Nomenclature

Background (total) velocity

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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

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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{ P }[/math]	The production of turbulence kinetic energy, in a steady uniform stratified shear flow, equals the product of the Reynolds stress and the shear, for example [math]\displaystyle{ P = -\overline{u'w'}\frac{\partial U}{\partial z} }[/math] . The production is balanced by the rate of dissipation turbulence kinetic energy through viscous friction, [math]\displaystyle{ \varepsilon }[/math] and the production of potential energy by the buoyancy flux, [math]\displaystyle{ B=-\frac{g}{\rho} \overline{\rho'w'} }[/math].	[math]\displaystyle{ P = -\overline{u'w'}\frac{\partial U}{\partial z} = \varepsilon + B }[/math]	[math]\displaystyle{ \mathrm{m^2\, s^{-3}} = \mathrm{W\, kg^{-1}} }[/math]
[math]\displaystyle{ 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]\displaystyle{ R_f = \frac{-B}{P} }[/math]
[math]\displaystyle{ \Gamma }[/math]	"Mixing coefficient"; in a stratified turbulent shear flow (where the production of turbulent kinetic energy by shear and the Reynolds stress, [math]\displaystyle{ P }[/math], equals the rate of dissipation, [math]\displaystyle{ \varepsilon }[/math], plus the buoyancy production, [math]\displaystyle{ B }[/math]), it is the ratio of the rate of potential energy due to buoyancy production to the rate of loss of kinetic energy through viscous friction.	[math]\displaystyle{ \Gamma = \frac{B}{\varepsilon} = \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

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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

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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

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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]