This is highly technical post. No need to comment on that fact.

G.L. Browning’s paper, *The unique, well posed reduced system for atmospheric flows: Robustness in the presence of small scale surface irregularities*

was published in Dynamics of Atmospheres and Oceans

### About the article

In 1948, Jule Charney made a seminal contribution to atmospheric science by mathematically formalizing concepts related to large-scale atmospheric behavior. Specifically, he introduced a scaling of the atmospheric equations of motion. This scaling provided insights into the physical properties of the trough and ridge pressure systems, which are standard fixtures in daily television forecasts.

These systems possess distinct attributes, such as:

- Horizontal lengths of around 1000 km
- Vertical depths of about 10 km
- Horizontal velocities of 10 m/s and vertical velocities of 0.01 m/s
- Variations from their hydrostatic mean in horizontal pressure and density, usually around 1%

Charney’s approach highlighted the dominance of certain terms in the equations of motion. For instance, the horizontal pressure gradient terms and the Coriolis terms were an order of magnitude larger than other terms in the momentum equations. This suggested that these dominant terms must be nearly equal for the system to evolve according to its described properties. This balance, termed the “geostrophic balance,” remains fundamental in preparing meteorological data for forecast models today. Simultaneously, a finer balance, known as “hydrostatic balance,” is crucial for determining altitude from pressure for aviation purposes.

However, Charney’s proposition of the “quasi-geostrophic system” based on these balances faced challenges. Critics believed it inadequately represented the evolution of large-scale systems. The scientific community gravitated towards a model primarily hinged on hydrostatic balance, the so-called “primitive” or “hydrostatic system.” This model is predominant in most of today’s large-scale weather and climate modeling.

Later, in 1979, Heinz-Otto Kreiss enriched the field with a rigorous mathematical theory to better dissect hyperbolic systems with multiple time scales, encompassing atmospheric, oceanic, and plasma equations of motion. Kreiss’s theory elucidated increasingly accurate “reduced” systems that effectively captured the slow, large-scale motions of the atmosphere. Crucially, the popular hydrostatic system was found to misalign with some facets of Kreiss’s theory. He argued that a slight modification to the quasi-geostrophic system could have rendered it exceptionally precise.

One might pose a question: How does weather prediction using the hydrostatic system appear accurate? Scrutiny by researchers like Gravel et al. showed that significant errors arise from the boundary layer parameterization. Without ad hoc corrections, surface velocities can increase unrealistically. This error can proliferate and lead to a forecast deviation of about 50% within 36 hours. To circumvent this, fresh observational data is regularly fed into the model. However, such an approach isn’t directly translatable to climate models.

Of note, the hydrostatic system is less effective when restricted to a limited geographic scope, such as forecasting solely for the US. This challenge catalyzed deeper inquiry, culminating in the mathematical innovations pioneered by Professor Kreiss. His framework invariably ensures well-posed reduced systems for limited domains and has broader applicability, including mesoscale and equatorial atmospheric behaviors, oceanic dynamics, and plasma physics.

### Abstract

It is well known that the primitive equations (the atmospheric equations of motion under the additional assumption of hydrostatic equilibrium for large scale motions) are ill posed when used in a limited area on the globe. Yet the atmospheric equations of motion for large scale motions are essentially a hyperbolic system that with appropriate boundary conditions should lead to a well posed system in a limited area. This apparent paradox was resolved by Kreiss through the introduction of the mathematical Bounded Derivative Theory (BDT) for any symmetric hyperbolic system with multiple time scales (as is the case for the atmospheric equations of motion). The BDT uses norm estimation techniques from the mathematical theory of symmetric hyperbolic systems to prove that if the norms of the spatial and temporal derivatives of the ensuing solution are independent of the fast time scales (thus the concept of bounded derivatives), then the subsequent solution will only evolve on the advective space and time scales (slowly evolving in time in BDT parlance) for a period of time. The requirement that the norm of the time derivatives of the ensuing solution be independent of the fast time scales leads to a number of elliptic equations that must be satisfied by the initial conditions and ensuing solution. In the atmospheric case this results in a 2D elliptic equation for the pressure and a 3D equation for the vertical component of the velocity.

Utilizing those constraints with an equation for the slowly evolving in time vertical component of vorticity leads to a single time scale (reduced) system that accurately describes the slowly evolving in time solution of the atmospheric equations and is automatically well posed for a limited area domain. The 3D elliptic equation for the vertical component of velocity is not sensitive to small scale perturbations at the lower boundary so the equation can be used all of the way to the surface in the reduced system, eliminating the discontinuity between the equations for the boundary layer and troposphere and the problem of unrealistic growth in the horizontal velocity near the surface in the hydrostatic system.

For references relevant to this discussion see the article

Browning, G. L. “The unique, well posed reduced system for atmospheric flows: Robustness in the presence of small scale surface irregularities.” *Dynamics of Atmospheres and Oceans* 91 (2020): 101143.