Nomenclature

Background (total) velocity

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

Turbulence properties

Symbol	Description	Eqn	Units
[math]\displaystyle{ \varepsilon }[/math]	The rate of dissipation of turbulent kinetic energy per unit mass by viscosity		[math]\displaystyle{ \mathrm{W\, kg^{-1}} }[/math]
[math]\displaystyle{ R_i }[/math]	(Gradient) Richardson number; the ratio of buoyancy freqency squared to velocity shear squared	[math]\displaystyle{ R_i = \frac{N^2}{S^2} }[/math]
[math]\displaystyle{ R_f }[/math]	Flux Richardson number; the ratio of the buoyancy flux expended to change the potential energy to the shear production of turbulent kinetic energy. It is also referred to as "Mixing efficiency". Mixing efficiency is the ratio of the net change in potential energy due to mixing to the energy expended in producing the mixing.	[math]\displaystyle{ R_f = \frac{-B}{P} }[/math]
[math]\displaystyle{ \Gamma }[/math]	"Efficiency factor"; indicates the conversion efficiency of turbulent kinetic energy into potential energy of the system	[math]\displaystyle{ \Gamma = \frac{R_f}{1-R_f} }[/math]
[math]\displaystyle{ \kappa_{\rho} }[/math]	Turbulent eddy diffusivity via the Osborn (1980) model	[math]\displaystyle{ \kappa_{\rho} = \Gamma \varepsilon N^{-2} }[/math]	[math]\displaystyle{ \mathrm{m^2\, s^{-1}} }[/math]
[math]\displaystyle{ D_{LL} }[/math]	Second-order longitudinal structure function	[math]\displaystyle{ D_{LL} = \big\langle[b^{\prime}(r) - b^{\prime}(r+n\delta{r})]^2\big\rangle }[/math]	[math]\displaystyle{ \mathrm{m^2\, s^{-2}} }[/math]

Fluid properties and background gradients for turbulence calculations

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

Theoretical Length and Time Scales

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

Turbulence Spectrum

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

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