Atmospheric dynamics and meteorology B. Legras, http://www.lmd.ens.fr/legras II Potential vorticity, tropopause and baroclinic instability (supposed to be known: notions on the conservation of potential vorticity, simple model of the baroclinic instability (Eady / Phillips)) Recommended documents: - Hoskins, McIntyre & Robertson, 1985, On the use and significance of isentropic potential vorticity maps, Quart. J. Met. Soc., 111, 877-946
Outline 1 The thermal tropopause The potential vorticity 3 The dynamic tropopause 4 The anomalies of potential vorticity 5 Mutual amplification 6 The baroclinic instabillity revisited
1The thermal tropopause Simple or double tropopause 3
Meridional variation of the tropopause Airborne measurement by radiometer. tropopause Aircraft track 4 Region of the double tropopause in the subtropics
Wind and temperature at the tropopause Temperature minima at the tropical tropopause Subtropical jet winds associated with tropopause drop B. Randell ERA-40 Atlas, 005 100 Hpa 00 Hpa Isentropic surfaces crossing the tropopause in the subtropics Potential temperature and 5 the tropopause
The potential vorticity Ertel potential vorticity θ D ζ a =0 ρ Dt ( ) for a flow without friction or heating. Simplified forms:. 6
Instantaneous height-latitude cross section of potential vorticity along a single longitude (55W), with the tropopause marked (in black) as the PVU contour. Courtesy of H. Wernli, ETH Zurich 7 PV increases in the stratosphere due to increase of static stability. It increases in latitude due to f. PV definition of the tropopause
1 PVU = 10-6 K kg-1 m s-1 8
Relation between the double tropopause and the PV distribution tropopause from aircraft profiler measurements zonal wind (from analysis) potential vorticity (from analysis) from Pan et al., JGR, 004 9
As the tropopause is a material surface, it can also be seen as a discontinuity (or a location of high gradients) in long-lived tracers. aircraft ozone measurements B. Randel tropopause fold associated with stratospheric intrusion 10 See also Bethan et al., QJRMS, 1996
3 Potential vorticity anomalies Cut-off Lows and Blocking Highs IPV on 330K for consecutive days. 1-PVU filled in black ADVECTION OF LOW IPV IPV A BETTER WAY OF DEFINING CUT-OFF THAN 500hPa HEIGHT CUT-OFF LOWS AND BLOCKING HIGHS HAVE SIMILAR (OPPOSITE) STRUCTURES 11 CONVECTIVE HEATING DAMPS CYCLONE FASTER THAN RADIATIVE COOLING DAMPS ANTICYCLONE Hoskins, McIntyre and Robertson (1985) Fig. 11 BLOCKING HIGH CUT-OFF LOW (HIGH PV)
Effect of an axisymmetric PV anomaly localized at the tropopause. Upper panel : positive anomaly Lower panel : negative anomaly Section of the PV distribution + wind (contours) warm Augmented stability cold PV + Cyclone cold Reduced stability 1 Inside the colored area: PV anomaly. Outside : no anomaly.de PV. Background PV : uniform stratification in the horizontal with a discontinuity at the tropopause. Surfaces are drawn with an interval Δθ=5K PV warm Anticyclone Hoskins, McIntyre and Robertson, 1985
Vertical or lateral propagation of a wave Growth and westward displacement Decay and westward displacement 13
Rossby waves in a PV gradient Ondes de Rossby dans un gradient de PV 14
Effect of a ground temperature anomaly Upper panel: warm anomaly, <> positive PV anomaly Lower panel : cold anomaly, <> negative PV anomaly Inside colored areas: PV anomaly. Outside : no PV anomaly. Potential temperature surfaces every 5K. Contours : azimuthal horizontal wind. Reduced stability warm Thermal cyclone Augmented stability cold 15 Thermal anticyclone
Amplification of a PV anomaly by mutual advection The warm surface anomalies acts as positive PV anomalies near the surface PV anomalies induce a flow in the same way an electric charge induces an electric field. The vertical penetration and the magnitude of the flow induced by a PV anomaly grow with its horizontal scale. The flows induced by PV anomalies can be approximately superimposed The induced flows can generate mutual amplification if the positions of the two anomalies are shifted with the upper one on the west of the lower one and if they get locked. meridional PV gradient at the tropopause UPPER-LEVEL +PV ADVECTED OVER A LOW PV +v REGION PV =0 LEVEL CAN LEAD TO topbaroclinic h t y MOIST PROCESSES MUTUAL AMPLIFICATION. TTHE T =0 COULD ENHANCE bottom +v b SURFACE t y DEVELOPMENT T surface gradient 16 ( )IPV g p + PV advection T advection Hoskins, McIntyre and Robertson (1985) Fig. 1 4. Mutual amplification
Analysis of winter storm Lothar 18Z, 5 DEC1999 PV= SURFACE LOW-LEVEL CYCLONE LOTHAR V850 Wernli et al. (00) Fig. 7a 17 TROPOPAUSE FOLDING ASSOCIATED WITH KURT AND ISENTROPIC DOWN-GLIDING(?) CYCLONE KURT
Analysis of winter storm Lothar 0Z, 6 DEC1999 PV= SURFACE LOW-LEVEL CYCLONE LOTHAR V850 Wernli et al. (00) Fig. 7b 18 TROPOPAUSE FOLD NOW ALSO ASSOCIATED WITH LOTHAR CYCLONE KURT
Analysis of winter storm Lothar 6Z, 6 DEC1999 CYCLONE KURT PV= SURFACE LOW-LEVEL CYCLONE LOTHAR V850 UPPER AND LOWER PV ANOMALIES NEARLY JOIN INTENSE WINDS KILL 50 Wernli et al. (00) Fig. 7c 19
Growth of a barotropic or baroclinic mode coupling two waves within opposite gradients of PV. Mutual amplificaton Locking by mutual increase of the phase speed. Us c0 c > 0 Ui + c0 c < 0 Figure in the referential frame moving at speed c in which the two waves are stationary. 0 5. Baroclinic instability dp/dy > 0 dp/dy > 0 dp/dy < 0 ou dt/dy < 0 The instability requires PV gradients of opposite sign. Westward tilt. U > c > U s i
Simplified Phillips layer model (1) z4 z3 z z1 z0 ( ) D ψ ψ+μ +β y =0 Dt z ψ =0 en z=z 0, z 4 z Layer thickness z i +1 z i =h Equations on levels 1 et 3 + ψ ψ +β y+s (ψ ψ ) =0 ( 1 1 3 1 ) t + ψ ψ +β y+s (ψ ψ ) =0 ( 3 3 1 3 ) t μ with S= 4h Basic state U 1, U 3, Ψ 1 = U 1 y, Ψ 3 = U 3 y Perturbed state ψ1 =Ψ 1 + ψ1 ', ψ3=ψ 3+ ψ3 ' Linearized equations ψ1 ' ψ1 ' O= +U 1 ( ψ1 ' +S (ψ3 ' ψ1 ' ))+ x S (U 1 U 3 )+β x t ψ3 ' ψ3 ' O= +U 3 ( ψ3 ' +S (ψ1 ' ψ3 ' ))+ x S (U 3 U 1)+β x t ( ( ( ( 1 ) ) ) )
Simplified Phillips layer model () Linearized equations 0= +U 1 t ψ1 ' ψ1 ' S (U 1 U 3 )+β ψ3 ' ψ3 ' 0= +U 3 ( ψ3 ' +S (ψ1 ' ψ3 ' )) + S (U 3 U 1 )+β t 1 1 1 1 We define ψm = (ψ1 ' + ψ3 ' ), ψt = (ψ3 ' ψ1 ' ), U m = (U 1 +U 3 ), U T = (U 3 U 1 ) ψm ψt such that 0= +U m ψm +β +U T t ψt 0= +U m ( ψt S ψt )+β +U T ( ψm + S ψm ) t i k ( x c t ) i k ( x c t ) We choose ψm = A e cosl y et ψt =B e cos l y avec K =k +l ( ( ) ) ( ψ1 ' +S (ψ3 ' ψ1 ' ))+ ( ( ) ) Thus i k ( (c U m ) K +β) A i k K U T B=0 i k U T (K S ) A+i k ((c U m )(K + S )+β ) B=0 After calculation, the dispersion relation is 4 (c U m ) (K + S K )+β(k +S )(c U m )+β +U T ( S K )=0 β( K + S ) In other words c=u m + ±δ K ( K + S ) β S S K with δ = 4 U T K ( K + S ) K + S
Simplified Phillips layer model (3) β( K + S ) Dispersion relation c=u m + ±δ K ( K + S ) β S S K with δ = 4 U T K ( K + S ) K + S a) U T =0, in this case c is real (pure propagation) with possible values β β c 1=U m (external mode) and c =U m (internal mode) K K + S b) β=0, in this case c=u m±u T ( K S K + S 1/ ) Instability occurs for K < S c) general case, the instability threshold δ=0 is then given by ( 1/ ) K4 β =1± 1 S 4S UT Notice that β produces a stabilizing effect With U T =4 m s 1 between 750 and 50 hpa. Most unstable mode: K =1/ 4000 km 3 S UT unstable stable 1 K S
Simplified Phillips layer model (4) The basic flow vorticity is Q 3 =β y+s y (U 3 U 1 ) Q 1 =β y+s y (U 1 U 3 ) Knowing that U 1<U 3, the necessary instability condition on the potential vorticity gradients is Q1 Q3 <0< y y that is S (U 1 U 3 ) < β < S (U 3 U 1 ) S UT or in other words >1 β This is identical to the instability criterion established in the previous slides Since the wind is uniform over each level, the gradient is only due to the planetary vorticity and the thermal structure. 4