A probabilistic theory for balance dynamics

A theory for balance dynamics is proposed in a probabilistic framework, assuming that the state variables and the master, or control, variables are random variables described by continuous probability density functions. Balance inversion, defined in the usual way as recovering the state variables from the control variables, is achieved through Bayes theorem. Balance dynamics is defined by the propagation of the joint probability density of the state and control variables through the Liouville equation. Assuming Gaussian statistics, balance inversion reduces to linear regression of the state variables onto the control variables, and balance dynamics reduces to the propagation of the mean and covariance of the state and control variables. Example solutions will be used to illustrate the recovery of gravity waves by balance inversion and the limitations of potential vorticity as the control variable in this framework.