For testing, I ran a POP T62_g37 case for three days and compared to a baseline. I did this in three steps:

1. On master, I verified that removing ALK from tracer_restore_vars only changed the ALK_RESTORE_TEND field (as pointed out in tracer restoring tendency is only added to tracer tendencies for NO3, SiO3, PO4 #213 we do not apply the restoring tendency to ALK)

2. Then I applied the changes in this pull request but still left 'ALK' out of 'tracer_restore_vars'. I saw small changes in PO4, NO3, and SiO3 but a larger difference in ALK (which I attribute to seeing a different NO3 tendency):

   Variable: PO4 ...
   ... variable is double ...
   ... Biggest (absolute) diff = 2.69784194983913e-14
   ... Biggest (relative) diff = 3.399820284831256e-15
   
   Variable: NO3 ...
   ... variable is double ...
   ... Biggest (absolute) diff = 2.869116055848053e-10
   ... Biggest (relative) diff = 5.955086504735507e-12
   
   Variable: SiO3 ...
   ... variable is double ...
   ... Biggest (absolute) diff = 7.149836278586008e-14
   ... Biggest (relative) diff = 3.26385282545721e-16
   
   Variable: ALK ...
   ... variable is double ...
   ... Biggest (absolute) diff = 0.03164020566555337
   ... Biggest (relative) diff = 6.834567472502646e-06
   

3. Then I turned ALK restoring back on, to verify that this PR does apply the restoring:

   Variable: PO4 ...
   ... variable is double ...
   ... Biggest (absolute) diff = 6.961764498214507e-12
   ... Biggest (relative) diff = 8.773215258462089e-13
   
   Variable: NO3 ...
   ... variable is double ...
   ... Biggest (absolute) diff = 1.086510242886973e-07
   ... Biggest (relative) diff = 2.255141429878699e-09
   
   Variable: SiO3 ...
   ... variable is double ...
   ... Biggest (absolute) diff = 2.843769664195861e-11
   ... Biggest (relative) diff = 1.298161984664459e-13
   
   Variable: ALK ...
   ... variable is double ...
   ... Biggest (absolute) diff = 3.556573223929718
   ... Biggest (relative) diff = 0.0007682516329629172
   

One question that has come up is when to add restoring to the tendencies. The ALK tendency, for example, is

dtracers(alk_ind) = -dtracers(no3_ind) + dtracers(nh4_ind) + c2 * P_CaCO3_remin

do auto_ind = 1, auto_cnt
   if (marbl_tracer_indices%auto_inds(auto_ind)%CaCO3_ind > 0) then
      dtracers(alk_ind) = dtracers(alk_ind) &
           + c2 * (f_graze_CaCO3_REMIN * auto_graze(auto_ind) * QCaCO3(auto_ind) - CaCO3_form(auto_ind))
   end if
end do

Should the dtracers(no3_ind) and dtracers(nh4_ind) terms include NO3 and NH4 restoring, respectively? (The default POP setup restores NO3 but not NH4). The current implementation in this PR does not include restoring, but there are a couple different ways to change the code so it does if that's a better way to do it.

As expected, 6b59dca only had round-off level effect on a few of the budget terms:

Variable: Jint_Ntot ...
... Biggest (absolute) diff = 1.539567084929416e-19

Variable: Jint_100m_Ntot ...
... Biggest (relative) diff = 4.15157868043084e-17

Variable: Jint_Ptot ...
... Biggest (absolute) diff = 1.572093150103981e-20

Variable: Jint_100m_Ptot ...
... Biggest (relative) diff = 2.82969226809634e-17

Variable: Jint_Sitot ...
... Biggest (absolute) diff = 8.483881999699072e-20

Variable: Jint_100m_Sitot ...
... Biggest (relative) diff = 3.374337348949718e-18

This came from my one-off tests, will re-run POP tests to make sure nothing breaks unexpectedly and then bring these changes to master.