The Rise of a Strong Inversion 

 Caused by Heating at the Ground 



Robert R. Long and Lakshmi H. Kantha 

 The Johns Hopkins University 

 Baltimore, Maryland 



ABSTRACT 



A theory is offered for the rise of a strong inver- 

 sion in the atmosphere caused by heating at the 

 ground. The heating, specified by the buoyancy 

 flux, qj , near the ground, causes turbulence in a 

 growing layer of depth, D, above the ground with an 

 inversion or interfacial layer of thickness, h, 

 separating the mixed layer from the non-turbulent 

 air above. There is a buoyancy jump, Ab, across 

 the interfacial layer and the air above the inver- 

 sion has a buoyancy gradient, N . 



The lower surface of the inversion layer rises 

 (at a speed, Ug = dD/dt) because of two processes. 

 One is related to the mean temperature rise of the 

 mixed layer which, in the present model, leaves h + 

 D unaffected but which causes the interfacial thick- 

 ness, h, to decrease and therefore D to increase at 

 a rate proportional to Ri~ , where Ri = DAb/w| is 

 the Richardson number and w* = (qiD) ' is the con- 

 vective velocity typical of the rms velocities in 

 the main portion of the mixed layer. The second 

 process, increasing both h and D, is the erosion of 

 the stable fluid by the turbulence in the mixed 

 layer and the intermittent turbulence in the inter- 

 facial layer. This causes D to increase at a rate 

 proportional to Ri~ ' . The total effect is con- 

 tained in the equation 



u„ 



aRi' 



■1 



+ cRi 



-7/1* 



where a and c are universal constants. Other re- 

 sults are presented, notably the ratio, | qz/ll 1 ' where 

 q2 is the (negative) buoyancy flux near the level 

 z = D. This ratio decreases with increase of sta- 

 bility as observed in experiments of Willis and Dear- 

 dorff. |q2/qi | - Ri'S/"*. 



1 . INTRODUCTION 



When the sun rises and begins to heat the ground, 

 the atmosphere is normally in a stable state (po- 



tential temperature increases with height) . If we 

 neglect the effect of mean wind for the moment, the 

 heating creates instability and turbulence near the 

 ground and a mixed layer of depth, D, appears, capped 

 by an inversion. This phenomenon is called penetra- 

 tive convection. The potential temperature of the 

 mixed layer is nearly constant with height except 

 very close to the ground, where a superadiabatic 

 lapse rate exists in a thin layer, and just below 

 the inversion base where there is weak stability. 

 The inversion base rises because of two processes. 

 The first is heating alone which tends to decrease 

 the thickness, h, of the inversion layer, (IL) , and 

 so increase D. The second is the entrainment effect 

 of the turbulent eddies just below the inversion 

 base. We do not have a detailed understanding of 

 this erosion process but laboratory experiments with 

 mechanical stirring [Moore and Long (1971) , Linden 

 (1973) ] suggest that the eddies in the mixed layer 

 deflect the IL upward storing potential energy. When 

 this is released by downward motion, a portion of 

 the lighter fluid in the IL is ejected into the 

 homogeneous layer where it is carried away by the 

 turbulent eddies , leaving the lower surface of the 

 IL sharp again. 



If there is no mean wind, the energy for the tur- 

 bulence comes from the energy flux divergence term 

 and from the buoyancy flux term in the energy equa- 

 tion, where q = -w'b' is the buoyancy flux . When 

 there is a mean wind, as is usual in the atmosphere, 

 the shear yields another energy source. This serves 

 to increase the turbulence energy and thus to in- 

 crease the entrainment effect through greater agita- 

 tion of the IL. In addition, the shear may cause 

 Kelvin-Helmholtz instability and consequent wave 

 breaking at the interface and thereby enhance ero- 

 sion. 



On the other hand, the effect of shear should be 



Buoyancy in an incompressible fluid is defined as b = 

 g(p - Po'/PO where g is gravity, p is density and pg is a 

 representative density. In the atmosphere, p and po are 

 potential densities. We may also write b = g(e -9o)/Bo 

 where 6 is a potential temperature . 



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