23/9/05

Dear Mike,

I've finally found some promising results with the GCR test. See the page at:

www.met.rdg.ac.uk/~ross/DARC/PVcv/GCR_InitialTests_12/Text.html

(alternatively, you can get to this page by clicking on run "12" from the table via

www.met.rdg.ac.uk/~ross/DARC/PVcv/PVcv.html

then you will see the page duplicated in two frames to allow on-screen comparison between different parts of the document).

On this page there is a clickable list above each group of plots allowing you to jump straight to each diagnostic.

In this test, the PV that is implied from the 'solution' has the same structure as the RHS PV.

Details of this test

The test reported here has the following features:

• There are no longer any extra PV-like modes in this version. The PV is entirely grid point (I have computed PV at the top and bottom with Neumann boundary conditions - see below).
• Diagonal preconditioning has been used (I found a bug in my diagonal preconditioner while I was adapting it for use with the entirely grid point calculation).
• 50 GCR iterations, each with a restart value of 5.
• I got the vertical regression to work on the first-guess field - it removes noise in the stratosphere, including in the potential temperature diagnostic (even though the regression is actually designed for and not ).

The Neumann boundary conditions

The PV that I am using is written in terms of and , and has the following form,

(the hat or overbar means that the quantity needs vertical interpolation from to levels, or vice-versa).

Instead of applying the Neumann boundary conditions to in the vertical, I have applied them to , as this leads to simpler code to start with. I cannot think of a good reason why this boundary condition is any worse than the Neumann boundary condition for . In fact, I think that the b.c. that I have used means that the solution will have zero pot. temp. increment at the top and bottom p-levels. Is that reasonable?

I needed to make an approximation for the second derivative term in the above formula. In finite difference form the second derivative at the bottom is,

where and are height on pressure and theta levels respectively (the levels are staggered using the following guide (right-hand labels)),

The Neumann boundary condition means that,

Putting this into the formula for the second derivative, and turning the handle gives,

We don't know the quantity , and so the following approximation is made,

which assumes that the distance between the and levels is the same as the distance between the and levels.

Forthcoming tests

I shall do a couple of further tests:

• Run this test with a few thousand iterations instead of just 50 to see how close we can get to the required solution.
• Run the code with the preconditioner and with the hybrid grid+extra PV-like modes version of the PV. The bug that I found in the preconditioner might allow us to make progress with this version of the code.

If I still find that it is difficult to make progress with the grid+extra PV-like modes version of the PV, would the grid-only version suffice?

Thanks,

Ross