The scaling of turbulence in the presence of stable stratification

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The force of gravity singles out the vertical direction in a stably stratified fluid. Since buoyancy forces remove energy preferentially from the vertical component of velocity, there is a pervasive belief that turbulence in such systems may be significantly anisotropic, with horizontal length and velocity scales greatly exceeding their vertical counterparts. This paper examines the scaling of the equations of motion of a stratified fluid, assuming that the energy‐containing scales of the motion are significantly anisotropic and that nonlinear terms are important in the equations of motion (an assumption of the existence of “turbulence”). Relevant nondimensional parameters are found to be the vertical Reynolds number Rew ≡ wh/v, where w and h are vertical velocity and length scales respectively, and the turbulence Froude number Fr ≡ u/Nl, where u and l are the horizontal velocity and length scales, respectively, and N is the buoyancy frequency. Two determinate scalings are found. The first case, termed buoyancy‐affected high Rew turbulence and characterized by Rew ≫ 1 and Fr ∼ 1, is necessarily isotropic. This scaling has its roots in the work of Dougherty (1961); its use is widespread in recent literature. It is important to recognize this as the appropriate scaling for isotropic turbulence in which the importance of buoyancy forces is in limiting the vertical (hence through isotropy, also the horizontal) scale of the energy‐containing eddies which irreversibly transfer energy to smaller scales, while noting that in a stratified fluid there are other possible modes of motion which can transfer energy reversibly (linear waves) or to larger scales (vortical modes). The second, termed buoyancy‐affected low Rew turbulence and characterized by Rew ∼ 1 and Fr ∼ 1, is truly anisotropic, with the possibility of u ≫ w and h ≪ 1. The field observations of Gargett et al. (1984) and the laboratory observations of Stillinger et al. (1983) and Itsweire et al. (1986) are examined in the context of these scalings. It is concluded that the field observations are characteristic of the buoyancy‐affected high Rew regime, while the laboratory regime may be buoyancy‐affected low Rew throughout its reported evolution. A critical test for the existence of the latter regime is suggested.