Nomenclature

Background (total) velocity

Symbol	Description	Units
$u$	zonal or longitudinal component of velocity	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
$v$	meridional or transverse component of velocity	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
$w$	vertical component of 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	$\mathrm{m}$
$\delta r_0$	Along-beam bin size for acoustic Doppler sensor	$\mathrm{m}$
$\delta r$	Along-beam bin separation 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
$\epsilon$	The rate of dissipation of turbulent kinetic energy per unit mass by viscosity		$\mathrm{W}\mathrm{k}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}$
$B$	Buoyancy production -- the rate of production of potential energy by turbulence in a stratified flow through the vertical flux of buoyancy.	$B=\frac{g}{\rho }\overline{{\rho }^{\prime }{w}^{\prime }}$	$\mathrm{W}\mathrm{k}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}$
$P$	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 $P=-\overline{u^{\prime }{w}^{\prime }}\frac{\partial U}{\partial z}$ . The production is balanced by the rate of dissipation turbulence kinetic energy, $\epsilon$, and the production of potential energy by the buoyancy flux, $B$.	$P=-\overline{u^{\prime }{w}^{\prime }}\frac{\partial U}{\partial z}=\epsilon +B$	$\mathrm{W}\mathrm{k}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}$
$R_f$	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.	$R_f=\frac{B}{P}$
$\Gamma$	"Mixing coefficient"; The ratio of the rate of production of potential energy, $B$, to the rate of dissipation of kinetic energy, $\epsilon$.	$\Gamma =\frac{B}{\epsilon }=\frac{R_f}{1-R_f}$
$R_i$	(Gradient) Richardson number; the ratio of buoyancy freqency squared to velocity shear squared	$R_i=\frac{N^2}{S^2}$
${\kappa }_{\rho }$	Turbulent eddy diffusivity via the Osborn (1980) model	${\kappa }_{\rho }=\Gamma \epsilon N^{-2}$	${\mathrm{m}}^{\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
$D_{ll}$	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	$\beta =\frac{1}{\rho }\frac{\partial \rho }{\partial S_P}$
$S$	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	$\sim 1\times 10^{-6}$	${\mathrm{m}}^{\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
${\Gamma }_a$	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	$N^2=g[\alpha ({\Gamma }_a+\frac{\partial T}{\partial z})-\beta \frac{\partial S_P}{\partial z}]$	$\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$

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)	$L_E=\frac{⟨\rho {\text{'}}^2{⟩}^{1/2}}{\partial \bar{\rho }/\partial z}$	$\mathrm{m}$
$L_Z$	Boundary (law of the wall) length scale	$L_Z=0.39z_w$ with 0.39 being von Kármán's constant	$\mathrm{m}$
$L_S$	Corssin length scale	$L_S=\sqrt{\epsilon /S^3}$	$\mathrm{m}$
$L_K$	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	$L_o={\left(\frac{\epsilon }{N^3}\right)}^{1/2}$	$\mathrm{m}$
$L_T$	Thorpe length scale	$L_T$	$\mathrm{m}$
$z_w$	Distance from a boundary	$z_w$	$\mathrm{m}$

Turbulence Spectrum

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

Symbol	Description	Eqn	Units
$\Delta t$	Sampling interval	$\frac{1}{f_s}$	$\mathrm{s}$
$f_s$	Sampling rate	$f_s=\frac{1}{\Delta t}$	${\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
$\Delta s$	Sample spacing	$\Delta s=U_P\Delta t$	$\mathrm{m}$
$\Delta l$	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	$\hat{k}=\frac{\omega }{U_P}=2\pi k$	$\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}$
$\tilde{k}$	Normalized wavenumber	e.g., $\tilde{k}=k{L}_K,L_K={({\nu }^3/\epsilon )}^{1/4}$	-
$\tilde{\Phi }$	Normalized velocity spectrum	e.g., ${\tilde{\Phi }}_u(\tilde{k})={\left(ϵ{\nu }^5\right)}^{-1/4}{\Phi }_u(k)$	-
$\tilde{\Psi }$	Normalized shear spectrum	e.g., $\tilde{\Psi }(\tilde{k})=L_K^2{\left(ϵ{\nu }^5\right)}^{-1/4}\Psi (k)$	-
$k_{\Delta }$	Nyquist wavenumber, based on sampling volume size $\Delta l$	$k_{\Delta }=\frac{0.5}{\Delta l}$	$\mathrm{c}\mathrm{p}\mathrm{m}$
$k_N$	Nyquist wavenumber, via Taylor's hypothesis	$k_N=\frac{f_N}{U_P}$	$\mathrm{c}\mathrm{p}\mathrm{m}$
$\Psi (k)$	Shear spectrum. Use ${\Psi }_1$, ${\Psi }_2$ to distinguish the orthogonal components of the shear. Use ${\Psi }_N$ for the Nasmyth spectrum, ${\Psi }_{PK}$ for the Panchev-Kesich spectrum and ${\Psi }_L$ for the Lueck spectrum.		${\mathrm{s}}^{\mathrm{-}\mathrm{2}}\mathrm{c}\mathrm{p}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}$
$\Phi (k)$	Velocity spectrum. Use ${\Phi }_u$, ${\Phi }_v$, ${\Phi }_v$, or ${\Phi }_1$, ${\Phi }_2$ , ${\Phi }_3$ for the different orthogonal components of the velocity. Use ${\Phi }_K$ for the Kolmogorov spectrum.		${\mathrm{m}}^{\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}\mathrm{c}\mathrm{p}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}$