On the Shallow Motion Approximations

Abstract

{T}he approximate equations for shallow motions are derived thatare valid for the vertical scale being much smaller than the horizontal,and the local time scale being much greater than the minimum of horizontaland vertical advective time scale, respectively, times the ratioof perturbation and basic state specific volume. {T}he {B}oussinesqapproximations form a subclass of shallow motions. {T}hey do notapply to near-neutral conditions. {T}he square of the {F}roude numberhas to be much smaller than 1 and {R}/g(d{T}0/dz) also has to bemuch smaller than 1 for those approximations to be valid. {A}lsothe use of the shallow motion approximations together with {R}eynoldsaveraging is discussed. - all results summarized in {T}able1 - {REQUIRED}for {B}oussinesq approx: flow must be separable into a basic stateand perturbations - 2 advantages of {B}oussinesq approximation: *incompressible mass continuity equation can be used ("shallow motion")plus a linearized version of the ideal gas law * buoyancy term canbe expressed in terms of temperature instead of density ("shallowconvection"); requires more restrictive conditions! - when can incompressiblemass continuity be used? * magnitude of perturbations << basic statevalues * {D} << {H} ({D} ... vertical scale of perturbation densitychange, {H} ... vertical state of basic state density change, seebelow for formulae and values) * scale of local time change >> 0.01* min({L}/{V}, {D}/{W}) (0.01 comes from alpha'/alpha, where alpha'is the perturbation specific volume) {T}his means that density changeshave to come mostly by mass transport {T}his excludes strong diabaticheating with little motion and compression waves - when can temperatureonly (ie without pressure) be used for density? * {T}he followingconditions are {ADDITIONAL} to the ones for incompr. mass continuity!* {V}^2/{D} << 1- remarks on {H} * {H} = alpha0 / ( d alpha0 / d z ) * {H} = {H}i/(1 + {R}/g * d{T}0/dz ) (scale height {H}i={R} {T}0/g) * when {T}decreases with z: {H} > {H}i; {H}max for adiabatic lapse rate = 41m/{K}* {T}0 * for inversions {H}