  A new horizontally explicit/vertically implicit (HEVI) time splitting scheme
for atmospheric modelling is introduced, for which the horizontal divergence
terms are applied within the implicit vertical substep. The new HEVI scheme is
implemented in conjunction with a mixed mimetic spectral element spatial
discretisation and semi-implicit vertical time stepping scheme that both
preserve the skew-symmetric structure of the non-canonical Hamiltonian form of
the equations of motion. Within this context the new HEVI scheme allows for the
exact balance of all energetic exchanges in space and time. However since the
choice of horizontal fluxes for which this balance is satisfied is not
consistent with the horizontal velocity at the end of the time level the scheme
still admits a temporal energy conservation error. Linearised eigenvalue
analysis shows that similar to a fully implicit method, the new HEVI scheme is
neutrally stable for all buoyancy modes, and unlike a second order trapezoidal
HEVI scheme is stable for all acoustic modes below a certain horizontal CFL
number. The scheme is validated against standard test cases for both planetary
and nonhydrostatic regimes. For the planetary scale baroclinic instability test
case, the new formulation exhibits a secondary oscillation in the potential to
kinetic energy power exchanges, with a temporal frequency approximately four
times that exhibited by a horizontally third order, vertically second order
trapezoidal scheme. For the nonhydrostatic test case, the vertical upwinding of
the potential temperature diagnostic equation is shown to reduce spurious
oscillations without altering the energetics of the solution, since this
upwinding is performed in an energetically consistent manner. For this test
case, which is configured on an affine geometry, the exact balance of energy
exchanges allows the model to run stably without any form of dissipation.
