# 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.

EDIT 1: My question is strongly related to the concept of geostrophic adjustment, or rather gradient-wind adjustment, for that matter. The question can then be rephrased as: "Why is it acceptable to assume in the vortex adjustment calculation that the final state is in gradient-wind balance?"

EDIT 2: In Appendix A of Vallis 1992 (cited above), the author shows that if the same calculation is done to quadratic order instead of linearly, the result is, according to the author, gradient wind balance up to two additional terms.

• good question by the way. I am dealing with non linear balance numerically but i totally can relate to the absence of a non analytical solution of NS. I presume you are aware of the approximations that NWS folks make and that really hurts the physics behind the phenomena concerned – gansub Jan 14 '19 at 2:16
• Thanks. By NWS, you mean the National Weather Service? I mean, all weather forecasting models have extensive parametrisations of all kinds of processes which are by definition approximations, since they do not resolve all the action. In particular, since internal gravity waves can be very small-scale they are usually parametrised. What approximation were you referring to precisely though? – Cyclone Jan 21 '19 at 9:49
• Apologies. I meant NWP instead of NWS. Yes I mean the entire gamut of approximations made by NWP models. For instance in calculating PV anomalies using the NEB the definition of what an anomaly is subjective. Then the various parameterizations are all approximate. So the true physics behind say a tropical cyclone cannot be easily evaluated. – gansub Jan 21 '19 at 13:05
• Yes, that's right, NWP is far from resolving all physics involved. Existing models are necessarily incomplete (and will be in the foreseeable future), but chances are a complete model would be just complex as reality itself and not very useful. On this point I agree with Isaac Held that model hierarchies are the only way to tackle these problems: journals.ametsoc.org/doi/pdf/10.1175/BAMS-86-11-1609. The best intuitive understanding may arise from simple models which don't require including all the elements of the phenomenon. On the other hand, quantitative modelling does require this. – Cyclone Feb 4 '19 at 9:45
• thanks for the link. From reading about you offline you are a subject matter expert. It would be great if you could spend some time on earthscience.stackexchange.com there are several unanswered tropical cyclone and indeed meteorology questions. – gansub Feb 4 '19 at 10:08

Well, this equation

$$\frac{v_{gr}^2}{r} + fv_{gr} = \frac{1}{\rho}\frac{\partial p}{\partial r}$$

Pretty much seams like a Froude number, written in the form where $$p=\rho g h$$ and when ignoring the coriolis parameter $$f$$ as then it writes in a plain form approx;

$$\frac{v_{gr}^2}{r} = \frac{1}{\rho}\frac{\rho g h}{r}$$

Which end's up being;

$$\frac{v_{gr}^2}{gh} = 1$$

And this is pretty much the Froude number equation;

$$Fr=\frac{v^2}{gL}$$

And this $$Fr=1$$ describes the minimum energy condition (maximum flow with the available energy) in open channel flow. This Froude number has been widely examined in open channel flows, and I propose the book of Hubert Chanson "The Hydraulics of Open Channel Flow: An Introduction" for reading about this.

I my self btw. consider that the athmospheric phenomenon called "morning glory cloud" is also explainable through these same principles; To me it's "just" a hydraulic jump, which is fully defined through froude-number.

I am pretty sure that this extremum principle defined through Froude-number applies and works also here.