Wallace and Hobbs 2.7 Static Stability

by Joe Archive on August 6, 2008

2.7.1 Unsaturated air.br /br /If the lapse rate of the atmosphere in a region is less than the dry adiabatic lapse rate and a parcel is lifted adiabatically from the bottom of the region to the top (assuming that the parcel is never saturated in this path), whent he parcel reaches the top, it will have cooled according to the dry adiabatic lapse rate and thus it will be cooler than its surrounds.br /br /Cooler air at the same pressure will be more dense and thus the parcel will experience a downwards restoring force due to its negative buoyancy. The reverse also holds for a parcel moved from the top downwards. br /br /Generally, if Gamma Gamma_d, the atmosphere is “positively stable” (for unsaturated air) and vertical mixing is inhibited.br /br /Example 2.12 Derive the formula for the restoring force on a parcel displaced a small distance away from rest in a stable atmosphere.br /br /let primes denote the parcel’s variables.br /br /for the atmosphere, in hydrostatic equilibriumbr /br /dp/dz = – rho gbr /br /the acceleration of the parcel when it is raised upwards isbr /br /a = -(rho’-rho)*g/rhobr / br /(negative sign indicates downward accel.)br /br /a = -g(p/R) * (1/T’ – 1/T)/[(p/R)*(1/T)]br /a = -gT(1/T’ – 1/T) = -g(T/T’ -1) br /a = -g/T’ * (T-T’) = -g/T’ * [(T_0 – Gamma) – (T_0 -Gamma_d)]*delta zbr /a ~ -g/T (Gamma_d – Gamma)*delta zbr /br /Harmonic oscillation!br /br /A Layer of air with an increasing temperature (Gamma 1) is an inversion – a very stable layer of air.br /br /Gamma = Gamma_d — neutral stability – a displaced parcel remains neutrally buoyant, as long as it never becomes saturated!br /br /if Gamma Gamma_d, the air becomes unstable – a parcel displaced upwards experiences an upwards force, because it becomes warmer than its environment. These conditions do not last, as the instability causes a redistribution of heat bringing the atmosphere back to a neutral stability. br /br /Instability can persist int he presence of strong heating from below, eg close to a surface.br /br /Example 2.13 Show that dtheta/dz 0 in the environment is equivalent to positive static stability.br /br /This can be shown mathematically, but logic should suffice – when dtheta/dz 0, an upwards displaced parcel, moving adiabatically, and thus conserving theta, will have a lower theta than its surrounds. But, both the parcel and the environment are at the same pressure, so this is equivalent to the parcel being cooler than its surrounds in actual temperature, and thus negatively buoyant.br /br /QEDbr /br /2.7.2 – Saturated air.br /br /When a parcel becomes saturated as it moves, it no longer follows a dry adiabat – instead it follows a moist adiabat – the stability of a saturated atmosphere is then detemrined by moist adiabatsbr /br /2.7.2 – Conditional can convective instabilitybr /br /When the actual lapse rate of the atmosphere is somewhere between the dry and moist lapse rates, a parcel of air might be stable for small displacements, but large enought displacements to cause the parcel to become saturated can result in the parcel becoming warmer than its surrounds.br /br /The lifted parcel cools at the dry adiabat until the LCL is reached. At this point, it begins to follow a moist adaibat – cooling slower. If it is lifted far enough, it reaches the LFC (level of free convection) where it is positively buoyant. br /br /In a typical atmosphere, there is a point near the tropopause where the parcel ceases to be buoyant – the LNB, indicating the approximate height reached by the convection.br /br /In practice, mixing with air from the surrounds tends to reduce the buoyancy of the parcel, limiting the height it reaches.br /br /An atmosphere where Gamma is between the dry and moist lapse rates is referred to as conditionally unstable.

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