Arakawa(1966)'s Jacobian(his eq. 46) is a mean of 3 components: J++(eq.36), J+x(eq.37), and Jx+(eq.38), which have different conservation properties. I am trying to show the difference when using different version of Jacobian in the simulation of two dimensional turbulence, which can be formulated as the stream funciton(\psi)-relative vorticity(\zeta) equation:
\frac{\partial \psi}{\partial \t}=-J(\psi,\zeta)
The model was initialized with \zeta(\psi), such that the relative vorticity is a function of the stream funciton and J(\psi,\zeta)=0, therefore the flow field does not change with time. My result shows that energy/enstrophy remains almost conserved for a while but blows up anyway(for the accumulation of round-off error?), no matter which Jacobian implementation was employed, I am confused.
The Fourier-spectral method was emplyed for spatial differencing( so continuous form of J++ , J+x , and Jx+ were used in the code) and I know that the conservation property of Arakawa Jacobian, 1/3*( J++ + J+x + Jx+), should be independent of the sptial differencing technique. So I supposed that 1/3*( J++ + J+x + Jx+) should be better than using any of the 3 alone, in terms of energy/enstrophy conservation. However little difference was observed when different initial condition was used. In the simulation, horizontal diffusion in contained in a wavenumber filter to remove the shot waves beyond 2/3 of the maximum wavenumber. Highly possiblely it is an uneducated question, but would be appreciated if someone can explain.
I never worked directly using the Arakawa scheme, however I know that it was developed in the framework of a specific finite difference scheme in such a way to imply conservation of kinetic energy. I see you are using a spectral method in space and that makes me dubious about the similar property of the resulting method. Unfortunately, I have no experience on this and I can just suppose some check:
1) perform the spectra of the velocity/vorticity field, if you have a numerical instability it should appear as a pile-up at wavenumbers close to the cut-off.
2) Check the solution with and without de-aliasing.
3) Check the solution for different time integration methods.
4) what about the grid resolution in terms of the physical viscosity?
I am not sure we are understanding each other, the conservation property is expressed in discrete meaning. What is fulfilled for a certain FD formula is not fulfilled for a finite spectral representation. That does not depend on the order of accuracy.
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