DOI: 10.5061/dryad.70rxwdcc9
title: Data from: A method of separating linear internal wave and vortical mode energies using shipboard ADCP velocity measurements

description: We present a method to quantify total, horizontal kinetic and available
 potential energies of linear internal waves (IW) and vortical mode (VM)
 using only two-dimensional (2D) (depth, along-track distance) measurements
 of horizontal velocity, such as those commonly taken by oceanic shipboard
 ADCP (SADCP). Previous IW and VM energy decomposition methods (Bühler et
 al. 2014, 2017, and their extensions) require both velocity and buoyancy
 measurements. Applying Helmholtz decomposition, 2D horizontal kinetic
 energy wavenumber (kx , kz) spectra are projected onto divergent Kdiv and
 rotational Krot components. IW total energy spectrum is EIW = 2Kdiv . VM
 total energy is EVM=1/Bu[(1+Bu)Krot-Kdiv], where Bu is Burger number with
 N and f buoyancy and inertial frequencies, kh horizontal wavenumber
 magnitude and kz vertical wavenumber. IW and VM horizontal kinetic energy
 (K) and available potential energy (P) can be inferred from EVM and EIW as
 functions of Bu. The proposed method, derived directly from IW and VM
 theoretical polarization relations, is demonstrated using two sets of
 velocity and density data. EVM derived by this new method agrees with
 results computed using the Bühler et al. (2014, 2017) method at Bu~O(1)
 and within a factor of ~ 2–3 elsewhere, confirming that IW and VM energy
 can be separated using only velocity data. At Bu ≪ O (0.1), EVM is
 dominated by PVM, with KVM/PVM=Bu, and using K alone to extract EVM
 through the proposed method is challenging due to inherent uncertainty in
 spectral measurements. This method could be applied to global SADCP
 datasets to separate upper ocean IW and VM energy contributions in
 different dynamical regions at horizontal scales O (100m) – O (100km) and
 vertical scales O (10m) – O (100m).
description: # Data from: A method of separating linear internal wave and vortical mode
 energies using shipboard ADCP velocity measurements Dataset DOI:
 [10.5061/dryad.70rxwdcc9](https://doi.org/10.5061/dryad.70rxwdcc9) ##
 Description of the data and file structure Data to accompany the article
 "A Method of Separating Linear Internal Wave and Vortical Mode
 Energies Using Shipboard ADCP Velocity Measurements" (Journal of
 Atmospheric and Oceanic Technology, DOI: 10.1175/JTECH-D-25-0076.1) by
 Anda Vladoiu and Ren-Chieh Lien. ### File: SupportingData.zip
 **Description:** The .zip archive contains .mat processed data files used
 in creating the figures in the article. Please see article for detailed
 data description and processing methods. Descriptions of each .mat file,
 corresponding to distinct figures, are below, with units in square
 brackets. Each file contains structures pertaining to variables plotted in
 each figure panel, named in alphabetical order. Contact Anda Vladoiu
 ([avladoiu@uw.edu](mailto:avladoiu@uw.edu)) for additional information.
 #### file: Fig1.mat **Fig. 1.** Model (k~h~, k~z~) spectra for IW based on
 GM IW model spectra: (a) K^IW^~div~, (b) K^IW^~rot~, (c) P^IW^, and (d)
 E^IW^=K^IW^~div~ +K^IW^~rot~ +P^IW^, and for VM based on mean, smoothed
 spectrum from Vladoiu et al. (2024): (f) K^VM^~rot~ , (f) P^VM^, and (g)
 E^VM^=K^VM^~div~+K^VM^~rot~+P^VM^, where K^VM^~div~=0. (h) Total inferred
 K~div~=K^IW^~div~+K^VM^~div~ , (i) total K~rot~=K^IW^~rot~+K^VM^~rot , and
 (j) the total energy of VM derived from the K energy decomposition method
 E~*= (1/Bu)[(1 + Bu)K~rot~ - K~div~]. (k) K~div^~^~ and (l) K~rot^~^~ are
 realizations of K~div~ and K~rot~ spectra following chi^2^ distribution
 with 20 DOF. (m) E^VM~~~^ is the VM total energy derived from the K energy
 decomposition method using K~div^~^~ and K~rot^~^~. **Structure
 "Fig1a" contains variables:** "kh" = horizontal
 wavenumber magnitude [cpm] "kz" = vertical wavenumber [cpm]
 "KdivIW" = log10 internal wave divergent component of kinetic
 energy 2D spectrum [m^2/s^2/cpm^2] **Structure "Fig1b" contains
 variables:** "kh" = horizontal wavenumber magnitude [cpm]
 "kz" = vertical wavenumber [cpm] "KrotIW" = log10
 internal wave rotational component of kinetic energy 2D spectrum
 [m^2/s^2/cpm^2] **Structure "Fig1c" contains variables:**
 "kh" = horizontal wavenumber magnitude [cpm] "kz" =
 vertical wavenumber [cpm] "PIW" = log10 internal wave potential
 energy 2D spectrum [m^2/s^2/cpm^2] **Structure "Fig1d" contains
 variables:** "kh" = horizontal wavenumber magnitude [cpm]
 "kz" = vertical wavenumber [cpm] "EIW" = log10
 internal wave total energy 2D spectrum [m^2/s^2/cpm^2] **Structure
 "Fig1e" contains variables:** "kh" = horizontal
 wavenumber magnitude [cpm] "kz" = vertical wavenumber [cpm]
 "KrotVM" = log10 vortical mode rotational component of kinetic
 energy 2D spectrum [m^2/s^2/cpm^2] **Structure "Fig1f" contains
 variables:** "kh" = horizontal wavenumber magnitude [cpm]
 "kz" = vertical wavenumber [cpm] "PVM" = log10
 vortical mode potential energy 2D spectrum [m^2/s^2/cpm^2] **Structure
 "Fig1g" contains variables:** "kh" = horizontal
 wavenumber magnitude [cpm] "kz" = vertical wavenumber [cpm]
 "EVM" = log10 vortical mode total energy 2D spectrum
 [m^2/s^2/cpm^2] **Structure "Fig1h" contains variables:**
 "kh" = horizontal wavenumber magnitude [cpm] "kz" =
 vertical wavenumber [cpm] "Kdiv" = log10 total divergent kinetic
 energy 2D spectrum [m^2/s^2/cpm^2] **Structure "Fig1i" contains
 variables:** "kh" = horizontal wavenumber magnitude [cpm]
 "kz" = vertical wavenumber [cpm] "Krot" = log10 total
 rotational kinetic energy 2D spectrum [m^2/s^2/cpm^2] **Structure
 "Fig1j" contains variables:** "kh" = horizontal
 wavenumber magnitude [cpm] "kz" = vertical wavenumber [cpm]
 "EVMstar" = log10 vortical mode total energy 2D spectrum,
 inferred from the K energy decomposition method [m^2/s^2/cpm^2]
 **Structure "Fig1k" contains variables:** "kh" =
 horizontal wavenumber magnitude [cpm] "kz" = vertical wavenumber
 [cpm] "Kdivtilde" = log10 total divergent kinetic energy 2D
 spectrum with chi-squared uncertainty [m^2/s^2/cpm^2] **Structure
 "Fig1l" contains variables:** "kh" = horizontal
 wavenumber magnitude [cpm] "kz" = vertical wavenumber [cpm]
 "Krottilde" = log10 total rotational kinetic energy 2D spectrum
 with chi-squared uncertainty [m^2/s^2/cpm^2] **Structure "Fig1j"
 contains variables:** "kh" = horizontal wavenumber magnitude
 [cpm] "kz" = vertical wavenumber [cpm] "EVMstartilde"
 = log10 vortical mode total energy 2D spectrum, inferred from the K energy
 decomposition method with chi-squared uncertainty [m^2/s^2/cpm^2] ####
 file: Fig2.mat **Fig. 2.** Dependence on Bu of spectra shown in Fig. 1 for
 (a) total IW and VM energy, (b) IW and VM horizontal kinetic energy, and
 (c) IW and VM available potential energy. Curves represent spectra
 averaged over Burger number bins. **Structure "Fig2a" contains
 variables:** "Bu" = Burger number "EIW" = log10
 internal wave total energy 2D spectrum [m^2/s^2/cpm^2] "EVM" =
 log10 vortical mode total energy 2D spectrum [m^2/s^2/cpm^2]
 "EVMstar" = log10 vortical mode total energy 2D spectrum,
 inferred from the K energy decomposition method [m^2/s^2/cpm^2]
 "EVMstartilde" = log10 vortical mode total energy 2D spectrum,
 inferred from the K energy decomposition method with chi-squared
 uncertainty [m^2/s^2/cpm^2] "BuPDF" = Burger number
 "PDF" = PDF for all resolved (kh,kz)
 "PDF_positiveEVMstartilde" = PDF for (kh,kz) where EVMstartilde
 &gt; 0 **Structure "Fig2b" contains variables:** "Bu"
 = Burger number "KIW" = log10 internal wave kinetic energy 2D
 spectrum [m^2/s^2/cpm^2] "KVM" = log10 vortical mode kinetic
 energy 2D spectrum [m^2/s^2/cpm^2] "KVMstar" = log10 vortical
 mode kinetic energy 2D spectrum, inferred from the K energy decomposition
 method [m^2/s^2/cpm^2] "KVMstartilde" = log10 vortical mode
 kinetic energy 2D spectrum, inferred from the K energy decomposition
 method with chi-squared uncertainty [m^2/s^2/cpm^2] **Structure
 "Fig2c" contains variables:** "Bu" = Burger number
 "PIW" = log10 internal wave potential energy 2D spectrum
 [m^2/s^2/cpm^2] "PVM" = log10 vortical mode potential energy 2D
 spectrum [m^2/s^2/cpm^2] "PVMstar" = log10 vortical mode
 potential energy 2D spectrum, inferred from the K energy decomposition
 method [m^2/s^2/cpm^2] "PVMstartilde" = log10 vortical mode
 potential energy 2D spectrum, inferred from the K energy decomposition
 method with chi-squared uncertainty [m^2/s^2/cpm^2] #### file: Fig3.mat
 **Fig. 3.** FIG. 3. Ratio between Burger number bin-averaged VM energy
 (Bu) from K~div^~^~ and K~rot^~^~ with chi^2^ spectral uncertainty and VM
 energy (Bu) from K~div~ and K~rot~ without the uncertainty for DOF between
 4 and 120, assuming each 2D spectral estimate has 4 degrees of freedom.
 **Structure "Fig3" contains variables:** "Bu" = Burger
 number "ratio" = ratio of 2D spectra EVMstartilde/EVMstar
 "DOF" = degrees of freedom #### file: Fig4.mat **Fig.
 4.** Location, date, instruments providing velocity and density
 measurements, depth range, resolved horizontal and vertical wavelengths,
 number of sections, mean N and f, and trajectories (colors from blue to
 yellow indicate the time elapsed, and black circles indicate the beginning
 of each section) for the (a) Sargasso Sea and (b) eastern North Pacific
 datasets. **Structure "Fig4a" contains variables:**
 "lon" = longitude [degW] for Sargasso Sea sections
 "lat" = latitude [degN] for Sargasso Sea sections **Structure
 "Fig4b" contains variables:** "lon" = longitude [degW]
 for eastern North Pacific sections "lat" = latitude [degN] for
 eastern North Pacific sections "time" = Matlab time [days] ####
 file: Fig5.mat **Fig. 5.** Mean eastern North Pacific (k~h~, k~z~)
 spectrum for (a) E^VM^ derived from the K-P energy decomposition using
 both velocity and density measurements and (b) E^VM^~*~ derived from the K
 energy decomposition using velocity measurements only. (c) Ratio of the
 two VM energy estimates and (d) their scatterplot. **Structure
 "Fig5a" contains variables:** "kh" = horizontal
 wavenumber magnitude [cpm] "kz" = vertical wavenumber [cpm]
 "EVM" = log10 vortical mode total energy 2D spectrum
 [m^2/s^2/cpm^2] **Structure "Fig5b" contains variables:**
 "kh" = horizontal wavenumber magnitude [cpm] "kz" =
 vertical wavenumber [cpm] "EVMstar" = log10 vortical mode total
 energy 2D spectrum, inferred from the K energy decomposition method
 [m^2/s^2/cpm^2] **Structure "Fig5c" contains variables:**
 "kh" = horizontal wavenumber magnitude [cpm] "kz" =
 vertical wavenumber [cpm] "ratio" = log10 ratio of vortical mode
 total energy 2D spectrum inferred from the K-P energy decomposition method
 to the vortical mode total energy 2D spectrum inferred from the K energy
 decomposition method **Structure "Fig5d" contains variables:**
 "EVM" = vortical mode total energy 2D spectrum [m^2/s^2/cpm^2]
 "EVMstar" = vortical mode total energy 2D spectrum, inferred
 from the K energy decomposition method [m^2/s^2/cpm^2] "Bu" =
 log10 Burger number #### file: Fig6.mat **Fig. 6.** Dependence on Bu of
 mean eastern North Pacific (k~h~, k~z~) spectrum for (a) total IW and VM
 energy, (b) IW and VM horizontal kinetic energy, and (c) IW and VM
 available potential energy. Curves represent spectra averaged over Burger
 number bins. **Structure "Fig6a" contains variables:**
 "Bu" = Burger number "EIW" = log10 internal wave total
 energy 2D spectrum [m^2/s^2/cpm^2] "EVM" = log10 vortical mode
 total energy 2D spectrum [m^2/s^2/cpm^2] "EVMstar" = log10
 vortical mode total energy 2D spectrum, inferred from the K energy
 decomposition method [m^2/s^2/cpm^2] "BuPDF" = Burger number
 "PDF" = PDF for all resolved (kh,kz)
 "PDF_positiveEVMstartilde" = PDF for (kh,kz) where EVMstartilde
 &gt; 0 **Structure "Fig6b" contains variables:** "Bu"
 = Burger number "KIW" = log10 internal wave kinetic energy 2D
 spectrum [m^2/s^2/cpm^2] "KVM" = log10 vortical mode kinetic
 energy 2D spectrum [m^2/s^2/cpm^2] "KVMstar" = log10 vortical
 mode kinetic energy 2D spectrum, inferred from the K energy decomposition
 method [m^2/s^2/cpm^2] **Structure "Fig6c" contains variables:**
 "Bu" = Burger number "PIW" = log10 internal wave
 potential energy 2D spectrum [m^2/s^2/cpm^2] "PVM" = log10
 vortical mode potential energy 2D spectrum [m^2/s^2/cpm^2]
 "PVMstar" = log10 vortical mode potential energy 2D spectrum,
 inferred from the K energy decomposition method [m^2/s^2/cpm^2] #### file:
 Fig7.mat **Fig. 7.** Mean Sargasso Sea (k~h~, k~z~) spectrum for (a) E^VM^
 derived from the K-P energy decomposition using both velocity and density
 measurements and (b) E^VM^~*~ derived from the K energy decomposition
 using velocity measurements only. (c) Ratio of the two VM energy estimates
 and (d) their scatterplot. **Structure "Fig7a" contains
 variables:** "kh" = horizontal wavenumber magnitude [cpm]
 "kz" = vertical wavenumber [cpm] "EVM" = log10
 vortical mode total energy 2D spectrum [m^2/s^2/cpm^2] **Structure
 "Fig7b" contains variables:** "kh" = horizontal
 wavenumber magnitude [cpm] "kz" = vertical wavenumber [cpm]
 "EVMstar" = log10 vortical mode total energy 2D spectrum,
 inferred from the K energy decomposition method [m^2/s^2/cpm^2]
 **Structure "Fig7c" contains variables:** "kh" =
 horizontal wavenumber magnitude [cpm] "kz" = vertical wavenumber
 [cpm] "ratio" = log10 ratio of vortical mode total energy 2D
 spectrum inferred from the K-P energy decomposition method to the vortical
 mode total energy 2D spectrum inferred from the K energy decomposition
 method **Structure "Fig7d" contains variables:** "EVM"
 = vortical mode total energy 2D spectrum [m^2/s^2/cpm^2]
 "EVMstar" = vortical mode total energy 2D spectrum, inferred
 from the K energy decomposition method [m^2/s^2/cpm^2] "Bu" =
 log10 Burger number #### file: Fig8.mat **Fig. 8.** Dependence on Bu of
 Sargasso Sea (k~h~, k~z~) spectrum for (a) total IW and VM energy, (b) IW
 and VM horizontal kinetic energy, and (c) IW and VM available potential
 energy. Curves represent spectra averaged over Burger number bins.
 **Structure "Fig8a" contains variables:** "Bu" =
 Burger number "EIW" = log10 internal wave total energy 2D
 spectrum [m^2/s^2/cpm^2] "EVM" = log10 vortical mode total
 energy 2D spectrum [m^2/s^2/cpm^2] "EVMstar" = log10 vortical
 mode total energy 2D spectrum, inferred from the K energy decomposition
 method [m^2/s^2/cpm^2] "BuPDF" = Burger number "PDF" =
 PDF for all resolved (kh,kz) "PDF_positiveEVMstartilde" = PDF
 for (kh,kz) where EVMstartilde &gt; 0 **Structure "Fig8b"
 contains variables:** "Bu" = Burger number "KIW" =
 log10 internal wave kinetic energy 2D spectrum [m^2/s^2/cpm^2]
 "KVM" = log10 vortical mode kinetic energy 2D spectrum
 [m^2/s^2/cpm^2] "KVMstar" = log10 vortical mode kinetic energy
 2D spectrum, inferred from the K energy decomposition method
 [m^2/s^2/cpm^2] **Structure "Fig8c" contains variables:**
 "Bu" = Burger number "PIW" = log10 internal wave
 potential energy 2D spectrum [m^2/s^2/cpm^2] "PVM" = log10
 vortical mode potential energy 2D spectrum [m^2/s^2/cpm^2]
 "PVMstar" = log10 vortical mode potential energy 2D spectrum,
 inferred from the K energy decomposition method [m^2/s^2/cpm^2] #### file:
 Fig9.mat **Fig. 9.** PDFs of IW to VM spectral ratios for (a) total
 energy, (b) horizontal kinetic energy, and (c) available potential energy.
 Spectra were computed from GO-SHIP P02W SADCP velocity data collected
 during May 2022, between 75- and 225-m depth, and averaged between
 0.5