Magnetic fields are detected in a few percent of white dwarfs .
The number of such magnetic white dwarfs known is now some hundreds .
Fields range in strength from a few kG to several hundred MG .
Almost all the known magnetic white dwarfs have a mean field modulus \geq 1 MG. We are trying to fill a major gap in observational knowledge at the low field limit ( \leq 200 kG ) using circular spectro-polarimetry .

In this paper we report the discovery and monitoring of strong , periodic magnetic variability in two previously discovered “ super-weak field ” magnetic white dwarfs , WD 2047+372 and WD 2359-434 .
WD 2047+372 has a mean longitudinal field that reverses between about -12 and +15 kG , with a period of 0.243 d , while its mean field modulus appears nearly constant at 60 kG .
The observations can be intepreted in terms of a dipolar field tilted with respect to the stellar rotation axis .
WD 2359-434 always shows a weak positive longitudinal field with values between about 0 and +12 kG , varying only weakly with stellar rotation , while the mean field modulus varies between about 50 and 100 kG .
The rotation period is found to be 0.112 d using the variable shape of the H \alpha line core , consistent with available photometry .
The field of this star appears to be much more complex than a dipole , and is probably not axisymmetric .

Available photometry shows that WD 2359-434 is a light variable with an amplitude of only 0.005 mag ; our own photometry shows that if WD 2047+372 is photometrically variable , the amplitude is below about 0.01 mag .

These are the first models for magnetic white dwarfs with fields below about 100 kG based on magnetic measurements through the full stellar rotation .
They reveal two very different magnetic surface configurations , and that , contrary to simple ohmic decay theory , WD 2359-434 has a much more complex surface field than the much younger WD 2047+372 .