By Shi M., Yuan X., Cai M.

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**Additional resources for (3,k)-Factor-Critical Graphs and Toughness**

**Sample text**

40) can be expressed as first calculating b! 43) and then calculating u' and v' from h' using the geostrophic relation. This is a simple form of potential vorticity inversion where all the flow variables can be derived from the potential vorticity together with diagnostic relations between the variables. In the nonlinear case, we will often write the slow solutions corresponding to Rossby waves in the form of an evolution equation for potential vorticity, together with diagnostic equations allowing the other variables to be calculated.

We will use these additional approximations to illustrate various aspects of the behaviour of the solutions. 20 Large-scale atmosphere flow (ii) It appears strange that only the horizontal momentum is approximated, not the trajectory. This choice retains energetic consistency, which is essential in proving that the equations can be solved for large times. (iii) It is possible to make the geostrophic momentum approximation without making the shallow atmosphere approximation. 2. (iv) The solutions will only make sense if the horizontal pressure gradient V r II' goes to zero at the equator.

An important application of this linear analysis is to extract the Rossby wave solution from general data (u,v,h). This is called solving the 'Rossby adjustment' problem. Write u' = u(t)expi (kx + ly),v' = i)(t) exp i (kx + ly), h' = h(t) exp i (kx + ly). 26) can be rewritten as +i i=a *(i) u, Let E be the matrix whose columns are the eigenfunctions of L calculated above. Then a general state vector x = I v \ can be projected on to a \hj basis of eigenfunctions by setting x = E-1*. 39) The Rossby wave component xo is isolated by setting the second and third components of x to zero.

### (3,k)-Factor-Critical Graphs and Toughness by Shi M., Yuan X., Cai M.

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