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.

The annual mean net surface heat fluxes (NSHFs) from the ocean to the atmosphere play an important role in driving both atmospheric circulations and oceanic meridional overturning circulations. Those generated by historical forcing simulations using the UK HadGEM3-GC3.1 coupled climate model are shown to be relatively independent of resolution, for model horizontal grid spacings between 1 and 1/12 degree, and to agree well with those based on the DEEPC analyses for the period 2000-2009. Interpretations of the geographical patterns of the NSHFs are suggested that are based on relatively simple dynamical ideas. As a step toward investigation of their validity, we examine the contributions to the rate of change of the active tracers (potential temperature, salinity and potential density) from the main terms in their prognostic equations as a function of the active tracer and latitude. We find that the main contributions from vertical mixing occur in “near surface” layers and that, except at high latitudes, the time-mean advection of potential temperature and density is well anti-correlated with the sum of the surface fluxes and vertical diffusion. By contrast, the tracer budget for the salinity has at least four terms of comparable magnitude. The heat input by latitude bands is shown to be dominated by the NSHFs, the time-mean advection, and the equatorial Pacific. Expressions for global integrals of the salt and heat content tendencies due to advection as a function of salinity and potential temperature respectively are derived and shown to make contributions that should not be neglected.