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Some discussion of a new paper:

At present computer simulations can predict the occurrence of an El Niño event at best three seasons in advance. Climate modeling centers worldwide generate and disseminate these forecasts on an operational basis. Scientists have assumed that the skill and reliability of such tropical climate forecasts drop rapidly for lead times longer than one year.

The new findings of predictable climate variations up to three years in advance are based on a series of hindcast computer modeling experiments, which included observed ocean temperature and salinity data. The results are presented in the April 21, 2015, online issue of Nature Communications.

"We found that, even three to four years after starting the prediction, the model was still tracking the observations well," says Yoshimitsu Chikamoto at the University of Hawaii at Manoa International Pacific Research Center and lead author of the study. "This implies that central Pacific climate conditions can be predicted over several years ahead."

"The mechanism is simple," states co-author Shang-Ping Xie from the University of California San Diego. "Warmer water in the Atlantic heats up the atmosphere. Rising air and increased precipitation drive a large atmospheric circulation cell, which then sinks over the Central Pacific. The relatively dry air feeds surface winds back into the Atlantic and the Indian Ocean. These winds cool the Central Pacific leading to conditions, which are similar to a La Niña Modoki event. The central Pacific cooling then strengthens the global atmospheric circulation anomalies."

"Our results present a paradigm shift," explains co-author Axel Timmermann, climate scientist and professor at the University of Hawaii. "Whereas the Pacific was previously considered the main driver of tropical climate variability and the Atlantic and Indian Ocean its slaves, our results document a much more active role for the Atlantic Ocean in determining conditions in the other two ocean basins. The coupling between the oceans is established by a massive reorganization of the atmospheric circulation."

## Comments

I think that the most significant forcing function for ENSO is QBO, which is a global oscillation in wind direction.

This is an excerpt from the Chikamoto paper:

So what causes the QBO to reverse direction every 2.3 years? Is this a strong resonance condition in the upper atmosphere, possibly being pumped by the Walker circulation as they show:

I glanced at the paper and found this figure representing their "hindcasting and forecasting ensemble mean". It looks as if it is very conservative based on the sparse number of simulation points given and the size of the error bars:

As usual, I can't comment too much because I have no feel for how their simulations work.

`I think that the most significant forcing function for ENSO is QBO, which is a global oscillation in wind direction. This is an excerpt from the Chikamoto paper: > "This enhanced multi-year predictability in central Pacific SLP can be explained by the fact that SLP, and in particular wind variability, are more sensitive to variations in the zonal gradient of SST rather than to the nearly homogeneous SST warming 22 induced by GHGs increases. We find that, unlike in the case of SST, natural variability explains 4 80% of the total variance of the SLP anomalies over major parts of the tropics (Supplementary Fig. 1). In the central tropical Pacific, therefore, the natural atmosphere–ocean variability is the major contributor for multi-year predictability" So what causes the QBO to reverse direction every 2.3 years? Is this a strong resonance condition in the upper atmosphere, possibly being pumped by the Walker circulation as they show: ![walker](http://images.sciencedaily.com/2015/04/150421084810-large.jpg) I glanced at the paper and found this figure representing their "hindcasting and forecasting ensemble mean". It looks as if it is very conservative based on the sparse number of simulation points given and the size of the error bars: ![fc](http://imageshack.com/a/img538/5905/6f4G5R.gif) As usual, I can't comment too much because I have no feel for how their simulations work.`

at least for some aspects it seems you rather don't need simulations. they wrote ( in the blog post)

that is at least I found last july that already the NOAA images document a much more active role for the Atlantic Ocean in determining conditions, that is I wrote:

....even though this was based on an oversimplifying wind image on Wikipedia I find though their explanation (at least the excerpts I see here) of the ENSO mechanism not convincing and the above image rightous misleading.

I also think that QBO plays an important role in understanding the ENSO mechanism. The QBO itself seems to be driven by quaterly changes of westerlies and easterlies in the stratosphere (just watch this movie in the link, I think you had posted once: http://www.ugamp.nerc.ac.uk/hot/ajh/qbo.htm), which are as I understood basically the same for all longitudes (? at least thats what the ugamp site suggests in my interpretation) so the QBO period seems rather

strictly2 years with some outliers which produce those average of 2.3 years (....as I keep saying). The reason for this "strictly biannual period" might be some kind of period doubling of the quaterly/annual period or some planetary biannual thing. The ENSO will probably be mostly influenced by the lower parts of the QBO (which seem roughly still rather located at the upper boundary of the troposphere at 40-90mB). What I have sofar understood from the ugamp images is that the reason for an ENSO are so to say "overstrong lower easterlies" (like in Fig1 lower easterlies peak roughly in JUL 92,94,96 and 98) and as you can see the QBO easterlies in 1994 was rather strong (dark blue around 10mB) and indeed it looks as if there was some kind of El nino in 1994 (I don't know though if this was officially an El Nino by those SOI Nino3.4 or what do I know indicators) so in terms of the Atlantic: apart from the westerlies from north in autumn 94 (fig1) the westerlies from north 2 years before the El Nino (here Fig. 1 westerlies autumn 1992 and how (un)cool these are (and in particular wether their temperatures are different when coming from northern Atlantic or Pacific (see chimney explanation)) may also play a role. But actually one would need much better and more image material for any sound deduction, which I don't have access to. Moreover I can't spend too much time investigating this (I have currently a rather many strenous hours in front of computer sitting job)-so this reasoning here should be for now taken with a huge grain of salt.`>As usual, I can't comment too much because I have no feel for how their simulations work. at least for some aspects it seems you rather don't need simulations. they wrote ( in the blog post) >our results document a much more active role for the Atlantic Ocean in determining conditions in the other two ocean basins. and >Seeing the Atlantic as an important contributor to these rainfall shifts, which happen as far away as Australia, came to us as a great surprise. that is at least I found <a href="https://forum.azimuthproject.org/discussion/comment/11557/#Comment_11557">last july</a> that already the NOAA images document a much more active role for the Atlantic Ocean in determining conditions, that is I wrote: > it looks that the heat transfer of temperatures over the northern atlantic to the basin in front of northern south America seem to play a bigger role. ....even though this was based on an oversimplifying <a href="http://en.wikipedia.org/wiki/Atmospheric_circulation#/media/File:AtmosphCirc2.png">wind image on Wikipedia</a> I find though their explanation (at least the excerpts I see here) of the ENSO mechanism not convincing and the above image rightous misleading. > I think that the most significant forcing function for ENSO is QBO, which is a global oscillation in wind direction. I also think that QBO plays an important role in understanding the ENSO mechanism. The QBO itself seems to be driven by quaterly changes of westerlies and easterlies in the stratosphere (just watch this movie in the link, I think you had posted once: http://www.ugamp.nerc.ac.uk/hot/ajh/qbo.htm), which are as I understood basically the same for all longitudes (? at least thats what the ugamp site suggests in my interpretation) so the QBO period seems rather <em>strictly</em> 2 years with some outliers which produce those average of 2.3 years (....as I keep saying). The reason for this "strictly biannual period" might be some kind of period doubling of the quaterly/annual period or some planetary biannual thing. The ENSO will probably be mostly influenced by the lower parts of the QBO (which seem roughly still rather located at the upper boundary of the troposphere at 40-90mB). What I have sofar understood from the ugamp images is that the reason for an ENSO are so to say "overstrong lower easterlies" (like in <a href="http://www.ugamp.nerc.ac.uk/hot/ajh/qbo.htm">Fig1</a> lower easterlies peak roughly in JUL 92,94,96 and 98) and as you can see the QBO easterlies in 1994 was rather strong (dark blue around 10mB) and <a href="http://www.ospo.noaa.gov/data/sst/mean_anom/October.94.anomaly.gif">indeed</a> it looks as if there was some kind of El nino in 1994 (I don't know though if this was officially an El Nino by those SOI Nino3.4 or what do I know indicators) so in terms of the Atlantic: apart from the westerlies from north in autumn 94 (fig1) the westerlies from north 2 years before the El Nino (here Fig. 1 westerlies autumn 1992 and how (un)cool these are (and in particular wether their temperatures are different when coming from northern Atlantic or Pacific (see chimney explanation)) may also play a role. But actually one would need much better and more image material for any sound deduction, which I don't have access to. Moreover I can't spend too much time investigating this (I have currently a rather many strenous hours in front of computer sitting job)-so this reasoning here should be for now taken with a huge grain of salt.`

nad said:

Possible, but I have been using a period of 2.33 years, which gives a locking in of 3 cycles per 7 years. But then again, there is clearly jitter in the period about the 2.33 years, just as you say. If a period is longer in one cycle, it will catch up later. This is also known as frequency modulation (FM) about the mean period.

I have taken to use the 70 hPa measure of QBO because the jitter is stronger and so when I have to modulate the applied QBO frequency from cycle-to-cycle, it gives a much better fit in the model. Prior to that, I had been using the QBO at 20 hPa, which had less jitter, but also provided a poorer fit.

I will continue to better understand what the Chikamoto paper is stating, but in the meantime, I am doing my own thing. :)

`nad said: > " so the QBO period seems rather strictly 2 years with some outliers which produce those average of 2.3 years (....as I keep saying)." Possible, but I have been using a period of 2.33 years, which gives a locking in of 3 cycles per 7 years. But then again, there is clearly [jitter](http://en.wikipedia.org/wiki/Jitter) in the period about the 2.33 years, just as you say. If a period is longer in one cycle, it will catch up later. This is also known as frequency modulation (FM) about the mean period. I have taken to use the 70 hPa measure of QBO because the jitter is stronger and so when I have to modulate the applied QBO frequency from cycle-to-cycle, it gives a much better fit in the model. Prior to that, I had been using the QBO at 20 hPa, which had less jitter, but also provided a poorer fit. I will continue to better understand what the Chikamoto paper is stating, but in the meantime, I am doing my own thing. :)`

With strictly 2 year period I don't mean frequency modulation, but a constant frequency (1/(2 years)) with "breaks" or "outliers" in between, may be this can also be called jitter, I don't understand the Wikipedia explanations of jitter.

`>This is also known as frequency modulation (FM) about the mean period. With strictly 2 year period I don't mean frequency modulation, but a constant frequency (1/(2 years)) with "breaks" or "outliers" in between, may be this can also be called jitter, I don't understand the Wikipedia explanations of jitter.`

If this was frequency modulation then a waveform may look like

$ sin((\omega_0+f(t)) t )$

There is some aspect to this in the QBO, where the f(t) is a sinusoidal perturbation around the fundamental frequency.

Yet there is a portion of the waveform where the f(t) itself appears more random such as the case with classical jitter.

If volcanoes impacted the QBO, that would be random, but with something like tides or sunspots, that would be more of a frequency modulation I would suspect.

`If this was frequency modulation then a waveform may look like $ sin((\omega_0+f(t)) t )$ There is some aspect to this in the QBO, where the f(t) is a sinusoidal perturbation around the fundamental frequency. Yet there is a portion of the waveform where the f(t) itself appears more random such as the case with classical jitter. If volcanoes impacted the QBO, that would be random, but with something like tides or sunspots, that would be more of a frequency modulation I would suspect.`

yes. well the QBO oscillations have of course a little fuzzy frequency component, but I wouldn't call this "frequency modulation." And strictly speaking the QBO is even not really periodic, but only somewhat and the amplitude goes also quite wild. What I mean is, as said here that it looks as if the signal is mostly having a strict two year period (i.e. I mean by that f(t) comparatively small) but once in a while not - i.e. it gets out of sync. Moreover it looks as if it is "forced back" into the biannual rythmn. I could imagine that this "out of sync" (disturbations, breaks) and forced back behaviour in an oscillation is rather well distinguishable from a more random behaviour via a fourier transform but may be not and even if this would be so I don't know how the corresponding typical forms would look like.

`>If this was frequency modulation then a waveform may look like >$ sin((\omega_0+f(t)) t )$ yes. well the QBO oscillations have of course a little fuzzy frequency component, but I wouldn't call this "frequency modulation." And strictly speaking the QBO is even not really periodic, but only somewhat and the amplitude goes also quite wild. What I mean is, as <a href="https://forum.azimuthproject.org/discussion/comment/12446/#Comment_12446">said here</a> that it looks as if the signal is mostly having a strict two year period (i.e. I mean by that f(t) comparatively small) but once in a while not - i.e. it gets out of sync. Moreover it looks as if it is "forced back" into the biannual rythmn. I could imagine that this "out of sync" (disturbations, breaks) and forced back behaviour in an oscillation is rather well distinguishable from a more random behaviour via a <a href="http://en.wikipedia.org/wiki/Fourier_transform">fourier transform</a> but may be not and even if this would be so I don't know how the corresponding typical forms would look like.`

I don't understand what you are saying. I downloaded the QBO time-series from the data repository and then match it to a Fourier series with some jitter and frequency modulation applied. The metric showing agreement between the model and data, e.g. correlation coefficient, is quite high.

I wouldn't go by what the data "looks" like, but at a quantifiable objective deconstruction of the Fourier components.

A while back I tried this DiffEq solution to a frequency-modulated wave equation. This has forcing components at 1 year and 1/2 year periods and some from TSI variability

The CC is 0.88, which is quite good for such a waveform.

`I don't understand what you are saying. I downloaded the QBO time-series from the data repository and then match it to a Fourier series with some jitter and frequency modulation applied. The metric showing agreement between the model and data, e.g. correlation coefficient, is quite high. I wouldn't go by what the data "looks" like, but at a quantifiable objective deconstruction of the Fourier components. A while back I tried this DiffEq solution to a frequency-modulated wave equation. This has forcing components at 1 year and 1/2 year periods and some from TSI variability ![qbo](http://imageshack.com/a/img909/8920/mhfplY.gif) The CC is 0.88, which is quite good for such a waveform.`

I wrote:

I just checked Wikipedia's NOAA chart on El Nino - according to this chart there was apparently no El Nino in 1994, but there was one in 1992 but unfortunately there are no anomalies in 1992 as they were -as NOAA writes-

The SOI looks however for 1992 and 1994 rather elninoesk. For 1992 the winds in Fig1 didn't get that strong in terms of velocity as in 1994 (dark blue) but then the blue tongue is in 1992 much broader as in the other years at around the equator, so the easterlies in 1992 were apparently longer present around the equator.....which could also be interpreted as an "overstrong lower easterlies".

`I wrote: >and indeed it looks as if there was some kind of El nino in 1994 (I don't know though if this was officially an El Nino by those SOI Nino3.4 or what do I know indicators) I just checked <a href="http://en.wikipedia.org/wiki/El_Ni%C3%B1o_Southern_Oscillation#/media/File:Enso-global-temp-anomalies.png">Wikipedia's NOAA chart on El Nino</a> - according to this chart there was apparently no El Nino in 1994, but there was one in 1992 but unfortunately there are <a href="http://www.ospo.noaa.gov/Products/ocean/sst/monthly_mean_anom.html">no anomalies in 1992</a> as they were -as NOAA writes- >"eliminated due to volcanic aerosol contamination." The <a href="http://en.wikipedia.org/wiki/El_Ni%C3%B1o_Southern_Oscillation#/media/File:Soi.svg">SOI</a> looks however for 1992 and 1994 rather elninoesk. For 1992 the winds in <a href="http://www.ugamp.nerc.ac.uk/hot/ajh/qbo.htm">Fig1</a> didn't get that strong in terms of velocity as in 1994 (dark blue) but then the blue tongue is in 1992 much broader as in the other years at around the equator, so the easterlies in 1992 were apparently longer present around the equator.....which could also be interpreted as an "overstrong lower easterlies".`

I don't see why the above is supposed to be a TSI variability. The matching looks again impressive, but then again this is only a finite stripe you are looking at. From where do all those coefficients and the diffeq itself come? I have no mathematica and no feeling what the solutions to this weird diffeq could look like for a small variation of the parameters.

`>A while back I tried this DiffEq solution to a frequency-modulated wave equation. This has forcing components at 1 year and 1/2 year periods and some from TSI variability I don't see why the above is supposed to be a TSI variability. The matching looks again impressive, but then again this is only a finite stripe you are looking at. From where do all those coefficients and the diffeq itself come? I have no mathematica and no feeling what the solutions to this weird diffeq could look like for a small variation of the parameters.`

This is just a solution to a second-order differential wave equation. What is being modified is the characteristic frequency of the wave equation. Here is a simpler one:

This is cast as a variant of a Hill or Mathieu differential equation where the modulation of the characteristic frequency $\omega^2$ is shown in the lower panel. Of course this is a contrived example but it illustrates how a perturbation will speed up or slow down the frequency of the oscillation transiently.

To me, this is a no-brainer to do this kind of analysis. If you ever did audio electronics this is just a LC resonant circuit with a variable capacitor which will create a variable frequency tone.

I didn't mean to make it so complicated, but I have to make it quantitative -- otherwise it is just hand-waving.

`This is just a solution to a second-order differential wave equation. What is being modified is the characteristic frequency of the wave equation. Here is a simpler one: ![qbo hill](http://imageshack.com/a/img540/79/vt2JmA.gif) This is cast as a variant of a Hill or Mathieu differential equation where the modulation of the characteristic frequency $\omega^2$ is shown in the lower panel. Of course this is a contrived example but it illustrates how a perturbation will speed up or slow down the frequency of the oscillation transiently. To me, this is a no-brainer to do this kind of analysis. If you ever did audio electronics this is just a LC resonant circuit with a variable capacitor which will create a variable frequency tone. I didn't mean to make it so complicated, but I have to make it quantitative -- otherwise it is just hand-waving.`

All the variable capacitors I know of are trimmable via mechanical steering, so this is news to me that this is now standard in audio electronics.

`> If you ever did audio electronics this is just a LC resonant circuit with a variable capacitor which will create a variable frequency tone. All the variable capacitors I know of are trimmable via mechanical steering, so this is news to me that this is now standard in audio electronics.`

Of course! This is stuff I did when I was in high school years ago ! In college, I learned about active synthesis which is creating equivalent LC circuits out of OpAmps and RC elements -- using a potentiometer to modulate frequencies. And now everything of course is digitally synthesized. That is no longer audio electronics, it is more like programming,

`> "All the variable capacitors I know of are trimmable via mechanical steering, so this is news to me that this is now standard in audio electronics." Of course! This is stuff I did when I was in high school years ago ! In college, I learned about active synthesis which is creating equivalent LC circuits out of OpAmps and RC elements -- using a potentiometer to modulate frequencies. And now everything of course is digitally synthesized. That is no longer audio electronics, it is more like programming,`

I meant variable capacitors in their actual meaning and not analogs, so I guess I wouldn't have even call a varicap a variable capacitor, but well it seems this is now the convention. Yes I think one can use a potentiometer to modulate frequencies with OpAms and RC elements (my husband does actually some circuits like these) and even alternating voltage sources but whatsoever even if you do frequency modulation and triggering in an LC circuit analog setting I am not sure wether the Mathieu equation models a generic behaviour of such a circuit (in particular if you look at the derivation of the LC circuit equation I see problems) and at a first glance I would think that the Mathieu equation would anyways not model the behaviour I was describing, but I would need to look at this longer and I don't have the time for that.

`>Of course! This is stuff I did when I was in high school years ago ! In college, I learned about active synthesis which is creating equivalent LC circuits out of OpAmps and RC elements -- using a potentiometer to modulate frequencies. I meant variable capacitors in their actual meaning and not analogs, so I guess I wouldn't have even call a varicap a variable capacitor, but well it seems this is now the convention. Yes I think one can use a potentiometer to modulate frequencies with OpAms and RC elements (my husband does actually some circuits like these) and even alternating voltage sources but whatsoever even if you do frequency modulation and triggering in an LC circuit analog setting I am not sure wether the Mathieu equation models a generic behaviour of such a circuit (in particular if you look at the derivation of the <a href="http://en.wikipedia.org/wiki/LC_circuit#Differential_equation">LC circuit equation</a> I see problems) and at a first glance I would think that the Mathieu equation would anyways not model the behaviour I was describing, but I would need to look at this longer and I don't have the time for that.`

nad said:

Why wouldn't it model a frequency modulation ?

The Mathieu equation is actually closely related to the generalized wave equation.

Let's look at the network equations for an LC circuit

$ V(t) =- L \frac{di}{dt} $

and,

$ i(t) = C \frac{dV}{dt} $

But now we consider the case that C is variable with time C(t), which places a frequency modulation on the resonance. Substituting

$ V(t) = -L \frac{d}{dt}({C(t) \frac{dV}{dt}} ) $

via the chain rule this will expand into a regular wave equation term plus a modulation that is quite similar to the Mathieu equation.

$ V(t) = -L \frac{dC(t)}{dt} \frac{dV}{dt} - L C(t) \frac{d^2V}{dt^2} $

It is not exactly the same in terms of identical factors (an extra first-order damping term for one) but close enough for understanding.

I would be tempted to use this as a simple model for ENSO sloshing if the literature didn't actually say that Mathieu is the better model.

EDIT: get the signs right

.

`nad said: > " I would think that the Mathieu equation would anyways not model the behaviour I was describing, but I would need to look at this longer and I don't have the time for that." Why wouldn't it model a frequency modulation ? The Mathieu equation is actually closely related to the generalized wave equation. Let's look at the network equations for an LC circuit $ V(t) =- L \frac{di}{dt} $ and, $ i(t) = C \frac{dV}{dt} $ But now we consider the case that C is variable with time C(t), which places a frequency modulation on the resonance. Substituting $ V(t) = -L \frac{d}{dt}({C(t) \frac{dV}{dt}} ) $ via the chain rule this will expand into a regular wave equation term plus a modulation that is quite similar to the Mathieu equation. $ V(t) = -L \frac{dC(t)}{dt} \frac{dV}{dt} - L C(t) \frac{d^2V}{dt^2} $ It is not exactly the same in terms of identical factors (an extra first-order damping term for one) but close enough for understanding. I would be tempted to use this as a simple model for ENSO sloshing if the literature didn't actually say that Mathieu is the better model. EDIT: get the signs right .`

As far as I remember from high school the law: $ i(t) = C \frac{dV}{dt} $ was observed at a plate capacitor, so I am even not sure wether you can just replace the C by a function, but let's assume one can do that, then your calculation seems right, but now if you look at your equation $ V(t) = -L \frac{dC(t)}{dt} \frac{dV}{dt} - L C(t) \frac{d^2V}{dt^2} $ then you see that the variable capacitance C(t) is in front of the second derivative term, so if like if your capacitance varies with a cosine (which would be in audio so to say what I would call a generic application) then you would have a 1/cosine in front of the V(t) term, which seems quite different from the Mathieu equation. That's what I meant when I said I see problems yesterday, since I did this calculation in my head and on a first glance it doesn't seem to me that one can find a nifty substitution to get your equation (with C(t)=cosine) into the Mathieu equation form. But maybe I oversee something.

`>It is not exactly the same in terms of identical factors (an extra first-order damping term for one) but close enough for understanding. As far as I remember from high school the law: $ i(t) = C \frac{dV}{dt} $ was observed at a plate capacitor, so I am even not sure wether you can just replace the C by a function, but let's assume one can do that, then your calculation seems right, but now if you look at your equation $ V(t) = -L \frac{dC(t)}{dt} \frac{dV}{dt} - L C(t) \frac{d^2V}{dt^2} $ then you see that the variable capacitance C(t) is in front of the second derivative term, so if like if your capacitance varies with a cosine (which would be in audio so to say what I would call a generic application) then you would have a 1/cosine in front of the V(t) term, which seems quite different from the <a href="http://en.wikipedia.org/wiki/Mathieu_function">Mathieu equation</a>. That's what I meant when I said I see problems yesterday, since I did this calculation in my head and on a first glance it doesn't seem to me that one can find a nifty substitution to get your equation (with C(t)=cosine) into the Mathieu equation form. But maybe I oversee something.`

That's what I said. The Mathieu equation is not exactly the same as a variable capacitance resonant equation but it is close.

say that C(t)=C0*(1+a cos(t)). Dividing through by (1+a cos(t)) is close to multiplying by (1-a cos(t)) for small a. And there you have the Mathieu equation, apart from the C' * V' term. That cross-term does not show up in the derivations for sloshing dynamics in any of the papers and books that I have read on the subject. Faltinsen[1] adds a first-order damping term which may override that factor.

[1]O. M. Faltinsen and A. N. Timokha, Sloshing. Cambridge University Press, 2009.

`That's what I said. The Mathieu equation is not exactly the same as a variable capacitance resonant equation but it is close. say that C(t)=C0*(1+a cos(t)). Dividing through by (1+a cos(t)) is close to multiplying by (1-a cos(t)) for small a. And there you have the Mathieu equation, apart from the C' * V' term. That cross-term does not show up in the derivations for sloshing dynamics in any of the papers and books that I have read on the subject. Faltinsen[1] adds a first-order damping term which may override that factor. [1]O. M. Faltinsen and A. N. Timokha, Sloshing. Cambridge University Press, 2009.`

Ok let's look at this $C(t)=C0*(1+a cos t)$. I think you are right that if one does Taylor expansion in a then up to first order $(1/(1+a cos(t)) \simeq 1- a cos(t)$, moreover $$C´(t)/C(t)= -a sin(t)/(1+a cos(t)) \simeq -\frac{sin(t)(1+a cos(t))-a sin(t)cos(t)}{(1+a cos(t))^2}|_{a=0} a = -a sin(t)$$, so one has up to first order in a the equation: $$ (a cos(t)-1) v + a sin (t) v' = v'' $$ So if a is small this is almost a simple oscillator equation and maybe by some continuity argument one can show that the solutions of this equation are close to solutions to the oscillator, but then this is rather more close to a simple oscillator then to a solution of the Mathieu equation (but which of course has the simple oscillator for q small as a kind of limit equation). Moreover you can't even really say that this is a oscillator plus a Mathieulike term because in the end the $v'$ term has to be taken into account. So I see closeness to the simple oscillator but not really to the Mathieu equation.

`>...but it is close. Ok let's look at this $C(t)=C0*(1+a cos t)$. I think you are right that if one does Taylor expansion in a then up to first order $(1/(1+a cos(t)) \simeq 1- a cos(t)$, moreover $$C´(t)/C(t)= -a sin(t)/(1+a cos(t)) \simeq -\frac{sin(t)(1+a cos(t))-a sin(t)cos(t)}{(1+a cos(t))^2}|_{a=0} a = -a sin(t)$$, so one has up to first order in a the equation: $$ (a cos(t)-1) v + a sin (t) v' = v'' $$ So if a is small this is almost a simple oscillator equation and maybe by some continuity argument one can show that the solutions of this equation are close to solutions to the oscillator, but then this is rather more close to a simple oscillator then to a solution of the Mathieu equation (but which of course has the simple oscillator for <a href="http://en.wikipedia.org/wiki/Mathieu_function">q small</a> as a kind of limit equation). Moreover you can't even really say that this is a oscillator plus a Mathieulike term because in the end the $v'$ term has to be taken into account. So I see closeness to the simple oscillator but not really to the Mathieu equation.`

oops forgot to mention that I set $C0$ and $L$ to one for simplicity.

`oops forgot to mention that I set $C0$ and $L$ to one for simplicity.`

It is indeed close to the Mathieu equation. The intent is to demonstrate how a wave equation when modified can produce a frequency modulation. The mechanism behind sloshing is that not only can the forcing function generate the necessary energy at the frequency to trigger a resonance, but that it can also perturb the parameters that define the resonance condition. This has the effect of modifying the resonance condition both subtly or strongly depending on the compliance of the fluid volume. That is the underlying physics basis for the application of the Mathieu equation.

One would think that it would take a massive forcing to get this effetc in motion. But in fact when one is dealing with sloshing with respect to the thermocline, then the small differences in density of water above and below the thermocline become very sensitive to momentum transfer, as the gravitational differences are small between layers of slightly differing density. See this paper:

Valentine, Daniel T., and Jannette B. Frandsen. "Nonlinear Free-Surface and Viscous-Internal Sloshing." Journal of Offshore Mechanics and Arctic Engineering 127.2 (2005): 141-149.

`It is indeed close to the Mathieu equation. The intent is to demonstrate how a wave equation when modified can produce a frequency modulation. The mechanism behind sloshing is that not only can the forcing function generate the necessary energy at the frequency to trigger a resonance, but that it can also perturb the parameters that define the resonance condition. This has the effect of modifying the resonance condition both subtly or strongly depending on the compliance of the fluid volume. That is the underlying physics basis for the application of the Mathieu equation. One would think that it would take a massive forcing to get this effetc in motion. But in fact when one is dealing with sloshing with respect to the thermocline, then the small differences in density of water above and below the thermocline become very sensitive to momentum transfer, as the gravitational differences are small between layers of slightly differing density. See this paper: Valentine, Daniel T., and Jannette B. Frandsen. ["Nonlinear Free-Surface and Viscous-Internal Sloshing."](http://www.researchgate.net/profile/Jannette_Frandsen/publication/236156827_Nonlinear_free-surface_and_viscous-internal_sloshing/links/004635167fb76594af000000.pdf) Journal of Offshore Mechanics and Arctic Engineering 127.2 (2005): 141-149.`

now you are talking again about sloshing, while the previous discussion was about electric circuits. Just to make it clear - I don't want to deny that the Mathieu equation describes some type of movement, which could eventually be seen as some kind of oscillatory behaviour - finally as said a trivial special case of the Mathieu equation is the simple oscillator. The question is however how exactly are these "movements" connected and I tried to explain that I don't see a closeness of this above Ansatz of "L-variable linear capacitor circuits" with the Mathieu Equation. Where at this point one should remark that it is not clear wether the ansatz that one can extend $ i(t) = C \frac{dV}{dt} $ in this way as outlined above is realistic.

Anyways I don't have the time to study articles in fluid dynamics, I just glanced at the paper you cited above and pinpointed the words

which gave me some frowns, as applying finite difference methods to nonlinear equations (and they talk about Bousinesq equations etc.) might not be appropriate. But as said I only glanced at the paper i.e. I even couldn't see in their not easy to read notation wether they actually apply that method. I just wanted to warn you, since I don't know how familiar you are with the discretization of nonlinear equations.

`>It is indeed close to the Mathieu equation. The intent is to demonstrate how a wave equation when modified can produce a frequency modulation. The mechanism behind sloshing is that not only can the forcing function now you are talking again about sloshing, while the previous discussion was about electric circuits. Just to make it clear - I don't want to deny that the Mathieu equation describes some type of movement, which could eventually be seen as some kind of oscillatory behaviour - finally as said a trivial special case of the Mathieu equation is the simple oscillator. The question is however how exactly are these "movements" connected and I tried to explain that I don't see a closeness of this above Ansatz of "L-variable linear capacitor circuits" with the Mathieu Equation. Where at this point one should remark that it is not clear wether the ansatz that one can extend $ i(t) = C \frac{dV}{dt} $ in this way as outlined above is realistic. Anyways I don't have the time to study articles in fluid dynamics, I just glanced at the paper you cited above and pinpointed the words >The method applied to solve the system of Eqs...is the ETUDE finite difference method described in detail by Valentine ... which gave me some frowns, as applying finite difference methods to nonlinear equations (and they talk about Bousinesq equations etc.) might not be appropriate. But as said I only glanced at the paper i.e. I even couldn't see in their not easy to read notation wether they actually apply that method. I just wanted to warn you, since I don't know how familiar you are with the discretization of nonlinear equations.`

The electric/hydrodynamic analogues can be deduced by comparing to the wave equation formulated in this paper by the ENSO researcher Allan Clarke [1]

[1]A. J. Clarke, S. Van Gorder, and G. Colantuono, “Wind stress curl and ENSO discharge/recharge in the equatorial Pacific,” Journal of physical oceanography, vol. 37, no. 4, pp. 1077–1091, 2007.

Look at his equations (4.1), (4.5), and (4.6) where $D$ (the 20°C isotherm depth anomaly) and $T$ (the sea surface temperature anomaly) act as hydrological mathematical analogues for the voltage/current & capacitance/inductance formulation of electric circuits.

$ \frac{\partial D}{\partial t} = -\mu T $

$ \frac{\partial T}{\partial t} = \nu D$

then substituting one into the other, he derives (4.6)

$ T_{tt} + \omega^2 T = 0$

where the

ttsubscript notation is the second derivative with respect to time.To finalize the analogy the resonant condition relates to the LC electrical resonance.

$ \sqrt{\mu \nu} = \omega^2 $

The unpredictability of sloshing is due to the fact that $\omega$ effectively changes cyclically as the forcing changes due to the massive weight of the fluid volume will effectively modify the factors $\nu$ and $\mu$ in some regular pattern.

This effect well known in the sloshing literature, and I am simply applying it to a larger volume than sloshing researchers such as Frandsen and Faltinsen apply it to, which is large fluid volumes held within sea-faring tankers. They are simply applying the math of hydrodynamics to an engineering problem, while I am extending it to understanding the dynamics of ocean sloshing.

I don't think that Clarke (or any other ENSO researcher) has put 2 and 2 together and thought to extend the wave equation to match the rather obvious sloshing dynamics that the hydrological engineers have been applying for several years now.

This is the breakthrough idea that ENSO research needs.

I have seen your concerns echoed by Boeing engineers that work fluid dynamics problems for flight simulators. I just don't buy it. The papers and books by Frandsen and Faltinsen go though the validation of the hydrodynamic formulation against the numerical results. Other researchers further confirmed what they have found. The validation is actually pretty simple. They run their hydrodynamics codes through a numerical solver and find that the standing wave patterns match closely to the Mathieu function. The Mathieu function is the transcendental solution to the Mathieu equation, which is well-documented and found for example in the Mathematica library. This is in fact a perfect way to validate that your numerical integration schemes are solid -- no different than numerically solving the linear wave equation and finding that the result is a Sin or Cos.

Does that address your concerns?

`> "now you are talking again about sloshing, while the previous discussion was about electric circuits. " The electric/hydrodynamic analogues can be deduced by comparing to the wave equation formulated in this paper by the ENSO researcher Allan Clarke [1] [1]A. J. Clarke, S. Van Gorder, and G. Colantuono, [“Wind stress curl and ENSO discharge/recharge in the equatorial Pacific,”](http://yly-mac.gps.caltech.edu/AGU/AGU_2008/Zz_Others/Li_agu08/Clarke2007.pdf) Journal of physical oceanography, vol. 37, no. 4, pp. 1077–1091, 2007. Look at his equations (4.1), (4.5), and (4.6) where $D$ (the 20°C isotherm depth anomaly) and $T$ (the sea surface temperature anomaly) act as hydrological mathematical analogues for the voltage/current & capacitance/inductance formulation of electric circuits. $ \frac{\partial D}{\partial t} = -\mu T $ $ \frac{\partial T}{\partial t} = \nu D$ then substituting one into the other, he derives (4.6) $ T_{tt} + \omega^2 T = 0$ where the *tt* subscript notation is the second derivative with respect to time. To finalize the analogy the resonant condition relates to the LC electrical resonance. $ \sqrt{\mu \nu} = \omega^2 $ The unpredictability of sloshing is due to the fact that $\omega$ effectively changes cyclically as the forcing changes due to the massive weight of the fluid volume will effectively modify the factors $\nu$ and $\mu$ in some regular pattern. This effect well known in the sloshing literature, and I am simply applying it to a larger volume than sloshing researchers such as Frandsen and Faltinsen apply it to, which is large fluid volumes held within sea-faring tankers. They are simply applying the math of hydrodynamics to an engineering problem, while I am extending it to understanding the dynamics of ocean sloshing. I don't think that Clarke (or any other ENSO researcher) has put 2 and 2 together and thought to extend the wave equation to match the rather obvious sloshing dynamics that the hydrological engineers have been applying for several years now. This is the breakthrough idea that ENSO research needs. > "which gave me some frowns, as applying finite difference methods to nonlinear equations (and they talk about Bousinesq equations etc.) might not be appropriate. But as said I only glanced at the paper i.e. I even couldn't see in their not easy to read notation wether they actually apply that method. I just wanted to warn you, since I don't know how familiar you are with the discretization of nonlinear equations." I have seen your concerns echoed by Boeing engineers that work fluid dynamics problems for flight simulators. I just don't buy it. The papers and books by Frandsen and Faltinsen go though the validation of the hydrodynamic formulation against the numerical results. Other researchers further confirmed what they have found. The validation is actually pretty simple. They run their hydrodynamics codes through a numerical solver and find that the standing wave patterns match closely to the Mathieu function. The Mathieu function is the transcendental solution to the Mathieu equation, which is well-documented and found for example in the [Mathematica library](http://mathworld.wolfram.com/MathieuFunction.html). This is in fact a perfect way to validate that your numerical integration schemes are solid -- no different than numerically solving the linear wave equation and finding that the result is a Sin or Cos. Does that address your concerns?`

(4.6) is what I called the simple oscillator.

As said I have no time for studying this, so I didn't look at the article. I hope someone else here on the forum might adress your questions concerning sloshing. Finally I am definitely not an expert for sloshing modelling or modelling in general.

Like intuitively for describing forced oscillations I would have rather started with an ansatz where the periodic forcing is in the constant term, like one learns in school i.e. something like $T_{tt}+\omega 2T= F(t)$, where $F(t)$ is periodic.

`>then substituting one into the other, he derives (4.6) (4.6) is what I called the simple oscillator. As said I have no time for studying this, so I didn't look at the article. I hope someone else here on the forum might adress your questions concerning sloshing. Finally I am definitely not an expert for sloshing modelling or modelling in general. Like intuitively for describing forced oscillations I would have rather started with an ansatz where the periodic forcing is in the constant term, like one learns in school i.e. something like $T_{tt}+\omega 2T= F(t)$, where $F(t)$ is periodic.`

That's what I did in comment #7. I set the forcing terms as annual and biannual sinusoids. That's the starting point for any of this discussion. Without a forcing, it becomes an initial conditions problem. With the forcing, it becomes more of a boundary value problem, where the continuous forcing drives the solution.

`> "Like intuitively for describing forced oscillations I would have rather started with an ansatz where the periodic forcing is in the constant term" That's what I did in comment #7. I set the forcing terms as annual and biannual sinusoids. That's the starting point for any of this discussion. Without a forcing, it becomes an initial conditions problem. With the forcing, it becomes more of a boundary value problem, where the continuous forcing drives the solution.`

But the omega in your $T_{tt}+ω^2T=F(t)$ is not constant and the $F(t)$ is not periodic, as there are e.g. terms $-0.001 t$ and $-0.00007 t^2$ and what you call TSI looks even more strange.

`But the omega in your $T_{tt}+ω^2T=F(t)$ is not constant and the $F(t)$ is not periodic, as there are e.g. terms $-0.001 t$ and $-0.00007 t^2$ and what you call TSI looks even more strange.`

Those are drift terms to compensate for any long term trends, inconsequential really. The TSI is a periodic function that serves to replicate solar/sunspot activity.

So the forcing is essentially periodic, just as I stated when I set up the initial premise.

`> "But the omega in your Ttt+ω2T=F(t) is not constant and the F(t) is not periodic, as there are e.g. terms −0.001t and −0.00007t2 and what you call TSI looks even more strange." Those are drift terms to compensate for any long term trends, inconsequential really. The TSI is a periodic function that serves to replicate solar/sunspot activity. So the forcing is essentially periodic, just as I stated when I set up the initial premise.`

What is SOL ?

`>The TSI is a periodic function that serves to replicate solar/sunspot activity. What is SOL ?`

SOL is set at close to the angular frequency of the average solar cycle -- 10 to 12 years in period. For a more detailed look at the QBO modulation, the analysis of comment #10 indicates that this might be closer to a 15 year period. I am simply trying to characterize the behavior in terms of other known phenomenon.

The solar cycle as it impacts the earth has 3 main periods. (1) a yearly cycle corresponding to seasons (2) a semiannual cycle as the sun crosses the equator twice a year and (3) solar activity corresponding to sunspot cycles. As the QBO is measured higher in the stratosphere, the average cycle transitions from a quasi-periodic 28 months to a strong 6 month period. How does this transition come about is the question. It may be resonantly "pumped" by a combination of these 3 known periods. That's all I was trying to do with the figure in comment #7.

`SOL is set at close to the angular frequency of the average solar cycle -- 10 to 12 years in period. For a more detailed look at the QBO modulation, the analysis of comment #10 indicates that this might be closer to a 15 year period. I am simply trying to characterize the behavior in terms of other known phenomenon. The solar cycle as it impacts the earth has 3 main periods. (1) a yearly cycle corresponding to seasons (2) a semiannual cycle as the sun crosses the equator twice a year and (3) solar activity corresponding to sunspot cycles. As the QBO is measured higher in the stratosphere, the average cycle transitions from a quasi-periodic 28 months to a strong 6 month period. How does this transition come about is the question. It may be resonantly "pumped" by a combination of these 3 known periods. That's all I was trying to do with the figure in comment #7.`