Nomenclature
Background (total) velocity
| Symbol | Description | Units |
|---|---|---|
| [math]\displaystyle{ u }[/math] | zonal or longitudinal component of velocity | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| [math]\displaystyle{ v }[/math] | meridional or transverse component of velocity | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| [math]\displaystyle{ w }[/math] | vertical component of velocity | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| [math]\displaystyle{ u_e }[/math] | error velocity | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| V | velocity perpendicular to mean flow | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| [math]\displaystyle{ W_d }[/math] | Profiler fall speed | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| [math]\displaystyle{ U_P }[/math] | Flow speed past sensor | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| b | Along-beam velocity from acoustic Doppler sensor | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| [math]\displaystyle{ b^{\prime} }[/math] | Along-beam velocity from acoustic Doppler sensor with background flow deducted | [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] |
| [math]\displaystyle{ \delta{z} }[/math] | Vertical size of measurement bin for acoustic Doppler sensor | [math]\displaystyle{ \mathrm{m} }[/math] |
| r | Along-beam distance from acoustic Doppler sensor | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ \delta{r}_0 }[/math] | Along-beam bin size for acoustic Doppler sensor | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ \delta{r} }[/math] | Along-beam bin separation for acoustic Doppler sensor | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ \theta }[/math] | Beam transmit and receive angle relative to instrument axis for acoustic Doppler sensor | [math]\displaystyle{ ^{\circ} }[/math] |
Turbulence properties
| Symbol | Description | Eqn | Units |
|---|---|---|---|
| [math]\displaystyle{ \varepsilon }[/math] | The rate of dissipation of turbulent kinetic energy per unit mass by viscosity | [math]\displaystyle{ \mathrm{W\, kg^{-1}} }[/math] | |
| [math]\displaystyle{ B }[/math] | Buoyancy production -- the rate of production of potential energy by turbulence in a stratified flow through the vertical flux of buoyancy. | [math]\displaystyle{ B= \frac{g}{\rho} \overline{\rho'w'} }[/math] | [math]\displaystyle{ \mathrm{W\, kg^{-1}} }[/math] |
| [math]\displaystyle{ P }[/math] | 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 [math]\displaystyle{ P = -\overline{u'w'}\frac{\partial U}{\partial z} }[/math] . The production is balanced by the rate of dissipation turbulence kinetic energy, [math]\displaystyle{ \varepsilon }[/math], and the production of potential energy by the buoyancy flux, [math]\displaystyle{ B }[/math]. | [math]\displaystyle{ P = -\overline{u'w'}\frac{\partial U}{\partial z} = \varepsilon + B }[/math] | [math]\displaystyle{ \mathrm{W\, kg^{-1}} }[/math] |
| [math]\displaystyle{ R_f }[/math] | 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. | [math]\displaystyle{ R_f = \frac{B}{P} }[/math] | |
| [math]\displaystyle{ \Gamma }[/math] | "Mixing coefficient"; The ratio of the rate of production of potential energy, [math]\displaystyle{ B }[/math], to the rate of dissipation of kinetic energy, [math]\displaystyle{ \varepsilon }[/math]. | [math]\displaystyle{ \Gamma = \frac{B}{\varepsilon} = \frac{R_f}{1-R_f} }[/math] | |
| [math]\displaystyle{ R_i }[/math] | (Gradient) Richardson number; the ratio of buoyancy freqency squared to velocity shear squared | [math]\displaystyle{ R_i = \frac{N^2}{S^2} }[/math] | |
| [math]\displaystyle{ \kappa_{\rho} }[/math] | Turbulent eddy diffusivity via the Osborn (1980) model | [math]\displaystyle{ \kappa_{\rho} = \Gamma \varepsilon N^{-2} }[/math] | [math]\displaystyle{ \mathrm{m^2\, s^{-1}} }[/math] |
| [math]\displaystyle{ D_{ll} }[/math] | Second-order longitudinal structure function | [math]\displaystyle{ D_{ll} = \big\langle[b^{\prime}(r) - b^{\prime}(r+n\delta{r})]^2\big\rangle }[/math] | [math]\displaystyle{ \mathrm{m^2\, s^{-2}} }[/math] |
Fluid properties and background gradients for turbulence calculations
| Symbol | Description | Eqn | Units |
|---|---|---|---|
| [math]\displaystyle{ S_P }[/math] | Practical salinity | [math]\displaystyle{ - }[/math] | |
| [math]\displaystyle{ T }[/math] | Temperature | [math]\displaystyle{ \mathrm{^{\circ}C } }[/math] | |
| [math]\displaystyle{ P }[/math] | Pressure | [math]\displaystyle{ \mathrm{dbar} }[/math] | |
| [math]\displaystyle{ \rho }[/math] | Density of water | [math]\displaystyle{ \rho = \rho\left(T,S_a,P \right) }[/math] | [math]\displaystyle{ \mathrm{kg\, m^{-3}} }[/math] |
| [math]\displaystyle{ \alpha }[/math] | Temperature coefficient of expansion | [math]\displaystyle{ \alpha = \frac{1}{\rho} \frac{\partial\rho}{\partial T} }[/math] | [math]\displaystyle{ \mathrm{K^{-1}} }[/math] |
| [math]\displaystyle{ \beta }[/math] | Saline coefficient of contraction | [math]\displaystyle{ \beta = \frac{1}{\rho} \frac{\partial\rho}{\partial S_P} }[/math] | |
| [math]\displaystyle{ S }[/math] | Background velocity shear | [math]\displaystyle{ S = \left[ \left( \frac{\partial U}{\partial z}\right)^2 + \left( \frac{\partial V}{\partial z}\right)^2 \right]^{1/2} }[/math] | [math]\displaystyle{ \mathrm{s^{-1}} }[/math] |
| [math]\displaystyle{ \nu_{35} }[/math] | Temperature dependent kinematic viscosity of seawater at a practical salinity of 35 | [math]\displaystyle{ \sim 1\times 10^{-6} }[/math] | [math]\displaystyle{ \mathrm{m^2\, s^{-1} } }[/math] |
| [math]\displaystyle{ \nu_{00} }[/math] | Temperature dependent kinematic viscosity of freshwater | [math]\displaystyle{ \sim 1\times 10^{-6} }[/math] | [math]\displaystyle{ \mathrm{m^2\, s^{-1} } }[/math] |
| [math]\displaystyle{ \Gamma_a }[/math] | Adiabatic temperature gradient -- salinity, temperature and pressure dependent | [math]\displaystyle{ \sim 1\times 10^{-4} }[/math] | [math]\displaystyle{ \mathrm{K\, dbar^{-1} } }[/math] |
| [math]\displaystyle{ N }[/math] | Background stratification, i.e buoyancy frequency | [math]\displaystyle{ N^2 = g\left[ \alpha\left(\Gamma_a + \frac{\partial T}{\partial z} \right) - \beta \frac{\partial S_P}{\partial z} \right] }[/math] | [math]\displaystyle{ \mathrm{rad\, s^{-1} } }[/math] |
Theoretical Length and Time Scales
| Symbol | Description | Eqn | Units |
|---|---|---|---|
| [math]\displaystyle{ \tau_N }[/math] | Buoyancy timescale | [math]\displaystyle{ \tau_N = \frac{1}{N} }[/math] | [math]\displaystyle{ \mathrm{s} }[/math] |
| [math]\displaystyle{ T_N }[/math] | Buoyancy period | [math]\displaystyle{ T_N = \frac{2\pi}{N} }[/math] | [math]\displaystyle{ \mathrm{s} }[/math] |
| [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] | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ L_Z }[/math] | Boundary (law of the wall) length scale | [math]\displaystyle{ L_Z=0.39z_w }[/math] with 0.39 being von Kármán's constant | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ L_S }[/math] | Corssin length scale | [math]\displaystyle{ L_S = \sqrt{\varepsilon/S^3} }[/math] | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ L_K }[/math] | Kolmogorov length scale (smallest overturns) | [math]\displaystyle{ L_K=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4} }[/math] | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ L_o }[/math] | Ozmidov length scale, measure of largest overturns in a stratified fluid | [math]\displaystyle{ L_o=\left(\frac{\varepsilon}{N^3}\right)^{1/2} }[/math] | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ L_T }[/math] | Thorpe length scale | [math]\displaystyle{ L_T }[/math] | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ z_w }[/math] | Distance from a boundary | [math]\displaystyle{ z_w }[/math] | [math]\displaystyle{ \mathrm{m} }[/math] |
Turbulence Spectrum
These variables are used to express the Turbulence spectrum expected shapes.
| Symbol | Description | Eqn | Units |
|---|---|---|---|
| [math]\displaystyle{ \Delta t }[/math] | Sampling interval | [math]\displaystyle{ \frac{1}{f_s} }[/math] | [math]\displaystyle{ \mathrm{s} }[/math] |
| [math]\displaystyle{ f_s }[/math] | Sampling rate | [math]\displaystyle{ f_s=\frac{1}{\Delta t} }[/math] | [math]\displaystyle{ \mathrm{s^{-1}} }[/math] |
| [math]\displaystyle{ \Delta s }[/math] | Sample spacing | [math]\displaystyle{ \Delta s = U_P \Delta t }[/math] | [math]\displaystyle{ \mathrm{m} }[/math] |
| [math]\displaystyle{ \Delta l }[/math] | Linear dimension of sampling volume (instrument dependent) | [math]\displaystyle{ \mathrm{m} }[/math] | |
| [math]\displaystyle{ f }[/math] | Cyclic frequency | [math]\displaystyle{ f=\frac{\omega}{2\pi} }[/math] | [math]\displaystyle{ \mathrm{Hz} }[/math] |
| [math]\displaystyle{ \omega }[/math] | Angular frequency | [math]\displaystyle{ \omega = 2\pi f }[/math] | [math]\displaystyle{ \mathrm{rad\, s^{-1}} }[/math] |
| [math]\displaystyle{ f_N }[/math] | Nyquist frequency | [math]\displaystyle{ f_N=0.5f_s }[/math] | [math]\displaystyle{ \mathrm{Hz} }[/math] |
| [math]\displaystyle{ k }[/math] | Cyclic wavenumber | [math]\displaystyle{ k=\frac{f}{U_P} }[/math] | [math]\displaystyle{ \mathrm{cpm} }[/math] |
| [math]\displaystyle{ \hat{k} }[/math] | Angular wavenumber | [math]\displaystyle{ \hat{k}=\frac{\omega}{U_P} = 2\pi k }[/math] | [math]\displaystyle{ \mathrm{rad\, m^{-1}} }[/math] |
| [math]\displaystyle{ \tilde{k} }[/math] | Normalized wavenumber | e.g., [math]\displaystyle{ \tilde{k}=k L_K, L_K = \left(\nu^3/\varepsilon \right)^{1/4} }[/math] | - |
| [math]\displaystyle{ \tilde{\Phi} }[/math] | Normalized velocity spectrum | e.g., [math]\displaystyle{ \tilde{\Phi}_u(\tilde{k}) = \left(\epsilon \nu^5\right)^{-1/4} \Phi_u(k) }[/math] | - |
| [math]\displaystyle{ \tilde{\Psi} }[/math] | Normalized shear spectrum | e.g., [math]\displaystyle{ \tilde{\Psi}(\tilde{k}) = L_K^2 \left(\epsilon \nu^5\right)^{-1/4} \Psi(k) }[/math] | - |
| [math]\displaystyle{ k_\Delta }[/math] | Nyquist wavenumber, based on sampling volume size [math]\displaystyle{ \Delta l }[/math] | [math]\displaystyle{ k_\Delta=\frac{0.5}{\Delta l} }[/math] | [math]\displaystyle{ \mathrm{cpm} }[/math] |
| [math]\displaystyle{ k_N }[/math] | Nyquist wavenumber, via Taylor's hypothesis | [math]\displaystyle{ k_N=\frac{f_N}{U_P} }[/math] | [math]\displaystyle{ \mathrm{cpm} }[/math] |
| [math]\displaystyle{ \Psi(k) }[/math] | Shear spectrum. Use [math]\displaystyle{ \Psi_1 }[/math], [math]\displaystyle{ \Psi_2 }[/math] to distinguish the orthogonal components of the shear. Use [math]\displaystyle{ \Psi_N }[/math] for the Nasmyth spectrum, [math]\displaystyle{ \Psi_{PK} }[/math] for the Panchev-Kesich spectrum and [math]\displaystyle{ \Psi_L }[/math] for the Lueck spectrum. | [math]\displaystyle{ \mathrm{s^{-2}\, cpm^{-1}} }[/math] | |
| [math]\displaystyle{ \Phi(k) }[/math] | Velocity spectrum. Use [math]\displaystyle{ \Phi_u }[/math], [math]\displaystyle{ \Phi_v }[/math], [math]\displaystyle{ \Phi_v }[/math], or [math]\displaystyle{ \Phi_1 }[/math], [math]\displaystyle{ \Phi_2 }[/math] , [math]\displaystyle{ \Phi_3 }[/math] for the different orthogonal components of the velocity. Use [math]\displaystyle{ \Phi_K }[/math] for the Kolmogorov spectrum. | [math]\displaystyle{ \mathrm{m^2\, s^{-2}\, cpm^{-1}} }[/math] |