Coronal heating by the resonant absorption of Alfven waves: The effect of viscous stress tensor

The time-dependent linearized magnetohydrodynamics (MHD) equations for a fully compressible, low-beta, viscoresistive plasma are solved numerically using an implicit integration scheme. The full viscosity stress tensor (Braginskii 1965) is included with the five parameters eta(sub i) i = 0 to 4. In agreement with previous studies, the numerical simulations demonstrate that the dissipation on inhomogeneities in the background Alfven speed occurs in a narrow resonant layer. For an active region in the solar corona the values of eta(sub i) are eta(sub o) = 0.65 g/cm/s, eta(sub 1) = 3.7 x 10(exp -12) g/cm/s, eta(sub 2) = 4 eta(sub 1), eta(sub 3) = 1.4 x 10(exp -6) g/cm/s, eta(sub 4) = 2 eta(sub 3), with n = 10(exp 10)/cu cm, T = 2 x 10(exp 6) K, and B = 100 G. When the Lundquist number S = 10(exp 4) and R(sub 1) much greater than S (where R(sub 1) is the dimensionless shear viscous number) the width of the resistive dissipation layer d(sub r) is 0.22a (where a is the density gradient length scale) and d(sub r) approximately S(exp -1/3). When S much greater than R(sub 1) the shear viscous dissipation layer width d(sub r) scales as R(sub 1)(exp -1/3). The shear viscous and the resistive dissipation occurs in an overlapping narrow region, and the total heating rate is independent of the value of the dissipation parameters in agreement with previous studies. Consequently, the maximum values of the perpendicular velocity and perpendicular magnetic field scale as R(sub 1)(exp -1/3). It is evident from the simulations that for solar parameters the heating due to the compressive viscosity (R(sub 0) = 560) is negligible compared to the resistive and the shear viscous (R(sub 1)) dissipation and it occurs in a broad layer of order a in width. In the solar corona with S approximately equals 10(exp 4) and R(sub 1) approximately equals 10(exp 14) (as calculated from the Braginskii expressions), the shear viscous resonant heating is of comparable magnitude to the resistive resonant heating.