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; Fe ""'pv~, where pis mass density and v8 is the velocity of the energy bearing turbulent eddies. Acoustic waves with ro > roac and gravity waves with ro < 2khHirob propagate in the isothermal layer whose acoustic cutoff frequency, roaco and Brunt-Vliisiilii frequency, rob, satisfy ro:C = ygf4H; and ro: = (y1)gfyH;, where y and H; 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 ro2 ~ ; gk{ n + ~) , for ro < roac. 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 /-modes. Wave generation is concentrated at the top of the convection zone since the turbulent Mach number, M = v8 jc, peaks there; we assume M, ~ 1. The dimensionless efficiency, '1· for the conversion of the energy carried by convection into wave energy is calculated to be '1 ""'Mf 512 for p-modes,J-modes, and propagating acoustic waves, and '1 ""' M, for propagating gravity waves. Most of the energy going into p-modes, /-modes, and propagating acoustic waves is emitted by inertial range eddies of size h""' M:12H, at ro""' roac and kh""' 1/H,. The energy emission into propagating gravity waves is dominated by energy bearing eddies of size ""'H, and is concentrated at ro""' v,/H,""' M,roac and kh""' 1/H,. We find the power input to individual p-modes, E,, to vary as ro<2m2 +7m3Ji<m+ 3l at frequencies ro ~ v,/H,. Libbrecht has shown that the amplitudes and linewidths of the solar p-modes imply E, oc ro8 for ro ~ 2 x 102 s1 • The theoretical exponent matches the observational one form~ 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. Subject headings: convection Sun: atmosphere Sun: oscillations turbulence wave motions