SSH assimilation – No balance but UV still update?

[lmp4]

For the last four months I’ve had this issue on my to do list and I think I’ve finally come to grips with what’s going on.
So I have previously noted that the current ROMS-4DVAR formulation does not have an adjustment for near-equatorial geostrophic balance for the UV momentum balance [viewtopic.php?f=42&t=4026]. This unfortunately means I experience blow ups when I have the UV momentum switched on for my domain (1S 21S).

Since my research is focusing on the impact of assimilating different observations on ocean currents the issue of the importance of this UV balance for my results is of great interest to me.

So I’ll take you through my chain of thoughts to where I am now.
It was my first understanding that if the UV balance is turned off then the increment will only be affected by unbalanced component i.e. if I’m assimilating u and v directly.

δu =K_up δP + δu_U
So when the balance if off K_up δP = 0

So with this in mind if I assimilate any other variables then UV should not change?
After a few experiments I think this 'previous thinking' would only apply for a 'filter' type DA method (ENKF?) and this is actually not the case for ROMS 4D-Var a variational 'smoother' method.

Now let’s look at my experiment to explore this;

The figure above has two assimilation cases where in both the UV momentum balance is turned OFF. The left hand 4 figures shows the ocean currents (colour represents speed with black direction vectors super imposed) where I assimilate SST only, the right hand 4 figures shows SST and SSH assimilation. Within the figure frames; top left is the prior i.e. the model before assimilation, the top right the posterior i.e. the model analysis after assimilation. Bottom left is the difference between the posterior and prior i.e. the increments and the bottom right is the posterior SSH field.

In these figures it is clear that the assimilation is changing the UV fields even when the UV momentum balance is turned off.
Thinking about variational DA methods you can think of the cost function minimization finding an optimal trajectory not just optimal ICs. So when a field is adjusting such as the SST other variables have to adjust accordingly in order to produce the optimal trajectory.

Now focusing on the right hand experiment you can now see that the SSH assimilation is affecting the UV fields much more than the SST only assimilation again without the UV balance. Remember the UV balance in simply geostrophic balance and is turned off for these experiments.

Using the same logic as before it seems the ‘smoother’ nature of variational methods is actually providing the balance itself. This is especially the case for the SSH assimilation. It has been mentioned numerous times in this forum and in the literature (Shcepetkin and McWilliams 2005), that the equation for the free surface is a function of the divergence of the vertically integrated 2D-momentum div (ubar,vbar). So with this in mind if one changes zeta through assimilation then u and v must change regardless of this geostrophic balance. And you can see in the zeta plots how the currents have shifted to be consistent with zeta from the SST assim to the SSH assim.

Ultimately I have decided that the UV balance shouldn’t give me much more benefit than what the model is adjusting to anyways. Having said that this is still just a guess since I cannot recover currents near the equator from geostrophy. If my currents are adjusting to a changing SSH isn't this in a sense geostrophic adjustment of the currents?

If anyone has any inputs or thoughts about this I would grateful to dicsuss. Thanks.