  For an axisymmetric tokamak plasma, Hamiltonian theory predicts that the
orbits of charged particles must stay on invariant tori of conserved energy in
the moving frame of reference of a wave that propagates along the torus with a
fixed angular phase velocity. In principle, this is true for arbitrary mode
structures in the poloidal plane, but only if the fluctuations are expressed in
terms of potentials $\Phi$ and ${\bf A}$, which satisfy Faraday's law by
definition. Here, we use the physical fields ${\bf E}$ and ${\bf B}$, where
Faraday's law may be violated by errors introduced during the process of
computing or designing the wave field through numerical inaccuracies,
approximations, or gross negligence. Numerical heating caused by noise-like
artifacts on the grid scale can to some extent be reduced via shorter time
steps. In contrast, coherent inconsistencies between ${\bf E}$ and ${\bf B}$
cause spurious acceleration that is independent of time steps or numerical
methods, but can be sensitive to geometry. In particular, we show that secular
acceleration is enhanced when one imposes nonnormal modes that possess strong
up-down asymmetry instead of the usual in-out asymmetry of normal toroidal
(eigen)modes. Our tests are performed for full gyroorbit and guiding center
(GC) models, which give similar results. In addition, we show that $N$-point
gyroaveraging is not a recommendable method to enhance the realism of GC
simulations. Besides breaking conservation laws, $N$-point gyroaveraging in our
example makes the GC results deviate further from the full orbit results,
showing that this method can even give the wrong trend.
