In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the ocean, which gives rise to wind waves.
Flight dynamics is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of gravity (cg), known as pitch, roll and yaw.
In common usage, wind gradient, more specifically wind speed gradient or wind velocity gradient, or alternatively shear wind, is the vertical gradient of the mean horizontal wind speed in the lower atmosphere. It is the rate of increase of wind strength with unit increase in height above ground level. In metric units, it is often measured in units of meters per second of speed, per kilometer of height (m/s/km), which reduces to the standard unit of shear rate, inverse seconds (s−1).
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.
The Rayleigh–Taylor instability, or RT instability, is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.
In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.
Atmospheric dispersion modeling is the mathematical simulation of how air pollutants disperse in the ambient atmosphere. It is performed with computer programs that include algorithms to solve the mathematical equations that govern the pollutant dispersion. The dispersion models are used to estimate the downwind ambient concentration of air pollutants or toxins emitted from sources such as industrial plants, vehicular traffic or accidental chemical releases. They can also be used to predict future concentrations under specific scenarios. Therefore, they are the dominant type of model used in air quality policy making. They are most useful for pollutants that are dispersed over large distances and that may react in the atmosphere. For pollutants that have a very high spatio-temporal variability and for epidemiological studies statistical land-use regression models are also used.
The log wind profile is a semi-empirical relationship commonly used to describe the vertical distribution of horizontal mean wind speeds within the lowest portion of the planetary boundary layer. The relationship is well described in the literature.
In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.
The Fenske equation in continuous fractional distillation is an equation used for calculating the minimum number of theoretical plates required for the separation of a binary feed stream by a fractionation column that is being operated at total reflux.
In fluid dynamics, the von Kármán constant, named for Theodore von Kármán, is a dimensionless constant involved in the logarithmic law describing the distribution of the longitudinal velocity in the wall-normal direction of a turbulent fluid flow near a boundary with a no-slip condition. The equation for such boundary layer flow profiles is:
Wind engineering is a subset of mechanical engineering, structural engineering, meteorology, and applied physics that analyzes the effects of wind in the natural and the built environment and studies the possible damage, inconvenience or benefits which may result from wind. In the field of engineering it includes strong winds, which may cause discomfort, as well as extreme winds, such as in a tornado, hurricane or heavy storm, which may cause widespread destruction. In the fields of wind energy and air pollution it also includes low and moderate winds as these are relevant to electricity production resp. dispersion of contaminants.
The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Sommerfeld equation determines precisely what the conditions for hydrodynamic stability are.
Atmospheric tides are global-scale periodic oscillations of the atmosphere. In many ways they are analogous to ocean tides. Atmospheric tides can be excited by:
In flight dynamics, longitudinal static stability is the stability of an aircraft in the longitudinal, or pitching, plane under steady flight conditions. This characteristic is important in determining whether a human pilot will be able to control the aircraft in the pitching plane without requiring excessive attention or excessive strength.
For a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation.
The Einasto profile is a mathematical function that describes how the density of a spherical stellar system varies with distance from its center. Jaan Einasto introduced his model at a 1963 conference in Alma-Ata, Kazakhstan.
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x4 ≡ p to that of x4 ≡ q.
Monin–Obukhov (M–O) similarity theory describes non-dimensionalized mean flow and mean temperature in the surface layer under non-neutral conditions as a function of the dimensionless height parameter, named after Russian scientists A. S. Monin and A. M. Obukhov. Similarity theory is an empirical method which describes universal relationships between non-dimensionalized variables of fluids based on the Buckingham Pi theorem. Similarity theory is extensively used in boundary layer meteorology, since relations in turbulent processes are not always resolvable from first principles.
