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I'm trying to understand tides evolution of planets, and I often come across the Prinicipal Tidal Fourier Modes expressed as:

$$ \omega_{lmpq}$$

For example, see the discussion after Eq. A15 in Numerical simulation of tidal evolution of a viscoelastic body modelled with a mass-spring network.

I was wondering what is the physical meaning of these modes and what is the physical meaning of the integers $lmpq$. I do understand that $l$ and $m$ are the spherical harmonics integers that arise from solving the Laplace equation in spherical coordinates, but what are $p$ and $q$?

Any physical intuition behind these parameters and their relation to tides could help a lot!

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    $\begingroup$ "I often come across the Principal Tidal Fourier Modes" can you give an example of where you came across this. A quick google search finds no result for that phrase (in quotes) -- and without quotes there are lots of pages about sea tides, not about planetary tides. $\endgroup$
    – James K
    Nov 29, 2021 at 21:18
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    $\begingroup$ The term is mentioned as "Principal Tidal Fourier Mode" for example in: doi.org/10.1093/mnras/stw491 $\endgroup$ Nov 29, 2021 at 21:23

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In that paper, we actually applied the term "principal" to the semidiurnal mode $\omega_{2200}$. When the planet is not synchronised (i.e., when it is not showing the same side to the orbiter), it is this mode that provides most to the tidal effects. Hence the name "principal". Look, e.g., at eqn (145) in this paper. It shows how both partners (assumed nonsynchronous) provide their main inputs into $da/dt$. We see that each input is proportional to the value of the appropriate partner's quality function evaluated at the principal tidal mode: $K_2(\omega_{2200})$ for the primary's input, and $K^{\,\prime}_2(\omega^{\,\prime}_{2200})$ for the secondary's input.

The situation changes when both or one of the partners is not synchronised. See Section 4.4 in that paper, and Appendix G for more detail.

To understand why the quantities $\omega_{lmpq}$ were christened as tidal modes or Fourier tidal modes, please have a look at equations (11 - 13) in this work. Also, according to equation (14) the absolute values of these modes are the physical forcing frequencies exerted by tides in the perturbed body.

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