More on NASA/GISS ModelE Viscous Dissipation: The Units
This post is a continuation of discussions of the modeling of viscous dissipation in GCMs. The issue has recently come up again over at Lucia’s Blog in this thread. The thread at Lucia’s is related to the posts on this Blog here, here, and here. I have also in the past commented on the issue at Climate Audit, Professor Pielke Sr.’s Climate Science, and Real Climate. The dissipation questions come up now and then in several Blogs.
Generally, viscous dissipation is the irreversible conversion of mechanical energy to thermal energy by the flow working against viscous stresses. This occurs at all lengths scales but for turbulent flows the conversion is dominated by the contribution from the Kolmogorov scales which are the smallest and most intense. Bird, Stewart, and Lightfoot, has a complete discussion of the various forms of the energy balance equation for fluid flow, especially accounting of the reversible and irreversible conversions between the various forms of energy.
A very important aspect is that the conversion is irreversible and so always appears as a positive quantity on the right-hand side of the thermal-energy forms of the energy balance equation. The process always increase the thermal-energy content of the fluid.
The units of dissipation, when the dynamic viscosity is used, are given by the dynamic viscosity times the quadratic function of spatial derivatives of components of fluid velocity. The dynamic viscosity units are mu [=] kg/m s and the square of velocity gradients, for example (du/dz)^2 [=] 1/s^2. The product of the dynamic viscosity and square of the velocity gradient, which is the dissipation, has units eps [=] kg/m s^3. Given that Joules has units J [=] N m, where Newtons has units N [=] kg m/s^2, and Watts has units W [=] J/s, the units for eps are also eps [=] W/kg; a power-per-unit-mass quantity.
The kinematic viscosity, the dynamic viscosity divided by the density, mu/rho, has units nu [=] m^2/s. Carrying out the process in the previous paragraph gives the dissipation to have units eps [=] m^2/s^3, or eps [=] W/m^3; a power-per-unit-volume quantity and an un-surprising result given the previous results and the definition of kinematic viscosity.
The units used in the ModelE atmosphere routines for Total Potential Energy (TPE) and Change in Kinetic Energy (KE) which is denoted DKE, are mentioned several times in the code. For example in routine ATMDYN.f is this comment:
!@var DKE change in kinetic energy (m^2/s^2)
and in routine ATMDYN_COM.f this comment where DKE is defined:
!@var DKE change in KE due to dissipation (SURF/DC/MC) (m^2/s^2)
REAL*8, ALLOCATABLE, DIMENSION(:,:,:) :: DKE
So it is clear that the units for DKE are DKE [=] m^2/s^3, and in fact the places in the code at which DKE is calculated, the term is simply a change in the square of a velocity components.
Note that the units of DKE are not the units of viscous dissipation. The viscous dissipation must be multiplied by a quantity that has units of seconds, s, in order to get the units of DKE. Yet the comment where DKE is defined it is characterized to be the change in KE due to dissipation. Such a change in units means that a power, W, density is changed into an energy, J, density; either per-unit-mass or per-unit-volume.
So, now to the original post on this blog and the NASA/GISS ModelE coding that has led to these discussions. That coding is in versions of the ATMDYN routine and appears as follows:
!$OMP PARALLEL DO PRIVATE(I,J,L,ediff,K)
DO K=1,KMAXJ(J) ! loop over surrounding vel points
!$OMP END PARALLEL DO
I have not yet been successful in tracking down all the units for RAPJ(K,J), SHA, and PK(L,I,J).
To me, the most important aspect of this coding is related to the basic concept of viscous dissipation mentioned far above. That is, the viscous dissipation must always represent and increase in the thermal energy of the fluid. My question is, How can changes (possibly with time) in the square of the velocity components always be such that the quantity ‘ediff’ is always negative?
Much of this would not be necessary if we could find documentation for the NASA/GISS ModelE viscous dissipation model.