# some questions about buoyancy in module_big_step_utilities_em.F

#### Lanzhi Tang

##### New member
we used the WRF output ((w, u, v, P, AL, and ALB) to calculate the vertical velocity equation and found that it was not conserved. Then we further output the buoyancy in subroutine pg_buoy_w of module_big_step_utilities_em.F, and compared the perturbed pressure gradient force, buoyancy and other terms from WRF model and using WRF output data. we found that the result we calculated was similar to the WRF model except for buoyancy, which is almost an order of magnitude larger than WRF model. We wonder why the buoyancy is so different between calculation and WRF model. Beside, we find 1/ ALB is always larger than RHO, is that reasonable?
vertical speed equation:

buoyancy calculation:

Hi,
I first would like to apologize for the delay in response. We've been a bit short-staffed and I wanted to ask someone else who would be able to answer better than I can. They said the model buoyancy has a different form from the one shown in your equation, so it it's difficult to compare terms. ALB is horizontally uniform in the domain, so some areas will generally have higher or lower values relative to the base state profile.

Hi,
Thanks for your help. In WRF V4_technote, the perturbation form of vertical momentum equation is as:
(1)​

In z coordinates, the vertical velocity perturbation equation is as:
(2)​
We think
(3)
(4)​
In Eqs. (3), (4), the left side is the direct output in subroutine pg_buoy_w, and the right side is the diagnosis by using wrfout variables. Actually, the direct output and the diagnosis are almost the same in Eq. (3); but there is an obvious difference in Eq. (4) between the direct output and diagnosis. We wonder why the buoyancy force (Eq.4) is so different when the vertically perturbed pressure gradient force (Eq.3) is equal.

Since the reference state in WRF is in dry hydrostatic balance, we try to use (or derived from the hydrostatic pressure, ) as the reference state of dry air density. It seems that the treatment of dry air density reference state is wrong. Or is there any other reason why the Eq. (4) is not equal here?

Hi,
I just want to let you know I haven't forgotten about you. A couple of my dynamicist colleagues are looking at it and discussing it. I'll keep you posted when I get some clarity. Thank you for your patience.

Hi,
After speaking to our dynamicists, they determined your decomposition of the pressure gradient/buoyancy terms in (1) into a piece for the pressure gradient (3) and a piece for the buoyancy (4) doesn't look right. The (alpha/alpha_d)*mubar*qt term is included as part of the buoyancy contribution (4), but that term originated from the dp/dz term and should instead be included as part of (3). The buoyancy should be equivalent to rho'/rho.

Hi,
That's to say, the buoyancy can be treated as ? But, we found that its result is significantly different from the common buoyancy formulation as , where, denote the domain mean values. The figure shows a precipitation example that occurred in China in July 2021. Our intention here is to quantitatively analyze the vertical momentum budget. But which buoyancy is right or more reasonable?

Figure. Vertical cross sections of (left) WRF outputted buoyancy and (right) diagnosed buoyancy by potential temperature. The black contours represent the air vertical velocity (the solid lines represent upward velocity）.

Hi,
My colleague doesn't believe it's correct to subtract the domain average. The balance is with the local g*al/alb where al = alpha'

Thank you very much for your patient reply. We also believe that it is inappropriate to use the domain-averaged potential temperature to calculate buoyancy. It returns to the original question, i.e., how to calculate the buoyancy term?
According to your suggestion, we calculated g*al/alb. However, its result is quite different from the WRF output buoyancy，as shown in the figure below. We think the WRF reference state is based on the dry hydrostatic balance, al = alt-alb, where alt and alb are the inverses of total dry air density and base dry air density.

Figure. Vertical cross sections of (left) WRF output buoyancy, (middle) diagnosed buoyancy by potential temperature,
and (right) diagnosed buoyancy by g*al/alb. (unit: m s^-2)