## 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 L=1,LM

DO J=J_0,J_1

DO I=1,IMAXJ(J)

ediff=0.

DO K=1,KMAXJ(J) ! loop over surrounding vel points

ediff=ediff+DKE(IDIJ(K,I,J),IDJJ(K,J),L)*RAPJ(K,J)

END DO

T(I,J,L)=T(I,J,L)-ediff/(SHA*PK(L,I,J))

END DO

END DO

END DO

!$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.

Dan,

I don’t know Fortran, but this appears to be a different version of ediff than in the other post. In this case, it looks like ediff is an iterative sum. Is DKE a local variable?

Comment by Ryan O | January 13, 2009 |

Sorry . . . should have included this above . . . but what physically does array T represent?

Comment by Ryan O | January 13, 2009 |

Ryan,

I’ve been busy trying to get to the bottom of the theoretical issues. I have concluded that the concept used in ModelE is wrong.

In general it is very difficult to be sure about anything when backing equations out of coding and I have not reconciled the the two. If we could find even a single equation we could be more sure of what the code should be doing.

I might try to figure out the code, but that is not high priority right now.

Comment by Dan Hughes | January 13, 2009 |

Dan,

Thanks for your work. It’s fascinating to read, though some of it goes over my head. No worries on the code . . . I was just curious. 🙂

Comment by Ryan O | January 13, 2009 |