Energy Conversion, Mixing Energy, and Neutral Surfaces with a Nonlinear Equation of State

A local neutral plane is defined so that a water parcel that is displaced adiabatically a small distance along the plane continues to have the same density as the surrounding water. Since such a displacement does not change the density field or the gravitational 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 energy E and gravitational potential energy U, and an equal conversion between U and kinetic energy K. While there is thus no net change of U, K does change. This implies that a force is in fact required for the displacement. It is further shown that displacements that are orthogonal to a vector P do not induce conversion between U and K, and therefore do not require a force. Analogously to neutral surfaces, which are defined to be approximately orthogonal to the dianeutral vector N, one may define “P surfaces” to be approximately orthogonal to P. These P surfaces are intermediate between neutral surfaces and surfaces of constant σ0 (potential density reference to the surface). If the equation of state is linear, there exists a well-known expression for the mixing energy in terms of the diapycnal flow. This expression is here generalized for a general nonlinear equation of state. The generalized expression involves the velocity component along P. Since P is not orthogonal to neutral surfaces, this means that stationary flow along neutral surfaces in general requires mixing energy.