<p>I'm currently working on a (relatively-simple) fluid-dynamics model of the tidal resonance at the Bay of Fundy, a place which is known for his world-breaking tidal range (15 meters on average, while the historical record is 21.6 meters - at year 1869). After a lot of effort, I succeeded in calculating the <span class="math-container">$M2$</span> periodic-component (semi-diurnal component) of the component of the horizontal tidal force along the north-east; from looking at maps of the Bay of Fundy, I gained the impression that the longitudinal axis of its body of water is directed approximately at 45 degrees north to the eastern direction, so it's in the north-east direction (which is 45 degrees to both the north and the east).</p>
<p>In the process of calculation I took the following values for the relevant sizes:</p>
<ul>
<li>Declination of the Moon's orbit relative to the ecliptic - <span class="math-container">$5$</span> degrees.</li>
<li>Declination of the Earth's axis relative to the perpendicular to the ecliptic - <span class="math-container">$23.5$</span> degrees.</li>
<li>Latitude of the Bay of Fundy - <span class="math-container">$45$</span> degrees.</li>
</ul>
<p>Taking into account only the lunar tidal force (without the solar), the result is:</p>
<p><span class="math-container">$$T_\text{NE}=\frac{3GM_\text{moon}R_\text{earth}}{2L^3}\cdot 0.473\cos(\frac{2\pi}{\Delta T_\text{tide}}t),$$</span></p>
<p>where <span class="math-container">$\Delta T_\text{tide} = 12 \text{h} 25 \text{m}$</span>.</p>
<p>Now, I wanted to estimate the horizontal movement of a vertical slice of water at the Bay of Fundy using the model of a forced harmonic oscillator with the northern-eastern tidal force as the forcing function. Under the assumption of negligible damping parameter, the result is supposed to be (on basis of the solution to the forced harmonic oscillator):</p>
<p><span class="math-container">$$\Delta x = \frac{\frac{3GM_\text{moon}R_\text{earth}}{2L^3}\cdot 0.473}{\omega^2_\text{tide}-\omega^2_\text{seiche}}$$</span></p>
<p>The horizontal translation of the water slice together with its depth and width at the entry to the bay determines the amount of water flowing into the bay, and together with the length of the bay enables one to get a rough approximation of the tidal range at the narrow part of the bay (where the tides reach their peak).</p>
<p>My problem is about getting accurate data on the geometric dimensions of the bay as well as its resonance period - taking the estimate <span class="math-container">$\Delta T_\text{seich}= 12\text{h} 40 \text{m}$</span> gives too low values for the the tidal range - it amounts to only about <span class="math-container">$3$</span> meters! In addition, according to Wikipedia, in each tide cycle an amount of water of approximately <span class="math-container">$100,000,000,000\ m^3$</span> flows into the bay; if we combine this with the depth of water at the entrance of the bay (<span class="math-container">$200\ m$</span>) and the width of its mouth (<span class="math-container">$52\ \text{km}$</span>) we can conclude that the horizontal movement of a vertical slice of water (which is twice the amplitude) is about <span class="math-container">$9.5\ \text{km}$</span>, much more than the result of my formula. Also, the peak depth-averaged tidal current velocity at the bay (at its narrow part) is about <span class="math-container">$4.5\ m/s $</span>, and if we compare it with <span class="math-container">$\Delta x \cdot \omega_{tide}$</span> than it's about 50-60 times larger (and still significantly larger if we also take into account the fact that the channel becomes narrower and shallower).</p>
<p>Therefore, as a first step  of checking this model, I'd like to get accurate data on the geometric dimensions of the Bay of Fundy:</p>
<ul>
<li>What is the exact length of the Bay of Fundy? I mean the length of that part of the bay with the resonant properties. I ask because there are inconsistencies between the different sources - Encyclopedia Britannica says it's 151 km long, another source states 220 km, another 290 km, and I even found a source claiming 400 km.</li>
<li>What is the depth of water at the entrance to the bay? What is the average depth at the bay?</li>
<li>I'm aware that this isn't an easy question to answer experimentally, but what is the resonance period (seiche period) at the bay? I'll be glad if anyone will redirect me to good sources on the resonance period.</li>
</ul>
<p>Finally, if the problem isn't with the accuracy of the data available on web, I want to know what is the flaw in my model.</p>

CC BY-SA 4.0
2021-05-30T21:15:55.690Z
Does your model include location and tides levels from ports around the bay?  *Actual* recordings may be of course better, than «just» the forecast of a tide chart like [this](https://www.tideschart.com/Canada/Nova-Scotia/Port-Lorne/) about Port Lorne.
Buttonwood
CC BY-SA 4.0
2021-05-30T21:34:12.930Z
@Buttonwood - in my model, i divided the total tidal range into two parts: 1. "global" tidal range - the water rise around the bay (without the resonance effect), which i assumed is about 1 meter, and the resonant part which is calculated by the procedure outlined in my post, and is equal to only about 2 meters. If we take into account that the bay becomes narrower (at a ratio of about 1 to 3) it might add a factor of two (so tidal range is supposed to be 5 meters), but its still far from regular measurements there (about 14 meters).
user2554
CC BY-SA 4.0
2021-06-01T05:07:37.150Z
Just a comment: Tsunami wave in open ocean are not that high, sometimes 1-2 meters, when they arrive on the coast then they rise to the spectacular 10/15 meters.
Is your simplified model providing 3 meters taking into account such an effect?

Ps: if you can provide in the comment pictures of such a tide I would be happy :) !
EarlGrey
CC BY-SA 4.0
2021-06-01T12:47:57.640Z
@EarlGrey - in my model i didn't take into the account the effect of shallowing of the water in the bay, and thanks for the comment. Anyway,  i believe it to be a marginal effect - there are many shallow water coasts in the world, but only those with the resonant effect experience amplification of tides. I think this effect might amplify the tides a bit more. Meanwhile, i found an article which says that the M2 component of tidal force isn't the only one with resonance with the bay - there is an additional N2 component.
user2554
CC BY-SA 4.0
2021-06-01T12:49:49.713Z
And a pic of such a tide can be found in wikipedia article "tidal resonance" - https://en.m.wikipedia.org/wiki/Tidal_resonance.
user2554
CC BY-SA 4.0
2021-06-02T07:45:16.790Z
Thanks, I hoped to see some brutal videos of the Bay of Fundy, but I google myself. Regarding components M2 and N2, you can move this comment as an answer (on SE it is acceptable to answer your own questions) and then pick your answer as best answer (so in the future someone having similar issue will find your question and your answer)
EarlGrey
CC BY-SA 4.0
2021-06-06T15:10:24.220Z
http://www8.umoncton.ca/umcm-climat/en/uved/grain/4_8_the_bay_of_fundy_and_the_tides_of_climate_change/page:3
arkaia
CC BY-SA 4.0
2021-06-06T15:14:06.543Z
Also, remember that you are just modeling the M2 tidal constituents, which is not nearly the entire story. You can check https://www.bio.gc.ca/science/research-recherche/ocean/modelling-modelisation/coastalembayments-cotieresdesbaies/documents/dupont2003.pdf for instance
arkaia
CC BY-SA 4.0
2021-06-06T15:16:45.487Z
Additionally, the length of resonance should probably include the entire Gulf of Maine. The tide is enhanced in the entire GoM+BoF system
arkaia
<oceanography><tides>
Questions about the validity of a certain model for tidal resonance at the Bay of Fundy
user2554
# Questions about the validity of a certain model for tidal resonance at the Bay of Fundy

(Asked by: username_0 on 2021-05-29)

I'm currently working on a (relatively-simple) fluid-dynamics model of the tidal resonance at the Bay of Fundy, a place which is known for his world-breaking tidal range (15 meters on average, while the historical record is 21.6 meters - at year 1869). After a lot of effort, I succeeded in calculating the $M2$ periodic-component (semi-diurnal component) of the component of the horizontal tidal force along the north-east; from looking at maps of the Bay of Fundy, I gained the impression that the longitudinal axis of its body of water is directed approximately at 45 degrees north to the eastern direction, so it's in the north-east direction (which is 45 degrees to both the north and the east).

In the process of calculation I took the following values for the relevant sizes:

* Declination of the Moon's orbit relative to the ecliptic - $5$ degrees.
* Declination of the Earth's axis relative to the perpendicular to the ecliptic - $23.5$ degrees.
* Latitude of the Bay of Fundy - $45$ degrees.

Taking into account only the lunar tidal force (without the solar), the result is:

$$T\_\text{NE}=\frac{3GM\_\text{moon}R\_\text{earth}}{2L^3}\cdot 0.473\cos(\frac{2\pi}{\Delta T\_\text{tide}}t),$$

where $\Delta T\_\text{tide} = 12 \text{h} 25 \text{m}$.

Now, I wanted to estimate the horizontal movement of a vertical slice of water at the Bay of Fundy using the model of a forced harmonic oscillator with the northern-eastern tidal force as the forcing function. Under the assumption of negligible damping parameter, the result is supposed to be (on basis of the solution to the forced harmonic oscillator):

$$\Delta x = \frac{\frac{3GM\_\text{moon}R\_\text{earth}}{2L^3}\cdot 0.473}{\omega^2\_\text{tide}-\omega^2\_\text{seiche}}$$

The horizontal translation of the water slice together with its depth and width at the entry to the bay determines the amount of water flowing into the bay, and together with the length of the bay enables one to get a rough approximation of the tidal range at the narrow part of the bay (where the tides reach their peak).

My problem is about getting accurate data on the geometric dimensions of the bay as well as its resonance period - taking the estimate $\Delta T\_\text{seich}= 12\text{h} 40 \text{m}$ gives too low values for the the tidal range - it amounts to only about $3$ meters! In addition, according to Wikipedia, in each tide cycle an amount of water of approximately $100,000,000,000\ m^3$ flows into the bay; if we combine this with the depth of water at the entrance of the bay ($200\ m$) and the width of its mouth ($52\ \text{km}$) we can conclude that the horizontal movement of a vertical slice of water (which is twice the amplitude) is about $9.5\ \text{km}$, much more than the result of my formula. Also, the peak depth-averaged tidal current velocity at the bay (at its narrow part) is about $4.5\ m/s $, and if we compare it with $\Delta x \cdot \omega\_{tide}$ than it's about 50-60 times larger (and still significantly larger if we also take into account the fact that the channel becomes narrower and shallower).

Therefore, as a first step of checking this model, I'd like to get accurate data on the geometric dimensions of the Bay of Fundy:

* What is the exact length of the Bay of Fundy? I mean the length of that part of the bay with the resonant properties. I ask because there are inconsistencies between the different sources - Encyclopedia Britannica says it's 151 km long, another source states 220 km, another 290 km, and I even found a source claiming 400 km.
* What is the depth of water at the entrance to the bay? What is the average depth at the bay?
* I'm aware that this isn't an easy question to answer experimentally, but what is the resonance period (seiche period) at the bay? I'll be glad if anyone will redirect me to good sources on the resonance period.

Finally, if the problem isn't with the accuracy of the data available on web, I want to know what is the flaw in my model.

**Tags:** oceanography, tides

> **Comments:**
> **username_1**: Does your model include location and tides levels from ports around the bay?  *Actual* recordings may be of course better, than «just» the forecast of a tide chart like [this](https://www.tideschart.com/Canada/Nova-Scotia/Port-Lorne/) about Port Lorne.  
> **username_0**: @username_1 - in my model, i divided the total tidal range into two parts: 1. "global" tidal range - the water rise around the bay (without the resonance effect), which i assumed is about 1 meter, and the resonant part which is calculated by the procedure outlined in my post, and is equal to only about 2 meters. If we take into account that the bay becomes narrower (at a ratio of about 1 to 3) it might add a factor of two (so tidal range is supposed to be 5 meters), but its still far from regular measurements there (about 14 meters).  
> **username_2**: Just a comment: Tsunami wave in open ocean are not that high, sometimes 1-2 meters, when they arrive on the coast then they rise to the spectacular 10/15 meters. Is your simplified model providing 3 meters taking into account such an effect?  Ps: if you can provide in the comment pictures of such a tide I would be happy :) !  
> **username_0**: @username_2 - in my model i didn't take into the account the effect of shallowing of the water in the bay, and thanks for the comment. Anyway,  i believe it to be a marginal effect - there are many shallow water coasts in the world, but only those with the resonant effect experience amplification of tides. I think this effect might amplify the tides a bit more. Meanwhile, i found an article which says that the M2 component of tidal force isn't the only one with resonance with the bay - there is an additional N2 component.  
> **username_0**: And a pic of such a tide can be found in wikipedia article "tidal resonance" - https://en.m.wikipedia.org/wiki/Tidal_resonance.  
> **username_2**: Thanks, I hoped to see some brutal videos of the Bay of Fundy, but I google myself. Regarding components M2 and N2, you can move this comment as an answer (on SE it is acceptable to answer your own questions) and then pick your answer as best answer (so in the future someone having similar issue will find your question and your answer)  
> **username_3**: http://www8.umoncton.ca/umcm-climat/en/uved/grain/4_8_the_bay_of_fundy_and_the_tides_of_climate_change/page:3  
> **username_3**: Also, remember that you are just modeling the M2 tidal constituents, which is not nearly the entire story. You can check https://www.bio.gc.ca/science/research-recherche/ocean/modelling-modelisation/coastalembayments-cotieresdesbaies/documents/dupont2003.pdf for instance  
> **username_3**: Additionally, the length of resonance should probably include the entire Gulf of Maine. The tide is enhanced in the entire GoM+BoF system  

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## 0 answers

