2.1 The solution at first order

In the first-order equations we neglect the small buoyancy force $b_{1}\hat{\boldsymbol{z}}$ on the right of the momentum equation (2.2). With this approximation the classic first-order solution is irrotational and is expressed using the familiar velocity potential $\unicode[STIX]{x1D719}(x,y,z,t)$ :

The problem then reduces to the solution of the 3-D Laplace equation $\unicode[STIX]{x1D719}_{xx}+\unicode[STIX]{x1D719}_{yy}+\unicode[STIX]{x1D719}_{zz}=0$ with the surface boundary condition $\unicode[STIX]{x1D719}_{tt}+g\unicode[STIX]{x1D719}_{z}=0$ at $z=0$ and the deep water condition that $\unicode[STIX]{x1D719}(x,y,-\infty ,t)=0$ . Correct to leading order in the scale separation parameter $\unicode[STIX]{x1D707}$ , the first-order solution for an SW group is

where $a(\tilde{x},y,z)$ is the slowly varying envelope, $c=\sqrt{g/k}$ is the SW phase speed, c.c. denotes the complex conjugate, and

is the wave group coordinate; $c/2$ in (2.12) is the deep water group velocity. We neglect dispersive spreading of the group (see van den Bremer & Taylor Reference van den Bremer and Taylor2016).

2.2 Quadratic properties of the first-order solution

At next order we need several quadratic properties of the first-order solution, some of which are expressed most easily using the first-order wave displacement $\unicode[STIX]{x1D743}_{1}$ defined via

At $z=0$ the vertical displacement $\unicode[STIX]{x1D701}_{1}(x,y,z,t)=\hat{\boldsymbol{z}}\boldsymbol{\cdot }\unicode[STIX]{x1D743}_{1}$ is the same as the first-order displacement of the free surface $h_{1}(x,y,t)$ ; the dynamically negligible buoyancy perturbation in the wave zone is diagnosed as $b_{1}=-N^{2}\unicode[STIX]{x1D701}_{1}$ .

Denoting a running phase average over the fast oscillation of the primary wave by an overbar, the Stokes drift in the $x$ -direction is

The mean wave momentum per unit area in the $x$ -direction is $\unicode[STIX]{x1D70C}_{0}M$ , where

with $|_{0}$ indicating evaluation at $z=0$ . The identity

shows that $u^{S}$ can be interpreted as the vertical distribution of the mean wave momentum (e.g. Phillips Reference Phillips1977). The chain of identities

is useful at next order. The identities in (2.17) express all important wave-averaged quantities in terms of the Stokes drift in (2.14).

2.3 The second-order wave-averaged equations of motion

To write the phase-averaged second-order equations compactly we introduce the Bernoulli function

Using $\unicode[STIX]{x1D71B}$ , the phase-averaged second-order equations are

We have neglected the term $\boldsymbol{u}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b_{1}$ in (2.20) because this term decays exponentially with depth over a layer of depth $(2k)^{-1}$ . This term is important for mixed layer dynamics, but not for the deep return flow or wake of radiated IWs. Although the buoyancy force $b\hat{\boldsymbol{z}}$ has a negligible effect on the SW group it does affect the deep flow and therefore it is essential to retain $\bar{b}_{2}\hat{\boldsymbol{z}}$ in (2.19). Because of the buoyancy force, the deep flow beneath the SW group is not irrotational and thus it is not possible to use a velocity potential to represent the solution of (2.19)–(2.21). Instead the system can be combined to obtain the IW equation for the second-order wave-averaged vertical velocity

Combining (2.20) with the vertical part of (2.19), the pressure $\unicode[STIX]{x1D71B}$ can be expressed in terms of the second-order vertical velocity as

The wave-averaged second-order surface boundary conditions following from (2.6) and (2.7) are

and

Identity (2.17) has been used to express the surface boundary condition (2.25) in terms of $\unicode[STIX]{x1D71B}$ . Following earlier authors (e.g. Longuet-Higgins & Stewart Reference Longuet-Higgins and Stewart1964; van den Bremer & Taylor Reference van den Bremer and Taylor2015), we make the ‘rigid-lid approximation’ by neglecting $\bar{h}_{2t}$ in (2.24) so that the surface boundary condition simplifies to

Note that neglecting the mean sea surface displacement, $\bar{h}_{2}$ , is only valid in deep water (see van den Bremer & Taylor Reference van den Bremer and Taylor2015). If required, the mean surface displacement $\bar{h}_{2}$ can be diagnosed from (2.25) as $\bar{h}_{2}\approx g^{-1}\unicode[STIX]{x1D71B}|_{0}$ . The bottom boundary condition is $\bar{w}_{2}=0$ .

3.1 Projection onto vertical modes and Fourier transform

With uniform $N$ we express the solution of the IW equation (2.22) and the pressure equation (2.23) as a sum of orthonormal vertical modes:

where the vertical wavenumber is

The modal amplitudes in (3.2) are given by

Projecting the IW equation (2.22) onto the sine modes we obtain

The forcing on the right-hand side of (3.5) comes from handling the $z$ -derivatives in (2.22) with integration by parts.

Moving with the SW group, we look for a solution of (3.5) of the form $\text{e}^{\unicode[STIX]{x1D6FF}t}w_{n}(\tilde{x},y)$ where $\tilde{x}$ is the group coordinate defined in (2.12). Using the Fourier transform

the solution of (3.5) is

where

Projecting the pressure equation (2.23) onto the sine basis functions and then Fourier transforming, one finds that the modal amplitudes of the pressure field in (3.2) are

The inverse Fourier transform of (3.7)

In passing from (3.10) to (3.11) we have exploited the symmetries ${\hat{w}}_{n}(q,s)={\hat{w}}_{n}(q,-s)$ and ${\hat{w}}_{n}(q,s)={\hat{w}}_{n}^{\star }(-q,s)$ , where $^{\star }$ indicates the complex conjugate, to write the inverse Fourier transform as an integral over the first quarter of the wavenumber plane. Here we have assumed that $\hat{M}$ has these same symmetries.

Table 1.

Numerical values characteristic of swell and stratification; $g=9.81~\text{m}~\text{s}^{-2}$ and $\unicode[STIX]{x1D70C}_{0}=1000~\text{kg}~\text{m}^{-3}$ . Here we have picked parameters for short period, 1 m amplitude swell as a typical example, and long period 4 m amplitude swell as an example of extreme forcing. The depth-averaged stratification, $N$ , is consistent with the phase speed of the first baroclinic mode for a depth $d=2000$  m. Figure 5 of Chelton et al. (Reference Chelton, Deszoeke, Schlax, El Naggar and Siwertz1998) shows that the average phase speed of the first baroclinic mode varies with latitude between approximately $1.5~\text{m}~\text{s}^{-1}$ – $3~\text{m}~\text{s}^{-1}$ . The surface group length, $\ell _{x}$ , is based on the assumption that a group is comprised of $n_{SW}$ consecutive waves that are at least half as high as the tallest in the group. While observations of wave group statistics in the North Sea (Battjes & Van Vledder Reference Battjes and Van Vledder1984, figure 3b) and in coastal regions (Elgar, Guza & Seymour Reference Elgar, Guza and Seymour1984) show that groups of one or two waves are far more likely, five wave groups are possible.

The vertical velocity is obtained by numerical integration of (3.11) and for illustration we use the typical SW and stratification parameters given in table 1, and choose $\unicode[STIX]{x1D6FF}=3.1\times 10^{-5}~\text{s}^{-1}$ for our slow growth parameter. The horizontal structure of the first four vertical modes is shown in figure 2. Each mode shows a wake of IWs trailing behind the SW group, which is centred at the origin. Figure 3 shows the full vertical velocity for IWs radiated from an SW group passing over a uniformly stratified ocean. Figure 3 shows a strong Eulerian return flow in the forcing region with weaker vertical velocities associated with IWs trailing behind the forcing region. The solution is dominated by the first vertical mode.

Figure 2.

Common logarithm of $|w_{n}|$ for the first four modes. Dashed contours indicate negative vertical velocities. The SW group is centred at the origin. The SW and stratification parameters are given for the typical case in table 1. The thin black line is the contour where the vertical velocity vanishes. The red dash-dot line is the theoretical prediction for maximum wake angle given by (3.18).

Figure 3.

Common logarithm of $|\bar{w}_{2}|$ . The SW group is centred at the origin and propagates in the positive $x$ -direction. Dashed contours indicate negative vertical velocities. The typical SW and stratification parameters used are given in table 1. The solution shown is a numerical solution of (3.11) with $\unicode[STIX]{x1D6FF}=3.1\times 10^{-5}~\text{s}^{-1}$ , with 200 vertical modes near the forcing region $[-0.1q_{max}x:0.1q_{max}x]$ and 20 vertical modes in the rest of the domain.

Figure 4.

The singular curves defined by $\unicode[STIX]{x1D6FE}(q,s)=0$ in the first quadrant of the $(q_{\ast },s_{\ast })$ -plane, where $q_{\ast }=q/q_{max}$ , $s_{\ast }=s/q_{max}$ , and $m_{n\ast }=m_{n}/q_{max}$ . If $m_{n\ast }\geqslant 1$ the curve passes through the origin. The curve for $m_{n\ast }=1$ is the thick black line.

3.2 The singular curves

Returning to (3.7), we see the problem of zero divisors if we set $\unicode[STIX]{x1D702}=0$ . With small non-zero $\unicode[STIX]{x1D702}$ the solution is concentrated in wavenumber space on the ‘singular curve’ where the denominator of (3.7) is close to zero. These curves are shown in the first quadrant of the $(q,s)$ -plane in figure 4 for a few values of stratification, depth, and SW group speed. The singular curves are defined by the zeros of the function

The structure of these curves depends on the crucial non-dimensional parameter

which is the ratio of the SW group speed to the approximate IW phase speed of vertical mode $n$ . Figure 4 shows that the case where $m_{n\ast }\leqslant 1$ is very different from the case $m_{n\ast }>1$ . In particular, in 2-D ( $s=0$ ), only cases with $m_{n\ast }<1$ will radiate IWs, because only those singular curves intersect the $q$ -axis. Further discussion of the 2-D problem is in § 5. The solution shown in figures 2 and 3 has $m_{n\ast }>1$ for all vertical mode numbers $n$ .

Using the definition of $q_{max}$ , the condition $\unicode[STIX]{x1D6FE}(q,s)=0$ can be re-arranged as

Recalling the IW dispersion relation for vertical mode $n$ , namely $\unicode[STIX]{x1D714}^{2}=N^{2}(q^{2}+s^{2})/(q^{2}+s^{2}+m_{n}^{2})$ , we identify the right-hand side of (3.14) as the IW phase speed in the $x$ -direction. Thus the singular curve (3.14) is the resonance condition that the $x$ -phase speed of the IWs matches the group velocity $c/2$ of the SWs. Combining the IW dispersion relation and (3.14) we see that

This relation shows there is an upper bound on the values of $q$ relevant for IW radiation: because $\unicode[STIX]{x1D714}<N$ , wavenumbers with $q>q_{max}$ cannot radiate.

3.3 The wake angle

The stratification in the ocean is not sufficiently strong to support IWs that propagate at the SW group speed. However, if the radiated IWs propagate obliquely to the direction of SW group propagation then the point of intersection between the IW crest and the SW group can move with the SW group speed. To illustrate this, we will put ourselves in the reference frame moving with the SW group. Then the surrounding water is flowing backward at speed $c/2$ . As shown in figure 5, the component of this backward flow that is parallel to the IW phase velocity must be equal in magnitude to the IW phase velocity. This geometric condition recovers the resonance condition

The wake angle $\unicode[STIX]{x1D703}_{n}$ is related to the direction of the phase velocity, $\unicode[STIX]{x1D712}_{n}$ , by $\unicode[STIX]{x1D703}_{n}=\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D712}_{n}$ , so that $\cos \unicode[STIX]{x1D712}_{n}=\sin \unicode[STIX]{x1D703}_{n}$ . Therefore the wake angle is given by

where we have used the IW dispersion relation for $\unicode[STIX]{x1D714}(q,s)$ . To find the widest possible wake angle $\unicode[STIX]{x1D703}_{n}$ , we maximize the right-hand side of (3.17) over all wavenumbers by taking $q=s=0$ to find

The maximum wake angle is determined by the stratification, depth, and SW group speed. The first four modal constituents of vertical velocity $w_{n}(\tilde{x},y)$ are shown in figure 2. The dashed red line at the angle determined by (3.18) is parallel to the zero contour (the solid, black line), indicating that wake angle predicted by (3.18) is accurate. (Recall in figure 2 that $m_{n\ast }>1$ for all $n$ .)

Figure 5.

Schematic of the IW wake in the reference frame where the SW group is stationary. The surrounding water rushes backward (to the left) at the group speed ( $c/2$ ). The IW crest intersects the $x$ -axis at an angle $\unicode[STIX]{x1D703}_{n}$ , while the phase velocity of the IW, which is perpendicular to its crest, is at an angle of $\unicode[STIX]{x1D712}_{n}=\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D703}_{n}$ to the $x$ -axis. The condition in (3.16) is that the IW crest is stationary because the advance of phase normal to the crest is halted by the normal component, $(c/2)\cos \unicode[STIX]{x1D712}_{n}$ , of the water velocity.

The maximum wake angle in (3.17) is undefined if $m_{n\ast }<1$ . This is the case if the stratification is strong so the fastest radiated IWs are moving at the SW group speed. Choosing even smaller values of $m_{n\ast }$ (larger $N$ , larger $d$ , or smaller $c/2$ ) does not change the wake angle beyond $\unicode[STIX]{x1D703}_{n}=\unicode[STIX]{x03C0}/2$ .

4.1 The radiation integral

From (2.19)–(2.21) we obtain the second-order energy conservation equation

Thus the vertical component of the energy flux at the surface is $\unicode[STIX]{x1D71B}\bar{w}_{2}|_{0}$ ; the total flux of energy out of the SW group and into the IW field in Watts is therefore $\unicode[STIX]{x1D70C}_{0}J$ where

In (4.2) the surface boundary condition in (2.26) has been used for $\bar{w}_{2}|_{0}$ . Using the modal expansion for $\unicode[STIX]{x1D71B}$ in (3.2) the right-hand side of (4.2) is

where $J_{n}$ is the energy flux into vertical mode $n$ .

With Parseval’s theorem, we can express $J_{n}$ in terms of the Fourier transforms $\hat{M}$ and $\hat{\unicode[STIX]{x1D71B}}_{n}$ as

The Fourier transform $\hat{\unicode[STIX]{x1D71B}}_{n}$ is given by (3.9) and thus the energy flux into vertical mode $n$ is $\unicode[STIX]{x1D70C}_{0}J_{n}$ where

It is helpful to define

so that the partial fraction decomposition of (4.5) can be written as

We can drop the term $q+\text{i}\unicode[STIX]{x1D702}$ in the square bracket above: the $q$ is an odd function that integrates to zero and the $\text{i}\unicode[STIX]{x1D702}$ is a non-singular term that vanishes in the limit $\unicode[STIX]{x1D702}\rightarrow 0$ . The remaining terms in the integral are even in both $q$ and $s$ and taking advantage of these symmetries we can restrict the integration to the first quadrant to obtain

Taking the limit $\unicode[STIX]{x1D702}\rightarrow 0$ in (4.8) using the result in appendix B, we find that the double integral is concentrated on the singular curves given by the zeros of $\unicode[STIX]{x1D6FE}(q,s)$ in (3.12), and shown in figure 4. Thus, after the limit $\unicode[STIX]{x1D702}\rightarrow 0$ the double integral in (4.8) is reduced to a single integral with respect to $q$ :

In the second argument of $\hat{M}$ in (4.9), the singular curves defined by $\unicode[STIX]{x1D6FE}(s,q)=0$ , with $\unicode[STIX]{x1D6FE}(s,q)$ in (3.12), are parameterized by $q$ as $s=s_{n}(q)$ , where

The upper limit of integration in (4.9) is $q_{max}=2N/c$ and the lower limit is

The definition of $q_{min}$ above corresponds to the distinction between the curves in figure 4 corresponding to $m_{n\ast }\geqslant 1$ , which pass through the origin, and those curves with $m_{n\ast }<1$ which cross the $q$ axis at $\sqrt{q_{max}^{2}-m_{n}^{2}}$ . Both types of curves asymptote to $s=\infty$ as $q\rightarrow q_{max}$ : this asymptote corresponds to the upper limit of integration in (4.9). In physical terms there is a ‘cutoff’ wavenumber $q_{max}$ because IWs have frequencies less than $N$ : wavenumbers $q>q_{max}$ correspond the non-existent IWs with frequencies greater than $N$ .

The ‘radiation integral’ on the right of (4.9) is our most general expression for the energy lost to mode- $n$ internal gravity waves from the SW group. We make approximations to (4.9) assuming realistic SW and stratification parameters in § 4.3.

4.2 Energy transfer between a Gaussian SW group and the IW wake

To make a simple estimate of the energy loss from surface gravity waves we adopt the Gaussian model from appendix A and use numerical values in table 1. Since energy in the SWs is partitioned equally between kinetic and potential, the energy density of an SW train is given by $gh_{1}^{2}$ (Phillips Reference Phillips1977). Then using $h_{1}$ in (2.10) and the Gaussian wave envelope in (A 1) the total energy of the packet is $\unicode[STIX]{x1D70C}_{0}E$ , in Joules, where

Next we non-dimensionalize (4.9) as before using $q_{\ast }=q/q_{max}$ and $m_{n\ast }=m_{n}/q_{max}$ and replace $|\hat{M}|^{2}$ by the Gaussian expression (A 3). Using $c^{2}k=g$ , we find

where

Using the values in table 1, $q_{min\ast }=0$ .

Now we consider the slow time evolution of the SWs and assume that only the IW radiation affects the energy tendency:

In (4.16) we have expressed the energy flux in terms of the energy in the wave group, and the factor $\unicode[STIX]{x1D6FC}$ . We assume that $\unicode[STIX]{x1D6FC}$ is constant as $E\propto a_{max}^{2}$ slowly decreases due to radiative damping. Then the solution of the differential equation (4.16) is

where $E_{0}$ is the initial energy corresponding to the initial $a_{max}$ . Therefore the half-life of a group of SWs is given by

where $a_{max}$ and $J_{n}$ above are evaluated at $t=0$ . Equation (4.18) highlights that faster waves in a deeper ocean survive longer, while larger waves, larger wave groups and stronger stratification damp the wave group more quickly.

Computing the energy flux from SWs to IWs by numerical evaluation of the integral for $J_{n}$ in (4.13) we find that the radiated energy into the first $5$ vertical modes is approximately 0.2 W for typical forcing and approximately 100 W for extreme forcing in table 1. In the extreme forcing case, it would take over 100 000 days (and much more for the typical case) for the SWs to lose half their energy. In this time the wave group would have travelled more than 1000 times around the Earth. We can safely say that this is a small loss of energy from the SWs.

Figure 6.

The energy flux spectral density for the typical SW and stratification parameters in table 1. The solid lines show the exact integrand in (4.13), and the dashed lines show the approximation in (4.20) for the first three vertical modes as a function of normalized horizontal wavenumber, $q/q_{max}$ , which is equivalent to the IW frequency, $\unicode[STIX]{x1D714}$ , normalized by the buoyancy frequency $N$ .

Might this small energy flux be a significant energy source for the IW field? The ocean IW spectrum has two peaks: the near-inertial spectral peak and a secondary peak at frequencies close to the buoyancy frequency $N$ (Pinkel Reference Pinkel1975). As Olbers & Eden (Reference Olbers and Eden2016) note, the energy flux from SWs to IWs is small compared to the energy flux from the wind into near-inertial waves. However, figure 6 (which is discussed in more detail in § 4.3) shows that the majority of the energy radiated from SWs goes into near- $N$ IWs. The amplitude of the near- $N$ spectral peak is much smaller than that of the near-inertial spectral peak, so the energy flux estimated above may be significant for near- $N$ IWs while also negligible compared to the energy flux due to wind forcing. From figure 6 of Pinkel (Reference Pinkel1975) we estimate this peak in isotherm displacement squared $\unicode[STIX]{x1D701}^{\prime 2}$ to be $0.5~\text{m}^{2}~(\text{cycles per hour})^{-1}$ in amplitude, and $2$ cycles per hour wide. Then we can estimate the near- $N$ IW energy density as $\unicode[STIX]{x1D70C}N_{pyc}^{2}\unicode[STIX]{x1D701}^{\prime 2}d$ , where $N_{pyc}=2\unicode[STIX]{x03C0}/(900~\text{s})$ is a typical stratification in the pycnocline. The power density of IW radiation from SW forcing is $\unicode[STIX]{x1D70C}_{0}\sum _{n=1}^{5}J_{n}/\ell _{x}\ell _{y}$ . Then a time scale of forcing, $\unicode[STIX]{x1D70F}$ , can be obtained by dividing observations of energy density by the power density to find $\unicode[STIX]{x1D70F}\approx 500$ days in the typical case and $\unicode[STIX]{x1D70F}\approx 1$ day in the extreme case. This implies that one day of extreme forcing by SW groups with these parameters would produce the measured amount of near- $N$ IW energy. Therefore, this mechanism is likely not a source of near- $N$ IWs under typical SW forcing, but may be significant during extreme forcing.

4.3 The limit of weak stratification and fast SWs

Figure 2 of Chelton et al. (Reference Chelton, Deszoeke, Schlax, El Naggar and Siwertz1998), shows a global map of the fastest possible IW phase speed based on observations of the vertically integrated stratification, $\int _{-d}^{0}N(z)\,\text{d}z$ and assuming hydrostatic IWs. The hydrostatic approximation results in an overestimate of the IW phase speed, so these estimates are used as an upper bound. The map shows that IW phase speeds never exceed $3.2~\text{m}~\text{s}^{-1}$ , and generally range from $1$ to $3~\text{m}~\text{s}^{-1}$ for the first baroclinic mode. If we consider swell with periods of $8$ or larger, then

In other words, for SWs with periods of $8$ or longer, the SW group speed is always significantly greater than the IW phase speed. In this sense the ocean stratification is weak, IWs are slow, and the SWs are fast. Thus the relevant case in (4.14) is $m_{n\ast }>1$ and $q_{min\ast }=0$ . The other case, $m_{n\ast }<1$ and $q_{min\ast }=\sqrt{1-m_{n\ast }^{2}}$ , is not relevant for Earth’s oceans.

Equation (4.13) is opaque: the dependence of $J_{n}$ on key parameters is buried inside a difficult integral. We will exploit the weak stratification and fast SWs to approximate (4.13), and reveal the dependence of $J_{n}$ on the stratification and SW parameters. After systematic simplification of the integrand in (4.13) assuming $m_{n\ast }^{2}\gg 1$ , and $\ell _{y}>\ell _{x}$ , we have

Above, ${\mathcal{K}}_{n}$ is an $m_{n\ast }^{2}\gg 1$ approximation to the integral $m_{n\ast }{\mathcal{J}}_{n}$ in (4.13). Using the typical wave and stratification parameters assumed in table 1, the approximate integrand in (4.20), is very close to the exact integrand in (4.13): see figure 6. Note that while the exact ${\mathcal{J}}_{n}$ in (4.13) depends on three parameters, the approximation ${\mathcal{K}}_{n}$ in (4.20) contains only the single parameter $m_{n}\ell _{y}$ , which is equal to the product of the three parameters in ${\mathcal{J}}_{n}$ . Therefore

provided that $m_{n\ast }^{2}\gg 1$ and $\ell _{y}/\ell _{x}>1$ .

Figure 7.

The approximate dimensionless integral ${\mathcal{K}}_{n}$ (dashed) compared with exact solutions ${\mathcal{J}}_{n}m_{n\ast }$ (solid) over a range of the control parameters $m_{n}\ell _{y}$ . The approximation agrees well in two cases where the assumptions ( $m_{n\ast }^{2}\gg 1$ and $\ell _{y}>\ell _{x}$ ) are met (blue and green). The approximation is very poor when the assumptions are not met (orange and red).

Figure 7 shows the dependence of the integral ${\mathcal{K}}_{n}$ on the control parameter $m_{n}\ell _{y}$ . The dashed line is ${\mathcal{K}}_{n}$ , and the solid lines are exact solutions ${\mathcal{J}}_{n}m_{n\ast }$ from (4.13). The approximation agrees well with the two exact solutions with parameters that satisfy the assumptions $m_{n\ast }^{2}\gg 1$ and $\ell _{y}>\ell _{x}$ , and very poorly with cases that do not satisfy these assumptions.

We see from (4.21) and from figure 7 that the energy flux decreases quickly with increasing $n$ . This is also clear in figure 6. Therefore, neglecting all but the first mode, the energy flux becomes

where we have replaced the wavenumber $k$ and phase speed $c$ with an expression including $g$ and the SW period $T$ using the deep water dispersion relation. Using a spectrum of waves, Olbers & Eden (Reference Olbers and Eden2016) recover the same dependence on stratification, SW period and nearly the same dependence on SW amplitude. (Olbers & Eden (Reference Olbers and Eden2016) present their results in terms of the wind speed. The energy flux in their equation (35) varies as wind speed to the seventh power, and they assume a parametrization in which the wave amplitude is proportional to the wind speed squared. Not included in their paper is a non-dimensional form of the radiative transfer which recovers the $T^{-5}$ dependence as well, Dirk Olbers 2017, personal communication.) From (4.22) it is clear that the SW amplitude, stratification, and wave period are the strongest controlling parameters. Thus even small changes in $T$ , $a_{max}$ , or $N$ can drastically change the rate of energy flux from SWs to IWs.

4.4 Long-crested SWs

An even simpler approximation is obtained if $m_{n}\ell _{y}$ is large: as $m_{n}\ell _{y}\rightarrow \infty$ we have ${\mathcal{K}}_{n}\sim 2^{-1/2}\unicode[STIX]{x03C0}^{-5/2}(d/n\ell _{y})^{3}$ . In this case, with long-crested wave groups, (4.21) simplifies to

provided $m_{n\ast }^{2}\gg 1$ , $\ell _{y}>\ell _{x}$ and $m_{n}\ell _{y}\gg 1$ . Note that (4.23) is a very poor approximation for our assumed typical parameters ( $m_{n}\ell _{y}\approx 1$ ), but the range of validity of this approximation is not far from the assumed parameter values. Nevertheless, the $m_{n}\ell _{y}\gg 1$ approximation is valid for high vertical modes $n\gg 1$ . This approximation highlights the weak dependence of the energy flux on the width of the group $\ell _{y}$ and the strong dependence on the depth $d$ .

5.1 Solution of the 2-D problem

With no $y$ dependence the IW equation for vertical velocity (2.22) is

It will be convenient to introduce the streamfunction $\unicode[STIX]{x1D713}$ , such that $\bar{w}_{2}=\unicode[STIX]{x1D713}_{x}$ , and $\bar{u}_{2}=-\unicode[STIX]{x1D713}_{z}$ . Replacing $\bar{w}_{2}$ by $\unicode[STIX]{x1D713}$ and moving into the wave group frame, we have

where recall $q_{max}=2N/c$ . The surface boundary condition in (2.26) is equivalent to $\unicode[STIX]{x1D713}(\tilde{x},0)=M(\tilde{x})$ . Then projecting (5.2) onto the sine modes defined in (3.4), we find that the evolution equation for amplitude of mode $n$ is

with $m_{n^{\ast }}=m_{n}/q_{max}$ and $M^{1D}(\tilde{x})$ is the one-dimensional Gaussian envelope in (A 4). In this 2-D case we do not need to consider a slowly growing wave group in order to implement the radiation condition i.e. $M^{1D}$ on the right of (5.3) is not growing.

Notice that (5.3) further elucidates the importance of the critical parameter $m_{n\ast }$ . In two dimensions, when $m_{n\ast }<1$ , solutions to (5.3) are propagating waves. This case corresponds to the singular curves that intersect the $q$ -axis at $q=q_{min}$ in figure 4. On the other hand, if $m_{n\ast }>1$ then solutions of (5.3) are evanescent i.e. exponentially trapped around the forcing $M^{1D}$ . This case corresponds to a return flow comprised of the evanescent modes similar to the unstratified case depicted in figure 1(a). Because $m_{n\ast }\propto n$ , in two dimensions there is a vertical mode number $n_{rad}$ above which $m_{n\ast }>1$ i.e. modes with $n>n_{rad}$ are evanescent and contribute to the return flow and not to radiation of IWs. If the stratification is weak (and the stratification of the ocean is weak in this sense) then $n_{rad}<1$ and all vertical modes are evanescent. (Whereas in three dimensions all modes are radiate.)

To solve (5.3) we employ Green’s functions. First consider the radiating modes with $q_{min}^{2}>0$ . The Green’s function $G(\tilde{x})$ is defined by

with solution

where $H(-\tilde{x})$ is Heaviside step function. The factor $H(-\tilde{x})$ ensures that the solution in (5.5) satisfies the radiation condition: radiated IWs trail behind ( $\tilde{x}<0$ ) the SW forcing. This choice of Heaviside function is analogous to choosing the sign of $\unicode[STIX]{x1D6FF}$ in (3.1). Using this Green’s function the streamfunction for radiating modes with $1\leqslant n\leqslant n_{rad}$ is

Turning to the evanescent modes with $q_{min}^{2}<0$ , we introduce $\unicode[STIX]{x1D6FD}_{n}^{2}=-q_{min}^{2}>0$ . Then the relevant solution of (5.4) is

and the streamfunction for evanescent modes with $n>n_{rad}$ is

To compute the energy flux, we first consider the solutions in (5.6) and (5.8) far in the wake of the SW packet i.e. as $\tilde{x}\rightarrow -\infty$ . In the wake, the evanescent modal amplitudes in (5.8) have decayed to zero and one is left with only the radiating modes with $n\leqslant n_{rad}$ . Moreover, the integral in (5.6) is simplified by taking the lower limit to $-\infty$ so that

where $\hat{M}^{1D}(q_{min})$ is the Fourier transform of $M^{1D}(\tilde{x})$ evaluated at the wavenumber $q_{min}$ and $\text{Im}$ is the imaginary part. Thus in the wake

where $\langle ~\rangle$ denotes a phase average of the IW i.e. an integral in $\tilde{x}$ over one wavelength $2\unicode[STIX]{x03C0}/q_{min}$ . The potential energy density is $b^{2}/N^{2}=q_{max}^{2}\unicode[STIX]{x1D713}^{2}$ , so the IW energy density in the wake of the SW packet is therefore

where the energy density of mode $n$ is

Figure 8.

The flow as in figure 1(b) is used here to illustrate the energy flux calculation (5.13)–(5.14). The thick rectangle defines a ‘control box’. The left edge of this box (at $x/d\approx -4$ ) is fixed far from the SW group. The right edge of the box moves with the SW group at speed $c/2$ . Energy from previously generated IWs is carried into the box through the left edge at a rate $E_{n}\unicode[STIX]{x1D710}_{n}$ . Energy also enters the box via a downward flux from the SW group.

We can relate the energy density of the $n$ th mode to the radiative flux using an argument given by Lamb (Reference Lamb1932) in the context of SW resistance to a moving body. Figure 8 shows the wake of IWs behind an SW group as in figure 1(b). To compute the energy flux we first consider the total energy inside a box with left edge fixed far from the SW forcing region, and with the right edge of the box just in front of, and moving along with, the SW group at speed $c/2$ . The energy in this ‘control box’ for the $n$ th mode is $E_{n}L(t)$ , where $L(t)$ is the length of the box. Then the energy flux into this box can be written as a sum of the energy flux from the SW forcing $J_{n}^{2D}$ , and the energy flux from IWs as they propagate through the left side of the box with their group speed $\unicode[STIX]{x1D710}_{n}$ . Thus the energy conservation equation for the control box is

Now $L=ct/2$ , where $t$ is the time since the SW group entered the box by passing through the left edge and so from (5.13) we obtain Lamb’s result

The IW group velocity $\unicode[STIX]{x1D710}_{n}$ in (5.14) can be expressed in terms of the IW phase speed for a radiating wave, $c/2$ , and the critical parameter $m_{n\ast }$ as

Using expression (5.15) for the group velocity $\unicode[STIX]{x1D710}_{n}$ , the radiative flux for the $n$ th vertical mode given by (5.14) is

In the illustrative case of a Gaussian envelope in (A 4) and (A 5), the radiation flux in (5.16) is

5.2 The $\ell _{y}\rightarrow \infty$ limit of the 3-D solution illustrated with a Gaussian wave packet

Throughout this section, we assume that $m_{n\ast }<1$ so that $q_{min}$ in (5.3) is real. This condition is necessary for the radiation of IW energy: with $s=0$ the resonance condition for vertical mode $n$ is that $c/2$ is equal to $N/\sqrt{q^{2}+m_{n}^{2}}$ , or equivalently that $q$ is equal to $q_{min}$ in (4.9) and figure 4. The results of Chelton et al. (Reference Chelton, Deszoeke, Schlax, El Naggar and Siwertz1998) indicate that throughout Earth’s ocean the SW group speed, $c/2$ , for swell is likely greater than the IW phase speed $N/m_{n}$ for all $n$ . In which case $q_{min}$ in (5.3) is imaginary and there is no radiation.

Nonetheless, as a consistency check on the previous 3-D calculations, it is interesting to suppose that $q_{min}$ is real and show that the 2-D result in (5.17) is recovered by taking the limit $\ell _{y}\rightarrow \infty$ in the 3-D radiation integral (4.9). The $\ell _{y}\rightarrow \infty$ limit is tricky e.g. as $\ell _{y}\rightarrow \infty$ , $\hat{M}(q,s)\propto \unicode[STIX]{x1D6FF}(s)$ and a straightforward calculation using (4.9) requires consideration of $|\hat{M}|^{2}\propto \unicode[STIX]{x1D6FF}[s_{n}(q)]^{2}$ . The argument of the squared $\unicode[STIX]{x1D6FF}$ -function, $s_{n}(q)$ , is zero at $q=q_{min}$ . But at this same point the big square root in (4.9) has an integrable singularity.

To avoid involvement with these duelling singularities, we first insert $|M|^{2}$ in (A 3) into (4.9) and then consider $\ell _{y}\rightarrow \infty$ :

The factor $\ell _{y}$ left outside the integral in (5.18) is expected on physical grounds: the radiation should be linearly proportional to the length in $y$ of the SW packet.

Now recall that $s_{n}(q)$ is defined in (4.10), and observe that $s_{n}(q_{min})=0$ . We can now simplify the integrand in (5.18) using a standard results for $\unicode[STIX]{x1D6FF}$ -functions:

Evaluating (5.18) with the $\unicode[STIX]{x1D6FF}$ -function in (5.20) we obtain

where $J_{n}^{2D}$ is given by (5.17) and $\sqrt{\unicode[STIX]{x03C0}}\ell _{y}$ is the effective length in $y$ of the SW packet.