Magnetic fields have been widely used in industry to enhance the performance of crystal growth processes. However, no attempts have been made at applying optimization strategies to effect optimal enhancements. Here, a mathematical formulation and computational techniques are presented to describe optimal control and design strategies for the suppression of turbulent motions in the melt and the minimization of temperature gradients in the crystal in Czochralski crystal growth processes. It is shown that an axial magnetic field can effectively suppress convection in Czochralski growth of silicon. Other control parameters such as crystal and crucible rotation rates are found to be less effective.