Basing on currently defined conductivity, 115 years observed evolution of the geomagnetic dipole, Faraday's and Ohm's laws I estimate averaged radial derivatives of the vortex magnetic field hidden just below the surface of the Earth's core. This allows to formulate a simple model of vortex field beneath the surface of the core and to evaluate typical scale of the field, which determines the major geodynamo parameters and the adequacy range of the proposed simple model. Estimated scale of the vortex field (about 60 km) is much less than the typical scale resulting from the extrapolation of the observed field to the core-mantle boundary. This agrees well with the modern planetary dynamo theory, allowing direct observational estimation of the typical velocity field just beneath the surface of the Earth’s core. The proposed new approach to determine the subsurface characteristics of the hidden in the depths of the physical object of the vortex magnetic field and velocity from the observed evolution of the potential field can be used for both astrophysical and for technical objects with hardly accessible electric current systems. Scaling laws for MHD dynamo in fast rotating planets and stars express the characteristic energy, hydrodynamic and magnetic values through the primary values, such as the size of the conductive core of the planet, the angular rotation rate, electrical conductivity and energy flows. Most of the earlier proposed scaling laws based only on observations and assumptions about force balances. Recent and my new approaches to fully take into account the energy and induction balance has additionally expressed here in terms of primary values such important characteristics as forces, magnitudes, energies, scales and orientations of hydromagnetic fields. The direct numerical simulation of the dynamos and modeling ability in a fairly wide range of parameters for the first time allowed direct test such laws. The obtained numerical geodynamo-like results for the Earth, Jupiter and partially Saturn postulated previously not identified analytically simplest law that predicts the field strength is only depended on the specific energy density of convection. This already widely used law was original way analytically grounded here along with other previously known and new laws. This analytic identifies the physics determining geomagnetic periodicities for jerk, secular variations and inversions/excursions. Mean period between the inversions is roughly proportional to the intensity of the geomagnetic field that is confirmed by some paleomagnetic researches. Possible dynamos in Mercury, Ganymede, Uranus and Neptune are also discussed. 

This work was partly supported by Russian RFBR grant No 16-05-00507.