Mathematics of circulation in arbitrary fluid property spaces

Projecting fluid systems onto coordinates defined by fluid properties
(e.g., pressure, temperature, tracer concentration) can reveal deep
insights, for example into the thermodynamics and energetics of the
ocean and atmosphere. We present a mathematical formalism for fluid flow
in such coordinates. We formulate mass conservation, streamfunction,
tracer conservation, and tracer angular momentum within fluid property
space (q-space) defined by an arbitrary number of continuous fluid
properties. Points in geometric position space (x-space) do not
generally correspond in a 1-to-1 manner to points in q-space. We
therefore formulate q-space as a differentiable manifold, which allows
differential and integral calculus but lacks a metric, thus requiring
exterior algebra and exterior calculus. The Jacobian, as the ratio of
volumes in x-space and q-space, is central to our theory. When x-space
is not 1-to-1 with q-space, we define a generalized Jacobian either by
patching x-space regions that are 1-to-1 with q-space, or by integrating
a Dirac delta to select all x-space points corresponding to a given q
value. The latter method discretises to a binning algorithm, providing a
practical framework for analysis of fluid motion in arbitrary
coordinates. Considering q-space defined by tracers, we show that tracer
diffusion and tracer sources drive motion in q-space, analogously to how
internal stresses and external forces drive motion in x-space. Just as
the classical angular momentum of a body is unaffected by internal
stresses, the globally integrated tracer angular momentum is unaffected
by tracer diffusion — unless different tracers are diffused
differently, as in double diffusion.