Hello, I’ve just been looking back at the 1960 paper by Longuet-Higgins and Stewart, specifically the part where the derive the 2 component solution at second order
(eq. 2.22), and they give an interpretation of the result as a modulation of the amplitude with a P factor and a modulation of the wavelength with a Q factor… which makes sense if P and Q are << 1. Except that for most situations of pratical interest Q > 1 … and the 2nd order short wave is larger than the 1st order … which looks suspicious.
Before I jump to the next (3rd) order, has anybody looked into this?
The asymptotics are such that P,Q are slowly varying compared to $\psi_1$, eg the second term in parentheses in 2.25. Does this imply that they themselves have to be small?
I think the language is a bit inconsistent, i.e. “if P, Q are any small quantities” and “varying slowly compared to \psi_1”… not necessarily the same thing.
It’s not obvious to me why Q (a_2 k_1) needs to be small. For a_2 it makes sense because it affects the geometry of the surface, but k_1 should be allowed to be very large (but maybe not capillary kind of large in this derivation).
A related question that I have is about the amplitude modulation. If we use the wave action balance, we can derive the same solution as the L-H&S (2.27), except it’s for wave action and not amplitude:
\dfrac{N'}{N_1} = 1 + P
But N \sim a^2, so I expect (ignoring intrinsic frequency modulation for simplicity, but it can be factored in):
\dfrac{a'}{a_1} \sim \sqrt{1 + P}
which is clearly inconsistent with L-H&S. Why?
One more: What makes this a 2nd order theory? The solutions are 1st order in a_2k_2.
For the second point, if N\equiv E/\omega and \omega =\sqrt{gk}, with “the wavenumber increased by the same factor”, then you would get the same answer, right?
The result for a' is O(a_1a_2) , I think that’s what they are referring to. It makes sense to count a_1 in the ordering, as it catalyzes the computations (eg if it were zero none of this would arise).
Is this forum now open? My student Aidan has reproduced most of these computations over the last few years and would likely be useful to have here. In which case I can invite him.
Good catch and at first I thought so, but now that I wrote it out it doesn’t seem so. \omega in the denominator makes things different rather than cancel out.
N' = N (1 + P)
E' = E \dfrac{\omega'}{\omega} (1 + P) = E \sqrt{\dfrac{g'k'}{gk}} (1 + P)
k' = k (1 + P)
If we assume g' = g (no gravity modulation), we get
E' = E \sqrt{1 + P} (1 + P) = E (1 + P)^{1.5}
so
a' = a (1 + P)^{0.75}
If we take gravity modulation to the 1st order as
g' = g (1 - P)
then I get
a' = a (1 - P)^{0.25} (1 + P)^{0.75}
So I wonder if there’s something assumption breaking when using the wave action balance (or there’s an error elsewhere in this math). I arrive at N' = N (1 + P) from different starting equations than L-H&S 1960.
L-H&S 1960 didn’t consider modulation of gravity but L-H 1987 did (in addition to higher order waves which yield much larger modulation magnitudes).
Yes, please invite Aidan and other close colleagues. For now we’re seeding the forum with personal referrals and @Fabrice will invite on the mailing list when ready.
Thanks for the reference to the 1987 paper, that clarifies things considerably. The notation changes between the papers (in particular the phase of the long wave), so it’s worth being mindful of that.
where g' is the reduced gravity, the short wave frequency is \sigma', and the short wave amplitude is a'. Now, a'=(1+P)a for a the long wave amplitude, k'=(1+P)k for k the long wave wavenumber, g'=(1-P)g and P=ak\cos \psi is as defined in LH1987.
Putting this together, we have \frac{E'}{\sigma'} = \frac{(1-P)(1+P)^2}{\sqrt{(1-P)(1+P)}} \frac{E}{\sigma}.
Letting N'\equiv \frac{E'}{\sigma} and N\equiv \frac{E}{\sigma} we have
That is only one solution of the wave action equation… and it is in fact unstable, as we argued in Peureux et al; (2021) https://doi.org/10.1029/2020JC016735 … about which Milan showed that our choice of initial conditions was a bit special.
Now, back to the LH-S solution with P&Q, indeed the large Q effect does not appear in the 1987 solution. Is that just the consequence of going to higher order?
Finally these are without wind or dissipation. What is the latest on this problem including generation and dissipation? I’ve just review a paper by Addona et al. with lab experiments, going back to Hasselmann (1971, JFM)?
