# Estimating Eulerian Energy Spectra from Drifters

## Abstract

**:**

## 1. Introduction

## 2. Spectra and Structure Functions

## 3. Idealized Spectra

## 4. The Nastrom and Gage (1985) Spectrum

## 5. Turbulence Spectra

## 6. Discussion and Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The high pass filter, used in converting the energy spectra function (in

**red**). The filter of Bennett (1984) is shown for comparison (in

**blue**). Also shown is the approximate filter (17) (the dashed curve), based on the asymptotic limits of the full filter.

**Figure 2.**Idealized enstrophy spectra ((

**a**) panel) and the corresponding second-order structure functions ((

**b**) panel). The dashed curve in the right panel was calculated using the approximate filter function in (17) and plotted in Figure 1.

**Figure 3.**The zonal, meridional and total energy spectra extracted from Figure 3 of [18] ((

**a**) panel). The 2D longitudinal structure function calculated from the zonal energy spectrum is shown by the solid curve in the (

**b**) panel. Also shown is the structure function (dotted curve) calculated from data, obtained from Figure 3 of [26]. The dashed horizontal line is the twice the zonal energy, calculated from integrating the zonal spectrum. Reproduced with permission.

**Figure 4.**Streamfunction (

**a**) and particle trajectories (

**b**) from a two-dimensional turbulence simulation forced in the range $K=[30,35]$. The trajectories have been “unwrapped” from the doubly periodic domain, which explains why their extent exceeds the domain scale of $2\pi \times 2\pi $.

**Figure 6.**Kinetic energy spectrum from the turbulence simulation with the reconstruction from the structure function based on (15). The structure function was obtained from the spectrum using (14).

**Figure 7.**The “filter” function used in the spectrum reconstruction in (15) (dashed curve). Also shown is the function used in the alternate reconstruction in (4) (solid curve).

**Figure 8.**As in Figure 6, but with the reconstruction from the structure function based on (4).

**Figure 9.**Kinetic energy spectra using the reconstruction based on (4), for the particle-derived second-order structure function (

**red**dots). Also shown is the reconstruction based on a sixth-order polynomial fit of the structure function (

**yellow**dotted curve).

**Figure 10.**Second-order structure function for the particles with a sixth-order polynomial fit ((

**a**) panel). The two-point velocity correlations calculated using (8) and (13), for the raw particle data (

**red**), from the polynomial fit of ${S}_{2l}$ (

**yellow**) and derived from the theoretical structure function ((

**b**) panel).

**Figure 11.**Spectral reconstructions from a simulation of a 2D inverse cascade, forced in the range $K=[100,120]$. The original spectrum is in

**black**and the reconstruction from the theoretical structure function is in

**red**. The

**green**dots show the spectrum from the raw particle ${S}_{2l}$ and the

**blue**curve shows that obtained with the polynomial fit ${S}_{2l}$.

**Figure 12.**As in Figure 11, but with a simulation of a 2D enstrophy cascade, forced in the range $K=[1,5]$.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

LaCasce, J.H.
Estimating Eulerian Energy Spectra from Drifters. *Fluids* **2016**, *1*, 33.
https://doi.org/10.3390/fluids1040033

**AMA Style**

LaCasce JH.
Estimating Eulerian Energy Spectra from Drifters. *Fluids*. 2016; 1(4):33.
https://doi.org/10.3390/fluids1040033

**Chicago/Turabian Style**

LaCasce, J. H.
2016. "Estimating Eulerian Energy Spectra from Drifters" *Fluids* 1, no. 4: 33.
https://doi.org/10.3390/fluids1040033