Let us consider an idealized streamfunction on a midlati- tude β -plane comprising a zonally averaged part and a wave disturbance Φ = Φ 0 − f 0 ¯ uy + f 0 A sin kx cos ℓy The parameters Φ 0 , ¯ u and A depend only on pressure and the wavenumbers are k = 2 π/L x and ℓ = 2 π/L y . 9
Let us consider an idealized streamfunction on a midlati- tude β -plane comprising a zonally averaged part and a wave disturbance Φ = Φ 0 − f 0 ¯ uy + f 0 A sin kx cos ℓy The parameters Φ 0 , ¯ u and A depend only on pressure and the wavenumbers are k = 2 π/L x and ℓ = 2 π/L y . The geostrophic winds are u g = − 1 ∂ Φ u + u ′ ∂y = ¯ g = ¯ u + ℓA sin kx sin ℓy f 0 v g = + 1 ∂ Φ + v ′ ∂x = g = + kA cos kx cos ℓy f 0 9
Let us consider an idealized streamfunction on a midlati- tude β -plane comprising a zonally averaged part and a wave disturbance Φ = Φ 0 − f 0 ¯ uy + f 0 A sin kx cos ℓy The parameters Φ 0 , ¯ u and A depend only on pressure and the wavenumbers are k = 2 π/L x and ℓ = 2 π/L y . The geostrophic winds are u g = − 1 ∂ Φ u + u ′ ∂y = ¯ g = ¯ u + ℓA sin kx sin ℓy f 0 v g = + 1 ∂ Φ + v ′ ∂x = g = + kA cos kx cos ℓy f 0 The geostrophic vorticity is ζ g = 1 ∇ 2 Φ = − ( k 2 + ℓ 2 ) A sin kx cos ℓy f 0 9
It is easily shown that the advection of relative vorticity by the wave component vanishes, ∂ζ g ∂ζ g u ′ ∂x + v ′ ∂y = 0 g g 10
It is easily shown that the advection of relative vorticity by the wave component vanishes, ∂ζ g ∂ζ g u ′ ∂x + v ′ ∂y = 0 g g Thus, the advection of relative vorticity reduces to u ∂ζ g u ( k 2 + ℓ 2 ) A cos kx cos ℓy − V g ·∇ ζ g = − ¯ ∂x = k ¯ 10
It is easily shown that the advection of relative vorticity by the wave component vanishes, ∂ζ g ∂ζ g u ′ ∂x + v ′ ∂y = 0 g g Thus, the advection of relative vorticity reduces to u ∂ζ g u ( k 2 + ℓ 2 ) A cos kx cos ℓy − V g ·∇ ζ g = − ¯ ∂x = k ¯ The advection of planetary vorticity is − βv g = − βkA cos kx cos ℓy 10
It is easily shown that the advection of relative vorticity by the wave component vanishes, ∂ζ g ∂ζ g u ′ ∂x + v ′ ∂y = 0 g g Thus, the advection of relative vorticity reduces to u ∂ζ g u ( k 2 + ℓ 2 ) A cos kx cos ℓy − V g ·∇ ζ g = − ¯ ∂x = k ¯ The advection of planetary vorticity is − βv g = − βkA cos kx cos ℓy The total vorticity advection is u ( k 2 + ℓ 2 ) − β ] kA cos kx cos ℓy − V g ·∇ ( ζ g + f ) = [¯ 10
Repeat: The total vorticity advection is u ( k 2 + ℓ 2 ) − β ] kA cos kx cos ℓy − V g ·∇ ( ζ g + f ) = [¯ 11
Repeat: The total vorticity advection is u ( k 2 + ℓ 2 ) − β ] kA cos kx cos ℓy − V g ·∇ ( ζ g + f ) = [¯ For relatively short wavelengths ( L ≪ 3 , 000 km) the advec- tion of relative vorticity dominates. For planetary-scale waves ( L ∼ 10 , 000 km) the β -term dominates and the waves regress. 11
Repeat: The total vorticity advection is u ( k 2 + ℓ 2 ) − β ] kA cos kx cos ℓy − V g ·∇ ( ζ g + f ) = [¯ For relatively short wavelengths ( L ≪ 3 , 000 km) the advec- tion of relative vorticity dominates. For planetary-scale waves ( L ∼ 10 , 000 km) the β -term dominates and the waves regress. Thus, as a general rule, short-wavelength synoptic-scale dis- turbances should move eastward in a westerly flow. Long planetary waves regress or remain stationary. 11
The Tendency Equation 12
The Tendency Equation Although the vertical velocity plays an essential role in the dynamics, the evolution of the geostrophic circulation can be determined without explicitly determining the distribu- tion of ω . 12
The Tendency Equation Although the vertical velocity plays an essential role in the dynamics, the evolution of the geostrophic circulation can be determined without explicitly determining the distribu- tion of ω . The vorticity equation is ∂ζ g ∂ω ∂t = − V g ·∇ ( ζ g + f ) + f 0 ∂p 12
The Tendency Equation Although the vertical velocity plays an essential role in the dynamics, the evolution of the geostrophic circulation can be determined without explicitly determining the distribu- tion of ω . The vorticity equation is ∂ζ g ∂ω ∂t = − V g ·∇ ( ζ g + f ) + f 0 ∂p Recalling that the vorticity and geopotential are related by ζ g = (1 /f 0 ) ∇ 2 Φ and reversing the order of differentiation, we get � 1 � 1 ∂ω ∇ 2 Φ t = − V g ·∇ ∇ 2 Φ + f + f 0 f 0 f 0 ∂p Note: Φ t ≡ ∂ Φ /∂t . Holton uses χ . 12
The thermodynamic equation is � ∂ � � ∂ Φ � + σω = − κ ˙ Q ∂t + V g · ∇ ∂p p 13
The thermodynamic equation is � ∂ � � ∂ Φ � + σω = − κ ˙ Q ∂t + V g · ∇ ∂p p Let us multiply by f 0 /σ and differentiate with respect to p : � � � f 0 � � f 0 � ∂ Φ �� κ ˙ ∂ ∂ Φ t = − ∂ ∂ω ∂ Q σ V g · ∇ − f 0 ∂p − f 0 ∂p σ ∂p ∂p ∂p ∂p σp 13
The thermodynamic equation is � ∂ � � ∂ Φ � + σω = − κ ˙ Q ∂t + V g · ∇ ∂p p Let us multiply by f 0 /σ and differentiate with respect to p : � � � f 0 � � f 0 � ∂ Φ �� κ ˙ ∂ ∂ Φ t = − ∂ ∂ω ∂ Q σ V g · ∇ − f 0 ∂p − f 0 ∂p σ ∂p ∂p ∂p ∂p σp We now ignore the effects of diabatic heating and set ˙ Q = 0 . 13
The thermodynamic equation is � ∂ � � ∂ Φ � + σω = − κ ˙ Q ∂t + V g · ∇ ∂p p Let us multiply by f 0 /σ and differentiate with respect to p : � � � f 0 � � f 0 � ∂ Φ �� κ ˙ ∂ ∂ Φ t = − ∂ ∂ω ∂ Q σ V g · ∇ − f 0 ∂p − f 0 ∂p σ ∂p ∂p ∂p ∂p σp We now ignore the effects of diabatic heating and set ˙ Q = 0 . It is simple to eliminate ω by addition of the thermodynamic and vorticity equations as expressed above. We then get � 1 � � �� � f 2 ∇ 2 + ∂ ∂ 0 ∇ 2 Φ + f Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� � � �� � B A � �� � f 2 + ∂ − ∂ Φ 0 σ V g · ∇ ∂p ∂p � �� � C 13
Again, � � 1 � �� � f 2 ∇ 2 + ∂ ∂ 0 ∇ 2 Φ + f Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� � � �� � B A � �� � f 2 + ∂ − ∂ Φ 0 σ V g · ∇ ∂p ∂p � �� � C 14
Again, � � 1 � �� � f 2 ∇ 2 + ∂ ∂ 0 ∇ 2 Φ + f Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� � � �� � B A � �� � f 2 + ∂ − ∂ Φ 0 σ V g · ∇ ∂p ∂p � �� � C This is the geopotential tendency equation . It provides a relationship between 14
Again, � � 1 � �� � f 2 ∇ 2 + ∂ ∂ 0 ∇ 2 Φ + f Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� � � �� � B A � �� � f 2 + ∂ − ∂ Φ 0 σ V g · ∇ ∂p ∂p � �� � C This is the geopotential tendency equation . It provides a relationship between Term A: The local geopotential tendency Φ t 14
Again, � � 1 � �� � f 2 ∇ 2 + ∂ ∂ 0 ∇ 2 Φ + f Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� � � �� � B A � �� � f 2 + ∂ − ∂ Φ 0 σ V g · ∇ ∂p ∂p � �� � C This is the geopotential tendency equation . It provides a relationship between Term A: The local geopotential tendency Φ t Term B: The advection of vorticity 14
Again, � � 1 � �� � f 2 ∇ 2 + ∂ ∂ 0 ∇ 2 Φ + f Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� � � �� � B A � �� � f 2 + ∂ − ∂ Φ 0 σ V g · ∇ ∂p ∂p � �� � C This is the geopotential tendency equation . It provides a relationship between Term A: The local geopotential tendency Φ t Term B: The advection of vorticity Term C: The vertical shear of temperature advection. 14
Term (A) involves second derivatives with respect to spatial variables of the geopotential tendency Φ t . For sinusoidal variations, this is typically proportional to − Φ t . 15
Term (A) involves second derivatives with respect to spatial variables of the geopotential tendency Φ t . For sinusoidal variations, this is typically proportional to − Φ t . Term (B) is proportional to the advection of absolute vor- ticity. For the upper troposphere it is usually the dominant term. 15
Term (A) involves second derivatives with respect to spatial variables of the geopotential tendency Φ t . For sinusoidal variations, this is typically proportional to − Φ t . Term (B) is proportional to the advection of absolute vor- ticity. For the upper troposphere it is usually the dominant term. For short waves we have seen that the relative vorticity ad- vection dominates the planetary vorticity advection. With a ridge to the west and a trough to the east, this term is then negative. 15
Term (A) involves second derivatives with respect to spatial variables of the geopotential tendency Φ t . For sinusoidal variations, this is typically proportional to − Φ t . Term (B) is proportional to the advection of absolute vor- ticity. For the upper troposphere it is usually the dominant term. For short waves we have seen that the relative vorticity ad- vection dominates the planetary vorticity advection. With a ridge to the west and a trough to the east, this term is then negative. Thus, Term (B) makes Φ t positive, so that a ridge tends to develop and, associated with this, the vorticity becomes negative. 15
Term (A) involves second derivatives with respect to spatial variables of the geopotential tendency Φ t . For sinusoidal variations, this is typically proportional to − Φ t . Term (B) is proportional to the advection of absolute vor- ticity. For the upper troposphere it is usually the dominant term. For short waves we have seen that the relative vorticity ad- vection dominates the planetary vorticity advection. With a ridge to the west and a trough to the east, this term is then negative. Thus, Term (B) makes Φ t positive, so that a ridge tends to develop and, associated with this, the vorticity becomes negative. Term (B) acts to transport the pattern of geopotential. However, since V g ·∇ ζ = 0 on the trough and ridge axes, this term does not cause the wave to amplify or decay . 15
The means of amplification or decay of midlatitude waves is contained in Term (C). This term is proportional to minus the rate of change of temperature advection with respect to pressure. 16
The means of amplification or decay of midlatitude waves is contained in Term (C). This term is proportional to minus the rate of change of temperature advection with respect to pressure. It is therefore related to plus the rate of change of temper- ature advection with respect to height. This is called the differential temperature advection . 16
The means of amplification or decay of midlatitude waves is contained in Term (C). This term is proportional to minus the rate of change of temperature advection with respect to pressure. It is therefore related to plus the rate of change of temper- ature advection with respect to height. This is called the differential temperature advection . The magnitude of the temperature (or thickness) advection tends to be largest in the lower troposphere, beneath the 500 hPa trough and ridge lines in a developing baroclinic wave. 16
17
• Below the 500 hPa ridge, there is warm advection as- sociated with the advancing warm front. This increases thickness and builds the upper level ridge. 18
• Below the 500 hPa ridge, there is warm advection as- sociated with the advancing warm front. This increases thickness and builds the upper level ridge. • Below the 500 hPa trough, there is cold advection as- sociated with the advancing cold front. This decreases thickness and deepens the upper level trough. 18
• Below the 500 hPa ridge, there is warm advection as- sociated with the advancing warm front. This increases thickness and builds the upper level ridge. • Below the 500 hPa trough, there is cold advection as- sociated with the advancing cold front. This decreases thickness and deepens the upper level trough. Thus in contrast to term (B), term (C) is dominant in the lower troposphere; but its effect is felt at higher levels. 18
• Below the 500 hPa ridge, there is warm advection as- sociated with the advancing warm front. This increases thickness and builds the upper level ridge. • Below the 500 hPa trough, there is cold advection as- sociated with the advancing cold front. This decreases thickness and deepens the upper level trough. Thus in contrast to term (B), term (C) is dominant in the lower troposphere; but its effect is felt at higher levels. In words, we may write the geopotential tendency equation: � Falling � � � � � Positive Differential + ∝ Pressure Vorticity Advection Temperature Advec’n 18
Tendence due to diff’l Tendency due to vorticity temperature advection advection 19
The Omega Equation 20
The Omega Equation We will now eliminate the geopotential tendency by com- bining the momentum and thermodynamic equations, and obtain an equation for the vertical velocity ω . 20
The Omega Equation We will now eliminate the geopotential tendency by com- bining the momentum and thermodynamic equations, and obtain an equation for the vertical velocity ω . The thermodynamic equation for adiabatic flow is � ∂ � � ∂ Φ � ∂t + V g · ∇ + σω = 0 ∂p 20
The Omega Equation We will now eliminate the geopotential tendency by com- bining the momentum and thermodynamic equations, and obtain an equation for the vertical velocity ω . The thermodynamic equation for adiabatic flow is � ∂ � � ∂ Φ � ∂t + V g · ∇ + σω = 0 ∂p We now write it as � ∂ Φ � ∂ Φ t ∂p = − V g · ∇ − σω ∂p 20
The Omega Equation We will now eliminate the geopotential tendency by com- bining the momentum and thermodynamic equations, and obtain an equation for the vertical velocity ω . The thermodynamic equation for adiabatic flow is � ∂ � � ∂ Φ � ∂t + V g · ∇ + σω = 0 ∂p We now write it as � ∂ Φ � ∂ Φ t ∂p = − V g · ∇ − σω ∂p We take the Laplacian of this and obtain � � ∂ Φ �� ∇ 2 ∂ Φ t ∂p = −∇ 2 − σ ∇ 2 ω V g · ∇ ∂p 20
Recall that the vorticity equation may be written ∂ζ g ∂ω ∂t = − V g ·∇ ( ζ g + f ) + f 0 ∂p 21
Recall that the vorticity equation may be written ∂ζ g ∂ω ∂t = − V g ·∇ ( ζ g + f ) + f 0 ∂p Multiply by f 0 and use f 0 ζ g = ∇ 2 Φ : ∇ 2 Φ t = − f 0 V g ·∇ ( 1 ∂ω ∇ 2 Φ + f ) + f 2 0 f 0 ∂p 21
Recall that the vorticity equation may be written ∂ζ g ∂ω ∂t = − V g ·∇ ( ζ g + f ) + f 0 ∂p Multiply by f 0 and use f 0 ζ g = ∇ 2 Φ : ∇ 2 Φ t = − f 0 V g ·∇ ( 1 ∂ω ∇ 2 Φ + f ) + f 2 0 f 0 ∂p Now differentiate with respect to pressure: � 1 � �� ∂ 2 ω ∇ 2 ∂ Φ t ∂ ∇ 2 Φ + f + f 2 ∂p = − f 0 V g ·∇ 0 ∂p 2 ∂p f 0 21
Recall that the vorticity equation may be written ∂ζ g ∂ω ∂t = − V g ·∇ ( ζ g + f ) + f 0 ∂p Multiply by f 0 and use f 0 ζ g = ∇ 2 Φ : ∇ 2 Φ t = − f 0 V g ·∇ ( 1 ∂ω ∇ 2 Φ + f ) + f 2 0 f 0 ∂p Now differentiate with respect to pressure: � 1 � �� ∂ 2 ω ∇ 2 ∂ Φ t ∂ ∇ 2 Φ + f + f 2 ∂p = − f 0 V g ·∇ 0 ∂p 2 ∂p f 0 We now have two equations with identical expressions for the tendency Φ t . So we can subtract one from the other to obtain a diagnostic equation for the vertical velocity. 21
� 1 � � � �� ∂ 2 ∂ σ ∇ 2 + f 2 ∇ 2 Φ + f ω = − f 0 − V g ·∇ 0 ∂p 2 ∂p f 0 � �� � � �� � B A � � �� − ∂ Φ + ∇ 2 V g · ∇ ∂p � �� � C 22
� 1 � � � �� ∂ 2 ∂ σ ∇ 2 + f 2 ∇ 2 Φ + f ω = − f 0 − V g ·∇ 0 ∂p 2 ∂p f 0 � �� � � �� � B A � � �� − ∂ Φ + ∇ 2 V g · ∇ ∂p � �� � C This is the omega equation , a diagnostic relationship for the vertical velocity ω . 22
� 1 � � � �� ∂ 2 ∂ σ ∇ 2 + f 2 ∇ 2 Φ + f ω = − f 0 − V g ·∇ 0 ∂p 2 ∂p f 0 � �� � � �� � B A � � �� − ∂ Φ + ∇ 2 V g · ∇ ∂p � �� � C This is the omega equation , a diagnostic relationship for the vertical velocity ω . It provides a relationship between 22
� 1 � � � �� ∂ 2 ∂ σ ∇ 2 + f 2 ∇ 2 Φ + f ω = − f 0 − V g ·∇ 0 ∂p 2 ∂p f 0 � �� � � �� � B A � � �� − ∂ Φ + ∇ 2 V g · ∇ ∂p � �� � C This is the omega equation , a diagnostic relationship for the vertical velocity ω . It provides a relationship between Term A: The vertical velocity 22
� 1 � � � �� ∂ 2 ∂ σ ∇ 2 + f 2 ∇ 2 Φ + f ω = − f 0 − V g ·∇ 0 ∂p 2 ∂p f 0 � �� � � �� � B A � � �� − ∂ Φ + ∇ 2 V g · ∇ ∂p � �� � C This is the omega equation , a diagnostic relationship for the vertical velocity ω . It provides a relationship between Term A: The vertical velocity Term B: The differential advection of vorticity 22
� 1 � � � �� ∂ 2 ∂ σ ∇ 2 + f 2 ∇ 2 Φ + f ω = − f 0 − V g ·∇ 0 ∂p 2 ∂p f 0 � �� � � �� � B A � � �� − ∂ Φ + ∇ 2 V g · ∇ ∂p � �� � C This is the omega equation , a diagnostic relationship for the vertical velocity ω . It provides a relationship between Term A: The vertical velocity Term B: The differential advection of vorticity Term C: The temperature advection. 22
Term (A), the left side of the equation, involves spatial sec- ond derivatives of ω . For sinusoidal variations it is proportional to the negative of ω and is thus related directly to the vertical velocity w . 23
Term (A), the left side of the equation, involves spatial sec- ond derivatives of ω . For sinusoidal variations it is proportional to the negative of ω and is thus related directly to the vertical velocity w . Term (B) is the change with pressure of the advection of absolute vorticity, that is, the differential vorticity advec- tion . 23
Term (A), the left side of the equation, involves spatial sec- ond derivatives of ω . For sinusoidal variations it is proportional to the negative of ω and is thus related directly to the vertical velocity w . Term (B) is the change with pressure of the advection of absolute vorticity, that is, the differential vorticity advec- tion . Term (C) is the Laplacian of minus the temperature advec- tion, and is thus proportional to the advection of tempera- ture . 23
Term (A), the left side of the equation, involves spatial sec- ond derivatives of ω . For sinusoidal variations it is proportional to the negative of ω and is thus related directly to the vertical velocity w . Term (B) is the change with pressure of the advection of absolute vorticity, that is, the differential vorticity advec- tion . Term (C) is the Laplacian of minus the temperature advec- tion, and is thus proportional to the advection of tempera- ture . In words, we may write the omega equation as follows � Rising � � � � Temperature � Differential + ∝ Motion Vorticity Advection Advection 23
Idealized baroclinic wave. Solid: 500 hPa geopotential con- tours. Dashed: 1000 hPa contours. Regions of strong ver- tical motion due to differential vorticity advection are indi- cated. 24
Term (B) is � 1 � �� ∂ ∝ ∂ � � �� ∇ 2 Φ + f − f 0 − V g ·∇ − V g ·∇ ζ g + f ∂p f 0 ∂z so it is proportional to differential vorticity advection. 25
Term (B) is � 1 � �� ∂ ∝ ∂ � � �� ∇ 2 Φ + f − f 0 − V g ·∇ − V g ·∇ ζ g + f ∂p f 0 ∂z so it is proportional to differential vorticity advection. • At the surface Low , the advection of vorticity is small 25
Term (B) is � 1 � �� ∂ ∝ ∂ � � �� ∇ 2 Φ + f − f 0 − V g ·∇ − V g ·∇ ζ g + f ∂p f 0 ∂z so it is proportional to differential vorticity advection. • At the surface Low , the advection of vorticity is small • Above this, there is strong positive vorticity advection at 500 hPa 25
Term (B) is � 1 � �� ∂ ∝ ∂ � � �� ∇ 2 Φ + f − f 0 − V g ·∇ − V g ·∇ ζ g + f ∂p f 0 ∂z so it is proportional to differential vorticity advection. • At the surface Low , the advection of vorticity is small • Above this, there is strong positive vorticity advection at 500 hPa • Therefore, the differential vorticity advection is positive 25
Term (B) is � 1 � �� ∂ ∝ ∂ � � �� ∇ 2 Φ + f − f 0 − V g ·∇ − V g ·∇ ζ g + f ∂p f 0 ∂z so it is proportional to differential vorticity advection. • At the surface Low , the advection of vorticity is small • Above this, there is strong positive vorticity advection at 500 hPa • Therefore, the differential vorticity advection is positive • This indices an upward vertical velocity 25
Term (B) is � 1 � �� ∂ ∝ ∂ � � �� ∇ 2 Φ + f − f 0 − V g ·∇ − V g ·∇ ζ g + f ∂p f 0 ∂z so it is proportional to differential vorticity advection. • At the surface Low , the advection of vorticity is small • Above this, there is strong positive vorticity advection at 500 hPa • Therefore, the differential vorticity advection is positive • This indices an upward vertical velocity • Correspondingly, w < 0 above the surface High Pressure 25
Term (B) is � 1 � �� ∂ ∝ ∂ � � �� ∇ 2 Φ + f − f 0 − V g ·∇ − V g ·∇ ζ g + f ∂p f 0 ∂z so it is proportional to differential vorticity advection. • At the surface Low , the advection of vorticity is small • Above this, there is strong positive vorticity advection at 500 hPa • Therefore, the differential vorticity advection is positive • This indices an upward vertical velocity • Correspondingly, w < 0 above the surface High Pressure • We assume the scale is short enough that relative vortic- ity advection dominates planetary vorticity advection. Conclusion: Differential vorticity advection implies: Rising motion above the surface low Subsidence above the surface High. 25
Idealized baroclinic wave. Solid: 500 hPa geopotential con- tours. Dashed: 1000 hPa contours. Regions of strong ver- tical motion due to temperature advection are indicated. 26
Term (C) is � � �� − ∂ Φ + ∇ 2 V g · ∇ ∝ − V g · ∇ T ∂p so it is propotrional to the temperature advection. 27
Term (C) is � � �� − ∂ Φ + ∇ 2 V g · ∇ ∝ − V g · ∇ T ∂p so it is propotrional to the temperature advection. • Ahead of the surface Low there is warm advection 27
Term (C) is � � �� − ∂ Φ + ∇ 2 V g · ∇ ∝ − V g · ∇ T ∂p so it is propotrional to the temperature advection. • Ahead of the surface Low there is warm advection • Therefore, Term (C) is positive 27
Term (C) is � � �� − ∂ Φ + ∇ 2 V g · ∇ ∝ − V g · ∇ T ∂p so it is propotrional to the temperature advection. • Ahead of the surface Low there is warm advection • Therefore, Term (C) is positive • So, there is Rising Motion ahead of the Low centre. 27
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