<p>In <a href="https://academic.oup.com/mnras/article/458/3/2890/2589321" rel="nofollow noreferrer">that paper</a>, we actually applied the term &quot;principal&quot; to the semidiurnal mode <span class="math-container">$\omega_{2200}$</span>. 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 &quot;principal&quot;. Look, e.g., at eqn (145) in <a href="https://arxiv.org/abs/1904.02253" rel="nofollow noreferrer">this paper</a>. It shows how both partners (assumed nonsynchronous) provide their main inputs into <span class="math-container">$da/dt$</span>. We see that each input is proportional to the value of the appropriate partner's quality function evaluated at the principal tidal mode: <span class="math-container">$K_2(\omega_{2200})$</span> for the primary's input, and <span class="math-container">$K^{\,\prime}_2(\omega^{\,\prime}_{2200})$</span> for the secondary's input.</p>
<p>The situation changes when both or one of the partners is synchronised. See Section 4.4 in <a href="https://arxiv.org/abs/1904.02253" rel="nofollow noreferrer">that paper</a>, and Appendix G for more detail.</p>
<p>To understand why the quantities <span class="math-container">$\omega_{lmpq}$</span> were christened as <em>tidal modes</em> or <em>Fourier tidal modes</em>, please have a look at equations (11 - 13) in <a href="https://iopscience.iop.org/article/10.1088/0004-637X/764/1/26" rel="nofollow noreferrer">this work</a>. Also, according to equation (14), the absolute values of these modes are the physical forcing frequencies exerted by tides in the perturbed body.</p>

CC BY-SA 4.0
2022-06-20T16:42:47.867Z
2023-04-02T00:58:35.163Z
2023-04-02T00:58:35.163Z
Michael_1812
<p>I'm trying to understand tides evolution of planets, and I often come across the Prinicipal Tidal Fourier Modes expressed as:</p>
<p><span class="math-container">$$ \omega_{lmpq}$$</span></p>
<p>For example, see the discussion after Eq. A15 in <a href="https://academic.oup.com/mnras/article/458/3/2890/2589321" rel="noreferrer">Numerical simulation of tidal evolution of a viscoelastic body modelled with a mass-spring network</a>.</p>
<p>I was wondering what is the physical meaning of these modes and what is the physical meaning of the integers <span class="math-container">$lmpq$</span>. I do understand that <span class="math-container">$l$</span> and <span class="math-container">$m$</span> are the spherical harmonics integers that arise from solving the Laplace equation in spherical coordinates, but what are <span class="math-container">$p$</span> and <span class="math-container">$q$</span>?</p>
<p>Any physical intuition behind these parameters and their relation to tides could help a lot!</p>

CC BY-SA 4.0
2021-11-29T21:18:09.443Z
"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.
James K
CC BY-SA 4.0
2021-11-29T21:23:26.903Z
The term is mentioned as "Principal Tidal Fourier Mode" for example in: https://doi.org/10.1093/mnras/stw491
ValientProcess
<gravity><mathematics><tidal-forces><modeling>
Physical Meaning of the Principal Tidal Fourier Modes and Their Integers
ValientProcess
# Physical Meaning of the Principal Tidal Fourier Modes and Their Integers

(Asked by: username_0 on 2021-11-29)

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](https://academic.oup.com/mnras/article/458/3/2890/2589321).

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!

**Tags:** gravity, mathematics, tidal-forces, modeling
> **Comments:**
> **username_1**: "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.  
> **username_0**: The term is mentioned as "Principal Tidal Fourier Mode" for example in: https://doi.org/10.1093/mnras/stw491  

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## 1 answers

### Answer by username_2 on 2022-06-20. Score: 3

In [that paper](https://academic.oup.com/mnras/article/458/3/2890/2589321), 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](https://arxiv.org/abs/1904.02253). 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 synchronised. See Section 4.4 in [that paper](https://arxiv.org/abs/1904.02253), 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](https://iopscience.iop.org/article/10.1088/0004-637X/764/1/26). 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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