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Energy conservation in mapped reference frame: Nonlinear Schrodinger equation

+ 2 like - 0 dislike
81 views

This question might be a bit off topic for this forum. If so, let me know and I will remove it. 

$\textbf{General overview}$: I am studying surface gravity waves and am mapping reference frames, from a laboratory frame to a so called fetch frame, and am finding that energy is not conserved in the new frame. My question is, is there some form of work in the new reference frame that I am overlooking, that could lead to this discrepancy? Or perhaps I'm missing something else. 

Now the details: This question is about 2 dimensional deep-water irrotational inviscid surface gravity waves, where $x$ is the direction of wave propagation, and $z$ depth. These waves obey the laplacian in the interior, i.e. 

$$\nabla^2 \phi =0,$$

with $\phi$ the velocity potential. The boundary conditions are 

$$\eta_t+\phi_x\eta_x =\phi_z; \quad \phi_t+\frac{1}{2}(\nabla \phi)^2 + gz=0; \quad @z=\eta$$

with $\eta$ the free surface displacement. Finally, we have no flow at the bottom, i.e $\phi_z = 0 \ @z=-h$ with the depth h taken to infinity. There are several integrals of these equations (Whitham 1962, Benjamin and Olver 1982). Let's consider one, namely the energy. It can be shown that 

$$\frac{\partial}{\partial t} \left(\int_{-h}^{\eta} \frac{1}{2}(\nabla \phi)^2 \ dz +\frac{1}{2}g \eta^2 \right)+\frac{\partial}{\partial x} \left(\int_{-h}^{\eta}\phi_x(p+\frac{1}{2}(\nabla \phi)^2 + gz) \ dz  \right)=0$$

Now, due to the complexity in the equations, we consider an asymptotic model, namely, the nonlinear Schrodinger equation (NLSE). This is a weakly nonlinear narrow-banded asymptotic model, and can be derived from the governing equations in a variety of ways. The most straightforward is to take an asymptotic series of $(\eta,\phi)$, in the small parameter $\epsilon =ak$, the slope of the waves, i.e. 

$$\eta =\bar{\eta} +\frac{1}{2}\sum_{n,m: n\ge m}^N \eta_{nm} \epsilon^n e^{i m\theta}+c.c; \quad \phi =\bar{\phi} +\frac{1}{2}\sum_{n,m: n\ge m}^N \phi_{nm} \epsilon^n e^{i m\theta}e^{mz}+c.c.$$

where $\theta =x-t$ (we're going to nondimensionalize space by $k$ the wavenumber, time by $\omega$ with $\omega^2 = gk$), and the coefficients $\eta_{nm},\phi_{nm}$ are slow functions of space and time, with overbars representing the phase average of a quantity and c.c. denoting complex conjugate. We let $N=3$, that is we want to derive equations accurate to order 3 and finally let $\phi_{11}=A$. We can now find the other coefficients, as a function of $A$. Grinding through quite a bit of algebra, one finds

$$\phi =\epsilon^2\bar{\phi}+\frac{1}{2}([\epsilon A-iz\epsilon^2 A_x -\left(\frac{1}{16}+\frac{z^2}{2}\right) \epsilon^3 A_{xx} -\frac{1}{2}|A|^2A]e^{i\theta}e^z +c.c)$$

$$\eta =\epsilon^3 \bar{\eta}+\frac{1}{2}\left([i\epsilon A+\frac{\epsilon^2}{2}A_x+\frac{i}{16}\epsilon^3 A_{xx}-\frac{3i}{8}\epsilon^3 |A|^2A]e^{i\theta} +\left[-\frac{\epsilon^2}{2}A^2 +i\epsilon^3 AA_{x}\right]e^{2i\theta}\right.$$

$$\left.-\frac{3i}{8}\epsilon^3 A^3e^{3i\theta} +c.c.\right),$$

$$\left. \epsilon^3 \phi_z\right|_{z=0} = \frac{\epsilon^3}{2}\frac{\partial |A|^2}{\partial x}; \quad \quad \epsilon^3 \bar{\eta}+\epsilon^3 \left.\bar{\phi}_t\right|_{z=0}=0$$

$$\epsilon^2 A_t+\epsilon^2 \frac{1}{2}A_x +\epsilon^3 \frac{i}{8}A_{xx} +\epsilon^3 \frac{i}{2}|A|^2A =0$$

The last equation is the nonlinear Schrodinger equation (NLSE) for water waves. Now, this equation has an infinite number of conserved integrals. The three of interest here are the simplest:

$$I_1= \int\epsilon^2 |A|^2 \ dx; \quad I_2 =i\int\epsilon^3 (AA^*_x-A^*A_x) dx; \quad I_3 = \epsilon^4\int |A|^4-\frac{1}{2}|A_x|^2 \ dx,$$

where the integrals are taken over all of space (and the wave envelopes A are taken to be compact through this discussion). 

Now, the energy I would like to consider is that corresponding to the phase averaged quantities, and will be defined as 

$$E =\int \frac{1}{2\pi}\int_0^{2\pi}\left(\int_{-h}^{\eta} \frac{1}{2}(\nabla \phi)^2 \ dz +\frac{1}{2} \eta^2 \right)\ d\theta \ dx $$

Putting in the expansions for $(\phi,\eta)$ I find 

$$E= \frac{1}{2}I_o+\frac{1}{4}I_1+\frac{1}{8} I_2$$

which is exactly conserved. 

Now, for easiest comparison of the predictions of this equation with laboratory experiments, one maps into so called fetch coordinates, given by $\chi = x$, $\tau =2x-t$. For fixed $x$,  $\tau$ goes like minus time. 

In these coordinates, the NLSE becomes 

$$\epsilon^3(A_{\chi}+iA_{\tau \tau}+i|A|^2A)=0$$

and the integrals $I_o,I_1$ stay the same (with $x\to \tau$, and we denote these with a prime) while 

$$I_2 \to I_2'=\epsilon^4 \int \frac{1}{2}|A|^4-|A_{\tau}|^2 d \tau.$$

However, when I compute the energy in this new reference frame (call it E'), I find (either by transforming $E$ given above, or by transforming $(\phi,\eta)$ and then the energy integral)

$$E' = \frac{1}{2}I_o'+\frac{1}{2}I_1'+\frac{1}{4}I_2'+i \int (AA^*_{\chi}-A^*A_{\chi}) d \tau $$

This last term is my source of confusion. If one substitutes in the NLSE, then they see $E'$ is not conserved. This makes me think that there is some form of work being done in this reference frame that I did not account for, but I have not been able to figure out where it could be coming from. Note, a simple test example is the compact soliton solution $A= sech \sqrt{2} \tau e^{-i\chi}$. Clearly, in this example, the energy should be conserved regardless of whether one is considering canonical time evolution equations, or the spatial evolution equations. 

Does anyone have any comments or suggestions? Also, please let me know if something needs to be clarified. 

asked Aug 16, 2015 in Theoretical Physics by Nick Pizzo [ no revision ]
recategorized Aug 16, 2015 by dimension10

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