4.1. The $O(1)$ balance

At leading order in $\epsilon$ we recover the usual baroclinic Kelvin wave system (Wang & Mooers Reference Wang and Mooers1976)

(4.12)\begin{equation} \mathcal{A}_0(1+\mathcal{A}_0^2) \frac{\partial }{\partial z}\left[\frac{1}{\mathcal{N}^2}\frac{\partial \phi_0}{\partial z}\right] + \mathcal{A}_0\left(-k^2 + \frac{\partial ^2}{\partial \lambda^2}\right)\phi_0 = 0, \end{equation}

subject to

(4.13)\begin{equation} \frac{\partial \phi_0}{\partial z}=0\quad \textrm{at} \ z=-1,0, \end{equation}

and

(4.14)\begin{equation} ik\phi_0 - \mathcal{A}_0\frac{\partial \phi_0}{\partial \lambda} = 0\quad \textrm{at} \ \lambda = 0. \end{equation}

Solving for $\phi _0$ and $\omega _0$ we have

(4.15)\begin{equation} \phi_0^{(n)} = \exp\left[ -\frac{\lambda}{c_n}\right]\mathcal{Z}_n(z), \end{equation}

and

(4.16)\begin{equation} \omega_0^{(n)} = k(U+c_n), \end{equation}

where we recall that a full solution for the pressure can be constructed by summing over all vertical modes, $n$, and integrating over all along-shore wavenumbers, $k$. Here, the vertical structure functions $\mathcal {Z}_n(z)$ and wave speeds $c_n$ satisfy

(4.17)\begin{equation} \frac{\partial }{\partial z}\left[\frac{1}{\mathcal{N}^2}\frac{\partial \mathcal{Z}_n}{\partial z}\right] = -\frac{1}{c_n^2}\mathcal{Z}_n, \end{equation}

and

(4.18)\begin{equation} \frac{\partial \mathcal{Z}_n}{\partial z} = 0\quad \textrm{at} \ z = -1,0. \end{equation}

We note that for $\mathcal {N}^2 = 1$ we have $\mathcal {Z}_n = \cos [n{\rm \pi} z]$ and $c_n = 1/(n{\rm \pi} )$. The leading-order pressure is then given by

(4.19)\begin{equation} p_0 = \sum_{n=1}^\infty \left[\frac{1}{2{\rm \pi} }\int_{-\infty}^\infty \tilde{A}_n(k) \exp\left[\textrm{i} k\left(x-(U + c_n)t\right)\right]\,\textrm{d} k\right] \exp\left[-\frac{\lambda}{c_n}\right]\mathcal{Z}_n(z), \end{equation}

where $A_n$ can be determined from the initial conditions and $\tilde {f}$ denotes the Fourier transform of $f$. Note that this result can be written as

(4.20)\begin{equation} p_0 = \sum_{n=1}^\infty A_n(x-(U+c_n)t) \exp\left[-\frac{\lambda}{c_n}\right]\mathcal{Z}_n(z), \end{equation}

using the frequency shift property of the Fourier transform.

4.2. The $O(\epsilon )$ balance

Having determined the leading-order solution, we can now solve for the $O(\epsilon )$ correction. By linearity we will assume that

(4.21)\begin{equation} \phi_0 = \phi_0^{(n)} = \exp\left[ -\frac{\lambda}{c_n}\right]\mathcal{Z}_n(z), \end{equation}

and solve for $(\phi _1^{(n)},\omega _1^{(n)})$, the solution for $(\phi _1,\omega _1)$ forced by $\phi _0^{(n)}$. A full solution for $\phi _1$ can then be formed by summing over all $n$ as before.

The $O(\epsilon )$ system is

(4.22)\begin{align} &\mathcal{A}_0\left(1+\mathcal{A}_0^2\right)\frac{\partial }{\partial z}\left[\frac{1}{\mathcal{N}^2}\frac{\partial \phi_1^{(n)}}{\partial z}\right] + \mathcal{A}_0\left(-k^2 + \frac{\partial ^2}{\partial \lambda^2}\right)\phi_1^{(n)} \nonumber\\ &\quad = \mathcal{A}_0\left(1+\mathcal{A}_0^2\right)\left[ \frac{\partial }{\partial z}\left( \frac{\delta'}{\mathcal{N}^2} \right) \frac{\partial \phi_0^{(n)}}{\partial \lambda} +2\frac{\delta'}{\mathcal{N}^2}\frac{\partial ^2\phi_0^{(n)}}{\partial z\partial\lambda} \right] + \frac{\mathcal{E}}{{Pr}}\left(1+\mathcal{A}_0^2\right) \frac{\partial }{\partial z}\left[ \frac{1}{\mathcal{N}^2} \mathcal{D}_b \frac{\partial \phi_0^{(n)}}{\partial z} \right] \nonumber\\ &\qquad + 2\mathcal{A}_0^2\left[\textrm{i}\omega_1^{(n)}+\mathcal{E}\mathcal{D}_u\right] \frac{\partial }{\partial z}\left[\frac{1}{\mathcal{N}^2} \frac{\partial \phi_0^{(n)}}{\partial z} \right] + \mathcal{E}\mathcal{D}_u \left(-k^2 + \frac{\partial ^2}{\partial \lambda^2}\right)\phi_0^{(n)}, \end{align}

subject to

(4.23)\begin{equation} \frac{\partial \phi_1^{(n)}}{\partial z} = \delta' \frac{\partial \phi_0^{(n)}}{\partial \lambda}\quad \textrm{at} \ z = -1,0, \end{equation}

and

(4.24)\begin{align} \left[\textrm{i}k - \mathcal{A}_0\frac{\partial }{\partial \lambda}\right]\phi_1^{(n)} = -\left[\textrm{i}\omega_1^{(n)}+\mathcal{E}\mathcal{D}_u\right] \frac{\partial \phi_0^{(n)}}{\partial \lambda} -\frac{\delta'}{\mathcal{N}^2}\mathcal{A}_0\left(1+\mathcal{A}_0^2\right) \frac{\partial \phi_0^{(n)}}{\partial z}\quad \textrm{at} \ \lambda = 0. \end{align}

A consequence of our coordinate transformation (3.10) is that the vertical boundary condition, (4.23), is not homogeneous. This can be treated following Romea & Allen (Reference Romea and Allen1984) by substituting

(4.25)\begin{equation} \phi_1^{(n)} = \psi_1^{(n)}+\delta\frac{\partial \phi_0^{(n)}}{\partial \lambda}, \end{equation}

where $\psi _1^{(n)}$ describes the correction to $\phi$ in the case of a vertical wall, $\delta = 0$. The effect of this substitution is to give a system with homogeneous boundary conditions on $\psi _1^{(n)}$ at the top and bottom boundary allowing us to expand $\psi ^{(n)}$ as a sum of vertical modes. Substituting for $\phi _0^{(n)}$ using (4.21) gives the system for $\psi _1^{(n)}$

(4.26)\begin{align} &\mathcal{A}_0\left(1+\mathcal{A}_0^2\right)\frac{\partial }{\partial z}\left[\frac{1}{\mathcal{N}^2}\frac{\partial \psi_1^{(n)}}{\partial z}\right] + \mathcal{A}_0\left(-k^2 + \frac{\partial ^2}{\partial \lambda^2}\right)\psi_1^{(n)} = \left[ 2\textrm{i} k^2\omega_1^{(n)}\mathcal{Z}_n \right.\nonumber\\ &\quad + \mathcal{E}(1-k^2c_n^2)\frac{\partial }{\partial z} \left. \left[ \frac{1}{\mathcal{N}^2} \frac{\partial }{\partial z}\left(\frac{D_b}{{Pr}}\frac{\partial ^2\mathcal{Z}_n}{\partial z^2}\right) \right] + \frac{\mathcal{E}(1+k^2c_n^2)}{c_n^2}\frac{\partial }{\partial z}\left(D_u\frac{\partial \mathcal{Z}_n}{\partial z}\right)\right] \exp\left[-\frac{\lambda}{c_n}\right], \end{align}

subject to

(4.27)\begin{equation} \frac{\partial \psi_1^{(n)}}{\partial z} = 0\quad \textrm{at} \ z = -1,0, \end{equation}

and

(4.28)\begin{align} &\left[\textrm{i}k - \mathcal{A}_0\frac{\partial }{\partial \lambda}\right]\psi_1^{(n)} = \frac{\textrm{i}\omega_1^{(n)}\mathcal{Z}_n}{c_n} \nonumber\\ &\quad +\frac{\mathcal{E}}{c_n}\frac{\partial }{\partial z}\left(D_u\frac{\partial \mathcal{Z}_n}{\partial z}\right) +\textrm{i}kc_n\left(1-k^2c_n^2\right) \frac{\delta'}{\mathcal{N}^2} \frac{\partial \mathcal{Z}_n}{\partial z}\quad \textrm{at} \ \lambda = 0. \end{align}

Since $\psi _1^{(n)}$ is forced by a range of vertical modes unless $\mathcal {D}_u\mathcal {N}^2 = \mathcal {D}_b\mathcal {N}^2 = \mathrm {const.}$ (Martinsen & Weber Reference Martinsen and Weber1981) and $\delta '=0$ (Romea & Allen Reference Romea and Allen1984), we expand as a sum of vertical modes

(4.29)\begin{equation} \psi_1^{(n)} = \sum_{m=0}^\infty \psi_{1m}^{(n)}(\lambda) \mathcal{Z}_m(z), \end{equation}

where $\mathcal {Z}_0 = 1$ describes the barotropic mode with corresponding eigenvalue $c_m^{-2} = 0$. In order to determine the $\psi _{1m}^{(n)}$ we now multiply our system by $\mathcal {Z}_m$ and integrate vertically between $-1$ and $0$. Note that we have the orthogonality relation

(4.30)\begin{equation} \int_{-1}^0 \mathcal{Z}_n \mathcal{Z}_m\,\textrm{d} z = z_m^2 \delta_{nm}, \end{equation}

where the $z_m$ are normalisation constants. We obtain

(4.31)\begin{align} &\left[\frac{\partial ^2}{\partial \lambda^2} - \mu_{mn}^2\right]\psi_{1m}^{(n)} = \left[ -\frac{2 k\omega_1^{(n)}}{c_n}\delta_{mn} \right.\nonumber\\ & \quad \left.+\frac{2\textrm{i} \mathcal{E}\left(1-k^2c_n^2\right)}{{Pr}\, kc_n^3}\sigma_{mn} - \frac{2\textrm{i} \mathcal{E}\left(1+k^2c_n^2\right)}{kc_n^3}\varepsilon_{mn}\right] \exp\left[-\frac{\lambda}{c_n}\right], \end{align}

subject to

(4.32)\begin{equation} \left[1 + c_n\frac{\partial }{\partial \lambda}\right]\psi_{1m}^{(n)} = \frac{\omega_1^{(n)}}{kc_n}\delta_{mn} +\frac{2\textrm{i} \mathcal{E}}{kc_n}\varepsilon_{mn} +\frac{1}{c_n}\left(1-k^2c_n^2\right)\gamma_{mn}\quad \textrm{at} \ \lambda = 0, \end{equation}

where

(4.33)\begin{equation} \varepsilon_{mn} = -\frac{1}{2z_m^2} \int_{-1}^0 \frac{\partial }{\partial z}\left(D_u\frac{\partial \mathcal{Z}_n}{\partial z}\right) \mathcal{Z}_m\,\textrm{d} z, \end{equation}

and

(4.34)\begin{equation} \sigma_{mn} = \frac{c_n^2}{2z_m^2} \int_{-1}^0 \frac{\partial }{\partial z}\left[ \frac{1}{\mathcal{N}^2} \frac{\partial }{\partial z}\left(D_b\frac{\partial ^2\mathcal{Z}_n}{\partial z^2}\right) \right] \mathcal{Z}_m\,\textrm{d} z, \end{equation}

describe the effects of vertical mixing of momentum and buoyancy respectively,

(4.35)\begin{equation} \gamma_{mn} = \frac{c_n^2}{z_m^2}\int_{-1}^0 \left[ \frac{\delta'}{\mathcal{N}^2} \frac{\partial \mathcal{Z}_n}{\partial z}\right]\mathcal{Z}_m\,\textrm{d} z, \end{equation}

describes the effects of a sloping boundary and

(4.36)\begin{equation} \mu_{mn}^2(k) = \frac{1-k^2(c_n^2-c_m^2)}{c_m^2}, \end{equation}

describes the offshore decay scale.

Integration by parts allows $\varepsilon _{mn}$ to be written as

(4.37)\begin{equation} \varepsilon_{mn} = \frac{1}{2z_m^2} \int_{-1}^0 D_u\frac{\partial \mathcal{Z}_n}{\partial z}\frac{\partial \mathcal{Z}_m}{\partial z}\,\textrm{d} z, \end{equation}

showing that $\varepsilon _{nn}$ is non-negative assuming a positive viscosity profile $D_u$. Similarly, $\gamma _{nn}$ may be written as

(4.38)\begin{equation} \gamma_{nn} = \frac{c_n^2}{2} \int_{-1}^0 \delta'' \left[ \frac{1}{\mathcal{N}^2}\frac{\partial \mathcal{Z}_n}{\partial z}\right]^2 \, \textrm{d} z, \end{equation}

so $\gamma _{nn} = 0$ for a linear slope profile with constant $\delta'$ and is positive/negative for a slope with positive/negative curvature respectively. The coefficient $\sigma _{nn}$ is not sign definite but will typically be positive, assuming a stable stratification, $\mathcal {N}^2$, and positive diffusivity profile, $D_b$. Negative values of $\sigma _{nn}$ may occur if $D_b$ varies over scales smaller than those of the oscillations of $\mathcal {Z}_n$.

We now solve (4.31) and (4.32) for $\psi _{1m}^{(n)}$ by considering three possible cases for different values of $m$.

4.2.1. Case $m = 0$: the $O(\epsilon )$ barotropic mode

We begin by considering the barotropic correction to the leading-order mode. The barotropic correction has $m = 0$ so

(4.39)\begin{equation} \mu_{0n}^2 = k^2, \end{equation}

and our system becomes

(4.40)\begin{equation} \left[\frac{\partial ^2}{\partial \lambda^2} - k^2\right]\psi_{10}^{(n)} = \frac{2\textrm{i} \mathcal{E}}{kc_n^2}\left[ \frac{1}{{Pr}}\left(1-k^2c_n^2\right)\sigma_{0n} - \left(1+k^2c_n^2\right)\varepsilon_{0n}\right]\exp\left[-\frac{\lambda}{c_n}\right], \end{equation}

subject to

(4.41)\begin{equation} \left[1 + c_n\frac{\partial }{\partial \lambda}\right]\psi_{10}^{(n)} = \frac{2\textrm{i} \mathcal{E}}{kc_n}\varepsilon_{0n} +\frac{1}{c_n}\left(1-k^2c_n^2\right)\gamma_{0n}\quad \textrm{at} \ \lambda = 0, \end{equation}

with solution

(4.42)\begin{align} \psi_{10}^{(n)} &= \left[ \frac{1+|k|c_n}{c_n}\gamma_{0n} + \frac{2\textrm{i} \mathcal{E}}{kc_n(1-|k|c_n)}\varepsilon_{0n}\right] \exp\left[-|k|\lambda\right] \nonumber\\ & \quad + \frac{2\textrm{i} \mathcal{E}}{k c_n} \left[ \frac{1}{{Pr}} \sigma_{0n} - \frac{1+k^2c_n^2}{1-k^2c_n^2} \varepsilon_{0n}\right] \exp\left[ -\frac{\lambda}{c_n}\right]. \end{align}

4.2.2. Case $m=n$: the $O(\epsilon )$ correction to the $n$th baroclinic mode

Here, we consider the pressure correction with the same vertical structure as the leading-order mode with which it is associated. In this case $m=n$ and we have

(4.43)\begin{equation} \mu_{nn}^2 = \frac{1}{c_n^2}, \end{equation}

so

(4.44)\begin{align} \left[\frac{\partial ^2}{\partial \lambda^2} - \frac{1}{c_n^2}\right]\psi_{1n}^{(n)} = \left[ -\frac{2 k\omega_1^{(n)}}{c_n} +\frac{2\textrm{i} \mathcal{E}\left(1-k^2c_n^2\right)}{{Pr}\, kc_n^3}\sigma_{nn} - \frac{2\textrm{i} \mathcal{E} \left(1+k^2c_n^2\right)}{kc_n^3}\varepsilon_{nn}\right]\exp\left[-\frac{\lambda}{c_n}\right], \end{align}

subject to

(4.45)\begin{equation} \left[1 + c_n\frac{\partial }{\partial \lambda}\right]\psi_{1n}^{(n)} = \frac{\omega_1^{(n)}}{kc_n} +\frac{2\textrm{i} \mathcal{E}}{kc_n}\varepsilon_{nn} +\frac{1}{c_n}\left(1-k^2c_n^2\right)\gamma_{nn}\quad \textrm{at} \ \lambda = 0. \end{equation}

This system is forced at its resonant frequency and hence has solution of the form

(4.46)\begin{equation} \psi_{1n}^{(n)} = C_n \lambda \exp\left[-\frac{\lambda}{c_n}\right], \end{equation}

where we have two equations for $C_n$ allowing us to determine $\omega _1^{(n)}$ through a consistency condition. We have

(4.47)\begin{equation} -\frac{2C_n}{c_n} = -\frac{2 k\omega_1^{(n)}}{c_n} +\frac{2\textrm{i} \mathcal{E}\left(1-k^2c_n^2\right)}{{Pr}\,kc_n^3}\sigma_{nn} - \frac{2\textrm{i} \mathcal{E}\left(1+k^2c_n^2\right)}{kc_n^3}\varepsilon_{nn}, \end{equation}

and

(4.48)\begin{equation} c_n C_n = \frac{\omega_1^{(n)}}{kc_n} +\frac{2\textrm{i} \mathcal{E}}{kc_n}\varepsilon_{nn} +\frac{1}{c_n}\left(1-k^2c_n^2\right)\gamma_{nn}, \end{equation}

therefore

(4.49)\begin{equation} \omega_1^{(n)} = -k\gamma_{nn}-\textrm{i}\mathcal{E}\left(\varepsilon_{nn}+\frac{1}{{Pr}}\sigma_{nn}\right), \end{equation}

and

(4.50)\begin{equation} \psi_{1n}^{(n)} = \left[-k^2\gamma_{nn} + \frac{\textrm{i}\mathcal{E}}{kc_n^2}\left(\varepsilon_{nn}-\frac{1}{{Pr}}\sigma_{nn}\right)\right]\lambda \exp\left[-\frac{\lambda}{c_n}\right]. \end{equation}

4.2.3. Case $m\neq n, m>0$: the $O(\epsilon )$ scattered modes

Finally, we consider the scattered modes. We define the scattered modes to be those baroclinic correction terms with a different vertical structure to the leading-order mode with which they are associated. In this case $m\neq n, m>0$ so we have

(4.51)\begin{equation} \left[\frac{\partial ^2}{\partial \lambda^2} - \mu_{mn}^2\right]\psi_{1m}^{(n)} = \frac{2\textrm{i} \mathcal{E}}{kc_n^3}\left[\frac{1}{{Pr}}\left(1-k^2c_n^2\right)\sigma_{mn} - \left(1+k^2c_n^2\right)\varepsilon_{mn}\right]\exp\left[-\frac{\lambda}{c_n}\right], \end{equation}

subject to

(4.52)\begin{equation} \left[1 + c_n\frac{\partial }{\partial \lambda}\right]\psi_{1m}^{(n)} = \frac{2\textrm{i} \mathcal{E}}{kc_n}\varepsilon_{mn} +\frac{1}{c_n}\left(1-k^2c_n^2\right)\gamma_{mn}\quad \textrm{at} \ \lambda = 0. \end{equation}

This can be solved similarly to the case of $m=0$ to obtain the solution

(4.53)\begin{align} \psi_{1m}^{(n)} &= \frac{2\textrm{i} \mathcal{E}}{kc_n}\left[ \frac{\left(1-k^2c_n^2\right)\sigma_{mn} - {Pr}\left(1+k^2c_n^2\right)\varepsilon_{mn}}{{Pr}\left(1-\mu_{mn}^2 c_n^2\right)} \right]\exp\left[-\frac{\lambda}{c_n}\right] \nonumber\\ &\quad + \frac{1}{k c_n}\left[ \frac{2\textrm{i} \mathcal{E} \varepsilon_{mn}+k(1-k^2c_n^2)\gamma_{mn}}{1-c_n\mu_{mn}}\right] \exp\left[ -\mu_{mn} \lambda \right], \end{align}

for $\mu _{mn}^2 > 0$ and

(4.54)\begin{align} \psi_{1m}^{(n)} &= \frac{2\textrm{i} \mathcal{E}}{kc_n}\left[ \frac{\left(1-k^2c_n^2\right)\sigma_{mn} - {Pr}\left(1+k^2c_n^2\right)\varepsilon_{mn}}{{Pr} \left(1+|\mu_{mn}^2| c_n^2\right)} \right]\exp\left[-\frac{\lambda}{c_n}\right] \nonumber\\ &\quad + \frac{1}{k c_n}\left[ \frac{2\textrm{i} \mathcal{E} \varepsilon_{mn}+k(1-k^2c_n^2)\gamma_{mn}}{1+i c_n|\mu_{mn}|}\right] \exp\left[ \textrm{i} |\mu_{mn}| \lambda \right], \end{align}

for $\mu _{mn}^2 < 0$. We note that when $\mu _{mn}^2 < 0$ the waves can propagate offshore and into the interior ocean due to the wavelike term $\exp [\textrm {i}|\mu _{mn}|\lambda ]$. Offshore propagation due to mixing was discussed by Davey et al. (Reference Davey, Hsieh and Wajsowicz1983) and is also present in our solution (see § 4.5). The propagation we observe here can result, however, from either mixing or topography so is likely due to a different mechanism. It requires $\mu _{mn}^2 < 0$ so is only possible for superinertial waves ($k c_n > 1$) and will be examined in future work. This effect was not observed by Romea & Allen (Reference Romea and Allen1984) due to their making the long wave (small $k$) approximation.

4.3. Summary

We now present our full linear solution for the case of small Rossby number and discuss some interesting limits with links to previous work. The full pressure field can be written as

(4.55)\begin{align} p &= \sum_{n=1}^\infty \left[\int_{-\infty}^\infty \frac{1}{2{\rm \pi} }\tilde{A}_n(k)\left[\left(1-\frac{\epsilon\delta}{c_n}\right)\phi_0^{(n)} + \epsilon\psi_1^{(n)} \right] \right. \nonumber\\ & \quad \times \left.\exp\left[ \textrm{i} k\left(x-\left(U+c_n-\epsilon\gamma_{nn}\right)t\right) - \epsilon\mathcal{E}\left(\varepsilon_{nn}+\frac{1}{{Pr}}\sigma_{nn}\right)t \right]\, \textrm{d} k \right]+O(\epsilon^2), \end{align}

where

(4.56)\begin{equation} \phi_0^{(n)} = \exp\left[ -\frac{\lambda}{c_n}\right]\mathcal{Z}_n(z), \end{equation}

and

(4.57)\begin{equation} \psi_1^{(n)} = \sum_{m=0}^\infty \psi_{1m}^{(n)}(\lambda) \mathcal{Z}_m(z), \end{equation}

for $\psi _{1m}^{(n)}$ given in § 4.2. Vertical mixing acts to damp the modes by reducing the amplitude exponentially with time as observed by Martinsen & Weber (Reference Martinsen and Weber1981) and Davey et al. (Reference Davey, Hsieh and Wajsowicz1983). Additionally, vertical mixing modifies the shape of the modes by adding terms with decay scale $\lambda \sim c_n$ to the order-$\epsilon$ pressure correction, these terms have imaginary coefficients corresponding to spatially dependent phase shifts. These effects are very similar to the those observed for the case of an Ekman layer as studied by Romea & Allen (Reference Romea and Allen1984).

From (4.55), the vertical and offshore structure for a given along-shore wavenumber, $k$, is given by

(4.58)\begin{equation} \phi^{(n)} = \left(1-\frac{\epsilon\delta}{c_n}\right)\phi_0^{(n)} + \epsilon\psi_1^{(n)}, \end{equation}

correct to $O(\epsilon )$. We note that the real part of $\phi ^{(n)}$ is independent of $\mathcal {E}$ while the imaginary part of $\phi ^{(n)}$ is directly proportional to $\mathcal {E}$. Therefore we need only consider $\textrm {Re}[\phi ^{(n)}]$ and $\textrm {Im}[\phi ^{(m)}]/\mathcal {E}$ for various $\epsilon$, $\delta$, $\mathcal {N}^2$, $Pr$, $D_u$ and $D_b$. To illustrate these results we present a simple example where the forms of $D_u$ and $D_b$ are chosen to represent elevated turbulence within the layer (Large et al. Reference Large, McWilliams and Doney2011) while the form of $\mathcal {N}^2$ is taken as an exponential stratification where $\mathcal {N}^2$ decays by approximately $60\,\%$ over the layer depth (Wang & Mooers Reference Wang and Mooers1976). Specifically, we take

(4.59)\begin{equation} \mathcal{N}^2 = \exp(z), \end{equation}

and

(4.60)\begin{equation} D_u = D_b = -({\rm \pi} /2) \sin({\rm \pi} z). \end{equation}

We also choose the small parameter $\epsilon = 0.1$ and set $k = 1$, corresponding to a wavelength on the scale of the deformation radius, $L_R$. Additionally we take $Pr = 1$ such that the turbulent mixing of momentum and buoyancy occur at similar rate. While this approximation is not necessarily true everywhere, it is often the case that ${Pr} \approx 1$ (Venayagamoorthy & Stretch Reference Venayagamoorthy and Stretch2010). We now present $\textrm {Re}[\phi ^{(n)}]$ and $\textrm {Im}[\phi ^{(m)}]/\mathcal {E}$ for a range of vertical mode numbers $n$ and slope profiles, $\delta$.

Figure 3 shows $\textrm {Re}[\phi ^{(n)}]$ for the first four mode numbers and a linear slope $\delta (z) = -z$. Near the boundary, disturbances decay approximately with the perpendicular distance from the wall while over larger distances the stratification flattens the phase lines.

Figure 3.

Value of $\textrm {Re}[\phi ^{(n)}]$ for $\mathcal {N}^2 = \exp (z)$, $D_u = D_b = -({\rm \pi} /2) \sin ({\rm \pi} z)$, $\delta = -z$, $\epsilon = 0.1$, $k = 1$ and $Pr = 1$. We show the first four modes $n=1,2,3,4$. (a) $n= 1$, (b) $n= 2$, (c) $n= 3$ and (d) $n= 4$.

Furthermore, the reduction in magnitude for larger $n$ suggests that the slope acts to suppress the amplitude of pressure field and hence the along-shore velocity for high mode numbers. We note that an effect of the slope is to induce an $O(\epsilon )$ offshore velocity ($v$) which is likely related to the decrease in along-shore velocity. Figure 4 shows $\textrm {Im}[\phi ^{(n)}]/\mathcal {E}$ for the same mode number and slope. We can see that vertical mixing induces higher mode number corrections to the $n$th mode. Additionally, the large amplitudes for higher $n$ suggests that vertical mixing acts more strongly on higher modes. This is consistent with our observations in § 7 that vertical mixing preferentially damps the highest-order modes.

Figure 4.

Value of $\textrm {Im}[\phi ^{(n)}]/\mathcal {E}$ for $\mathcal {N}^2 = \exp (z)$, $D_u = D_b = -({\rm \pi} /2) \sin ({\rm \pi} z)$, $\delta = -z$, $\epsilon = 0.1$, $k = 1$ and $Pr = 1$. We show the first four modes $n=1,2,3,4$. (a) $n= 1$, (b) $n= 2$, (c) $n= 3$ and (d) $n= 4$.

Figure 5 shows the first mode, $\textrm {Re}[\phi ^{(1)}]$, for a range of steep slope topographies. A notable feature of the distorted modes is that the zero contour can intersect with the top surface, this can significantly affect the form of the along-shore surface velocity by inducing sign changes along the $y$ direction. Note that since $\textrm {Im}[\phi ^{(n)}]/\mathcal {E}$ only depends on $\delta$ though $\lambda$, topographic effects appear only as a depth-dependent horizontal translation. Therefore, since these effects appear predominately in the real part of $\phi ^{(n)}$, $\textrm {Im}[\phi ^{(1)}]/\mathcal {E}$ is not shown here.

Figure 5.

The $n=1$ mode $\textrm {Re}[\phi ^{(1)}]$ for $\mathcal {N}^2 = \exp (z)$, $D_u = D_b = -({\rm \pi} /2) \sin ({\rm \pi} z)$, $\epsilon = 0.1$, $k = 1$ and $Pr = 1$. We show a range of slope geometries $\delta (z) = 0, -z, 3z^2/2, 3\sqrt {|z|}/4$. (a) $\delta (z) = 0$, (b) $\delta (z) = -z$, (c) $\delta (z) = 3z^2/2$ and (d) $\delta (z) = 3\sqrt {|z|}/4$.

4.4. The amplitude evolution

Using the dispersion relation

(4.61)\begin{equation} \omega^{(n)} = k(U+c_n-\epsilon \gamma_{nn}) - \textrm{i} \epsilon \mathcal{E} \left(\varepsilon_{nn} + \frac{1}{{Pr}}\sigma_{nn}\right), \end{equation}

correct to $O(\epsilon )$ we can determine an evolution equation for the amplitude, $A_n$, as

(4.62)\begin{equation} \frac{\partial A_n}{\partial t} + (U+c_n-\epsilon\gamma_{nn}) \frac{\partial A_n}{\partial x} = -{E}\left(\varepsilon_{nn}+\frac{1}{{Pr}}\sigma_{nn}\right)A_n, \end{equation}

where we have substituted ${E} = \epsilon \,\mathcal {E}$. This equation can be solved as the product of a travelling wave and exponential decay

(4.63)\begin{equation} A_n = B_n ( x - (U+c_n-\epsilon\gamma_{nn})t) \exp\left[-{E}\left(\varepsilon_{nn}+\frac{1}{{Pr}}\sigma_{nn}\right)t\right], \end{equation}

for some function $B_n$ determined by initial conditions. We note that perturbations decay on a time scale of

(4.64)\begin{equation} t \sim \frac{1}{{E}(\varepsilon_{nn}+\sigma_{nn}/{Pr})}, \end{equation}

corresponding to a distance of

(4.65)\begin{equation} x \sim \frac{U+c_n-\epsilon\gamma_{nn}}{{E}(\varepsilon_{nn}+\sigma_{nn}/{Pr})}. \end{equation}

4.5. The decoupled modes case

One particularly simple example which illustrates some of the effects of vertical mixing is the case of $\delta = 0$ and $D_u\mathcal {N}^2 = D_b\mathcal {N}^2 = 1$. Here, we have $\gamma _{mn} = 0$ and

(4.66)\begin{equation} \varepsilon_{mn} = \sigma_{mn} = \frac{1}{2c_n^2}\delta_{mn}, \end{equation}

so the vertical modes decouple giving that

(4.67)\begin{equation} \phi_{1}^{(n)} = \frac{\textrm{i}\mathcal{E}}{2kc_n^4}\left(1-\frac{1}{{Pr}}\right)y \exp\left[-\frac{y}{c_n}\right]\mathcal{Z}_n(z), \end{equation}

and hence

(4.68)\begin{equation} \phi^{(n)} = \phi_0^{(n)} + \epsilon\phi_1^{(n)} + O(\epsilon^2) = \left[ 1 + \frac{i\epsilon\mathcal{E}}{2kc_n^4}\left(1-\frac{1}{{Pr}}\right)y\right] \exp\left[-\frac{y}{c_n}\right]\mathcal{Z}_n(z) + O(\epsilon^2). \end{equation}

We note that, due to the decoupling of the modes, we do not observe the high mode number corrections from mixing effects seen in figure 4. Equation (4.68) can be rewritten as

(4.69)\begin{equation} \phi^{(n)} = \exp\left[-\frac{y}{c_n}\left(1-\frac{\textrm{i}{E}}{2kc_n^3} \left[1-\frac{1}{{Pr}}\right]\right)\right]\mathcal{Z}_n(z) + O(\epsilon^2), \end{equation}

with the complex exponential dependence corresponding to decaying offshore oscillations as discussed in Davey et al. (Reference Davey, Hsieh and Wajsowicz1983) and Martinsen & Weber (Reference Martinsen and Weber1981). The full pressure field can be written as

(4.70)\begin{align} p &= \sum_{n=0}^\infty \left[\int_{-\infty}^\infty \frac{1}{2{\rm \pi} }\tilde{A}_n(k)\exp\left[-\frac{y}{c_n}\left(1-\frac{\textrm{i}{E}}{2kc_n^3}\left[1-\frac{1}{{Pr}}\right]\right)\right]\mathcal{Z}_n(z) \right. \nonumber\\ &\quad \times \left.\exp\left[ \textrm{i} k\left(x-\left(U+c_n\right)t\right) - \frac{{E}}{2c_n^2}\left(1+\frac{1}{{Pr}}\right)t \right]\, \textrm{d} k \right]+O(\epsilon^2), \end{align}

where we note that all wavenumbers for a given vertical mode number decay at the same rate, given by

(4.71)\begin{equation} t \sim \frac{2c_n^2}{{E}}\left(1+\frac{1}{{Pr}}\right)^{-1}. \end{equation}

In the case of ${Pr} = \infty$ we recover the weak mixing limit of Davey et al. (Reference Davey, Hsieh and Wajsowicz1983) and Martinsen & Weber (Reference Martinsen and Weber1981). Note that it was also shown (Martinsen & Weber Reference Martinsen and Weber1981) that the wave speed is modified by a term proportional to the vertical Ekman number squared, however, in our formulation this effect would be $O(\epsilon ^2)$ so is not seen here. For the case of ${Pr} = 1$ there are no offshore oscillations and we can write

(4.72)\begin{equation} p = \sum_{n=1}^\infty \left[ A_n(x-(U+c_n)t)\exp\left[-\frac{y}{c_n}- \frac{{E}}{c_n^2 } t \right] \mathcal{Z}_n(z) \right] +O(\epsilon^2), \end{equation}

consistent with the observations of Davey et al. (Reference Davey, Hsieh and Wajsowicz1983) that mixing with ${Pr} = 1$ does not modify the speed or spatial structure of the wave.

5.1. The $O(\epsilon )$ balance

Similarly to the linear case, we can now use leading-order solution to determine the $O(\epsilon )$ correction. While we will present the governing equations for all vertical modes of this correction term, we will only give the solution for the case where the vertical structure of the correction matches that of the leading-order mode. In this case a secularity condition will allow us to determine the slow evolution of the amplitude, $A_n$. The $O(\epsilon )$ system is

(5.12)\begin{align} &\mathcal{A}_0\left(1+\mathcal{A}_0^2\right)\frac{\partial }{\partial z}\left[\frac{1}{\mathcal{N}^2}\frac{\partial p_1^{(n)}}{\partial z}\right] + \mathcal{A}_0\left(\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial \lambda^2}\right)p_1^{(n)} \nonumber\\ &\quad = \mathcal{A}_0\left(1+\mathcal{A}_0^2\right)\left[ \frac{\partial }{\partial z}\left( \frac{\delta'}{\mathcal{N}^2} \right) \frac{\partial p_0^{(n)}}{\partial \lambda}+2\frac{\delta'}{\mathcal{N}^2}\frac{\partial ^2p_0^{(n)}}{\partial z\partial\lambda} \right] + \frac{\mathcal{E}}{{Pr}}\left(1+\mathcal{A}_0^2\right) \frac{\partial }{\partial z}\left[ \frac{1}{\mathcal{N}^2} \mathcal{D}_b \frac{\partial p_0^{(n)}}{\partial z} \right] \nonumber\\ & \qquad +2\mathcal{A}_0^2\left[-\frac{\partial }{\partial T}+\mathcal{E}\mathcal{D}_u\right] \frac{\partial }{\partial z}\left[\frac{1}{\mathcal{N}^2} \frac{\partial p_0^{(n)}}{\partial z} \right] + \mathcal{E}\mathcal{D}_u \left(\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial \lambda^2}\right)p_0^{(n)} \nonumber\\ &\qquad -\mathcal{R} \left[ -\frac{\partial F_u}{\partial \lambda} + \mathcal{A}_0 \frac{\partial F_u}{\partial x} + \left(1+\mathcal{A}_0^2\right) \frac{\partial }{\partial z}\left[ \frac{F_b}{\mathcal{N}^2}\right]\right], \end{align}

subject to

(5.13)\begin{equation} \frac{\partial p_1^{(n)}}{\partial z} = \delta' \frac{\partial p_0^{(n)}}{\partial \lambda}\quad \textrm{at} \ z = -1,0, \end{equation}

and

(5.14)\begin{align} \left[\frac{\partial }{\partial x} - \mathcal{A}_0\frac{\partial }{\partial \lambda}\right]p_1^{(n)} = \left[\frac{\partial }{\partial T}-\mathcal{E}\mathcal{D}_u\right] \frac{\partial p_0^{(n)}}{\partial \lambda} -\frac{\delta'}{\mathcal{N}^2}\mathcal{A}_0\left(1+\mathcal{A}_0^2\right) \frac{\partial p_0^{(n)}}{\partial z} - \mathcal{R} F_u\quad \textrm{at} \ \lambda = 0. \end{align}

Similarly to the linear case, this can be simplified by substituting

(5.15)\begin{equation} p_1^{(n)} = q_1^{(n)} + \delta\frac{\partial p_0^{(n)}}{\partial \lambda}, \end{equation}

to get a system for $q_1^{(n)}$. Given the functional dependence of the forcing we now take

(5.16)\begin{equation} q_1^{(n)} = q_1^{(n)}( x - (U+c_n)t, \lambda,z,T), \end{equation}

so

(5.17)\begin{equation} \mathcal{A}_0 = -c_n\frac{\partial }{\partial x}, \end{equation}

and expand $q_1$ as

(5.18)\begin{equation} q_1^{(n)} = \sum_{m=0}^\infty q_{1m}^{(n)} \mathcal{Z}_m. \end{equation}

Finally, we substitute for the right-hand side terms in (5.12) to give the following system for $q_{1m}^{(n)}$

(5.19)\begin{align} &\left[\frac{\partial ^2}{\partial \lambda^2}-\frac{1}{c_m^2}-\frac{c_n^2-c_m^2}{c_m^2}\frac{\partial ^2}{\partial x^2} \right]\frac{\partial q_{1m}^{(n)}}{\partial x}\nonumber\\ &\quad\! =\!\left[-\frac{2\delta_{mn}}{c_n}\frac{\partial ^2}{\partial x^2}\frac{\partial A_n}{\partial T}- \frac{2\mathcal{E}\sigma_{mn}}{{Pr}\,c_n^3}\left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)A_n + \frac{2\mathcal{E}\varepsilon_{mn}}{c_n^3}\left(1-c_n^2\frac{\partial ^2}{\partial x^2}\right)A_n \right]\exp\left[-\frac{\lambda}{c_n}\right]\nonumber\\ & \qquad \!+ \mathcal{R} \left[ \frac{3\alpha_{mn}}{c_n^3}\left[2-c_n^2\frac{\partial ^2}{\partial x^2}\right](A_nA_n') - \frac{3\beta_{mn}}{c_n^3} \left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)(A_nA_n') \right]\exp\left[-\frac{2\lambda}{c_n}\right], \end{align}

subject to

(5.20)\begin{align} &\left[1 + c_n\frac{\partial }{\partial \lambda}\right]\frac{\partial q_{1m}^{(n)}}{\partial x} = -\frac{\delta_{mn}}{c_n}\frac{\partial A_n}{\partial T}-\frac{2\mathcal{E}\varepsilon_{mn}}{c_n}A_n \nonumber\\ &\quad +\frac{\gamma_{mn}}{c_n} \left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)\frac{\partial A_n}{\partial x} - \frac{3\mathcal{R}\alpha_{mn}}{c_n} (A_n A_n')\quad \textrm{at} \ \lambda = 0, \end{align}

where

(5.21)\begin{equation} \alpha_{mn} = \frac{1}{3c_n z_m^2}\int_{-1}^0 \left[ \mathcal{Z}_n^2 + \frac{c_n^2}{\mathcal{N}^2} \mathcal{Z}_n'^2 \right] \mathcal{Z}_m\,\textrm{d} z, \end{equation}

and

(5.22)\begin{equation} \beta_{mn} = -\frac{1}{3c_n z_m^2}\int_{-1}^0 \frac{\partial }{\partial z}\left[ \frac{c_n^2}{\mathcal{N}^2}\mathcal{Z}_n \mathcal{Z}_n' + \frac{c_n^4}{\mathcal{N}^4} \mathcal{Z}_n' \mathcal{Z}_n'' \right]\mathcal{Z}_m\,\textrm{d} z, \end{equation}

describe the effects of nonlinearity. We note that $\varepsilon _{mn}$, $\sigma _{mn}$ and $\gamma _{mn}$ are the same as defined in § 4.2. This system can be solved for any $m$ using Fourier transforms in $x$. For simplicity, we consider only the case of $m=n$ in order to determine the evolution of the amplitude on the slow time scale $T$.

5.1.1. Case $m=n$: the $O(\epsilon )$ correction to the $n$th baroclinic mode

Here, we consider the correction term with the same vertical structure as the leading-order mode, hence $m=n$. We have

(5.23)\begin{align} &\left[\frac{\partial ^2}{\partial \lambda^2}-\frac{1}{c_n^2}\right]\frac{\partial q_{1n}^{(n)}}{\partial x} \nonumber\\ &\quad = \left[-\frac{2}{c_n}\frac{\partial ^2}{\partial x^2}\frac{\partial A_n}{\partial T}- \frac{2\mathcal{E}\sigma_{nn}}{{Pr}\,c_n^3}\left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)A_n + \frac{2\mathcal{E}\varepsilon_{nn}}{c_n^3}\left(1-c_n^2\frac{\partial ^2}{\partial x^2}\right)A_n \right]\exp\left[-\frac{\lambda}{c_n}\right] \nonumber\\ &\qquad + 3\mathcal{R} \left[ \frac{\alpha_{nn}}{c_n^3}\left[2-c_n^2\frac{\partial ^2}{\partial x^2}\right](A_nA_n') - \frac{\beta_{nn}}{c_n^3} \left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)(A_nA_n') \right]\exp\left[-\frac{2\lambda}{c_n}\right], \end{align}

subject to

(5.24)\begin{align} &\left[1 + c_n\frac{\partial }{\partial \lambda}\right]\frac{\partial q_{1n}^{(n)}}{\partial x} = -\frac{1}{c_n}\frac{\partial A_n}{\partial T}-\frac{2\mathcal{E}\varepsilon_{nn}}{c_n}A_n \nonumber\\ &\quad + \frac{\gamma_{nn}}{c_n} \left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)\frac{\partial A_n}{\partial x} - \frac{3\mathcal{R}\alpha_{nn}}{c_n} (A_n A_n')\quad \textrm{at} \ \lambda = 0, \end{align}

with solution of the form

(5.25)\begin{equation} \frac{\partial q_{1n}^{(n)}}{\partial x} = C_{1n} \lambda \exp\left[-\frac{\lambda}{c_n}\right] +C_{2n} \exp\left[-\frac{2\lambda}{c_n}\right]. \end{equation}

Here, $C_{1n}$ and $C_{2n}$ are both set by (5.23) while (5.24) gives a consistency condition which allows us to determine the slow evolution of $A_n$. We have

(5.26)\begin{align} C_{1n} &= \frac{\partial ^2}{\partial x^2}\frac{\partial A_n}{\partial T}+ \frac{\mathcal{E}\sigma_{nn}}{{Pr}\,c_n^2}\left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)A_n - \frac{\mathcal{E}\varepsilon_{nn}}{c_n^2}\left(1-c_n^2\frac{\partial ^2}{\partial x^2}\right)A_n, \end{align}

(5.27)\begin{align} C_{2n} &= \mathcal{R} \left[ \frac{\alpha_{nn}}{c_n}\left[2-c_n^2\frac{\partial ^2}{\partial x^2}\right](A_nA_n') - \frac{\beta_{nn}}{c_n} \left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)(A_nA_n') \right], \end{align}

and

(5.28)\begin{equation} c_n C_{1n} - C_{2n} = -\frac{1}{c_n}\frac{\partial A_n}{\partial T}-\frac{2\mathcal{E}\varepsilon_{nn}}{c_n}A_n+ \frac{\gamma_{nn}}{c_n} \left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right)\frac{\partial A_n}{\partial x} - \frac{3\mathcal{R}\alpha_{nn}}{c_n}A_n A_n'. \end{equation}

Substituting for $C_{1n}$ and $C_{2n}$ in (5.28) we have

(5.29)\begin{align} \left(1+c_n^2\frac{\partial ^2}{\partial x^2}\right) \left[ \frac{\partial A_n}{\partial T} + \mathcal{E}\left( \varepsilon_{nn}+\frac{1}{{Pr}}\sigma_{nn} \right) A_n +\mathcal{R}(\alpha_{nn}+\beta_{nn})A_nA_n' - \gamma_{nn}A_n' \right] = 0, \end{align}

and cancelling the leading operator gives

(5.30)\begin{equation} \frac{\partial A_n}{\partial T} = - \mathcal{E}\left(\varepsilon_{nn}+\frac{1}{{Pr}}\sigma_{nn}\right) A_n -\mathcal{R}(\alpha_{nn}+\beta_{nn})A_nA_n' + \gamma_{nn}A_n', \end{equation}

which describes the slow evolution of $A_n$. In the case of $\mathcal {R} = 0$, this evolution is equivalent to the frequency correction, $\omega _1$, in the linear problem as given by (4.49).

5.2. Summary

Similarly to the linear case, our solution for $p$ is given by

(5.31)\begin{equation} p = \left(1-\frac{\epsilon\delta}{c_n}\right) A_n\left(x-(U+c_n)t, T\right) \exp \left[-\frac{\lambda}{c_n}\right] \mathcal{Z}_n(z) + \epsilon \sum_{m=0}^\infty q_{1m}^{(n)}(\lambda) \mathcal{Z}_m(z) + O(\epsilon^2), \end{equation}

where $q_{1m}^{(n)}$ is given by the solution of (5.19) and (5.20) and we have assumed that $p_0$ consists only of a single vertical mode with mode number $n$. In the general case, where $p_0$ consists of a sum of modes, the solution will be much more complicated due to the appearance of new terms corresponding to the nonlinear interactions between each pair of leading-order modes. Therefore, the general solution will consist of a sum over $n$ of (5.31) plus these new interaction terms.

We now recall that the full time evolution is given by the multiple scale expansion

(5.32)\begin{equation} \frac{\partial A_n}{\partial t} = \left.\frac{\partial A_n}{\partial t}\right|_T + \epsilon \left.\frac{\partial A_n}{\partial T}\right|_t, \end{equation}

so combine our slow evolution result (5.30) with the leading-order advection to give an equation for the full time evolution of the wave amplitude

(5.33)\begin{equation} \frac{\partial A_n}{\partial t} + (U+c_n-\epsilon\gamma_{nn})\frac{\partial A_n}{\partial x} +{Ro}(\alpha_{nn}+\beta_{nn})A_n\frac{\partial A_n}{\partial x}= - {E}\left(\varepsilon_{nn}+\frac{1}{{Pr}}\sigma_{nn}\right) A_n. \end{equation}

This is Hopf's equation with linear friction and in the case of ${Ro}\, (\alpha _{nn}+\beta _{nn}) = 0$ we recover (4.62) with solutions given by (4.63). We now define

(5.34)\begin{gather} U_n = U+c_n-\epsilon\gamma_{nn}, \end{gather}

(5.35)\begin{gather}a_n = {Ro}(\alpha_{nn}+\beta_{nn}), \end{gather}

and

(5.36)\begin{equation} \kappa_n = {E}\left(\varepsilon_{nn}+\frac{1}{{Pr}}\sigma_{nn}\right), \end{equation}

to write

(5.37)\begin{equation} \frac{\partial A_n}{\partial t} + U_n \frac{\partial A_n}{\partial x} + a_n A_n\frac{\partial A_n}{\partial x}= - \kappa_n A_n. \end{equation}

Here, $U_n$ describes advection with a constant speed, $a_n$ describes the effects of nonlinearity and $\kappa _n$ describes the damping of the amplitude by vertical mixing.

6.1. Nonlinear solution

We begin by making the coordinate transformation

(6.1a–c)\begin{equation} \tau = \kappa_n t,\quad \eta = x - U_n t,\quad F = \frac{a_n}{\kappa_n} A_n, \end{equation}

so (5.37) becomes

(6.2)\begin{equation} \frac{\partial F}{\partial \tau} + F\frac{\partial F}{\partial \eta} = -F, \end{equation}

and we impose initial condition

(6.3)\begin{equation} F(\eta,\tau = 0) = F_0(\eta) = \frac{a_n}{\kappa_n}A_n(\eta,t=0). \end{equation}

Motivated by the form of the linear solution, (4.63), we now substitute

(6.4)\begin{equation} F = G(\eta,\tau)\, \textrm{e}^{-\tau}, \end{equation}

into (6.2) to obtain

(6.5)\begin{equation} \frac{\partial G}{\partial \tau} + \textrm{e}^{-\tau} G \frac{\partial G}{\partial \eta} = 0, \end{equation}

subject to

(6.6)\begin{equation} G(\eta,\tau = 0) = F_0(\eta). \end{equation}

This system can be solved using the method of characteristics and the solution for $G$ is given by

(6.7)\begin{equation} G(\eta,\tau) = F_0(r), \end{equation}

where $r = r(\eta ,\tau )$ is defined implicitly by

(6.8)\begin{equation} \eta = r + F_0(r)(1-\textrm{e}^{-\tau}). \end{equation}

Therefore our solution for $F$ can be written as

(6.9)\begin{equation} F(\eta,\tau) = F_0(r(\eta,\tau))\, \textrm{e}^{-\tau}. \end{equation}

We note that nonlinear effects enter through $r$ and the exponential factor corresponds to the decay of the amplitude due to the effects of vertical mixing. The solution will break if $r$ becomes multi-valued for any $(\eta ,\tau )$.

6.2. Wave breaking

Here we examine the possibility of wave breaking and determine conditions for when breaking will occur. We proceed by substituting

(6.10)\begin{equation} \theta = 1 - \textrm{e}^{-\tau}, \end{equation}

into (6.5) to obtain Burger's equation

(6.11)\begin{equation} \frac{\partial G}{\partial \theta} + G \frac{\partial G}{\partial \eta} = 0, \end{equation}

and note that solutions to Burger's equation will always break for a smooth initial profile if there is a region of negative gradient. Assuming that the amplitude vanishes as $x \to \pm \infty$ we note that all non-zero initial profiles must therefore break in $\eta -\theta$ space. Since $\tau \to \infty$ corresponds to $\theta \to 1$ we concluded that solutions to (6.5) will break at some finite time, $\tau ^*$, if solutions to Burger's equation, (6.11), would break for some $\theta ^*<1$. It can be shown that solutions to Burger's equation first break at

(6.12)\begin{equation} \theta = -\frac{1}{F_0'(r(\eta,\theta))}, \end{equation}

hence there will only be breaking solutions for $F(\eta ,\tau )$ if

(6.13)\begin{equation} -F_0' > 1, \end{equation}

for some $\eta$. This corresponds to $r_\infty (\eta )$ being multi-valued where $r_\infty$ the value of $r$ when $\theta = 1$ (corresponding to $\tau = \infty$) and is given implicitly by

(6.14)\begin{equation} \eta = r_\infty + F_0(r_\infty). \end{equation}

Substituting for $F_0$ we have that solutions will break if

(6.15)\begin{equation} -\frac{a_n}{\kappa_n} \frac{\partial A_n}{\partial x}(x,0) > 1, \end{equation}

for some $x$. If $r$ does become multi-valued at say $(\eta ,\tau ) = (\eta ^*,\tau ^*)$ then our solution will break at

(6.16)\begin{equation} (x,t) = (\eta^*+U_n \tau^*/\kappa_n,\tau^*/\kappa_n), \end{equation}

where $(\eta ^*,\tau ^*)$ depends on the ratio $a_n/\kappa _n$ through the initial condition since $F_0(\eta ) = a_n A_n(\eta ,0)/\kappa _n$. We note that waves will travel further before breaking for larger $U_n$ hence topography can modify the distance travelled by the wave before breaking by modifying the speed, $U_n$.

6.3. Summary

Combining our results we have full solution

(6.17)\begin{equation} A_n = G( r(x-U_n t, \kappa_n t) )\, \textrm{e}^{-\kappa_n t}, \end{equation}

where

(6.18)\begin{equation} \eta = r(\eta,\tau) + \frac{a_n}{\kappa_n} G(r(\eta,\tau))(1-\textrm{e}^{-\tau}), \end{equation}

with wave breaking if

(6.19)\begin{equation} \max_x\left[-\frac{a_n}{\kappa_n} \frac{\partial G}{\partial x}\right] > 1. \end{equation}

For a given initial amplitude, $G$, we can see that wave breaking will occur only if the ratio $a_n/\kappa _n$ is sufficiently large. Therefore although vertical mixing acts to suppress wave breaking it cannot prevent breaking if the nonlinearity is strong enough. We note that $a_n$ may be negative in which case any breaking will occur of the side of the wave with positive gradient.

In the case of weak mixing we can expand $\textrm {e}^{-\tau }$ for small $\tau$ to get

(6.20)\begin{equation} \eta = r(\eta,t) + a_n G(r(\eta,t))(t + O(\kappa_n t^2)), \end{equation}

and

(6.21)\begin{equation} A_n = G( r(x-U_n t, t) ) (1 + O(\kappa_n t) ). \end{equation}

Taking $\kappa _n \to 0$ recovers the solution to Burger's equation and we note that all initial profiles will break in the limit of small $\kappa _n$.

For weak nonlinearity we have

(6.22)\begin{equation} \eta = r(\eta,\tau) + O(a_n), \end{equation}

hence in the limit of $a_n \to 0$ we have

(6.23)\begin{equation} A_n = G(x-U_n t)\, \textrm{e}^{-\kappa_n t}, \end{equation}

which matches the linear result of (4.63). In this case there are no breaking solutions and our solution is a self-similar travelling wave with exponentially decaying amplitude.

Figure 6 shows solutions for $A_n$ for a range of values of $(a_n,\kappa _n)$ including cases of small $a_n$ and small $\kappa _n$. Solutions are calculated by numerically solving (6.18) for the characteristic variable $r$ and using (6.17) to determine $A_n$ from the initial condition

(6.24)\begin{equation} G(x) = \exp[ -(x-1)^2 ]. \end{equation}

We note that the characteristics do not form straight lines in $(x,t)$ space; although they do in $(x,\theta )$ space. For this Gaussian initial condition, we expect solutions to break if

(6.25)\begin{equation} \frac{a_n}{\kappa_n} > \sqrt{\frac{e}{2}}, \end{equation}

which is consistent with figure 6 in which we observe breaking for $(a_n,\kappa _n) = (0.2,0.1)$ (b) and $(0.2,0.01)$ (c) but not for $(a_n,\kappa _n) = (0.1,0.1)$ (a) and $(0.01,0.1)$ (d). We say that a numerical solution breaks if the characteristic curves, shown as white lines in figure 6, touch at any point in the domain. The solutions in figure 6 which do not break, panels (a) and (d), both exhibit different behaviour due to a very different ratio between nonlinearity and mixing. In figure 6(a) we have $a_n/\kappa _n = 1$ so we expect the solution to initially sharpen before the mixing damps the effects of nonlinearity sufficiently that breaking is avoided. Conversely, in figure 6(d), $a_n \ll \kappa _n$ so mixing dominates nonlinearity from the start. There is therefore very little nonlinear evolution and the solution remains self-similar, retaining its initial shape while exponentially decaying in time.

Figure 6.

Solution for $A_n$ (6.17) as a function of $(x,t)$ for $U_n = 0.5$ and $(a_n,\kappa _n) = (0.1,0.1)$ (a), $(0.2,0.1)$ (b), $(0.2,0.01)$ (c) and $(0.01,0.1)$ (d). Characteristic curves are shown as white lines and we take Gaussian initial profile, $G(x) = \exp [-(x-1)^2]$. We can see that cases (b) and (c) break while case (a) sharpens towards a smooth profile and case (d) remains approximately self-similar.