# Glossary

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## Contents

### Glossary of Terms for Fluid Dynamics

Add as you feel necessary. When needed, provide a link to a reference page or other terms.

Purpose: Many of the terms on this list have multiple definitions depending on context. The context for these definitions is geophysical and environmental fluid dynamics.

Disclaimer: this list is mostly the result of googling, and as such should not be referenced directly.

Note: The AMS Glossary is a good source for definitions, should the definition that you seek not be available below.

## A-B

The inertia added to a system due to the fact that an accelerating or decelerating body must move some volume of surrounding fluid with it as it moves.
Available Potential Energy
The potential energy available for conversion into other forms of energy. For intuitive purposes one may think of this as $APE = \int_\Omega \left(\rho-\rho^*\right)dV$ where $\rho$ is the density field, and $\rho^*$ is the redistributed density field such that the potential energy is minimized.
Azimuth
The horizontal angle. In polar coordinates, the angular direction.
Barotropic fluid
A fluid in which $\nabla p\times \nabla\rho = 0$. This means that surfaces of constant pressure and constant density are parallel. Fluids in which $p = p(\rho)$ or the density is constant ( $\rho = \rho_0$) are necessarily barotropic.
Baroclinic motion
Motion caused by the misalignment of the surfaces of constant pressure and constant density (i.e. $\nabla p\times \nabla\rho \neq 0$).
Boundary Layer
Region near a boundary in which viscosity becomes important.
Buoyancy Frequency
The frequency at which an infinitesimally perturbed fluid parcel oscillates around its rest state in the absence of friction. Formally given as $N^2(z) = -\frac{g}{\rho_0}\frac{d\overline{\rho}(z)}{dz}$, where $\overline{\rho}(z)$ is the background stratification. $\beta$ - plane
The $\beta$ - plane approximation assumes that the Coriolis frequency varies linearly with latitude i.e. $f=f_0+\beta y$. $f_0=2\Omega\sin(\theta_0)$ and $\beta=\frac{2\Omega}{a}\cos(\theta_0)$ where $\Omega$ is the period of Earth's rotation, $\theta_0$ is the reference latitude, and $a$ is the mean radius of the Earth. Wikipedia's entry on this is a good one. cf. $f$-plane

## C-D

Capillary Wave
Waves in which the dominant restoring force is due to surface tension. Typical length scales are under 7cm (Kundu, 4th ed.).
The advection of particles under the flow map of a chaotic dynamical system.
Characteristic Scale
This scale is context dependent. In an engineering situation like a jet out of a small hole one scale is given by the size of the hole, and another, less easily quantifiable scale will be the length over which the jet mixes with the ambient fluid.
Circulation
The circulation of a flow is the integral of the vorticity over a surface, $\Gamma=\int_{\Omega}\vec{\omega} dA$, and represents the vorticity flux through the surface $\Omega$. By Stokes' theorem, $\Gamma=\int_{\partial\Omega}\vec{u}\cdot ds$.
Coriolis Frequency $f = 2\Omega\sin\theta$, where $\Omega$ is the rate of earth's rotation, and $\theta$ is the latitude (with the northern hemisphere to be positive). See also $f$-plane, $\beta$-plane.
Correlation Time
The time it takes for the auto correlation function of a process to decrease by a given amount.
Diffusivity
Rate of diffusion (unit: m $^2$/s). Mass diffusivity: the rate at which mass (molecular) of the substance diffuses through a unit surface in a unit time. Thermal diffusivity: $\kappa = \frac{k}{\rho C_p}$, where $k$ is the thermal conductivity, $\rho$ is the density, and $C_p$ is the specific heat. Momentum diffusivity: see kinematic viscosity.
Direct Numerical Simulation (DNS)
Simulation in which you make no assumption on turbulence, and typically attempt to resolve as much as possible. cf. LES
Dispersion Relation
For a given wave, the dispersion relation is the relationship between wavenumber, $k$, and (angular) wave frequency, $\omega$; typically written as $\omega = \omega(k)$. See also Phase Velocity, Group Velocity, and Dispersive Waves.
Dispersive Wave
A wave for which the group velocity is dependent on wavenumber, so energy in different wavelengths propagates at different velocities.
Dubreil-Jacotin-Long (DJL) equation
A scalar equation equivalent to the steady, incompressible, stratified Euler equations.
Dynamical System
A set of differential equations describing a classical mechanical system. There are many generalizations and formalisms associated with this concept, but physically the most important point is that the solution to a dynamical system is a time evolution function for the system. There are numerous examples of dynamical systems in classical mechanics, but simple examples include pendulums and predator prey models. See also fixed point, stable manifold, and unstable manifold.

## E-G

When energy is supplied at large scales and then passed to smaller and smaller scales until viscosity causes dissipation.
Enstrophy $\frac{1}{2}\omega^2$, the norm squared of the vorticity.
Euler equations
The Euler equations are the Navier-Stokes equations with the viscous term neglected.
Eulerian measurement
A measurement taken at a fixed position in space. See also Lagrangian measurement.
Fixed Point
For a time-invariant dynamical system $\dot{x} = \vec{v}(\vec{x}), \vec{x}(0) = \vec{x_0}$, a fixed point is a point $\vec{x_f}$ in the domain such that $\vec{v}(\vec{x_f}) = \vec{0}$. Fixed points are important in the analysis of time-invariant systems because they define both stable and unstable manifolds, which divide the domain into regions of different dynamics.
Fluid Parcel
A volume of fluid which (see chapter 1 of Kundu.):
• deforms under outside forces
• is non-diffusive, such that molecules do not cross the boundary
• is large enough to be well defined by thermodynamic quantities
• is small enough to reach internal equilibrium much faster than the background flow $f$ - plane
The $f$ - plane approximation assumes that the Coriolis frequency is constant in latitude i.e. $f=f_0$. Where $f_0=2\Omega\sin(\theta_0)$, $\Omega$ is the period of Earth's rotation, and $\theta_0$ is the reference latitude. cf. $\beta$-plane
Geostrophic Balance
Geostrophic balance occurs when the Coriolis pseudo-force balances the pressure forces. Under geostrophic balance, flow is along lines of constant pressure.
Gravity Wave
A wave in which the dominant restoring force is due to gravity acting to restore displaced mass.
Group Velocity
The velocity with which the energy of a wave packet propagates. Mathematically, the group velocity $\vec{c_g}= \nabla_{\vec{k}} \omega =\left(\frac{\partial \omega}{\partial k},\frac{\partial \omega}{\partial l},\frac{\partial \omega}{\partial m} \right)$, where $\vec{k}=(k,l,m)$ is the wavenumber. The group velocity is closely related to the velocity of a wave envelope .
Gyre
A vortex, a region dominated by a coherent rotating structure.

## H-J

Halocline
Internal Solitary Wave
A single, internal, travelling wave, often located at a pycnocline.
Internal Tide
Internal waves generated at a tidal frequency.
Internal Wave
A wave whose displaced quantity is interior to the fluid. These can include gravity waves and Rossby waves.
Isentropic
Of or having constant entropy. An isentropic surface is a surface of constant entropy.
Isobaric
Of or having constant pressure. An isobaric surface is a surface of constant pressure.
Isohaline
Of or having constant salinity. An isohaline surface is a surface of constant salinity.
Isopycnal
Of or having constant density. An isopycnal surface is a surface of constant density.
Isothermal
Of or having constant temperature. An isothermal surface is a surface of constant temperature.

## K-Q

Korteweg–de Vries (KdV) equation
A nonlinear equation used to describe long internal waves (among other things).
Kelvin Wave
Gravity waves that are boundary trapped and are in geostrophic balance in the direction orthogonal to the boundary. Kelvin waves require the presence of a boundary such as a coastline, channel wall, or the equator. Like Poincaré waves, Kelvin waves are rotating gravity waves. In the Northern (Southern) hemisphere, Kelvin waves propagate with the boundary on the right (left) with respect to the direction of propagation.
Lagrangian Coherent Structures
An invariant manifold which separates the flow of a time-varying system into regions of qualitatively distinct dynamics. They will typically be time-dependent curves, and their definition requires a choice of integration time. A full description of the construction of LCS is beyond the scope of this glossary. The important point for an intuitive understanding is that LCS are time-dependent analogues of separatrices.
Lagrangian measurement
A measurement taken while moving with a particle, i.e., while moving with the fluid. See also Eulerian measurement.
Large Scale Flow
In geophysical fluid dynamics this refers to the flow dominated by the Earth's rotation, so almost geostrophic flow.
Large Eddy Simulation (LES)
A simulation in which a turbulence model has been included to approximate small scale motion and thus reduce the complexity of the problem.
Material Derivative
A Lagrangian quantity, the material derivative describes the rate of change of a function from the perspective of a fluid particle moving with the flow. If $f$ is the function in question, the material derivative is defined as $\frac{Df}{Dt} = \frac{\partial f}{\partial t}+ \vec{u} \cdot \nabla f$. This derivative has many names, including but not limited to advective derivative, hydrodynamic derivative, Lagrangian derivative, and total derivative.
Material Volume
A fixed piece of matter which moves with the flow. It is comprised of the same particles for all time.
Meridional
Along a north-south direction; or along a meridian.
Mixing
This has the standard English language meaning, but there are many more connotations in fluid mechanics.
Navier-Stokes equations
The equations of motion for a Newtonian fluid.
Nepheloid Layer
A layer of water with a high concentration of suspended sediment.
Normal Mode
For a linear PDE, the normal modes are the functions which describe the spatial structure of the standing waves that solve that PDE. We can then approximate any wave that solves the PDE, including non-standing waves, by using the normal modes as a basis.
Particle Image Velocimetry (PIV)
A non-invasive method used to measure whole velocity fields by taking two images shortly after each other, and calculating the distance individual tracer particles have travelled within the time interval. From the known time interval and the measured displacement, the velocity is calculated.
Phase Velocity
The velocity at which a wave crest or trough propagates. Mathematically, $\vec{c_p}=\frac{\omega}{|k|}\hat{k}$ where $c$ is the phase velocity, $|k|$ is the magnitude of the wavenumber, and $\hat{k}$ is the unit vector corresponding to $\vec{k}$. cf. group velocity. The magnitude of the phase velocity, the phase speed, is sometimes called the celerity of the wave.
Poincaré Map
If a dynamical system has a periodic orbit (in either time or space), the trajectories of the particles in that region can be studied using a Poincaré map. Define a plane $S$ which is normal to the flow, which we call a Poincaré section. The Poincaré map is defined as a first return map: given $x \in S$, the flow will map $x$ back onto $S$ after one period, say to the point $x'$. We then define the Poincaré map as $P(x) = x'$. The Poincaré map, when it can be defined, allows us to study trajectories by studying sequences of points on a lower dimensional surface than the original system.
Poincaré Wave
Gravity waves that are slow/large enough to feel the effects of rotation, but for which gravity remains the dominant restoring force. Under the linear Shallow water equations with an f-plane, the dispersion relation for Poincaré waves is $\omega^2 = f_0^2 + c_0^2K^2$, where $f_0$ is the Coriolis frequency, $c_0^2=gH$ is the shallow-water gravity-wave speed, and $K=\frac{2\pi}{\lambda}$ is the horizontal wavenumber.
Pycnocline
Q and R
Two invariants of the velocity gradient tensor which are useful for defining the flow.

## R-Z

Reynolds Decomposition
When studying turbulent flows, quantities can be decomposed into a mean part and a deviation from the mean (also called the turbulent fluctuations). For example, the horizontal velocity $u$ is written as $u = \bar u + u^{\prime}$, where $\bar{u}$ is the mean part and $u^{\prime}$ is the fluctuation. The fluctuations have zero mean, i.e., $ = 0$. Also, if $u$ is stationary in time, then $\bar u$ is the time average of $u$.
Rossby Wave
Waves in which the dominant restoring force is to due the conservation of potential vorticity. These waves are a result of gradient in potential vorticity imparted by the latitudinal variation of the Coriolis frequency. Waves for which the gradient in potential vorticity is provided by topography are termed topographic Rossby waves.
Saltation
One of the ways in which sediment in a bed-load layer may be transported. If the near-bed flow velocity exceeds some critical value, the sediment particles jump downstream and this is termed saltation. The particles stay close to the surface and move in smooth, approximately parabolic trajectories.
Separatrix
A curve in a time-invariant dynamical system which separates the phase space into regions of distinct dynamics. Points which start on different sides of a separatrix will be separated as they follow their trajectories. See also Lagrangian coherent structures.
Shallow water equations
The equations of motion for a fluid where horizontal scales are much larger than the depth.
Soliton
A single wave crest or trough that propagates at a constant speed, without changing its shape, even after interacting with other waves.
Stable Manifold
The stable manifold of a fixed point is the set of all points which are attracted to the fixed point over time. Mathematically, if $\vec{x_f}$ is the fixed point, and $\phi_t$ is the flow map for the dynamical system, then if $\vec{x}$ is a point on the stable manifold, then $\lim \limits_{t \to \infty} \phi_t(\vec{x})= \vec{x_f}$. See also unstable manifold.
Stratification
The way in which a fluids density varies with depth.
Surface Wave
Waves in which the displaced quantity is a water-air interface. These can include gravity waves, Rossby waves, and capillary waves.
Thermocline
Thermohaline Flow
Flow in the ocean due to density gradients which are caused by surface heat (thermal fluxes) and freshwater or saline fluxes.
Tracer
A quantity transported by the flow.
Turbulence Intensity
A quantity that gives a measure of the level or strength of turbulence. It is defined as the ratio of the root-mean-square of the turbulent velocity fluctuations, $u'_{rms}$, to the mean velocity, $\bar u$: Turbulence intensity = $u'_{rms}/\bar u$ = $\frac{\sqrt{\frac{1}{3}\left( {u'^2}_x + {u'^2}_y + {u'^2}_z \right)} }{\sqrt{ {{\overline{u}^2}_x} + {{\overline{u}^2}_y} + {{\overline{u}^2}_z}} }$
Unstable Manifold
The unstable manifold of a fixed point is the set of all points which are repelled by the fixed point over time. Mathematically, if $\vec{x_f}$ is the fixed point, and $\phi_t$ is the flow map for the dynamical system, then if $\vec{x}$ is a point on the unstable manifold, then $\lim \limits_{t \to -\infty} \phi_t(\vec{x})= \vec{x_f}$. See also stable manifold.
A useful quantity in formulating fluid deformation and rotation rates. The strain rate tensor and rotation tensor are defined in terms of this quantity.
Viscosity
A quantity measuring the magnitude of internal friction. Dynamic viscosity: $\mu$ (unit: $\text{kg}\cdot \text{m}^{-1} \text{s}^{-1}$). Kinematic viscosity: $\nu = \mu/\rho$ (unit: $\text{m}^2/\text{s}$), also called momentum diffusivity.
Viscous dissipation rate
The irreversible rate of kinetic energy being converted into internal (or thermal) energy through viscosity. Often defined as $2\mu S_{ij}S_{ij}$ where $S_{ij}$ is the strain rate tensor.
Wave Envelope
The curve which outlines the extremes of a wave packet.
Wave equation
An equation to describe the vibrations of a medium.
Wavelet
A wavelet is a localized wave that are near zero outside of a specific region. Wavelets are used to decompose a given signal into frequency and time. Similar to a wave packet.
Wave Packet
A group of waves which travel together, often as the result of a burst of energy. This leads to the concept of a wave envelope. Also called a wave train.
Zonal
Along an east-west direction; or along a latitude circle.