  Instabilities of the dust layer in a protoplanetary disk are investigated. It
is known that the streaming instability develops and dust density concentration
occurs in a situation where the initial dust density is uniform. This work
considers the effect of initial dust density gradient vertical to the midplane.
Dust and gas are treated as different fluids. Pressure of dust fluid is assumed
to be zero. The gas friction time is assumed to be constant. Axisymmetric
two-dimensional numerical simulation was performed using the spectral method.
We found that an instability develops with a growth rate on the order of the
Keplerian angular velocity even if the gas friction time multiplied by the
Keplerian angular velocity is as small as 0.001.
  This instability is powered by two sources: (1) the vertical shear of the
azimuthal velocity, and (2) the relative motion of dust and gas coupled with
the dust density fluctuation due to advection. This instability diffuses dust
by turbulent advection and the maximum dust density decreases. This means that
the dust concentration by the streaming instability which is seen in the case
of a uniform initial dust density becomes ineffective as dust density gradient
increases by the dust settling toward the midplane.
