The Taylor–Goldstein equation is an
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
used in the fields of
geophysical fluid dynamics
Geophysical fluid dynamics, in its broadest meaning, refers to the fluid dynamics of naturally occurring flows, such as lava flows, oceans, and planetary atmospheres, on Earth and other planets.
Two physical features that are common to many of th ...
, and more generally in
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
, in presence of quasi-
2D flows.
It describes the
dynamics of the
Kelvin–Helmholtz instability
The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids. ...
, subject to
buoyancy
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the p ...
forces (e.g. gravity), for stably stratified fluids in the
dissipation-less limit. Or, more generally, the dynamics of
internal waves in the presence of a (continuous)
density stratification
Lake stratification is the tendency of lakes to form separate and distinct thermal layers during warm weather. Typically stratified lakes show three distinct layers, the Epilimnion comprising the top warm layer, the thermocline (or Metalimnion) ...
and
shear flow The term shear flow is used in solid mechanics as well as in fluid dynamics. The expression ''shear flow'' is used to indicate:
* a shear stress over a distance in a thin-walled structure (in solid mechanics);Higdon, Ohlsen, Stiles and Weese (1960) ...
. The Taylor–Goldstein equation is derived from the 2D
Euler equations
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler includ ...
, using the
Boussinesq approximation.
The equation is named after
G.I. Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as " ...
and
S. Goldstein, who derived the equation independently from each other in 1931. The third independent derivation, also in 1931, was made by B. Haurwitz.
Formulation

The equation is derived by solving a
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
ized version of the
Navier–Stokes equation, in presence of gravity
and a mean density gradient (with gradient-length
), for the perturbation velocity field
:
where
is the unperturbed or basic flow. The perturbation velocity has the
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...
-like solution
(
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
understood). Using this knowledge, and the
streamfunction representation
for the flow, the following dimensional form of the Taylor–Goldstein equation is obtained:
:
where
denotes the
Brunt–Väisälä frequency
In atmospheric dynamics, oceanography, asteroseismology and geophysics, the Brunt–Väisälä frequency, or buoyancy frequency, is a measure of the stability of a fluid to vertical displacements such as those caused by convection. More precisely ...
. The
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
parameter of the problem is
. If the imaginary part of the
wave speed is positive, then the flow is unstable, and the small perturbation introduced to the system is amplified in time.
Note that a
purely imaginary Brunt–Väisälä frequency
results in a flow which is always unstable. This instability is known as the
Rayleigh–Taylor instability
The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.
Draz ...
.
No-slip boundary conditions
The relevant boundary conditions are, in case of the
no-slip boundary conditions at the channel top and bottom
and
:
Notes
References
*
{{DEFAULTSORT:Taylor-Goldstein equation
Atmospheric thermodynamics
Atmospheric dynamics
Equations of fluid dynamics
Fluid dynamics
Oceanography
Buoyancy