2 Shallow-water dynamics

The shallow-water equations are

where $\boldsymbol{u}(\boldsymbol{x},t)=(u,v)$ is the fluid velocity at location $\boldsymbol{x}=(x,y)$ and time $t$ , ${\hat{h}}=h/h_{0}$ is the depth divided by its constant mean value $h_{0}$ , $c^{2}=gh_{0}$ with gravity constant $g$ , $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D735}\times \boldsymbol{u}\equiv v_{x}-u_{y}$ is the vertical component of vorticity and $f_{0}$ is the constant Coriolis parameter. Subscripts denote partial derivatives and $\unicode[STIX]{x1D735}=(\unicode[STIX]{x2202}_{x},\unicode[STIX]{x2202}_{y})$ . For definiteness, we suppose that the fluid is horizontally unbounded and quiescent at infinity. The vorticity and divergence equations corresponding to (2.1)–(2.2) are

and

The shallow-water equations conserve energy in the form

and the potential vorticity

on fluid particles,

In this section, we generalize the variational principle of S14 to full shallow-water dynamics (2.1)–(2.2) with continuously distributed vorticity and constant Coriolis parameter $f_{0}$ . We start by representing the physical variables

in terms of the potentials $\unicode[STIX]{x1D713}(\boldsymbol{x},t)$ and $\unicode[STIX]{x1D6FE}(\boldsymbol{x},t)$ . Compared with the method followed in S14, the representation (2.8)–(2.10) corresponds to the immediate adoption of the Coulomb gauge, as explained in appendix A. For present purposes, it is sufficient to note that (2.8)–(2.10) is a general representation that automatically satisfies the shallow-water continuity equation (2.2).

We motivate the derivation of the variational principle by first imposing – and then gradually lifting – a strong constraint: we assume that the potential vorticity (2.6) is uniform. If the potential vorticity is uniform, then

By (2.8)–(2.10), equation (2.11) is equivalent to

Both (2.12) and the corresponding form of (2.4), namely

result from the variational principle $\unicode[STIX]{x1D6FF}L_{1}[\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FE}]=0$ , where

Now, we partly relax the constraint: we allow the potential vorticity to be non-uniform, but we assume that the departures from (2.11) are concentrated in delta functions. That is, we assume that

This is a consistent restriction on shallow-water dynamics in the sense that (2.15) is compatible with the conservation of total vorticity, $\iint \,\text{d}\boldsymbol{x}hq$ . The sum in (2.15) is the product of the fluid depth ${\hat{h}}$ and what Wagner & Young (Reference Wagner and Young2015) call the ‘available potential vorticity’ – the difference between the potential vorticity defined by (2.6) and its uniform value in the state of rest. This difference is, like the potential vorticity itself, conserved on fluid particles. In a slight departure from the terminology of Wagner & Young (Reference Wagner and Young2015), we shall refer to the sum itself as the available potential vorticity (APV). This sum obeys the flux-form conservation law (2.40).

Since the delta functions in (2.15) represent singularities in potential vorticity, they must, according to (2.7), move at the fluid velocity $\boldsymbol{u}$ . That is, the velocities of the point vortices must be given by $\dot{\boldsymbol{x}}_{i}(t)=\boldsymbol{u}(\boldsymbol{x}_{i}(t),t)$ , or, using (2.8)–(2.10),

where the overdot denotes time differentiation. The Lagrangian corresponding to (2.16) is

That is, $\unicode[STIX]{x1D6FF}L_{2}/\unicode[STIX]{x1D6FF}\boldsymbol{x}_{i}=0$ implies (2.16).

We obtain the Lagrangian for shallow-water dynamics by combining (2.14) and (2.17) in the form

As we shall show, shallow-water dynamics with point vortices of APV is equivalent to the requirement that (2.18) be stationary with respect to arbitrary independent variations of the fields $\unicode[STIX]{x1D713}(\boldsymbol{x},t)$ and $\unicode[STIX]{x1D6FE}(\boldsymbol{x},t)$ and the vortex locations $\boldsymbol{x}_{i}(t)$ . The last line of (2.18) is an alternative way of writing (2.17) that is useful for performing the field variations. The form (2.17) is preferable for performing the vortex-location variations. Before demonstrating the equivalence between (2.18) and shallow-water dynamics, we fully relax the constraint on potential vorticity.

Suppose that the APV is not confined to delta functions but varies continuously in space. It must still be advected at the fluid velocity $\boldsymbol{u}$ . We replace the ansatz (2.15) by

where $\unicode[STIX]{x1D6FC}(\boldsymbol{x},t)$ and $\unicode[STIX]{x1D6FD}(\boldsymbol{x},t)$ are a set of fluid particle labels that measure APV and move with the fluid. Thus,

where $\unicode[STIX]{x1D736}=(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$ . Compare (2.20)–(2.16). The labels $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ replace the subscript $i$ on $\unicode[STIX]{x1D6E4}_{i}$ in (2.15). Just as the last term in (2.15) integrates to $\sum _{i}\unicode[STIX]{x1D6E4}_{i}$ , the last term in (2.19) integrates to $\iint \,\text{d}\unicode[STIX]{x1D6FC}\,\text{d}\unicode[STIX]{x1D6FD}$ ; the total APV is a constant.

We obtain the general variational principle for shallow-water dynamics by replacing

and $\dot{\boldsymbol{x}}_{i}\rightarrow \unicode[STIX]{x2202}\boldsymbol{x}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}$ in (2.18), where $\boldsymbol{x}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D70F})$ is the location at time $\unicode[STIX]{x1D70F}$ of the fluid particle labelled $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$ . Then,

By the chain rule,

Solving (2.23) for $\unicode[STIX]{x2202}\boldsymbol{x}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}$ , we obtain

Then, using (2.24)–(2.25) to eliminate $\unicode[STIX]{x2202}x/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x2202}y/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}$ from (2.22), and noting that the first term in (2.22) simplifies as

we obtain the generalization of (2.18) to the case of continuously varying vorticity, namely

We shall demonstrate that the requirement that (2.27) be stationary with respect to arbitrary variations in its four fields is equivalent to shallow-water dynamics.

The equations resulting from the requirement that (2.27) be stationary are

where ${\hat{h}}$ and $\boldsymbol{u}$ are given by (2.8)–(2.10),

is the APV and

From (2.24)–(2.25) and (2.31), it follows that

Thus, $\boldsymbol{J}$ is the flux of APV. In terms of the physical variables defined by (2.8)–(2.10), equations (2.28) and (2.29) take the forms

and

Eliminating $Q$ between (2.34) and (2.35), we obtain the shallow-water divergence equation (2.4). To see that (2.28)–(2.30) imply the shallow-water vorticity equation (2.3), we make use of the identity

which holds for any two functions $\unicode[STIX]{x1D6FC}(x,y,t)$ and $\unicode[STIX]{x1D6FD}(x,y,t)$ . By (2.31)–(2.33), equation (2.36) is equivalent to

Substituting (2.34) into (2.37) and using (2.2), we obtain (2.3). This concludes the proof that stationarity of (2.27) is equivalent to shallow-water dynamics.

Before proceeding, we briefly return to the Lagrangian (2.18) for point vortices of APV. The equations resulting from the requirement that (2.18) be stationary with respect to variations of $\unicode[STIX]{x1D713}(\boldsymbol{x},t)$ , $\unicode[STIX]{x1D6FE}(\boldsymbol{x},t)$ and $\boldsymbol{x}_{i}(t)$ are (2.28) with

equation (2.29) with

and (2.16). The definitions (2.38) and (2.39) imply

If the fluid depth is nearly constant, then we may replace the denominators $1-\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6FE}$ in (2.14) by unity, obtaining the quadratic approximation

to $L_{1}$ . If $f_{0}=0$ , then the Lagrangian $L_{1}^{0}+L_{2}$ , where $L_{2}$ is given by (2.17), is equivalent to the Lagrangian given in S14, which was shown in that paper to be closely analogous to the Lagrangian for classical electrodynamics. In this approximation, the waves are linear non-dispersive waves that interact with the point vortices but not with each other. The point vortices are analogous to electrons. The term $Q$ is analogous to electric charge density and $\boldsymbol{J}$ is analogous to electric current density. Equation (2.16) is analogous to the Lorentz force law. Further details on the electrodynamic analogy are given in appendix A.

In the present paper, we have greater use for the continuous-vorticity Lagrangian (2.27) than for its discrete-vorticity counterpart (2.18). The latter remains a valuable thinking tool (as illustrated in § 5) and may be useful for numerical studies. However, it is worth pointing out that point-vortex dynamics corresponding to (2.18) cannot be obtained as a special case of (2.27). This is obvious from the fact that, in applying point-vortex dynamics, we must omit the influence of each vortex on itself. Furthermore, any attempt to derive the continuous-vorticity dynamics by averaging over point vortices would introduce terms analogous to the molecular viscosity terms that arise when one attempts to derive ideal fluid dynamics by averaging over molecular motions.

The continuous-vorticity Lagrangian (2.27) seems to be a good starting point for theories describing the interactions between inertia–gravity waves and quasi-geostrophic flow. Three reasons for this are readily apparent. First, the Eulerian average of (2.9) and (2.10) corresponds to the Lagrangian mean velocity, often denoted $\bar{\boldsymbol{u}}^{L}$ . Thus, the average values of $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D6FE}_{t}$ represent the streamfunction and velocity potential for the Lagrangian mean flow. The automatic satisfaction of (2.2) by the representation (2.8)–(2.10) corresponds to the fact that, in the Lagrangian mean, no Reynolds fluxes occur in the average of (2.2).

Second, the Lagrangian $L_{1}$ , by itself, describes the fluid motion that results when waves propagate into a region that is initially at rest, and in which the APV therefore vanishes. That is, equation (2.14) is the Lagrangian for nonlinear inertia–gravity waves uncontaminated by APV. If $L_{1}$ is approximated as (2.41), then the equations resulting from $\unicode[STIX]{x1D6FF}L_{1}^{0}=0$ combine to yield the Klein–Gordon equation,

for $\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6FE}$ , which describes linear shallow-water inertia–gravity waves. The fully nonlinear form (2.14) will be used in § 4 to determine the mean flow that arises when inertia–gravity waves propagate into an initially quiescent region.

Third, there is an intimate connection between (2.27) and the Lagrangian for quasi-geostrophic motion. To see this most directly, suppose that all of the $\unicode[STIX]{x1D6FE}$ -terms except $f_{0}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FE}$ and $c^{2}(\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6FE})^{2}$ are simply dropped from (2.27). (This can be justified by scaling the variables in the manner appropriate for geostrophic flow.) The resulting Lagrangian is

where $J(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})\equiv \unicode[STIX]{x1D6FC}_{x}\unicode[STIX]{x1D6FD}_{y}-\unicode[STIX]{x1D6FD}_{x}\unicode[STIX]{x1D6FC}_{y}$ . If we use

hence

to eliminate $\unicode[STIX]{x1D6FE}$ from (2.43), we obtain the Lagrangian

for quasi-geostrophic dynamics. The condition (2.45) is a statement of geostrophic balance. Stationarity of (2.46) implies

Using the Jacobi identity,

we find that (2.47)–(2.49) imply the quasi-geostrophic potential vorticity equation,

where $r_{d}=c/f_{0}$ is the deformation radius.

A more systematic treatment of (2.27) yields theories that describe the interactions between motions that obey (2.51) at leading order and those that obey (2.42) at leading order. The use of the Lagrangian formulation guarantees that all conservation laws will be automatically maintained. However, instead of applying this strategy to shallow-water dynamics, we generalize our results to hydrostatic Boussinesq dynamics, which are of much greater interest to meteorologists and oceanographers. That such a generalization must exist is obvious from the close correspondence between the shallow-water equations and the hydrostatic Boussinesq equations in buoyancy coordinates.

3 Hydrostatic Boussinesq dynamics

In buoyancy coordinates, the hydrostatic Boussinesq equations take the form

Here, $\boldsymbol{u}(x,y,\unicode[STIX]{x1D703},t)=\boldsymbol{u}(\boldsymbol{x},\unicode[STIX]{x1D703},t)=(u,v)$ is the horizontal velocity, $\unicode[STIX]{x1D703}$ is the buoyancy, $z(\boldsymbol{x},\unicode[STIX]{x1D703},t)$ is the height of the isobuoyancy surface corresponding to $\unicode[STIX]{x1D703}$ at the horizontal location $\boldsymbol{x}$ and $B=p-z\unicode[STIX]{x1D703}$ , with $p$ equal to the pressure divided by the constant mass density. As before, $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D735}\times \boldsymbol{u}\equiv v_{x}-u_{y}$ , but now all horizontal derivatives are taken with $\unicode[STIX]{x1D703}$ fixed. For simplicity, we consider a horizontally unbounded fluid with flat rigid boundaries at $z_{1}$ and $z_{2}=z_{1}+h_{0}$ . At these boundaries, the buoyancy is (and remains) uniform at the values $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ . Thus, the boundary conditions on (3.1)–(3.3) are

The associated vorticity and divergence equations are

and

where now $\unicode[STIX]{x1D735}=(\unicode[STIX]{x2202}_{x},\unicode[STIX]{x2202}_{y})$ is the horizontal gradient operator with $\unicode[STIX]{x1D703}$ held fixed. The potential vorticity equation is

where

For derivations of these equations, see, for example, Salmon (Reference Salmon1998, pp. 105–107).

We seek a variational principle analogous to that found for shallow-water dynamics. First, as in § 2, we suppose that the fluid was, at some time, in the state of rest. In this state, the potential vorticity depends only on $\unicode[STIX]{x1D703}$ . However, since the potential vorticity and buoyancy are conserved on fluid particles, it must be true that

at all times, for some function $F$ . Evaluating (3.9) in the state of rest, we find that $F(\unicode[STIX]{x1D703})=f_{0}{N_{0}}^{2}(\unicode[STIX]{x1D703})$ , where $N_{0}(\unicode[STIX]{x1D703})$ is the Vaisala frequency associated with the state of rest. Thus,

We introduce the potential representation

Just as (2.8)–(2.10) automatically satisfies (2.2), the representation (3.11)–(3.13) satisfies (3.3). The prescribed function ${N_{0}}^{2}(\unicode[STIX]{x1D703})$ is analogous to $h_{0}^{-1}$ in § 2. In terms of $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D6FE}$ , the ansatz (3.10) takes the same form,

as in shallow-water dynamics. The $\unicode[STIX]{x1D703}$ -derivative of (3.2) is

Let

where $B_{0}(\unicode[STIX]{x1D703})$ is associated with the state of rest. Then, the hydrostatic equation (3.15) becomes

and the divergence equation (3.6) becomes

Equations (3.14), (3.17) and (3.18) result from the variational principle $\unicode[STIX]{x1D6FF}L_{1}[\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FE},\hat{B}]=0$ , where

The Lagrangian (3.19) is analogous to (2.14).

Before proceeding, we slightly adjust the vertical coordinate. We define a new buoyancy coordinate $Z(\unicode[STIX]{x1D703})$ by

Thus, $Z(\unicode[STIX]{x1D703})$ is the height of buoyancy surface $\unicode[STIX]{x1D703}$ in the state of rest. With this definition, (3.19) takes the form

The integration limits on $Z$ are $z_{1}$ and $z_{2}$ , and we henceforth regard $N_{0}=N_{0}(Z)$ . From now on, we work in the $(x,y,Z,t)$ system. Equation (3.10) takes the form

and (3.17) takes the form

Following the same path as in § 2 (but skipping the step corresponding to point vortices), we generalize the ansatz (3.22) to

where $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$ are labels that measure APV and move along buoyancy surfaces at the horizontal velocity $\boldsymbol{u}$ . Thus,

We define

The Lagrangian (3.21) is analogous to (2.14), and (3.26) is analogous to the last line of (2.27). As we now demonstrate, the variational principle $\unicode[STIX]{x1D6FF}(L_{1}+L_{2})=0$ is equivalent to hydrostatic Boussinesq dynamics.

The variational principle $\unicode[STIX]{x1D6FF}(L_{1}+L_{2})=0$ implies

and the boundary conditions $\hat{B}_{Z}=0$ at $Z=z_{1}$ and $Z=z_{2}$ , where, by (3.11)–(3.13),

Solving (3.30) and (3.31) for $u$ and $v$ , we obtain

where

is the APV. Then, (3.27) may be rewritten as

and (3.28) may be rewritten as

By (3.32) and (3.35), the right-hand side of (3.36) is

Thus, (3.36) is equivalent to the divergence equation (3.6). By (3.33a,b ) and (3.34), the mathematical identity (2.36) takes the same form,

as in § 2. Substituting (3.32) and (3.35) into (3.38), we obtain the vorticity equation (3.5). This concludes the proof that $\unicode[STIX]{x1D6FF}(L_{1}+L_{2})=0$ is equivalent to hydrostatic Boussinesq dynamics.

In the rest of this paper, we replace $Z$ by $z$ to simplify notation, but it is important to remember that $z$ is a disguised buoyancy coordinate. It is in fact the same coordinate introduced by Young (Reference Young2012).

To further simplify matters, we adopt the standard field-theory convention that all variations vanish at the extremities of the fluid, both in time and in space. That is, we do not attempt to incorporate boundary conditions (which are in some sense arbitrary) into the variational principle. Our focus is on the equations themselves.

4 Bretherton flow

As a first example of the use of the variational principle discovered in the previous section, we consider the flow that develops when waves enter a region that is initially at rest. Bühler & McIntyre (Reference Bühler and McIntyre2005) call this ‘Bretherton flow’ after the pioneering work of Bretherton (Reference Bretherton1969). If the fluid is initially at rest, then the APV vanishes, and the dynamics is completely described by $\unicode[STIX]{x1D6FF}L_{1}[\unicode[STIX]{x1D713},\unicode[STIX]{x1D6FE},\hat{B}]=0$ , where $L_{1}$ is given by (3.21). If the waves are sufficiently weak, we may write

where

contains the quadratic terms in $L_{1}$ , and

contains the higher-order corrections. By (3.11), small $\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6FE}$ corresponds to a nearly undisturbed stratification.

If the wave amplitude is small, then, to leading order, the waves are governed by $\unicode[STIX]{x1D6FF}L_{1}^{0}=0$ , which implies

Using (4.4) and (4.5) to eliminate $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D6FE}$ , we obtain a single equation for $\hat{B}$ ,

If $N_{0}(z)$ varies slowly, then (4.7) supports inertia–gravity waves of the form $\hat{B}\propto \cos (kx+ly+mz-\unicode[STIX]{x1D714}t)$ with dispersion relation

where $K^{2}=k^{2}+l^{2}$ . The dispersion relation (4.8) is correct for hydrostatic inertia–gravity waves, which have frequencies much smaller than $N_{0}$ .

If the wave amplitude $a$ is small, then the waves, which are approximately governed by (4.4)–(4.6), induce an $O(a^{2})$ mean flow that may be computed from $L_{1}$ . Let

where a bar denotes the mean flow and a prime denotes the wave. We assume that the wave is slowly varying in the sense that its amplitude, wavenumbers and frequency all vary on scales that are large compared with its wavelength and period. We follow Whitham’s (Reference Whitham1965, Reference Whitham1974) procedure. The averaged Lagrangian is

where $L_{1}^{0}$ and $L_{1C}$ are defined by (4.2) and (4.3). The angle brackets denote the average over wave phase. Thus, $\langle \bar{\unicode[STIX]{x1D713}}\rangle =\bar{\unicode[STIX]{x1D713}}$ and $\langle \unicode[STIX]{x1D713}^{\prime }\rangle =0$ . We approximate the last term in (4.10) by keeping only the cubic contributions to (4.3) that are quadratic in the primes. Then,

where

and

The enormous advantage of Whitham’s method is that terms in the Lagrangian may be simplified before the variations are taken to determine the equations. It turns out that all of the wave-forcing terms on the right-hand sides of (4.12) and (4.13) are negligible except for the first term in (4.12). Let

where $A$ is the slowly varying amplitude and $\unicode[STIX]{x1D719}$ is the rapidly varying phase. The frequency and wavenumbers,

all vary slowly. First, we consider the two terms on the right-hand side of (4.12). At leading order,

where $K^{2}=k^{2}+l^{2}$ . For the second term, we have

where we have used the fact that $\unicode[STIX]{x1D713}^{\prime }=-f_{0}\unicode[STIX]{x1D6FE}^{\prime }$ at leading order; see (4.4). The phase average in (4.17) is over two factors, $\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6FE}^{\prime }$ and $\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FE}^{\prime }$ , that are $90^{\circ }$ out of phase. Thus, (4.17) is smaller than (4.16) by a factor $\unicode[STIX]{x1D716}$ , where $\unicode[STIX]{x1D716}\ll 1$ is the ‘scale-separation’ parameter – the ratio of the wavelength to the length scale of slow variation. Similar remarks apply to all of the terms in (4.13): in contrast to (4.16), all involve at least two derivatives of the slowly varying wave amplitude, frequency and wavenumbers. Thus, the approximation (4.11) becomes

and the complete Lagrangian (4.10) becomes

where the relative frequency $\unicode[STIX]{x1D714}_{r}$ is defined by (4.8). The equations resulting from $\unicode[STIX]{x1D6FF}\langle L_{1}\rangle =0$ are

where $E\equiv (1/2)K^{2}\unicode[STIX]{x1D714}^{2}A^{2}$ , $\bar{\boldsymbol{U}}\equiv (-\bar{\unicode[STIX]{x1D713}}_{y},\bar{\unicode[STIX]{x1D713}}_{x})$ , $\unicode[STIX]{x1D735}_{3}\equiv (\unicode[STIX]{x2202}_{x},\unicode[STIX]{x2202}_{y},\unicode[STIX]{x2202}_{z})$ and

is the relative group velocity. The only forcing term in the equations (4.20)–(4.22) determining the mean flow is the curl of the pseudomomentum,

This $O(a^{2})$ term induces an $O(a^{2})$ mean flow that contributes the small final term to the dispersion relation (4.23). Equation (4.23) differs from the expected Doppler-shifted result $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{r}+\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{k}$ by an even smaller $O(a^{4})$ term, namely $(\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{k})^{2}$ , because the approximation (4.18) excludes terms that are quadratic in the mean flow. Similar remarks apply to (4.24). If we keep only the largest terms in each equation, then (4.20)–(4.22) are unchanged, (4.23) reduces to $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{r}$ and (4.24) takes the familiar form

of action conservation. In this limit, we may consider the slowly varying wavetrain to be a prescribed solution of the linear equations (4.4)–(4.6). The only challenge is to determine the induced mean flow from (4.20)–(4.22). However, it is worth pointing out that (4.20)–(4.24) are consistent with all conservation laws (including total energy conservation) because they have been obtained from a variational principle, whereas a posteriori approximations like (4.27) generally destroy exact conservation.

The general solution of the mean flow equations (4.20)–(4.22) includes free inertia–gravity waves, but to include them would be redundant: we are solely interested in the forced response to $\unicode[STIX]{x1D735}\times \boldsymbol{p}$ . If $f_{0}=0$ , then $\bar{\unicode[STIX]{x1D6FE}}$ and $\bar{B}$ vanish, and

determines the Lagrangian mean velocity. If $f_{0}\neq 0$ , then (4.20)–(4.22) may be combined to give

Suppose that the pseudomomentum $\boldsymbol{p}$ corresponds to a wavepacket. The wavepacket propagates at the group velocity corresponding to its ‘carrier wavenumber’ $\boldsymbol{k}$ . The forced solution of (4.29) propagates at this same group velocity. Hence, we may solve (4.29) in the reference frame moving with the packet by replacing $\unicode[STIX]{x2202}_{t}\rightarrow -\boldsymbol{c}_{gr}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{3}$ in (4.29). If the group velocity is sufficiently small, then the $\unicode[STIX]{x2202}_{tt}$ -term in (4.29) is negligible, and (4.29) reduces to

In this same limit, equation (4.22) implies $\bar{B}=f_{0}\bar{\unicode[STIX]{x1D713}}$ , so that (4.30) is equivalent to

The left-hand side of (4.31) is the quasi-geostrophic potential vorticity. Thus, if the inertia–gravity wavepacket propagates slowly enough, the mean flow response is quasi-geostrophic.

The WKB ansatz (4.14), which assumes a scale separation in all four of the dependent variables $(x,y,z,t)$ , is overly restrictive, because inertia–gravity waves with frequencies near $f_{0}$ correspond to large horizontal length scales. If we assume that only the time scale separates the mean flow from the waves, then only the last two terms in (4.13) may be neglected; all of the other terms in (4.12) and (4.13) must be retained. However, before proceeding with less restrictive assumptions, we generalize the dynamics to the case of non-vanishing APV.

5 Non-vanishing available potential vorticity

If the APV does not vanish, then the full Lagrangian includes $L_{2}$ , which couples the inertia–gravity waves to the APV. The introduction of APV is analogous to the introduction of electric charge. In this section, we develop approximations to $L_{2}$ analogous to those developed for $L_{1}$ in the previous section. We start by considering the form (2.17) of $L_{2}$ corresponding to shallow-water dynamics with point vortices of APV. Let

where $\bar{\boldsymbol{x}}_{i}(t)=\langle \boldsymbol{x}_{i}(t)\rangle$ is the average location of the $i$ th point vortex and $\unicode[STIX]{x1D743}_{i}(t)=(\unicode[STIX]{x1D709}_{i}(t),\unicode[STIX]{x1D702}_{i}(t))$ is the departure therefrom. Thus, $\unicode[STIX]{x1D709}_{i}(t)$ and $\unicode[STIX]{x1D702}_{i}(t)$ are rapidly fluctuating variables like $\unicode[STIX]{x1D6FE}^{\prime }(\boldsymbol{x},t)$ and $\unicode[STIX]{x1D713}^{\prime }(\boldsymbol{x},t)$ . Substituting (5.1) into (2.17) and averaging, we obtain

We proceed by Taylor expansion of the functions $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D6FE}_{y}$ and $\unicode[STIX]{x1D6FE}_{x}$ about the argument $\bar{\boldsymbol{x}}_{i}$ . Consider the single term

The function $\bar{\unicode[STIX]{x1D713}}$ depends slowly on its argument no matter what the character of the argument itself. Similarly, the function $\unicode[STIX]{x1D713}^{\prime }$ depends rapidly on its argument. Proceeding with the Taylor expansion, and keeping only the quadratic terms in the fast variables, we obtain

In the preceding section, we approximated $L_{1}$ by keeping only the cubic terms of the form bar–prime–prime. Since the constant $\unicode[STIX]{x1D6E4}_{i}$ counts as a ‘bar’ variable, the middle term on the right-hand side of (5.4) is of the order bar–bar–prime–prime. Thus, if we treat $L_{2}$ in the same manner as we treated $L_{1}$ in the previous section, we can consistently neglect this middle term. (We come back to this point in § 8.) Our approximation becomes

The last two terms in (5.2) may be approximated in a similar manner, and the result is

The $\bar{\unicode[STIX]{x1D6FE}}$ -terms in (5.6) involve only the mean flow and represent higher-order corrections to quasi-geostrophy. Neglecting these terms, and defining $\unicode[STIX]{x1D743}(\boldsymbol{x},t)$ to be the fluctuating displacement of the fluid particle currently located at $\boldsymbol{x}$ (whether or not there is any APV there), we rewrite (5.6) as

Because of the delta function, all of the terms in the integrand of (5.7) can be considered to be functions of $(\boldsymbol{x},t)$ . We define

In contrast to the $Q$ defined by (2.38), $\tilde{Q}$ depends slowly on time; it evolves with the average locations of the point vortices. However, $\tilde{Q}$ is not the average of $Q$ ; in that way it resembles the quantity $\tilde{\unicode[STIX]{x1D70C}}$ introduced by Bühler & McIntyre (Reference Bühler and McIntyre1998). Just as their $\tilde{\unicode[STIX]{x1D70C}}$ represents the mass density that would occur if the fluid moved at the average velocity, $\tilde{Q}$ represents the APV that would be present if the point vortices moved at their average velocity $\dot{\bar{\boldsymbol{x}}}_{i}$ .

We pass to the limit of continuous APV by replacing

where $(\tilde{\unicode[STIX]{x1D6FC}},\tilde{\unicode[STIX]{x1D6FD}})$ are labelling variables for $\tilde{Q}$ . Compare (5.9)–(2.21). Then, following steps similar to those between (2.21) and (2.27), we arrive at

for the shallow-water case. The $\langle L_{2}\rangle$ for hydrostatic Boussinesq dynamics requires only an additional integration with respect to $z$ and the reinterpretation of all of the derivatives in (5.10) as derivatives with $z$ held fixed.

The Lagrangian (5.10) depends on the four fast variables $\unicode[STIX]{x1D713}^{\prime }$ , $\unicode[STIX]{x1D6FE}^{\prime }$ , $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D702}$ . However, we may use the equations

to eliminate $\unicode[STIX]{x1D713}^{\prime }$ and $\unicode[STIX]{x1D6FE}^{\prime }$ in favour of $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D702}$ . Note that the variations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D743}$ are performed inside the averaging symbol in (5.10), and that the time integrations by parts needed to establish (5.11)–(5.12) ignore the slow time variations of the Jacobian in (5.10). Equations (5.11)–(5.12) equate the time derivatives of $\unicode[STIX]{x1D743}$ to the fluctuating velocity. Substituting (5.11)–(5.12) back into (5.10) (and adding the $z$ -integration required for three-dimensional Boussinesq dynamics), we obtain

for hydrostatic Boussinesq dynamics.

6 The complete Lagrangian

Now, we combine the results of §§ 4 and 5 to obtain the complete Lagrangian. To obtain a form of $\langle L_{1}\rangle$ that is compatible with (5.13), we must eliminate the variables $\unicode[STIX]{x1D713}^{\prime }$ and $\unicode[STIX]{x1D6FE}^{\prime }$ from (4.10)–(4.13). Using (5.11)–(5.12), we obtain

where

and

In writing (6.1), we have dropped the last two terms in (4.13): since the average is an average over the fast time dependence of the waves, we have $\unicode[STIX]{x2202}_{t}\langle \rangle =0$ .

Now, we embark on a series of simplifications. First, assuming that the mean flow response will be slow, we drop the $(1/2)\unicode[STIX]{x1D735}\bar{\unicode[STIX]{x1D6FE}}_{t}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\bar{\unicode[STIX]{x1D6FE}}_{t}$ -term from (6.1). This neglect is equivalent to the neglect of $\bar{B}_{tt}$ in (4.29), which led to (4.31). By a priori neglect of $(1/2)\unicode[STIX]{x1D735}\bar{\unicode[STIX]{x1D6FE}}_{t}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\bar{\unicode[STIX]{x1D6FE}}_{t}$ , we prevent the mean flow from developing its own inertia–gravity waves.

Next, noting that the slow variables $\bar{B}$ and $\bar{\unicode[STIX]{x1D6FE}}$ appear only in $\langle L_{1}\rangle$ , we use the equation

to remove $\bar{B}$ and $\bar{\unicode[STIX]{x1D6FE}}$ from (6.1). That is, by substituting

into the terms

we obtain

after neglecting terms that are quartic in the wave amplitudes. With these simplifications, equation (6.1) takes the form

Finally, we note that $\unicode[STIX]{x1D6FF}(\langle L_{1}\rangle +\langle L_{2}\rangle )=0$ implies

To consistent order, we may use (6.9) to eliminate the Jacobian on the right-hand side of (5.13) from its product with $\langle \unicode[STIX]{x1D709}\unicode[STIX]{x1D702}_{t}\rangle$ . Neglecting terms quartic in the wave amplitudes, we obtain

where

is the familiar potential vorticity operator.

Combining (6.8) and (6.10), we obtain the averaged Lagrangian for the complete system as

where

is the Lagrangian for quasi-geostrophic dynamics in the absence of inertia–gravity waves,

is the Lagrangian for inertia–gravity waves in the absence of a quasi-geostrophic mean flow and

is the Lagrangian that couples the inertia–gravity waves to the quasi-geostrophic motion. The quadratic wave averages $M$ and $T$ are defined by (6.2) and (6.3). The terms involving $M$ and $T$ occur in (6.1) and would be present even if there were no APV. The last term in (6.15) represents the coupling between the waves and the APV.

The Lagrangian (6.12)–(6.15) is our fundamental result. To derive it, we have assumed that the quasi-geostrophic motion, now represented by $(\bar{\unicode[STIX]{x1D713}},\tilde{\unicode[STIX]{x1D6FC}},\tilde{\unicode[STIX]{x1D6FD}})$ , evolves slowly in comparison with the inertia–gravity waves, now represented by $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},B^{\prime })$ , and we have identified the average with an average over fast time. This assumption of a time scale separation between the mean flow and the waves would seem to be the weakest assumption on which to base any wave–mean theory: the waves cannot have frequencies less than $f_{0}$ , and the quasi-geostrophic flow evolves on a time scale longer than $1/f_{0}$ . Stronger assumptions lead to simplifications of (6.15) that yield simpler equations. In the next section, we explore a spectrum of possibilities.

7 Wave–mean equations

We work with the Lagrangian (6.12)–(6.15), which assumes that the waves and the mean flow are separated only by their time scales. We obtain the equations for the mean flow by varying $\bar{\unicode[STIX]{x1D713}}$ , $\tilde{\unicode[STIX]{x1D6FC}}$ and $\tilde{\unicode[STIX]{x1D6FD}}$ . By the Jacobi identity (2.50) and the definition

the equations

imply

The $\unicode[STIX]{x1D6FF}\bar{\unicode[STIX]{x1D713}}$ -variation yields the defining equation for $\tilde{Q}$ . The variations of $\unicode[STIX]{x1D709}$ , $\unicode[STIX]{x1D702}$ and $B^{\prime }$ yield the equations for the waves. The precise form of these equations will depend on additional assumptions that we may wish to make. At leading order, $\unicode[STIX]{x1D6FF}L_{IG}=0$ implies the linear dynamics

which are equivalent to (4.4)–(4.6). We may consistently use (7.4)–(7.6) to simplify the coupling Lagrangian (6.15).

The strongest possible assumption appears to be the WKB assumption of a separation in scale with respect to all four of the independent variables $x,y,z,t$ . Then, the averaging operator corresponds to an average over wave phase. Let $\unicode[STIX]{x1D716}$ be the scale-separation parameter, as in § 4. Then, the last two terms in (6.15) are $O(\unicode[STIX]{x1D716}^{2})$ because they involve two spatial derivatives of phase averages, whereas the $\bar{\unicode[STIX]{x1D713}}M$ -term in (6.15) is $O(\unicode[STIX]{x1D716})$ , as already noted in § 4. Thus, under WKB scaling,

where

The second approximation in (7.7) uses the fact that the variables $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D702}$ are $90^{\circ }$ out of phase, as may be shown from (7.4)–(7.6). Using (7.4)–(7.6) and $\unicode[STIX]{x2202}_{x}\langle \rangle =0$ , etc., we can rewrite the right-hand side of (7.8) in several ways, including the one preferred by Bühler & McIntyre (Reference Bühler and McIntyre1998). In the case of a slowly varying wavetrain, we find that $\boldsymbol{p}=E\boldsymbol{k}/\unicode[STIX]{x1D714}$ , as in § 4. The variational principle generalizes the results obtained in § 4. The equation

replaces (4.31), and we now have (7.3) for advection of APV. The dispersion relation (4.23) and the action equation (4.24) are unchanged. These results are consistent with Bühler & McIntyre (Reference Bühler and McIntyre1998). They obtain (7.3) and (7.9), but their wave equations differ from (4.23) and (4.24) because they assume that mean flow is $O(1)$ . For reasons that are further explained in § 8, we have assumed that the mean flow is $O(a^{2})$ .

A weaker assumption than the WKB assumption is that the waves have frequencies near the inertial frequency $f_{0}$ . Such waves vary rapidly in $t$ and $z$ , but not in $x$ or $y$ . Assuming that the average corresponds to an average over fast time and fast $z$ , we can consistently neglect the $z$ -derivatives of averages in the coupling Lagrangian (6.15). Then,

We use (7.6) to eliminate $B^{\prime }$ from (6.14). Then, (6.14) takes the form

In near-inertial motion, the last term in (7.11) is small. With this in mind, we rewrite the complete Lagrangian as

where $L_{QG}$ is given by (6.13),

is the Lagrangian for inertia waves and

is the sum of (7.10) and the last term in (7.11). We shall treat all of the terms in (7.14) as small terms.

Following Young & Ben Jelloul (Reference Young and Ben Jelloul1997) and Xie & Vanneste (Reference Xie and Vanneste2015, hereafter XV), we set

The exponential factor in (7.15) represents the fast time dependence of inertial motion. The complex coefficient $\unicode[STIX]{x1D712}(x,y,z,t)$ varies slowly in $x,y,t$ but rapidly in $z$ . From (7.15), we have

and

where c.c. denotes the complex conjugate. We substitute (7.16) and (7.17) back into (7.13) and (7.14), computing the averages as averages over the fast time dependence of the exponential factors. In (7.13), we must keep terms of the first and second order. In (7.14), we keep only the leading-order terms. The Coriolis parameter $f_{0}$ serves as a convenient ordering parameter. Thus, in (7.13) we set

and

and in (7.14) we set

and

Here, $^{\ast }$ denotes complex conjugation and the averaging symbols enclose the rapid $z$ -dependence of $\unicode[STIX]{x1D712}$ . In simplifying the terms, we have freely used integrations by parts with respect to the rapid variations in $z$ and $t$ , and the fact that the averages are averages over these rapid variations. With these substitutions, (7.13) takes the form

and (7.14) takes the form

The complete Lagrangian for near-inertial motion is

The variations of $\tilde{\unicode[STIX]{x1D6FC}}$ and $\tilde{\unicode[STIX]{x1D6FD}}$ lead to (7.3), as in general. The variations of $\bar{\unicode[STIX]{x1D713}}$ and $\unicode[STIX]{x1D712}$ yield

and

(The variations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712}$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712}^{\ast }$ may be taken independently, but they lead to the same result.) Note that the terms in (7.27) vary rapidly in $z$ . Equations (7.3), (7.26) and (7.27) are very close to the equations derived by XV. In our notation, the XV equations comprise (7.3),

and

Equation (7.29) was previously derived by Young & Ben Jelloul (Reference Young and Ben Jelloul1997), who did not consider the effect of the waves on the mean flow. The XV equations correspond to a Lagrangian, namely

that differs from (7.25) by a term with a factor of the form $\unicode[STIX]{x2202}_{z}\langle \rangle$ . This term is negligible if, as we have assumed, there is a vertical scale separation between the quasi-geostrophic mean flow and the near-inertial waves. Similarly, the differences between (7.26) and (7.28), and between (7.27) and (7.29), are insignificant under this assumption. The important point is that both sets of equations – ours and those of XV – being equivalent to variational principles, correspond to a consistent set of conservation laws. This fact was used to great advantage by XV.

Finally, we consider the case in which we make no additional assumptions beyond the separation in time scales used to derive (6.12)–(6.15). Then, all of the terms in (6.15) must be kept. If we keep all of the terms in (6.15), then the requirement that (6.12) be stationary with respect to variations in $\bar{\unicode[STIX]{x1D713}}$ yields the generalization

of (7.9) and (7.26), where

Using (7.4)–(7.6) and $\unicode[STIX]{x2202}_{t}\langle \rangle =0$ , we find that

By similar manipulations,

Thus, (7.32) may be written as

Assuming only the separation in time scales, Wagner & Young (Reference Wagner and Young2015, hereafter WY) derive (7.31) with

We must show that our (7.35) agrees with their (7.36). WY work in Cartesian coordinates with fluid particle displacements $(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2},\unicode[STIX]{x1D709}_{3})=(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})$ . (The vertical displacement $\unicode[STIX]{x1D701}$ must not be confused with the relative vorticity denoted by the same symbol earlier in the paper.) Although the first three terms on the right-hand side of (7.36) involve only $x$ - and $y$ -derivatives, the last term in (7.36) contains index summations from 1 to 3. Thus, the connection between (7.35) and (7.36) is not obvious. In appendix B, we show that (7.35) and (7.36) are in fact equivalent.