Nomenclature

Background (total) velocity

Symbol	Description	Units
<math>u</math>	zonal or longitudinal component of velocity	<math> \mathrm{m\, s^{-1}}</math>
<math>v</math>	meridional or transverse component of velocity	<math>\mathrm{m\, s^{-1}}</math>
<math>w</math>	vertical component of velocity	<math> \mathrm{m\, s^{-1}}</math>
<math>u_e</math>	error velocity	<math>\mathrm{m\, s^{-1}}</math>
V	velocity perpendicular to mean flow	<math>\mathrm{m\, s^{-1}}</math>
<math>W_d</math>	Profiler fall speed	<math>\mathrm{m\, s^{-1}}</math>
<math>U_P</math>	Flow speed past sensor	<math>\mathrm{m\, s^{-1}}</math>
b	Along-beam velocity from acoustic Doppler sensor	<math>\mathrm{m\, s^{-1}}</math>
<math> b^{\prime}</math>	Along-beam velocity from acoustic Doppler sensor with background flow deducted	<math>\mathrm{m\, s^{-1}}</math>
<math> \delta{z}</math>	Vertical size of measurement bin for acoustic Doppler sensor	<math>\mathrm{m}</math>
r	Along-beam distance from acoustic Doppler sensor	<math>\mathrm{m}</math>
<math> \delta{r}</math>	Along-beam bin size for acoustic Doppler sensor	<math>\mathrm{m}</math>
<math> \theta</math>	Beam transmit and receive angle relative to instrument axis for acoustic Doppler sensor	<math>^{\circ}</math>

Turbulence properties

Symbol	Description	Eqn	Units
<math>\varepsilon</math>	The rate of dissipation of turbulent kinetic energy per unit mass by viscosity		<math>\mathrm{W\, kg^{-1}}</math>
<math>B</math>	Buoyancy production -- the rate of production of potential energy by turbulence in a stratified flow through the vertical flux of buoyancy.	<math>B= \frac{g}{\rho} \overline{\rho'w'} </math>	<math>\mathrm{W\, kg^{-1}}</math>
<math>P</math>	The production of turbulence kinetic energy. In a steady, spatially uniform and stratified shear flow, turbulence kinetic energy is produced by the product of the Reynolds stress and the shear, for example <math>P = -\overline{u'w'}\frac{\partial U}{\partial z} </math> . The production is balanced by the rate of dissipation turbulence kinetic energy, <math>\varepsilon</math>, and the production of potential energy by the buoyancy flux, <math>B</math>.	<math>P = -\overline{u'w'}\frac{\partial U}{\partial z} = \varepsilon + B</math>	<math>\mathrm{W\, kg^{-1}}</math>
<math>R_f</math>	Flux Richardson number; the ratio of the buoyancy flux expended for the net change in potential energy (i.e., mixing) to the shear production of turbulent kinetic energy.	<math>R_f = \frac{B}{P}</math>
<math>\Gamma</math>	"Mixing coefficient"; The ratio of the rate of production of potential energy, <math>B</math>, to the rate of dissipation of kinetic energy, <math>\varepsilon</math>.	<math>\Gamma = \frac{B}{\varepsilon} = \frac{R_f}{1-R_f}</math>
<math>R_i</math>	(Gradient) Richardson number; the ratio of buoyancy freqency squared to velocity shear squared	<math>R_i = \frac{N^2}{S^2} </math>
<math>\kappa_{\rho}</math>	Turbulent eddy diffusivity via the Osborn (1980) model	<math>\kappa_{\rho} = \Gamma \varepsilon N^{-2}</math>	<math>\mathrm{m^2\, s^{-1}}</math>
<math>D_{ll}</math>	Second-order longitudinal structure function	<math>D_{ll} = \big\langle[b^{\prime}(r) - b^{\prime}(r+n\delta{r})]^2\big\rangle</math>	<math>\mathrm{m^2\, s^{-2}}</math>

Fluid properties and background gradients for turbulence calculations

Symbol	Description	Eqn	Units
<math>S_P</math>	Practical salinity		<math> - </math>
<math>T</math>	Temperature		<math> \mathrm{^{\circ}C } </math>
<math>P</math>	Pressure		<math>\mathrm{dbar} </math>
<math>\rho</math>	Density of water	<math> \rho = \rho\left(T,S_a,P \right)</math>	<math>\mathrm{kg\, m^{-3}} </math>
<math>\alpha</math>	Temperature coefficient of expansion	<math> \alpha = \frac{1}{\rho} \frac{\partial\rho}{\partial T}</math>	<math> \mathrm{K^{-1}}</math>
<math>\beta</math>	Saline coefficient of contraction	<math> \beta = \frac{1}{\rho} \frac{\partial\rho}{\partial S_P}</math>
<math>S</math>	Background velocity shear	<math> S = \left[ \left( \frac{\partial U}{\partial z}\right)^2 + \left( \frac{\partial V}{\partial z}\right)^2 \right]^{1/2} </math>	<math> \mathrm{s^{-1}} </math>
<math> \nu_{35} </math>	Temperature dependent kinematic viscosity of seawater at a practical salinity of 35	<math> \sim 1\times 10^{-6} </math>	<math> \mathrm{m^2\, s^{-1} } </math>
<math>\nu_{00}</math>	Temperature dependent kinematic viscosity of freshwater	<math>\sim 1\times 10^{-6} </math>	<math>\mathrm{m^2\, s^{-1} } </math>
<math>\Gamma_a </math>	Adiabatic temperature gradient -- salinity, temperature and pressure dependent	<math>\sim 1\times 10^{-4}</math>	<math>\mathrm{K\, dbar^{-1} } </math>
<math>N </math>	Background stratification, i.e buoyancy frequency	<math>N^2 = g\left[ \alpha\left(\Gamma_a + \frac{\partial T}{\partial z} \right) - \beta \frac{\partial S_P}{\partial z} \right] </math>	<math>\mathrm{rad\, s^{-1} } </math>

Theoretical Length and Time Scales

Symbol	Description	Eqn	Units
<math>\tau_N</math>	Buoyancy timescale	<math> \tau_N = \frac{1}{N}</math>	<math> \mathrm{s} </math>
<math>T_N</math>	Buoyancy period	<math> T_N = \frac{2\pi}{N}</math>	<math> \mathrm{s} </math>
<math>L_E</math>	Ellison length scale (limit of vertical displacement without irreversible mixing)	<math>L_E=\frac {\langle \rho'^2\rangle^{1/2}}{\partial \overline{\rho}/\partial z}</math>	<math> \mathrm{m} </math>
<math> L_Z</math>	Boundary (law of the wall) length scale	<math> L_Z=0.39z_w </math> with 0.39 being von Kármán's constant	<math> \mathrm{m} </math>
<math>L_S</math>	Corssin length scale	<math> L_S = \sqrt{\varepsilon/S^3} </math>	<math> \mathrm{m} </math>
<math>L_K</math>	Kolmogorov length scale (smallest overturns)	<math>L_K=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4}</math>	<math> \mathrm{m} </math>
<math>L_o</math>	Ozmidov length scale, measure of largest overturns in a stratified fluid	<math>L_o=\left(\frac{\varepsilon}{N^3}\right)^{1/2}</math>	<math> \mathrm{m} </math>
<math>L_T</math>	Thorpe length scale	<math>L_T</math>	<math> \mathrm{m} </math>
<math>z_w</math>	Distance from a boundary	<math>z_w</math>	<math> \mathrm{m} </math>

Turbulence Spectrum

These variables are used to express the Turbulence spectrum expected shapes.

Symbol	Description	Eqn	Units
<math>\Delta t</math>	Sampling interval	<math> \frac{1}{f_s} </math>	<math> \mathrm{s} </math>
<math>f_s</math>	Sampling rate	<math>f_s=\frac{1}{\Delta t} </math>	<math> \mathrm{s^{-1}} </math>
<math>\Delta s</math>	Sample spacing	<math> \Delta s = U_P \Delta t </math>	<math> \mathrm{m} </math>
<math>\Delta l</math>	Linear dimension of sampling volume (instrument dependent)		<math> \mathrm{m} </math>
<math>f</math>	Cyclic frequency	<math>f=\frac{\omega}{2\pi}</math>	<math> \mathrm{Hz} </math>
<math>\omega</math>	Angular frequency	<math>\omega = 2\pi f</math>	<math> \mathrm{rad\, s^{-1}} </math>
<math>f_N</math>	Nyquist frequency	<math>f_N=0.5f_s</math>	<math> \mathrm{Hz} </math>
<math>k</math>	Cyclic wavenumber	<math>k=\frac{f}{U_P}</math>	<math> \mathrm{cpm} </math>
<math>\hat{k}</math>	Angular wavenumber	<math>\hat{k}=\frac{\omega}{U_P} = 2\pi k</math>	<math> \mathrm{rad\, m^{-1}} </math>
<math>\tilde{k}</math>	Normalized wavenumber	e.g., <math>\tilde{k}=k L_K, L_K = \left(\nu^3/\varepsilon \right)^{1/4}</math>	-
<math>\tilde{\Phi}</math>	Normalized velocity spectrum	e.g., <math>\tilde{\Phi}_u(\tilde{k}) = \left(\epsilon \nu^5\right)^{-1/4} \Phi_u(k)</math>	-
<math>\tilde{\Psi}</math>	Normalized shear spectrum	e.g., <math>\tilde{\Psi}(\tilde{k}) = L_K^2 \left(\epsilon \nu^5\right)^{-1/4} \Psi(k)</math>	-
<math>k_\Delta</math>	Nyquist wavenumber, based on sampling volume size <math>\Delta l</math>	<math>k_\Delta=\frac{0.5}{\Delta l}</math>	<math> \mathrm{cpm} </math>
<math>k_N</math>	Nyquist wavenumber, via Taylor's hypothesis	<math>k_N=\frac{f_N}{U_P}</math>	<math> \mathrm{cpm} </math>
<math>\Psi(k)</math>	Shear spectrum. Use <math>\Psi_1</math>, <math>\Psi_2</math> to distinguish the orthogonal components of the shear. Use <math>\Psi_N</math> for the Nasmyth spectrum, <math>\Psi_{PK}</math> for the Panchev-Kesich spectrum and <math>\Psi_L</math> for the Lueck spectrum.		<math> \mathrm{s^{-2}\, cpm^{-1}}</math>
<math>\Phi(k)</math>	Velocity spectrum. Use <math>\Phi_u</math>, <math>\Phi_v</math>, <math>\Phi_v</math>, or <math>\Phi_1</math>, <math>\Phi_2</math> , <math>\Phi_3</math> for the different orthogonal components of the velocity. Use <math>\Phi_K</math> for the Kolmogorov spectrum.		<math> \mathrm{m^2\, s^{-2}\, cpm^{-1}} </math>