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; F_c ~ pv^3_H, where p is mass density and v_H is the velocity of the energy bearing turbulent eddies. Acoustic waves with ω > ω_(ɑc) and gravity waves with ω < 2k_h H_iωb propagate in the isothermal layer whose acoustic cutoff frequency, ω_(ac), and Brunt-Vaisala frequency, ω_b, satisfy ω^2_(ɑc) = yg/4H_i and ω^2_b = (y-1)g/yH_i, where y and H_i 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 ω^2≈2/m gk_h(n + m/2), for ω < ω_(ɑc). Here, m is the polytropic index, k_h 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 = v_H/c, peaks there; we assume M_t « 1. The dimensionless efficiency, η, for the conversion of the energy carried by convection into wave energy is calculated to be η~M_t^(5/12) for p-modes,f-modes, and propagating acoustic waves, and η~M, 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 ~ M_t^(3/2)H_t, at ω ~ ω_(ɑc) and k_h ~ 1/H_t. The energy emission into propagating gravity waves is dominated by energy bearing eddies of size ~ H_t and is concentrated at ω ~ v_t/H_t ~ M_t ω_(ɑc) and k_h ~ 1/H_t. We find the power input to individual p-modes, E_p, to vary as ω<^(2m^2+7m-3)/(m+3) at frequencies ω « v_t/H_t. Libbrecht has shown that the amplitudes and linewidths of the solar p-modes imply E_p ∝ ω^8 for ω « 2 x 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.