[MITgcm-support] Balance of momentum

Jean-Michel Campin jmc at ocean.mit.edu
Fri Dec 6 17:48:15 EST 2013 

Hi Marco, Dave, Martin and Ryan,

I would like to clarify few things regarding diagnostics of the
horizontal momentum equation. I did not find very easy to follow
the discussion started ~ a week ago (the subject has changed, not 
always easy to distinguish quickly the answer from the quoting of 
the original message and long lines are not easy to read from the 
mitgcm-support archive web-site). But I hope this will not add more
confusion.

The different horizontal momentum terms are split into different
diagnostics, mainly to reflect the time-stepping options (forward
in time = explicit / backward in time = implicit, and going
through the Adams-Bashforth or not). A finer decomposition can
be obtained with a 2nd (larger) set of diagnostics that are specific
to the formulation used (flux-form or vector-invariant).

1) the surface pressure horizontal gradient needs to be added.
this was mentioned in the Emily's message that Martin (on Nov 28) pointed to:
 http://mitgcm.org/pipermail/mitgcm-support/2010-December/006921.html
and it's needed whether or not the free-surface or the rigid-lid is used
(in this later case, this is the pressure from the lid).

To Ryan: Having many diagnostics slows down the code (it's not 
clear how much, but it does) and when some term can be easily and
precisely recomputed from other diags, it's a matter of choice
if we decide to add a diagnostics or not. 
In this case, from a single 2-D diagnostics output (ETAN), one can 
compute the horizontal gradient in both direction (DXC & DYC are 
standard grid output); but we would need two 3-D diagnostics (to get it 
properly masked) for the contribution at each level of the du/dt 
and dv/dt equations. So, it's not very clear to me that these two
3-D diagnostics should be added (may be we can continue this discussion
elsewhere). 

2) Diagnostics Um_Advec and Um_Advec already contain the Coriolis
 term, unless one uses the CD-Scheme (and in this case, 
 Um_Cori and Vm_Cori needs to be added). 
 The main reason for this is that when using the vector-invariant form, 
 Coriolis term can be incorporated to the absolute vorticity (and then 
 treated with the advection), and in order to make the diagnostics more 
 similar between the 2 momentum formulations, the same convention was 
 retained also for the flux-form.
 Note that, for the same reasons, the metrics-terms are also part
 of Um_Advec and Um_Advec.

Now, here is where I get confused: In Emily's message, the CD-Scheme is 
not used, but momentum budget is closed by adding Um_Cori to to Um_Advec ?

3) Diagnostics Um_Diss and Vm_Diss contain all the explicit (forward
 in time) dissipation terms (horizontal viscosity, explicit vertical
 viscosity, side drag, bottom friction).
 If one uses implicit vertical viscosity (implicitViscosity=.TRUE.,),
 the vertical viscosity tendency needs to be added (as explained in
 Emily's message).
 
4) forcing (diagnostics Um_Ext, Vm_Ext) and baroclinic pressure gradient
 (diagnostics Um_dPHdx and Vm_dPHdy) needs to be added separately.
 If filters are used (Shapiro, FFT), their contribution needs to be
 diagnosed separately (e.g., using SHAP_dU and SHAP_dV ).
 
5) The Adams-Bashforth will alter the momentum budget (although over
 a long period, most cancel in time-averaging) and, depending on the 
 particular set of parameter (e.g., staggerTimeStep, forcing_In_AB, 
 momDissip_In_AB ) it's not always negligible. This AB effect can
 be diagnosed using the 2 diagnostics AB_gU and AB_gV (added 2 years ago).

So that the momentum budget for the U componentic can be diagnosed with:

[du/dt] = - gravity * ( ETAN(i) - ETAN(i-1) ) / DXC
        + Um_dPHdx
        + Um_Advec   (+ Um_Cori : but only if using CD-Scheme )
        + Um_Diss    (+ Implicit vertical viscosity tendency)
        + Um_Ext
        + AB_gU

Some remarks:
a) When using the flux-form formulation, the non-linear free-surface can introduce 
   a small difference (rescaling) which is not currently diagnosed.
b) the 3-D Coriolis ( in 2 Omega cos(Phi) between U and W) is not currently
  diagnosed separately (and is not part of Um_Cori) but is part of
  Um_Advec. Also using the quasiHydrostatic option might alter some of the 
  diagnostics (e.g., what is described as the hydrostatic pressure is no 
  longer strictly hydrostatic).
c) when trying to do a budget over just 1 time-step, one needs to be aware 
  of the precise time-discretisation of the the first term which is evaluated 
  in the model as 
   - gravity * {  implicSurfPress * (d.EtaN/dx)^n+1 
                +(1-implicSurfPress)*(d.EtaN/dx)^n }
  (using the default implicSurfPress=1 , it might be be easier to use 
   snap-shot output Eta.[iter+1] to get EtaN^n+1 )

Cheers,
Jean-Michel

On Thu, Dec 05, 2013 at 10:23:04AM +0000, David Munday wrote:
> Hi Marco,
> 
> Um_HydroP isn't in my available_diagnostics, I can't find it anywhere in the documentation, and grepping the code turns up nothing. Is it the same as Um_dPHdx?
> 
> In general, I'd expect Um_Advec and Um_dPHdx to be similar in magnitude and of opposite sign, due to the dominance of geostrophic balance. Assuming Um_HydroP is Um_dPHdx by another name, then its magnitude is about right, but its sign is wrong (or Um_Advec's is). Taking the difference and adding Um_Diss gives me -4.3197E-9, so you're still missing a term.
> 
> Have you output every single Um_* diagnostic available, including the ones you think are zero, the implicit/explicit viscosity, and the gradient in SSH?
> 
> Best wishes,
> 
> Dave
> 
> 
> On 4 Dec 2013, at 21:44, Marco Reale wrote:
> 
> Hi David,
> 
> and thanks a lot for your response: I followed your suggestion.
> 
> So finally  the equation will be :
> 
> TOTUTEND/86400=Um_Advec+Um_diss+Um_HydroP;
> 
> each of these members of the equation are 4D arrays : following a point to check the balance at t = 30, level=12, i=9, j=30 and I got the following results:
> 
> TOTUTEND/86400= -8.0561e-10
> Um_Advec= 2.5398e-07
> Um_diss= 7.7003e-09
> Um_HydroP=2.4196e-07
> 
> the balance of this term is 5.0365e-07
> 
> 
> So I have done something wrong: I’m working with rough data without taking into account volume of cells, etc.
> 
> Any suggestions?
> 
> Marco
> 
> 

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