tokens: 1999 ). The cycle duration is believed to be controlled by the meridional flow. Fig. 7 shows also the ratio of the cycle period to advection time. The ratio – though not constant – is of the order one and varies moderately. Figure 7. Cycle period for marginal dynamos (a) and the period ratio to the advection time (b). V b is the meridional velocity at the inner boundary r i at the latitude of 45°. Fig. 8 shows estimations of the Rossby and dynamo numbers. The Rossby number is position-dependent. The estimations of Fig. 8 were done for the radius r one pressure scaleheight above the base of the convection zone, i.e. for the radius r where r − r i = H p . The Rossby number Ro ∼ 2 of Fig. 8 agrees with estimation of van Saders et al. ( 2016 ). The two numbers in Fig. 8 are often used to parametrize observations or theoretical computations of magnetic activity. The two parameters of this figure vary moderately but are not constant. This implies that stellar activity can be only crudely parametrized by a single parameter. Figure 8. (a) The Rossby number estimated for the rotation periods of equation ( 2 ). (b) The marginal dynamo number ( 4 ) estimated with the isotropic part η of the eddy diffusivity at middle depth in the convection zone. 5 CONCLUSIONS Asteroseismic measurements of stellar ages revealed a dramatic decrease in the rate of stellar spin-down and magnetic activity for ages beyond about a half of the main-sequence life (Metcalfe et al. 2016 ; van Saders et al. 2016 ). This paper suggests an explanation for the violation of gyrochronology by a switch-off of the global dynamo for a rotation period exceeding the critical value estimated by equation ( 2 ). The switch-off should not be abrupt. Dynamo theory explains the observed variations in amplitudes and periods of solar cycles by fluctuations in dynamo parameters (cf. Choudhuri 2011 ). The fluctuations will cause a star to wander between the parameter regions of (supercritical) dynamo of ‘normal’ magnetic activity and (subcritical) dynamo of fading activity if the rotation period is close to its critical value. The dynamo models with fluctuating parameters reproduce the statistics of grand solar minima (Usoskin, Solanki Kovaltsov 2007 ) only if the dynamo number is close to its marginal value (Moss et al. 2008 ; Olemskoy Kitchatinov 2013 ; Karak, Kitchatinov Brandenburg 2015 ). It may be appropriate to note that observations of total solar eclipses in the Maunder minimum did not find an extended solar corona (Eddy 1976 ) indicating a relatively small co-rotation radius and inefficient spin-down. Unfortunately, the concept of dynamo switch-off at the critical rotation period cannot be supported by dynamo modelling because of uncertainties in key dynamo parameters. The uncertainties can, however, be reduced using this concept. Equation ( 5 ) for α-parameter or similar equation for the dynamo number can be used to estimate the dynamo parameter values for a star of given rotation
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