PRIMITIVE EQUATIONS Primitive Equations Definition The so called primitive equations are those that govern the evolution of the large scale motions In other words are the equations that describe the horizontal and vertical movement of the atmosphere and changes in temperature They are easiest to interpret when we transform the z coordinate into p coordinate that is x y z x y p Vertical movement in p coordinates The vertical velocity component in x y p coordinate is V horizontal wind dp p p V p w dt t z Substituting p z g from the Hydrostatic equation 1 week for a 10hPa day Note that w and 10hPa day 100hPa day parcel to have opposite sign move from p the lower to V p gw ascending the upper descending t troposphere movements gw negative positive How to interpret Pressure dp dt 600mb 700mb 800mb 900mb 1000mb Comparing w with 100hPa day is equivalent to 1km day or 1cm s in the lower troposphere and twice that value in the midtroposphere see the example shown before the distance between two pressure levels increases with height Hydrostatic balance We saw before Joel s classes that the vertical component of the movement could be described as dw 1 p g Cz Fz dt z Where Cz are the vertical components of the Coriolis and Frictional forces respectively Vertical velocities are very small and we can assume to within 1 that the upward gradient force balances the downward pull of gravity also for large scale motions this approach is not valid for cloud scale motions though The Hydrostatic Balance can be assumed Thermodynamic Energy Equation The evolution of the weather systems is governed by dynamical Newton s Laws AND THERMODYNAMIC PROCESSES First and second law of thermodynamics The first law of the Thermodynamics which represents changes and heat expansion contraction increase decrease in temperature etc is a prognostic equation for the parcel of air moving in the atmosphere Changes in temperature cause changes in the gradient of geopotential height with implications for the winds indicated by the relationships between pressure gradient and geopotential gradient First Law of Thermodynamics The first Law of the Thermodynamics can be written as J dt cpdT dp Where J represents the DIABATIC HEATING RATE Joules kg 1 s 1 and dt is the infinitesimal time interval Dividing by dt and rearranging the terms we obtain dT dp cp J dt dt Using the state equation for a substitution of and replacing dp dt by we obtain the thermodynamic energy equation dT T J dt p cp R cp 0 286 Interpretations dT T J dt p cp 1 1 Rate of change of temperature due to ADIABATIC EXPANSION OR COMPRESSION In a typical midlatitude disturbance air parcels in the middle troposphere 500hPa undergo vertical displacements of 100hPa day Assuming T 250K the resulting adiabatic temperature change is T 250 K 100 hPa 0 286 14 3o day p 500 hPa day Interpretations dT T J dt p cp 2 2 represents the effects of DIABATIC heat sources and sinks absorption of solar radiation absorption emission of longwave radiation latent heat release and in the upper atmosphere heat absorbed or liberated in chemical and photochemical reactions as in the ozone layer Exchange of mass with the environment due to convection and turbulence can also affect 2 Cancelations among these contributions occur throughout the atmosphere and the net radiative heating rates are less than 1 o day How horizontal advection and stability contribute to changes in Temperature Let s first expand the thermodynamic equation to let the horizontal advection appear explicitly and separated from the vertical advection T T T J V T w t z p cp T T p T J V T w t p z p cp T T T J V T t p p cp T p T T w gw p z p p Interpretation T T T J V T t p p cp I II i Horizontal Advection term ii combined effect of adiabatic compression and vertical advection It can be shown Ex 7 37 by using the state equation and the fact that the adiabatic lapse rate d g cp and the environmental lapse rate is T T g T T T w d w c z p p p Let s interpret these terms T T T J V T t p p cp T T d w p p If the observed lapse rate is equal to the dry adiabatic lapse rate then this term vanishes In a stably stratified atmosphere d and if there is subsidence w 0 or 0 the contribution of this term is to increase temperature contribution The opposite occurs in ascending movements The magnitude of this term depends on the stability of the layer difference between and d Other terms T T T J V T t p p cp If the motion is adiabatic and the atmosphere stably stratified then J 0 and parcels will conserve potential temperature as they move along their tridimensional trajectories Example of the importance of advection of temperature in association with extratropical cyclones T T T J V T t p p cp Ex 7 4 During the time that a frontal zone passes over a station the temperature falls at a rate of 2oC h The wind is blowing from the north at 40km h and the temperature is decreasing with latitude at a rate of 10oC 100km Estimate the terms in the eq above neglecting the Diabatic heating i e J 0 y v 0 T 10 C 1 V T v 40kmh 40 C h 1 y 100km x Therefore subsidence must be warming the air at a rate of 2oC h as it moves southward Tropical Regions little variations in temperature from day to day T extratropi c t Weak temperature gradients T T T J V T t p p cp Tropics horizontal temperature gradients are much weaker than in the extratropics horizontal advection is unimportant temperatures at fixed points vary little from day to day Frontal systems ITCZ Anticyclone High temperature gradients Anticyclone Frontal systems Tropical regions Little variations in Daily Temperature T T T J V T t p p cp Diabatic heating rates in regions of tropical convection ex Release of latent heat absorption of solar radiation etc are larger than those in the extratropics Ascents are concentrated in narrow rain bands ITCZ Warming due to the release of Latent Heat is compensated by cooling due to upward movement ITCZ Anticyclone Anticyclone Tropical regions Little variations in Daily Temperature T T T J V T t p p cp Slow Subsidence dominates T p moist adiabatic which means that vertical movements in tropical regions that follow the moist adiabatic lapse rate do not add much to variations in temperature even in regions where is negative and large upward movement In regions with slow subsidence over tropical oceans warming due to adiabatic compression is balanced by weak radiative cooling Anticyclone Anticyclone Inference of the vertical motion field the continuity equation Consider an