[ecco-support] [EXTERNAL] Re: Change-of-variable formulas for adjoint sensitivities

[Christopher Wolfe]

Hi Paola and Ircho:

Assuming I did my calculations right, it looks like it *is* possible to get the sensitivity to wind stress curl from the sensitivities to wind stress, at least in a (doubly) periodic domain. You can use a Helmholtz decomposition to write the wind stress in terms of its divergence and curl. The result ends up being the convolution of the curl of the sensitivities with the Green’s function for the Poisson equation. It doesn’t seem like the result depends on the actual value of the stress divergence. (See attached PDF for details.)

The problem comes in bounded domains, since you need to pick boundary conditions for the Helmholtz decomposition which, in turn, determine the boundary conditions on the Green’s function. I’m not sure what those boundary conditions should be, or if they can even be uniquely specified. So the problem may still turn out to be indeterminate, as Ichiro suggests.

Oh well.
Christopher



> On Jul 7, 2020, at 6:42 PM, Fukumori, Ichiro (US 329B) <ichiro.fukumori at jpl.nasa.gov> wrote:
> 
> Hi Christopher,
>  
> I don't think the sensitivity to wind stress curl can be derived from
> sensitivity to wind stress alone. The problem stems from the
> indeterminate nature of deriving wind stress from its curl.
>  
> For instance, consider two wind stress perturbations that have
> identical curls but different irrotational components. A
> wind-stress-forced model's sensitivity to the perturbations would in
> general be different between the two; i.e., the sensitivity to wind
> stress curl is indeterminate. Expressing this sensitivity from
> sensitivity to wind stress alone amounts to implicitly specifying
> allowable irrotational perturbations to the wind.
>  


> On Jul 6, 2020, at 9:16 PM, Paola Cessi <pcessi at ucsd.edu> wrote:
> 
> Hi Christopher,
> 
> I don’t think it is possible. You need to know the derivative of the cost of function wrt to the derivatives of the wind-stress (or the curl), and it is not possible to get those from the cost function derivatives wrt to wind-stress alone. For example, in the shallow water case, you only need to know the momentum eqns. to get the sensitivity to the wind stress, but for the sensitivity of the curl, you need to massage the mom. eqns. AND add the continuity.

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