The Math (EP)

Overview

We need to go through third order in the wave slope $\epsilon$ to get the modified growth rate.

$O(\epsilon)$

Linear problem (Miles)

Derive the Rayleigh equation in Lagrangian coordinates for $\varphi(b)$.

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Use linear boundary conditions to get a dispersion relation.

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Recover $c^{(0)}(k)$ and Miles-(esque) growth rate $\omega^{(0)}(k)$.

$O(\epsilon^2)$

Insufficient to get growth rate modification

Apply mass and vorticity conservation at second order.

↓

$c^{(1)}(k)=0$

&

Identify the wave-induced mean flow $U^{(2)}(b)$.

$O(\epsilon^3)$

Get growth rate modification in response to $U^{(2)}(b)$

Include $U^{(2)}$ in third-order boundary conditions at $b=0$.

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Obtain a third-order dispersion relation for $c^{(2)}(k)$.

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Get an $\epsilon^2$ correction to Miles’ growth rate via $\mathrm{Im}\,c^{(2)}(k)$.

1. The Eulerian frame vs. the Lagrangian frame

Eulerian view

As usual, we express the horizontal velocity $u(x,z,t)$, the vertical velocity $w(x,z,t)$, and the pressure $p(x,z,t)$ at fixed spatial points. We can write the Euler equations,

\[ \rho\left(\partial_t u + u\,u_x + w\,u_z\right) + p_x = 0, \] \[ \rho\left(\partial_t w + u\,w_x + w\,w_z\right) + p_z + \rho g = 0, \]

Incompressibility is written as

\[ u_x + w_z = 0. \]

The free surface is a moving interface $z = \eta(x,t)$, and the critical layer is defined at the height $z_c$ where $U(z_c) = c$ (since $c$ really has small imaginary part, this is really $U(z_c) = c_r$).

Lagrangian view

We instead let the spatial locations be the dependent variables, where particle labels $(a,b)$ are the independent variables (along with time). We choose the particle labels to be based on the initial position of the particles. We can define a mapping from $(a,b) \mapsto (x,z)$ via

\[ \mathcal{J} = x_a z_b - x_b z_a. \]

The Euler equations can then be written as

\[ \mathcal{J}\,\ddot x + \frac{1}{\rho}\left(p_a z_b - p_b z_a\right) = 0, \] \[ \mathcal{J}\,\ddot z + \frac{1}{\rho}\left(p_b x_a - p_a x_b + \mathcal{J} g\right) = 0, \]

and incompressibility as $\dot{\mathcal{J}}=0$. In these coordinates, the free surface is simply $b = 0$.

2. Asymptotic expansions

We consider a permanent, progressive, monochromatic, spatially periodic 2D wave of wavenumber $k$ and phase speed $c$. First, we write the coordinates $x$ and $z$ in terms of unordered expansions:

\[ x(a,b,t) = a + U(b)\,t + \sum_{n\ge1} x_n(b)\,\sin(\theta_n), \] \[ z(a,b,t) = b + z_0(b) + \sum_{n\ge1} z_n(b)\,\cos(\theta_n), \] \[ \theta_n = n k\,[a - (c - U(b))t]. \]

After defining a small parameter $\epsilon$, the wave slope, we can convert these to ordered expansions via

\[ x = x^{(0)} + \epsilon\,x^{(1)} + \epsilon^2 x^{(2)} + \epsilon^3 x^{(3)} + \dots, \] \[ z = z^{(0)} + \epsilon\,z^{(1)} + \epsilon^2 z^{(2)} + \epsilon^3 z^{(3)} + \dots, \] \[ p = p^{(0)} + \epsilon\,p^{(1)} + \epsilon^2 p^{(2)} + \epsilon^3 p^{(3)} + \dots \]

The background/mean velocity (this is $\overline{u}_L$) and phase speed are also expanded in powers of $\epsilon$:

\[ U(b) = U^{(0)}(b) + \epsilon^2 U^{(2)}(b) + \dots, \qquad c = c^{(0)} + \epsilon\,c^{(1)} + \epsilon^2 c^{(2)} + \dots. \]

The term $U^{(2)}(b)$ corresponds to a wave-induced mean flow, and we will see that while $c^{(2)}(k)$ gives the first correction to the phase speed (and growth rate).

3. Rayleigh equation in Lagrangian coordinates

At first order in $\epsilon$, we use the interesting change of variables (due to Bennett, Lagrangian Fluid Dynamics)

\[ x^{(1)}(a,b,t) = A(a + U^{(0)}(b)t,\,b,\,t), \quad A(a,b,t) = \xi(b)\,\sin\big(k(a - c^{(0)} t)\big), \] \[ z^{(1)}(a,b,t) = B(a + U^{(0)}(b)t,\,b,\,t), \quad B(a,b,t) = \frac{\varphi(b)}{k\big(c^{(0)} - U^{(0)}(b)\big)} \cos\big(k(a - c^{(0)} t)\big). \]

Substituting these into the linearized Lagrangian momentum equations and mass conservation yields the Rayleigh equation in the label coordinate $b$:

\[ \varphi_{bb} - \left( k^2 - \frac{U^{(0)}_{bb}}{c^{(0)} - U^{(0)}} \right)\varphi = 0. \]

This equation not only motivates the definition of the critical level, but will be used to get from the dispersion relation to the analytical solution for the growth rate. Notice there is a singularity when $c^{(0)} = U^{(0)}$. To avoid this, we assume $c^{(0)}$ has a small imaginary part, $c^{(0)} = c^{(0)}_r + i c^{(0)}_i$ with $|c^{(0)}_i| \ll 1$.

4. Dispersion relation and Miles-type growth rate

We will use a dispersion relation to solve for $c^{(0)}$ (both the imaginary part, $c^{(0)}_i$, which will give the growth rate, and the real part, $c^{(0)}_r$). Using the linearized dynamic boundary condition at $b = 0$, we obtain a dispersion relation. To actually solve for $c^{(0)}$, we must make a second asymptotic expansion in the density ratio. We then obtain a dispersion relation of the form

\[ D(c^{(0)},k;\,\epsilon_\rho) = D_0(c^{(0)},k) + \epsilon_\rho\,D_1(c^{(0)},k) = 0, \]

where $\epsilon_\rho = \rho_{\text{air}}/(\rho_{\text{air}}+\rho_{\text{water}})$, and $D_0, D_1$ are known functions of the background profile $U^{(0)}(b)$ and the amplitude $\varphi(b)$.

To solve, we also expand the phase speed $c^{(0)}$ in $\epsilon_\rho$:

\[ c^{(0)}(k,\epsilon_\rho) = c^{(0)}_0(k) + \epsilon_\rho\,c^{(0)}_1(k) + O(\epsilon_\rho^2). \]

Then $c^{(0)}_0(k)$ satisfies $D_0(c^{(0)}_0,k) = 0$ and

\[ c^{(0)}_1(k) = -\,\frac{D_1(c^{(0)}_0,k)}{\partial_{c^{(0)}} D_0(c^{(0)}_0,k)}. \]

One can then use the Rayleigh equation and integration by parts to obtain

\[ \mathrm{Im}\,c^{(0)}_1(k) \propto (c_0^{(0)} - U^{(0)}_s)^2\pi \frac{\frac{d^2}{db^2}U^{(0)}_c}{\left|\frac{d}{db}U^{(0)}_c\right|} \,\frac{|\varphi_c|^2}{|\varphi_s|^2}, \]

where $\varphi_c = \varphi(b_c)$ and $\varphi_s = \varphi(0)$.

Up to this point we have rederived the classical Miles growth rate, but entirely in Lagrangian coordinates. This does explicitly and concisely show why the critical level is wavy though, and other interesting differences can be seen even at this order upon transforming the Lagrangian coordinates back into the Eulerian frame.

5. Why we need higher order: $c^{(1)} = 0$

Recall we expanded the phase speed in terms of the wave slope $\epsilon$,

\[ c(k,\epsilon) = c^{(0)}(k) + \epsilon\,c^{(1)}(k) + \epsilon^2 c^{(2)}(k) + \dots. \]

At second-order, imposing vorticity conservation yields that

\[ c^{(1)}(k) = 0. \]

Consequently, the first modification to the growth rate appears at $O(\epsilon^2)$, through the term $c^{(2)}(k)$, and we must carry the analysis through third order in $\epsilon$.

6. Structure of the third-order calculation

At third order, we

Analyze the dynamic boundary condition, which involves the contributions of $U^{(2)}$ and higher harmonics.
Use conservation laws (and a lot of algebra) to simplify into a solvable form for $c^{(2)}$.

The outcome is a third-order dispersion relation of the schematic form

\[ F(0^+) + c^{(2)}\,G(0^+) = F(0^-) + c^{(2)}\,G(0^-), \]

where $F$ and $G$ are explicit functionals of the background profile $U^{(0)}(b)$, the first-order amplitude $\varphi(b)$, and their derivatives. Note that it is a linear equation in $c^{(2)}$ , so unlike the leading-order case, no asymptotic expansion in the density ratio is needed here.

7. The wave-induced mean flow in the modified growth rate

Wave-induced mean flow (Lagrangian drift)

The wave-induced mean flow, $U^{(2)}(b)$, can be written in terms of the background profile $U^{(0)}$ and the first-order amplitude $\varphi(b)$:

\[ U^{(2)}(b) = \frac{1}{4}\,\big(c^{(0)} - U^{(0)}(b)\big)\, \frac{d^2}{db^2} \left[ \frac{\varphi(b)^2}{ k^2 \big(c^{(0)} - U^{(0)}(b)\big)^2 } \right]. \]

This is a Lagrangian Stokes-drift-like current.

Modified growth rate

The corrected growth rate can be written as

\[ \omega(k;\epsilon) = k\Big[ \epsilon_\rho\,\mathrm{Im}\,c^{(0)}_1(k) + \epsilon^2\,\mathrm{Im}\,c^{(2)}(k) \Big]. \]

The first term is the classical Miles growth rate. The second term is the leading-order modification, which we have analytically shown depends on the wave-induced mean flow $U^{(2)}$. It scales like $\epsilon^2$, and calculating it for realistic background profiles shows that its sign is opposite to the first term, so it suppresses growth as steepness increases.