> "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?

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?