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Abstract: We introduce a new method, the piecewise Reynolds mean (PREM), for decomposing the flow velocity into the mean-flow and eddy-flow parts in the time domain for subsequent calculation of the mean and eddy kinetic energies (MKE and EKE). Compared with conventional methods like the Reynolds mean (REM) and running mean (RUM), PREM has the advantage of exact balance between the MKE and EKE, without the additional residual kinetic energy (RKE), while retaining time-dependent mean-flow. It is mathematically simple and computationally lightweight, depending on a pre-defined separation scale for the mean-flow and eddies. Based on satellite observations and the separation scale of 1 year, we compare PREM with RUM, as well as another newly proposed method, the eddy detection and extraction (EDEX). The latter is based on objective identification of mesoscale eddies and eddy anomaly extraction algorithms, and is therefore only suitable for mesoscale eddy energetics, but independent of separation scales. It is shown that compared with RUM, PREM gives larger mean EKE and stronger interannual variability. In strong-current and eddy-rich regions, the two methods differ the most (max: Kuroshio Extension, RMSD = 60.3 J/m3); but in areas with weak current and eddy, the difference accounts for the largest fraction of total EKE (max: south of the Aleutian Islands, 208%). EKE estimated by the two methods is out of phase (min correlation = 0.38). The mean EKE and standard deviation from the EDEX method resemble the PREM with 1-year separation scale, but is generally smaller in magnitude.
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Key words:
- mesoscale eddy /
- kinetic energy /
- piecewise Reynolds mean
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Figure 1. Schematic diagram of the PREM (a) and RUM methods (b). Red curves are the original time series of velocity, and blue cures are the running-averaged velocity. Black bars with dots denote the perturbation velocity (difference between original and running-average) used for calculating the EKE within each window (in RUM the window size is unity).
Figure 2. Comparison of 28-year averaged energy terms (J/m3) and conversion rate (J/m3/d) estimated by the PREM and RUM methods. a. MKE of the PREM method; b. MKE difference of PREM-REM; c. EKE of the PREM method; d. EKE difference of PREM-REM; e. RKE of the RUM method; f. Ratio of RKE/EKE of the RUM method; g. BTR of the PREM method]; h. BTR difference of PREM-REM.
Figure 3. Comparison of interannual standard deviation of energy terms (J/m3) and conversion rate (J/m3/d) estimated by the PREM and RUM methods. a. MKE of the PREM method; b. MKE difference of PREM-REM; c. EKE of the PREM method; d. EKE difference of PREM-REM; e. RKE of the RUM method; f. ratio of RKE/EKE of the RUM method; g. BTR of the PREM method; h. BTR difference of PREM-REM.
Figure 4. Root-mean-square (RMS) difference between the energy terms (J/m3) and conversion rate (J/m3/d) estimated by the PREM and RUM methods. Shown areRMS difference (left column); RMS difference (right column) divided by standard deviation of the energy term estimated by the RUM method. The energy terms are MKE (top row), EKE (middle row), and BTR (bottom row).
Figure 5. Correlation between the energy terms (J/m3) and conversion rate (J/m3/d) estimated by the PREM and RUM methods. The energy terms are MKE (a), EKE (b), and BTR (c).
Figure 6. 28-year averaged EKE (J/m3) estimated by the EDEX method (a), fraction of 28-year averaged EKE of the EDEX method relative to the PREM method (b), interannual standard deviation of EKE of the EDEX method (c), fraction of interannual standard deviation of the EDEX method relative to the PREM method (d), and correlation of EKE of the EDEX and PREM methods (e).
Table 1. Formulae of the mean energy terms and conversion rate using the REM, PREM, and RUM methods
Energy terms Reynolds mean (REM) Piecewise Reynolds mean (PREM) Running mean (RUM) $ \dfrac{2}{{\rho }_{0}}\mathrm{T}\mathrm{K}\mathrm{E} $ $ \left[{u}^{2}\right]+\left[{v}^{2}\right] $ $ \langle{u}^{2}\rangle+\left\langle{{v}^{2}}\right\rangle $ $ \langle{u}^{2}\rangle+\left\langle{{v}^{2}}\right\rangle $ $ \dfrac{2}{{\rho }_{0}}\mathrm{M}\mathrm{K}\mathrm{E} $ $ {\left[u\right]}^{2}+{\left[v\right]}^{2} $ $ {\langle u\rangle}^{2}+{\left\langle{v}\right\rangle}^{2} $ $ \langle{\langle u\rangle}^{2}\rangle+\langle{\langle v\rangle}^{2}\rangle $ $ \dfrac{2}{{\rho }_{0}}\mathrm{E}\mathrm{K}\mathrm{E} $ $ \left[{u}^{2}\right]-{\left[u\right]}^{2}+\left[{v}^{2}\right]-{\left[v\right]}^{2} $ $ \langle{u}^{2}\rangle-{\langle u\rangle}^{2}+\langle{v}^{2}\rangle-{\langle v\rangle}^{2} $ $ \langle{u}^{2}\rangle+\langle{\langle u\rangle}^{2}\rangle-2\langle u\langle u\rangle\rangle+ $$ \langle{v}^{2}\rangle+\langle{\langle v\rangle}^{2}\rangle-2\langle v\langle v\rangle\rangle $ $ \dfrac{1}{{\rho }_{0}}\mathrm{R}\mathrm{K}\mathrm{E} $ 0 0 $ \langle u\langle u\rangle\rangle-\langle{\langle u\rangle}^{2}\rangle+ $$ \langle v\langle v\rangle\rangle-\langle{\langle v\rangle}^{2}\rangle $ $ \dfrac{1}{{\rho }_{0}}\mathrm{B}\mathrm{T}\mathrm{R} $ $ \left(\left[{u}^{2}\right]-{\left[u\right]}^{2}\right)\dfrac{\partial \left[u\right]}{\partial x}+ $
$ \left(\left[uv\right]-\left[u\right]\left[v\right]\right)\left(\dfrac{\partial \left[u\right]}{\partial y}+\dfrac{\partial \left[v\right]}{\partial x}\right)+ $
$ \left(\left[{v}^{2}\right]-{\left[v\right]}^{2}\right)\dfrac{\partial \left[v\right]}{\partial y} $$ \left(\left\langle{{u}^{2}}\right\rangle-{\left\langle{u}\right\rangle}^{2}\right)\dfrac{\partial \left\langle{u}\right\rangle}{\partial x}+ $
$ \left(\left\langle{uv}\right\rangle-\left\langle{u}\right\rangle\left\langle{v}\right\rangle\right)\left(\dfrac{\partial \left\langle{u}\right\rangle}{\partial y}+\dfrac{\partial \left\langle{v}\right\rangle}{\partial x}\right)+ $
$ \left(\left\langle{{v}^{2}}\right\rangle-{\left\langle{v}\right\rangle}^{2}\right)\dfrac{\partial \left\langle{v}\right\rangle}{\partial y} $$ \left\langle{\left\langle{{\left(u-\left\langle{u}\right\rangle\right)}^{2}}\right\rangle\dfrac{\partial \left\langle{u}\right\rangle}{\partial x}}\right\rangle+ $
$ \left\langle{\left\langle{\left(u-\left\langle{u}\right\rangle\right)\left(v-\left\langle{v}\right\rangle\right)}\right\rangle\dfrac{\partial \left\langle{u}\right\rangle}{\partial y}}\right\rangle+ $
$ \left\langle{\left\langle{\left(u-\left\langle{u}\right\rangle\right)\left(v-\left\langle{v}\right\rangle\right)}\right\rangle\dfrac{\partial \left\langle{v}\right\rangle}{\partial x}}\right\rangle+ $
$ \left\langle{\left\langle{{\left(v-\left\langle{v}\right\rangle\right)}^{2}}\right\rangle\dfrac{\partial \left\langle{v}\right\rangle}{\partial y}}\right\rangle $Note: The terms in the first column are averaged over the entire record in REM and are therefore time-independent (i.e., the [] operator is applied); and running-aveaged over a moving window of a pre-defined length in PREM and RUM (i.e., the $\langle \rangle $ operator is applied). 
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Table 2. Comparison between EKE estimated by the PREM and RUM methods in various regions shown in Fig. 7
Region MEAN STD RMSD MIN
CORRABS REL ABS REL ABS REL SPNP 1.38 0.44 1.63 1.81 2.47 2.08 0.38 KE 33.13 0.33 36.7 1.08 60.30 1.54 0.66 EWP 7.26 0.25 13.36 1.59 17.57 1.90 0.52 ESI 3.92 0.17 5.80 0.90 8.01 1.18 0.73 GS 17.69 0.16 17.35 0.49 33.52 0.87 0.69 SPNA 2.53 0.30 2.88 0.67 4.70 1.09 0.55 SPSA 1.51 0.33 2.17 1.08 3.23 1.55 0.62 ARC 16.93 0.15 16.59 0.55 29.82 0.90 0.73 BC 7.38 0.12 6.36 0.42 16.94 1.19 0.47 EA 6.46 0.05 7.49 0.19 15.74 0.49 0.85 SC 5.49 0.06 4.79 0.26 16.34 0.54 0.84 SPSP 3.56 0.26 3.58 0.61 6.42 0.88 0.73 Note: The compared quantities include: difference of time-mean EKE (MEAN), difference of interannual standard deviation (STD), and root-mean-square difference (RMSD), each of these are presented in terms of the absolute difference (ABS) in J/m3, and the relative difference (REL). MEAN REL is relative to the mean EKE of RUM. STD REL and RMSD REL are relative to the standard deviation of RUM. The numbers are the maximum values in each region. The last column shows the minimal correlation coefficient (MIN CORR) between PREM and RUM EKEs.
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