Nomenclature

Background (total) velocity

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

Turbulence properties

Symbol	Description	Eqn	Units
$\epsilon$	Viscous dissipation rate of turbulent kinetic energy per unit mass		$\mathrm{W}\mathrm{k}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}$
Ri	(Gradient) Richardson number; the ratio of buoyancy freqency squared to velocity shear squared	$Ri=\frac{N^2}{S^2}$
Ri${}_f$	Flux Richardson number; the ratio of the buoyancy flux to the shear production of turbulent kinetic energy	$\frac{B}{P}$
${\kappa }_{\rho }$	Turbulent eddy diffusivity via the Osborn (1980) model	$\kappa =\mathrm{\Gamma }\epsilon N^{-2}$	${\mathrm{m}}^{\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
$D_{LL}$	Second-order longitudinal structure function	$D_{LL}=⟨[b^{\prime }(r)-b^{\prime }(r+n\delta r){]}^2⟩$	${\mathrm{m}}^{\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}$

Fluid properties and background gradients for turbulence calculations

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

Theoretical Length and Time Scales

Symbol	Description	Eqn	Units
${\tau }_N$	Buoyancy timescale	${\tau }_N=\frac{1}{N}$	$\mathrm{s}$
$T_N$	Buoyancy period	$T_N=\frac{2\pi }{N}$	$\mathrm{s}$
$L_E$	Ellison length scale (limit of vertical displacement without irreversible mixing)	$L_E=\frac{⟨\rho {\text{'}}^2{⟩}^{1/2}}{\partial \bar{\rho }/\partial z}$	$\mathrm{m}$
$L_{\rho }$	Density length scale	$L_{\rho }$	$\mathrm{m}$
$L_S$	Corssin length scale	$L_S=\sqrt{\epsilon /S^3}$	$\mathrm{m}$
$\eta$	Kolmogorov length scale (smallest overturns)	$\eta ={\left(\frac{{\nu }^3}{\epsilon }\right)}^{1/4}$	$\mathrm{m}$
$L_K$	Kolmogorov length scale (smallest overturns)	$L_K={\left(\frac{{\nu }^3}{\epsilon }\right)}^{1/4}$	$\mathrm{m}$
$L_o$	Ozmidov length scale, measure of largest overturns in a stratified fluid	$L_o={\left(\frac{\epsilon }{N^3}\right)}^{1/2}$	$\mathrm{m}$
$L_T$	Thorp length scale	$L_T$	$\mathrm{m}$

Turbulence Spectrum

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

Symbol	Description	Eqn	Units
$\mathrm{\Delta }t$	Sampling interval	$\frac{1}{f_s}$	$\mathrm{s}$
$f_s$	Sampling rate	$f_s=\frac{1}{\mathrm{\Delta }t}$	${\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
$\mathrm{\Delta }s$	Sample spacing	$\mathrm{\Delta }s=U_P\mathrm{\Delta }t$	$\mathrm{m}$
$\mathrm{\Delta }l$	Linear dimension of sampling volume (instrument dependent)		$\mathrm{m}$
$f$	Cyclic frequency	$f=\frac{\omega }{2\pi }$	$\mathrm{H}\mathrm{z}$
$\omega$	Angular frequency	$\omega =2\pi f$	$\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
$f_N$	Nyquist frequency	$f_N=0.5f_s$	$\mathrm{H}\mathrm{z}$
$k$	Cyclic wavenumber	$k=\frac{f}{U_P}$	$\mathrm{c}\mathrm{p}\mathrm{m}$
$\hat{k}$	Angular wavenumber	$\hat{k}=\frac{\omega }{U_P}=2\pi k$	$\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}$
$k_{\mathrm{\Delta }}$	Nyquist wavenumber, based on sampling volume size $\mathrm{\Delta }l$	$k_{\mathrm{\Delta }}=\frac{0.5}{\mathrm{\Delta }l}$	$\mathrm{c}\mathrm{p}\mathrm{m}$
$k_N$	Nyquist wavenumber, via Taylor's hypothesis	$k_N=\frac{f_N}{U_P}$	$\mathrm{c}\mathrm{p}\mathrm{m}$
${\mathrm{\Psi }}_{\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}}(k)$	Theoretical, empirical or model spectrum of any given variable		units of the variable squared/units of k (i.e.,cpm)
${\mathrm{\Phi }}_{\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}}(k)$	Observed spectrum of any given variable		units of the variable squared/units of k (i.e.,cpm)