  Compatible finite element discretisations for the atmospheric equations of
motion have recently attracted considerable interest. Semi-implicit
timestepping methods require the repeated solution of a large saddle-point
system of linear equations. Preconditioning this system is challenging since
the velocity mass matrix is non-diagonal, leading to a dense Schur complement.
Hybridisable discretisations overcome this issue: weakly enforcing continuity
of the velocity field with Lagrange multipliers leads to a sparse system of
equations, which has a similar structure to the pressure Schur complement in
traditional approaches. We describe how the hybridised sparse system can be
preconditioned with a non-nested two-level preconditioner. To solve the coarse
system, we use the multigrid pressure solver that is employed in the
approximate Schur complement method previously proposed by the some of the
authors. Our approach significantly reduces the number of solver iterations.
The method shows excellent performance and scales to large numbers of cores in
the Met Office next-generation climate- and weather prediction model LFRic.
