2.1 Characteristic scales

The characteristic scales used throughout S19 are $k$ and $c^{[0]}$ , where $k$ is the magnitude of the wavenumber and $c^{[0]}$ is the leading-order term of the intrinsic phase speed $c$ . Some typical example values are $(c^{[0]},k)\approx (3~\text{m}~\text{s}^{-1},1~\text{rad}~\text{m}^{-1})$ , $(9~\text{m}~\text{s}^{-1},0.1~\text{rad}~\text{m}^{-1})$ and $(22~\text{m}~\text{s}^{-1},0.02~\text{rad}~\text{m}^{-1})$ for a wave whose wavelength is 5 m, 50 m and 300 m, respectively. For any variable $\unicode[STIX]{x1D711}$ , its perturbation series is denoted as

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D711}=\mathop{\sum }_{n}\unicode[STIX]{x1D711}^{[n]},\end{eqnarray}$$

where $n$ is an integer index and the superscript $[n]$ denotes the $n$ th term of the series. When the $n$ th term is non-dimensionalised with $k$ and $c^{[0]}$ , it is at most of $O(\unicode[STIX]{x1D716}^{n})$ , where $\unicode[STIX]{x1D716}\equiv 0.1$ in S19. Note that $\unicode[STIX]{x1D716}$ is defined as 0.1 for two reasons. Firstly, specifying a value of $\unicode[STIX]{x1D716}$ is necessary to evaluate the orders of the coefficients contained in some terms of the governing equations. Secondly, this definition keeps the notation simple because it then allows the order of any quantity appearing in S19 to be expressed in terms of $\unicode[STIX]{x1D716}$ . In general, one may wish to introduce multiple scaling symbols: e.g. one for the wave amplitude and another for the current speeds. However, this is unnecessary here because S19 considers only a (typical oceanic) condition where the wave amplitude is of $O(0.1k^{-1})$ and the current speeds are, at most, of $O(0.1^{2}c^{[0]})$ . (Note that common current speeds relevant to oceanic submesoscale and Langmuir circulations are of order of $1{-}10~\text{cm}~\text{s}^{-1}$ (e.g. Suzuki et al. Reference Suzuki, Fox-Kemper, Hamlington and Van Roekel2016), neglecting the intermittent currents that may be produced by wave breaking.) More details are given in § 3.1.

2.2 Flow components

Throughout this paper, the following subscript indices are used for the tensor indices: $H=1,2$ ; $h=1,2$ ; $i=1,2,3$ ; and $\ell =1,2,3,t$ . Here, 1 and 2 are for the horizontal dimensions, 3 is for the vertical dimension, and $t$ is for the time dimension. The Einstein summation convention is used throughout. At any position in the water, the fluid velocity $(u_{1},u_{2},u_{3})\equiv (u,v,w)$ consists solely of the current velocity $(u_{1}^{c},u_{2}^{c},u_{3}^{c})\equiv (u^{c},v^{c},w^{c})$ and the wave velocity $(u_{1}^{w},u_{2}^{w},u_{3}^{w})\equiv (u^{w},v^{w},w^{w})$ : that is, $u_{i}=u_{i}^{c}+u_{i}^{w}$ . Every point of the physical Euclidean space is given both a Cartesian coordinate $(x,y,z)$ and a surface-following coordinate $(x,y,\unicode[STIX]{x1D701})$ , as illustrated in figure 1. The horizontal coordinate system of the surface-following coordinate system is identical to the Cartesian one. The surface labelled with a constant value of $\unicode[STIX]{x1D701}$ follows the vertical displacement due to the wave motion, as detailed in § 2.3.

All velocities are measured with respect to the Cartesian coordinate system, and the directions of $u$ , $v$ and $w$ are the same as the directions of $x$ , $y$ and $z$ , respectively, even when the position of a velocity is indicated with $(x,y,\unicode[STIX]{x1D701})$ . That is, $u_{i}(x,y,\unicode[STIX]{x1D701},t)=u_{i}(x,y,z(x,y,\unicode[STIX]{x1D701},t),t)$ , where $z(x,y,\unicode[STIX]{x1D701},t)$ is the Cartesian vertical coordinate of the position indicated by $(x,y,\unicode[STIX]{x1D701},t)$ . In other words, the velocity of a fluid parcel is described using the Cartesian velocity components, and its position is indicated using a surface-following curvilinear coordinate system. This approach is commonly used in interfacial or boundary-layer problems (e.g. Hsu et al. Reference Hsu, Hsu and Street1981; Hunt, Leibovich & Richards Reference Hunt, Leibovich and Richards1988; Belcher & Hunt Reference Belcher and Hunt1993) involving a surface undulation, when it is desirable to distinguish the momentum (e.g. current) extrinsic to the surface undulation from that (e.g. wave) intrinsic to the surface undulation. This distinction can be often most meaningfully made with respect to a Cartesian basis while taking into account the material-surface displacement induced by the flow intrinsic to the surface undulation. Moreover, this description facilitates interpretation of observational data (Hsu et al. Reference Hsu, Hsu and Street1981). A detailed discussion on the advantages of this approach over other coordinate systems or other velocity descriptions is given in Hsu et al. (Reference Hsu, Hsu and Street1981).

Consider a deep-water surface gravity wave field whose leading-order displacement of the water surface is given by $a(x,y,t)\cos \unicode[STIX]{x1D712}$ . Here, $a$ is the amplitude and $\unicode[STIX]{x1D712}$ is defined as

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D712}\equiv s(x,y,t)-\unicode[STIX]{x1D6E9},\end{eqnarray}$$

where $s(x,y,t)$ is the phase function and $\unicode[STIX]{x1D6E9}$ is the phase shift parameter. The phase shift parameter is a real variable and is independent of the coordinate variables. The wavenumber $k\hat{k}_{h}$ and the apparent frequency $k(c+\hat{k}_{h}u_{h}^{D})$ are defined as

(2.3a,b ) $$\begin{eqnarray}k\hat{k}_{h}\equiv \unicode[STIX]{x2202}_{h}s=\unicode[STIX]{x2202}_{h}\unicode[STIX]{x1D712},\quad k(c+\hat{k}_{h}u_{h}^{D})\equiv -\unicode[STIX]{x2202}_{t}s=-\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D712},\end{eqnarray}$$

where $(\hat{k}_{1},\hat{k}_{2})$ is the unit vector of the wavenumber and $(u_{1}^{D},u_{2}^{D})$ is the Doppler shift velocity. From these definitions, it is evident that $a$ , $k$ , $\hat{k}_{h}$ , $c$ and $u_{h}^{D}$ are independent of $\unicode[STIX]{x1D6E9}$ , $z$ and $\unicode[STIX]{x1D701}$ . The apparent frequency or equivalently $c$ and $u_{h}^{D}$ are unknown variables to be determined as part of the wave solutions.

The wave solutions sought correct to $O(\unicode[STIX]{x1D716}^{3})$ in § 5 are functions of $(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})$ and periodic in $\unicode[STIX]{x1D6E9}$ , that is,

(2.4) $$\begin{eqnarray}u_{i}^{w}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})=u_{i}^{w}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9}+2\unicode[STIX]{x03C0}).\end{eqnarray}$$

Importantly, a periodicity in time or space is not required. Hence, the amplitude or wavenumber can change in time and space. Because the wave solutions at any value of $\unicode[STIX]{x1D6E9}$ satisfy the governing equations, $u_{i}^{w}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})$ represents a one-parameter family of solutions which are all valid at $(x,y,\unicode[STIX]{x1D701},t)$ . Denote the averaging over this solution family at a coordinate $(x,y,\unicode[STIX]{x1D701},t)$ by

(2.5) $$\begin{eqnarray}\langle \unicode[STIX]{x1D711}\rangle (x,y,\unicode[STIX]{x1D701},t)\equiv \frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D711}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})\,\text{d}\unicode[STIX]{x1D6E9}\end{eqnarray}$$

for any function $\unicode[STIX]{x1D711}$ . Crucially, $\langle ~\rangle$ exactly commutes with the spatial and temporal partial differentiation operators with respect to $(x,y,\unicode[STIX]{x1D701},t)$ . (In contrast, $\langle ~\rangle$ may not commute with the partial differentiation operators with respect to $(x,y,z,t)$ because the averaging $\langle ~\rangle$ is not done at a fixed $(x,y,z,t)$ .) Moreover, $\langle ~\rangle$ is idempotent: $\langle \langle \unicode[STIX]{x1D711}\rangle \rangle =\langle \unicode[STIX]{x1D711}\rangle$ . If $\unicode[STIX]{x1D711}$ is independent of $\unicode[STIX]{x1D6E9}$ , then $\langle \unicode[STIX]{x1D711}\rangle =\unicode[STIX]{x1D711}$ . These properties are essential for mathematical rigour. Physically, $\langle ~\rangle$ represents the ensemble averaging at a given $(x,y,\unicode[STIX]{x1D701},t)$ over the ensemble of states having the same current and wave properties (i.e.  $a$ and $s$ ) but different values of the phase shift parameter $\unicode[STIX]{x1D6E9}$ . This use of $\unicode[STIX]{x1D6E9}$ follows that of Hayes (Reference Hayes1970) and Grimshaw (Reference Grimshaw1984). The wave motion consists of the ensemble average $\mathfrak{U}_{i}\equiv \langle u_{i}^{w}\rangle$ and the oscillatory deviations from it, that is,

(2.6) $$\begin{eqnarray}u_{i}^{w}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})=\mathfrak{U}_{i}(x,y,\unicode[STIX]{x1D701},t)+[u_{i}^{w}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})-\mathfrak{U}_{i}(x,y,\unicode[STIX]{x1D701},t)].\end{eqnarray}$$

2.3 Coordinate systems

Let us call the surface-following coordinate system the $\unicode[STIX]{x1D701}$ -coordinate system. Recall that $(x,y,t,\unicode[STIX]{x1D6E9})$ are identical between the Cartesian and $\unicode[STIX]{x1D701}$ -coordinate systems. Also recall that the $z$ -coordinate – i.e. the height – of a point $(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})$ is $z(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})$ . Hereafter, let us use the following notation: for any variable $\unicode[STIX]{x1D711}$ ,

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{\ell }^{z}\unicode[STIX]{x1D711}\equiv \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}(x,y,z,t,\unicode[STIX]{x1D6E9})}{\unicode[STIX]{x2202}x_{\ell }}\quad \text{where }(x_{1},x_{2},x_{3},x_{t})\equiv (x,y,z,t), & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D711}\equiv \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})}{\unicode[STIX]{x2202}x_{\ell }}\quad \text{where }(x_{1},x_{2},x_{3},x_{t})\equiv (x,y,\unicode[STIX]{x1D701},t). & \displaystyle\end{eqnarray}$$

Multiple operations are denoted by $(\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}})^{n}$ , e.g.  $(\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}})^{2}\equiv \unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}$ . Because $a$ , $k$ , $\hat{k}_{h}$ , $c$ and $u_{h}^{D}$ are functions only of $(x,y,t)$ , their derivatives do not have the superscript $z$ or $\unicode[STIX]{x1D701}$ on $\unicode[STIX]{x2202}_{\ell }$ .

Now, let us more precisely define the constant- $\unicode[STIX]{x1D701}$ surfaces to be used as the $\unicode[STIX]{x1D701}$ -coordinates. First, let the water surface be the constant- $\unicode[STIX]{x1D701}$ surface labelled with $\unicode[STIX]{x1D701}=0$ . Then, the height of the water surface – namely, $z(x,y,\unicode[STIX]{x1D701}=0,t,\unicode[STIX]{x1D6E9})$ – satisfies $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}z=w-u_{h}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}z$ . This is the standard kinematic boundary condition for the free surface. In S19, let us limit our consideration to only those currents whose vertical velocities $w^{c}$ are zero at the water surface. That is, $w=w^{w}$ at the water surface. Therefore, the water surface satisfies $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}z=w^{w}-u_{h}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}z$ . Now, let us define every constant- $\unicode[STIX]{x1D701}$ surface in the interior by the following equation for its $z$ -coordinate $z(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})$ :

(2.9) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}z=w^{w}-u_{h}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}z.\end{eqnarray}$$

Here any initial condition or constant of integration should be chosen so that the constant- $\unicode[STIX]{x1D701}$ surfaces become identical to the constant- $z$ surfaces in the absence of waves (or any other disturbances such as geostrophically balanced surface tilts, which are negligible at the orders concerned in S19). If $w^{w}$ in (2.9) is replaced with $w$ , then the constant- $\unicode[STIX]{x1D701}$ surfaces defined in this way are the interior material surfaces. However, (2.9) uses $w^{w}$ so that the constant- $\unicode[STIX]{x1D701}$ surfaces follow the displacement due to the wave vertical velocities, rather than the fluid vertical velocities. Thus, current vertical velocity $w^{c}$ , which may be of $O(\unicode[STIX]{x1D716}^{2}c^{[0]})$ in the interior, goes through (i.e. does not move) the constant- $\unicode[STIX]{x1D701}$ surfaces. Note that the kinematic boundary condition for the water surface is the same as (2.9) at $\unicode[STIX]{x1D701}=0$ .

In S19, the amplitude is of first order, i.e.  $a=O(\unicode[STIX]{x1D716}k^{-1})$ . Therefore, the perturbation series of $z(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})$ is

(2.10) $$\begin{eqnarray}z(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})=\unicode[STIX]{x1D701}+\mathop{\sum }_{n\geqslant 1}z^{[n]}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9}),\end{eqnarray}$$

where $z^{[n]}\leqslant O(\unicode[STIX]{x1D716}^{n}k^{-1})$ . The value of $\unicode[STIX]{x1D701}$ is equal to the unperturbed height and $\sum z^{[n]}$ is the perturbation.

Let us define the constant- $\unicode[STIX]{x1D701}$ layer containing a point $(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})$ as the layer containing the point and bounded by two infinitesimally close constant- $\unicode[STIX]{x1D701}$ surfaces. The concept of a layer is useful because the wave motion undulates both the slope and thickness of a material layer. The constant- $\unicode[STIX]{x1D701}$ surface and layer at $(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})$ have the following properties:

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \text{vertical motion of the surface,}\quad S_{t}\equiv \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}z=\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\mathop{\sum }_{n\geqslant 1}z^{[n]}, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \text{slope of the surface,}\quad S_{h}\equiv \unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}z=\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}\mathop{\sum }_{n\geqslant 1}z^{[n]}, & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \text{layer-thickness perturbation,}\quad S_{3}\equiv \unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\mathop{\sum }_{n\geqslant 1}z^{[n]}, & \displaystyle\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle \text{normalised layer thickness,}\quad J\equiv \unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}z=1+S_{3}. & \displaystyle\end{eqnarray}$$

Here $J$ is the layer thickness normalised with the undeformed thickness and is also the Jacobian determinant (i.e.  $\text{d}x\,\text{d}y\,\text{d}z=J\,\text{d}x\,\text{d}y\,\text{d}\unicode[STIX]{x1D701}$ ). The commutation of the differential operators yields the following identities: $\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}J=\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}S_{3}$ and $\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}S_{m}=\unicode[STIX]{x2202}_{m}^{\unicode[STIX]{x1D701}}S_{\ell }$ for $m=1,2,3,t$ .

Since $\unicode[STIX]{x2202}_{3}^{z}x$ , $\unicode[STIX]{x2202}_{3}^{z}y$ and $\unicode[STIX]{x2202}_{3}^{z}t$ are all zero, the standard chain rule for $\unicode[STIX]{x2202}_{3}^{z}$ yields $1=\unicode[STIX]{x2202}_{3}^{z}z=(\unicode[STIX]{x2202}_{3}^{z}\unicode[STIX]{x1D701})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}z$ . Therefore, $J^{-1}=\unicode[STIX]{x2202}_{3}^{z}\unicode[STIX]{x1D701}$ . Then, the chain rules for $\unicode[STIX]{x2202}_{3}^{z}$ and $\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}$ yield

(2.15a,b ) $$\begin{eqnarray}\unicode[STIX]{x2202}_{3}^{z}\unicode[STIX]{x1D711}=(\unicode[STIX]{x2202}_{3}^{z}\unicode[STIX]{x1D701})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D711}=J^{-1}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D711}\quad \text{and}\quad \unicode[STIX]{x2202}_{\ell }^{z}=\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}-S_{\ell }J^{-1}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}.\end{eqnarray}$$

2.4 Equations of motion

Hereafter, consider an inviscid and incompressible fluid having a uniform density $\unicode[STIX]{x1D70C}_{0}$ . Thus, the equations of motion with respect to the Cartesian coordinate system are

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{z}u_{i}+u_{h}\unicode[STIX]{x2202}_{h}^{z}u_{i}+w\unicode[STIX]{x2202}_{3}^{z}u_{i}=-\unicode[STIX]{x2202}_{i}^{z}P, & \displaystyle\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{i}^{z}u_{i}=0, & \displaystyle\end{eqnarray}$$

where $P\equiv p/\unicode[STIX]{x1D70C}_{0}+gz$ , with $p$ the pressure and $g$ the gravitational acceleration. According to (2.15) and $w=w^{c}+w^{w}$ , (2.16) and (2.17) are equivalent to

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle \text{momentum equations,}\quad \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}u_{i}+u_{h}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}u_{i}+w^{c}J^{-1}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{i}=-\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P+S_{i}J^{-1}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P, & \displaystyle\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle \text{incompressibility,}\quad \unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}u_{i}-S_{i}J^{-1}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{i}=0. & \displaystyle\end{eqnarray}$$

The lower and upper boundary conditions for deep-water waves without wind forcing are

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle u_{i}^{w}=0\quad \text{at }\unicode[STIX]{x1D701}=-\infty , & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{g}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P-w^{w}+u_{h}S_{h}=0\quad \text{at }\unicode[STIX]{x1D701}=0, & \displaystyle\end{eqnarray}$$

where $w^{c}=0$ at $\unicode[STIX]{x1D701}=0$ in S19. Equation (2.21) is the joint condition between the kinematic upper boundary condition (i.e.  $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}z=w-u_{h}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}z$ ) and the dynamic upper boundary condition (i.e.  $p=\text{const.}$ at $\unicode[STIX]{x1D701}=0$ ). The definition of $P$ and the dynamic upper boundary condition give $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P=g\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}z$ at $\unicode[STIX]{x1D701}=0$ . Combining this and the kinematic upper boundary condition yields (2.21).

3.1 Scaling conditions

S19 considers the following conditions: $a=O(\unicode[STIX]{x1D716}k^{-1})$ , to consider a typical non-breaking wave field; $\hat{k}_{1}=O(1)$ and $\hat{k}_{2}\leqslant O(\unicode[STIX]{x1D716})$ , because the wave propagation is roughly in the streamwise direction; $\mathfrak{U}_{h}\leqslant O(\unicode[STIX]{x1D716}^{2}\hat{k}_{h}c^{[0]})$ and $\mathfrak{U}_{3}\leqslant O(\unicode[STIX]{x1D716}^{4}c^{[0]})$ , being consistent with a largely uniform and first-order-irrotational wave field; and $u_{i}^{c}\leqslant O(\unicode[STIX]{x1D716}^{2}c^{[0]})$ , like the CL theory. (Higher-order irrotationality of the wave motion is not assumed in S19.) However, unlike the CL theory, S19 does not assume that the current velocity field is non-divergent. Therefore, S19 anticipates a condition $\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}u_{i}^{c}\leqslant O(\unicode[STIX]{x1D716}^{4}c^{[0]}k)$ , being consistent with the order of $\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{i}$ . The orders of all other quantities are listed in appendix A. Many of these conditions are roughly depicted in figure 1. Importantly, S19 considers only those current structures whose spanwise widths are wider than roughly one wavelength so that the fourth and higher spanwise derivatives of $u^{c}$ and $w^{c}$ are insignificant compared to their lower spanwise derivatives (see table 4). The anticipated gradients of the wave properties due to the wave–current interaction are specified later in table 5. Note that these scaling conditions are validly possible; that is, these scales are self-consistent with the scales implied by the equations of the wave properties and of the wave-averaged circulation derived later in §§ 5 and 6. An example of the self-consistency conditions is $O(\unicode[STIX]{x2202}_{\ell }\unicode[STIX]{x1D711})\geqslant O([c^{[0]}k]^{-1}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{\ell }\unicode[STIX]{x1D711})$ for any quantity $\unicode[STIX]{x1D711}$ . This is a statement that $\unicode[STIX]{x2202}_{\ell }\unicode[STIX]{x1D711}$ remains negligible at least during one characteristic period $(c^{[0]}k)^{-1}$ , if it is initially negligible. To achieve self-consistency, the scales specified here and the results in §§ 5 and 6 are iteratively derived. (This, of course, does not eliminate a possibility of having another validly possible set of scaling conditions.)

The aforementioned scaling conditions imply an important relationship between the current velocity and its ensemble average (see § A.2 for derivation), namely,

(3.1) $$\begin{eqnarray}u_{i}^{c}=\langle u_{i}^{c}\rangle +O(\unicode[STIX]{x1D716}^{4}c^{[0]}).\end{eqnarray}$$

3.2 The class of the current structures assumed in § 5

In § 5, the wave solutions are obtained for a particular class of current structures having the form $u^{c}=\sum _{n\geqslant 1}u_{(n)}^{c}$ and $w^{c}=\sum _{n\geqslant 1}w_{(n)}^{c}$ where

(3.2) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}u_{(1)}^{c}=C_{\mathbb{A}(1)}{\mathcal{U}}_{(1)}(x,\unicode[STIX]{x1D701},t),\\ u_{(n)}^{c}=B_{\mathbb{A}(n)}\cos \left(\sqrt{K_{\mathbb{A}(n)}^{2}-k^{2}}y+C_{\mathbb{A}(n)}\right){\mathcal{U}}_{(n)}(x,\unicode[STIX]{x1D701},t)\quad \text{for }n\geqslant 2,\end{array}\right\}\end{eqnarray}$$

and

(3.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}w_{(1)}^{c}=C_{\mathbb{B}(1)}{\mathcal{W}}_{(1)}(x,\unicode[STIX]{x1D701},t),\\ w_{(n)}^{c}=B_{\mathbb{B}(n)}\cos \left(\sqrt{K_{\mathbb{B}(n)}^{2}-k^{2}}y+C_{\mathbb{B}(n)}\right){\mathcal{W}}_{(n)}(x,\unicode[STIX]{x1D701},t)\quad \text{for }n\geqslant 2.\end{array}\right\}\end{eqnarray}$$

The subscript indices are surrounded with parentheses to avoid possible confusion with the tensor indices. In (3.2) and (3.3), $B_{\mathbb{A}(n)}$ , $C_{\mathbb{A}(n)}$ , $B_{\mathbb{B}(n)}$ and $C_{\mathbb{B}(n)}$ are real constants; $K_{\mathbb{A}(n)}$ and $K_{\mathbb{B}(n)}$ (for $n\geqslant 2$ ) are real parameters in the range of $k<K_{\mathbb{A}(n)}\leqslant 1.15k$ and $k<K_{\mathbb{B}(n)}\leqslant 1.15k$ (see § A.3 for their scaling conditions); and ${\mathcal{U}}_{(n)}$ and ${\mathcal{W}}_{(n)}$ may be any functions of $(x,\unicode[STIX]{x1D701},t)$ as long as they satisfy the scaling conditions. The upper bound (i.e.  $1.15k$ ) of $K_{\mathbb{A}(n)}$ and $K_{\mathbb{B}(n)}$ gives the narrowest (i.e. 88 % of the wavelength $2\unicode[STIX]{x03C0}k^{-1}$ ) current structure considered in S19. For theoretical consistency, $u_{(n)}^{c}$ and $w_{(n)}^{c}$ should individually satisfy the scaling conditions. Figure 2 shows an example.

Figure 2.

An example of the current structures satisfying (3.2) and (3.3). The wave solutions in § 5 are derived assuming that the current field satisfies (3.2) and (3.3). The background colour shows $u^{c}/c^{[0]}$ , and the contours show the streamfunction of $(v^{c},w^{c})$ . The sense of rotation of each roll is indicated by the arrows. In this example, $K_{\mathbb{A}(2)}=K_{\mathbb{B}(2)}=1.15k$ , $u^{c}=u_{(1)}^{c}+u_{(2)}^{c}={\mathcal{U}}_{(1)}-\cos (\sqrt{K_{\mathbb{A}(2)}^{2}-k^{2}}y+\unicode[STIX]{x03C0}){\mathcal{U}}_{(2)}$ , the streamfunction of $(v^{c},w^{c})=(v^{c},w_{(2)}^{c})$ is $-(K_{\mathbb{B}(2)}^{2}-k^{2})^{-1/2}\cos (\sqrt{K_{\mathbb{B}(2)}^{2}-k^{2}}y+0.5\unicode[STIX]{x03C0}){\mathcal{W}}_{(2)}$ , ${\mathcal{U}}_{(1)}={\mathcal{U}}_{(2)}=1.5\text{e}^{0.6k\unicode[STIX]{x1D701}}\unicode[STIX]{x1D716}^{2}c^{[0]}$ and ${\mathcal{W}}_{(2)}=-3k\unicode[STIX]{x1D701}\text{e}^{0.3k\unicode[STIX]{x1D701}}\unicode[STIX]{x1D716}^{2}c^{[0]}$ .

5.1 The wave solutions correct to third order

Using (2.1), the governing equations (2.9) and (2.18)–(2.21) can be written as

(5.1) $$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{m,n,q=0}^{\infty }(\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}u_{i}^{[m]}+u_{h}^{[n]}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}u_{i}^{[m]}+w^{c}(J^{-1})^{[q]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{i}^{[m]}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P^{[m]}-S_{i}^{[n]}(J^{-1})^{[q]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[m]})=0, & \displaystyle\end{eqnarray}$$

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{m,n,q=0}^{\infty }(\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}u_{i}^{[m]}-S_{i}^{[n]}(J^{-1})^{[q]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{i}^{[m]})=0, & \displaystyle\end{eqnarray}$$

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{m,n=0}^{\infty }(S_{t}^{[m]}-{w^{w}}^{[m]}+u_{h}^{[m]}S_{h}^{[n]})=0, & \displaystyle\end{eqnarray}$$

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{m=0}^{\infty }{u_{i}^{w}}^{[m]}=0\quad \text{at }\unicode[STIX]{x1D701}=-\infty , & \displaystyle\end{eqnarray}$$

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{m,n=0}^{\infty }\left(\frac{1}{g}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P^{[m]}-{w^{w}}^{[m]}+u_{h}^{[m]}S_{h}^{[n]}\right)=0\quad \text{at }\unicode[STIX]{x1D701}=0, & \displaystyle\end{eqnarray}$$

where $m$ , $n$ and $q$ are integers, and $S_{\ell }^{[n]}=(\sum _{m=1}^{n}\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}z^{[m]})^{[n]}$ . Note that $S_{\ell }^{[0]}=0$ , $u_{i}^{[0]}=0$ , $P^{[0]}=0$ and $J^{[0]}=1$ because $ak=O(\unicode[STIX]{x1D716})$ . Also note that $(J^{-1})^{[0]}=1$ , $(J^{-1})^{[1]}=-{S_{3}}^{[1]}$ , $(J^{-1})^{[2]}=({S_{3}}^{[1]})^{2}-{S_{3}}^{[2]}$ and $(J^{-1})^{[3]}=2{S_{3}}^{[1]}{S_{3}}^{[2]}-({S_{3}}^{[1]})^{3}-{S_{3}}^{[3]}$ . This follows from (2.14) and the fact that $J^{-1}$ can be expressed in series form as $J^{-1}=1-S_{3}+S_{3}^{2}-S_{3}^{3}+\cdots \,$ because $J=1+S_{3}$ .

Now, as mentioned in § 2.2, we look for a local wave solution satisfying $z^{[1]}=a(x,y,t)\cos \unicode[STIX]{x1D712}$ at the water surface ( $\unicode[STIX]{x1D701}=0$ ). According to the scaling conditions (§ 3), the governing equations (5.1)–(5.5) yield the first-order equations as

(5.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[1]}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P^{[1]}=O(\unicode[STIX]{x1D716}^{2}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

(5.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[1]}=O(\unicode[STIX]{x1D716}^{2}c^{[0]}k), & \displaystyle\end{eqnarray}$$

(5.6c ) $$\begin{eqnarray}\displaystyle & \displaystyle {S_{t}}^{[1]}-{w^{w}}^{[1]}=O(\unicode[STIX]{x1D716}^{2}c^{[0]}), & \displaystyle\end{eqnarray}$$

(5.6d ) $$\begin{eqnarray}\displaystyle & \displaystyle {u_{i}^{w}}^{[1]}=O(\unicode[STIX]{x1D716}^{2}c^{[0]})\quad \text{at }\unicode[STIX]{x1D701}=-\infty , & \displaystyle\end{eqnarray}$$

(5.6e ) $$\begin{eqnarray}\displaystyle & \displaystyle (1/g)\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P^{[1]}-{w^{w}}^{[1]}=O(\unicode[STIX]{x1D716}^{2}c^{[0]})\quad \text{at }\unicode[STIX]{x1D701}=0. & \displaystyle\end{eqnarray}$$

(5.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle {u_{h}^{w}}^{[1]}=\hat{k}_{h}c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}}\cos \unicode[STIX]{x1D712}, & \displaystyle\end{eqnarray}$$

(5.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle {w^{w}}^{[1]}=c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}}\sin \unicode[STIX]{x1D712}, & \displaystyle\end{eqnarray}$$

(5.7c ) $$\begin{eqnarray}\displaystyle & \displaystyle P^{[1]}={c^{[0]}}^{2}ak\text{e}^{k\unicode[STIX]{x1D701}}\cos \unicode[STIX]{x1D712}, & \displaystyle\end{eqnarray}$$

(5.7d ) $$\begin{eqnarray}\displaystyle & \displaystyle z^{[1]}=a\text{e}^{k\unicode[STIX]{x1D701}}\cos \unicode[STIX]{x1D712}, & \displaystyle\end{eqnarray}$$

(5.7e ) $$\begin{eqnarray}\displaystyle & \displaystyle {c^{[0]}}^{2}=g/k. & \displaystyle\end{eqnarray}$$

$\text{e}^{kz}$

Substitution of (5.7) in (5.1)–(5.5) yields the second-order equations as

(5.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{h}^{w}}^{[2]}+\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}P^{[2]}+\hat{k}_{h}c^{[0]}c^{[1]}ak^{2}\text{e}^{k\unicode[STIX]{x1D701}}\sin \unicode[STIX]{x1D712}=O(\unicode[STIX]{x1D716}^{3}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

(5.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{w^{w}}^{[2]}+\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[2]}-c^{[0]}c^{[1]}ak^{2}\text{e}^{k\unicode[STIX]{x1D701}}\cos \unicode[STIX]{x1D712}=O(\unicode[STIX]{x1D716}^{3}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

(5.8c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[2]}=O(\unicode[STIX]{x1D716}^{3}c^{[0]}k), & \displaystyle\end{eqnarray}$$

(5.8d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}z^{[2]}-{w^{w}}^{[2]}+c^{[1]}ak\text{e}^{k\unicode[STIX]{x1D701}}\sin \unicode[STIX]{x1D712}-{\textstyle \frac{1}{2}}c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\sin (2\unicode[STIX]{x1D712})=O(\unicode[STIX]{x1D716}^{3}c^{[0]}), & \displaystyle\end{eqnarray}$$

(5.8e ) $$\begin{eqnarray}\displaystyle & \displaystyle {u_{i}^{w}}^{[2]}=O(\unicode[STIX]{x1D716}^{3}c^{[0]})\quad \text{at }\unicode[STIX]{x1D701}=-\infty , & \displaystyle\end{eqnarray}$$

(5.8f ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{g}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P^{[2]}-{w^{w}}^{[2]}-\frac{1}{2}c^{[0]}a^{2}k^{2}\sin (2\unicode[STIX]{x1D712})+c^{[1]}ak\sin \unicode[STIX]{x1D712}=O(\unicode[STIX]{x1D716}^{3}c^{[0]})\quad \text{at }\unicode[STIX]{x1D701}=0. & \displaystyle\end{eqnarray}$$

(5.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle {u_{h}^{w}}^{[2]}={\textstyle \frac{1}{2}}\hat{k}_{h}c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\cos (2\unicode[STIX]{x1D712})+\mathfrak{U}_{h}, & \displaystyle\end{eqnarray}$$

(5.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle {w^{w}}^{[2]}={\textstyle \frac{1}{2}}c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\sin (2\unicode[STIX]{x1D712}), & \displaystyle\end{eqnarray}$$

(5.9c ) $$\begin{eqnarray}\displaystyle & \displaystyle P^{[2]}={\textstyle \frac{1}{2}}{c^{[0]}}^{2}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\cos (2\unicode[STIX]{x1D712}), & \displaystyle\end{eqnarray}$$

(5.9d ) $$\begin{eqnarray}\displaystyle & \displaystyle z^{[2]}={\textstyle \frac{1}{2}}a^{2}k\text{e}^{2k\unicode[STIX]{x1D701}}\cos (2\unicode[STIX]{x1D712}), & \displaystyle\end{eqnarray}$$

(5.9e ) $$\begin{eqnarray}\displaystyle & \displaystyle c^{[1]}=0, & \displaystyle\end{eqnarray}$$

$\mathfrak{U}_{h}$

$\mathfrak{U}_{h}=0$

$\unicode[STIX]{x1D701}=-\infty$

$\mathfrak{U}_{h}$

$\mathfrak{U}_{h}=(1/2)c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\hat{k}_{h}+O(\unicode[STIX]{x1D716}^{3}c^{[0]})$

$\unicode[STIX]{x1D735}\times \boldsymbol{u}^{w}$

$\mathfrak{U}_{h}$

$P^{[1]}+P^{[2]}$

${c^{[0]}}^{2}ak\text{e}^{kz}\cos \unicode[STIX]{x1D712}-(1/2)(c^{[0]})^{2}a^{2}k^{2}\text{e}^{2kz}$

${c^{[0]}}^{2}ak\text{e}^{kz}\cos \unicode[STIX]{x1D712}$

$-(1/2)(c^{[0]})^{2}a^{2}k^{2}\text{e}^{2kz}$

$P^{[1]}$

Figure 4.

The second-order errors (at $\unicode[STIX]{x1D701}=0$ ) in the first-order pressure solutions of the Eulerian theory and of S19 are shown by the difference between $P_{plot}$ and S19’s second-order-accurate solution $P^{[1]}+P^{[2]}$ shown in (5.7c ) and (5.9c ). For the dotted line, $P_{plot}$ is the first-order solution of the Eulerian theory, ${c^{[0]}}^{2}ak\text{e}^{kz}\cos \unicode[STIX]{x1D712}$ . For the dashed line, $P_{plot}$ is the first-order solution of S19, $P^{[1]}$ . For the solid line, $P_{plot}$ is the second-order-accurate solution of the Eulerian theory, ${c^{[0]}}^{2}ak\text{e}^{kz}\cos \unicode[STIX]{x1D712}-(1/2){c^{[0]}}^{2}a^{2}k^{2}\text{e}^{2kz}$ .

Substitution of the first-order and second-order wave solutions (5.7) and (5.9) in the governing equations (5.1)–(5.5) yields the third-order governing equations as listed in appendix B. The third-order wave solutions can be obtained by using the complex Fourier series

(5.10a-c ) $$\begin{eqnarray}{u_{i}^{w}}^{[3]}=\mathop{\sum }_{n=1}^{3}u_{i}^{\{n\}}\text{e}^{\text{i}n\unicode[STIX]{x1D712}},\quad P^{[3]}=\mathop{\sum }_{n=1}^{3}P^{\{n\}}\text{e}^{\text{i}n\unicode[STIX]{x1D712}},\quad z^{[3]}=\mathop{\sum }_{n=1}^{3}z^{\{n\}}\text{e}^{\text{i}n\unicode[STIX]{x1D712}},\end{eqnarray}$$

where $u_{i}^{\{n\}}$ , $P^{\{n\}}$ and $z^{\{n\}}$ are third-order complex functions. Do not confuse the superscripts $[n]$ and $\{n\}$ ; the former denotes the $n$ th-order term of the perturbation series, and the latter denotes the $n$ th harmonic of the third-order solutions. The harmonics other than $n=1,2,3$ are not needed because the terms appearing in the third-order equations do not contain the other harmonics.

In addition to the governing equations in appendix B, the wave solutions being sought are constrained by the phase function properties (2.3) and the lower-order dispersion relations (5.7e ) and (5.9e ). In particular, these equations imply that

(5.11) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{\ell }c^{[0]}=-\frac{c^{[0]}}{2k}\unicode[STIX]{x2202}_{\ell }k, & \displaystyle\end{eqnarray}$$

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(k\hat{k}_{h})=\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{h}\unicode[STIX]{x1D712}=\unicode[STIX]{x2202}_{h}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D712}=-\unicode[STIX]{x2202}_{h}\left(kc^{[0]}+k\mathop{\sum }_{n\geqslant 2}c^{[n]}+k\hat{k}_{H}u_{H}^{D}\right), & \displaystyle\end{eqnarray}$$

(5.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}k=-\frac{1}{2}c^{[0]}\hat{k}_{h}\unicode[STIX]{x2202}_{h}k-\hat{k}_{h}\unicode[STIX]{x2202}_{h}\left(k\mathop{\sum }_{n\geqslant 2}c^{[n]}+k\hat{k}_{H}u_{H}^{D}\right). & \displaystyle\end{eqnarray}$$

Here, (5.7e ) yields (5.11); (2.3) and (5.9e ) yield (5.12); and (5.11), (5.12) and $\hat{k}_{h}\unicode[STIX]{x2202}_{t}\hat{k}_{h}=0$ yield (5.13).

The solutions of ${u_{i}^{w}}^{[3]}$ , $P^{[3]}$ and $z^{[3]}$ are derived in appendix D and presented in a physically understandable way in § 5.2. Crucially, the third-order dynamics determines not only ${u_{i}^{w}}^{[3]}$ , $P^{[3]}$ and $z^{[3]}$ but also the solution of $\hat{k}_{1}u_{1}^{D}+c^{[2]}$ and the evolution of $a$ . Namely, substituting (D 1) and (D 2c ) into the upper boundary condition (B 6a ) yields the equations for $\unicode[STIX]{x2202}_{t}a$ and $\hat{k}_{1}u_{1}^{D}+c^{[2]}$ . The one for $\unicode[STIX]{x2202}_{t}a$ is

(5.14a ) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}a+c_{h}^{g}\unicode[STIX]{x2202}_{h}a=-\frac{a}{2}D^{g}-\frac{1}{4c^{[0]}k}\text{i}\unicode[STIX]{x2202}_{t}\,\text{Im}(G)-\frac{1}{2}\,\text{Re}(G)+O(\unicode[STIX]{x1D716}^{4}c^{[0]}),\end{eqnarray}$$

where

(5.14b ) $$\begin{eqnarray}\displaystyle & \displaystyle c_{h}^{g}\equiv \frac{c^{[0]}}{2}\hat{k}_{h}, & \displaystyle\end{eqnarray}$$

(5.14c ) $$\begin{eqnarray}\displaystyle & \displaystyle D^{g}\equiv \unicode[STIX]{x2202}_{h}c_{h}^{g}-\mathop{\sum }_{n\geqslant 1}\frac{k(K_{\mathbb{B}(n)}+k)}{K_{\mathbb{B}(n)}}\int _{-\infty }^{0}\text{e}^{(K_{\mathbb{B}(n)}+k)\unicode[STIX]{x1D701}^{\prime }}(\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w_{(n)}^{c})^{\prime }\,\text{d}\unicode[STIX]{x1D701}^{\prime }, & \displaystyle\end{eqnarray}$$

(5.14d ) $$\begin{eqnarray}\displaystyle & \displaystyle G\equiv \frac{2\unicode[STIX]{x2202}_{t}C_{(2)}}{g}+\mathop{\sum }_{n\geqslant 1}\cos \left(\sqrt{K_{(n)}^{2}-k^{2}}y+C_{(n,4)}\right)\left[\frac{K_{(n)}-k}{c^{[0]}k}\text{i}C_{(n,3)}+\frac{K_{(n)}+k}{gk}\unicode[STIX]{x2202}_{t}C_{(n,3)}\right]. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here, $(\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w_{(n)}^{c})^{\prime }$ is $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w_{(n)}^{c}$ at $(x,y,\unicode[STIX]{x1D701}^{\prime },t)$ and $K_{\mathbb{B}(n)}$ , $C_{(2)}$ , $C_{(n,3)}$ , $C_{(n,4)}$ , and $K_{(n)}$ are defined in table 1; $c_{h}^{g}$ is the leading-order group velocity; $D^{g}$ is the horizontal divergence of $c_{h}^{g}$ and of the surface water (that is, $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w^{c}=-\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}v^{c}+O(\unicode[STIX]{x1D716}^{3}c^{[0]}k)$ ); and $G$ is the contribution to the upper boundary condition made by the pressure homogeneous solution $P_{\mathbb{H}}^{\{1\}}$ shown later in (5.23a ). Importantly, the orders of $\unicode[STIX]{x2202}_{2}^{4}G$ , $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{2}G$ , $\unicode[STIX]{x2202}_{t}^{2}G$ and $\unicode[STIX]{x2202}_{1}G$ are fourth or higher, according to the scaling conditions in § A.4. Note that the advective effect of the current – that is, $u_{h}^{D}\unicode[STIX]{x2202}_{h}a$ – is negligible at this order. The most crucial result of (5.14a ) is that the roll (i.e.  $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w^{c}$ ) affects the amplitude evolution at its leading-order balance.

The solution of $\hat{k}_{1}u_{1}^{D}+c^{[2]}$ may be split (according to the dependence on $u^{c}$ ) as

(5.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{1}^{D}=u^{c}(\unicode[STIX]{x1D701}=0)-\mathop{\sum }_{n\geqslant 1}\frac{K_{\mathbb{A}(n)}+k}{2K_{\mathbb{A}(n)}}\int _{-\infty }^{0}\text{e}^{(K_{\mathbb{A}(n)}+k)\unicode[STIX]{x1D701}^{\prime }}(\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{(n)}^{c})^{\prime }\,\text{d}\unicode[STIX]{x1D701}^{\prime }+O(\unicode[STIX]{x1D716}^{3}c^{[0]}), & \displaystyle\end{eqnarray}$$

(5.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle c^{[2]}=\frac{c^{[0]}a^{2}k^{2}}{4}+2k\int _{-\infty }^{0}\text{e}^{2k\unicode[STIX]{x1D701}^{\prime }}\hat{k}_{1}\mathfrak{U}_{1}^{\prime }\,\text{d}\unicode[STIX]{x1D701}^{\prime }-\frac{c^{[0]}\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}a}{4ak^{2}}-\frac{\text{i}\,\text{Im}(G)}{2ak}+\frac{\unicode[STIX]{x2202}_{t}\,\text{Re}(G)}{4c^{[0]}ak^{2}}+O(\unicode[STIX]{x1D716}^{3}c^{[0]}), & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

$u^{c}(\unicode[STIX]{x1D701}=0)\equiv u^{c}(x,y,\unicode[STIX]{x1D701}=0,t)$

$(\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{(n)}^{c})^{\prime }\equiv (\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{(n)}^{c})(x,y,\unicode[STIX]{x1D701}^{\prime },t)$

$\mathfrak{U}_{1}^{\prime }\equiv \mathfrak{U}_{1}(x,y,\unicode[STIX]{x1D701}^{\prime },t)$

$w^{c}=0$

$\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}a=0$

$G=0$

$\mathfrak{U}_{h}$

$\mathfrak{U}_{h}=(1/2)c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\hat{k}_{h}+O(\unicode[STIX]{x1D716}^{3}c^{[0]})$

$\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}a=0$

$G=0$

$c^{[2]}=c^{[0]}a^{2}k^{2}/2$

Table 1.

Definitions of some variables.

5.2 Physical interpretation of the third-order wave solutions: seven wave types

The third-order solutions derived in appendix D consist of seven types of independent wave motions, hereafter called $\mathbb{H}$ -, $\mathbb{A}$ -, $\mathbb{B}$ -, $\mathbb{C}$ -, $\mathbb{D}$ -, $\mathbb{E}$ - and $\mathbb{F}$ -waves. Each is characterised by a unique divergence and force balance. To see this, write the incompressibility equation (2.17) as $\unicode[STIX]{x2202}_{i}^{z}({u_{i}^{w}}^{[1]}+{u_{i}^{w}}^{[2]}-\mathfrak{U}_{i})+\unicode[STIX]{x2202}_{i}^{z}\mathfrak{U}_{i}+\unicode[STIX]{x2202}_{i}^{z}u_{i}^{c}=-\unicode[STIX]{x2202}_{i}^{z}{u_{i}^{w}}^{[3]}+O(\unicode[STIX]{x1D716}^{4}c^{[0]}k)$ . Each term on the left-hand side is divergent and therefore requires divergent third-order waves that cancel their divergence. In addition to the divergent waves, there are divergence-free waves required by the boundary conditions or momentum conservation. To see this, write the momentum equations (2.18) as

(5.16) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[3]}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P^{[3]}+F_{\mathbb{ A}i}+F_{\mathbb{B}i}+F_{\mathbb{C}i}+F_{\mathbb{D}i}+F_{\mathbb{E}i}+F_{\mathbb{F}i}=O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k),\end{eqnarray}$$

where

(5.17) $$\begin{eqnarray}\displaystyle & \displaystyle F_{\mathbb{A}i}\equiv (u_{h}^{c}-u_{h}^{D})\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[1]}, & \displaystyle\end{eqnarray}$$

(5.18) $$\begin{eqnarray}\displaystyle & \displaystyle F_{\mathbb{B}i}\equiv w^{c}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[1]}, & \displaystyle\end{eqnarray}$$

(5.19) $$\begin{eqnarray}\displaystyle & \displaystyle F_{\mathbb{C}i}\equiv [u_{h}^{w}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}u_{i}^{w}-S_{i}J^{-1}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P]-c^{[2]}\hat{k}_{h}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[1]}, & \displaystyle\end{eqnarray}$$

(5.20) $$\begin{eqnarray}\displaystyle & \displaystyle (F_{\mathbb{D}1},F_{\mathbb{D}2},F_{\mathbb{D}3})\equiv ((\unicode[STIX]{x2202}_{t}\hat{k}_{1})c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}+g\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}(a\text{e}^{k\unicode[STIX]{x1D701}})\text{e}^{\text{i}\unicode[STIX]{x1D712}},(\unicode[STIX]{x2202}_{t}\hat{k}_{2})c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}}\text{e}^{\text{i}\unicode[STIX]{x1D712}},0), & \displaystyle\end{eqnarray}$$

(5.21) $$\begin{eqnarray}\displaystyle & \displaystyle (F_{\mathbb{E}1},F_{\mathbb{E}2},F_{\mathbb{E}3})\equiv (0,g\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}(a\text{e}^{k\unicode[STIX]{x1D701}})\text{e}^{\text{i}\unicode[STIX]{x1D712}},0), & \displaystyle\end{eqnarray}$$

(5.22) $$\begin{eqnarray}\displaystyle & \displaystyle (F_{\mathbb{F}1},F_{\mathbb{F}2},F_{\mathbb{F}3})\equiv (\hat{k}_{1}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}(c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}})\text{e}^{\text{i}\unicode[STIX]{x1D712}},0,-\text{i}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}(c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}})\text{e}^{\text{i}\unicode[STIX]{x1D712}}). & \displaystyle\end{eqnarray}$$

Note that, according to the lower-order wave solutions (5.7) and (5.9), the sum $F_{\mathbb{D}i}+F_{\mathbb{E}i}+F_{\mathbb{F}i}$ is the third-order imbalance in the Doppler-shifted linear balance of the lower-order waves; that is, $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}({u_{i}^{w}}^{[1]}+{u_{i}^{w}}^{[2]})+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}(P^{[1]}+P^{[2]})+(u_{h}^{D}+c^{[2]}\hat{k}_{h})\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[1]}=F_{\mathbb{D}i}+F_{\mathbb{E}i}+F_{\mathbb{F}i}+O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k)$ . This imbalance becomes significant when, for example, the current causes a non-zero $\unicode[STIX]{x2202}_{t}a$ or $\unicode[STIX]{x2202}_{t}\hat{k}_{2}$ . Based on these divergence balance and momentum balance, we can decompose the derived third-order solutions in the following way.

5.2.1 $\mathbb{H}$ -wave

The $\mathbb{H}$ -wave corresponds to the homogeneous pressure solution. It exists to satisfy the upper boundary condition for the third harmonics and the horizontal boundary conditions for all harmonics. Define the $\mathbb{H}$ -wave as $u_{\mathbb{H}i}\equiv u_{\mathbb{H}i}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}+\sum _{n=2}^{3}u_{i}^{\{n\}}\text{e}^{\text{i}n\unicode[STIX]{x1D712}}$ , $P_{\mathbb{H}}\equiv P_{\mathbb{H}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}+\sum _{n=2}^{3}P^{\{n\}}\text{e}^{\text{i}n\unicode[STIX]{x1D712}}$ and $z_{\mathbb{H}}\equiv z_{\mathbb{H}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}+\sum _{n=2}^{3}z^{\{n\}}\text{e}^{\text{i}n\unicode[STIX]{x1D712}}$ , where

(5.23a ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\mathbb{H}}^{\{1\}}=C_{(2)}\text{e}^{k\unicode[STIX]{x1D701}}+\mathop{\sum }_{n\geqslant 1}C_{(n,3)}\cos \left(\sqrt{K_{(n)}^{2}-k^{2}}y+C_{(n,4)}\right)\text{e}^{K_{(n)}\unicode[STIX]{x1D701}}, & \displaystyle\end{eqnarray}$$

(5.23b ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\mathbb{H}}^{\{1\}}=\frac{\hat{k}_{1}}{c^{[0]}}P_{\mathbb{H}}^{\{1\}}-\frac{\text{i}\hat{k}_{1}}{g}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ H}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.23c ) $$\begin{eqnarray}\displaystyle & \displaystyle v_{\mathbb{H}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}P_{\mathbb{H}}^{\{1\}}-\frac{1}{gk}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ H}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.23d ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{\mathbb{H}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{H}}^{\{1\}}-\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ H}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.23e ) $$\begin{eqnarray}\displaystyle & \displaystyle z_{\mathbb{H}}^{\{1\}}=\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{ H}}^{\{1\}}-\text{i}\frac{2}{gc^{[0]}k^{2}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{H}}^{\{1\}}. & \displaystyle\end{eqnarray}$$

Here $P^{\{2\}}$ , $u_{i}^{\{2\}}$ , $z^{\{2\}}$ , $P^{\{3\}}$ , $u_{i}^{\{3\}}$ and $z^{\{3\}}$ are given by (D 4) and (D 5), and $C_{(2)}$ , $C_{(n,3)}$ , $C_{(n,4)}$ and $K_{(n)}$ are defined in § A.4. The $\mathbb{H}$ -wave is the only wave type that contains the second and third harmonics. The $\mathbb{H}$ -wave satisfies

(5.24) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{\mathbb{H}}=\unicode[STIX]{x2202}_{i}^{z}{u_{\mathbb{H}}}_{i}=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k), & \displaystyle\end{eqnarray}$$

(5.25) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{\mathbb{H}}}_{i}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P_{\mathbb{H}}=O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k). & \displaystyle\end{eqnarray}$$

Therefore, the $\mathbb{H}$ -wave is divergence-free to third order, and it is driven solely by the boundary conditions.

5.2.2 $\mathbb{A}$ -wave

The $\mathbb{A}$ -wave occurs due to the coexistence of the leading-order wave ${u_{i}^{w}}^{[1]}$ and the vertical shear of the along-roll jet $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{1}^{c}$ . Define the $\mathbb{A}$ -wave as $u_{\mathbb{A}i}\equiv u_{\mathbb{A}i}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , $P_{\mathbb{A}}\equiv P_{\mathbb{A}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ and $z_{\mathbb{A}}\equiv z_{\mathbb{A}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , where $P_{\mathbb{A}}^{\{1\}}=\sum _{n\geqslant 1}P_{\mathbb{A}(n)}^{\{1\}}$ and

(5.26a ) $$\begin{eqnarray}\displaystyle P_{\mathbb{A}(n)}^{\{1\}} & = & \displaystyle \int _{\unicode[STIX]{x1D701}^{\prime }=\unicode[STIX]{x1D701}}^{0}\text{e}^{K_{\mathbb{A}(n)}(\unicode[STIX]{x1D701}-\unicode[STIX]{x1D701}^{\prime })}\frac{\mathbb{A}_{(n)}(x,y,\unicode[STIX]{x1D701}^{\prime },t)}{2K_{\mathbb{A}(n)}}\,\text{d}\unicode[STIX]{x1D701}^{\prime }\nonumber\\ \displaystyle & & \displaystyle +\,\int _{\unicode[STIX]{x1D701}^{\prime }=-\infty }^{\unicode[STIX]{x1D701}}\text{e}^{K_{\mathbb{A}(n)}(\unicode[STIX]{x1D701}^{\prime }-\unicode[STIX]{x1D701})}\frac{\mathbb{A}_{(n)}(x,y,\unicode[STIX]{x1D701}^{\prime },t)}{2K_{\mathbb{A}(n)}}\,\text{d}\unicode[STIX]{x1D701}^{\prime },\end{eqnarray}$$

(5.26b ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\mathbb{A}}^{\{1\}}=\frac{\hat{k}_{1}}{c^{[0]}}P_{\mathbb{A}}^{\{1\}}+\hat{k}_{1}ak\text{e}^{k\unicode[STIX]{x1D701}}\hat{k}_{1}(u_{1}^{c}-u_{1}^{D}), & \displaystyle\end{eqnarray}$$

(5.26c ) $$\begin{eqnarray}\displaystyle & \displaystyle v_{\mathbb{A}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}P_{\mathbb{A}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.26d ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{\mathbb{A}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{A}}^{\{1\}}-\text{i}ak\text{e}^{k\unicode[STIX]{x1D701}}\hat{k}_{1}(u_{1}^{c}-u_{1}^{D}), & \displaystyle\end{eqnarray}$$

(5.26e ) $$\begin{eqnarray}\displaystyle & \displaystyle z_{\mathbb{A}}^{\{1\}}=\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{ A}}^{\{1\}}+\frac{2a}{c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}\hat{k}_{1}(u_{1}^{c}-u_{1}^{D}). & \displaystyle\end{eqnarray}$$

Here $u_{1}^{D}$ is given by (5.15a ), and $\mathbb{A}_{(n)}$ and $K_{\mathbb{A}(n)}$ are given in table 1. The $\mathbb{A}$ -wave is divergent and cancels the undulation-induced divergence of the current shown in (4.1) and figure 3(a), namely,

(5.27) $$\begin{eqnarray}\unicode[STIX]{x2202}_{i}^{z}{u_{\mathbb{ A}}}_{i}-S_{h}^{[1]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}u_{h}^{c}=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k).\end{eqnarray}$$

The $\mathbb{A}$ -wave satisfies the momentum balance as

(5.28) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{\mathbb{ A}}}_{i}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P_{\mathbb{ A}}+F_{\mathbb{A}i}=O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k).\end{eqnarray}$$

Note that $u_{h}^{c}-u_{h}^{D}$ in (5.17) represents the vertically sheared part of the horizontal current. Therefore, $F_{\mathbb{A}i}$ is the advection of the leading-order wave velocities ${u_{i}^{w}}^{[1]}$ by the vertically sheared part of the horizontal current. Figure 5 shows the velocities of the $\mathbb{A}$ -wave due to the current shown in figure 2 and the leading-order wave having $(\hat{k}_{1},\hat{k}_{2})=(1,0)$ . Note that $u_{\mathbb{A}}$ is in phase with $\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}{w^{w}}^{[1]}$ . This correlation is crucial because it converts the oscillatory motion ${w^{w}}^{[1]}$ into a non-oscillatory vertical motion via the advection $u_{\mathbb{A}}\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}{w^{w}}^{[1]}$ at the fourth-order balance. The $\mathbb{A}$ -wave is divergent at the places corresponding to contours 2 and 4 of figure 3(a) to cancel the undulation-induced divergence of the current. The $\mathbb{A}$ -wave ceases towards the upwelling section because the vertical shear of $u_{1}^{c}$ ceases towards the upwelling section in this example (figure 2). We have that $w_{\mathbb{A}}$ is negligible because the first and second terms on the right-hand side of (5.26d ) largely cancel each other. On the other hand, $u_{\mathbb{A}}$ is approximately twice as large as each term on the right-hand side of (5.26b ) because the two terms on the right-hand side of (5.26b ) are similar to each other. The profile of $u_{1}^{c}$ in this example and the corresponding $u_{1}^{D}$ computed from (5.15a ) are shown in figure 6.

Figure 5.

The $\mathbb{A}$ -wave at the downwelling section ( $ky=0$ ), at the middle section ( $ky=0.9\unicode[STIX]{x03C0}$ ) and at the upwelling section ( $ky=1.8\unicode[STIX]{x03C0}$ ) shown in figure 2. The colour shows $u_{\mathbb{A}i}/(\unicode[STIX]{x1D716}^{3}c^{[0]})$ .

Figure 6.

Plots of $u_{1}^{c}$ (solid) and $u_{1}^{D}$ (dashed) at the downwelling section ( $ky=0$ ), at the middle section ( $ky=0.9\unicode[STIX]{x03C0}$ ) and at the upwelling section ( $ky=1.8\unicode[STIX]{x03C0}$ ) in figure 2. The velocities in (c) are zero.

Figure 7.

The $\mathbb{B}$ -wave at the downwelling section ( $ky=0$ ), at the middle section ( $ky=0.9\unicode[STIX]{x03C0}$ ) and at the upwelling section ( $ky=1.8\unicode[STIX]{x03C0}$ ) shown in figure 2. The colour shows $u_{\mathbb{B}i}/(\unicode[STIX]{x1D716}^{3}c^{[0]})$ .

5.2.3 $\mathbb{B}$ -wave

The $\mathbb{B}$ -wave occurs due to the coexistence of the leading-order wave and the roll. Define the $\mathbb{B}$ -wave as $u_{\mathbb{B}i}\equiv u_{\mathbb{B}i}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , $P_{\mathbb{B}}\equiv P_{\mathbb{B}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ and $z_{\mathbb{B}}\equiv z_{\mathbb{B}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , where $P_{\mathbb{B}}^{\{1\}}=\sum _{n\geqslant 1}P_{\mathbb{B}(n)}^{\{1\}}$ and

(5.29a ) $$\begin{eqnarray}\displaystyle P_{\mathbb{B}(n)}^{\{1\}} & = & \displaystyle \int _{\unicode[STIX]{x1D701}^{\prime }=\unicode[STIX]{x1D701}}^{0}\text{e}^{K_{\mathbb{B}(n)}(\unicode[STIX]{x1D701}-\unicode[STIX]{x1D701}^{\prime })}\frac{\mathbb{B}_{(n)}(x,y,\unicode[STIX]{x1D701}^{\prime },t)}{2K_{\mathbb{B}(n)}}\,\text{d}\unicode[STIX]{x1D701}^{\prime }\nonumber\\ \displaystyle & & \displaystyle +\,\int _{\unicode[STIX]{x1D701}^{\prime }=-\infty }^{\unicode[STIX]{x1D701}}\text{e}^{K_{\mathbb{B}(n)}(\unicode[STIX]{x1D701}^{\prime }-\unicode[STIX]{x1D701})}\frac{\mathbb{B}_{(n)}(x,y,\unicode[STIX]{x1D701}^{\prime },t)}{2K_{\mathbb{B}(n)}}\,\text{d}\unicode[STIX]{x1D701}^{\prime },\end{eqnarray}$$

(5.29b ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\mathbb{B}}^{\{1\}}=\frac{\hat{k}_{1}}{c^{[0]}}P_{\mathbb{B}}^{\{1\}}-\text{i}\hat{k}_{1}ak\text{e}^{k\unicode[STIX]{x1D701}}w^{c}, & \displaystyle\end{eqnarray}$$

(5.29c ) $$\begin{eqnarray}\displaystyle & \displaystyle v_{\mathbb{B}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}P_{\mathbb{B}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.29d ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{\mathbb{B}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{B}}^{\{1\}}-ak\text{e}^{k\unicode[STIX]{x1D701}}w^{c}, & \displaystyle\end{eqnarray}$$

(5.29e ) $$\begin{eqnarray}\displaystyle & \displaystyle z_{\mathbb{B}}^{\{1\}}=\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{ B}}^{\{1\}}-\text{i}\frac{a}{c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}w^{c}. & \displaystyle\end{eqnarray}$$

The $\mathbb{B}_{(n)}$ and $K_{\mathbb{B}(n)}$ are given in table 1. The $\mathbb{B}$ -wave exists to cancel the undulation-induced divergence of the current due to $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w^{c}$ and $S_{3}$ shown in (4.1) and figure 3(b), namely,

(5.30) $$\begin{eqnarray}\unicode[STIX]{x2202}_{i}^{z}{u_{\mathbb{ B}}}_{i}-S_{3}^{[1]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w^{c}=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k).\end{eqnarray}$$

Therefore, according to (4.1), (5.27) and (5.30), the $\mathbb{A}$ - and $\mathbb{B}$ -waves cancel the current divergence to third order: $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}^{c}+\boldsymbol{u}_{\mathbb{A}}+\boldsymbol{u}_{\mathbb{B}})=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k)$ . The $\mathbb{B}$ -wave satisfies

(5.31) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{\mathbb{ B}}}_{i}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P_{\mathbb{ B}}+F_{\mathbb{B}i}=O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k).\end{eqnarray}$$

The $F_{\mathbb{B}i}$ defined in (5.18) is the vertical advection of the leading-order wave momentum by the vertical current. Figure 7 shows the velocities of the $\mathbb{B}$ -wave due to the current shown in figure 2 and the leading-order wave having $(\hat{k}_{1},\hat{k}_{2})=(1,0)$ . The $\mathbb{B}$ -wave is divergent at the places corresponding to contours 1 and 3 of figure 3(b). Unlike the $\mathbb{A}$ -wave, the $\mathbb{B}$ -wave has significant $v_{\mathbb{B}}$ and $w_{\mathbb{B}}$ .

5.2.4 $\mathbb{C}$ -wave

The $\mathbb{C}$ -wave exists for the same reason as the $\mathbb{A}$ -wave except that the relevant flow is $u_{1}^{c}$ for the $\mathbb{A}$ -wave and $\mathfrak{U}_{1}$ for the $\mathbb{C}$ -wave. Define the $\mathbb{C}$ -wave as $u_{\mathbb{C}i}\equiv u_{\mathbb{C}i}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , $P_{\mathbb{C}}\equiv P_{\mathbb{C}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ and $z_{\mathbb{C}}\equiv z_{\mathbb{C}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , where

(5.32a ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\mathbb{C}}^{\{1\}}=\int _{\unicode[STIX]{x1D701}^{\prime }=\unicode[STIX]{x1D701}}^{0}\text{e}^{k(\unicode[STIX]{x1D701}-\unicode[STIX]{x1D701}^{\prime })}\frac{\mathbb{C}(x,y,\unicode[STIX]{x1D701}^{\prime },t)}{2k}\,\text{d}\unicode[STIX]{x1D701}^{\prime }+\int _{\unicode[STIX]{x1D701}^{\prime }=-\infty }^{\unicode[STIX]{x1D701}}\text{e}^{k(\unicode[STIX]{x1D701}^{\prime }-\unicode[STIX]{x1D701})}\frac{\mathbb{C}(x,y,\unicode[STIX]{x1D701}^{\prime },t)}{2k}\,\text{d}\unicode[STIX]{x1D701}^{\prime },\quad & \displaystyle\end{eqnarray}$$

(5.32b ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\mathbb{C}}^{\{1\}}=\frac{\hat{k}_{1}}{c^{[0]}}P_{\mathbb{C}}^{\{1\}}+\hat{k}_{1}ak\text{e}^{k\unicode[STIX]{x1D701}}\left(\hat{k}_{1}\mathfrak{U}_{1}-c^{[2]}+\frac{1}{2}c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\right)+\text{i}\hat{k}_{1}\frac{a}{c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}c^{[2]},\quad & \displaystyle\end{eqnarray}$$

(5.32c ) $$\begin{eqnarray}\displaystyle & \displaystyle v_{\mathbb{C}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}P_{\mathbb{C}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.32d ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{\mathbb{C}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{C}}^{\{1\}}-\text{i}ak\text{e}^{k\unicode[STIX]{x1D701}}\left(\hat{k}_{1}\mathfrak{U}_{1}-c^{[2]}+\frac{1}{2}c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\right)+\frac{a}{c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}c^{[2]},\quad & \displaystyle\end{eqnarray}$$

(5.32e ) $$\begin{eqnarray}\displaystyle & \displaystyle z_{\mathbb{C}}^{\{1\}}=\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{ C}}^{\{1\}}+\frac{2a}{c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}\left(\hat{k}_{1}\mathfrak{U}_{1}-c^{[2]}+\frac{3}{8}c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\right)+\text{i}\frac{3a}{g}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}c^{[2]}. & \displaystyle\end{eqnarray}$$

Here, $\mathbb{C}$ is defined in table 1, and $c^{[2]}$ and $\unicode[STIX]{x2202}_{t}c^{[2]}$ are given by (5.15b ) and (D 3b ). The $\mathbb{C}$ -wave is divergent and cancels the undulation-induced divergence of $\mathfrak{U}_{h}$ , namely,

(5.33) $$\begin{eqnarray}\unicode[STIX]{x2202}_{i}^{z}u_{\mathbb{ C}i}-S_{h}^{[1]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{h}=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k).\end{eqnarray}$$

Then, because $\unicode[STIX]{x2202}_{i}^{z}\mathfrak{U}_{i}=-S_{h}^{[1]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{h}+O(\unicode[STIX]{x1D716}^{4}c^{[0]}k)$ according to the chain rule (2.15) and the scaling conditions in § 3, the $\mathbb{C}$ -wave cancels the divergence of $\mathfrak{U}_{i}$ to third order: that is, $\unicode[STIX]{x2202}_{i}^{z}(u_{\mathbb{C}i}+\mathfrak{U}_{i})=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k)$ . The $\mathbb{C}$ -wave is driven by $F_{\mathbb{C}i}$ as

(5.34) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{\mathbb{ C}}}_{i}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P_{\mathbb{ C}}+F_{\mathbb{C}i}=O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k).\end{eqnarray}$$

According to the first- and second-order wave solutions (5.7) and (5.9) as well as the scaling conditions in § 3, the term surrounded by the square brackets in (5.19) satisfies

(5.35) $$\begin{eqnarray}[u_{h}^{w}\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}u_{i}^{w}-S_{i}J^{-1}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P]=(\mathfrak{U}_{h}+{\textstyle \frac{1}{2}}c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\hat{k}_{h})\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}{u_{i}^{w}}^{[1]}+O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k).\end{eqnarray}$$

Therefore, $F_{\mathbb{C}i}$ is analogous to $F_{\mathbb{A}i}$ , with $u_{h}^{c}$ corresponding to $\mathfrak{U}_{h}+(1/2)c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\hat{k}_{h}$ and $u_{h}^{D}$ corresponding to $c^{[2]}\hat{k}_{h}$ .

5.2.5 $\mathbb{D}$ -wave

The $\mathbb{D}$ -wave exists due to the divergence of the $\unicode[STIX]{x1D6E9}$ -dependent motions of the lower-order waves (i.e.  ${u_{i}^{w}}^{[1]}+{u_{i}^{w}}^{[2]}-\mathfrak{U}_{i}$ ). This divergence becomes non-zero to third order when the wave properties ( $a$ , $k$ , $\hat{k}_{h}$ ) vary. Define the $\mathbb{D}$ -wave as $u_{\mathbb{D}i}\equiv u_{\mathbb{D}i}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , $P_{\mathbb{D}}\equiv P_{\mathbb{D}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ and $z_{\mathbb{D}}\equiv z_{\mathbb{D}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , where

(5.36a ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\mathbb{D}}^{\{1\}}=\unicode[STIX]{x1D6FC}_{\mathbb{D}}\unicode[STIX]{x1D701}^{2}\text{e}^{k\unicode[STIX]{x1D701}}+\unicode[STIX]{x1D6FD}_{\mathbb{D}}\unicode[STIX]{x1D701}\text{e}^{k\unicode[STIX]{x1D701}}+\unicode[STIX]{x1D6FE}_{\mathbb{D}}\text{e}^{k\unicode[STIX]{x1D701}}, & \displaystyle\end{eqnarray}$$

(5.36b ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\mathbb{D}}^{\{1\}}=\frac{\hat{k}_{1}}{c^{[0]}}P_{\mathbb{D}}^{\{1\}}-\frac{\text{i}\hat{k}_{1}}{g}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ D}}^{\{1\}}-\text{i}a\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}\hat{k}_{1}-\text{i}c^{[0]}\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}(a\text{e}^{k\unicode[STIX]{x1D701}}), & \displaystyle\end{eqnarray}$$

(5.36c ) $$\begin{eqnarray}\displaystyle & \displaystyle v_{\mathbb{D}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}P_{\mathbb{D}}^{\{1\}}-\frac{1}{gk}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ D}}^{\{1\}}-\text{i}a\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}\hat{k}_{2}-\frac{a}{c^{[0]}k}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}\hat{k}_{2}, & \displaystyle\end{eqnarray}$$

(5.36d ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{\mathbb{D}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{D}}^{\{1\}}-\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ D}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.36e ) $$\begin{eqnarray}\displaystyle & \displaystyle z_{\mathbb{D}}^{\{1\}}=\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{ D}}^{\{1\}}-\text{i}\frac{2}{gc^{[0]}k^{2}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{D}}^{\{1\}}. & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D6FC}_{\mathbb{D}}$ , $\unicode[STIX]{x1D6FD}_{\mathbb{D}}$ and $\unicode[STIX]{x1D6FE}_{\mathbb{D}}$ are defined in table 1; $\unicode[STIX]{x2202}_{t}\hat{k}_{1}$ , $\unicode[STIX]{x2202}_{t}\hat{k}_{2}$ and $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}\hat{k}_{2}$ are given by (D 3c ), (C 5) and (D 3d ); and $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{D}}^{\{1\}}$ to third order is given by (5.36a ) and (C 5). The $\mathbb{D}$ -wave satisfies

(5.37) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{i}^{z}({u_{\mathbb{D}}}_{i}+{u_{i}^{w}}^{[1]}+{u_{i}^{w}}^{[2]}-\mathfrak{U}_{i})=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k), & \displaystyle\end{eqnarray}$$

(5.38) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{\mathbb{D}}}_{i}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P_{\mathbb{D}}+F_{\mathbb{D}i}=O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k). & \displaystyle\end{eqnarray}$$

Therefore, the $\mathbb{D}$ -wave cancels the divergence of the lower-order $\unicode[STIX]{x1D6E9}$ -dependent motions, and it is forced by the imbalance $F_{\mathbb{D}i}$ due to a non-zero $\unicode[STIX]{x2202}_{t}\hat{k}_{h}$ , $\unicode[STIX]{x2202}_{1}a$ or $\unicode[STIX]{x2202}_{1}k$ in (5.20).

5.2.6 $\mathbb{E}$ -wave

The $\mathbb{E}$ -wave exists due to a non-zero $\unicode[STIX]{x2202}_{2}a$ or $\unicode[STIX]{x2202}_{2}k$ . Define the $\mathbb{E}$ -wave as $u_{\mathbb{E}i}\equiv u_{\mathbb{E}i}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , $P_{\mathbb{E}}\equiv P_{\mathbb{E}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ and $z_{\mathbb{E}}\equiv z_{\mathbb{E}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , where

(5.39a ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\mathbb{E}}^{\{1\}}=-\frac{g}{2k}(\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}a)\unicode[STIX]{x1D701}\text{e}^{k\unicode[STIX]{x1D701}}+\frac{g}{8k^{2}}(\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}a)\text{e}^{k\unicode[STIX]{x1D701}}, & \displaystyle\end{eqnarray}$$

(5.39b ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\mathbb{E}}^{\{1\}}=\frac{\hat{k}_{1}}{c^{[0]}}P_{\mathbb{E}}^{\{1\}}-\frac{\text{i}\hat{k}_{1}}{g}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ E}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.39c ) $$\begin{eqnarray}\displaystyle v_{\mathbb{E}}^{\{1\}} & = & \displaystyle -\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}P_{\mathbb{E}}^{\{1\}}-\frac{1}{gk}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ E}}^{\{1\}}-\text{i}c^{[0]}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}(a\text{e}^{k\unicode[STIX]{x1D701}})\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{k}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{t}a+\frac{\text{i}}{c^{[0]}k^{2}}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a,\end{eqnarray}$$

(5.39d ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{\mathbb{E}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{E}}^{\{1\}}-\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{ E}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.39e ) $$\begin{eqnarray}\displaystyle & \displaystyle z_{\mathbb{E}}^{\{1\}}=\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{ E}}^{\{1\}}-\text{i}\frac{2}{gc^{[0]}k^{2}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{E}}^{\{1\}}. & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x2202}_{t}a$ and $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a$ are given by (5.14a ) and (D 3a ). Equation (5.14a ) and (5.39a ) give $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P_{\mathbb{E}}^{\{1\}}$ to third order. The $\mathbb{E}$ -wave satisfies

(5.40) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{i}^{z}{u_{\mathbb{E}}}_{i}=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k), & \displaystyle\end{eqnarray}$$

(5.41) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{\mathbb{E}}}_{i}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P_{\mathbb{E}}+F_{\mathbb{E}i}=O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k). & \displaystyle\end{eqnarray}$$

Therefore, the $\mathbb{E}$ -wave is divergence-free to third order, and it is forced by the imbalance $F_{\mathbb{E}i}$ due to a non-zero $\unicode[STIX]{x2202}_{2}a$ or $\unicode[STIX]{x2202}_{2}k$ in (5.21).

5.2.7 $\mathbb{F}$ -wave

The $\mathbb{F}$ -wave exists due to a temporal change in $a$ or $k$ . Define the $\mathbb{F}$ -wave as $u_{\mathbb{F}i}\equiv u_{\mathbb{F}i}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , $P_{\mathbb{F}}\equiv P_{\mathbb{F}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ and $z_{\mathbb{F}}\equiv z_{\mathbb{F}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}$ , where

(5.42a ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\mathbb{F}}^{\{1\}}=\text{i}\frac{c^{[0]}a}{2}(\unicode[STIX]{x2202}_{t}k)\unicode[STIX]{x1D701}\text{e}^{k\unicode[STIX]{x1D701}}, & \displaystyle\end{eqnarray}$$

(5.42b ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\mathbb{F}}^{\{1\}}=\frac{\hat{k}_{1}}{c^{[0]}}P_{\mathbb{F}}^{\{1\}}-\text{i}\frac{\hat{k}_{1}}{c^{[0]}k}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}(c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}})-\frac{\hat{k}_{1}}{c^{[0]}k}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a, & \displaystyle\end{eqnarray}$$

(5.42c ) $$\begin{eqnarray}\displaystyle & \displaystyle v_{\mathbb{F}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}P_{\mathbb{F}}^{\{1\}}, & \displaystyle\end{eqnarray}$$

(5.42d ) $$\begin{eqnarray}\displaystyle & \displaystyle w_{\mathbb{F}}^{\{1\}}=-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{F}}^{\{1\}}-\frac{1}{c^{[0]}k}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}(c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}})+\text{i}\frac{1}{c^{[0]}k}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a, & \displaystyle\end{eqnarray}$$

(5.42e ) $$\begin{eqnarray}\displaystyle & \displaystyle z_{\mathbb{F}}^{\{1\}}=\frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P_{\mathbb{ F}}^{\{1\}}-\frac{\text{i}}{c^{[0]}k}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}(a\text{e}^{k\unicode[STIX]{x1D701}})-\frac{\text{i}}{gk}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}(c^{[0]}ak\text{e}^{k\unicode[STIX]{x1D701}})-\frac{3}{gk}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a. & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x2202}_{t}a$ and $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a$ are given by (5.14a ) and (D 3a ). The $\mathbb{F}$ -wave satisfies

(5.43) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{i}^{z}{u_{\mathbb{F}}}_{i}=O(\unicode[STIX]{x1D716}^{4}c^{[0]}k), & \displaystyle\end{eqnarray}$$

(5.44) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}{u_{\mathbb{F}}}_{i}+\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}P_{\mathbb{F}}+F_{\mathbb{F}i}=O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k). & \displaystyle\end{eqnarray}$$

Therefore, the $\mathbb{F}$ -wave is divergence-free to third order, and it is forced by the imbalance $F_{\mathbb{F}i}$ due to a non-zero $\unicode[STIX]{x2202}_{t}a$ or $\unicode[STIX]{x2202}_{t}k$ in (5.22).

5.3 Wave action density conservation

The dispersion relations (5.7e ) and (5.9e ) and the phase function (2.2) determine $\unicode[STIX]{x2202}_{t}c^{[0]}$ and $\unicode[STIX]{x2202}_{t}k$ as (5.11) and (5.13), respectively. Meanwhile, $\unicode[STIX]{x2202}_{t}a$ is required by the governing equations to satisfy (5.14). These equations then yield the time derivative of $c^{[0]}a^{2}/2$ as

(5.45) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\frac{c^{[0]}a^{2}}{2}+c_{h}^{g}\unicode[STIX]{x2202}_{h}\frac{c^{[0]}a^{2}}{2}=-\frac{c^{[0]}a^{2}}{2}D^{g}-\frac{c^{[0]}a}{2}\,\text{Re}(G)-\frac{a}{4k}\text{i}\unicode[STIX]{x2202}_{t}\,\text{Im}(G)+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k^{-1}),\end{eqnarray}$$

where $c_{h}^{g}$ , $D^{g}$ and $G$ are defined by (5.14b )–(5.14d ). Because of (5.7e ), the quantity $c^{[0]}a^{2}/2$ is equal to the wave energy density (divided by the water density) $ga^{2}/2$ divided by the intrinsic frequency $\sqrt{gk}$ . Thus, $c^{[0]}a^{2}/2$ can be recognised as the leading-order wave action density.

If $w^{c}=w_{(2)}^{c}$ in (3.3), then $w_{(n)}^{c}=0$ for $n\neq 2$ and $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w_{(2)}^{c}=-\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}v^{c}+O(\unicode[STIX]{x1D716}^{3}c^{[0]}k)$ according to the current’s scales listed in § 3. Thus, in this case, (5.14c ) reduces to

(5.46) $$\begin{eqnarray}D^{g}=\unicode[STIX]{x2202}_{h}(c_{h}^{g}+u_{h}^{g})+O(\unicode[STIX]{x1D716}^{3}c^{[0]}k),\end{eqnarray}$$

where

(5.47) $$\begin{eqnarray}u_{h}^{g}\equiv \frac{k(K_{\mathbb{B}(2)}+k)}{K_{\mathbb{B}(2)}}\int _{\unicode[STIX]{x1D701}^{\prime }=-\infty }^{0}\text{e}^{(K_{\mathbb{B}(2)}+k)\unicode[STIX]{x1D701}^{\prime }}u_{h}^{c}(x,y,\unicode[STIX]{x1D701}^{\prime },t)\,\text{d}\unicode[STIX]{x1D701}^{\prime }.\end{eqnarray}$$

Therefore, in this case, (5.45) and (5.46) yield

(5.48) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\frac{c^{[0]}a^{2}}{2}+\unicode[STIX]{x2202}_{h}\left((c_{h}^{g}+u_{h}^{g})\frac{c^{[0]}a^{2}}{2}\right)=-\frac{c^{[0]}a}{2}\,\text{Re}(G)-\frac{a}{4k}\text{i}\unicode[STIX]{x2202}_{t}\,\text{Im}(G)+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k^{-1}).\end{eqnarray}$$

Then, $c_{h}^{g}+u_{h}^{g}$ is the velocity appropriate for the flux of the action in the sheared currents.

When $G$ and $w^{c}$ are negligible, equation (5.45) reduces to the leading-order expression of the conventional equation for an inviscid, deep-water wave without wind forcing (Bretherton & Garrett Reference Bretherton and Garrett1968). Therefore, equation (5.45) generalises the wave action density conservation to include the effect of the roll and the along-roll jet. It shows that the upwelling (i.e.  $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w^{c}<0$ in (5.14c )) of the roll has an effect of reducing the action density, and the downwelling ( $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}w^{c}>0$ ) of the roll has an effect of increasing the action density.

5.4 Evolution of the wave energy and $U_{h}^{S}$

According to (5.14), the wave energy density (divided by the water density) evolves as

(5.49) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}E+c_{h}^{g}\unicode[STIX]{x2202}_{h}E=-ED^{g}-\frac{ga}{2}\,\text{Re}(G)-\frac{c^{[0]}a}{4}\text{i}\unicode[STIX]{x2202}_{t}\,\text{Im}(G)+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{3}),\end{eqnarray}$$

where $E\equiv ga^{2}/2$ . Note that the advection of $E$ by the current is of sixth order.

Combining (5.12), (5.13), (5.45) and the identities listed in appendix C yields

(5.50a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{1}^{S}+c_{H}^{g}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}U_{1}^{S} & = & \displaystyle -U_{1}^{S}D^{g}-c^{[0]}ak^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\hat{k}_{1}\,\text{Re}(G)\nonumber\\ \displaystyle & & \displaystyle -\,\frac{ak}{2}\text{e}^{2k\unicode[STIX]{x1D701}}\hat{k}_{1}\text{i}\unicode[STIX]{x2202}_{t}\,\text{Im}(G)+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k),\end{eqnarray}$$

(5.50b ) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{2}^{S}+c_{H}^{g}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}U_{2}^{S}=-U_{1}^{S}\unicode[STIX]{x2202}_{2}(u_{1}^{D}+c^{[2]}\hat{k}_{1})+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k),\end{eqnarray}$$

$U_{h}^{S}\equiv c^{[0]}a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\hat{k}_{h}$

$\unicode[STIX]{x2202}_{t}E$

$\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{1}^{S}$

$D^{g}$

5.5 Thickness-weighted ensemble average of the wave momentum

According to (2.5), (2.6) and (2.14), the thickness-weighted ensemble average of the wave momentum is

(5.51) $$\begin{eqnarray}u_{i}^{S}(x,y,\unicode[STIX]{x1D701},t)\equiv \frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}J(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})u_{i}^{w}(x,y,\unicode[STIX]{x1D701},t,\unicode[STIX]{x1D6E9})\,\text{d}\unicode[STIX]{x1D6E9}\equiv \langle Ju_{i}^{w}\rangle =\mathfrak{U}_{i}+\langle S_{3}u_{i}^{w}\rangle .\end{eqnarray}$$

Note that $Ju_{h}^{w}$ is equal to the flux of volume through the relevant sidewall of the volume element of the $\unicode[STIX]{x1D701}$ -coordinate system (see figures 1 and 3 for the volume elements). The values of $\langle S_{3}u_{h}^{w}\rangle$ computed using the wave solutions in §§ 5.1 and 5.2 and appendices D and E are shown in (F 1) and (F 2) in appendix F. These results (F 1) and (F 2) show that $u_{h}^{S}$ is $u_{h}^{S}=\mathfrak{U}_{h}+(1/2)U_{h}^{S}$ to third order but has a complex expression at fourth order. The terms $(1/2)U_{h}^{S}$ in (F 1) and (F 2) result from the correlation between ${u_{h}^{w}}^{[1]}$ and $J^{[1]}={S_{3}}^{[1]}$ .

6.1 The pseudo-incompressibility necessary for the wave-averaged governing equations

In general, it is possible to define different types of wave-averaged flows such as the ensemble average $\langle \boldsymbol{u}\rangle$ (as defined in (2.5)), the thickness-weighted ensemble average $\langle J\boldsymbol{u}\rangle$ , an Eulerian average and the generalised Lagrangian mean. Regardless of the averaging method used in a given theory, the wave-averaged flow (say, $\bar{\boldsymbol{u}}$ ) in the given theory is forced by the corresponding wave-averaged pressure (say, $\bar{p}$ ) typically in the form $\bar{\unicode[STIX]{x2202}}_{t}\bar{u}_{i}+\bar{\unicode[STIX]{x2202}}_{i}\bar{p}=\bar{F}_{i}$ , where $\bar{\unicode[STIX]{x2202}}_{\ell }$ and $\bar{F}_{i}$ are the gradient operators and forces in the given theory. For an incompressible fluid, $\bar{p}$ is typically an unknown variable, and its solution must be obtained by solving the Poisson equation typically formulated as $\bar{\unicode[STIX]{x2202}}_{i}\bar{\unicode[STIX]{x2202}}_{t}\bar{u}_{i}+\bar{\unicode[STIX]{x2202}}_{i}\bar{\unicode[STIX]{x2202}}_{i}\bar{p}=\bar{\unicode[STIX]{x2202}}_{i}\bar{F}_{i}$ . Therefore, the Poisson equation – hence, the governing equations of $\bar{\boldsymbol{u}}$ and $\bar{p}$ – can be solved only if the unwanted unknown variable $\bar{\unicode[STIX]{x2202}}_{i}\bar{\unicode[STIX]{x2202}}_{t}\bar{u}_{i}$ is zero (or takes a known value). However, in general, the property that $\bar{\unicode[STIX]{x2202}}_{i}\bar{u}_{i}=0$ – hence, $\bar{\unicode[STIX]{x2202}}_{i}\bar{\unicode[STIX]{x2202}}_{t}\bar{u}_{i}=0$ – may not be satisfied even if the fluid is incompressible (i.e.  $\unicode[STIX]{x2202}_{i}^{z}u_{i}=0$ ). Therefore, it is crucial that the theory is built for a solvable wave-averaged flow, which satisfies $\bar{\unicode[STIX]{x2202}}_{i}\bar{u}_{i}=0$ . Hereafter, let us call this property (i.e.  $\bar{\unicode[STIX]{x2202}}_{i}\bar{u}_{i}=0$ ) the pseudo-incompressibility of $\bar{\boldsymbol{u}}$ , as opposed to the incompressibility of $\bar{\boldsymbol{u}}$ (i.e.  $\unicode[STIX]{x2202}_{i}^{z}\bar{u}_{i}=0$ ). The solvability for $\bar{\boldsymbol{u}}$ requires the pseudo-incompressibility of $\bar{\boldsymbol{u}}$ , rather than its incompressibility.

To this end, consider any velocity field $\vec{\mathbb{U}}$ and its thickness-weighted divergence $J\unicode[STIX]{x2202}_{i}^{z}\mathbb{U}_{i}$ . According to the chain rule (2.15) and the identities $\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}J=\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}S_{3}$ and $\unicode[STIX]{x2202}_{\ell }^{\unicode[STIX]{x1D701}}S_{m}=\unicode[STIX]{x2202}_{m}^{\unicode[STIX]{x1D701}}S_{\ell }$ (for $m=1,2,3,t$ ), the thickness-weighted divergence of $\vec{\mathbb{U}}$ satisfies the following identity:

(6.1) $$\begin{eqnarray}J\unicode[STIX]{x2202}_{i}^{z}\mathbb{U}_{i}=J\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}\mathbb{U}_{h}-S_{h}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\mathbb{U}_{h}+\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\mathbb{U}_{3}=\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}(J\mathbb{U}_{h})+\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}(\mathbb{U}_{3}-S_{h}\mathbb{U}_{h}).\end{eqnarray}$$

The right-hand side is the net mass transfer per second (divided by the mass of the undeformed element $\unicode[STIX]{x1D70C}_{0}\,\text{d}x\,\text{d}y\,\text{d}\unicode[STIX]{x1D701}$ ) due to $\vec{\mathbb{U}}$ across the surfaces of the instantaneous (or frozen) volume element of the $\unicode[STIX]{x1D701}$ -coordinate system. (See figures 1 and 3 for the volume elements of the $\unicode[STIX]{x1D701}$ -coordinate system.) Physically, $J\mathbb{U}_{1}$ and $J\mathbb{U}_{2}$ are the mass fluxes through the sidewalls of the volume element. On the other hand, $\mathbb{U}_{3}-S_{h}\mathbb{U}_{h}$ is the mass flux through the tilted top or bottom wall of the volume element. Here, $\mathbb{U}_{3}$ is the mass flux through the horizontal-plane projection of the tilted surface and $-S_{h}\mathbb{U}_{h}$ is the sum of the mass fluxes through the vertical-plane projections of the tilted top or bottom surface. The term $\mathbb{U}_{3}-S_{h}\mathbb{U}_{h}$ is also the vertical velocity of the tilted surface due to $\vec{\mathbb{U}}$ (cf. (2.9)). Now, the ensemble average (2.5) of (6.1) yields $\langle J\unicode[STIX]{x2202}_{i}^{z}\mathbb{U}_{i}\rangle =\unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}\langle J\mathbb{U}_{h}\rangle +\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\langle \mathbb{U}_{3}-S_{h}\mathbb{U}_{h}\rangle$ . Therefore,

(6.2) $$\begin{eqnarray}\langle J\unicode[STIX]{x2202}_{i}^{z}\mathbb{U}_{i}\rangle =0\quad \Longleftrightarrow \quad \unicode[STIX]{x2202}_{h}^{\unicode[STIX]{x1D701}}\langle J\mathbb{U}_{h}\rangle +\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\langle \mathbb{U}_{3}-S_{h}\mathbb{U}_{h}\rangle =0.\end{eqnarray}$$

That is, the pseudo-incompressibility of the velocity field $(\langle J\mathbb{U}_{h}\rangle ,\langle \mathbb{U}_{3}-S_{h}\mathbb{U}_{h}\rangle )$ is equivalent to the property that the thickness-weighted ensemble average of $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\vec{\mathbb{U}}$ is zero.

Because the fluid velocity $\boldsymbol{u}$ is incompressible, equation (6.2) for $\vec{\mathbb{U}}=\boldsymbol{u}$ yields

(6.3) $$\begin{eqnarray}\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}U_{i}=0,\end{eqnarray}$$

where $\boldsymbol{U}\equiv (U_{1},U_{2},U_{3})\equiv (U,V,W)$ and

(6.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle U\equiv \langle Ju\rangle =\langle u\rangle +\langle S_{3}u\rangle =\langle u^{c}\rangle +\mathfrak{U}_{1}+\langle S_{3}u\rangle , & \displaystyle\end{eqnarray}$$

(6.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle V\equiv \langle Jv\rangle =\langle v\rangle +\langle S_{3}v\rangle =\langle v^{c}\rangle +\mathfrak{U}_{2}+\langle S_{3}v\rangle , & \displaystyle\end{eqnarray}$$

(6.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle W\equiv \langle w-S_{h}u_{h}\rangle =\langle w\rangle -\langle S_{h}u_{h}\rangle =\langle w^{c}\rangle +\mathfrak{U}_{3}-\langle S_{h}u_{h}\rangle . & \displaystyle\end{eqnarray}$$

$\boldsymbol{U}$

$\langle u_{i}^{c}\rangle +\mathfrak{U}_{i}$

$\langle S_{3}u_{h}\rangle$

$\langle -S_{h}u_{h}\rangle$

$\boldsymbol{U}$

$\boldsymbol{U}$

$\boldsymbol{U}$

$\langle J\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\rangle =0$

$\boldsymbol{U}$

$\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}U_{i}$

$\langle J\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}\rangle$

$\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}$

Unlike the wave-averaged material flow $\boldsymbol{U}$ , other wave-averaged flows – namely, $\langle \boldsymbol{u}\rangle$ , $\langle \boldsymbol{u}^{c}\rangle$ and $\boldsymbol{u}^{Q}\equiv (\langle u_{h}\rangle -U_{h}^{S}/2,\langle w\rangle )$ – are not guaranteed to satisfy their pseudo-incompressibility (appendix G). Therefore, only $\boldsymbol{U}$ is appropriately solvable. This is a crucial point which differentiates S19 from the CL theory. In the latter, the Eulerian mean velocity (which is closely related to $\langle \boldsymbol{u}^{c}\rangle$ ) is assumed to be pseudo-incompressible (and also incompressible) and is thereby considered to govern the wave-averaged pressure. In contrast, in S19, $\langle \boldsymbol{u}^{c}\rangle$ is neither pseudo-incompressible nor incompressible, and the wave-averaged pressure is governed by $\boldsymbol{U}$ .

Table 2.

The non-zero terms in the ensemble-averaged horizontal momentum equations.

Table 3.

The non-zero terms in the ensemble-averaged vertical momentum equation.

Table 4.

The orders of the derivatives of $u_{i}^{c}$ and $\mathfrak{U}_{i}$ . The order of a higher derivative that is not listed in the table is equal to or higher than the highest order of the listed derivatives contained in the higher derivative: e.g.  $\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}u_{i}^{c}\leqslant O(\unicode[STIX]{x1D716}^{3}c^{[0]}k^{2})$ because $\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}u_{i}^{c}\leqslant O(\unicode[STIX]{x1D716}^{3}c^{[0]}k)$ .

Table 5.

The orders of the derivatives of $a$ , $k$ , $\hat{k}_{h}$ , $c^{[0]}$ and $c^{[2]}$ . For $a$ , any derivative not included in the table is of fourth or higher order. For $k$ , $\hat{k}_{h}$ , $c^{[0]}$ and $c^{[2]}$ , any derivative left blank or not included in the table is of third or higher order.

6.2 The momentum equations and upper boundary condition of $\boldsymbol{U}$

Substitution of the first- to third-order wave solutions presented in § 5 and appendix E into (5.1) yields the momentum equations of $\boldsymbol{u}$ at fourth order. These momentum equations have some terms that have a non-zero ensemble average to fourth order; all of these terms are listed in tables 2 and 3. According to these results, the momentum equations of $\boldsymbol{U}$ are (as derived in appendix H)

(6.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{1}+(U_{H}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}+W\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}})(U_{1}-U_{1}^{S})+\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}\langle P^{[4]}\rangle +\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D6F9}-\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{1}^{S}+\unicode[STIX]{x1D6EC}_{1}=O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k),\qquad & \displaystyle\end{eqnarray}$$

(6.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{2}+(U_{H}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}+W\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}})(U_{2}-U_{2}^{S})+\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}\langle P^{[4]}\rangle +\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D6F9}+c_{H}^{g}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}U_{2}^{S}=O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k),\qquad & \displaystyle\end{eqnarray}$$

(6.5c ) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}W+(U_{H}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}+W\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}})W+\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\langle P^{[4]}\rangle +\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D6F9}-2\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D719}\nonumber\\ \displaystyle & & \displaystyle \quad +\,(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}=O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k),\end{eqnarray}$$

where

(6.5d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7} & \equiv & \displaystyle \frac{1}{2}a\text{e}^{k\unicode[STIX]{x1D701}}\left(k\,\text{Re}(P^{\{1\}})-\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\,\text{Re}(P^{\{1\}})-\frac{1}{c^{[0]}}\text{i}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\,\text{Im}(P^{\{1\}})+\frac{2}{c^{[0]}k}\text{i}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\,\text{Im}(P^{\{1\}})\right)\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{1}{8}U_{H}^{S}U_{H}^{S}+a\text{e}^{2k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a,\end{eqnarray}$$

(6.5e ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}\equiv \frac{a}{2c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}(\text{i}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\,\text{Im}(P^{\{1\}})+c^{[0]}\text{e}^{k\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a), & \displaystyle\end{eqnarray}$$

(6.5f ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}\equiv \unicode[STIX]{x1D6F7}+\unicode[STIX]{x1D719}+\frac{1}{4}U_{H}^{S}U_{H}^{S}-U_{1}^{S}(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1}), & \displaystyle\end{eqnarray}$$

(6.5g ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}_{1} & \equiv & \displaystyle -\hat{k}_{1}\frac{ak}{2c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}\left(\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\,\text{Re}(P^{\{1\}})+\frac{2}{k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\,\text{Re}(P^{\{1\}})+\frac{1}{k^{2}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\,\text{Re}(P^{\{1\}})\right)\nonumber\\ \displaystyle & & \displaystyle +\,3a^{2}k^{2}\text{e}^{2k\unicode[STIX]{x1D701}}\hat{k}_{1}\unicode[STIX]{x2202}_{t}c^{[2]},\end{eqnarray}$$

and $\hat{k}_{1}u_{1}^{D}+c^{[2]}$ , $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ , $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a$ , $\unicode[STIX]{x2202}_{t}c^{[2]}$ and $P^{\{1\}}$ are given in (5.15), (5.50), (D 3) and (D 1). For additional information on the wave-related terms, note that $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}P^{\{1\}}=\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}(P_{\mathbb{D}}^{\{1\}}+P_{\mathbb{E}}^{\{1\}}+P_{\mathbb{H}}^{\{1\}})+O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{3}k)$ , and also see (5.11), (5.14), (C 3) and (C 5).

According to the wave solutions in § 5, the terms $\unicode[STIX]{x1D6EC}_{1}$ and $-2\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D719}$ depend mostly on $P_{\mathbb{H}}^{\{1\}}$ given by (5.23a ). Therefore, if there is no forcing of the third-order waves through the horizontal boundary condition (thus, $P_{\mathbb{H}}^{\{1\}}=0$ , as explained in § 5.2), then $\unicode[STIX]{x1D6EC}_{1}$ and $-2\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D719}$ become negligible for the fourth-order balance. In (6.5c ), the term $(-u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ derives from the fact that the waves are refracted and Doppler shifted according to (5.50b ). These equations are expressed in terms of the Stokes shear force (Suzuki & Fox-Kemper Reference Suzuki and Fox-Kemper2016). The equivalent equations expressed in terms of the vortex force are shown in appendix I. The Stokes shear force – rather than the vortex force – is useful for physical analysis because a large part of the vortex force is equal to a potential force. Since a potential force cannot inject vorticity or energy (except through the energy flux at the boundaries), the evolution of the roll is easier to understand in terms of a force that is not dominated by a potential force. In contrast, most potential force in the Stokes shear force is simply the horizontal mean of the Stokes shear force and easily removable with the same procedure as the familiar separation of the horizontal mean buoyancy and the buoyancy perturbation.

Finally, averaging (5.5) with (2.5) yields the upper boundary condition for $\boldsymbol{U}$ as

(6.6) $$\begin{eqnarray}\frac{1}{g}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\langle P\rangle -W=0\quad \text{at }\unicode[STIX]{x1D701}=0.\end{eqnarray}$$

Here, $\langle P\rangle =\langle P^{[4]}\rangle +O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2})$ according to the wave solutions given in § 5. Note that the averaged kinematic boundary condition is $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\langle z\rangle =\langle w\rangle -\langle S_{h}u_{h}\rangle =W$ . The value of $\langle S_{h}u_{h}\rangle =\langle S_{h}u_{h}^{w}\rangle +O(\unicode[STIX]{x1D716}^{5}c^{[0]})$ is shown in (F 3) in appendix F. An important point of (F 3) is that the averaged vertical motion of the water surface, $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\langle z\rangle$ , due to the correlation $\langle u_{h}S_{h}\rangle$ is of fourth order. Therefore, $\langle u_{h}S_{h}\rangle$ cannot be neglected for the wave-averaged governing equations.

6.3 Implications of (6.3) and (6.5)

The fundamental difference between the CL theory and S19 is the recognition that the wave-averaged mass conservation is satisfied by the wave-averaged material flow $\boldsymbol{U}$ (§ 6.1). This difference is crucial because it allows S19 to deal with the current-induced evolution of the wave field without neglecting the evolution of the wave-induced mass divergence and its effect on the wave-averaged pressure. This point is exemplified here by contrasting a case having time-dependent wave properties with another case having time-independent wave properties.

Before discussing the two cases, let us note the following essential facts. It can be shown (e.g. by taking the curl of (6.5)) that the rolls of typical Langmuir circulations are driven mainly by the term $c_{H}^{g}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}U_{2}^{S}$ in (6.5b ) and the term $(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ in (6.5c ). The former represents the propagation of $U_{2}^{S}$ at the group velocity. The latter is the counterpart of the Stokes shear force (hereafter, SSF), $U_{1}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ , discussed in Suzuki & Fox-Kemper (Reference Suzuki and Fox-Kemper2016), and it is different from the SSF by $(-u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ . This difference originates from the wave refraction effect explicit in the governing equations of $\boldsymbol{U}$ . The SSF of Suzuki & Fox-Kemper (Reference Suzuki and Fox-Kemper2016) lacks the refraction effect because the equations of Suzuki & Fox-Kemper (Reference Suzuki and Fox-Kemper2016) are equivalent to the CL equations, which lack the refraction effect. Hereafter, $(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ is called the refraction-affected Stokes shear force (RASSF). According to (6.4a ), RASSF is equal to $(\langle u_{1}^{c}\rangle +\mathfrak{U}_{1}+\langle S_{3}u_{1}\rangle -u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ .

Because the RASSF is a vertical force, its horizontal mean is hydrostatically balanced and dynamically unimportant (like the horizontal mean of buoyancy). Thus, what is dynamically important is the RASSF perturbation from the horizontal mean. Under the scaling conditions being considered (§ 3), the horizontal perturbation in $\langle u_{1}^{c}\rangle -u_{1}^{D}$ is much larger than the horizontal perturbation in $\mathfrak{U}_{1}+\langle S_{3}u_{1}\rangle -c^{[2]}\hat{k}_{1}$ or in $U_{1}^{S}$ . Therefore, the RASSF perturbation is approximately equal to $(\langle u_{1}^{c}\rangle -u_{1}^{D})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ and occurs at the along-roll jets. When the waves are propagating in the downstream direction of the along-roll jets, the signs of $\langle u_{1}^{c}\rangle -u_{1}^{D}$ and $U_{1}^{S}$ are the same at the jets. Therefore, in this situation, the RASSF perturbation at the jets is a downward force. This is why the RASSF perturbation drives the downwellings at the along-roll jets (figures 1 and 2). This effect of the RASSF perturbation is the same as that of the SSF perturbation $\langle u_{1}^{c}\rangle \unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ , as detailed in Suzuki & Fox-Kemper (Reference Suzuki and Fox-Kemper2016). Now, let us consider the aforementioned two cases.

6.3.1 Case I: $x$ -independent but $t$ -dependent wave properties

When the wave properties are independent of $x$ , (5.50b ) shows that $c_{H}^{g}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}U_{2}^{S}$ is negligible while $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{2}^{S}$ is significant due to the refraction. Hence, the wave properties change with time. Moreover, (5.50b ) shows that $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}U_{2}^{S}$ is of fourth order. Then, as shown in § 6.1 and appendix G, the evolution of the wave-induced mass divergence can affect the wave-averaged pressure at its leading order (i.e. fourth order). Therefore, the dynamics of $\boldsymbol{U}$ described by (6.3) and (6.5) is crucial. Then, because $c_{H}^{g}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}U_{2}^{S}$ in (6.5b ) is negligible, the rolls are driven solely by the RASSF perturbation. This is in contrast to the CL theory, in which the rolls are driven by the SSF perturbation.

Figure 8.

Comparison between the RASSF and the SSF.

For example, consider a current structure having $K_{\mathbb{A}(2)}^{2}-k^{2}=0.0051~\text{rad}~\text{m}^{-1}$ in (3.2) and an along-roll jet $\langle u_{1}^{c}\rangle$ shown in figure 8(c). In the same panel, the Doppler-shift velocities (5.15a ) of a wave whose wavelength $\unicode[STIX]{x1D706}$ is 10 m and of another wave whose $\unicode[STIX]{x1D706}$ is 50 m are also shown. In this example, both the shorter and longer waves have $ak=0.1$ and $(\hat{k}_{1},\hat{k}_{2})=(1,0)$ at the time of consideration. Because the shorter wave has a larger $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ near the surface, it produces a larger SSF perturbation (figure 8 a). However, as shown in figure 8(b), the refraction effect makes the following two differences: first, the longer wave becomes more important than the shorter wave because $\langle u_{1}^{c}\rangle -u_{1}^{D}$ near the surface is larger for the longer wave; and second, the magnitude of $(\langle u_{1}^{c}\rangle -u_{1}^{D})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ is much smaller than that of $\langle u_{1}^{c}\rangle \unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}$ because $u_{1}^{D}$ is close to $\langle u_{1}^{c}\rangle$ near the surface. More specifically, compared to the SSF perturbation, the RASSF perturbation is decreased by a factor of 10 for the shorter wave and 5 for the longer wave for this example current. Therefore, unlike the CL theory, S19 predicts that long waves are more critical than short waves in driving the rolls and thereby upper-ocean mixing. Meanwhile, the energy transfer from the waves to the rolls is likely to be overestimated by the CL theory in a situation like case I. Furthermore, because $u_{1}^{D}$ is determined essentially by the current profile at depths shallower than the wavelength, S19 is much more sensitive to the near-surface current profile.

6.4 Wave–wave nonlinear processes producing the RASSF

The RASSF results from the wave–wave nonlinearity due to the first-order wave and the sum of the $\mathbb{A}$ - and $\mathbb{C}$ -waves (§ 5.2). These waves horizontally displace each other’s vertical velocities via the advection terms ${u_{1}^{w}}^{[1]}\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}{w^{w}}^{[3]}$ and ${u_{1}^{w}}^{[3]}\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}{w^{w}}^{[1]}$ . The ensemble averages of these terms are non-zero as shown in table 3. In this way, they convert the oscillatory velocities ${w^{w}}^{[1]}$ and ${w^{w}}^{[3]}$ into the non-oscillatory velocity $W$ . Furthermore, the $\mathbb{A}$ - and $\mathbb{C}$ -waves also alter the gradients of the first-order wave pressure by undulating the constant- $\unicode[STIX]{x1D701}$ surfaces. The ensemble averages of these perturbations are non-zero as the presence of $S_{i}^{[3]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}$ in tables 2 and 3 shows. In this way, the first-order wave pressure also forces $\boldsymbol{U}$ . According to (5.26) and (5.32), the sum of the $\mathbb{A}$ - and $\mathbb{C}$ -waves is

(6.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle \,\text{Re}(u_{\mathbb{A}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}+u_{\mathbb{C}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}})=\frac{\hat{k}_{1}}{c^{[0]}}(P_{\mathbb{A}}^{\{1\}}+P_{\mathbb{C}}^{\{1\}})\cos \unicode[STIX]{x1D712}-\frac{\hat{k}_{1}a}{c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}(\unicode[STIX]{x2202}_{t}c^{[2]})\sin \unicode[STIX]{x1D712}+u^{\unicode[STIX]{x1D6E4}}, & \displaystyle\end{eqnarray}$$

(6.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle \,\text{Re}(v_{\mathbb{A}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}+v_{\mathbb{C}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}})=\frac{1}{c^{[0]}k}\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}(P_{\mathbb{A}}^{\{1\}}+P_{\mathbb{C}}^{\{1\}})\sin \unicode[STIX]{x1D712}, & \displaystyle\end{eqnarray}$$

(6.7c ) $$\begin{eqnarray}\displaystyle & \displaystyle \,\text{Re}(w_{\mathbb{A}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}+w_{\mathbb{C}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}})=\frac{1}{c^{[0]}k}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}(P_{\mathbb{A}}^{\{1\}}+P_{\mathbb{C}}^{\{1\}})\sin \unicode[STIX]{x1D712}+\frac{a}{c^{[0]}}\text{e}^{k\unicode[STIX]{x1D701}}(\unicode[STIX]{x2202}_{t}c^{[2]})\cos \unicode[STIX]{x1D712}+w^{\unicode[STIX]{x1D6E4}},\qquad & \displaystyle\end{eqnarray}$$

(6.7d ) $$\begin{eqnarray}\displaystyle \,\text{Re}(z_{\mathbb{A}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}+z_{\mathbb{C}}^{\{1\}}\text{e}^{\text{i}\unicode[STIX]{x1D712}}) & = & \displaystyle \frac{1}{gk}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}(P_{\mathbb{ A}}^{\{1\}}+P_{\mathbb{ C}}^{\{1\}})\cos \unicode[STIX]{x1D712}-\frac{1}{4c^{[0]}}a\text{e}^{k\unicode[STIX]{x1D701}}\hat{k}_{1}U_{1}^{S}\cos \unicode[STIX]{x1D712}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{3a}{g}\text{e}^{k\unicode[STIX]{x1D701}}(\unicode[STIX]{x2202}_{t}c^{[2]})\sin \unicode[STIX]{x1D712}+z^{\unicode[STIX]{x1D6E4}},\end{eqnarray}$$

(6.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle u^{\unicode[STIX]{x1D6E4}}\equiv \hat{k}_{1}ak\text{e}^{k\unicode[STIX]{x1D701}}\hat{k}_{1}(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\cos \unicode[STIX]{x1D712}, & \displaystyle\end{eqnarray}$$

(6.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle w^{\unicode[STIX]{x1D6E4}}\equiv ak\text{e}^{k\unicode[STIX]{x1D701}}\hat{k}_{1}(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\sin \unicode[STIX]{x1D712}, & \displaystyle\end{eqnarray}$$

(6.8c ) $$\begin{eqnarray}\displaystyle & \displaystyle z^{\unicode[STIX]{x1D6E4}}\equiv \frac{2}{c^{[0]}}a\text{e}^{k\unicode[STIX]{x1D701}}\hat{k}_{1}(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\cos \unicode[STIX]{x1D712}. & \displaystyle\end{eqnarray}$$

$\mathbb{A}$

$\mathbb{C}$

$u^{\unicode[STIX]{x1D6E4}}$

$w^{\unicode[STIX]{x1D6E4}}$

$z^{\unicode[STIX]{x1D6E4}}$

$W$

$w^{\unicode[STIX]{x1D6E4}}$

${u^{w}}^{[1]}$

${w^{w}}^{[1]}$

$u^{\unicode[STIX]{x1D6E4}}$

(6.9) $$\begin{eqnarray}\displaystyle & \displaystyle -\langle {u^{w}}^{[1]}\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}w^{\unicode[STIX]{x1D6E4}}\rangle =-{\textstyle \frac{1}{4}}(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

(6.10) $$\begin{eqnarray}\displaystyle & \displaystyle -\langle u^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}{w^{w}}^{[1]}\rangle =-{\textstyle \frac{1}{4}}(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

according to (2.5), (5.7a ), (5.7b ), (6.8a ) and (6.8b ). Here, the signs are chosen so that these terms appear on the right-hand side of (6.5c ). Equations (6.9) and (6.10) show that the sum of these two advection terms makes one half of the RASSF. The other half comes from the correlation between the vertical gradient of the first-order pressure and the displacement $z^{\unicode[STIX]{x1D6E4}}$ involved in the last term on the right-hand side of (2.18). This correlation produces a potential force and one half of the RASSF because

(6.11) $$\begin{eqnarray}\displaystyle & \displaystyle \langle S_{1}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle =\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}\langle z^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle +O(\unicode[STIX]{x1D716}^{6}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

(6.12) $$\begin{eqnarray}\displaystyle & \displaystyle \langle S_{2}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle =\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}\langle z^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle +O(\unicode[STIX]{x1D716}^{6}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

(6.13) $$\begin{eqnarray}\displaystyle & \displaystyle \langle S_{3}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle =\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\langle z^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle -\langle z^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle , & \displaystyle\end{eqnarray}$$

where $S_{i}^{\unicode[STIX]{x1D6E4}}\equiv \unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}z^{\unicode[STIX]{x1D6E4}}$ is the part of $S_{i}$ due to $z^{\unicode[STIX]{x1D6E4}}$ according to the definitions (2.12) and (2.13) of $S_{i}$ . Note that the first terms on the right-hand side of (6.11)–(6.13) form a potential force. Meanwhile, the last term on the right-hand side of (6.13) gives half the RASSF, that is,

(6.14) $$\begin{eqnarray}-\langle z^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle =-{\textstyle \frac{1}{2}}(U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1})\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}U_{1}^{S}+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k),\end{eqnarray}$$

according to (5.7c ) and (6.8c ). The physical meaning of $-z^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}$ is the following. Among many other forces, the fluid is forced by the pressure gradient force $-\unicode[STIX]{x2202}_{i}^{z}P$ . The chain rules (2.15) imply $-\langle \unicode[STIX]{x2202}_{i}^{z}P\rangle =-\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}\langle P\rangle +\langle S_{i}\unicode[STIX]{x2202}_{3}^{z}P\rangle$ . The term $-\unicode[STIX]{x2202}_{i}^{\unicode[STIX]{x1D701}}\langle P\rangle$ represents the pressure gradient force disregarding the effects of the undulation of the constant- $\unicode[STIX]{x1D701}$ surfaces and the fluctuation in $P$ . The effects of the surface undulation and the fluctuation in $P$ are represented by $\langle S_{i}\unicode[STIX]{x2202}_{3}^{z}P\rangle$ , and the leading-order effect of $z^{\unicode[STIX]{x1D6E4}}$ in this term is the left-hand side of (6.11)–(6.13). Figure 9 shows the directions of $\langle S_{i}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle$ . These directions imply that a significant portion of $\langle S_{i}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle$ is a potential force. According to (6.11)–(6.13), the separation of the potential force from $\langle S_{i}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle$ leaves half the RASSF.

Figure 9.

Diagram showing the directions of the forces $S_{2}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}$ (solid arrows) and $S_{3}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}$ (dashed arrows) at different phases indicated in the two panels. According to (5.7c ), the sign of $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}$ is the same as the sign of $\cos \unicode[STIX]{x1D712}$ . In these panels, there is an along-roll jet at the location indicated by $\odot$ . Thus, $U_{1}-u_{1}^{D}-c^{[2]}\hat{k}_{1}$ has a positive maximum there. The solid lines depict the corresponding $z^{\unicode[STIX]{x1D6E4}}$ in (6.8c ). Because the signs of $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}$ and $S_{i}^{\unicode[STIX]{x1D6E4}}$ change with $\cos \unicode[STIX]{x1D712}$ , the arrows always point towards $\odot$ and produce a non-zero wave average.

6.5 Is it necessary to determine $\mathfrak{U}_{i}$ separately from $\langle u_{i}^{c}\rangle$ ?

In § 5, $\mathfrak{U}_{i}$ and $u_{i}^{c}$ are separately treated in order to make the logical structure of the theory clearer. Then, the preceding sections show that the value of $\mathfrak{U}_{i}$ is not determined by the balance correct to third order, but it is determined by the balance at fourth order. However, the fourth-order balance does not dictate $\mathfrak{U}_{i}$ and $u_{i}^{c}$ separately. Therefore, separation of $\mathfrak{U}_{i}$ and $u_{i}^{c}$ is not possible unless we introduce an additional definition about how to distinguish $\mathfrak{U}_{i}$ from $u_{i}^{c}$ or an additional and unnecessary constraint such as the wave’s second-order irrotationality. Meanwhile, such separation of $\mathfrak{U}_{i}$ and $u_{i}^{c}$ is unnecessary unless one tries to separate the $\mathbb{A}$ - and $\mathbb{C}$ -waves or $\hat{k}_{1}u_{1}^{D}$ and $c^{[2]}$ . However, such separation is unnecessary because they always appear in pairs. For example, $\mathfrak{U}_{1}$ can be always treated together with $u_{(1)}^{c}$ in (3.2) as $\mathfrak{U}_{1}+u_{(1)}^{c}$ because they appear in parallel in the driving functions $\mathbb{A}$ and $\mathbb{C}$ (table 1 and appendix D) and in (5.26a ) and (5.32a ). In this way, the sum of the $\mathbb{A}$ - and $\mathbb{C}$ -waves can be computed without ever separating $\mathfrak{U}_{1}$ and $u_{1}^{c}$ . Likewise, the sum $\hat{k}_{1}u_{1}^{D}+c^{[2]}$ can be computed without ever separating $\mathfrak{U}_{i}$ and $u_{i}^{c}$ by using the identity $2k\int _{-\infty }^{0}\text{e}^{2k\unicode[STIX]{x1D701}^{\prime }}\hat{k}_{1}\mathfrak{U}_{1}(x,y,\unicode[STIX]{x1D701}^{\prime },t)\,\text{d}\unicode[STIX]{x1D701}^{\prime }=\hat{k}_{1}\mathfrak{U}_{1}(\unicode[STIX]{x1D701}=0)-\hat{k}_{1}\int _{-\infty }^{0}\text{e}^{2k\unicode[STIX]{x1D701}^{\prime }}(\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{1})^{\prime }\,\text{d}\unicode[STIX]{x1D701}^{\prime }$ in (5.15). Here, $(\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{1})^{\prime }$ is $\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{1}$ at $(x,y,\unicode[STIX]{x1D701}^{\prime },t)$ . Finally, note that $\unicode[STIX]{x2202}_{t}c^{[2]}=\unicode[STIX]{x2202}_{t}(\hat{k}_{1}u_{1}^{D}+c^{[2]})+O(\unicode[STIX]{x1D716}^{4}{c^{[0]}}^{2}k)$ .

6.6 Value of $\mathfrak{U}_{i}$ in the current’s absence

Consider a case in which $u_{i}^{c}$ and $P_{\mathbb{H}}^{\{1\}}$ are zero. In this case, $\langle u_{i}\rangle =\mathfrak{U}_{i}$ ; $u_{1}^{D}$ and $G$ are zero; $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{2}\hat{k}_{2}$ , $\unicode[STIX]{x2202}_{t}c^{[2]}$ , $\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}c^{[2]}$ and $\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}D^{g}$ are of third order; and $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{t}a$ , $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}a$ , $\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\,\text{Re}(P^{\{1\}})$ and $\text{i}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\,\text{Im}(P^{\{1\}})$ are of fourth order. Thus, (H 1) becomes

(6.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{1}={\textstyle \frac{1}{2}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{1}^{S}-\unicode[STIX]{x2202}_{1}^{\unicode[STIX]{x1D701}}\breve{\unicode[STIX]{x1D6F7}}+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

(6.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{2}={\textstyle \frac{1}{2}}\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{2}^{S}-\unicode[STIX]{x2202}_{2}^{\unicode[STIX]{x1D701}}\breve{\unicode[STIX]{x1D6F7}}+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

(6.15c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}\mathfrak{U}_{3}=-\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}\breve{\unicode[STIX]{x1D6F7}}+O(\unicode[STIX]{x1D716}^{5}{c^{[0]}}^{2}k), & \displaystyle\end{eqnarray}$$

$\breve{\unicode[STIX]{x1D6F7}}\equiv \langle P^{[4]}\rangle +(4k)^{-1}ga(\unicode[STIX]{x2202}_{2}\unicode[STIX]{x2202}_{2}a)\text{e}^{2k\unicode[STIX]{x1D701}}$

$\unicode[STIX]{x2202}_{t}^{\unicode[STIX]{x1D701}}U_{h}^{S}/2$

$\langle {u_{H}^{w}}^{[1]}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}{u_{h}^{w}}^{[1]}\rangle$

$\langle {u_{H}^{w}}^{[1]}\unicode[STIX]{x2202}_{H}^{\unicode[STIX]{x1D701}}{u_{h}^{w}}^{[3]}\rangle$

$\langle -S_{h}^{[3]}\unicode[STIX]{x2202}_{3}^{\unicode[STIX]{x1D701}}P^{[1]}\rangle$

$u_{i}^{c}$

$P_{\mathbb{H}}^{\{1\}}$

$\breve{\unicode[STIX]{x1D6F7}}$

$(\mathfrak{U}_{h},\mathfrak{U}_{3})$

$(U_{h}^{S}/2,0)$

$u_{h}^{S}$

$U_{h}^{S}$

$u_{h}^{S}$

$u_{i}^{c}$

$P_{\mathbb{H}}^{\{1\}}$

$\breve{\unicode[STIX]{x1D6F7}}$