Posted on June 20, 2020

by Gerald Browning

Climate model sensitivity to CO2 is heavily dependent on artificial parameterizations (e.g. clouds, convection) that are implemented in global climate models that utilize the wrong atmospheric dynamical system and excessive dissipation.

The peer reviewed manuscript entitled “The Unique, Well Posed Reduced System for Atmospheric Flows: Robustness In The Presence Of Small Scale Surface Irregularities” is in press at the journal Dynamics of Atmospheres and Oceans (DAO) [link] and the submitted version of the manuscript is available on this site, with some slight differences from the final published version. Link to paper is here: Manuscript

**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 equations of motions for large-scale atmospheric 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.

The mathematical arguments are based on the Bounded Derivative Theory (BDT) for symmetric hyperbolic systems introduced by Professor Heinz-Otto Kreiss over four decades ago and on the theory of numerical approximations of partial differential equations.

What is the relevance of this research for climate modeling? At a minimum, climate modelers must make the following assumptions:

1. The numerical climate model must accurately approximate the correct dynamical system of equations

Currently all global climate (and weather) numerical models are numerically approximating the primitive equations — the atmospheric equations of motion modified by the hydrostatic assumption. However this is not the system of equations that satisfies the mathematical estimates required by the BDT for the initial data and subsequent solution in order to evolve as the large scale motions in the atmosphere. The correct dynamical system is introduced in the new manuscript that goes into detail as to why the primitive equations are not the correct system.

Because the primitive equations use discontinuous columnar forcing (parameterizations), excessive energy is injected into the smallest scales of the model. This necessitates the use of unrealistically large dissipation to keep the model from blowing up. That means the fluid is behaving more like molasses than air. References are included in the new manuscript that show that this substantially reduces the accuracy of the numerical approximation.

2. The numerical climate model correctly approximates the transfer of energy between scales as in the actual atmosphere.

Because the dissipation in climate models is so large, the parameterizations must be tuned in order to try to artificially replicate the atmospheric spectrum. Mathematical theory based on the turbulence equations has shown that the use of the wrong amount or type of dissipation leads to the wrong solution. In the climate model case, this implies that no conclusions can be drawn about climate sensitivity because the numerical solution is not behaving as the real atmosphere.

3. The forcing (parameterizations) accurately approximate the corresponding processes in the atmosphere and there is no accumulation of error over hundreds of years of simulation.

It is well known that there are serious errors in the parameterizations, especially with respect to clouds and moisture that are crucial to the simulation of the real atmospheres. Pat Frank has addressed the accumulation of error in the climate models. In the new manuscript, even a small error in the system impacts the accuracy of the solution in a short period of time.

One might ask how can climate models apparently predict the large-scale motions of the atmosphere in the past given these issues. I have posted a simple example on Climate Audit (reproducible on request) that shows that given any time dependent system (even if it is not the correct one for the fluid being studied), if one is allowed to choose the forcing, one can reproduce any solution one wants. This is essentially what the climate modelers have done in order to match the previous climate given the wrong dynamical system and excessive dissipation.

I reference a study on the accuracy of a primitive equation global forecast model by Sylvie Gravel et al. [link]. She showed that the largest source of error in the initial stages of a forecast are from the excessive growth of the horizontal velocity near the lower boundary. Modelers have added a boundary layer drag/dissipation in an attempt to prevent this from happening. I note in the new manuscript that this problem does not occur with the correct dynamical system and that in fact the correct system is not sensitive to small-scale perturbations at the lower boundary.

**Biosketch:** I am an independent applied mathematician trained in partial differential equations and numerical analysis concerned about the loss of integrity in science through the abuse of rigorous mathematical theory by numerical modelers. I am not funded by any outside organization. My previous publications can be found on google scholar by searching for Browning and Kreiss.

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June 24, 2020 at 12:13PM