The expression (dB/dz) - (dM/dy) is identically zero - if wave is stationary and u is not 0. (Eliassen and Palm, 1961).
--> if critical level or dissipative effects not incorporated, RHS of (1) does not vanish.
Although conditions under which RHS of (1) vanishes are not fulfilled in winter stratosphere, C-D theorem valid to a first approximation in the following sense:
Planetary waves in general, transport heat and momentum, theorem would mean effects of eddy fluxes and mean meridional circulation cancel each other ... zonal mean field unchanged.
Hiroto and Sato (1969) found the actual change of zonal wind velocity is very poorly correlated with convergence of eddy momentum flux, suggested C-D theorem may hold in ordinary situations.
Where the C-D theorem can not be applied will explain SSW within the framework of planetary wave-zonal interaction.
Assumptions: beta plane, atmosphere bounded laterally by vertical walls at 2 lattitudes, extending infintely in vertical direction.
Wave has no phase variation in latitudinal direction. Wave transports no momentum, forcing to mean height change occurs only when heat transport varies with height.
First: consider a situation in which basic wind profile has critical level at height zc.
--> heat flux jumps from positive to 0 in crossing zc. Equation 1 becomes:
The zonal mean heights tend to rise in higher latitudes and fall in lower. Mean meridional circulation is implicit in the sollution.
Since upward propagating planetary waves accompany poleward heat transport (Eliassen and Palm, 1961), there is a heating tendency at higher latitudes and cooling at lower as a result of flux divergence.
Effect forces zonal mean upward motion at higher latitudes and downard motion at lower latitudes.
--> forced vertical motions diminish above critical level because heat transport vanishes there..
--> to satisfy continuity of mass flux, must be a flow from higher to lower latitudes near critical level.
The coriolis force acts on this 'wave-induced meridional circulation' and is the origin of easterly acceleration.
temperature change results from balance between eddy heating and effect of mean vertial motion.
--> below the critical level, the eddy heating exceeds the effect of the mean vertical motion.
--> above the critical level, only vertical motion causes temperature changes.
The solution to (2) is consistent with statement 'rossby waves incident on critical level force southward transport of potential vorticity' (Dickinson, 1970).
Consider the following situation: basic wind is westerly everywhere, planetary waves in transient state of upward propagation ... waves generated from far below and leading edge of waves has reached a certain height zf.
--> wave amplitude as well as heat flux due to waves (B) may decrease with height in vicinity of zf.
Conclusion: upward propagating planetary waves accompany easterly acceleration of zonal winds and warming of the air in the higher latitude side, when the waves are in a building up state.
Contrast: when waves are decaying, second-order effects may act in opposite way.
previous discussion predicated on cases with no momentum transport associated with waves)
If a horizontal flux of zonal momentum exists, can contribute to redistribution of momentum in horizontal layer, but can not cause changes to total momentum in layer.
The model of the sudden warming
Assume: zonally asymmetric circulation with normal winter profile, planetary-scale disturbances (wn 1 and 2) grow with time to reach large amplitude and persist for long time.
--> blocking is observed in parallel with breakdown of polar night jet, reasonable assumption.
Response of the stratosphere
Phase 1
Wave may propagate upward, giving rise to deceleration of westerly jets, weakening polar night jet at same time the disturbance increases its amplitude.
Total flow pattern appears as a deformation, then breakdown of PV.
--> temperature disturbance due to waves becomes significant and zonal mean temperature at high latitudes rises.
Phase 2
eASTERLY ACCELERATION INCREASES with increasing height due to wave amplitude increasing with decrease of air density. At a certain level, wave-induced acceleration destroys westerly jet, creates an easterly jet.
--> once easterly jet appears, critical layer interaction appears
--> planetary waves inhibited from propagating further upward and are absorbed, reducing their amplitude.
Intense warming occurs just below the critical level, easterly winds near critical level accelerated by the waves, lowering the level.
Warming and wind reversal shift downward.
Lindzen-Holton hypothesis (1968): process of absorption of waves at a critical level and consequent descent of the level is consistent with LH 1968 on QBO of equatorial winds.
--> different acceleration mechanisms.
LH scheme: zonal winds accelerated by deposition of zonal momentum transported vertically by equatorial waves.
SSWs: acceleration attributed to coriolis force due to secondary circulation induced by heaet transport by planetary waves.
(Discussions on these two types of zonal wind acceleration by waves have been made by Bretherton (1969), Lindzen (1970), Hayashi (1970).
Model used consists of 1) planetary wave propagation in a zonal wind system and 2) chagne of zonal winds forced by waves.
Adiabatic, geostrophic, potential vorticity equation split into zonal mean and deviation from it.
<<<<<<<<<<<<<<<<<<<<<
######################################################################
######################################################################
################### DEFINITIONS ######################################
######################################################################
######################################################################
The following definitions are used below, with the temperature condition turned on/off for sensitivity tests.
#MAJOR SSW: westerly winds at 60N, 10 hPa reverse (become easterly (positive to negative))
# *zonal mean temperature increases poleward from 60 degrees latitude
#MINOR SSW: westerly winds are slowed but do not reverse
# *significant temperature increase (25 degrees in a period of a week or less)
#FINAL SSW: becomes easterly for the summer
#
# https://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-16-0465.1
Based on reanalysis, an average of 0.46 SSW events occur during DJF when averaged over 50 years. During this 20 year analysis, we found an average of 0.50 SSW events during DJF.
#http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=8344&context=etd_theses
#Based on the reanalysis results, 26 SSW events were identified and occurred with an average of 0.46 events per
#winter over the 56 year period.
#at 0.50 events per year, control run is PERFECT
https://journals.ametsoc.org/doi/pdf/10.1175/JCLI-D-16-0465.1