  A field theory and the associated structure-preserving geometric
Particle-In-Cell (PIC) algorithm are developed to study low frequency
electrostatic perturbations with fully kinetic ions and adiabatic electrons in
magnetized plasmas. The algorithm is constructed by geometrically discretizing
the field theory using discrete exterior calculus, high-order Whitney
interpolation forms, and non-canonical Hamiltonian splitting method. The
discretization preserves the non-canonical symplectic structure of the
particle-field system, as well as the electromagnetic gauge symmetry. As a
result, the algorithm is charge-conserving and possesses long-term conservation
properties. Because drift wave turbulence and anomalous transport intrinsically
involve multi time-scales, simulation studies using fully kinetic particle
demand algorithms with long-term accuracy and fidelity. The
structure-preserving geometric PIC algorithm developed adequately servers this
purpose. The algorithm has been implemented in the \textsl{SymPIC} code, tested
and benchmarked using the examples of ion Bernstein waves and drift waves. We
apply the algorithm to study the Ion Temperature Gradient (ITG) instability and
turbulence in a 2D slab geometry. Simulation results show that at the early
stage of the turbulence, the energy diffusion is between the Bohm scaling and
gyro-Bohm scaling. At later time, the observed diffusion is closer to the
gyro-Bohm scaling, and density blobs generated by the rupture of unstable modes
are the prominent structures of the fully developed ITG turbulence.
