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.

Turbulence properties

Symbol	Description	Eqn	Units
[math]\displaystyle{ \epsilon }[/math]	Turbulent kinetic energy dissipation		W/kg
[math]\displaystyle{ Ri }[/math]	Richardson number	[math]\displaystyle{ Ri = \frac{N^2}{S^2} }[/math]
[math]\displaystyle{ Ri_f }[/math]	Flux gradient Richardson number	[math]\displaystyle{ \frac{B}{P} }[/math] or Ivey & Immerger?
[math]\displaystyle{ \kappa }[/math]	Turbulent diffusivity	[math]\displaystyle{ \kappa = \Gamma \epsilon N^{-2} }[/math]	m[math]\displaystyle{ ^2 }[/math]s[math]\displaystyle{ ^{-1} }[/math]

Fluid properties and background gradients for turbulence calculations

Symbol	Description	Eqn	Units
[math]\displaystyle{ S }[/math]	Background velocity shear	[math]\displaystyle{ S = \frac{\partial |U|}{\partial z} }[/math]	s[math]\displaystyle{ ^{-1} }[/math]
[math]\displaystyle{ \nu }[/math]	Viscosity of water for seawater at 35psu and 20 oC	[math]\displaystyle{ 1\times 10^{-6} }[/math]	m2/s
[math]\displaystyle{ N }[/math]	Background stratification, i.e buoyancy frequency	[math]\displaystyle{ N = \sqrt{\frac{-g}{\bar{\rho}} \frac{\partial\bar{\rho}}{\partial z}} }[/math]	rad/s

Theoretical Length and Time Scales

Symbol	Description	Eqn	Units
[math]\displaystyle{ \tau_N }[/math]	Buoyancy timescale	[math]\displaystyle{ \tau_N = \frac{2\pi}{N} }[/math]	s
[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]	m
[math]\displaystyle{ L_\rho }[/math]	Density length scale	[math]\displaystyle{ L_\rho }[/math]	m
[math]\displaystyle{ L_S }[/math]	Corssin length scale	[math]\displaystyle{ L_S = \sqrt{\epsilon/S^3} }[/math]	m
[math]\displaystyle{ \eta }[/math]	Kolmogorov length scale (smallest overturns)	[math]\displaystyle{ \eta=\left(\frac{\nu^3}{\epsilon}\right)^{1/4}=\frac{1}{2\pi\hat{k}_K} }[/math]	m
[math]\displaystyle{ L_o }[/math]	Ozmidov length scale, measure of largest overturns in a stratified fluid	[math]\displaystyle{ L_o=\left(\frac{\epsilon}{N^3}\right)^{1/2} }[/math]	m

Turbulence Spectrum

Taylor's Frozen Turbulence for converting temporal to spatial measurements [math]\displaystyle{ \left(\bar{u}_1\frac{\partial }{\partial{x}} = \frac{\partial}{\partial{t}}\right) }[/math]

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

Symbol	Description	Eqn	Units
[math]\displaystyle{ \Delta t }[/math]	Sampling interval	[math]\displaystyle{ \frac{1}{f_s} }[/math]	s
[math]\displaystyle{ \Delta s }[/math]	Sampling volume dimension		m
[math]\displaystyle{ f }[/math]	Frequency	[math]\displaystyle{ \frac{\omega}{2\pi} }[/math]	Hz
[math]\displaystyle{ f_n }[/math]	Nyquist frequency	[math]\displaystyle{ f_n=0.5f_s }[/math]	Hz
[math]\displaystyle{ f_s }[/math]	Sampling frequency	[math]\displaystyle{ f_s=\frac{1}{\Delta t} }[/math]	Hz
[math]\displaystyle{ k }[/math]	Wavenumbers (angular)	[math]\displaystyle{ k=\frac{f}{\bar{u}}=2\pi\hat{k} }[/math]	rad/m
[math]\displaystyle{ \hat{k} }[/math]	Wavenumbers	[math]\displaystyle{ \hat{k}=\frac{k}{2\pi} }[/math]	cpm
[math]\displaystyle{ \hat{k}_\Delta }[/math]	Nyquist wavenumber, based on sampling volume's size [math]\displaystyle{ \Delta l }[/math]	[math]\displaystyle{ \hat{k}_\Delta=\frac{0.5}{\Delta l} }[/math]	cpm
[math]\displaystyle{ \hat{k}_n }[/math]	Nyquist wavenumber, via Taylor's hypothesis (temporal measurements)	[math]\displaystyle{ \hat{k}_n=\frac{f_n}{u} }[/math]	cpm
[math]\displaystyle{ \omega }[/math]	Angular frequency	[math]\displaystyle{ 2\pi f }[/math]	rad/s