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Fluctuations -> Midplane data -> correlation functions

General Description

Plots toroidal angle and time-averaged spatial correlations at the outboard midplane. Using the definition

\delta f_{mid}(r,\varphi,t) =\sum_{n=-N}^N f_n(r,\theta=0,t) e^{-i n \varphi}

It is straightforward to show

\begin{matrix} C(r_0, \Delta r, \Delta \varphi) & = & \frac{<\delta f_{mid}(r_0,\varphi,t) \delta f_{mid}(r_0 + \Delta r,\varphi + \Delta \varphi,t)>_{\varphi,t}} {<\delta f_{mid}^2(r_0,\varphi,t)>_{\varphi,t} } \\ \ & = & \frac{Re(<f_0(r_0,t)f_0(r_0 + \Delta r,t>_t + 2 \sum_{n=1}^N <f_{-n}(r_0,t) f_n(r_0+\Delta r, t>_t)} {<f_0^2(r_0,t)>_t + 2 \sum_{n=1}^N <|f_n(r_0,t)|^2>_t } \end{matrix}

One can plot C(\Delta r, \Delta \varphi = 0), C(\Delta r = 0, \Delta \varphi ), and C(\Delta r, \Delta \varphi) using this widget, as well as the "box-averaged" value <C>_{r_0} for any of these quantities. When plotting, C(Δr) or C(\Delta \varphi )an exponential of the form exp( − | Δx | / lx) is fit to the correlation function and overplot as a dashed red curve. The value of lx is given in the top-left corner of the plot. The option for fitting a Gaussian rather than pure exponential will be added in a future release.


Note: becasue of an IDL bug, the colorbar on the right of the C(\Delta r, \Delta \varphi) plot is disabled by default. This bug does not appear to be a problem when running IDL in OS X, so the colorbar can be easily resotred by "uncommenting" the relevant code.


Sample Images

Image:Midplane rcorr.tiff Image:Midplane alphacorr.tiff Image:Midplane ralphacorr.tiff

Button Definitions

Plot 
Generate/regenerate plot
C(dr,da) droplist 
choose between C(\Delta r, \Delta \varphi = 0), C(\Delta r = 0, \Delta \varphi ), and C(\Delta r, \Delta \varphi)
Local/box-avg
Toggle whether to plot C(r_0, \Delta r, \Delta \varphi)or <C(\Delta r, \Delta \varphi)>_{r_0}
Finite-n only 
Toggle whether to include n = 0 fluctuations in calculation. Note that the autocorrelation function is usually defined with the mean value subtracted from the fluctuating quantity. Since the averaging includes toroidal angle as well as time, this would remove the n = 0 components from the field. This toggle allows the n = 0 components to be included (with their time average removed) in the calculation. No time averaging is done for the finite-n components (which should have very small means because of their finite real frequencies, but should probably update code to do this for completeness).
index up 
Cycles up through all possible fluctuation fields (electrostatic potential, ion and electron density, energy, and temperature)
index down 
Cycles down the fluctuation fields
i_r0++
increases value of r0 used in calculation
i_r0--
decreases value of r0 used in calculation
PS dump 
Create postscript file from screen image
Write to .sav file
Dump data to IDL savefile
Write to text file
Dump data to ASCII file
Done 
Kill plot window
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