We discuss the evolution of the magnetic field of an accreting white dwarf .
We calculate the ohmic decay modes for accreting white dwarfs , whose interiors are maintained in a liquid state by compressional heating .
We show that the lowest order ohmic decay time is ( 8 – 12 ) billion years for a dipole field , and ( 4 – 6 ) billion years for a quadrupole field .
We then compare the timescales for ohmic diffusion and accretion at different depths in the star , and for a simplified field structure and assuming spherical accretion , study the time-dependent evolution of the global magnetic field at different accretion rates .
We neglect mass loss by classical nova explosions and assume the white dwarf mass increases with time .
In this case , the field structure in the outer layers of the white dwarf is significantly modified for accretion rates above the critical rate \dot { M } _ { c } \approx ( 1 – 5 ) \times 10 ^ { -10 } M _ { \odot } { yr ^ { -1 } } .
We consider the implications of our results for observed systems .
We propose that accretion-induced magnetic field changes are the missing evolutionary link between AM Her systems and intermediate polars .
The shorter ohmic decay time for accreting white dwarfs provides a partial explanation of the lack of accreting systems with \approx 10 ^ { 9 } { G } fields .
In rapidly accreting systems such as supersoft X-ray sources , amplification of internal fields by compression may be important for Type Ia supernova ignition and explosion .
Finally , spreading matter in the polar cap may induce complexity in the surface magnetic field , and explain why the more strongly accreting pole in AM Her systems has a weaker field .
We conclude with speculations about the field evolution when classical nova explosions cause the white dwarf mass to decrease with time .