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

Reynold's Decomposition

• Variable names for Decomposition of total, mean, turbulent and waves.

Background (total) velocity

Parameter name	Symbol	Description	Standard long name	Units
EAST_VEL	$u$	zonal velocity	eastward_velocity	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
NORTH_VEL	$v$	meridional velocity	northward_velocity	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
UP_VEL	$W$	vertical velocity	upward_velocity	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
ERROR_VEL	$u_e$	error velocity	error_velocity	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
U_VEL	$U$	velocity parellel to mean flow	meanflow_velocity	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
V_VEL	$V$	velocity perpendicular to mean flow	crossflow_velocity	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
Drop_Speed	$W_d$	Profiler fall speed	mean_drop_speed	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
FlowPast_Speed	$U_P$	Flow speed past sensor	mean_velocity_past_turbulence_sensor	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
AlongBeam_Velocity	$b$	Along-beam velocity from acoustic Doppler sensor	observed_velocity_along_an_acoustic_beam	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
AlongBeam_Residual_Velocity	$b^{\prime }$	Along-beam velocity from acoustic Doppler sensor with background flow deducted	residual_velocity_along_an_acoustic_beam	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
Vertical_Bin_Size	$\delta z$	Vertical size of measurement bin for acoustic Doppler sensor	vertical_bin_size	$\mathrm{m}$
AlongBeam_Distance	$r$	Along-beam distance from acoustic Doppler sensor	distance_along_an_acoustic_beam	$\mathrm{m}$
AlongBeam_Bin_Size	$\delta r$	Along-beam bin size for acoustic Doppler sensor	bin_size_along_an_acoustic_beam	$\mathrm{m}$
Beam_Angle	$\theta$	Beam transmit and receive angle relative to instrument axis for acoustic Doppler sensor	acoustic_beam_angle	${\circ }^{}$

Turbulence properties

Parameter name	Symbol	Description	Standard long name	Eqn	Units
EPSI	$\epsilon$	Turbulent kinetic energy dissipation rate	tke_dissipation		$\mathrm{W}\mathrm{k}{\mathrm{g}}^{\mathrm{-}\mathrm{1}}$
RI	$Ri$	Richardson number	richardson_number	$Ri=\frac{N^2}{S^2}$
RI_F	$R{i}_f$	Flux gradient Richardson number	flux_grad_richardson_number	$\frac{B}{P}$ or Ivey & Immerger? Karan et cie
Krho	${\kappa }_{\rho }$	Turbulent diffusivity	turbulent_diffusivity	$\kappa =\mathrm{\Gamma }\epsilon N^{-2}$	${\mathrm{m}}^{\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
DLL	$D_{LL}$	Second-order longitudinal structure function	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

Parameter Name	Symbol	Description	Standard long name	Eqn	Units
Z	$z$	vertical coordinate -- positive upwards	vertical_coordinate		$\mathrm{m}$
G	$g$	acceleration of gravity	acceleration_of_gravity	$\sim 9.81$	$\mathrm{m}{\mathrm{s}}^{\mathrm{-}\mathrm{2}}$
SALINITY	$S_a$	Salinity	Salinity	$\sim 35$
TEMP	$T$	Temperature	Temperature	$\sim -2\to 40$	${}^{\mathrm{\circ }}\mathrm{C}$
PRES	$P$	Pressure	Pressure	$0\to \sim 1\times 10^4$	$\mathrm{d}\mathrm{b}\mathrm{a}\mathrm{r}$
DENSITY	$\rho$	Density of water	Density	$\rho =\rho (T,S_a,P)$	$\mathrm{k}\mathrm{g}{\mathrm{m}}^{\mathrm{-}\mathrm{3}}$
ALPHA	$\alpha$	Temperature coefficient of expansion	Temperature_coefficient_of_expansion	$\alpha =\frac{1}{\rho }\frac{\partial \rho }{\partial T}$	${\mathrm{K}}^{\mathrm{-}\mathrm{1}}$
BETA	$\beta$	Saline coefficient of contraction	Saline_coefficient_of_contraction	$\beta =\frac{1}{\rho }\frac{\partial \rho }{\partial S_a}$	${\displaystyle }$
S	$S$	Background velocity shear	background_velocity_shear	$S={({\left(\frac{\partial U}{\partial z}\right)}^2+{\left(\frac{\partial V}{\partial z}\right)}^2)}^{1/2}$	s${-1}^{}$
KVISC35	${\nu }_{35}$	Temperature dependent kinematic viscosity of seawater at a salinity of 35	seawater_kinematic_viscosity_at_35psu	$\sim 1\times 10^{-6}$	${\mathrm{m}}^{\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
KVISC00	${\nu }_{00}$	Temperature dependent kinematic viscosity of freshwater	freshwater_kinematic_viscosity	$\sim 1\times 10^{-6}$	${\mathrm{m}}^{\mathrm{2}}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$
GAMMA_A	$\mathrm{\Gamma }$	Adiabatic temperature gradient -- salinity, temperature and pressure dependent	Rate of change of temperature due to pressure	$\sim 1\times 10^{-4}$	$\mathrm{K}\mathrm{d}\mathrm{b}\mathrm{a}{\mathrm{r}}^{\mathrm{-}\mathrm{1}}$
N	$N$	Background stratification, i.e buoyancy frequency	background_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

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

Turbulence Spectrum

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

$\frac{\partial }{\partial x}=\frac{1}{U_P}\frac{\partial }{\partial t}$ .

Convert frequency spectra into wavenumber spectra using

$k=f/U_P$ and $\mathrm{\Psi }(k)=U_P\mathrm{\Psi }(f)$ .

• Missing the y-axi variable. CEB proposes:
  • ${\mathrm{\Psi }}_{variable}$ for model/theoretical spectrum of variable e.g., du/dx or u
  • ${\mathrm{\Phi }}_{variable}$ for observed spectrum of variable e.g., du/dx or u
• Lowest frequency and wavenumber resolvable

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	$k_N=\frac{f_N}{U_P}$	$\mathrm{c}\mathrm{p}\mathrm{m}$

Supplementary Data required for computing Turbulence

Channel	Shear Probes	ADCP	ADVs
Ax	x	x	x
Ay	x	x	x
Az	x	x	x