Nomenclature

Frame of reference

• Define frame of reference, and notation. Use u,v,w and x,y, and z?
• Dumping a sketch would be useful

---- MOVE THIS TO CONCEPT ---

Reynold's Decomposition

• Variable names for Decomposition of total, mean, turbulent and waves.
• Needs to be decided across the ADV/ADCP working groups

---- MOVE THIS TO FUNDAMENTALS ---

Background (total) velocity

---- MAKE SURE TO BE CONSISTENT WITH NETCDF TABLE --- ---- NETCDF TABLE will have own page (periodic copy&paste of excel sheet)---

Symbol	Description	Units
u	zonal velocity	<math> \mathrm{m\, s^{-1}}</math>
v	meridional velocity	<math>\mathrm{m\, s^{-1}}</math>
u_e	error velocity	<math>\mathrm{m\, s^{-1}}</math>
V	velocity perpendicular to mean flow	<math>\mathrm{m\, s^{-1}}</math>
W_d	Profiler fall speed	<math>\mathrm{m\, s^{-1}}</math>
U_P	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

Parameter name	Symbol	Description	Standard long name	Eqn	Units
EPSI	<math>\varepsilon</math>	Turbulent kinetic energy dissipation rate	tke_dissipation		<math> \mathrm{W\, kg^{-1}} </math>
RI	<math>Ri</math>	Richardson number	richardson_number	<math> Ri = \frac{N^2}{S^2}</math>
RI_F	<math>Ri_f</math>	Flux gradient Richardson number	flux_grad_richardson_number	<math> \frac{B}{P} </math> or Ivey & Immerger? Karan et cie
Krho	<math>\kappa_\rho</math>	Turbulent diffusivity	turbulent_diffusivity	<math> \kappa = \Gamma \varepsilon N^{-2} </math>	<math>\mathrm{m^2\, s^{-1}}</math>
DLL	<math>D_{LL}</math>	Second-order longitudinal structure function	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	Units	Eqn
S_a	Salinity		<math> \sim 35 </math>
T	Temperature	<math> \mathrm{^{\circ}C } </math>	<math>\sim -2 \rightarrow 40 </math>
P	Pressure	<math>\mathrm{dbar} </math>	<math>0\ \rightarrow\ \sim 1\times10^4</math>
\rho	Density of water	<math>\mathrm{kg\, m^{-3}} </math>	\rho = \rho\left(T,S_a,P \right)
\alpha	Temperature coefficient of expansion	\mathrm{K^{-1}}	\alpha = \frac{1}{\rho} \frac{\partial\rho}{\partial T}
\beta	Saline coefficient of contraction		\beta = \frac{1}{\rho} \frac{\partial\rho}{\partial S_a}
S	Background velocity shear	\mathrm{s^{-1}}	S = \left( \left( \frac{\partial U}{\partial z}\right)^2 + \left( \frac{\partial V}{\partial z}\right)^2 \right)^{1/2}
\nu_{35}	Temperature dependent kinematic viscosity of seawater at a salinity of 35	\mathrm{m^2\, s^{-1} }	\sim 1\times 10^{-6}
\nu_{00}	Temperature dependent kinematic viscosity of freshwater	\mathrm{m^2\, s^{-1} }	\sim 1\times 10^{-6}
\Gamma	Adiabatic temperature gradient -- salinity, temperature and pressure dependent	\mathrm{K\, dbar^{-1} }	\sim 1\times 10^{-4}
N	Background stratification, i.e buoyancy frequency	\mathrm{rad\, s^{-1} }	N^2 = g\left[ \alpha\left(\Gamma + \frac{\partial T}{\partial z} \right) - \beta \frac{\partial S_a}{\partial z} \right]

Theoretical Length and Time Scales

Parameter	Symbol	Description	Standard long name	Eqn	Units
T_N	<math>\tau_N</math>	Buoyancy timescale	buoyancy_time_scale	<math> \tau_N = \frac{1}{N}</math>	<math> \mathrm{s} </math>
T_P	<math>T_N</math>	Buoyancy period	buoyancy_period	<math> T_N = \frac{2\pi}{N}</math>	<math> \mathrm{s} </math>
L_E	<math>L_E</math>	Ellison length scale (limit of vertical displacement without irreversible mixing)	Eliison_lenght_scale	<math>L_E=\frac {\langle \rho'^2\rangle^{1/2}}{\partial \overline{\rho}/\partial z}</math>	<math> \mathrm{m} </math>
L_RHO	<math> L_\rho</math>	Density length scale	density_length_scale	<math> L_\rho </math>	<math> \mathrm{m} </math>
L_S	<math>L_S</math>	Corssin length scale	Corssin_shear_length_scale	<math> L_S = \sqrt{\varepsilon/S^3} </math>	<math> \mathrm{m} </math>
L_K	<math>\eta</math>	Kolmogorov length scale (smallest overturns)	Kolmogorov_length_scale	<math>\eta=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4}</math>	<math> \mathrm{m} </math>
L_K	<math>L_K</math>	Kolmogorov length scale (smallest overturns)	Kolmogorov_length_scale	<math>L_K=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4}</math>	<math> \mathrm{m} </math>
L_O	<math>L_o</math>	Ozmidov length scale, measure of largest overturns in a stratified fluid	Ozmidov_stratification_length_scale	<math>L_o=\left(\frac{\varepsilon}{N^3}\right)^{1/2}</math>	<math> \mathrm{m} </math>
L_T	<math>L_T</math>	Thorp length scale	Thorpe_stratification_length_scale	<math>L_T</math>	<math> \mathrm{m} </math>

Turbulence Spectrum

---- MERGE WITH THE SPECTRUM IN FUNDEMENTALS ---

Taylor's Frozen Turbulence for converting temporal to spatial measurements. Convert time derivatives to spatial gradients along the direction of profiling using

<math> \frac{\partial}{\partial x} = \frac{1}{U_P} \frac{\partial}{\partial t} </math> .

Convert frequency spectra into wavenumber spectra using

<math> k = f/U_P </math> and <math> \Psi(k) = U_P \Psi(f) </math> .

• Missing the y-axi variable. CEB proposes:
  • <math>\Psi_{variable}</math> for model/theoretical spectrum of variable e.g., du/dx or u
  • <math>\Phi_{variable}</math> for observed spectrum of variable e.g., du/dx or u
• Lowest frequency and wavenumber resolvable

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>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>