Introduction and theory¶
Along with updrafts, downdrafts—downward-moving masses of air—are important features in the dynamics of the Earth’s atmosphere; they transport mass, momentum, heat and moisture vertically and also generate and maintain storms [knupp_cotton_1985].
Indeed, one of the main objectives of present-day research into downdraft dynamics is to improve the predictions of global climate models [thayer-calder_2013], whose output informs the understanding of the larger-scale dynamics, including the pressing issue of anthropogenic climate change. Specifically, the high computational cost of running a global climate model over the necessarily large spatial domain and prediction timescales constrains their maximum resolution, which is still too coarse to describe convection. The models therefore employ schemes known as parametrisations which estimate the effect of convection on the state of the model using the information available at each time step; an accurate estimation requires a strong understanding of the factors that govern convection.
On a smaller scale, strong downdrafts that reach the Earth’s surface (downbursts) are known to cause significant damage to man-made structures and create hazardous, or even deadly, conditions for aircraft [thayer-calder_2013]. Another aim of downdraft research is therefore to understand the mechanisms that generate such extreme events and improve the ability to predict them in advance.
Considering these motivations, the goal of this work is to gain insight into which processes and conditions initiate, and which maintain or inhibit, downdrafts. The approach will be to construct a significantly simplified model of a downdraft using parcel theory.
An air parcel is a mass of air with an imaginary flexible (but usually closed) boundary; under the usual assumptions, its exact size and shape are irrelevant. The only force assumed to act on the parcel is the net buoyant force (per unit mass), given in accordance with Archimedes’ principle by
If the parcel is lowered in the atmosphere to a location with a higher pressure, the work done to compress it and any heat exchanged will manifest as a change in its internal energy in accordance with the first law of thermodynamics. The second key assumption of parcel theory is that this process is adiabatic; this is valid due to the low thermal conductivity of air.
The potential presence of water in gas, liquid and solid phases in the parcel is a major complication; under the assumption that the parcel remains in phase equilibrium (i.e., changes are slow enough for excess liquid to evaporate if the vapour pressure is below the saturation value), there are two modes of adiabatic descent the parcel may undergo. If no liquid is present, the descent is dry adiabatic and the rate of work on the parcel causes it to warm at an approximate rate of 9.8 K/km. If liquid is present, the descent is moist adiabatic: progressive warming of the parcel raises its saturation vapour pressure, allowing the liquid to progressively evaporate during descent, with the necessary transfer of latent heat from the air to the water creating an opposing cooling effect.
Moist adiabatic descent is commonly assumed to be either pseudoadiabatic, in which case liquid water does not contribute to the heat capacity of the parcel (as if it precipitates from the parcel immediately upon condensation), or reversible, in which case the liquid does contribute to the heat capacity. A reversibly descending parcel warms at a slightly slower rate than a pseudoadiabatically descending one due to its larger heat capacity [saunders_1957]. Both modes were investigated, but reversible descent was ultimately chosen as it is the more realistic case for a parcel known to retain liquid water.
If the pressure and temperature of the parcel are thus known at any point in its descent, its density may be calculated using the ideal gas law,
where \(T_v\) is the virtual temperature that contains a small correction to account for the different density of water vapour. If an mass \(l\) of liquid water, per unit total parcel mass, is also present, it is easily shown that (assuming the liquid occupies negligible volume) the corrected parcel density is
Knowledge of the parcel and environmental densities enable calculation of the buoyant force per unit mass on the parcel using (1), and its resulting displacement and velocity may be obtained by (numerically) solving the ODE