Consider a stratified state of the fluid in which 
(or 
) is a function of position x, y, z (with z as the
local vertical coordinate such that 
). Let
which has units of force per unit mass per unit length
or simply 
.
The vertical component of G, i.e. 
is a measure of the stability of the stratified fluid
For the stable state we can define a neutral surface
as one everywhere normal to the vector
field of G.
Consider a unit mass of fluid at point A which is displaced
adiabatically and without mixing to some point B. Let 
be the vector displacement A to B, assumed small enough that
G at B and A are nearly the same. For the stable state
a vertical restoring force 
will exist tending to accelerate the parcel back to the neutral
surface (for B above the surface the parcel is heavier than its
environment and is forced downwards; the opposite is true if B
lies below the neutral surface containing A). If B lies on
the surface containing A then no restoring force exists and
there is no work required in producing a displacement
along the neutral surface.
If one ignores all forces and accelerations (viscous or
Coriolis) other than that due to 
then for a unit mass with no change of 
or S
For stable conditions and constant G a solution is
where 
is the component of normal to the
neutral surface, while 
is the horizontal component.
Also
which is the Brunt-Väisälä frequency for the stably
stratified case.
For the very special case of 
= 0 the whole fluid
region for which this is true is neutral and d /dt =
constant is a solution.
An alternate form of (2.31) is
For the special case of S uniform then G is proportional
to 
and hence an isentropic surface (or equivalently
a surface of constant 
) is a neutral surface.
In particular it can be shown from (2.15), (2.16) and (2.21)
that
for the case of uniform S.
In fact for dry air the above simplifies to
.
For the more general case where S is not uniform and where S is
not a function of , then it is not possible exactly to
characterize the neutral surfaces by constancy of a particular
physical property.
However, there exist finite regions of the ocean which do
possess well correlated 
relations, and for these
regions the potential specific volume 
or the potential density
serve to approximately
characterize the neutral surfaces.
If one employs the hydrostatic approximation for 
in (2.31) and teh definition of the sound speed c (Eq. 2.19)
then
