A mathematical formalism for circulation in water mass configuration
space

We present a mathematical formalism for water mass analysis and
circulation by formulating mass continuity, tracer continuity,
circulation streamfunction, and tracer angular momentum within water
mass configuration space (q-space), which is defined by an arbitrary
number of continuous 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 for differential and integral calculus but lacks a metric,
with the use of exterior algebra and exterior calculus enabling us to
develop q-space mass and tracer budgets. The Jacobian, which measures
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 together 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 discretizes
to a binning algorithm, thus providing a practical framework for water
mass analysis. Considering q-space defined by tracers, we show that
diffusion is directly connected to local tracer space circulation and
angular momentum. We also show that diffusion, remarkably, cannot change
globally integrated tracer angular momentum (unless different tracers
are diffused differently, as in double diffusion), thus leaving only
boundary processes (e.g., air–sea or land–sea fluxes), or interior
sources to generate globally integrated tracer angular momentum.