[PPT] - The Potential Vorticity Equation The Potential Vorticity Equation PowerPoint Presentation

SLIDE 1

The Potential Vorticity Equation

SLIDE 2

The Potential Vorticity Equation

The geopotential tendency equation is

∇2 + ∂

∂p

f2

σ ∂ ∂p

Φt = − f0Vg·∇

1 f0 ∇2Φ + f

+ ∂

∂p

f2

σ Vg · ∇

−∂Φ

∂p

SLIDE 3

The Potential Vorticity Equation

The geopotential tendency equation is

∇2 + ∂

∂p

f2

σ ∂ ∂p

Φt = − f0Vg·∇

1 f0 ∇2Φ + f

+ ∂

∂p

f2

σ Vg · ∇

−∂Φ

∂p

The second term on the right (Term (C)) may be expanded:

−Vg · ∇ ∂ ∂p

f2

σ ∂Φ ∂p

− f2

σ ∂Vg ∂p ·∇∂Φ ∂p

SLIDE 4

The Potential Vorticity Equation

The geopotential tendency equation is

∇2 + ∂

∂p

f2

σ ∂ ∂p

Φt = − f0Vg·∇

1 f0 ∇2Φ + f

+ ∂

∂p

f2

σ Vg · ∇

−∂Φ

∂p

The second term on the right (Term (C)) may be expanded:

−Vg · ∇ ∂ ∂p

f2

σ ∂Φ ∂p

− f2

σ ∂Vg ∂p ·∇∂Φ ∂p But the thermal wind relationship is f0 ∂Vg ∂p = k × ∇∂Φ ∂p This is just the p-derivative of f0Vg = k × ∇Φ.

SLIDE 5

Thus, ∂Vg/∂p is perpendicular to ∇(∂Φ/∂p) and the second term above vanishes.

2

SLIDE 6

Thus, ∂Vg/∂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 RHS = −f0Vg·∇ 1 f0 ∇2Φ + f + ∂ ∂p f0 σ ∂Φ ∂p

= −f0Vg·∇q

2

SLIDE 7

Thus, ∂Vg/∂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 RHS = −f0Vg·∇ 1 f0 ∇2Φ + f + ∂ ∂p f0 σ ∂Φ ∂p

= −f0Vg·∇q

The quantity in square brackets is called the quasi-geostrophic potential vorticity q ≡ 1 f0 ∇2Φ + f + ∂ ∂p f0 σ ∂Φ ∂p

2

SLIDE 8

Thus, ∂Vg/∂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 RHS = −f0Vg·∇ 1 f0 ∇2Φ + f + ∂ ∂p f0 σ ∂Φ ∂p

= −f0Vg·∇q

The quantity in square brackets is called the quasi-geostrophic potential vorticity q ≡ 1 f0 ∇2Φ + f + ∂ ∂p f0 σ ∂Φ ∂p

The left side of the tendency equation may be written

LHS = f0 ∂ ∂t 1 f0 ∇2Φ + ∂ ∂p f0 σ ∂Φ ∂p

= f0

∂q ∂t since f does not vary with time.

2

SLIDE 9

The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: ∂ ∂t + Vg·∇

q = dgq

dt = 0 The potential vorticity q is comprised of three terms:

3

SLIDE 10

The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: ∂ ∂t + Vg·∇

q = dgq

dt = 0 The potential vorticity q is comprised of three terms:

The relative vorticity, ζg

3

SLIDE 11

The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: ∂ ∂t + Vg·∇

q = dgq

dt = 0 The potential vorticity q is comprised of three terms:

The relative vorticity, ζg
The planetary vorticity f

3

SLIDE 12

The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: ∂ ∂t + Vg·∇

q = dgq

dt = 0 The potential vorticity q is comprised of three terms:

The relative vorticity, ζg
The planetary vorticity f
The stretching vorticity, (∂/∂p)[(f0/σ)∂Φ/∂p].

3

SLIDE 13

The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: ∂ ∂t + Vg·∇

q = dgq

dt = 0 The potential vorticity q is comprised of three terms:

The relative vorticity, ζg
The planetary vorticity f
The stretching vorticity, (∂/∂p)[(f0/σ)∂Φ/∂p].

The equation states that q is conserved following the geostrophic flow.

3

SLIDE 14

The tendency equation may now be written in a conserv- ative form called the quasi-geostrophic potential vorticity equation or QGPV equation: ∂ ∂t + Vg·∇

q = dgq

dt = 0 The potential vorticity q is comprised of three terms:

The relative vorticity, ζg
The planetary vorticity f
The stretching vorticity, (∂/∂p)[(f0/σ)∂Φ/∂p].

The equation states that q is conserved following the geostrophic flow. Note that q is completely determined once the three-dim- ensional distribution of geopotential Φ is given.

3

SLIDE 15

The QGPV equation may be used to predict the evolution

f atmospheric flows in midlatitudes.

4

SLIDE 16

The QGPV equation may be used to predict the evolution

f atmospheric flows in midlatitudes.

Or, in plain language, to make Weather Forecasts.

4

SLIDE 17

The QGPV equation may be used to predict the evolution

f 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

SLIDE 18

The QGPV equation may be used to predict the evolution

f 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

SLIDE 19

Exercise

5

SLIDE 20

Exercise

An idealized geopotential field is given at time t = 0 by Φ = Φ0 − f0¯ uy + A sin(kx − mp) where Φ0, ¯ u and A are functions of p.

5

SLIDE 21

Exercise

An idealized geopotential field is given at time t = 0 by Φ = Φ0 − f0¯ uy + A sin(kx − mp) where Φ0, ¯ u and A are functions of p. This represents a mean westerly flow ¯ u > 0 distorted by a wave disturbance of amplitude A.

5

SLIDE 22

Exercise

An idealized geopotential field is given at time t = 0 by Φ = Φ0 − f0¯ uy + A sin(kx − mp) where Φ0, ¯ 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

f x and p.

5

SLIDE 23

Exercise

An idealized geopotential field is given at time t = 0 by Φ = Φ0 − f0¯ uy + A sin(kx − mp) where Φ0, ¯ 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

f x and p.

(b) Ignoring the β-effect, compute the geostrophic vorticity and divergence.

5

SLIDE 24

Exercise

An idealized geopotential field is given at time t = 0 by Φ = Φ0 − f0¯ uy + A sin(kx − mp) where Φ0, ¯ 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

f 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

SLIDE 25

(d) Compute the vorticity advection and temperature advection.

6

SLIDE 26

(d) Compute the vorticity advection and temperature advection. (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

SLIDE 27

(d) Compute the vorticity advection and temperature advection. (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

SLIDE 28

x

The ENIAC Integrations

(ENIAC: Electronic Numerical Integrator and Computer)

7

SLIDE 29

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

SLIDE 30

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

SLIDE 31

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

SLIDE 32

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”:

1. Entirely new methods of weather prediction by calcula-

tion will have been made possible;

8

SLIDE 33

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”:

1. Entirely new methods of weather prediction by calcula-

tion will have been made possible;

2. A new rational basis will have been secured for the plan-

ning of physical measurements and field observations;

8

SLIDE 34

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”:

1. Entirely new methods of weather prediction by calcula-

tion will have been made possible;

2. A new rational basis will have been secured for the plan-

ning of physical measurements and field observations;

3. The first step towards influencing the weather by rational

human intervention will have been made.

8

SLIDE 35