WAVE GENERATION BY TURBULENT CONVECTION

We consider wave generation by turbulent convection in a plane parallel, stratified atmosphere that sits in a gravitational field, g. The atmosphere consists of two semi-infinite layers, the lower adiabatic and polytropic and the upper isothermal. The adiabatic layer supports a convective energy flux given by mixing length theory; Fc ∼ ρυH3, where ρ is mass density and υH is the velocity of the energy bearing turbulent eddies. Acoustic waves with ω > ωac and gravity waves with ω < 2kh Hiωb propagate in the isothermal layer whose acoustic cutoff frequency, ωac, and Brunt-Väisälä frequency, ωb, satisfy ωac2, = γg/4Hi and ωb2 = (γ - 1)g/γHi, where γ and Hi denote the adiabatic index and scale height. The atmosphere traps acoustic waves in upper part of the adiabatic layer (p-modes) and gravity waves on the interface between the adiabatic and isothermal layers (f-modes). These modes obey the dispersion relation ω2 ≈ 2/m gkh(n + m/2) , for ω < ωac. Here, m is the polytropic index, kh is the magnitude of the horizontal wave vector, and n is the number of nodes in the radial displacement eigenfunction; n = 0 for f-modes. Wave generation is concentrated at the top of the convection zone since the turbulent Mach number, M = υH/c, peaks there; we assume Mt ≪ 1. The dimensionless efficiency, η, for the conversion of the energy carried by convection into wave energy is calculated to be η ∼ Mt15/2 for p-modes, f-modes, and propagating acoustic waves, and η ∼ Mt for propagating gravity waves. Most of the energy going into p-modes, f-modes, and propagating acoustic waves is emitted by inertial range eddies of size h ∼ Mt3/2Ht at ω ∼ ωac and kh ∼ 1/Ht. The energy emission into propagating gravity waves is dominated by energy bearing eddies of size ∼ Ht and is concentrated at ω ∼ υt/Ht ∼ Mtωac and kh, ∼ 1/Ht. We find the power input to individual p-modes, Ėp, to vary as ω(2m2+7m-3)/(m+3) at frequencies ω ≪ υt/Ht. Libbrecht has shown that the amplitudes and linewidths of the solar p-modes imply Ėp ∝ ω8 for ω ≪ 2 × 10-2 s-1. The theoretical exponent matches the observational one for m ≈ 4, a value obtained from the density profile in the upper part of the solar convection zone. This agreement supports the hypothesis that the solar p-modes are stochastically excited by turbulent convection.