We present fully three-dimensional local simulations of compressible MRI turbulence with the object of studying and elucidating the excitation of the non-axisymmetric spiral density waves that are observed to always be present in such simulations .
They are potentially important for affecting protoplanetary migration through the action of associated stochastic gravitational forces and producing residual transport in MHD inactive regions through which they may propagate .
The simulations we perform are with zero net flux and produce mean activity levels corresponding to the Shakura & Sunyaev \alpha { } \sim { } 5 \times { } 10 ^ { -3 } , being at the lower end of the range usually considered in accretion disc modelling .
We reveal the nature of the mechanism responsible for the excitation of these waves by determining the time dependent evolution of the Fourier transforms of the participating state variables .
The dominant waves are found to have no vertical structure and to be excited during periodically repeating swings in which they change from leading to trailing .
The initial phase of the evolution of such a swing is found to be in excellent agreement with that expected from the WKBJ theory developed in a preceding paper by Heinemann & Papaloizou .
However , shortly after the attainment of the expected maximum wave amplitude , the waves begin to be damped on account of the formation of weak shocks .
As expected from the theory the waves are seen to shorten in radial wavelength as they propagate .
This feature enables nonlinear dissipation to continue in spite of amplitude decrease .
As a consequence the waves are almost always seen to be in the non linear regime .

We demonstrate that the important source terms causing excitation of the waves are related to a quantity that reduces to the potential vorticity for small perturbations from the background state with no vertical dependence .
We find that the root mean square density fluctuations associated with the waves are positively correlated with both this quantity and the general level of hydromagnetic turbulence .
The mean angular momentum transport associated with spiral density waves generated in our simulations is estimated to be a significant fraction of that associated with the turbulent Reynolds stress .