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The aim of this study is to investigate a wave dynamics and size scaling ofavalanches which were created by the mathematical model {[}J. \v{C}ern\'akPhys. Rev. E \textbf{65}, 046141 (2002)]. Numerical simulations were carriedout on a two dimensional lattice $L\times L$ in which two constant thresholds$E_{c}^{I}=4$ and $E_{c}^{II}>E_{c}^{I}$ were randomly distributed. A densityof sites $c$ with the threshold $E_{c}^{II}$ and threshold $E_{c}^{II}$ areparameters of the model. I have determined autocorrelations of avalanche sizewaves, Hurst exponents, avalanche structures and avalanche size moments forseveral densities $c$ and thresholds $E_{c}^{II}$. I found correlated avalanchesize waves and multifractal scaling of avalanche sizes not only for specificconditions, densities $c=0.0$, 1.0 and thresholds $8\leq E_{c}^{II}\leq32$, inwhich relaxation rules were precisely balanced, but also for more generalconditions, densities $0.0<c<1.0$ and thresholds $8\leq E_{c}^{II}\leq3 inwhich relaxation rules were unbalanced. The results suggest that the hypothesisof a precise relaxation balance could be a specific case of a more generalrule.

Abelian deterministic self organized criticality model: Complex dynamics of avalanche waves