Nomenclature: Difference between revisions

Revision as of 15:00, 3 December 2021

Symbol	Description	Units
u	zonal velocity	$m s^{- 1}$
v	meridional velocity	$m s^{- 1}$
$u_{e}$	error velocity	$m s^{- 1}$
V	velocity perpendicular to mean flow	$m s^{- 1}$
$W_{d}$	Profiler fall speed	$m s^{- 1}$
$U_{P}$	Flow speed past sensor	$m s^{- 1}$
b	Along-beam velocity from acoustic Doppler sensor	$m s^{- 1}$
$b^{'}$	Along-beam velocity from acoustic Doppler sensor with background flow deducted	$m s^{- 1}$
$δ z$	Vertical size of measurement bin for acoustic Doppler sensor	$m$
r	Along-beam distance from acoustic Doppler sensor	$m$
$δ r$	Along-beam bin size for acoustic Doppler sensor	$m$
$θ$	Beam transmit and receive angle relative to instrument axis for acoustic Doppler sensor	$\circ$

Symbol	Description	Eqn	Units
$ε$	The rate of dissipation of turbulent kinetic energy per unit mass by viscosity		$W k g^{- 1}$
Ri	(Gradient) Richardson number; the ratio of buoyancy freqency squared to velocity shear squared	$R_{i} = \frac{N^{2}}{S^{2}}$
Ri $_{f}$	Flux Richardson number; the ratio of the buoyancy flux to the shear production of turbulent kinetic energy	$R_{f} = \frac{B}{P}$
$Γ$	Mixing efficiency; the ratio of the buoyancy flux to the rate of dissipation of turbulent kinetic energy	$Γ = \frac{B}{ε} = \frac{R_{f}}{1 - R_{f}}$
$κ_{ρ}$	Turbulent eddy diffusivity via the Osborn (1980) model	$κ_{ρ} = Γ ε N^{- 2}$	$m^{2} s^{- 1}$
$D_{L L}$	Second-order longitudinal structure function	$D_{L L} = ⟨ [b^{'} (r) - b^{'} (r + n δ r)]^{2} ⟩$	$m^{2} s^{- 2}$

Symbol	Description	Eqn	Units
$S_{P}$	Practical salinity		$-$
$T$	Temperature		$^{\circ} C$
$P$	Pressure		$d b a r$
$ρ$	Density of water	$ρ = ρ (T, S_{a}, P)$	$k g m^{- 3}$
$α$	Temperature coefficient of expansion	$α = \frac{1}{ρ} \frac{\partial ρ}{\partial T}$	$K^{- 1}$
$β$	Saline coefficient of contraction	$β = \frac{1}{ρ} \frac{\partial ρ}{\partial S_{a}}$
$S$	Background velocity shear	$S = {({(\frac{\partial U}{\partial z})}^{2} + {(\frac{\partial V}{\partial z})}^{2})}^{1 / 2}$	$s^{- 1}$
$ν_{35}$	Temperature dependent kinematic viscosity of seawater at a practical salinity of 35	$\sim 1 \times 1 0^{- 6}$	$m^{2} s^{- 1}$
$ν_{00}$	Temperature dependent kinematic viscosity of freshwater	$\sim 1 \times 1 0^{- 6}$	$m^{2} s^{- 1}$
$Γ_{a}$	Adiabatic temperature gradient -- salinity, temperature and pressure dependent	$\sim 1 \times 1 0^{- 4}$	$K d b a r^{- 1}$
$N$	Background stratification, i.e buoyancy frequency	$N^{2} = g [α (Γ_{a} + \frac{\partial T}{\partial z}) - β \frac{\partial S_{P}}{\partial z}]$	$r a d s^{- 1}$

Symbol	Description	Eqn	Units
$τ_{N}$	Buoyancy timescale	$τ_{N} = \frac{1}{N}$	$s$
$T_{N}$	Buoyancy period	$T_{N} = \frac{2 π}{N}$	$s$
$L_{E}$	Ellison length scale (limit of vertical displacement without irreversible mixing)	$L_{E} = \frac{⟨ ρ'^{2} ⟩^{1 / 2}}{\partial \overline{ρ} / \partial z}$	$m$
$L_{Z}$	Boundary (law of the wall) length scale	$L_{Z} = 0.39 z_{w}$ with 0.39 being von Kármán's constant	$m$
$L_{S}$	Corssin length scale	$L_{S} = \sqrt{ε / S^{3}}$	$m$
$η$	Kolmogorov length scale (smallest overturns)	$η = {(\frac{ν^{3}}{ε})}^{1 / 4}$	$m$
$L_{K}$	Kolmogorov length scale (smallest overturns)	$L_{K} = {(\frac{ν^{3}}{ε})}^{1 / 4}$	$m$
$L_{o}$	Ozmidov length scale, measure of largest overturns in a stratified fluid	$L_{o} = {(\frac{ε}{N^{3}})}^{1 / 2}$	$m$
$L_{T}$	Thorpe length scale	$L_{T}$	$m$
$z_{w}$	Distance from a boundary	$z_{w}$	$m$

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

Symbol	Description	Eqn	Units
$Δ t$	Sampling interval	$\frac{1}{f_{s}}$	$s$
$f_{s}$	Sampling rate	$f_{s} = \frac{1}{Δ t}$	$s^{- 1}$
$Δ s$	Sample spacing	$Δ s = U_{P} Δ t$	$m$
$Δ l$	Linear dimension of sampling volume (instrument dependent)		$m$
$f$	Cyclic frequency	$f = \frac{ω}{2 π}$	$H z$
$ω$	Angular frequency	$ω = 2 π f$	$r a d s^{- 1}$
$f_{N}$	Nyquist frequency	$f_{N} = 0.5 f_{s}$	$H z$
$k$	Cyclic wavenumber	$k = \frac{f}{U_{P}}$	$c p m$
$\hat{k}$	Angular wavenumber	$\hat{k} = \frac{ω}{U_{P}} = 2 π k$	$r a d m^{- 1}$
$\tilde{k}$	Normalized wavenumber	e.g., $\tilde{k} = k L_{K}, L_{K} = {(ν^{3} / ε)}^{1 / 4}$	-
$\tilde{Φ}$	Normalized veolcity spectrum	e.g., $\tilde{Φ_{u} (\tilde{k})} = {(ϵ ν^{5})}^{- 1 / 4} Φ_{u} (k)$	-	$k_{Δ}$	Nyquist wavenumber, based on sampling volume size $Δ l$	$k_{Δ} = \frac{0.5}{Δ l}$	$c p m$
$k_{N}$	Nyquist wavenumber, via Taylor's hypothesis	$k_{N} = \frac{f_{N}}{U_{P}}$	$c p m$
$Ψ (k)$	Shear spectrum		${(1 / s)}^{2} / c p m$
$Φ (k)$	Velocity spectrum. Use $Φ_{u}$ , $Φ_{v}$ , $Φ_{v}$ , or $Φ_{1}$ , $Φ_{2}$ , $Φ_{3}$ for the different orthogonal components of the velocity.		${(m / s)}^{2} / c p m$

@@ Line 293: / Line 293: @@
 | -
 |-
+|-
+| <math>\tilde{\Phi}</math>
+| Normalized veolcity spectrum
+| e.g., <math>\tilde{\Phi_u(\tilde{k})} = \left(\epsilon \nu^5\right)^{-1/4} \Phi_u(k)</math>
+| -
 | <math>k_\Delta</math>
 | Nyquist wavenumber, based on sampling volume size <math>\Delta l</math>
@@ Line 304: / Line 308: @@
 | <math> \mathrm{cpm} </math>
 |-
-| <math>\Psi_{\mathrm{variable}}(k)</math>
+| <math>\Psi(k)</math>
 | Shear spectrum
 |
-| (1/s)<math>^2</math>/cpm
+| <math> \mathrm{(1/s)}^2/\mathrm{cpm} </math>
 |-
-| <math>\Phi_{\mathrm{variable}}(k)</math>
+| <math>\Phi(k)</math>
 | Velocity spectrum. Use <math>\Phi_u</math>, <math>\Phi_v</math>, <math>\Phi_v</math>, or <math>\Phi_1</math>, <math>\Phi_2</math> , <math>\Phi_3</math> for the different orthogonal components of the velocity.
 |
-| (m/s)<math>^2</math>/cpm
+| <math> \mathrm{(m/s)}^2/\mathrm{cpm} </math>
 |}
 </div>