1D702}_{i}}{H_{i}}. & & \displaystyle\end{eqnarray}$$

This represents geostrophic balance for two layers in the linear approximation.

We now perform the transform (2.16) from layers to the vertical mode basis and find

(2.28a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}Q_{m}}{\unicode[STIX]{x2202}t}=0,\quad Q_{m}=\unicode[STIX]{x1D701}_{m}-\frac{fp_{m}}{c_{m}}. & & \displaystyle\end{eqnarray}$$

The direct transformation of linear PV is given by

(2.29) $$\begin{eqnarray}\displaystyle Q_{m}=L_{m}H_{1}^{2}Q_{1}+H_{2}^{2}Q_{2}. & & \displaystyle\end{eqnarray}$$

It follows that $Q_{1}=Q_{2}=0\;\Longleftrightarrow \;Q_{+}=Q_{-}=0$ .

Using (2.28b ) to calculate the PV associated with the eigenfunctions derived in (2.22a–f ), we find that the fast modes have zero linear PV; all PV for the system is contained in the slow modes. This simplifies the calculation of enstrophy in these cases.

2.3 Integral conservation laws

In addition to potential vorticity, there are two integrally conserved quantities: energy and enstrophy. The quadratic and cubic parts of these are used in § 3.3.

In the layer basis, enstrophy conservation in flux form is defined as

(2.30a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}Z_{i}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(Z_{i}\boldsymbol{u}_{i})=0,\quad Z_{i}=h_{i}q_{i}^{2}. & & \displaystyle\end{eqnarray}$$

This can be expanded as follows:

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle Z_{i}=Z_{i}^{(2)}+Z_{i}^{(3)}+\cdots \,, & \displaystyle\end{eqnarray}$$

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle Z_{i}^{(2)}=H_{i}Q_{i}^{2}, & \displaystyle\end{eqnarray}$$

(2.33) $$\begin{eqnarray}\displaystyle & \displaystyle Z_{i}^{(3)}=-\unicode[STIX]{x1D702}_{i}Q_{i}^{2}. & \displaystyle\end{eqnarray}$$

Here, if the linear PV in both layers is zero ( $Q_{i}=0$ ), we see that the enstrophy (up to third order) is also zero. Earlier, we observed that this also implies that the linear PV in the mode basis is zero ( $Q_{m}=0$ ). If we combine these two ideas, it follows that the enstrophy in the mode basis is also zero in this case.

We integrate this across the domain,

(2.34a,b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}{\mathcal{Z}}_{i}}{\text{d}t}=0,\quad {\mathcal{Z}}_{i}=\int _{D_{i}}h_{i}q_{i}^{2}\,\text{d}\boldsymbol{x}. & & \displaystyle\end{eqnarray}$$

Energy conservation can be derived from the initial equations by multiplying the momentum equations (2.1) and (2.4) by $h_{i}\boldsymbol{u}_{i}$ and manipulating into the form

(2.35a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}E}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\frac{1}{2}(r_{\unicode[STIX]{x1D70C}}h_{1}|\boldsymbol{u}_{1}|^{2}\boldsymbol{u}_{1}+h_{2}|\boldsymbol{u}_{2}|^{2}\boldsymbol{u}_{2}+(r_{\unicode[STIX]{x1D70C}}p_{1}h_{1}+p_{2}h_{2}))\right)=0, & \displaystyle\end{eqnarray}$$

(2.35b ) $$\begin{eqnarray}\displaystyle & \displaystyle E={\textstyle \frac{1}{2}}(r_{\unicode[STIX]{x1D70C}}h_{1}|\boldsymbol{u}_{1}|^{2}+h_{2}|\boldsymbol{u}_{2}|^{2}+g(r_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D702}_{1}^{2}+2r_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D702}_{1}\unicode[STIX]{x1D702}_{2}+\unicode[STIX]{x1D702}_{2}^{2})), & \displaystyle\end{eqnarray}$$

(2.36a,b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}{\mathcal{E}}}{\text{d}t}=0,\quad {\mathcal{E}}=\frac{1}{2}\int _{D}r_{\unicode[STIX]{x1D70C}}h_{1}|\boldsymbol{u}_{1}|^{2}+h_{2}|\boldsymbol{u}_{2}|^{2}+g(r_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D702}_{1}^{2}+2r_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D702}_{1}\unicode[STIX]{x1D702}_{2}+\unicode[STIX]{x1D702}_{2}^{2})\,\text{d}\boldsymbol{x}. & & \displaystyle\end{eqnarray}$$

2.4 Rigid lid limit

The rigid lid approximation is a specific case of the two-layer equations used in geophysical applications. The rigid lid equations are only physically realised when the top layer is at a fixed solid boundary; otherwise, they are an approximation based on the (unphysical) assumptions that the gravitational force is large compared with the Coriolis force, and the densities of the two layers are close. In this parameter regime, the sizes of waves on the external boundary are negligible compared with the size of the waves on the internal layer. The limit $g\rightarrow \infty$ is used to force rigidity in the upper layer when no physical boundary actually exists. This limit is taken separately to the asymptotic limit we are taking; this defines the basic system before any other assumptions are made.

We start the derivation (see Salmon Reference Salmon1998) by taking $r_{\unicode[STIX]{x1D70C}}\rightarrow 1$ and defining the reduced gravity $g^{\prime }=g(1-r_{\unicode[STIX]{x1D70C}})$ , which we then require to be finite in the limit. The transformation to external and internal modes then becomes

(2.37) $$\begin{eqnarray}\displaystyle L_{m} & = & \displaystyle \left\{\begin{array}{@{}ll@{}}1,\quad & m=+,\\ \displaystyle -\frac{H_{2}}{H_{1}},\quad & m=-,\end{array}\right.\end{eqnarray}$$

(2.38) $$\begin{eqnarray}\displaystyle c_{m}^{2} & = & \displaystyle \left\{\begin{array}{@{}ll@{}}g(H_{1}+H_{2}),\quad & m=+,\\ \displaystyle \frac{g^{\prime }H_{1}H_{2}}{H_{1}+H_{2}},\quad & m=-,\end{array}\right.\end{eqnarray}$$

by setting $r_{\unicode[STIX]{x1D70C}}=1-\unicode[STIX]{x1D6FF}$ , $g^{\prime }=g\unicode[STIX]{x1D6FF}$ , $\unicode[STIX]{x1D6FF}\ll 1$ and taking Taylor series in $\unicode[STIX]{x1D6FF}$ .

Because in this limit we take $g\rightarrow \infty$ , the external wave speed becomes infinite ( $c_{+}\rightarrow \infty$ ) and the corresponding external Rossby radius of deformation $\unicode[STIX]{x1D706}_{+}=c_{+}/f\rightarrow \infty$ as well. By definition, below the Rossby radius of deformation, the surface displacement, and hence fast modes, are negligible, and so for an infinite radius, all fast modes must be neglected. This corresponds to the external boundary becoming fixed in the long-wave limit. Effectively, waves cannot propagate on the external boundary and so it becomes ‘rigid’.

Another consequence of this limit is that the pressure in the external mode becomes undefined, changing the structure of the equations: this pressure can no longer evolve in time. This means that if the external mode is initially defined as motionless, the equations only describe the internal mode in a one-layer system similar to the usual one.

Neither the one-layer nor the rigid lid equations allow for resonant interactions in the fast modes. This is because no resonances can exist between fast-mode triads with the same characteristic velocity $c$ . Any interactions amongst fast modes will be a new phenomenon in the case of equations with two free layers.

2.5 Thin-layer limit

Another application relevant to the ocean is to make the upper layer thin compared with the lower layer. However, this will change the analysis completely: the thin layer/internal mode will be pushed to the next order of the expansion and so the leading-order effects will be equivalent to the one-layer case, with corrections at higher order.

To see this, we consider the amplitude ratio $D/H\sim \unicode[STIX]{x1D716}$ . If we choose one of our layer depths to be asymptotically small, to avoid violation of this condition, we require $D\sim \unicode[STIX]{x1D716}^{2}$ , and so all of the dynamics of this layer can only effect the $O(\unicode[STIX]{x1D716}^{2})$ terms and higher in the non-triad interactions.

Equivalently, on setting $H_{1}=H$ , $H_{2}=H\unicode[STIX]{x1D716}$ in (2.17),

(2.39) $$\begin{eqnarray}\displaystyle c_{m}^{2} & = & \displaystyle gH\left\{\begin{array}{@{}ll@{}}\displaystyle 1+\frac{r_{\unicode[STIX]{x1D70C}}}{4}\unicode[STIX]{x1D716}+O(\unicode[STIX]{x1D716}^{2}),\quad & m=+,\\ \displaystyle \left(1-\frac{r_{\unicode[STIX]{x1D70C}}}{4}\right)\unicode[STIX]{x1D716}+O(\unicode[STIX]{x1D716}^{2}),\quad & m=-,\end{array}\right.\end{eqnarray}$$

(2.40) $$\begin{eqnarray}\displaystyle L_{m} & = & \displaystyle \left\{\begin{array}{@{}ll@{}}\displaystyle 1-\left(\frac{r_{\unicode[STIX]{x1D70C}}}{4}-1\right)\unicode[STIX]{x1D716}+O(\unicode[STIX]{x1D716}^{2}),\quad & m=+,\\ \displaystyle -\frac{r_{\unicode[STIX]{x1D70C}}}{4}\unicode[STIX]{x1D716}+O(\unicode[STIX]{x1D716}^{2}),\quad & m=-,\end{array}\right.\end{eqnarray}$$

which shows that the internal mode will be asymptotically small and asymptotically slow in our calculations.

3.1 Multiscale asymptotic expansion

Much work already exists that analyses the triad resonances of the one-layer equations, for instance Warn (Reference Warn1986), Embid & Majda (Reference Embid and Majda1996), Babin et al. (Reference Babin, Mahalov and Nicolaenko1997) and Medvedev (Reference Medvedev1999). Our extension to two layers recovers similar results, although we will explore the differences in § 3.4. For the usual quasigeostrophic limit, it is required that the non-dimensional parameters Rossby number, Froude number and displacement/depth ratio are all proportional to $\unicode[STIX]{x1D716}$ , as previously stated in § 2. We can then reduce the system to the non-localised form,

(3.1) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\frac{1}{\unicode[STIX]{x1D716}}{\mathcal{L}}(\boldsymbol{u})+{\mathcal{N}}(\boldsymbol{u},\boldsymbol{u})=0, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}=(u_{+},v_{+},h_{+},u_{-},v_{-},h_{-})^{\text{T}}$ and ${\mathcal{L}}$ is the linear operator (2.20) defined above in Fourier space with the set of orthonormal eigenvectors in (2.22). Here, the nonlinear terms are represented by the bilinear operator ${\mathcal{N}}$ . This operator has two arguments, but only one input, $\boldsymbol{u}$ , and hence we have some freedom in its definition. We choose to define it symmetrically, so that it operates equivalently on the first and second arguments. Using (2.23) and (2.24) (Fourier transformation assumed), we write

(3.2) $$\begin{eqnarray}\displaystyle {\mathcal{N}}(\boldsymbol{a},\boldsymbol{b})=\frac{1}{2}\left(\begin{array}{@{}c@{}}N_{u}^{m}(a,b)+N_{u}^{m}(b,a)\\ N_{p}^{m}(a,b)+N_{p}^{m}(b,a)\end{array}\right). & & \displaystyle\end{eqnarray}$$

Unlike the usual quasigeostrophic limit, we retain a fast time scale $\unicode[STIX]{x1D70F}=t/\unicode[STIX]{x1D716}$ as in Embid & Majda (Reference Embid and Majda1996) or Thomas (Reference Thomas2016) and so write $\boldsymbol{u}=\boldsymbol{u}(\boldsymbol{x},\unicode[STIX]{x1D70F},t)$ as a function of two time scales. Then, (3.1) becomes

(3.3) $$\begin{eqnarray}\displaystyle \frac{1}{\unicode[STIX]{x1D716}}\left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+{\mathcal{L}}(\boldsymbol{u})\right)=-\left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+{\mathcal{N}}(\boldsymbol{u},\boldsymbol{u})\right). & & \displaystyle\end{eqnarray}$$

As previously mentioned, the different types of eigenfunctions for the operator ${\mathcal{L}}$ have different behaviour. One set represents fast gravity waves ( $\unicode[STIX]{x1D6FC}=\pm$ ) that are wave-like in fast time $\unicode[STIX]{x1D70F}$ with frequency $\unicode[STIX]{x1D714}$ . The other set represents slow PV modes ( $\unicode[STIX]{x1D6FC}=0$ ) that do not evolve on the $\unicode[STIX]{x1D70F}$ time scale due to the zero eigenvalue. The usual quasigeostrophic equations are derived by taking this limit without using a fast time so that only slow modes are present.

We expand the variable $\boldsymbol{u}$ as follows:

(3.4) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x},\unicode[STIX]{x1D70F},t)=\boldsymbol{u}_{0}(\boldsymbol{x},\unicode[STIX]{x1D70F},t)+\unicode[STIX]{x1D716}\boldsymbol{u}_{1}(\boldsymbol{x},\unicode[STIX]{x1D70F},t)+\cdots \,. & & \displaystyle\end{eqnarray}$$

Substitution into the non-local equation (3.3) gives the following at each order:

(3.5) $$\begin{eqnarray}\displaystyle O(\unicode[STIX]{x1D716}^{-1}) & & \displaystyle \quad \frac{\unicode[STIX]{x2202}\boldsymbol{u}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+{\mathcal{L}}(\boldsymbol{u}_{0})=0,\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle O(1) & & \displaystyle \quad \frac{\unicode[STIX]{x2202}\boldsymbol{u}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+{\mathcal{L}}(\boldsymbol{u}_{1})=-\left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{0}}{\unicode[STIX]{x2202}t}+{\mathcal{N}}(\boldsymbol{u}_{0},\boldsymbol{u}_{0})\right).\end{eqnarray}$$

At first order, the equation is linear and we can write the solution in terms of the exponential operator and an average over fast time $\bar{\boldsymbol{u}}$ ,

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}_{0}(\boldsymbol{x},\unicode[STIX]{x1D70F},t)=\bar{\boldsymbol{u}}(\boldsymbol{x},t)\text{e}^{-\unicode[STIX]{x1D70F}{\mathcal{L}}}, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{\boldsymbol{u}}=\lim _{\unicode[STIX]{x1D70F}\rightarrow \infty }\int _{0}^{\unicode[STIX]{x1D70F}}\boldsymbol{u}_{0}(\boldsymbol{x},s,t)\,\text{d}s. & \displaystyle\end{eqnarray}$$

Then, solving at the next order,

(3.9) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{1}\text{e}^{\unicode[STIX]{x1D70F}{\mathcal{L}}}=\boldsymbol{u}_{1}|_{\unicode[STIX]{x1D70F}=0}-\left(\unicode[STIX]{x1D70F}\frac{\unicode[STIX]{x2202}\bar{\boldsymbol{u}}}{\unicode[STIX]{x2202}t}+\int _{0}^{\unicode[STIX]{x1D70F}}{\mathcal{N}}(\bar{\boldsymbol{u}}\text{e}^{-s{\mathcal{L}}},\bar{\boldsymbol{u}}\text{e}^{-s{\mathcal{L}}})\text{e}^{s{\mathcal{L}}}\,\text{d}s\right). & & \displaystyle\end{eqnarray}$$

With the equation in this form, we can identify possible secular terms as any of $O(\unicode[STIX]{x1D70F})$ or higher: those in the round brackets. To maintain the separation of scales for the velocities/pressures as defined in (3.4), these terms must be zero in the limit $\unicode[STIX]{x1D70F}\rightarrow \infty$ . This is the ‘cancellation of oscillations’ concept, used by Schochet (Reference Schochet1994), where he used the concept to prove convergence for general hyperbolic equations. We will also assume that the vector $\bar{\boldsymbol{u}}$ is written in its eigenbasis, such that the matrix exponential is just the exponential of the frequency of the corresponding eigenvalue ( $\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}$ ),

(3.10) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\bar{\boldsymbol{u}}}{\unicode[STIX]{x2202}t} & = & \displaystyle -\lim _{\unicode[STIX]{x1D70F}\rightarrow \infty }\frac{1}{\unicode[STIX]{x1D70F}}\int _{0}^{\unicode[STIX]{x1D70F}}{\mathcal{N}}(\bar{\boldsymbol{u}}\text{e}^{-s{\mathcal{L}}},\bar{\boldsymbol{u}}\text{e}^{-s{\mathcal{L}}})\text{e}^{s{\mathcal{L}}}\,\text{d}s\nonumber\\ \displaystyle & = & \displaystyle -\lim _{\unicode[STIX]{x1D70F}\rightarrow \infty }\frac{1}{\unicode[STIX]{x1D70F}}\int _{0}^{\unicode[STIX]{x1D70F}}\mathop{\sum }_{\substack{ \boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2} \\ \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2} \\ \boldsymbol{k}=\boldsymbol{k}_{1}+\boldsymbol{k}_{2}}}C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}(t)\unicode[STIX]{x1D70E}_{\boldsymbol{ k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}(t)\boldsymbol{r}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D6FC}m}\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}}\text{e}^{\text{i}(\unicode[STIX]{x1D714}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}+\unicode[STIX]{x1D714}_{\boldsymbol{ k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}-\unicode[STIX]{x1D714}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D6FC}m})\unicode[STIX]{x1D70F}}\,\text{d}s,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{i}}^{\unicode[STIX]{x1D6FC}_{i}m_{i}}$ represents the wave amplitude of each eigenfunction and the interaction coefficient is defined as

(3.11) $$\begin{eqnarray}\displaystyle C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}} & = & \displaystyle \frac{\text{i}A_{m_{1}m_{2}}^{m}}{2}\left[(\boldsymbol{v}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}\boldsymbol{\cdot }\boldsymbol{k}_{2})(\boldsymbol{v}_{\boldsymbol{k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}\boldsymbol{\cdot }\boldsymbol{v}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D6FC}m})+(\boldsymbol{v}_{\boldsymbol{ k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}\boldsymbol{\cdot }\boldsymbol{k}_{1})(\boldsymbol{v}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}\boldsymbol{\cdot }\boldsymbol{v}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D6FC}m})\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\frac{c_{m}}{c_{m_{2}}}(\boldsymbol{v}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}\boldsymbol{\cdot }(\boldsymbol{k}_{1}+\boldsymbol{k}_{2}))p_{\boldsymbol{k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}p_{\boldsymbol{ k}}^{\unicode[STIX]{x1D6FC}m}+\frac{c_{m}}{c_{m_{1}}}(\boldsymbol{v}_{\boldsymbol{k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}\boldsymbol{\cdot }(\boldsymbol{k}_{1}+\boldsymbol{k}_{2}))p_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}p_{\boldsymbol{ k}}^{\unicode[STIX]{x1D6FC}m}\right],\qquad\end{eqnarray}$$

where $\boldsymbol{v}$ is the two-dimensional vector composed of the velocities, and we have expressed (3.2) explicitly in spectral space. Here, $\unicode[STIX]{x1D6FC}_{i},m_{i}$ define the modes being considered.

The integral simplifies further. In the limit, the integral of all oscillatory contributions exactly cancels to 0, and so the only contributions come from the non-oscillatory constant contributions, where

(3.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}+\unicode[STIX]{x1D714}_{\boldsymbol{ k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}-\unicode[STIX]{x1D714}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D6FC}m}=0. & & \displaystyle\end{eqnarray}$$

These are the resonant triads. This leaves the equations

(3.13) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\bar{\boldsymbol{u}}}{\unicode[STIX]{x2202}t}=\mathop{\sum }_{\substack{ \boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2} \\ \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2}}}C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}(t)\unicode[STIX]{x1D70E}_{\boldsymbol{ k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}(t)\boldsymbol{r}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D6FC}m}\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}}\unicode[STIX]{x1D6FF}_{\boldsymbol{k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{1}-\unicode[STIX]{x1D714}_{2}}, & & \displaystyle\end{eqnarray}$$

or in terms of only wave amplitudes,

(3.14) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{\unicode[STIX]{x1D6FC}m}=\mathop{\sum }_{\substack{ \boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2} \\ \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2}}}C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}(t)\unicode[STIX]{x1D70E}_{\boldsymbol{ k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}(t)\unicode[STIX]{x1D6FF}_{\boldsymbol{ k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{1}-\unicode[STIX]{x1D714}_{2}}. & & \displaystyle\end{eqnarray}$$

The only possible resonances are the combinations of modes $(\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2},\unicode[STIX]{x1D6FC})=$

(3.15a ) $$\begin{eqnarray}\displaystyle (0,\pm ,\pm ),(\pm ,0,\pm ), & & \displaystyle\end{eqnarray}$$

(3.15b ) $$\begin{eqnarray}\displaystyle (\pm ,\pm ,0), & & \displaystyle\end{eqnarray}$$

(3.15c ) $$\begin{eqnarray}\displaystyle (\pm ,\pm ,\pm ), & & \displaystyle\end{eqnarray}$$

(3.15d ) $$\begin{eqnarray}\displaystyle (0,0,0). & & \displaystyle\end{eqnarray}$$

The first and second in (3.15a ) are treated as equivalent due to the symmetry chosen in the interaction coefficient $C$ . Combination (3.15b ) leads to an interaction term of zero, as can be shown by direct substitution of the eigenvectors into $C$ in (3.11). This is done in § 3.2, with an alternate analysis in § 3.3 to discern the physical cause of this zero value.

Only three types of interaction remain: slow–slow–slow (3.15d ), which define the development of the PV modes over the longer time scale $t$ , and fast–slow–fast (3.15a ) and fast–fast–fast (3.15c ), which define the scattering of fast modes off a slow mode and interactions amongst themselves.

For the one-layer equations, it can be shown (see Warn (Reference Warn1986) or Embid & Majda (Reference Embid and Majda1996), for example) that there are no fast–fast–fast resonances, and, further, that the equations for the slow part are exactly equivalent to the quasigeostrophic approximation. We show that in the two-layer case, the slow part again evolves independently of the fast part, giving the quasigeostrophic equation (see § 3.2). However, unlike the one-layer case, we show, in § 3.4, that there are interactions among the fast waves for our two-layer system.

3.2 Interaction coefficients

In this section, we give the nonlinear interaction coefficient in (3.11) explicitly for the different possible combinations of modes. This allows us to examine in detail and categorise the different possible nonlinear interactions in this system. The vertical mode parameter $m_{i}$ is left general, and setting $m=m_{1}=m_{2}$ returns a comparable expression to the one derived by Ward & Dewar (Reference Ward and Dewar2010) for the one-layer case. For the slow modes,

(3.16a ) $$\begin{eqnarray}\displaystyle C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ 0,0,0 \\ m_{1}m_{2}m}} & = & \displaystyle \frac{A_{m_{1}m_{2}}^{m}c_{m}(\boldsymbol{k}_{2}\times \boldsymbol{k}_{1})}{2\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D714}_{2}}\left(c_{m_{1}}\frac{\unicode[STIX]{x1D714}_{2}^{2}}{c_{m_{2}}}-c_{m_{2}}\frac{\unicode[STIX]{x1D714}_{1}^{2}}{c_{m_{1}}}\right)\end{eqnarray}$$

(3.16b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \frac{\text{i}A_{m_{1}m_{2}}^{m}}{2}\frac{c_{m}}{\unicode[STIX]{x1D714}}\left(\frac{\unicode[STIX]{x1D714}_{2}}{c_{m_{2}}}(\boldsymbol{v}_{1}\boldsymbol{\cdot }\boldsymbol{k}_{2})+\frac{\unicode[STIX]{x1D714}_{1}}{c_{m_{1}}}(\boldsymbol{v}_{2}\boldsymbol{\cdot }\boldsymbol{k}_{1})\right).\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}0 \\ m_{1}m_{2}m}} & = & \displaystyle 0,\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}0\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}} & = & \displaystyle \frac{\text{i}A_{m_{1}m_{2}}^{m}}{4c_{m_{2}}\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D714}_{2}|\boldsymbol{k}||\boldsymbol{k}_{1}|}[(\text{i}f^{2}(c_{m}^{2}-c_{m_{2}}^{2})|\boldsymbol{k}|^{2}(\boldsymbol{k}\times \boldsymbol{k}_{1})_{z}+2\text{i}f^{2}c_{m_{2}}^{2}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{\boldsymbol{ 1}})(\boldsymbol{k}\times \boldsymbol{k}_{1})_{z})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1}f(c_{m}^{2}-c_{m_{2}}^{2})|\boldsymbol{k}|^{2}(\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{k})-\text{i}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D714}c_{m_{2}}^{2}(\boldsymbol{k}_{1}\times \boldsymbol{k})_{z}|\boldsymbol{k}_{1}|^{2}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1}fc_{m_{2}}^{2}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{\mathbf{1}})|\boldsymbol{k}_{1}|^{2}+2\text{i}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D714}c_{m_{2}}^{2}(\boldsymbol{k}_{1}\times \boldsymbol{k})_{z}(\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{k})\nonumber\\ \displaystyle & & \displaystyle +\,2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}c_{m_{2}}^{2}f(\boldsymbol{k}\times \boldsymbol{k}_{1})_{z}^{2}+\text{i}c_{m_{2}}^{2}c_{m}^{2}|\boldsymbol{k}_{1}|^{2}|\boldsymbol{k}|^{2}(\boldsymbol{k}\times \boldsymbol{k}_{1})_{z}].\end{eqnarray}$$

For three fast modes,

(3.19) $$\begin{eqnarray}\displaystyle C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}} & = & \displaystyle \frac{\text{i}A_{m_{1}m_{2}}^{m}}{4\sqrt{2}\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D714}_{2}|\boldsymbol{k}||\boldsymbol{k}_{1}||\boldsymbol{k}_{2}|}\nonumber\\ \displaystyle & & \displaystyle \times \,[(\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{2}(|\boldsymbol{k}_{1}|^{2}+|\boldsymbol{k}_{2}|^{2})+(\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1}|\boldsymbol{k}_{1}|^{2}+\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D714}_{2}|\boldsymbol{k}_{2}|^{2})f^{2})(\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{k}_{2})\nonumber\\ \displaystyle & & \displaystyle +\,(2\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{2}+(\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D714}_{2})f^{2})(\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{k}_{2})^{2}+f^{2}(\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D714}_{2}-\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D714}_{1})(\boldsymbol{k}_{1}\times \boldsymbol{k}_{2})_{z}^{2}\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}f(\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{2}+f^{2})|\boldsymbol{k}_{2}|^{2}(\boldsymbol{k}_{2}\times \boldsymbol{k}_{1})_{z}-\text{i}f(\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{1}+f^{2})|\boldsymbol{k}_{1}|^{2}(\boldsymbol{k}_{2}\times \boldsymbol{k}_{1})_{z}\nonumber\\ \displaystyle & & \displaystyle +\,c_{m}^{2}(\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D714}_{2}\unicode[STIX]{x1D6FC}_{2})|\boldsymbol{k}_{2}|^{2}|\boldsymbol{k}|^{2}|\boldsymbol{k}_{1}|^{2}+c_{m}^{2}(\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D6FC}_{1}|\boldsymbol{k}_{2}|^{2}+\unicode[STIX]{x1D714}_{2}\unicode[STIX]{x1D6FC}_{2}|\boldsymbol{k}_{1}|^{2})|\boldsymbol{k}|^{2}(\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{k}_{2})\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}fc_{m}^{2}(|\boldsymbol{k}_{2}|^{2}-|\boldsymbol{k}_{1}|^{2})|\boldsymbol{k}|^{2}(\boldsymbol{k}_{2}\times \boldsymbol{k}_{1})_{z}].\end{eqnarray}$$

We recover the second version of the slow–slow–slow interactions (3.16b ) from the usual quasigeostrophic equations, as demonstrated in the following. As the fast modes have zero linear PV, the restriction to slow modes is equivalent to the assumption that the flow, to first approximation, is solely the geostrophic part. The second part of the quasigeostrophic approximation is to assume that the advection of the flow is due only to this geostrophic part (the slow-mode interaction), and hence we would expect that the equation for the slow part,

(3.20) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{0m}}{\unicode[STIX]{x2202}t}+\mathop{\sum }_{1,2}C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ 0,0,0 \\ m_{1}m_{2}m}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{0m_{1}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{2}}^{0m_{2}}~\unicode[STIX]{x1D6FF}_{\boldsymbol{k}-\boldsymbol{k}_{1}+\boldsymbol{k}_{2}}=0, & & \displaystyle\end{eqnarray}$$

is equivalent to the quasigeostrophic equations,

(3.21) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}Q_{i}}{\unicode[STIX]{x2202}t}+((\boldsymbol{u}_{g})_{i}\boldsymbol{\cdot }\unicode[STIX]{x1D735})Q_{i}=0. & & \displaystyle\end{eqnarray}$$

To prove this equivalence between (3.20) and (3.21), we start by transforming (3.21) into the mode basis,

(3.22) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}Q_{m}}{\unicode[STIX]{x2202}t}+\mathop{\sum }_{m_{1}m_{2}}A_{m_{1}m_{2}}^{m}((\boldsymbol{u}_{g})_{m_{1}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})Q_{m_{2}}=0. & & \displaystyle\end{eqnarray}$$

For a general eigenvector $\boldsymbol{r}_{\boldsymbol{k}}^{0m}$ , we now consider the linear potential vorticity $Q_{\boldsymbol{k}}^{m}=-\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{m}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{0m}/c_{m}$ calculated directly from the form in (2.22a–f ) using (2.28b ). With this, we then write the quasigeostrophic equation symmetrically in Fourier space to see

(3.23) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{0m}}{\unicode[STIX]{x2202}t}+\mathop{\sum }_{1,2}\frac{\text{i}A_{m_{1}m_{2}}^{m}}{2}\frac{c_{m}}{\unicode[STIX]{x1D714}}\left(\frac{\unicode[STIX]{x1D714}_{2}}{c_{m_{2}}}(\boldsymbol{v}_{1}\boldsymbol{\cdot }\boldsymbol{k}_{2})+\frac{\unicode[STIX]{x1D714}_{1}}{c_{m_{1}}}(\boldsymbol{v}_{2}\boldsymbol{\cdot }\boldsymbol{k}_{1})\right)\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{0m_{1}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{2}}^{0m_{2}}\unicode[STIX]{x1D6FF}_{\boldsymbol{k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{0m}}{\unicode[STIX]{x2202}t}+\mathop{\sum }_{1,2}C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ 0,0,0 \\ m_{1}m_{2}m}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{0m_{1}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{2}}^{0m_{2}}\unicode[STIX]{x1D6FF}_{\boldsymbol{k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}}=0.\end{eqnarray}$$

This confirms that the two-layer quasigeostrophic equations are recovered as they were in the one-layer case (see Embid & Majda Reference Embid and Majda1996). These are still the usual quasigeostrophic equations in our limit, even though we have included a fast time scale. To give additional insight into why it is possible to separate the equations for the slow and fast evolution, in § 3.3, an additional method showing that this must happen is presented, using conservation laws.

The nonlinear interaction coefficients, calculated in § 3.4, must pair with possible resonances similar to those calculated by Warn (Reference Warn1986).

Before proceeding, we note that the triad interaction equations (3.10) can be extended to allow for near resonances, by replacing the resonance condition (3.12) with

(3.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D6FC}m}-\unicode[STIX]{x1D714}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}-\unicode[STIX]{x1D714}_{\boldsymbol{ k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}=\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FA}_{\boldsymbol{ k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}. & & \displaystyle\end{eqnarray}$$

At an exact resonance ( $\unicode[STIX]{x1D6FA}=0$ ), there are secular terms in the expansion proportional to $t$ , and their removal leads to (3.10). However, for $\unicode[STIX]{x1D6FA}\neq 0$ and of order unity, a different kind of secular term appears when the integral (equivalent to the one in (3.10)) is taken, namely proportional to $(\text{e}^{-\text{i}\unicode[STIX]{x1D6FA}t}-1)/\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FA}$ , and these terms also need to be removed. The outcome is

(3.25) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{\unicode[STIX]{x1D6FC}m}=\mathop{\sum }_{\substack{ \boldsymbol{k}_{1},\boldsymbol{k}_{2} \\ \unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2}}}\frac{-C_{\boldsymbol{k}_{1}\boldsymbol{ k}_{2}\boldsymbol{ k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}\text{e}^{-\text{i}\unicode[STIX]{x1D6FA}_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}t}}{\text{i}\unicode[STIX]{x1D6FA}_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}(t)\unicode[STIX]{x1D70E}_{\boldsymbol{ k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}(t)\unicode[STIX]{x1D6FF}_{\boldsymbol{ k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}}, & & \displaystyle\end{eqnarray}$$

where, as before, the summation is over all wavenumbers and modes. At an exact resonance, a single triad leads to a periodic exchange of energy between the three wave amplitudes, and the same outcome holds for such a near resonance. An early derivation of this can be found in § 6 of Bretherton (Reference Bretherton1964), and see also the recent review by Vanneste (Reference Vanneste2005). In the remaining sections, we shall focus on the exact resonance cases, but will return to this issue of near resonances in our discussion section.

3.3 Resonant triads and conservation laws

In this section, a different method, based on conservation laws, is used to find the interaction coefficients, with the aim of providing insight into the physical cause of the results of the previous section.

Following the argument in Vanneste (Reference Vanneste2005) (more details are given in Ripa (Reference Ripa1981) and Vanneste & Vial (Reference Vanneste and Vial1994)), we define the nonlinear interaction coefficient in terms of the quadratic part of the energy and/or enstrophy. This is done by defining quadratic forms that, as well as giving the first terms of the conservation laws, form an orthogonality condition over the different modes.

First, we consider the linearised problem. The dispersion relation is derived where the different branches give rise to different modes, as in § 2.1. We now write the quadratic part of the energy (or enstrophy) in the form

(3.26) $$\begin{eqnarray}\displaystyle {\mathcal{E}}^{(2)}=\frac{1}{2}\int _{D}\boldsymbol{u}^{\dagger }\unicode[STIX]{x1D640}\boldsymbol{u}\,\text{d}\boldsymbol{x}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}$ is the velocity, $\boldsymbol{u}^{\dagger }$ is its conjugate transpose and $\unicode[STIX]{x1D640}$ is a Hermitian matrix. It can be proven (Vanneste Reference Vanneste2005) that this must obey the following orthogonality relation:

(3.27) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{p}^{\dagger }\unicode[STIX]{x1D640}\boldsymbol{u}_{q}=E_{p}\unicode[STIX]{x1D6FF}_{pq}, & & \displaystyle\end{eqnarray}$$

where $p$ and $q$ define the mode ( $\unicode[STIX]{x1D6FC},m,\boldsymbol{k}$ ) of the velocities. This relation defines the constants $E_{p}$ .

We now solve the nonlinear problem by taking Fourier transforms and splitting into the eigenmodes from the linear problem. We can use this orthogonality relation to isolate the effect on the amplitude of each mode,

(3.28) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\unicode[STIX]{x1D70E}}_{p}=\frac{1}{2}\mathop{\sum }_{qr}C_{p}^{qr}\unicode[STIX]{x1D70E}_{q}^{\ast }\unicode[STIX]{x1D70E}_{r}^{\ast }\text{e}^{\text{i}\unicode[STIX]{x1D6FA}_{pqr}t}\unicode[STIX]{x1D6FF}_{\boldsymbol{k}_{p}+\boldsymbol{k}_{q}+\boldsymbol{k}_{r}}, & \displaystyle\end{eqnarray}$$

(3.29) $$\begin{eqnarray}\displaystyle & \displaystyle C_{p}^{qr}=\boldsymbol{u}_{p}^{\dagger }\unicode[STIX]{x1D640}[\boldsymbol{N}(\boldsymbol{u}_{q},\boldsymbol{u}_{r})+\boldsymbol{N}(\boldsymbol{u}_{r},\boldsymbol{u}_{q})]^{\ast }/E_{p}, & \displaystyle\end{eqnarray}$$

(3.30) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}_{pqr}=\unicode[STIX]{x1D714}_{p}+\unicode[STIX]{x1D714}_{q}+\unicode[STIX]{x1D714}_{r}, & \displaystyle\end{eqnarray}$$

where $\dot{\unicode[STIX]{x1D70E}}$ denotes differentiation of $\unicode[STIX]{x1D70E}$ with respect to time.

In this analysis, there is less emphasis on the exact form of the modes and so we have switched to a more concise notation that absorbs the details of each mode into a simpler form. The $\unicode[STIX]{x1D70E}_{a}$ are the coefficients of Fourier wave mode $a$ , where $a$ contains the information of wavenumber $\boldsymbol{k}$ and mode $(\unicode[STIX]{x1D6FC},m)$ . The $C$ and $\unicode[STIX]{x1D70E}$ are now defined as in the previous sections but in a more compact form,

(3.31) $$\begin{eqnarray}\displaystyle C_{p}^{qr}=C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}},\quad p=\{\boldsymbol{k},\unicode[STIX]{x1D6FC},m\},q=\{\boldsymbol{k}_{1},\unicode[STIX]{x1D6FC}_{1},m_{1}\},r=\{\boldsymbol{k}_{2},\unicode[STIX]{x1D6FC}_{2},m_{2}\}. & & \displaystyle\end{eqnarray}$$

So far, (3.28)–(3.30) are identical to Vanneste’s work, but also equivalent to Embid & Majda (Reference Embid and Majda1996); the selection of the orthonormal basis simply chooses the basis in which $\unicode[STIX]{x1D640}=\boldsymbol{I}$ and $E_{p}=1$ .

Using the conservation of energy and enstrophy laws, the energy (or enstrophy) can be expanded as

(3.32) $$\begin{eqnarray}\displaystyle {\mathcal{E}}={\mathcal{E}}^{(2)}+{\mathcal{E}}^{(3)}+\cdots =\frac{1}{2}\mathop{\sum }_{p}E_{p}|\unicode[STIX]{x1D70E}_{p}|^{2}+\frac{1}{6}\mathop{\sum }_{pqr}S_{pqr}\unicode[STIX]{x1D70E}_{p}^{\ast }\unicode[STIX]{x1D70E}_{q}^{\ast }\unicode[STIX]{x1D70E}_{r}^{\ast }\text{e}^{\text{i}\unicode[STIX]{x1D6FA}_{pqr}t}+\cdots \,. & & \displaystyle\end{eqnarray}$$

Here, the coefficient $S_{pqr}$ is symmetric in its arguments and can be derived from (2.33) for the enstrophy and similarly for the energy. However, as we shall see later, this term will be multiplied by zero at this order of the asymptotics, and so the exact form is not needed for our purpose and we omit it here. We differentiate this in time, and, because it is a conserved quantity, this derivative must be equal to zero at each order. The lowest-order terms are

(3.33) $$\begin{eqnarray}\displaystyle \dot{{\mathcal{E}}}^{(2)} & = & \displaystyle \frac{1}{2}\mathop{\sum }_{pqr}E_{p}C_{p}^{qr}\unicode[STIX]{x1D70E}_{p}^{\ast }\unicode[STIX]{x1D70E}_{q}^{\ast }\unicode[STIX]{x1D70E}_{r}^{\ast }\text{e}^{\text{i}\unicode[STIX]{x1D6FA}_{pqr}t}+\text{c.c.}+\cdots =0,\end{eqnarray}$$

(3.34) $$\begin{eqnarray}\displaystyle \dot{{\mathcal{E}}}^{(3)} & = & \displaystyle \frac{1}{6}\mathop{\sum }_{pqr}\text{i}\unicode[STIX]{x1D6FA}_{pqr}S_{pqr}\unicode[STIX]{x1D70E}_{p}^{\ast }\unicode[STIX]{x1D70E}_{q}^{\ast }\unicode[STIX]{x1D70E}_{r}^{\ast }\text{e}^{\text{i}\unicode[STIX]{x1D6FA}_{pqr}t}+\cdots =0.\end{eqnarray}$$

Here, we have used (3.28) to express the time derivative as $\unicode[STIX]{x1D70E}$ terms.

On summing the different permutations over a chosen triad (a, b, c) and then repeating the method for enstrophy,

(3.35) $$\begin{eqnarray}\displaystyle E_{a}C_{a}^{bc}+E_{b}C_{b}^{ac}+E_{c}C_{c}^{ba} & = & \displaystyle -\text{i}\unicode[STIX]{x1D6FA}_{abc}S_{abc},\end{eqnarray}$$

(3.36) $$\begin{eqnarray}\displaystyle Z_{a}C_{a}^{bc}+Z_{b}C_{b}^{ac}+Z_{c}C_{c}^{ba} & = & \displaystyle -\text{i}\unicode[STIX]{x1D6FA}_{abc}T_{abc}.\end{eqnarray}$$

Here, $T_{abc}$ is defined equivalently to $S_{abc}$ . We consider the resonant cases $\unicode[STIX]{x1D6FA}_{abc}=0$ , to the first-order time scale. If we initialise our ‘simulation’ with only one triad and we can show that no other triad is excited on our time scale (*), then each fundamental property (energy, enstrophy) must be conserved per triad.

Fast modes have zero enstrophy contribution, $Z_{i}=0$ . Therefore, in the case of two fast modes (b, c) combining to make a slow mode (a), all that remains in (3.36) is $C_{a}^{bc}=0$ .

We still need to prove (*). This is an exercise in how resonant triads affect each other. If we start with one resonant triad, this is equivalent to showing that another triad with at least one mode in common with the initialised triad does not grow in first-order time. All that remains is to consider the two possible cases.

Figure 2.

Schematic diagram to show the two options for overlapping triad interactions. (i) Case 1: contains five modes with one shared mode between the two triads. (ii) Case 2: contains four modes with two shared.

3.3.1 Case 1: triads with a single node overlap

In the first case, we consider two triads (with one common mode occurring in both), i.e. a total of five modes. We show that if the modes of the second triad are initially zero, they must remain zero on that time scale.

The first case is shown schematically in figure 2(i). We have a single mode (labelled c) belonging to both triads (labelled a, b, c and c, d, e). The set of five evolution equations from (3.28) in terms of wave amplitudes and interaction coefficients is as follows:

(3.37a-c ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{a}}{\text{d}t}=C_{a}^{bc}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{b}}{\text{d}t}=C_{b}^{ac}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{c}}{\text{d}t}=C_{c}^{ba}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{a}^{\ast }+C_{c}^{ed}\unicode[STIX]{x1D70E}_{e}^{\ast }\unicode[STIX]{x1D70E}_{d}^{\ast }, & & \displaystyle\end{eqnarray}$$

(3.37d,e ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{d}}{\text{d}t}=C_{d}^{ec}\unicode[STIX]{x1D70E}_{e}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{e}}{\text{d}t}=C_{e}^{dc}\unicode[STIX]{x1D70E}_{d}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast }. & & \displaystyle\end{eqnarray}$$

In the case where only the (a, b, c) triad is non-zero initially ( $\unicode[STIX]{x1D70E}_{d}^{\ast }=0$ , $\unicode[STIX]{x1D70E}_{e}^{\ast }=0$ ), we have

(3.38a-c ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{a}}{\text{d}t}=C_{a}^{bc}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{b}}{\text{d}t}=C_{b}^{ac}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{c}}{\text{d}t}=C_{c}^{ba}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{a}^{\ast }, & & \displaystyle\end{eqnarray}$$

(3.38d,e ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{d}}{\text{d}t}=0,\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{e}}{\text{d}t}=0, & & \displaystyle\end{eqnarray}$$

and so from (3.38a–e ) we see that only the initial triad is evolving on the time scale $t$ . The existence of this type of triad pairing does not affect our ability to isolate a triad.

3.3.2 Case 2: triads with a double node overlap

In the second case, we consider four modes formed into two triads (two modes must occur in both), and we show that if the mode of the second triad is initially zero, then that triad cannot be isolated on this time scale.

Here, two nodes (labelled a, b) overlap between triads labelled (a, b, c) and (a, b, d). These are shown schematically in figure 2(ii). We have a set of four evolution equations as follows:

(3.39a,b ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{a}}{\text{d}t}=C_{a}^{bc}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast }+C_{a}^{bd}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{d}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{b}}{\text{d}t}=C_{b}^{ac}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast }+C_{a}^{ad}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{d}^{\ast }, & & \displaystyle\end{eqnarray}$$

(3.39c,d ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{c}}{\text{d}t}=C_{c}^{ab}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{a}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{d}}{\text{d}t}=C_{d}^{ab}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{b}^{\ast }. & & \displaystyle\end{eqnarray}$$

In the case where only the (a, b, c) triad is non-zero initially ( $\unicode[STIX]{x1D70E}_{d}^{\ast }=0$ ), we have

(3.40a,b ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{a}}{\text{d}t}=C_{a}^{bc}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{b}}{\text{d}t}=C_{b}^{ac}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast }, & & \displaystyle\end{eqnarray}$$

(3.40c,d ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{c}}{\text{d}t}=C_{c}^{ab}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{a}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{d}}{\text{d}t}=C_{d}^{ab}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{b}^{\ast }. & & \displaystyle\end{eqnarray}$$

On the time scale $t$ , both triads are evolving, and so the four-member equation set (3.39a–d ) is needed to describe the motion in $t$ . This means that we cannot isolate a single triad in this scenario. Therefore, we are not able to use the argument (*) from above; a single triad cannot always be considered when triads like these exist, as energy will always pass to other wavenumbers on the time scale being considered.

3.3.3 Application to the two-layer equations

We now apply the theory of the last two subsections to the two-layer equations. The equations exhibit this unusual property of non-isolated (non-trivial) resonant triads because there are two branches of the slow mode ( $\unicode[STIX]{x1D714}_{\pm }=0$ ). We can say that any triad containing a slow mode has a ‘sister’ triad containing the other slow mode. By the argument in § 3.3.2, we therefore cannot isolate most triads for the two-layer analysis. Instead, we isolate the set of nodes that form triads with two overlapping nodes. For fast–fast–slow interactions, this is as above,

(3.41a,b ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{a}}{\text{d}t}=C_{a}^{bc}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast }+C_{a}^{bd}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{d}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{b}}{\text{d}t}=C_{b}^{ac}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{c}^{\ast }+C_{a}^{ad}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{d}^{\ast }, & & \displaystyle\end{eqnarray}$$

(3.41c,d ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{c}}{\text{d}t}=C_{c}^{ab}\unicode[STIX]{x1D70E}_{b}^{\ast }\unicode[STIX]{x1D70E}_{a}^{\ast },\quad \frac{\text{d}\unicode[STIX]{x1D70E}_{d}}{\text{d}t}=C_{d}^{ab}\unicode[STIX]{x1D70E}_{a}^{\ast }\unicode[STIX]{x1D70E}_{b}^{\ast }, & & \displaystyle\end{eqnarray}$$

where modes $c$ and $d$ are slow modes with the same wavenumber but different vertical mode $m$ .

On performing the energy/enstrophy analysis as above from (3.35) and (3.36), we obtain

(3.42) $$\begin{eqnarray}\displaystyle (E_{a}C_{a}^{bc}+E_{b}C_{b}^{ac}+E_{c}C_{c}^{ba})\unicode[STIX]{x1D70E}_{c}^{\ast }+(E_{a}C_{a}^{bd}+E_{b}C_{b}^{ad}+E_{d}C_{d}^{ba})\unicode[STIX]{x1D70E}_{d}^{\ast } & = & \displaystyle 0,\end{eqnarray}$$

(3.43) $$\begin{eqnarray}\displaystyle (Z_{a}^{+}C_{a}^{bc}+Z_{b}^{+}C_{b}^{ac}+Z_{c}^{+}C_{c}^{ba})\unicode[STIX]{x1D70E}_{c}^{\ast }+(Z_{a}^{+}C_{a}^{bd}+Z_{b}^{+}C_{b}^{ad}+Z_{d}^{+}C_{d}^{ba})\unicode[STIX]{x1D70E}_{d}^{\ast } & = & \displaystyle 0,\end{eqnarray}$$

(3.44) $$\begin{eqnarray}\displaystyle (Z_{a}^{-}C_{a}^{bc}+Z_{b}^{-}C_{b}^{ac}+Z_{c}^{-}C_{c}^{ba})\unicode[STIX]{x1D70E}_{c}^{\ast }+(Z_{a}^{-}C_{a}^{bd}+Z_{b}^{-}C_{b}^{ad}+Z_{d}^{-}C_{d}^{ba})\unicode[STIX]{x1D70E}_{d}^{\ast } & = & \displaystyle 0,\end{eqnarray}$$

where $Z_{i}^{m}$ is the quadratic part of the enstrophy in the $m$ vertical mode; we have an enstrophy conservation law for each of these.

Using $Z_{a}^{\pm }=0$ , $Z_{b}^{\pm }=0$ (fast modes have zero enstrophy), we have

(3.45) $$\begin{eqnarray}\displaystyle Z_{c}^{m_{c}}C_{c}^{ba}\unicode[STIX]{x1D70E}_{c}^{\ast }+Z_{d}^{m_{c}}C_{d}^{ba}\unicode[STIX]{x1D70E}_{d}^{\ast } & = & \displaystyle 0,\end{eqnarray}$$

(3.46) $$\begin{eqnarray}\displaystyle Z_{c}^{m_{d}}C_{c}^{ba}\unicode[STIX]{x1D70E}_{c}^{\ast }+Z_{d}^{m_{d}}C_{d}^{ba}\unicode[STIX]{x1D70E}_{d}^{\ast } & = & \displaystyle 0.\end{eqnarray}$$

Then, we solve this linear system recalling that $Z_{c}^{m_{i}}\neq 0$ , $Z_{d}^{m_{i}}\neq 0$ and $Z_{c}^{m_{i}}Z_{d}^{-m_{i}}-Z_{c}^{-m_{i}}Z_{d}^{m_{i}}\neq 0$ . It follows that the interaction coefficients here must then both be zero. This implies that the fast–fast–slow interaction coefficient value of zero is a direct result of conservation of quadratic enstrophy.

3.4 The fast–fast–fast interactions

We now return to a key point raised at the end of § 3.2: the fast–fast–fast resonances. These are a clear difference from the single-layer version of the shallow water equations, where they cannot occur. We now consider where these resonances are permitted by the dispersion relations. The fast–fast–fast resonances were originally considered by Ball (Reference Ball1964) for the simpler case with no Coriolis force. Figure 3 shows graphically how these resonances can exist (this graphical method was discovered independently by researchers in different fields, for example Ziman (Reference Ziman1960) and Ball (Reference Ball1964)). In addition to the graphical method, the full derivation of the resonances is shown in appendix A. Case (a) in the diagram shows that there are no resonances between waves of the same vertical mode; this is equivalent to the one-layer case.

Figure 3.

Graphical method to find possible resonant triads in the fast–fast–fast interactions. The dark hyperboloid is the manifold on which the $\unicode[STIX]{x1D714}_{1}$ may lie (defined by the dispersion relation for $\boldsymbol{k}_{1}$ ); the light hyperboloid is centred around a chosen $\unicode[STIX]{x1D714}_{1}$ in the first manifold. The light hyperboloid therefore represents all possible solutions $\unicode[STIX]{x1D714}~(=\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2})$ , and any intercept with the dark hyperboloid represents a possible resonance where $\unicode[STIX]{x1D714}$ has the same vertical mode type as $\unicode[STIX]{x1D714}_{1}$ .

We find that the resonances always exist for any combination of fast modes with different vertical modes; see cases (b), (c) and (d) in figure 3. In addition, there is another more unusual resonance (seen in the lower sheet of the light hyperboloid in case (d) in figure 3) where the ratio of the input and output wave speeds is less than 1 ( $c_{m_{2}}/c_{m}<1$ ). Where the output wave is an external wave and one of the inputs is internal, if we then consider sufficiently large values of the Burger number for the external mode ( $Bu=c_{+}^{2}|\boldsymbol{k}_{1}|^{2}/f^{2}=L_{r}^{2}/L^{2}$ ), this resonance will exist. This condition corresponds to wavelengths at least $\sqrt{3}$ times smaller than the radius of deformation (see appendix A).

Alternatively, the equivalent resonance also exists where the two input waves are of different type from the output. Here, the ratio of input to output wave speeds needs to be greater than 1, and so the input waves are both external modes. We then require that the wavelength of one input mode is such that the external Burger number is sufficiently large, $Bu=c_{+}^{2}|\boldsymbol{k}_{1}|^{2}/f^{2}$ .

These resonances are unusual in that they only exist for angles of incidence within a range around $\pm \unicode[STIX]{x03C0}$ . Figure 4 shows the intersections of the surfaces from figure 3(d) projected into the ( $k$ , $l$ ) plane, and shows more clearly the angle dependence of the resonance. It can be seen that without the Coriolis force, the angle of incidence is in a range ( $-\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x03C0}/2$ ), but as the Coriolis force becomes more dominant, the range of angles is limited to be closer to $-\unicode[STIX]{x03C0}$ .

Figure 4.

Projection onto wave space of the intersections of the surfaces in figure 3(d), showing the two sets of resonances. (a) The non-rotating case, equivalent to the diagram in Ball (Reference Ball1964). (b) The case of large external Burger number. (c) The case of Burger number close to its critical value, such that the angle of incidence in the resonance must be small.

Figure 5 shows a possible mechanism based on an input of fast modes at a high wavenumber in the external mode with background modes at all wavenumbers. Via the directional resonance with low-wavenumber external waves, this would excite internal fast modes at a similar wavenumber to the initial input. These two high-wavenumber modes are then able to interact with greater strength to stimulate the low-wavenumber external modes, reinforcing the mechanism. These three regions would interact similarly to an exact triad, with energy passed among themselves, with the lower-amplitude parts gaining energy in a similar mechanism to that in Hasselmann’s criterion for wave growth (Hasselmann Reference Hasselmann1967).

Figure 5.

Diagram to show heuristically an example of the directional resonance discussed in § 3.4, on the traces shown in figure 4(c). We consider initially that there is a higher proportion of wave energy in external waves of high wavenumber at 1. This can interact via the directional resonance to stimulate a region in the internal modes at 2. Regions 1 and 2 also resonate, and so the waves in region 3 are augmented. The resonances in all three areas stimulate each other, and a mechanism similar to Hasselmann’s criterion will control the dynamics between these regions until other resonances (such as the resonance shown in grey) spread the wave energy out to other areas of wave space.

The full equations describing the first closure are

(3.47) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{0m}}{\unicode[STIX]{x2202}t}+\mathop{\sum }_{1,2}C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ 0,0,0 \\ m_{1}m_{2}m}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{0m_{1}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{2}}^{0m_{2}}\unicode[STIX]{x1D6FF}_{\boldsymbol{k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}}=0,\end{eqnarray}$$

(3.48) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{\unicode[STIX]{x1D6FC}m}}{\unicode[STIX]{x2202}t}+\mathop{\sum }_{1,2}C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ 0,\unicode[STIX]{x1D6FC}_{2},\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{0m_{1}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}~\unicode[STIX]{x1D6FF}_{\boldsymbol{k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{2}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\mathop{\sum }_{1,2}C_{\boldsymbol{k}_{1}\boldsymbol{k}_{2}\boldsymbol{k}}^{\substack{ \unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2},\unicode[STIX]{x1D6FC} \\ m_{1}m_{2}m}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{1}}^{\unicode[STIX]{x1D6FC}_{1}m_{1}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}_{2}}^{\unicode[STIX]{x1D6FC}_{2}m_{2}}~\unicode[STIX]{x1D6FF}_{\boldsymbol{k}-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{1}-\unicode[STIX]{x1D714}_{2}}=0,\end{eqnarray}$$

with the interaction coefficients from § 3.2.

In order to examine the strength of the new resonance within the full equation (3.48), we position the fast–fast–fast resonance against the fast–slow–fast resonance by numerical evaluation of the size of the interaction coefficient in the case of each triad for the same given wavelengths. The parameters are chosen to be physically applicable in an oceanic context, with typical wave speeds of orders ${\sim}100~\text{m}~\text{s}^{-1},10~\text{m}~\text{s}^{-1}$ for the barotropic and baroclinic parts respectively. Wavenumbers are then chosen such that the fast–fast–fast triad is exactly at resonance and the fast–slow–fast triad is close to resonance. Sets of wavenumbers are chosen with $\unicode[STIX]{x1D703}\sim 1^{\circ }$ , $\unicode[STIX]{x1D703}\sim 45^{\circ }$ and $\unicode[STIX]{x1D703}\sim 176^{\circ }$ , where $\unicode[STIX]{x1D703}$ is the angle between the input and output barotropic modes. This is to check that this angle does not affect the size of the interaction, to pick out an interaction lying on each of the two branches of the traces previously sketched in figure 4 and to have examples with barotropic wavelengths larger and smaller than the baroclinic one. For reference, the wavelength of the $(500,0)$ barotropic mode would be ${\sim}200$  m, that of the $(-15.21,10)$ baroclinic/barotropic mode would be ${\sim}5.5$  km and that of the $(-967.25,30)$ mode would be ${\sim}100$  m.

In table 1, the maximum absolute values of the different interaction coefficients in any permutation of mixed vertical modes are given, for mixed vertical modes on lines 1, 3 and 5 and for a single vertical mode on lines 2, 4, 6. The $\unicode[STIX]{x1D6FC}$ parts of the modes correspond respectively to the wavenumbers given in the left-hand column, with the vertical mode information $m$ quoted with the wavenumber. Fast–fast–fast interactions within a single vertical mode are not displayed as this interaction does not occur. Assuming that all mode amplitudes are within an order of magnitude of one another, this should scale like the change in time of each part of the reduced equations (from (3.48)). We find that both interactions have similar orders of magnitude. However, it should be noted that if we choose a single vertical mode for all of the constituent modes (lines 2, 4 and 8), a larger interaction coefficient can occur for the catalytic case. This suggests that although the overall dynamics may be dominated by the same resonances as present in the one-layer case, the new resonance is important in evaluating energy exchange between the baroclinic and barotropic modes. The difference in amplitude between the single- and mixed-mode cases is almost entirely due to the $A_{m_{1}m_{2}}^{m}$ term (from (2.25)), and for this parameter set, the coupling between the modes is weak due to the density ratio $r_{\unicode[STIX]{x1D70C}}\sim 1$ . This relates to the rigid lid limit: from (2.37), we see that for similar densities $L_{-}\sim -H_{2}/H_{1}$ and so for the mixed-mode layers, substitution into $A_{m_{1}m_{2}}^{m}$ leads to almost exact cancellation. For applications with a smaller density ratio, it would be expected that the mixed-mode terms would become a more significant part of the overall flow.

Table 1.

Size of the interaction coefficients for the given wavenumbers, chosen to form a resonant triad of fast–fast–fast modes and a near-resonant triad for sets of fast–slow–fast modes. This shows that for the barotropic/baroclinic interactions, the fast–fast–fast resonance is of similar magnitude to the fast–slow–fast one. However, the dominant interactions will still be those occurring within a single vertical mode. The physical parameters used were as follows: $g=10~\text{m}~\text{s}^{-2}$ , $f=0.0001~\text{s}^{-1}$ , $H_{1}=500~\text{m}$ , $H_{2}=4000~\text{m}$ , $L=100/2\unicode[STIX]{x03C0}~\text{km}$ ; the non-dimensional wavenumbers are quoted as $\boldsymbol{k}$ , where the physical wavenumber is $\boldsymbol{k}/L$ and the physical wavelength is $2\unicode[STIX]{x03C0}L/|\boldsymbol{k}|$ .