Different in UREF in NS10 and NH18 about hn2016_falwa

[AminFazlKazemi]

Hi Dr. Huang
This is Amin Fazl kazemi from I.R.Iran. I've just downloaded your description of SOR solving method to find UREF. It is published February 2020 and refers to HN2018 paper, Equation S3. By the way I find no equation marked as "S3" in HN2018 , Atmospheric blocking as a traffic jam in the jet stream. Could you please Help.

By the way I tried to solve Equation 12 of Nakamura&Solomon (2010) to find delta_u , the zonal adjustment to zonal mean zonal wind to yield largrangian mean , UREF
By the way the result is apparently not conserved by time. I donot know what wrong I have done.

Thanks

[csyhuang]

@AminFazlKazemi Equation S3 is in the supplementary materials of the Science paper:

In realistic atmosphere, Uref is not conserved due to non-conservative processes (dissipation, diabatic heating etc). Do I answer your question?
By the way, please open a new issue when you raise questions. Thank you.

[AminFazlKazemi]

Thanks for description. My PHD thesis topic is about studying diabatic/disssipative processes effects using your LWA conservation law. By the way, I have tried to solve equation 12 Nakamura/Solomon 2010 paper to yield delta_U =UBAR- UREF (UBAR=zonal_mean_zonal_wind) but the result is different from equation S3 haung/Nakamura 2018 paper. of course S3 results apparently remains nearly conserved but my own result out of Nakamura/Solomon paper formula does not. I want to know whether both formulas are to result in the same pattern Thanks

…

On Fri, Aug 19, 2022 at 4:05 AM Clare S. Y. Huang ***@***.***> wrote: @AminFazlKazemi <https://github.com/AminFazlKazemi> Equation S3 is in the supplementary materials of the Science paper: [image: Screen Shot 2022-08-18 at 6 32 05 PM] <https://user-images.githubusercontent.com/14957917/185512771-1b2e9a7e-0b0a-4fe5-8a9f-c07ff1a63c99.gif> In realistic atmosphere, Uref is not conserved due to non-conservative processes (dissipation, diabatic heating etc). Do I answer your question? By the way, please open a new issue when you raise questions. Thank you. — Reply to this email directly, view it on GitHub <#47 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AMDBA3JFIFWHJ4LJOSHQM6LVZ3CD7ANCNFSM52EH56TQ> . You are receiving this because you were mentioned.Message ID: ***@***.***>

[csyhuang]

@AminFazlKazemi On top of the variable of the equations being different, NS10 and NH18 are using different boundary conditions. NS10 uses the two poles as boundary, while NH18 is using the equator and the north pole as boundaries. We switched to hemispheric boundary conditions because I encountered the issue that anomalies in the southern hemisphere is causing problematic gradient of Qref in northern hemisphere. This hemispheric boundary conditions suffers from sensitivity on equator field values though, and we switched to using absolute vorticity at $5^\circ$ N as equatorward boundary condition and use direct solver (instead of SOR), which gives a much better signal.

When we tried both boundary conditions, there are substantial difference in Uref at topics and subtropics, but at midlatitude, their patterns were similar. What's the difference you observed between the two cases? Did I answer your question?

[AminFazlKazemi]

Hi Dr. Huang
I used this code to solve UREF based on Nakamura_Solomon 2010:
` @numba.guvectorize(['(float64[:],int64,float64,float64[:,:],float64[:,:],float64[:],float64[:],float64[:],float64[:,:],int64,int64,float64,float64[:,:])']
, '(z),(),(),(m,z),(m,z),(m),(z),(z),(m,z),(),(),()->(m,z)',target='cpu', nopython=True)

def density_weighted_elliptic_solver(A,max_iter,acceptable_precision,first_guess,epsilon,dmu2,dz2,ru0,forcing,ny,nz,alfa,result):
dz=dz2**.5
dmu=dmu2**.5
precision=1000.
iter_=0
old=first_guess.copy()
for iter_ in range(1,max_iter+1):
for i in range(1,ny-1):
for j in range(int(A[0]),nz-1):

   #print(j,i)  
   if j==0: 
     coeff=1./(ru0[j]*epsilon[i,j]/(ru0[j]*dz2[j]))
     t=forcing[i,j]-( ((ru0[j+1]*epsilon[i,j+1]*(first_guess[i,j+2]-first_guess[i,j+1])-ru0[j]*epsilon[i,j]*first_guess[i,j+1])/(dz2[j]))/ru0[j])
     first_guess[i,j] = (1.-alfa)*first_guess[i,j]+  alfa*coeff*t
   else: 
     epsilon_p=.5*(epsilon[i,j]+epsilon[i,j+1])
     epsilon_m=.5*(epsilon[i,j]+epsilon[i,j-1])
     ru0_p=.5*(ru0[j]+ru0[j+1])
     ru0_m=.5*(ru0[j]+ru0[j-1])
     zarib=1./(-(ru0_p*epsilon_p+ru0_m*epsilon_m)/(ru0[j]*dz2[j])-2./((.25*(dmu[i+1]+dmu[i-1]+2.*dmu[i]))*dmu[i]))    
     t=forcing[i,j]-((first_guess[i+1,j]/(.5*(dmu[i]+dmu[i+1]))+first_guess[i-1,j]/(.5*(dmu[i]+dmu[i-1])))/dmu[i] + (ru0_p*epsilon_p*first_guess[i,j+1]+ru0_m*epsilon_m*first_guess[i,j-1])/(dz2[j])/ru0[j])
     first_guess[i,j] = (1.-alfa)*first_guess[i,j]+  alfa*coeff*t
     

#diff=numpy.abs(calc(first_guess,epsilon,ru0,zs,mus)-forcing)

precision=abs(first_guess[0,0]-old[0,0])
#print("p_",precision)

for i in range(1,ny-1):
  for j in range(int(A[0]),nz-1):
     n=abs(first_guess[i,j]-old[i,j]) 
     if n>precision:
         precision=n
if precision<=acceptable_precision:
    print(iter_,precision)
    break
else:
    print(iter_,precision)
    old=first_guess.copy()

for i in range(1,ny-1):
for j in range(int(A[0]),nz-1):
result[i,j]=first_guess[i,j]
`
I used mu=sin(latitude) directly to differentiate. The program converges but the result is very low amplitude. In fact the resulted UREF is much the same as UBAR and there is something wrong with it.

forcing is the right-hand side of nakamura-Solomon equation 12.

vector A is used to switch between no-slip and surface wave activity boundary condition .

I also coded OPTIMIZED SOR method for NH18-S3 the same way. the result is quite favorable and apparently conserved to a great extent. It is very similar to HN2016_FALWA package and is completely symmetrical around equator.

I would be thankful if you provide me with descriptions about your new boundary condition using absolute vorticity at 5 degrees North instead of ubar+FAWA at equator so that I can apply and test it in my code.

Doc2.pdf

[csyhuang]

Hi @AminFazlKazemi ,
in NHN22 we use direct solver instead of SOR since we have MKL installed. The numerical procedures are outlined in supplementary materials:
https://agupubs.onlinelibrary.wiley.com/action/downloadSupplement?doi=10.1029%2F2021GL097699&file=2021GL097699-sup-0001-Supporting+Information+SI-S01.pdf
Equation (16) is the boundary condition I mentioned.

The part of code using direct solver is already available in this package:

hn2016_falwa/scripts/nhn_grl2022/sample_run_script.py

Line 117 in e287b2a

qref, uref, tref, fawa, ubar, tbar = qgfield_object._compute_qref_fawa_and_bc()

The lines doing matrix inversion:

hn2016_falwa/hn2016_falwa/oopinterface.py

Lines 884 to 885 in e287b2a

	lu, piv, info = dgetrf(qjj)
	qjj, info = dgetri(lu, piv)

But this version only works for data with 1 deg x 1 deg lat-lon grid. I am going to generalize this to accommodate arbitrary grids in the upcoming release.

I suggest you use intel MKL version of numpy/scipy to do the matrix inversion. It is very fast and more accurate.