Phase (ω/k_x, upper panel) and group (∂ω/∂k_x, lower panel) speeds of Rossby-dynamo waves vs toroidal wavenumber for the toroidal magnetic field strength of 10 kG at the latitude 30⁰ of the solar tachocline according to Eq. (16). The negative (positive) speeds correspond to the retrograde (prograde) propagation. The speeds are normalized by the tachocline rotation speed ΩR, where Ω = 2.9 × 10⁻⁶ rad s⁻¹ is the sidereal angular velocity and R ∼ 5 × 10¹⁰ cm is the distance from the solar center. The toroidal wavenumber k_x is normalized by R. Blue and red lines indicate the phase and group speeds for the dynamo coefficient, α_z (normalized by Ω), as 0.003 and 0.3, respectively. The dimensionless value of the surface gravity speed, $c/\mathrm{\Omega }R=\sqrt{\mathit{gH}}/\mathrm{\Omega }R$, where g is the reduced gravity and H = 10⁷ m is the thickness of the overshoot layer, here is equal to 0.33. The reduced gravity is related with the fractional deference between actual and adiabatic temperature gradients, δ, as gH/Ω²R² ∼ 10³δ ∼ 10³|∇ − ∇_ad|, where ∇ = dlnT/dlnP and ∇_ad = (∂lnT/∂lnP)_ad. The high frequency inertia gravity waves are not shown in these plots. We note that the group speeds of Rossby and dynamo waves reverse the propagation for higher value of dynamo coefficient (red lines): one of the dynamo waves changes from prograde to retrograde propagation at k_x ∼ 1.5 and the Rossby wave changes from the retrograde to the prograde propagation at k_x ∼ 2. Here k_y = 0 is taken during computation.