# Extremum principle for gradient wind balance

Dear Physics StackExchange,

as is well known, analytic solutions of the Navier-Stokes equations are only available for very special flows. Therefore, in most fields concerned with fluid dynamics, a lot of work relies on numerical solutions. The problem is that it is often difficult in numerical solutions to discern the basic underlying physical mechanisms that determine the structure of the solution. Often, this is a consequence of having included terms in the equations which are irrelevant to the study of the phenomenon under consideration.

In the theoretical study of atmospheric and oceanic fluid dynamics it has proved fruitful to consider "balanced models" which are models where certain terms in the NS equations balance, while the other terms are neglected. One major advantage of such balanced models is that they filter high-frequency waves out of the dynamics, thus uncovering the so-called "slow quasi-manifold". A concise introduction to balanced flow is given here.

Two prominent such balanced models are the geostrophic balance (between Coriolis and pressure gradient force) and the gradient wind balance (between Coriolis, pressure gradient force and centrifugal force), formally

$$f \textbf{k} \times \textbf u_{geo} = - \frac{1}{\rho}\nabla p$$ and $$\frac{v_{gr}^2}{r} + fv_{gr} = \frac{1}{\rho}\frac{\partial p}{\partial r},$$

where in the first equation $$\textbf u_{geo}=(u_{geo},v_{geo}$$ is the geostrophic wind, $$f$$ is the Coriolis parameter, $$\textbf{k}$$ is the local vertical, $$\rho$$ density, $$p$$ is pressure and $$\nabla=(\partial_x,\partial_y)$$, i.e. the first equation is a vector-valued equation in Cartesian coordinates. The second equation is a scalar equation in cylindrical coordinates, $$r$$ is the radius coordinate and $$v_{gr}$$ is the so-called gradient wind which is azimuthal.

The geostrophic model is shown in (Vallis 1992) to be, to linear order, the state of minimum energy subject to the constraint that potential vorticity is conserved by the flow (at least for a shallow water model). The author gives an intuitive interpretation why this might be expected. As a side note: the geostrophic balance also follows by simple scale analysis of the Navier Stokes equations, but the fact that it may be derived from an extremum principle as the state of minimum energy, IMHO, adds significant understanding to the physical meaning of this balance.

I have been trying to find a similar extremum principle for the gradient wind balance along similar lines, but so far all my attempts failed. Typically, it is only stated that it follows from scale analysis of the equations of motion. In the literature I have indeed found extremum principles that lead to a generalised gradient wind balance (such as equation (7.161) here (link broken, to be fixed)), but the authors begin by assuming that deviations from gradient wind balance are small. Therefore, while such derivations formulate a Hamiltonian to derive the balance equations, they do not have the explanatory power of Vallis' demonstration.