3.1 The MMSE

If we impose on $L$ the constraints $\boldsymbol{q}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ in ${\mathcal{D}}$ and $g\unicode[STIX]{x1D702}=\text{i}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D719}$ on $z=0$, by substituting for $\boldsymbol{q}$ and $\unicode[STIX]{x1D702}$ in (3.4), then ${\mathcal{L}}$ reduces to a functional depending on $\unicode[STIX]{x1D719}$ only, namely

(3.5)$$\begin{eqnarray}{\mathcal{L}}_{1}(\unicode[STIX]{x1D719})=\unicode[STIX]{x1D70C}\iint _{{\mathcal{D}}}\left\{-\frac{1}{2}\int _{-h}^{0}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\,\text{d}z+\frac{1}{2}\unicode[STIX]{x1D705}[\unicode[STIX]{x1D719}^{2}]_{z=0}\right\}\,\text{d}x\,\text{d}y+\unicode[STIX]{x1D70C}\int _{{\mathcal{C}}}\boldsymbol{n}\boldsymbol{\cdot }\!\!\int _{-h}^{0}\unicode[STIX]{x1D719}\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\,\text{d}z\,\text{d}c,\end{eqnarray}$$

in which $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D70E}^{2}/g$. The natural conditions of $\unicode[STIX]{x1D6FF}{\mathcal{L}}_{1}=0$ for arbitrary variations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}$ are those elements of (2.2)–(2.4) not used as constraints, expressed in the familiar form

(3.6a-c)$$\begin{eqnarray}\unicode[STIX]{x1D735}^{2}\unicode[STIX]{x1D719}=0\quad (-h<z<0),\quad \unicode[STIX]{x1D719}_{z}-\unicode[STIX]{x1D705}\unicode[STIX]{x1D719}=0\quad (z=0),\quad \unicode[STIX]{x1D719}_{z}+\boldsymbol{q}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h=0\quad (z=-h),\end{eqnarray}$$

for $x,y\in {\mathcal{D}}$ (again disregarding the terms on ${\mathcal{C}}$). The velocity field and free surface elevation are recovered from the respective applied constraints, $\boldsymbol{q}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D702}=\text{i}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D719}/g$ on $z=0$. Apart from the term on ${\mathcal{C}}$, the functional $-{\mathcal{L}}_{1}$ was that used by Chamberlain & Porter (Reference Chamberlain and Porter1995) as the basis for deriving the MMSE.

That equation follows by using the approximation $\unicode[STIX]{x1D719}(x,y,z)\approx \hat{\unicode[STIX]{x1D719}}(x,y,z)$ in (3.5), where

(3.7)$$\begin{eqnarray}\hat{\unicode[STIX]{x1D719}}(x,y,z)=\unicode[STIX]{x1D719}_{1}(x,y)Z_{1}(h(x,y),z),\quad Z_{1}(h,z)=\frac{\cosh k(z+h)}{\cosh (kh)},\quad \unicode[STIX]{x1D705}=k\tanh (kh).\end{eqnarray}$$

This form is based on the case $h=\text{constant}$, in which $\hat{\unicode[STIX]{x1D719}}$ is the exact progressing wave solution for the two-dimensional problem in the $(x,z)$ plane with $\unicode[STIX]{x1D719}_{1}(x)=\exp (\pm \text{i}kx)$. For varying $h$, $\unicode[STIX]{x1D719}_{1}(x,y)$ represents travelling waves modulated by the depth variations, the dispersion relation in (3.7) now having the solution $k(x,y)=k(h(x,y))$. Optimising $\unicode[STIX]{x1D719}_{1}$ by using the variational principle $\unicode[STIX]{x1D6FF}{\mathcal{L}}_{1}(\hat{\unicode[STIX]{x1D719}})=0$ leads to the MMSE in the form

(3.8)$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }(||Z_{1}||^{2}\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D719}_{1})+\{k^{2}||Z_{1}||^{2}+\langle Z_{1},\!\widetilde{\unicode[STIX]{x1D735}}^{2}Z_{1}\rangle +\widetilde{\unicode[STIX]{x1D735}}h.[Z_{1}\widetilde{\unicode[STIX]{x1D735}}Z_{1}]_{z=-h}\}\unicode[STIX]{x1D719}_{1}=0.\end{eqnarray}$$

The version given by Chamberlain & Porter (Reference Chamberlain and Porter1995) follows on using $\widetilde{\unicode[STIX]{x1D735}}Z_{1}={\dot{Z}}_{1}\widetilde{\unicode[STIX]{x1D735}}h$ and $\widetilde{\unicode[STIX]{x1D735}}^{2}Z_{1}={\dot{Z}}_{1}\widetilde{\unicode[STIX]{x1D735}}^{2}h+\ddot{Z}_{1}(\widetilde{\unicode[STIX]{x1D735}}h)^{2}$ to evaluate the coefficients, the abbreviations

$$\begin{eqnarray}{\dot{Z}}(h,z)=\unicode[STIX]{x2202}Z/\unicode[STIX]{x2202}h,\quad \langle f,g\rangle =\int _{-h}^{0}f\overline{g}\,\text{d}z,\quad \langle f,f\rangle =||f||^{2},\end{eqnarray}$$

having been introduced. The inner products needed to make (3.8) explicit are given in appendix A.

The presence of $\widetilde{\unicode[STIX]{x1D735}}^{2}h$ in the coefficient of $\unicode[STIX]{x1D719}_{1}$ in (3.8) shows that jump conditions must replace (3.8) at locations where the bed slope $\widetilde{\unicode[STIX]{x1D735}}h$ is discontinuous. It is clear that the slope of $\unicode[STIX]{x1D719}_{1}$ must be discontinuous at such locations, implying that the free surface slope and horizontal velocity components are discontinuous there. The jump conditions are given in Porter & Staziker (Reference Porter and Staziker1995) where they are shown to be direct consequences of Hamilton’s principle, as one would expect. The local unphysical discontinuities are simply the result of the approximation.

3.2 The CMSE

Referring again to (2.2)–(2.4), we may impose constraints complementary to those that lead to ${\mathcal{L}}_{1}$ and the MMSE, that is, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}=0$ in ${\mathcal{D}}$, $\text{i}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D702}+w=0$ on $z=0$ and $w+\boldsymbol{q}\boldsymbol{\cdot }\unicode[STIX]{x1D735}h=0$ on $z=-h$. In this case the constraints cannot be applied simply by substitution but have to be satisfied by constructing the vector streamfunction

$$\begin{eqnarray}\unicode[STIX]{x1D733}(x,y,z)=\int _{-h}^{z}\boldsymbol{q}(x,y,\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709},\end{eqnarray}$$

with the properties $(u,v,0)=\unicode[STIX]{x2202}\unicode[STIX]{x1D733}/\unicode[STIX]{x2202}z$ and $w=-\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\unicode[STIX]{x1D733}$, and defined so that $z=-h$ is a streamline. The modified functional (3.2) in this case is

$$\begin{eqnarray}{\mathcal{L}}_{2}(\unicode[STIX]{x1D733})=\textstyle \frac{1}{2}\unicode[STIX]{x1D70C}\iint _{{\mathcal{D}}}\left\{\int _{-h}^{0}\{(\unicode[STIX]{x2202}\unicode[STIX]{x1D733}/\unicode[STIX]{x2202}z)^{2}+(\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\unicode[STIX]{x1D733})^{2}\}\,\text{d}z-\unicode[STIX]{x1D705}^{-1}[(\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\unicode[STIX]{x1D733})^{2}]_{z=0}\right\}\text{d}x\,\text{d}y\end{eqnarray}$$

and the natural conditions of $\unicode[STIX]{x1D6FF}{\mathcal{L}}_{2}=0$ for arbitrary variations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D733}$ are, of course, those not applied as constraints, expressed in terms of $\unicode[STIX]{x1D733}$ as

(3.9a,b)$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D735}}(\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\unicode[STIX]{x1D733})+(\unicode[STIX]{x2202}\unicode[STIX]{x1D733}/\unicode[STIX]{x2202}z)^{2}=0\quad (-h<z<0),\quad \unicode[STIX]{x1D705}(\unicode[STIX]{x2202}\unicode[STIX]{x1D733}/\unicode[STIX]{x2202}z)+\widetilde{\unicode[STIX]{x1D735}}(\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\unicode[STIX]{x1D733})=0\quad (z=0),\end{eqnarray}$$

ignoring the contributions on ${\mathcal{C}}$, as usual, and for $(x,y)\in {\mathcal{D}}$. The original variables may be recovered from $\unicode[STIX]{x1D733}$ and, in particular, $\unicode[STIX]{x1D702}=-\text{i}[\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\unicode[STIX]{x1D733}]_{z=0}/\unicode[STIX]{x1D70E}$ gives the free surface elevation. Kim & Bai (Reference Kim and Bai2004) used a real-valued version of the functional ${\mathcal{L}}_{2}$ to construct the CMSE.

In two dimensions with $\unicode[STIX]{x1D733}=(\unicode[STIX]{x1D713},0)$ and $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}(x,z)$ the equations (3.9) take the more familiar forms

(3.10a,b)$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D735}}^{2}\unicode[STIX]{x1D713}=0\quad (-h<z<0),\quad \unicode[STIX]{x1D705}\unicode[STIX]{x1D713}_{z}-\unicode[STIX]{x1D713}_{zz}=0\quad (z=0),\end{eqnarray}$$

and it follows that $\unicode[STIX]{x1D713}(x,z)=\exp (\pm \text{i}kx)\sinh k(h+z)$ with $\unicode[STIX]{x1D705}=k\tanh (kh)$ if $h$ is constant. On this basis Kim & Bai (Reference Kim and Bai2004) took the approximation $\unicode[STIX]{x1D733}(x,y,z)\approx \hat{\unicode[STIX]{x1D733}}(x,y,z)$ that parallels (3.7), in the form

(3.11a,b)$$\begin{eqnarray}\hat{\unicode[STIX]{x1D733}}(x,y,z)=\unicode[STIX]{x1D74D}_{1}(x,y)Z_{2}(h(x,y),z),\quad Z_{2}(h,z)=\frac{\sinh k(h+z)}{\sinh (kh)},\quad \unicode[STIX]{x1D705}=k\tanh (kh),\end{eqnarray}$$

and using $\unicode[STIX]{x1D6FF}{\mathcal{L}}_{2}(\hat{\unicode[STIX]{x1D733}})=\mathbf{0}$ leads to the equation

(3.12)$$\begin{eqnarray}\displaystyle & & \displaystyle \widetilde{\unicode[STIX]{x1D735}}\{(||Z_{2}||^{2}-\unicode[STIX]{x1D705}^{-1})\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\unicode[STIX]{x1D74D}_{1}+\langle Z_{2},{\dot{Z}}_{2}\rangle \widetilde{\unicode[STIX]{x1D735}}h\boldsymbol{\cdot }\unicode[STIX]{x1D74D}_{1}\}-\{\langle Z_{2},{\dot{Z}}_{2}\rangle \widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\unicode[STIX]{x1D74D}_{1}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\langle {\dot{Z}}_{2},{\dot{Z}}_{2}\rangle \widetilde{\unicode[STIX]{x1D735}}h.\unicode[STIX]{x1D74D}_{1}\}\widetilde{\unicode[STIX]{x1D735}}h+k^{2}(||Z_{2}||^{2}-\unicode[STIX]{x1D705}^{-1})\unicode[STIX]{x1D74D}_{1}=\mathbf{0},\end{eqnarray}$$

with jump conditions applying at discontinuities in the bed slope (which will be referred to later). The version (3.12) of the CMSE is that given by Toledo & Agnon (Reference Toledo and Agnon2010), correcting a misprint in Kim & Bai (Reference Kim and Bai2004).

The main contribution of Toledo & Agnon (Reference Toledo and Agnon2010) was to convert (3.12) into a scalar form which they described as the pseudo-potential mild-slope equation and labelled the PMSE, namely,

(3.13)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }B^{-1}\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D6FA}+\unicode[STIX]{x1D6FA}=0,\\ \\ \\ B^{-1}=\{k^{2}(k^{2}+\unicode[STIX]{x1D6FC}h_{x}^{2}+\unicode[STIX]{x1D6FC}h_{y}^{2})\}^{-1}\left(\begin{array}{@{}cc@{}}k^{2}+\unicode[STIX]{x1D6FC}h_{y}^{2} & -\unicode[STIX]{x1D6FC}h_{x}h_{y}\\ -\unicode[STIX]{x1D6FC}h_{x}h_{y} & k^{2}+\unicode[STIX]{x1D6FC}h_{x}^{2}\end{array}\right),\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}(h)$ is a known function which is given after (4.19). The corresponding approximation to the free surface elevation is given in terms of the pseudo-potential $\unicode[STIX]{x1D6FA}$ by

(3.14a,b)$$\begin{eqnarray}\unicode[STIX]{x1D702}\approx -\frac{1}{\text{i}\unicode[STIX]{x1D70E}}\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }sB^{-1}\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D6FA},\quad s(h)=\sqrt{\frac{\sinh (2kh)}{\sinh (2kh)+2kh}}.\end{eqnarray}$$

In addition to the obvious reduction in the computational cost of using the PMSE rather than the CMSE, Toledo & Agnon (Reference Toledo and Agnon2010) observed that their scalar equation removes an unresolved difficulty posed by the vector CMSE, which requires an additional unspecified condition on lateral boundaries in the three-dimensional case.

4.1 A basic mild-slope equation

To examine the effect of using the variational principle $\unicode[STIX]{x1D6FF}{\mathcal{L}}=0$ with free field equations, we first take the set of minimal approximations

(4.1a-e)$$\begin{eqnarray}\unicode[STIX]{x1D719}\approx \hat{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D719}_{1}Z_{1},\quad u\approx \hat{u} =u_{1}Z_{1},\quad v\approx \hat{v}=v_{1}Z_{1},\quad w\approx {\hat{w}}=w_{1}Z_{2},\quad \unicode[STIX]{x1D702}\approx \hat{\unicode[STIX]{x1D702}}=\unicode[STIX]{x1D702}_{1},\end{eqnarray}$$

where the functions $\unicode[STIX]{x1D719}_{1}$, $u_{1}$, $v_{1}$ and $w_{1}$ depend on $x$ and $y$ only (as do $u_{2}$, $v_{2}$ and $w_{2}$ later). In the spirit of the MMSE, we apply the constraint $(g\hat{\unicode[STIX]{x1D702}}-\text{i}\unicode[STIX]{x1D70E}\hat{\unicode[STIX]{x1D719}})_{z=0}=0$, implying that

(4.2)$$\begin{eqnarray}\unicode[STIX]{x1D702}\approx \unicode[STIX]{x1D702}_{1}=\text{i}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D719}_{1}/g\end{eqnarray}$$

gives the approximation to the surface elevation.

Using (3.3), the application of $\unicode[STIX]{x1D6FF}{\mathcal{L}}=0$ for arbitrary variations $\unicode[STIX]{x1D6FF}u_{1}$, $\unicode[STIX]{x1D6FF}v_{1}$, $\unicode[STIX]{x1D6FF}w_{1}$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}_{1}\,(=(g/\text{i}\unicode[STIX]{x1D70E})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}_{1})$, leads to the natural conditions

$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\langle \hat{u} -\hat{\unicode[STIX]{x1D719}}_{x},Z_{1}\rangle =0,\quad \langle \hat{v}-\hat{\unicode[STIX]{x1D719}}_{y},Z_{1}\rangle =0,\quad \langle {\hat{w}}-\hat{\unicode[STIX]{x1D719}}_{z},Z_{2}\rangle =0,\\ \langle \hat{u} _{x}+\hat{v}_{y}+{\hat{w}}_{z},Z_{1}\rangle +(h_{x}u_{1}+h_{y}v_{1})\,\text{sech}^{2}(kh)-\text{i}\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D702}_{1}+w_{1}/\text{i}\unicode[STIX]{x1D70E})=0.\end{array}\right\}\end{eqnarray}$$

Substituting from (4.1) and (4.2) reduces these equations to

(4.3)$$\begin{eqnarray}\displaystyle & \displaystyle w_{1}=\unicode[STIX]{x1D705}\unicode[STIX]{x1D719}_{1}, & \displaystyle\end{eqnarray}$$

(4.4)$$\begin{eqnarray}\displaystyle & \displaystyle \left.\begin{array}{@{}c@{}}||Z_{1}||^{2}\,{\unicode[STIX]{x1D719}_{1}}_{x}+\langle {Z_{1}}_{x},Z_{1}\rangle \unicode[STIX]{x1D719}_{1}=||Z_{1}||^{2}\,u_{1},\\ \\ \\ ||Z_{1}||^{2}\,{\unicode[STIX]{x1D719}_{1}}_{y}+\langle {Z_{1}}_{y},Z_{1}\rangle \unicode[STIX]{x1D719}_{1}=||Z_{1}||^{2}\,v_{1},\\ \end{array}\right\} & \displaystyle\end{eqnarray}$$

and

(4.5)$$\begin{eqnarray}\displaystyle & & \displaystyle ||Z_{1}||^{2}\,{u_{1}}_{x}+\{\langle {Z_{1}}_{x},Z_{1}\rangle +\text{sech}^{2}(kh)h_{x}\}u_{1}\nonumber\\ \displaystyle & & \displaystyle \quad +\,||Z_{1}||^{2}\,{v_{1}}_{y}+\{\langle {Z_{1}}_{y},Z_{1}\rangle +\text{sech}^{2}(kh)h_{y}\}v_{1}+k^{2}||Z_{1}||^{2}\,\unicode[STIX]{x1D719}_{1}=0,\end{eqnarray}$$

which has made use of (4.3).

Our objective now is to deduce an equation for $\unicode[STIX]{x1D719}_{1}$ by eliminating $u_{1}$ and $v_{1}$ between (4.4) and (4.5), and this is less onerous to achieve than it might appear to be. We first differentiate $||Z_{1}||^{2}=\langle Z_{1},Z_{1}\rangle$ and $\unicode[STIX]{x1D705}=k\tanh (kh)$ to give

$$\begin{eqnarray}2||Z_{1}||||Z_{1}||_{x,\,y}=2\langle {Z_{1}}_{x,\,y},Z_{1}\rangle +\text{sech}^{2}(kh)h_{x,\,y},\quad 2||Z_{1}||^{2}k_{x,\,y}=-k\text{sech}^{2}(kh)h_{x,\,y},\end{eqnarray}$$

respectively. These equations combine to provide the identity

(4.6)$$\begin{eqnarray}\frac{\langle Z_{1},{Z_{1}}_{x,y}\rangle }{||Z_{1}||^{2}}=\frac{||Z_{1}||_{x,y}}{||Z_{1}||}+\frac{k_{x,y}}{k}=\frac{(k||Z_{1}||)_{x,y}}{(k||Z_{1}||)},\end{eqnarray}$$

which can be used to generate integrating factors for the equations (4.4), converting them to

(4.7a,b)$$\begin{eqnarray}(k||Z_{1}||\unicode[STIX]{x1D719}_{1})_{x}=k||Z_{1}||u_{1},\quad (k||Z_{1}||\unicode[STIX]{x1D719}_{1})_{y}=k||Z_{1}||v_{1},\end{eqnarray}$$

and transforming (4.5) similarly into

$$\begin{eqnarray}(k^{-1}||Z_{1}||u_{1})_{x}+(k^{-1}||Z_{1}||v_{1})_{y}+k||Z_{1}||\unicode[STIX]{x1D719}_{1}=0.\end{eqnarray}$$

The elimination of $u_{1}$ and $v_{1}$ is now immediate, giving

(4.8)$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }(k^{-2}\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711})+\unicode[STIX]{x1D711}=0,\quad \text{where }\unicode[STIX]{x1D711}=k||Z_{1}||\unicode[STIX]{x1D719}_{1}.\end{eqnarray}$$

This simple equation of the mild-slope type has not previously been derived, evidently because the corresponding approximation satisfies neither $\hat{\boldsymbol{q}}=\unicode[STIX]{x1D735}\hat{\unicode[STIX]{x1D719}}$ nor $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{q}}=0$ for general values of $kh$, which can readily be confirmed. As (4.1) cannot be reduced in a meaningful way, there is an argument for regarding (4.8) as the fundamental mild-slope equation. We note from (4.3) that the approximation to the free surface elevation is given by

$$\begin{eqnarray}\unicode[STIX]{x1D702}\approx \unicode[STIX]{x1D702}_{1}=\frac{\text{i}\unicode[STIX]{x1D70E}}{g}\frac{\unicode[STIX]{x1D711}}{k||Z_{1}||},\quad \text{where }k||Z_{1}||=\sqrt{k(2kh+\sinh (2kh))}/2\cosh (kh).\end{eqnarray}$$

Although (4.8) is new, there is a connection between it and existing work, as Porter (Reference Porter2003) transformed the MMSE into an equation that may be written in the present notation as

(4.9)$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }(k^{-2}\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711})+\{1-v(h)(\widetilde{\unicode[STIX]{x1D735}}h)^{2}\}\,\unicode[STIX]{x1D711}=0.\end{eqnarray}$$

It is shown in Porter (Reference Porter2003) that

$$\begin{eqnarray}v(h)=(||Z_{1}||^{2}||{\dot{Z}}_{1}||^{2}-\langle Z_{1},{\dot{Z}}_{1}\rangle ^{2})/(k^{2}||Z_{1}||^{4}),\end{eqnarray}$$

satisfies $0\leqslant v(h)<0.030$, and therefore the final term in (4.9) is virtually negligible for slowly varying bedforms and the solutions of (4.8) and (4.9) are effectively identical. It also follows from the earlier work that $v(h)\rightarrow 0$ as $kh\rightarrow 0$ and that $v(h)\sim O(\text{e}^{-2kh})$ as $kh\rightarrow \infty$.

We note that the solutions of both (4.8) and (4.9) are continuous and have continuous derivatives where the bed slope is discontinuous, the discontinuity in the slope of $\unicode[STIX]{x1D719}_{1}$ there arising through the scaling term in $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D711}/k||Z_{1}||$.

4.2 The MMSE

To derive the MMSE using the present approach, that is, without applying $\boldsymbol{q}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ as a constraint in the variational principle, requires a modification of the approximation (4.1). We note from (4.3) that ${\hat{w}}=\hat{\unicode[STIX]{x1D719}}_{z}$ is a consequence of (4.1) but, by (4.4), $\hat{u}$ and $\hat{v}$ need to be extended in order to comply with the condition $\hat{\boldsymbol{q}}=\unicode[STIX]{x1D735}\hat{\unicode[STIX]{x1D719}}$. It is easily checked that ${\dot{Z}}_{1}$ satisfies the same free surface boundary condition as $Z_{1}$, given in (3.6), and therefore the appropriate basis for the required improved approximations to the horizontal velocity components should consist of $Z_{1}$ and ${\dot{Z}}_{1}$. We thus take

(4.10)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D719}\approx \hat{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D719}_{1}Z_{1},\,u\approx \hat{u} =u_{1}Z_{1}+u_{2}{\dot{Z}}_{1},\,v\approx \hat{v}=v_{1}Z_{1}+v_{2}{\dot{Z}}_{1},\\ w\approx {\hat{w}}=w_{1}Z_{2},\,\unicode[STIX]{x1D702}\approx \hat{\unicode[STIX]{x1D702}}=\unicode[STIX]{x1D702}_{1},\end{array}\right\}\end{eqnarray}$$

with $\unicode[STIX]{x1D702}_{1}=\text{i}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D719}_{1}/g$. An application of (3.3) then leads to

$$\begin{eqnarray}\hat{u} =\hat{\unicode[STIX]{x1D719}}_{x},\quad \hat{v}=\hat{\unicode[STIX]{x1D719}}_{y},\quad {\hat{w}}=\hat{\unicode[STIX]{x1D719}}_{z},\quad \langle \hat{u} _{x}+\hat{v}_{y}+{\hat{w}}_{z},Z_{1}\rangle +(\hat{u} h_{x}+\hat{v}h_{y})_{z=-h}\,\text{sech}(kh)=0.\end{eqnarray}$$

By making use of (4.6), these equations readily reduce directly to the MMSE in the concise form (4.9). This is the main virtue of the present approach with regard to the MMSE: it bypasses the previous need to apply the derivations of Chamberlain & Porter (Reference Chamberlain and Porter1995) and Porter (Reference Porter2003) in succession to obtain the practical version (4.9).

A further point to be made here is that the MMSE gives a superior approximation to (4.8) because (4.10) strictly includes (4.1). However, we know from the investigation of Porter (Reference Porter2003) leading to (4.9) that the superiority is effectively negligible.

We conclude that applying $\hat{\boldsymbol{q}}=\unicode[STIX]{x1D735}\hat{\unicode[STIX]{x1D719}}$ at the outset is not necessary to derive the MMSE, provided that the approximation is chosen to allow the condition to be satisfied within the variational process. Using functions of $z$ other than ${\dot{Z}}_{1}$ to extend the basis of $\hat{u}$ and $\hat{v}$ will improve the approximation (4.8) by expanding the approximating set, but $\hat{\boldsymbol{q}}=\unicode[STIX]{x1D735}\hat{\unicode[STIX]{x1D719}}$ will not be satisfied.

4.3 A complementary approximation

By withholding the constraint $\boldsymbol{q}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ but applying the pressure condition on the free surface, therefore, a basic approximation and straightforward calculation lead to the equation (4.8) that is simpler than the MMSE in both its derivation and use. We now follow through the same process for the complementary situation, applying the bed condition and the kinematic condition at the free surface, but not the field equation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}=0$, with the expectation of deriving a version of the CMSE.

We again take (4.1) as the starting point but now we must extend the representation of ${\hat{w}}$ given there by the addition of a term that does not vanish on $z=-h$, in order to satisfy $w+\boldsymbol{q}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h=0$ on $z=-h$. Following the argument just used in relation to the MMSE, we confirm that both $Z_{2}$ and ${\dot{Z}}_{2}$ satisfy the appropriate free surface condition, given in (3.10), and since $({\dot{Z}}_{2})_{z=-h}=k\text{cosech}(kh)$, they together form a natural basis for the approximation of ${\hat{w}}$.

We therefore take

(4.11a-e)$$\begin{eqnarray}\unicode[STIX]{x1D719}\approx \hat{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D719}_{1}Z_{1},\quad u\approx \hat{u} =u_{1}Z_{1},\quad v\approx \hat{v}=v_{1}Z_{1},\quad w\approx {\hat{w}}=w_{1}Z_{2}+w_{2}{\dot{Z}}_{2},\quad \unicode[STIX]{x1D702}\approx \hat{\unicode[STIX]{x1D702}}=\unicode[STIX]{x1D702}_{1},\end{eqnarray}$$

which differs from (4.1) only in the change to ${\hat{w}}$. Imposing the required bed condition ${\hat{w}}+h_{x}\hat{u} +h_{y}\hat{v}=0$ on $z=-h$ readily leads to

$$\begin{eqnarray}{\hat{w}}=w_{1}Z_{2}-\unicode[STIX]{x1D705}k^{-2}\widetilde{v}{\dot{Z}}_{2},\quad \text{where }\widetilde{v}=u_{1}h_{x}+v_{1}h_{y}.\end{eqnarray}$$

Satisfying the second complementary boundary condition requires that $\text{i}\unicode[STIX]{x1D70E}\hat{\unicode[STIX]{x1D702}}+{\hat{w}}=0$ on $z=0$ giving

(4.12)$$\begin{eqnarray}\unicode[STIX]{x1D702}_{1}=-w_{1}/\text{i}\unicode[STIX]{x1D70E},\end{eqnarray}$$

because $({\dot{Z}}_{2})_{z=0}=0$.

The application of $\unicode[STIX]{x1D6FF}{\mathcal{L}}=0$ for arbitrary variations $\unicode[STIX]{x1D6FF}u_{1}$, $\unicode[STIX]{x1D6FF}v_{1}$, $\unicode[STIX]{x1D6FF}w_{1}\,(=\!-\text{i}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}_{1})$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}_{1}$ provides the natural conditions

(4.13)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\langle \hat{u} -\hat{\unicode[STIX]{x1D719}}_{x},Z_{1}\rangle -\unicode[STIX]{x1D705}k^{-2}h_{x}\langle {\hat{w}}-\hat{\unicode[STIX]{x1D719}}_{z},{\dot{Z}}_{2}\rangle =0,\\ \langle \hat{v}-\hat{\unicode[STIX]{x1D719}}_{y},Z_{1}\rangle -\unicode[STIX]{x1D705}k^{-2}h_{y}\langle {\hat{w}}-\hat{\unicode[STIX]{x1D719}}_{z},{\dot{Z}}_{2}\rangle =0,\end{array}\right\}\end{eqnarray}$$

and

(4.14)$$\begin{eqnarray}\langle {\hat{w}}-\hat{\unicode[STIX]{x1D719}}_{z},Z_{2}\rangle -(g\hat{\unicode[STIX]{x1D702}}/\text{i}\unicode[STIX]{x1D70E}-\hat{\unicode[STIX]{x1D719}})_{z=0}=0,\quad \langle \hat{u} _{x}+\hat{v}_{y}+{\hat{w}}_{z},Z_{1}\rangle =0.\end{eqnarray}$$

Equation (4.12) may be used in the first of (4.14) to give

(4.15)$$\begin{eqnarray}a(w_{1}-\unicode[STIX]{x1D705}\unicode[STIX]{x1D719}_{1})-b\unicode[STIX]{x1D705}k^{-2}\widetilde{v}=0,\quad \text{where }a(h)=||Z_{2}||^{2}-\unicode[STIX]{x1D705}^{-1},b(h)=\langle Z_{2},{\dot{Z}}_{2}\rangle .\end{eqnarray}$$

We may now return to the equations (4.13) and, using (4.6) and (4.15), rearrange them in the forms

(4.16)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}(k||Z_{1}||\unicode[STIX]{x1D719}_{1})_{x}=k||Z_{1}||(u_{1}+\unicode[STIX]{x1D6FE}h_{x}\widetilde{v}),\\ (k||Z_{1}||\unicode[STIX]{x1D719}_{1})_{y}=k||Z_{1}||(v_{1}+\unicode[STIX]{x1D6FE}h_{y}\widetilde{v}),\end{array}\right\}\quad \text{where }\unicode[STIX]{x1D6FE}(h)=\frac{1}{k^{2}}\left(\frac{b^{2}-ac}{a^{2}}\right),\,c(h)=||{\dot{Z}}_{2}||^{2}.\end{eqnarray}$$

The identity $a\unicode[STIX]{x1D705}^{2}=-k^{2}||Z_{1}||^{2}$ has simplified $\unicode[STIX]{x1D6FE}(h)$. (The terms $a(h)$, $b(h)$ and $c(h)$ preserve the notation of Kim & Bai (Reference Kim and Bai2004) and Toledo & Agnon (Reference Toledo and Agnon2010); they are given explicitly in the appendix A.)

The second equation of (4.14) easily reduces to

$$\begin{eqnarray}u_{1x}||Z_{1}||^{2}+u_{1}\langle Z_{1},Z_{1x}\rangle +v_{1y}||Z_{1}||^{2}+v_{1}\langle Z_{1},Z_{1y}\rangle +\text{sech}^{2}(kh)\widetilde{v}+k^{2}||Z_{1}||^{2}\unicode[STIX]{x1D719}_{1}=0,\end{eqnarray}$$

which coincides with (4.5) and therefore leads to

(4.17)$$\begin{eqnarray}(||Z_{1}||u_{1}/k)_{x}+(||Z_{1}||v_{1}/k)_{y}+k||Z_{1}||\unicode[STIX]{x1D719}_{1}=0.\end{eqnarray}$$

Finally, we may solve (4.16) for $u_{1}$ and $v_{1}$ to give

$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}k||Z_{1}||u_{1}=\unicode[STIX]{x1D711}_{x}-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6E4}^{-1}(\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h)h_{x},\\ k||Z_{1}||v_{1}=\unicode[STIX]{x1D711}_{y}-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6E4}^{-1}(\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h)h_{y},\end{array}\right\}\quad \text{where }\unicode[STIX]{x1D711}=k||Z_{1}||\unicode[STIX]{x1D719}_{1},\quad \unicode[STIX]{x1D6E4}=1+\unicode[STIX]{x1D6FE}(\widetilde{\unicode[STIX]{x1D735}}h)^{2},\end{eqnarray}$$

and substitution into (4.17) reveals that

(4.18)$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\,k^{-2}\{\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}-\unicode[STIX]{x1D6FE}(h)\unicode[STIX]{x1D6E4}^{-1}(\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h)\widetilde{\unicode[STIX]{x1D735}}h\!)\}+\unicode[STIX]{x1D711}=0,\quad \text{where }\unicode[STIX]{x1D711}=k||Z_{1}||\unicode[STIX]{x1D719}_{1}.\end{eqnarray}$$

This is the complementary counterpart of (4.8), the additional terms here arising from the change in the representation of ${\hat{w}}$ given in (4.11) and the application of the complementary boundary conditions as constraints.

The corresponding approximation to free surface elevation also changes from the MMSE case. To evaluate it we refer back to (4.12) and (4.15) and note from (4.16) that

$$\begin{eqnarray}k||Z_{1}||\widetilde{v}=\unicode[STIX]{x1D6E4}^{-1}\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h.\end{eqnarray}$$

Forming $\unicode[STIX]{x1D702}_{1}$ we find that

(4.19)$$\begin{eqnarray}\unicode[STIX]{x1D702}\approx \unicode[STIX]{x1D702}_{1}=\frac{\text{i}\unicode[STIX]{x1D70E}}{g}\frac{1}{k||Z_{1}||}\left(\unicode[STIX]{x1D711}+\frac{b}{k^{2}a}\,\unicode[STIX]{x1D6E4}^{-1}\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h\right).\end{eqnarray}$$

The function $\unicode[STIX]{x1D6FC}(h)$ occurring in the Toledo & Agnon (Reference Toledo and Agnon2010) version (3.13) of the CMSE is given by $\unicode[STIX]{x1D6FC}(h)=k^{2}\unicode[STIX]{x1D6FE}(h)$, where $\unicode[STIX]{x1D6FE}(h)$ is defined in (4.16). This is the stimulus to showing that the PMSE (3.13) and equation (4.18) are, in fact, identical, the ‘pseudo-potential’ $\unicode[STIX]{x1D6FA}$ in the former being just the scaled version $\unicode[STIX]{x1D711}$ of the present $\unicode[STIX]{x1D719}_{1}$ in the latter. In other words, equation (4.18) is the CMSE itself in a new configuration.

The Toledo & Agnon (Reference Toledo and Agnon2010) approximation (3.14) to the free surface elevation can be shown to coincide with the present (4.19) to within a constant multiplier and it can also be confirmed that the vector $\hat{\boldsymbol{q}}$ resulting from the variational principle does indeed satisfy $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{q}}=0$. This completes the connection between the present work and that of Kim & Bai (Reference Kim and Bai2004), but it should perhaps be emphasised that the derivation of (4.18) does not depend on satisfying $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{q}}=0$ in advance: it is compliance with the kinematic boundary conditions coupled with the choice of $Z_{2}$ and ${\dot{Z}}_{2}$ to achieve this that leads to the CMSE.

The CMSE has apparently not been derived before without using the vector streamfunction as a starting point, although the form (3.13) given by Toledo & Agnon (Reference Toledo and Agnon2010) suggests that an alternative development is possible, nor has it been suspected that the approximations leading to the MMSE and CMSE are in fact closely related. The derivation given here replaces the combination of Kim & Bai (Reference Kim and Bai2004) followed by Toledo & Agnon (Reference Toledo and Agnon2010), which is a significant saving.

We remark for future reference that the coefficient $\unicode[STIX]{x1D6FE}(h)$ in (4.18) (given in expanded form in appendix A) can be written as the function $f(kh)$ and an investigation shows that $f$ is a non-negative, decreasing function satisfying $f(kh)\rightarrow 1/3$ and $f^{\prime }(kh)\rightarrow 0$ as $kh\rightarrow 0$, and $f\sim O(\text{e}^{-2kh})$ as $kh\rightarrow \infty$.

Finally, in this section, we remark that if $\widetilde{\unicode[STIX]{x1D735}}h$ is discontinuous across a simple, smooth curve $L$ within ${\mathcal{D}}$ then (4.18) has to be solved on either side of $L$ and the solutions linked by jump conditions equivalent to the differential equation, namely,

(4.20)$$\begin{eqnarray}[\unicode[STIX]{x1D711}]=0,\quad [\boldsymbol{n}_{L}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6E4}^{-1}(\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h)(\boldsymbol{n}_{L}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h)]=0,\end{eqnarray}$$

where $[\,\,\,]$ denotes the jump in the included quantity across $L$ and $\boldsymbol{n}$ the normal to $L$. We note that $\unicode[STIX]{x1D6E4}$ is discontinuous through its dependence on $\widetilde{\unicode[STIX]{x1D735}}h$. It is assumed that $\unicode[STIX]{x1D711}$ is continuous and the remaining element of (4.20) following from integrating (4.18) over a small domain including $L$, which is then allowed to shrink onto $L$. An alternative, but more laborious, procedure is to build the discontinuity into the variational principle at the outset.

It follows from (4.19) that $\unicode[STIX]{x1D702}_{1}$ is discontinuous across $L$, whereas $\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D702}_{1}$ is discontinuous there in the MMSE. These isolated physically unrealistic features are consequences of the approximation and serve as reminders that it is inconsistent to carry out vertical averaging and then seek local flaws in the vertical structure.

4.4 Further complementary approximations

We have shown that choosing $Z_{2}$ and ${\dot{Z}}_{2}$ as a basis for ${\hat{w}}$ and satisfying the kinematic boundary conditions leads to the CMSE, for which the approximation to the velocity field satisfies $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{q}}=0$. We can, of course, satisfy the kinematic boundary conditions by using other functions to approximate ${\hat{w}}$ and, in particular, we can replace ${\dot{Z}}_{2}$ by a function $Z_{3}$, say, in which case we obtain an equation having the same structure as (4.18) but with a different coefficient $\unicode[STIX]{x1D6FE}(h)$ and a velocity field such that $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{q}}\neq 0$. The form of (4.18) is thus the result of satisfying the bed condition and not the field equation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}=0$.

To add some substance to these remarks, let ${\hat{w}}=w_{1}Z_{2}+w_{2}Z_{3}$ for any $Z_{3}$ that is non-vanishing at $z=-h$, allowing the bed condition to be satisfied as required. We may normalise $Z_{3}$ by choosing $(Z_{3})_{z=0}=0$ (a condition satisfied by ${\dot{Z}}_{2}$) and $(Z_{3})_{z=-h}=1$. Applying $\unicode[STIX]{x1D6FF}{\mathcal{L}}=0$ for this more general approximation to ${\hat{w}}$ and following the procedure given in the previous subsection, we obtain the equation (4.18) with $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6E4}$ replaced by $\unicode[STIX]{x1D6FE}_{1}$ and $\unicode[STIX]{x1D6E4}_{1}$ respectively, where

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{1}(h)=\left(\frac{b_{1}^{2}-ac_{1}}{a^{2}}\right)\,\text{cosech}^{2}(kh),\quad b_{1}=\langle Z_{2},Z_{3}\rangle ,\quad c_{1}=||Z_{3}||^{2},\quad \unicode[STIX]{x1D6E4}_{1}=1+\unicode[STIX]{x1D6FE}_{1}(\widetilde{\unicode[STIX]{x1D735}}h)^{2}.\end{eqnarray}$$

The free surface approximation in this case is

$$\begin{eqnarray}\unicode[STIX]{x1D702}\approx \unicode[STIX]{x1D702}_{1}=\frac{\text{i}\unicode[STIX]{x1D70E}}{g}\frac{1}{k||Z_{1}||}\left(\unicode[STIX]{x1D711}+\frac{b_{1}\,\text{sech}(kh)}{\unicode[STIX]{x1D705}a}\,\unicode[STIX]{x1D6E4}_{1}^{-1}\widetilde{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D711}\boldsymbol{\cdot }\widetilde{\unicode[STIX]{x1D735}}h\right).\end{eqnarray}$$

We can show that the approximation $\hat{\boldsymbol{q}}$ to the velocity field in this case satisfies $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{q}}\neq 0$ unless $Z_{3}={\dot{Z}}_{2}$.

The simplest function we can take for $Z_{3}$ is $Z_{3}=-z/h$, which satisfies the free surface condition in (3.10) as well as the imposed normalisations. The resulting coefficients, given in appendix A, are somewhat simpler than those occurring in the CMSE but the essential form of that equation is unchanged.