In this paper, a unified theory of internal bores and
gravity currents is presented within the framework of
the one-dimensional two-layer shallow-water equations.
The equations represent four basic physical laws:
the theory is developed on the basis of these laws.
Though the first three of the four basic laws are apparent,
the forth basic law has been uncertain.
This paper shows first that this forth basic law
can be deduced from the law which is called
in this paper the conservation law of circulation.
It is then demonstrated that,
within the framework of the equations,
an internal bore is represented by a shock
satisfying the shock conditions
that follow from the four basic laws.
A gravity current can also be treated
within the framework of the equations if the front conditions,
i.e. the boundary conditions to be imposed
at the front of the current, are known.
Basically, the front conditions for a gravity current
also follow from the four basic laws.
When the gravity current
is advancing along a no-slip boundary, however,
it is necessary to take into account the influence
of the thin boundary layer formed on the boundary;
this paper describes how this influence can be evaluated.
It is verified that the theory
can satisfactorily explain the behaviour of internal bores
advancing into two stationary layers of fluid.
The theory also provides a formula for the rate of advance
of a gravity current along a no-slip lower boundary;
this formula proves to be consistent with some empirical formulae.
In addition, some well-known theoretical formulae
on gravity currents turn out to be obtainable
on the basis of the theory.
The motion of a gravity current in a channel
can be studied within the framework of
the one-dimensional two-layer shallow-water equations
if the front conditions, i.e. the boundary conditions to be imposed
at the front of the gravity current, are known in advance.
This paper presents the front conditions
for a gravity current advancing in a background flow
along a no-slip boundary.
The conditions take different forms
depending on whether or not the background flow is a ``headwind''.
It is also shown that the conditions are
directly applicable to a stationary gravity current.
In geophysical fluid dynamics,
it is believed that, under the Boussinesq approximation,
the equation of continuity need not be satisfied.
This note proves that the mass flux density
of a fluid divided by the density
is the momentum per unit mass of the fluid.
This indicates, since the velocity of a fluid is defined
as the momentum per unit mass,
that the density and the velocity of a fluid
must always satisfy the equation of continuity.
The belief in geophysical fluid dynamics
is thus shown to be groundless.
This paper presents a theory describing the energy budget of a fluid
under the Boussinesq approximation:
the theory is developed in a manner
consistent with the conservation law of mass.
It shows that no potential energy is available
under the Boussinesq approximation,
and also reveals that the work done by the buoyancy force
due to changes in temperature corresponds to
the conversion between kinetic and internal energy.
This energy conversion, however,
makes only an ignorable contribution to the distribution of temperature
under the approximation.
The Boussinesq approximation is, in physical oceanography,
extended so that the motion of seawater can be studied.
This paper considers this extended approximation as well.
Under the extended approximation,
the work done by the buoyancy force due to changes in salinity
corresponds to the conversion between
kinetic and potential energy.
It also turns out that the conservation law of mass
does not allow the condition that the fluid velocity is solenoidal
to be imposed under the extended approximation;
the condition to be imposed instead is presented.
This note derives the Boussinesq approximation
in a manner consistent with the conservation law of mass.
It is shown that the governing equations of a fluid
under the approximation can be obtained
on the basis of the assumption that the fluid is incompressible,
in the sense that the density of the fluid is constant.
The equation of motion, in particular, can be formulated
with the help of the conservation law of energy.
The conditions for the approximation to be valid are also discussed.
The work done by the buoyancy force in the Boussinesq approximation
is customarily interpreted as corresponding to the conversion
between potential and kinetic energy.
Studies on the energy balance
of the thermohaline circulation in the oceans
have been conducted on the basis of this interpretation;
the circulation, as a result, is believed
to revolve by consuming the potential energy of the oceans.
This document shows, however, that
the work done by the buoyancy force in the approximation
corresponds to the conversion between internal and kinetic energy;
the internal energy of the oceans is thus shown to be
the source of the kinetic energy of the thermohaline circulation.
This note extends the Boussinesq approximation
to a two-component fluid,
in a manner consistent with the conservation law of mass.
It is shown that the governing equations for a two-component fluid
can systematically be formulated,
with the aid of the conservation law of energy,
on the following assumption:
the density of the fluid is a function
solely of the concentration of one component,
but the thermal expansion coefficient of the fluid
does not vanish.
It is also shown that
the velocity of the fluid cannot in general be solenoidal
under the extended approximation.
This note deals with internal gravity waves
in a single-component fluid and those in a two-component fluid
in a physically reasonable manner.
As a result, it becomes apparent
that internal gravity waves can be classified, on the basis of
their source of kinetic energy, into two categories:
thermal and non-thermal internal gravity waves.
The source of the kinetic energy of a thermal internal gravity wave
is, in spite of its being called an internal ``gravity'' wave,
the internal energy of the fluid.
In contrast, that of a non-thermal internal gravity wave
is the gravitational potential energy of the fluid:
waves in this category can exist only in multi-component fluids.
This note analyzes the process of displacing
a fluid element vertically in a layer of fluid.
It is shown that,
when the thickness of the fluid layer is very small
compared with the height over which
the density of the fluid element changes significantly,
the fluid element may be regarded as incompressible.
However, it also turns out that the temperature of the element
changes, no matter how thin the fluid layer may be,
at the adiabatic lapse rate;
it becomes apparent, consequently, that
an incompressible fluid, as well as a compressible fluid,
undergoes adiabatic heating and cooling.
The potential energy density
of an internal gravity wave
in a thermally stratified fluid
is stored, at least partly,
in the internal energy of the fluid.
This note verifies this fact
on the basis of a simple thought experiment.
This paper reconstructs the anelastic approximation
in such a manner that it can be applied to any kind of fluid.
As a result, it turns out that the work done by the buoyancy force
in the approximation corresponds to the conversion
between kinetic and internal energy.
Also, the conditions for the applicability
of the approximation are clarified.
It is demonstrated, in addition, that the Boussinesq approximation
is not a limiting case of the anelastic approximation.
This paper constructs a variant of the anelastic approximation;
its energetics and the conditions for its applicability,
in addition, are elucidated.
It is shown that the Boussinesq approximation
is reproducible as a limiting case of this variant.
On the other hand, it also turns out
that the variant is inapplicable to an ideal gas.
This paper deduces the conditions necessary
for sound waves to be ignorable in a deep fluid layer.
On the basis of these conditions,
it is shown that there exists only one physically meaningful
soundproof approximation for a deep layer of ideal gas:
the original anelastic approximation of Ogura and Phillips (1962).
This paper extends the anelastic approximation
to a two-component fluid.
Under the extended approximation, a force arises owing to
changes in concentration of one component;
it is demonstrated that the work done by this force
corresponds to the conversion between kinetic and internal energy.
Furthermore, the conditions under which
the extended approximation is applicable are formulated.