| Neutral surfaces are defined so that a water parcel that is displaced adiabatically
along such a surface always has the same density as the surrounding water. Since
such a displacement does not change the density field or the potential energy, it is
generally assumed that it does not produce a restoring buoyancy force. However, it is
here shown that, because of the nonlinear character of the equation of state (in
particular the thermobaric effect), such a ‘neutral´ displacement is accompanied
by a conversion between internal and potential energy, and an equal conversion
between potential and kinetic energy. While there is thus no net change of potential
energy, the kinetic energy does change, implying that there is in fact a restoring
force.
It is further shown that displacements that are orthogonal to a vector P do not induce
conversion between potential and kinetic energy, and therefore do not produce a restoring
buoyancy force. Hence, the properties usually associated with neutral surfaces, which are
orthogonal to the dianeutral vector N, should instead be associated with ‘P -surfaces’, which
are orthogonal to P .
To define P , we must first define the specific potential energy Î (r,t). This is the energy
required to move a fluid parcel with unit volume from a reference level (e.g. the surface) to its
actual depth, taking into account the depth-dependence of the buoyancy force due to
the thermobaric effect. Integrating Î (r,t) over the entire fluid volume gives the
‘incompressible potential energy’ UB, which is different from the true potential
energy U. The sum of the kinetic energy and UB (but not U) is conserved by the
incompressible Boussinesq equations with a nonlinear equation of state in the absence of
dissipation.
P is defined so that a water parcel that is displaced adiabatically along a P -surface always
has the same specific potential energy Î as the surrounding water. Such a displacement does
not change the Î -field, and therefore also not the incompressible potential energy UB or the
kinetic energy.
The helicity of P is nonzero, as is the helicity of N. It is therefore impossible to find
global surfaces that are everywhere exactly orthogonal to P . Hence, it will be necessary to
find approximate P -surfaces by some optimization procedure, similarly as has been done for
neutral surfaces.
The vectors N and P are not parallel, because of the thermobaric effect. Moreover, the
difference between neutral surfaces and P -surfaces is of the same magnitude as the difference
between neutral surfaces and surfaces of constant potential density referenced to a widely
different depth than the local depth. |