4.1. Linearity of the leading-order solution

To derive wave-QG, we adopt a scaling that assumes small-amplitude flow and develop parallel expansions of the Boussinesq system (2.2)–(2.6) and the APV equation (3.5). We assume the balanced flow is weak, in that internal waves comprise the leading-order solution, while balanced flow is described only at next order alongside quadratic wave quantities.

We denote the characteristic horizontal velocity of the waves by $\tilde{U}$ , the characteristic length scale of the flow by $L$ , and assume that the characteristic time scale is given by the Coriolis frequency $f_{0}$ . The linearity of the wave field then requires that

is much less than unity. We use the small parameter ${\it\epsilon}$ , which is a measure of wave amplitude analogous to steepness for surface waves, to distinguish each level of approximation in the development of the Boussinesq and APV equations.

4.2. The Rossby number and ‘two-timing’

We use a common vertical scale $H$ and common horizontal length scale $L$ for both the internal waves and the balanced flow. While this scaling ultimately limits attention to hydrostatic internal waves, it otherwise retains generality in the derivation, allowing both for consideration of comparable wave-mean spatial scales as well as further approximation based on spatial-scale separation.

If we denote the characteristic velocity of the balanced flow by $\bar{U}$ , the assumption of weak balanced flow is expressed by the scaling $\bar{U}={\it\epsilon}\tilde{U}$ . The Rossby number of the balanced flow is then

The Rossby number is a measure of time-scale separation between fast wavy motions oscillating on $f_{0}^{-1}$ and the slower balanced-flow evolution over $L/\bar{U}$ . To construct a single system of equations that captures the fast wave oscillations as well as the slow evolution of balanced flow, we use a multiple-time-scale expansion with ‘fast’ time $\,\tilde{t}=f_{0}t$ and slow time $\bar{t}=t\bar{U}/L$ , so that

This ‘two-timing’ also necessitates the introduction of an ‘average over the fast time’, which we denote with an overbar. If ${\it\phi}(\boldsymbol{x},t)$ is any field, then

The wavy part of ${\it\phi}$ , denoted $\tilde{{\it\phi}}$ , is defined via

The averaging or filtering operation in (4.4) is not unique. Alternatively we can view the overbar as a filtering operation that, in principle, removes wave time scales from ${\it\phi}$ exactly.

We assume that $\bar{{\it\phi}}$ has no dependence on the fast time $\,\tilde{t}$ and that the average of the wavy fields is zero, or equivalently that $\bar{\bar{{\it\phi}}}=\bar{{\it\phi}}$ . In the context of the perturbation expansion, this amounts to an assumption that average quadratic properties of the wavy fields – for example the Stokes velocity or average wave energy – evolve on the slow time scale $L/\bar{U}$ . Our focus on mean flow evolution means that the multiple-scale expansion in (4.3) neglects the nonlinear wave evolution time scale $L/\tilde{U} =({\it\epsilon}f_{0})^{-1}$ , which is intermediate between $f_{0}^{-1}$ and $L/\bar{U}={\it\epsilon}^{2}f_{0}^{-1}$ .

4.3. The non-dimensional Boussinesq and APV equations

We non-dimensionalize the Boussinesq equations with the two time scales in (4.3), the horizontal scale $L$ and vertical scale $H$ such that

where the ‘hat’ decoration denotes a non-dimensional quantity. We assume that the vertical and horizontal scales are related by

and $\mathit{Bu}$ is the Burger number. In (4.7), $N_{0}$ is the characteristic magnitude of the buoyancy frequency $N(z)$ . $\mathit{Bu}=1$ is standard scaling in the quasi-geostrophic approximation. The flow variables are scaled with

${\it\beta}$ in the Coriolis frequency $f=f_{0}+{\it\beta}y$ is scaled with

The scaling in (4.9) ensures that the effect of ${\it\beta}$ arises first in the QGPV equation. The scaling in (4.7) restricts attention to hydrostatic internal waves, but otherwise does not restrict wave field spatial scales.

We use these definitions to non-dimensionalize the Boussinesq equations and lighten the notation by dropping all decorations except for those on the fast time scale $\,\tilde{t}$ and slow time scale $\bar{t}$ . The non-dimensionalized Boussinesq equations then become

where in the vertical momentum equation we have introduced ${\it\alpha}\,\stackrel{\text{def}}{=}\,H/({\it\epsilon}L)$ . To justify the hydrostatic approximation, ${\it\alpha}$ is fixed at order unity as ${\it\epsilon}\rightarrow 0$ .

APV is scaled with $N_{0}^{2}\tilde{U} /L$ , so that from (3.7) the non-dimensional APV becomes

where ${\it\Lambda}=\ln N^{2}$ and

is the vorticity. The scaled APV evolution equation from (3.5) is

Each field is expanded in powers of ${\it\epsilon}$ so that, for example, $u=u_{0}+{\it\epsilon}u_{1}+\cdots \,$ . We proceed order by order, using dimensional variables for clarity but employing the non-dimensional equations (4.10)–(4.17) to guide the development.

4.4. Leading order: internal waves

The leading-order system is linear and describes hydrostatic internal waves,

We eliminate quasi-steady solutions – the balanced vortical mode – by insisting that the average of all leading-order fields is zero:

The leading-order wave particle displacement, ${\bf\xi}_{0}={\it\xi}_{0}\,\hat{\boldsymbol{x}}+{\it\eta}_{0}\,\hat{\boldsymbol{y}}+{\it\zeta}_{0}\,\hat{\boldsymbol{z}}$ , is defined by

Some important identities involving the wave particle displacement follow from the leading-order system (4.18)–(4.22): the vertical vorticity equation, which is formed by subtracting $\partial _{y}$ of (4.18) from $\partial _{x}$ of (4.19), can be manipulated using $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\,{\bf\xi}_{0}=0$ and integrating over $\,\tilde{t}$ to find

Integration of the buoyancy equation (4.21) yields

And then, eliminating the vertical displacement ${\it\zeta}_{0}$ between (4.25) and (4.26), we find the leading-order APV is zero:

The conclusion that $Q_{0}=0$ follows alternatively by integrating the leading-order APV equation, $Q_{0\,\tilde{t}}=0$ , and applying (4.23) to determine that the constant of integration is zero. The leading-order fields thus constitute internal waves oscillating on the fast time scale $\,\tilde{t}$ and with no signature in the APV field.

We emphasize the importance of the fact that $Q_{0}=0$ . Note that the first-order Ertel PV, ${\it\Pi}_{1}=N^{2}(v_{0x}-u_{0y})+f_{0}b_{0z}$ , is not zero for internal waves described by (4.18)–(4.22) – unless $N_{z}=0$ and the background PV is therefore uniform. That the leading-order wave field has PV, but no APV, is the first indication of APV’s utility in this problem. Increasingly important but less obvious simplifications follow at subsequent orders in the APV equation expansion.

4.5. First order: balanced flow and quadratic wave terms

The first-order fields are governed by

Because the first-order fields are permitted to have non-zero time-averages, (4.29)–(4.33) provide the definition of wave-averaged quasi-geostrophic balance.

Before proceeding in the derivation of (1.1)–(1.3), we observe that (4.29)–(4.33) also describe slow, nonlinear wave evolution due to wave self-interaction. Such slow wave evolution occurs when the right-side forcing resonates with the left-side linear internal wave operator (Müller et al. Reference Müller, Holloway, Henyey and Pomphrey1986). As we do not describe wave evolution in this paper, we ignore this possibility. However, a consistent description of the coupled evolution of wave and balanced flow requires treatment of nonlinear wave field self-interaction and careful accounting of time scales involved. In particular, wave self-interaction produces a time scale $({\it\epsilon}f_{0})^{-1}$ , intermediate between the linear-wave and balanced-flow evolution scales $f_{0}^{-1}$ and ${\it\epsilon}^{2}f_{0}^{-1}$ accounted for here. Including the time scale $({\it\epsilon}f_{0})^{-1}$ does not significantly change the basic result of this paper, but would require filtering $({\it\epsilon}f_{0})^{-1}$ from (1.1) through (1.3) to produce a consistent description of balanced-flow evolution.

Averaging equations (4.29)–(4.33) over $f_{0}^{-1}$ and rearranging terms, we can suggestively write the first-order mean velocities and averaged quadratic wave quantities as

where the wave velocity $\boldsymbol{u}^{w}$ is defined by

In appendix A we show that $\boldsymbol{u}^{w}$ can be written in terms of more familiar wave-averaged properties as

where

are the Stokes corrections to mean velocity and pressure fields (Craik Reference Craik1988; Bühler Reference Bühler2009). Using (4.36) to eliminate $\boldsymbol{u}^{w}$ from (4.34), we obtain the wave-averaged geostrophic balance condition,

Notice that $\bar{w}=-w^{S}$ , such that $w^{L}=0$ . Like standard QG, the vertical component of the balanced velocity is zero.

The wave-averaged hydrostatic relation follows from (A 9) and (4.38):

The final term in (4.39) is a Stokes correction associated with the resting buoyancy distribution $B(z)$ in (2.1); note that $(N^{2})_{z}=B^{\prime \prime }$ . Equation (4.39) relates the Lagrangian-mean streamfunction to the wave-averaged buoyancy field through wave-averaged hydrostatic balance.

Equations (4.38) and (4.39) are the wave-averaged balance conditions. Our derivation of wave-averaged balance shows that the ordinary sense of geostrophic balance from wave-ignoring QG theory is retained after wave averaging only for the Lagrangian-mean flow, $\boldsymbol{u}^{L}$ . The Eulerian-mean flow is not balanced.

The appearance of half the Stokes pressure correction in the balance condition (4.38) is a distinctive feature of the wave-averaged balance equations. The factor $1/2$ enters these basic relations via the quadratic wave identities (A 7)–(A 9). As in the standard QG approximation, the balance condition in (4.38) is redundant with the continuity equation, and we must seek an equation for mean-flow evolution at higher orders of approximation.

We turn to the APV equation (4.17), which at first order is

Integrating over $\,\tilde{t}$ , we are compelled to conclude that the first-order APV, $Q_{1}$ , does not depend on the fast time $\,\tilde{t}$ . In other words, $\tilde{Q}_{1}=0$ and

This result – which follows directly from expansion of the APV conservation equation – produces major simplifications at next order and is not readily apparent from the first-order Boussinesq equations (4.29)–(4.33).

4.6. Second and third orders: an evolution equation for $Q_{1}$

We proceed to higher orders only in the APV equation (4.17). At second order, the APV equation is

Because $Q_{1}$ is independent of the fast time $\,\tilde{t}$ , we can integrate (4.42) to yield

where ${\bf\xi}_{0}$ is the wave particle displacement defined in (4.24) and $\bar{Q}_{2}(\boldsymbol{x},\bar{t})$ is an unknown and inconsequential function of integration.

At third order the APV equation (4.17) is

while its wave average is

Notice that $Q_{1}$ is independent of the fast time and therefore stays outside of the averaging operation in (4.45). To manipulate the third term in (4.45) we use integration by parts and indicial notation, where ${\it\phi}_{,i}$ denotes the $i\text{th}$ derivative of ${\it\phi}$ and summation over repeated indices is implied. Using the expression for $Q_{2}$ in (4.43) and $\bar{\boldsymbol{u}}_{0}=0$ , we find

In passing from (4.46) to (4.47) we have used the fact that

which follows from the antisymmetry of $\overline{u_{0i}{\it\xi}_{0j}}$ and the symmetry $Q_{1,ij}$ . Thus there is no ‘diffusive’ term in (4.48) and the wave-averaged third-order APV equation (4.45) is

where $\boldsymbol{u}^{L}$ is the Lagrangian-mean velocity in (4.38). In analogy with the standard and unaveraged QG theory in which potential vorticity is attached to particle trajectories, here the mean APV, $Q_{1}$ , is attached to mean particle trajectories determined by the balanced Lagrangian-mean velocity $\boldsymbol{u}^{L}$ .

4.7. Quasi-geostrophic potential vorticity

To make the connection between (4.50) and conservation of the familiar QGPV we introduce

and rewrite (4.50) as

Recalling the expression for $Q_{1}$ in (4.41), and using the balance conditions in (4.38) and (4.39) to replace $\bar{\boldsymbol{u}}_{1}$ by ${\it\psi}$ and $p^{S}$ , the wave-averaged QGPV is

where operators ${\rm\Delta}$ and $\text{L}$ are defined in (1.2). The wave contribution to $q$ in (4.53) is

A slew of quadratic wave identities implied by (4.18)–(4.22) allow $q^{\mathit{w}}$ to be written in many equivalent forms. Some are more compact than (4.54), and to make contact with BM we show in appendix B that

where $\unicode[STIX]{x1D665}$ , defined in (B 7), is the leading-order internal wave pseudomomentum introduced by Andrews & McIntyre (Reference Andrews and McIntyre1978).

The result in (4.55) indicates agreement between our Eulerian derivation and the BM GLM derivation. The main difference is that BM assumes a slowly varying wave field; in that case the ‘wave-averaged vortex stretching’ $f_{0}(\overline{{\it\xi}_{0i}{\it\xi}_{0j}})_{,ij}/2$ , on the right of (4.55) with two external derivatives, is smaller than $\hat{\boldsymbol{z}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\!\times \!\,\unicode[STIX]{x1D665}$ appearing in (4.55) as well as equations (1.4) and (9.29) in BM. If spatial-scale separation is not assumed, the GLM-derived formulation also contains $f_{0}(\overline{{\it\xi}_{0i}{\it\xi}_{0j}})_{,ij}/2$ (Holmes-Cerfon, Bühler & Ferrari Reference Holmes-Cerfon, Bühler and Ferrari2011).

We identify two distinct parts of $q^{\mathit{w}}$ : the ‘pseudovorticity’, $\hat{\boldsymbol{z}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\!\times \!\,\unicode[STIX]{x1D665}$ , and wave-averaged vortex stretching $f_{0}(\overline{{\it\xi}_{0i}{\it\xi}_{0j}})_{,ij}/2$ . The appearance of pseudovorticity, a relative vorticity term that appears in wave-averaged circulation integrals, is a subtle and purely kinematic consequence of wave averaging: total wave-averaged fluid vorticity is $\hat{\boldsymbol{z}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\!\times \!(\boldsymbol{u}^{L}-\unicode[STIX]{x1D665})$ , rather than $\hat{\boldsymbol{z}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\!\times \!\,\boldsymbol{u}^{L}$ or $\hat{\boldsymbol{z}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\!\times \!\,\bar{\boldsymbol{u}}$ . A demonstration of this kinematic fact is given in § 10.2.7 of Bühler (Reference Bühler2009) for non-rotating fluids and finite particle displacements.

The wave-averaged vortex stretching $f_{0}(\overline{{\it\xi}_{0i}{\it\xi}_{0j}})_{,ij}/2$ , on the other hand, is a vortex stretching term that depends on spatial gradients in the mean-square wave displacement tensor $\overline{{\it\xi}_{0i}{\it\xi}_{0j}}$ . Wave-averaged vortex stretching reflects the expansion and contraction of ‘wave-averaged fluid elements’ due to non-zero divergence of $\boldsymbol{u}^{L}$ and thus of wave-averaged particle trajectories in non-uniform wave fields (McIntyre Reference McIntyre1988). Such expansion and contraction contributes to the PV balance in rotating flow. Wave-averaged vortex stretching is the only wave contribution to $q$ in two-dimensional flow, and in § 5 we show that wave-averaged vortex stretching is the leading-order wave contribution to the PV balance for a mode-one, horizontally modulated internal wave.

4.8. Boundary conditions

Boundary conditions for the wave-averaged QG equation (4.52) follow from evaluation of the buoyancy equation (4.13) on the boundaries. We assume flat bounding surfaces in $z$ so that $w=0$ in (4.13). We then expand (4.13) in powers of ${\it\epsilon}$ and recapitulate the expansion of the APV equation (4.17). The leading-order buoyancy equation, $b_{0\,\tilde{t}}=0$ , implies that $b_{0}=0$ and ${\it\zeta}_{0}=0$ at the boundaries. At ${\it\epsilon}^{1}$ we find that $b_{1}$ does not depend on the fast time $\,\tilde{t}$ such that $b_{1}=\bar{b}_{1}$ . At order ${\it\epsilon}^{2}$ we integrate over the fast-time variable to obtain $b_{2}=-{\bf\xi}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\bar{b}_{1}$ . At order ${\it\epsilon}^{3}$ we find in analogy with the calculation surrounding (4.45) that $\bar{b}_{1}$ is advected by the Lagrangian-mean velocity $\boldsymbol{u}^{L}$ . Finally, because $b_{0}={\it\zeta}_{0}=0$ , the Stokes corrections in the wave-averaged hydrostatic relation (4.39) vanish on the boundaries, so that $f_{0}{\it\psi}_{z}=\bar{b}_{1}$ . Thus the wave-averaged QG boundary condition is

This is the standard QG boundary condition: there is no explicit wave-averaged contribution.

5.1. The Bretherton flow: mean motion induced by a vertically propagating plane wave

Bretherton (Reference Bretherton1969) considered the mean motion induced by a vertically propagating plane internal wave packet in a non-rotating fluid. Here, we consider the rotating case by solving (5.1). The pressure field associated with the plane-wave packet is

where $k$ and $m$ are horizontal and vertical wavenumbers, ${\it\sigma}$ is frequency and $a$ is a three-dimensional envelope function with horizontal scale $\ell$ and vertical scale $d$ . The scale-separation parameter is

We assume $a$ is slowly varying so that ${\it\mu}\ll 1$ and $(dm)^{-1}\sim {\it\mu}\ll 1$ .

Because ${\it\mu}\ll 1$ , we drop $y$ -derivative terms from (4.18) through (4.22) to compute $\boldsymbol{u}$ , ${\bf\xi}$ and $b$ associated with $p$ in (5.3). These expressions are accurate to $O({\it\mu})$ and listed in table 1. A particularly useful result is the reduction of (4.19) to

With $\boldsymbol{u}$ and ${\bf\xi}$ , we compute $q^{\mathit{w}}$ to leading order in ${\it\mu}$ . Assuming ${\it\sigma}/f_{0}=O(1)$ , the slow variation of $a$ in $x$ , $y$ and $z$ implies that

Using (5.5), the three Jacobian terms in (5.2) scale with

Thus, neglecting the eight $O({\it\mu})$ terms in (5.2), the wave-averaged PV contribution $q^{\mathit{w}}$ associated with (5.3) reduces to

This is the conclusion reached by BM in their equation (9.22).

We make the implications of (5.9) concrete by picking the envelope

We solve (5.1) for ${\it\psi}$ given (5.10) and (5.9) with a spectral method, using Fourier collocation in $(x,y)$ and modal collocation in $z$ with constant- $N$ vertical modes $\mathit{h}_{n}=\cos (n{\rm\pi}z/H)$ . Figure 1(a) visualizes the wave field associated with (5.10) and the caption of figure 1 lists the physical parameters used to make figures 1–3.

Figure 1.

Visualization of the vertically propagating plane wave (a) and horizontally propagating mode-one wave (b) with isosurfaces of pressure, $p$ , at 0.325 and $-0.325$ of its maximum value. Wave fields are listed in table 1. A surface in the $yz$ -plane show the magnitude of wave-induced $u^{L}$ plotted in figure 2. A surface in the $xy$ -plane shows streamlines of $\boldsymbol{u}^{L}$ plotted in figure 3. Grey arrows indicate the direction of wave group propagation. Physical parameters are $f_{0}=10^{-4}~\text{s}^{-1}$ , $N=2\times 10^{-3}~\text{s}^{-1}$ , ${\it\sigma}=2f_{0}$ , $H=4\times 10^{3}\,\text{m}$ . The plane-wave vertical wavenumber is $m=(16{\rm\pi}H)^{-1}$ , the horizontal wavenumbers are $k=mf_{0}\sqrt{3}/N$ for the plane wave and $k={\it\kappa}_{1}\sqrt{3}={\rm\pi}\sqrt{3}f_{0}/NH$ for the mode, and the scale-separation parameter is ${\it\mu}=(\ell k)^{-1}=(dm)^{-1}=1/4$ .

The mean motion implied by (5.10) and (5.9) is depicted in figures 2 and 3. Figure 2(a) plots $u^{L}$ on a vertical plane in $(y,z)$ which divides the plane-wave packet, revealing the dipolar horizontal structure of $u^{L}$ and its vertical coincidence with the wave envelope. Figure 3(a) plots streamlines of $\boldsymbol{u}^{L}$ in an $xy$ -plane at $z=-H/2$ , showing that the plane wave $\boldsymbol{u}^{L}$ resembles a vortex dipole in the horizontal. Colour-filled contours indicate the magnitude of $\boldsymbol{u}^{L}$ and a dotted line outlines the plane-wave envelope.

Figure 2.

Vertical structure of wave-induced mean flows at $x=0$ for a vertically propagating plane wave (a) and vertical mode-one wave (b). Colour-filled contours show $u^{L}=-{\it\psi}_{y}$ normalized by its extreme value, isopycnals are in light grey, and dark grey dashed lines show wave envelopes with contours of $a/2$ . The plane-wave packet induces a dipolar $u^{L}$ while the mode-one wave induces a monopolar, mode-two eddy-like $u^{L}$ . Parameters are listed in the caption of figure 1.

Figure 3.

Top-down $xy$ -view at $z=-H/2$ of mean flows induced by the vertically propagating plane wave (a,b) and vertical mode-one wave (c,d). (a,c) Solid grey lines are streamlines of $\boldsymbol{u}^{L}$ , colour-filled contours show normalized flow magnitude $|\boldsymbol{u}^{L}|$ , and dark grey dashed lines show wave envelopes with contours of $a/2$ . (b,d) $u^{L}$ and $u^{S}$ are plotted versus $y$ on a line at $(x,z)=(0,-H/2)$ , both normalized by the maximum magnitude of $u^{L}$ . The $x$ -axes of (b,d) are different for mode and plane wave: $u^{L}$ dominates for the plane wave, while $u^{S}$ dominates for the mode-one wave. Parameters are listed in the caption of figure 1.

Figure 3(b) compares the $x$ -components of the Lagrangian-mean $\boldsymbol{u}^{L}$ and Stokes velocity $\boldsymbol{u}^{S}$ on a line in $y$ through $(x,z)=(0,-H/2)$ . The $x$ -component of $\boldsymbol{u}^{S}$ defined in (4.37) is

Integration by parts and use of $u_{x}\approx -w_{z}$ implies that $\overline{{\it\xi}u_{x}}+\overline{{\it\zeta}u_{z}}\approx (\overline{u{\it\zeta}})_{\!z}$ , and $(\overline{u{\it\zeta}})_{\!z}\approx O({\it\mu}{\it\zeta}u_{z})$ follows from the quadrature of $u$ and ${\it\zeta}$ for the packet. Thus

Figure 3(b) indicates that $u^{L}\gg u^{S}$ at $(x,z)=(0,-H/2)$ . This result can be anticipated with a scaling argument. The scaling of $u^{S}$ is relatively simple: because ${\it\eta}\sim f_{0}u/{\it\sigma}^{2}$ and $u_{y}\sim u/\ell$ ,

The scaling for $u^{L}$ requires (5.1). Scaling terms on the left of (5.1) gives

where we have used both $\ell =({\it\mu}k)^{-1}$ and $d=({\it\mu}m)^{-1}$ to obtain the rightmost term. For moderately super-inertial waves with $(\,f_{0}m/Nk)^{2}\approx O(1)$ , ${\rm\Delta}{\it\psi}$ and $\text{L}{\it\psi}$ scale similarly, and from (5.1) we obtain ${\it\psi}\sim \ell ^{2}q^{\mathit{w}}$ and ${\it\psi}/\ell \sim u^{L}\sim \ell q^{\mathit{w}}$ . The scaling for $q^{\mathit{w}}$ follows more simply: with $u_{x}\sim ku$ and ${\it\xi}_{y}\sim u/{\it\sigma}\ell$ we deduce that

Putting the pieces together and remembering that $k\ell ={\it\mu}^{-1}$ yields

The plane-wave Lagrangian-mean flow is $O({\it\mu}^{-1})$ larger than the Stokes velocity and the Eulerian mean flow is $\bar{u}\approx u^{L}$ to leading order in ${\it\mu}$ .

5.2. Mean motion induced by a vertical mode-one internal wave

We contrast the plane-wave-induced mean motion with the flow induced by a domain-filling, vertical mode-one internal wave. In an ocean of depth $H$ , the vertical modes are the eigenfunctions $\mathit{h}_{n}(z)$ that satisfy

where ${\it\kappa}_{n}^{-1}$ is the Rossby deformation length for mode $n$ and $\text{L}$ is the second-order linear operator defined in (1.1). When $N$ is constant, the vertical modes are $\mathit{h}_{n}=\cos (n{\rm\pi}z/H)$ with deformation length ${\it\kappa}_{n}^{-1}=NH/n{\rm\pi}f_{0}$ . We consider a mode-one wave pressure field of the form

where $k$ is horizontal wavenumber, ${\it\sigma}$ is frequency and $a$ is a slowly varying envelope function with horizontal scale $\ell$ . We assume $1/k\ell ={\it\mu}\ll 1$ as in (5.4), which permits easy computation of $\boldsymbol{u}$ , ${\bf\xi}$ and $b$ given in table 1 from equations (4.18)–(4.22).

With $\boldsymbol{u}$ and ${\bf\xi}$ we compute the mode-one $q^{\mathit{w}}$ to leading order in ${\it\mu}$ . The mode-one vertical structure implies that the terms in (5.2) scale differently than for the plane wave. In particular,

Moreover, because (5.5) and (5.7) apply also for the mode, none of the Jacobian, pseudomomentum-associated terms contribute to $q^{\mathit{w}}$ at leading order. Among the remaining terms on the second line of (5.2), the assumptions ${\it\mu}\ll 1$ and ${\it\sigma}/f_{0}=O(1)$ imply

Finally, the quadrature of $({\it\xi},{\it\zeta})$ and $({\it\eta},{\it\xi})$ and the fact that ${\it\mu}\ll 1$ imply

The only survivor at leading order from $q^{\mathit{w}}$ in (5.2) is therefore $f_{0}(\overline{{\it\zeta}^{2}})_{zz}/2$ , and the mode-one $q^{\mathit{w}}$ is

The final expression in (5.24) is found using ${\it\zeta}$ from table 1 along with $\text{L}\mathit{h}_{1}=-{\it\kappa}_{1}^{2}\mathit{h}_{1}$ . For a slowly varying mode- $n$ wave, $q^{\mathit{w}}$ follows by replacing ‘1’ with ‘ $n$ ’ in (5.24).

We investigate the consequences of (5.24) by choosing the envelope

As for the vertically propagating plane wave, we solve (5.1) for ${\it\psi}$ with $q^{\mathit{w}}$ determined by (5.25) and (5.24), using a spectral method. For the mode-one wave with constant  $N$ , ${\it\psi}$ is mode-two and thus proportional to $\cos (2{\rm\pi}z/H)$ . The wave field associated with (5.25) is visualized in figure 1(b) and the mean motion it induces is illustrated in figures 2 and 3.

Figure 2(b) shows the mode-two vertical structure of $\boldsymbol{u}^{L}$ , and figure 3(c) reveals the horizontally compact and monopolar form of $\boldsymbol{u}^{L}$ . Figure 3(d) compares $u^{L}$ with the Stokes velocity correction $u^{S}$ for the mode-one wave, where $u^{S}$ is defined in (4.37) and (5.11). Unlike the plane wave $u^{S}$ in (5.13), in the mode-one wave field $(\overline{u{\it\zeta}})_{z}$ is larger than $\overline{{\it\eta}u_{y}}$ by $O({\it\mu})$ , and thus

The Stokes velocity correction does not involve spatial derivatives of the envelope $a(x,y,t)$ .

Figure 2(b) indicates that $u^{S}\gg u^{L}$ at $(x,z)=(0,-H/2)$ for the mode-one wave: the reverse of the relationship found for the plane-wave case. This fact can be deduced with a scaling argument. First, ${\it\xi}\sim u/{\it\sigma}$ and ${\it\zeta}_{z}\approx -{\it\xi}_{x}$ imply ${\it\zeta}\sim Hk{\it\xi}$ , such that

That the mode $u^{S}$ scales with $k$ rather than $1/\ell$ contrasts with the plane-wave case. Next, from (5.1),

Assuming moderately super-inertial waves for which $({\it\kappa}_{1}/k)^{2}\approx O(1)$ , we conclude that $\text{L}{\it\psi}$ is $O({\it\mu}^{-2})$ larger than ${\rm\Delta}{\it\psi}$ . Therefore, ${\it\psi}\sim ({\it\mu}\ell )^{2}q^{\mathit{w}}$ and ${\it\psi}/\ell \sim u^{L}\sim {\it\mu}^{2}\ell q^{\mathit{w}}$ . Again using the fact that ${\it\zeta}\sim Hk{\it\xi}$ , we then find

Dropping the parts into place yields

which means the Lagrangian-mean flow is $O({\it\mu})$ smaller than the Stokes velocity field. This implies further that, to leading order in ${\it\mu}$ , the Eulerian-mean flow is

This Eulerian-mean $\bar{u}$ is an ‘anti-Stokes flow’. The Lagrangian-mean flow, which is relevant for potential vorticity advection, is a small residual remaining after the large cancellation in (5.32) and is $O({\it\mu})$ smaller for the mode-one wave than for the plane wave.

The fact that $\text{L}{\it\psi}$ is much larger than ${\rm\Delta}{\it\psi}$ for the mode-one wave is striking and means that the primary averaged effect of slowly varying, vertical-mode waves is a slight displacement of isopycnals. The isopycnal displacement is associated with a balanced flow when the wave field is spatially non-uniform. Equivalent to this physical explanation is the statement that the APV equation (5.1) can be solved by neglecting ${\rm\Delta}{\it\psi}$ and ‘cancelling the $\text{L}$ ’ between $\text{L}{\it\psi}$ and $q^{\mathit{w}}$ in (5.24). We must subtract the barotropic part of $\mathit{h}_{1}^{2}$ , since the vertical average of (5.1) implies ${\it\psi}$ has no barotropic component. This yields

Equation (5.33) is valid for general stratification profiles $N(z)$ and vertical modes $\mathit{h}_{n}$ when the 1 are replaced by $n$ . For slowly varying vertical mode waves, the streamlines of the wave-induced mean motion follow the contours of $a^{2}$ , which explains the monopolar mode-induced motion evident in figure 3.