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.

 

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.

 

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.

 

 

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