Questions about Exner function variables

[mkavulich]

In my review/update process of the CCPP metadata, I've developed some confusion about the true definition of several Exner function variables in the CCPP physics. There are five such variables used in the physics:

• dimensionless_exner_function, locally prslk or exner or pi2d, widely used across most physics
• dimensionless_exner_function_at_interface, locally prsik, used in CONV/RAS
• surface_dimensionless_exner_function, locally psk or prsik_1, used in most PBL and SFC schemes, GFS sfc diag
• dimensionless_exner_function_at_surface_adjacent_layer, locally prslk_1, used in MYJ PBL and SFC, GFS sfc diag
• ratio_of_exner_function_between_midlayer_and_interface_at_lowest_model_layer, locally prslki, used in , UFS sfc diag

This last one, prslki/ratio_of_exner_function_between_midlayer_and_interface_at_lowest_model_layer, is the one that is causing me some confusion. This variable uses the standard name ratio_of_exner_function_between_midlayer_and_interface_at_lowest_model_layer, which is somewhat strange to start with: what utility or physical meaning does a ratio of exner functions have? So, trying to determine what exactly this variable is, I perused the source code, and unfortunately I found even more confusion.

In SFC_Models/Land/RUC/lsm_ruc.F90 and SFC_Models/Land/Noah/lsm_noah.f, the comments say that it prslki is “dimensionless exner function at layer 1” (though it doesn't seem to actually be used in RUC LSM). In the Noah LSM code, this seems to agree with what I see, as it's used to calculate potential temperature from physical temperature at level 1:

https://github.com/NCAR/ccpp-physics/blob/main/physics/SFC_Models/Land/Noah/lsm_noah.f#L368

This also seems to align with the usage seen in the GFS SST scheme:

https://github.com/NCAR/ccpp-physics/blob/main/physics/SFC_Layer/UFS/sfc_nst.f90#L305-L309

However (and this may be the source of the initial confusing name), in NoahMP LSM the comments label this variable as "Exner function bt midlayer and interface at 1st layer" (which again, doesn't really make sense to me; assuming "bt" means "between", what is the utility of calculating something between the midlayer and interface?). The logic of the scheme itself gives some hint of what this may mean: in this block of code, it appears to be used to calculate an "alternate" virtual potential temperature for "zlvl 1", using "local potential temperature".

https://github.com/NCAR/ccpp-physics/blob/main/physics/SFC_Models/Land/Noahmp/noahmpdrv.F90#L1493-L1497

I do not really understand what the "local potential temperature" is supposed to be...is it potential temperature with respect to surface pressure rather than 1000 hPa? My best guess is that this is the intended meaning: dimensionless_exner_function_at_surface_adjacent_layer is supposed to be the exner function at the lowest model layer, but with respect to the surface pressure rather than the typical 1000 hPa. In which case, a more accurate standard name would be dimensionless_exner_function_wrt_surface_pressure_at_surface_adjacent_layer.

So to sum this all up, I believe ratio_of_exner_function_between_midlayer_and_interface_at_lowest_model_layer is a problematic variable that is either described incorrectly, being used incorrectly, or both. It's possible this is not confusing at all to those with more deep knowledge of the physics schemes, but I hope someone can help shed some light on if this standard name is accurate/correct in all these contexts it is being used in, and if not, what should the correct name be?

[mkavulich]

I have just discovered some more layers of confusion:

• In Interstitials/UFS_SCM_NEPTUNE/scm_sfc_flux_spec.meta, the local variable exner_inverse is assigned this same confusing standard name, ratio_of_exner_function_between_midlayer_and_interface_at_lowest_model_layer, despite seeming to be an entirely different quantity: from both the name (exner_inverse) and the context of the source code, it appears to indeed be the inverse exner function, which makes no sense from the standard name assigned.
• In Interstitials/UFS_SCM_NEPTUNE/GFS_surface_generic_pre.meta, the local variable work3 is assigned this standard name ratio_of_exner_function_between_midlayer_and_interface_at_lowest_model_layer....and here it actually seems to be calculating that exact thing! I believe that since this is interstitial code, it was written after the fact based on the initially misleading standard name.

I'll again reiterate that it's entirely possible I just do not understand the physics code or even the calculations/theory, but I believe there is at least something wrong here.

[JongilHan66]

@mkavulich As far as I know, prsik_1=Ps^(R/cp) and prslk_1=P1^(R/cp), where Ps is model surface pressure, P1 is pressure at the model first layer, R is a gas constant (~287J/K), and cp is a heat capacity (~1005 J/(K kg).

exner_inverse=work3=prsik_1/prslk_1=(Ps/P1)^(R/cp)
th1 = t1 * exner_inverse, where th1 is potential temperature at the model first layer and t1 is temperature at the model first layer. So, exner_inverse is the inverse exner function at the model first layer.

Note that in GFS, Ps rather than 1000mb is used to compute exner function, which is (P/Ps)^(R/cp).

[mkavulich]

@JongilHan66 Thanks for the reply. If I'm understanding this correctly, the variables as you understand them are

exner_inverse = (Ps/P)^(R/cp) = reciprocal of dimensionless exner function with respect to surface pressure
prslk_1 = P^(R/cp) = numerator of dimensionless exner function for layer 1 wrt surface pressure
prsik_1 = Ps^(R/cp) = denominator of dimensionless exner function for layer 1 wrt surface pressure

prslk and prsik I assume are the same as above but for the full depth of the atmosphere

Is this in the context of the dycore, or some part of the physics?

[JongilHan66]

@mkavulich It is used in the physics.

[joeolson42]

As Jongil points out, in column-based physical parameterizations, Ps can be used in place of 1000mb in many cases since there is no lateral communication with neighboring profiles necessary for most/all calculations, but I think 1000mb is the standard. From a diagnostics perspective, you would NEVER use ps! For example, tracking frontal progression across mountainous terrain is generally easier when using the standard 1000mb because the horizontal temperature gradients on the 2-m potential temperature field (not the first field I would use, but...) will be more due to different air masses rather than terrain elevation. Personally, I use 1000mb in all column model applications too.