The Potential Vorticity Equation
The Potential Vorticity Equation The geopotential tendency equation is � 1 � � �� f 2 � ∇ 2 + ∂ ∂ ∇ 2 Φ + f 0 Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� f 2 + ∂ � − ∂ Φ 0 σ V g · ∇ ∂p ∂p
The Potential Vorticity Equation The geopotential tendency equation is � 1 � � �� f 2 � ∇ 2 + ∂ ∂ ∇ 2 Φ + f 0 Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� f 2 + ∂ � − ∂ Φ 0 σ V g · ∇ ∂p ∂p The second term on the right (Term (C)) may be expanded: � � f 2 − f 2 − V g · ∇ ∂ ∂ Φ ∂ V g ∂p ·∇ ∂ Φ 0 0 ∂p σ ∂p σ ∂p
The Potential Vorticity Equation The geopotential tendency equation is � 1 � � �� f 2 � ∇ 2 + ∂ ∂ ∇ 2 Φ + f 0 Φ t = − f 0 V g ·∇ ∂p σ ∂p f 0 � �� f 2 + ∂ � − ∂ Φ 0 σ V g · ∇ ∂p ∂p The second term on the right (Term (C)) may be expanded: � � f 2 − f 2 − V g · ∇ ∂ ∂ Φ ∂ V g ∂p ·∇ ∂ Φ 0 0 ∂p σ ∂p σ ∂p But the thermal wind relationship is ∂ V g ∂p = k × ∇ ∂ Φ f 0 ∂p This is just the p -derivative of f 0 V g = k × ∇ Φ .
Thus, ∂ V g /∂p is perpendicular to ∇ ( ∂ Φ /∂p ) and the second term above vanishes. 2
Thus, ∂ V g /∂p is perpendicular to ∇ ( ∂ Φ /∂p ) and the second term above vanishes. The remaining term can be combined with term (B) in the tendency equation to give � 1 � f 0 �� ∇ 2 Φ + f + ∂ ∂ Φ RHS = − f 0 V g ·∇ = − f 0 V g ·∇ q f 0 ∂p σ ∂p 2
Thus, ∂ V g /∂p is perpendicular to ∇ ( ∂ Φ /∂p ) and the second term above vanishes. The remaining term can be combined with term (B) in the tendency equation to give � 1 � f 0 �� ∇ 2 Φ + f + ∂ ∂ Φ RHS = − f 0 V g ·∇ = − f 0 V g ·∇ q f 0 ∂p σ ∂p The quantity in square brackets is called the quasi-geostrophic potential vorticity � 1 � f 0 �� ∇ 2 Φ + f + ∂ ∂ Φ q ≡ f 0 ∂p σ ∂p 2
Thus, ∂ V g /∂p is perpendicular to ∇ ( ∂ Φ /∂p ) and the second term above vanishes. The remaining term can be combined with term (B) in the tendency equation to give � 1 � f 0 �� ∇ 2 Φ + f + ∂ ∂ Φ RHS = − f 0 V g ·∇ = − f 0 V g ·∇ q f 0 ∂p σ ∂p The quantity in square brackets is called the quasi-geostrophic potential vorticity � 1 � f 0 �� ∇ 2 Φ + f + ∂ ∂ Φ q ≡ f 0 ∂p σ ∂p The left side of the tendency equation may be written � 1 ∂ ∇ 2 Φ + ∂ � f 0 ∂ Φ �� ∂q LHS = f 0 = f 0 ∂t f 0 ∂p σ ∂p ∂t since f does not vary with time. 2
The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: � ∂ � q = d g q ∂t + V g ·∇ dt = 0 The potential vorticity q is comprised of three terms: 3
The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: � ∂ � q = d g q ∂t + V g ·∇ dt = 0 The potential vorticity q is comprised of three terms: • The relative vorticity, ζ g 3
The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: � ∂ � q = d g q ∂t + V g ·∇ dt = 0 The potential vorticity q is comprised of three terms: • The relative vorticity, ζ g • The planetary vorticity f 3
The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: � ∂ � q = d g q ∂t + V g ·∇ dt = 0 The potential vorticity q is comprised of three terms: • The relative vorticity, ζ g • The planetary vorticity f • The stretching vorticity, ( ∂/∂p )[( f 0 /σ ) ∂ Φ /∂p ] . 3
The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: � ∂ � q = d g q ∂t + V g ·∇ dt = 0 The potential vorticity q is comprised of three terms: • The relative vorticity, ζ g • The planetary vorticity f • The stretching vorticity, ( ∂/∂p )[( f 0 /σ ) ∂ Φ /∂p ] . The equation states that q is conserved following the geo- strophic flow . 3
The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: � ∂ � q = d g q ∂t + V g ·∇ dt = 0 The potential vorticity q is comprised of three terms: • The relative vorticity, ζ g • The planetary vorticity f • The stretching vorticity, ( ∂/∂p )[( f 0 /σ ) ∂ Φ /∂p ] . The equation states that q is conserved following the geo- strophic flow . Note that q is completely determined once the three-dim- ensional distribution of geopotential Φ is given. 3
The QGPV equation may be used to predict the evolution of atmospheric flows in midlatitudes. 4
The QGPV equation may be used to predict the evolution of atmospheric flows in midlatitudes. Or, in plain language, to make Weather Forecasts . 4
The QGPV equation may be used to predict the evolution of atmospheric flows in midlatitudes. Or, in plain language, to make Weather Forecasts . Indeed an even simpler equation, ignoring the stretching term, was used for the first computer forecasts on the ENIAC computer (Electronic Numerical Integrator and Computer) in 1950. 4
The QGPV equation may be used to predict the evolution of atmospheric flows in midlatitudes. Or, in plain language, to make Weather Forecasts . Indeed an even simpler equation, ignoring the stretching term, was used for the first computer forecasts on the ENIAC computer (Electronic Numerical Integrator and Computer) in 1950. This was the equation d dt ( ζ + f ) = 0 for the conservation of absolute vorticity . 4
Exercise 5
Exercise An idealized geopotential field is given at time t = 0 by Φ = Φ 0 − f 0 ¯ uy + A sin( kx − mp ) where Φ 0 , ¯ u and A are functions of p . 5
Exercise An idealized geopotential field is given at time t = 0 by Φ = Φ 0 − f 0 ¯ uy + A sin( kx − mp ) where Φ 0 , ¯ u and A are functions of p . This represents a mean westerly flow ¯ u > 0 distorted by a wave disturbance of amplitude A . 5
Exercise An idealized geopotential field is given at time t = 0 by Φ = Φ 0 − f 0 ¯ uy + A sin( kx − mp ) where Φ 0 , ¯ u and A are functions of p . This represents a mean westerly flow ¯ u > 0 distorted by a wave disturbance of amplitude A . (a) Compute the geostrophic wind components as functions of x and p . 5
Exercise An idealized geopotential field is given at time t = 0 by Φ = Φ 0 − f 0 ¯ uy + A sin( kx − mp ) where Φ 0 , ¯ u and A are functions of p . This represents a mean westerly flow ¯ u > 0 distorted by a wave disturbance of amplitude A . (a) Compute the geostrophic wind components as functions of x and p . (b) Ignoring the β -effect, compute the geostrophic vorticity and divergence. 5
Exercise An idealized geopotential field is given at time t = 0 by Φ = Φ 0 − f 0 ¯ uy + A sin( kx − mp ) where Φ 0 , ¯ u and A are functions of p . This represents a mean westerly flow ¯ u > 0 distorted by a wave disturbance of amplitude A . (a) Compute the geostrophic wind components as functions of x and p . (b) Ignoring the β -effect, compute the geostrophic vorticity and divergence. (c) Compute the variations in the temperature field due to the wave. 5
(d) Compute the vorticity advection and temperature advec- tion. 6
(d) Compute the vorticity advection and temperature advec- tion. (e) Using the geopotential tendency equation, describe how the pressure at a point upstream from a trough and down- stream from a ridge is expected to change. 6
(d) Compute the vorticity advection and temperature advec- tion. (e) Using the geopotential tendency equation, describe how the pressure at a point upstream from a trough and down- stream from a ridge is expected to change. (f) Using the omega equation, describe the pattern of vertical velocity associated with the wave disturbance. 6
x The ENIAC Integrations (ENIAC: Electronic Numerical Integrator and Computer) 7
Electronic Computer Project, 1946 (under direction of John von Neumann) Von Neumann’s idea: Weather forecasting was a scientific problem par excellence for solution using a large computer. 8
Electronic Computer Project, 1946 (under direction of John von Neumann) Von Neumann’s idea: Weather forecasting was a scientific problem par excellence for solution using a large computer. The objective of the project was to study the problem of predicting the weather by simulating the dynamics of the atmosphere using a digital electronic computer. 8
Electronic Computer Project, 1946 (under direction of John von Neumann) Von Neumann’s idea: Weather forecasting was a scientific problem par excellence for solution using a large computer. The objective of the project was to study the problem of predicting the weather by simulating the dynamics of the atmosphere using a digital electronic computer. A Proposal for funding listed three “possibilities”: 8
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