Lorenz Energy Cycle - Linking Weather and Climate (MET 6155)

J-P Michael

Peter Pantina

November 22, 2009

Overview

The Lorenz energy cycle describes the interaction between zonal-mean and perturbations of po-

tential and kinetic energies. To obtain this simple model of scale interactions, we start by deriving

the Eulerian mean equations of motion in log-pressure coordinates. From these we derive a sep-

arate set of equations for the mean and eddy flows. We then employ a global averaging scheme

and other techniques to obtain equations for the planetary energy budget.

The resulting equations are comprised of energy transformation, source, or sink terms known as

the Lorenz Energy Cycle. We explore each term and its significance in this system. Finally, we

explain the flow of energy in the cycle for each step.

Mean Flow Equations

We start our discussion with x and y components of the momentum equation (1) and (2) , the

hydrostatic approximation (3), the mass continuity equation (4), and the thermodynamic equation

(5) (in log-pressure coordinates). We then derive the Eulerian mean equations. We show the steps

for the zonal momentum equation which serves as a guide for the others. For a complete derivation,

follow 10.2 in Holton.

We define the following quantities:

J = Diabatic heating rate

H = scale height

κ = R/C

p

F

x

; F

y

= drag force terms

N

2

=

R

H

(

∂T

∂z

+

κT

H

)

= Brunt-Vaisala frequency

∂u

∂t

+ u

∂u

∂x

+ v

∂u

∂y

+ w

∂u

∂z

− fv +

∂Φ

∂x

= F

x

(1)

∂v

∂t

+ u

∂v

∂x

+ v

∂v

∂y

+ w

∂v

∂z

+ fu +

∂Φ

∂y

= F

y

(2)

∂Φ

∂z

=

RT

H

(3)

∂u

∂x

+

∂v

∂y

+

1

ρ

0

∂(ρ

0

w)

∂z

= 0

(4)

( ∂

∂t

+ u

∂

∂x

+ v

∂

∂y

+ w

∂

∂z

) ∂Φ

∂z

+ wN

2

=

κJ

H

(5)

1

We begin by zonally averaging the above equations. First, we decompose each quantity into the

zonal mean and perturbation (a = a + a ).

∂(¯u + u )

∂t

+(¯u+u )

∂(¯u + u )

∂x

+(¯v+v )

∂(¯u + u )

∂y

+( ¯w+w )

∂(¯u + u )

∂z

−f

0

(¯v+v )+

∂(Φ + Φ )

∂x

= F

x

+F

x

By definition, all zonally averaged quantities are independent of x. We also note that the mean

of a perturbation is zero. Recall:

ab = ab + a b + ab + a b = ab + a b

∂(¯u +

u )

∂t

+

$

$

$

$

$

$

$

$

(¯u + u )

∂(¯u + u )

∂x

+(¯v + v )

∂(¯u + u )

∂y

+( ¯w + w )

∂(¯u + u )

∂z

−f

0

(¯v +

v )+

∂(Φ + Φ )

∂x

= F

x

+

F

x

∂¯u

∂t

+ ¯v

∂¯u

∂y

+ v

∂u

∂y

+ ¯w

∂¯u

∂z

+ w

∂u

∂z

− f

0

¯v = F

x

By introducing u times the mass continuity equation and using the chain rule, we can rewrite this

as seen below. Finally, we assume quasi-geostrophy and then neglect advection by ageostrophic

wind and vertical eddy fluxes. It was previously shown that f¯v =

∂Φ

∂x

= 0, so ¯v is also ageostrophic,

and advection by it can be neglected.

∂¯u

∂t

+

¯v

∂¯u

∂y

+

∂u v

∂y

+

¯w

∂¯u

∂z

+

∂u w

∂z

− f

0

¯v = F

x

∂¯u

∂t

− f

0

¯v = −

∂(u v )

∂y

+ F

x

(6)

The meridional momentum equation (2), the mass continuity equation (4), and the thermody-

namic equation (5) reduce likewise:

f

0

¯u = −

∂Φ

∂y

(7)

∂¯v

∂y

+

1

ρ

0

∂(ρ

0

¯w)

∂z

= 0

(8)

∂

∂t

(∂Φ

∂z

)

+ ¯wN

2

=

κJ

H

−

∂

∂y

(

v

∂Φ

∂z

)

(9)

Eddy Flow Equations

To compare the mean and the eddy flows we also need equations for the eddy motion. We derive

these by taking the full equations, subtracting the above mean equations and then linearizing for

ease of analysis.

Taking the decomposed form of (1):

∂(¯u + u )

∂t

+(¯u+u )

∂(¯u + u )

∂x

+(¯v+v )

∂(¯u + u )

∂y

+( ¯w+w )

∂(¯u + u )

∂z

−f

0

(¯v+v )+

∂(Φ + Φ )

∂x

= F

x

+F

x

2

subtracting the mean equation (6) and recalling that zonally averaged quantities are independent

of x:

∂u

∂t

+(¯u+u )

∂(

¡

¯u + u )

∂x

+(¯v+v )

∂(¯u + u )

∂y

+( ¯w +w )

∂(¯u + u )

∂z

−f

0

v +

∂(

Φ + Φ )

∂x

−

∂u v

∂y

= F

x

Linearizing allows us to neglect products of perturbation:

∂u

∂t

+ ¯u

∂u

∂x

+ ¯v

∂(¯u + u )

∂y

+ v

∂¯u

∂y

+ ¯w

∂(¯u + u )

∂z

+ w

∂¯u

∂z

− f

0

v +

∂Φ

∂x

= F

x

Employing the quasi-geostrophic assumption as before we are left with:

( ∂

∂t

+ ¯u

∂

∂x

)

u −

(

f

0

−

∂¯u

∂y

)

v = −

∂Φ

∂x

+ F

x

(10)

Likewise the other equations reduce to:

( ∂

∂t

+ ¯u

∂

∂x

)

v + f

0

u = −

∂Φ

∂y

+ F

y

(11)

( ∂

∂t

+ ¯u

∂

∂x

) ∂Φ

∂z

+ v

∂

∂y

(∂Φ

∂z

)

+ N

2

w =

κJ

H

(12)

∂u

∂x

+

∂v

∂y

+

1

ρ

0

∂(ρ

0

w )

∂z

= 0

(13)

where F

x

and F

y

are the zonally varying components of friction.

Mean Energy Equations

Mean Kinetic Energy Equation

To get an equation for kinetic energy we combine the mean zonal (1) and meridional (2) momentum

equations and multiply them by ρ

0

¯u and ρ

0

¯v, respectively. This gives us the time rate of change

for mean velocities squared, which when combined, is the mean kinetic energy tendency equation.

ρ

0

¯u

[∂¯u

∂t

+

∂u v

∂y

− f

0

¯v = F

x

]

ρ

0

¯v

[

f

0

¯u +

∂Φ

∂y

= 0

]

The first term on the LHS in the zonal equation is rewritten using the chain rule. Both equations

have a f

0

ρ

0

¯u¯v term which cancel.

ρ

0

2

∂¯u

2

∂t

+ ρ

0

¯u

∂u v

∂y

+ ρ

0

¯v

∂Φ

∂y

= ρ

0

¯uF

x

The following form is easier to interpret physically. The second and third terms on the LHS of

the above equation are rewritten using chain rule:

ρ

0

2

∂¯u

2

∂t

= −

∂

∂y

(ρ

0

¯vΦ) + ρ

0

Φ

∂¯v

∂y

−

∂

∂y

(ρ

0

¯uu v ) + ρ

0

u v

∂¯u

∂y

+ ρ

0

¯uF

x

(14)

3

Next, we define the global average function as:

〈 〉 ≡

1

A

∫

∞

0

∫

D

0

∫

L

0

( ) dx dy dz

which has the following properties:

〈

∂Ψ

∂x

〉 = 0

〈

∂Ψ

∂y

〉 = 0, if Ψ is zero at upper and lower y boundaries

〈

∂Ψ

∂z

〉 = 0, if Ψ is zero at upper and lower z boundaries

where Ψ is any quantity.

Now we apply the global average to equation (14). It is reasonable to assume that ¯v = 0 and

u v = 0 for y = ±D, so the first and third terms on the RHS are zero.

〈ρ

0

2

∂¯u

2

∂t

〉

=

〈

−

¨

¨

¨

¨

¨

∂

∂y

(ρ

0

¯vΦ) + ρ

0

Φ

∂¯v

∂y

−

¨

¨

¨

¨

¨

¨

∂

∂y

(ρ

0

¯uu v ) + ρ

0

u v

∂¯u

∂y

+ ρ

0

¯uF

x

〉

d

dt

〈ρ

0

¯u

2

2

〉

=

〈

ρ

0

Φ

∂¯v

∂y

〉

+

〈

ρ

0

u v

∂¯u

∂y

〉

+

〈

ρ

0

¯uF

x

〉

(15)

Mean Potential Energy Equation

Next we derive the equation for zonal-mean potential energy starting with the definition of zonal-

mean potential energy in terms of differential thickness:

P ≡

1

2

〈

ρ

0

N

2

(∂Φ

∂z

)

2

〉

We take mean thermodynamic equation (9) and multiply by ρ

0

(

∂Φ

∂z

)/N

2

. This gives us the time

rate of change for mean differential thickness squared, which is the mean potential energy tendency

equation.

ρ

0

N

2

∂Φ

∂z

[ ∂

∂t

∂Φ

∂z

+

∂v

∂y

∂Φ

∂z

+ ¯wN

2

=

κJ

H

]

ρ

0

2N

2

∂

∂t

(∂Φ

∂z

)

2

= −ρ

0

¯w

∂Φ

∂z

+

ρ

0

κJ

N

2

H

∂Φ

∂z

−

ρ

0

N

2

∂Φ

∂z

∂

∂y

(

v

∂Φ

∂z

)

Finally, we take the global average:

d

dt

〈

ρ

0

2N

2

(∂Φ

∂z

)

2

〉

= −

〈

ρ

0

¯w

∂Φ

∂z

〉

+

〈 ρ

0

κJ

N

2

H

∂Φ

∂z

〉

−

〈 ρ

0

N

2

∂Φ

∂z

∂

∂y

(

v

∂Φ

∂z

)〉

(16)

4

Conversion of Mean Energies

We notice that the first terms on the RHS of both (15) and (16) are equal and opposite by noting

mass continuity (8):

〈

ρ

0

Φ

∂¯v

∂y

〉

= −

〈

Φ

∂ρ

0

¯w

∂z

〉

Using the chain rule, and noting the properties of the global average function:

〈 ∂ρ

0

¯wΦ

∂z

〉

−

〈

Φ

∂ρ

0

¯w

∂z

〉

=

〈

ρ

0

¯w

∂Φ

∂z

〉

Therefore:

〈

ρ

0

Φ

∂¯v

∂y

〉

︸

︷︷

︸

(15)

= −

〈

Φ

∂ρ

0

¯w

∂z

〉

=

〈

ρ

0

¯w

∂Φ

∂z

〉

︸

︷︷

︸

(16)

=

R

H

〈

ρ

0

¯wT

〉

Finally, using hydrostatics we see that this term can be written as the correlation between vertical

mass flux and temperature. Thus, this term common to both KE and PE equations confirms that

there is a conversion from potential to kinetic energy in the energy cycle.

Eddy Energy Equations

Eddy Kinetic Energy Equation

Next, we derive the eddy energy equations so we can complete the Lorenz energy cycle.

We derive the eddy kinetic energy equation by following the steps we used for the mean kinetic

energy equation. However, here we begin with the eddy momentum equations (10) and (11), and

take the zonal average. The zonal average of the terms crossed out below is zero.

Multiplying the zonal eddy momentum equation by ρ

0

u :

ρ

0

u

[( ∂

∂t

+ ¯u

∂

∂x

)

u −

(

f

0

−

∂¯u

∂y

)

v = −

∂Φ

∂x

+ F

x

]

ρ

0

u

∂u

∂t

+ ρ

0

u ¯u

∂u

∂x

− f

0

ρ

0

u v + ρ

0

u v

∂¯u

∂y

= −ρ

0

u

∂Φ

∂x

+ ρ

0

u F

x

Zonally averaging the equation:

ρ

0

u

∂u

∂t +

ρ

0

u ¯u

∂u

∂x

− f

0

ρ

0

u v + ρ

0

u v

∂¯u

∂y

= −ρ

0

u

∂Φ

∂x

+ ρ

0

u F

x

Multiplying the meridional eddy momentum equation by ρ

0

v :

ρ

0

v

[( ∂

∂t

+ ¯u

∂

∂x

)

v + f

0

u = −

∂Φ

∂y

+ F

y

]

ρ

0

v

∂v

∂t

+ ρ

0

v ¯u

∂v

∂x

+ f

0

ρ

0

v u = −ρ

0

v

∂Φ

∂y

+ ρ

0

v F

y

5

Zonally averaging the equation:

ρ

0

v

∂v

∂t +

ρ

0

v ¯u

∂v

∂x

+ f

0

ρ

0

v u = −ρ

0

v

∂Φ

∂y

+ ρ

0

v F

y

Now, we combine the above equations and take the global average. Note the f

0

ρ

0

v u terms

cancel:

d

dt

〈

ρ

0

u

2

+ v

2

2

〉

= −

〈

ρ

0

(

u

∂Φ

∂x

+ v

∂Φ

∂y

)

〉

−

〈

ρ

0

u v

∂¯u

∂y

〉

+

〈

ρ

0

(

u F

x

+ v F

y

)〉

The first term on the RHS can be rewritten using the chain rule and the properties of the global

average function, as we have done previously:

d

dt

〈

ρ

0

u

2

+ v

2

2

〉

=

〈

ρ

0

Φ

(∂u

∂x

+

∂v

∂y

)

〉

−

〈

ρ

0

u v

∂¯u

∂y

〉

+

〈

ρ

0

(

u F

x

+ v F

y

)〉

(17)

Eddy Potential Energy Equation

We find the eddy potential energy equation by same principles. Here we manipulate the eddy

thermodynamic equation (12).

d

dt

〈

ρ

0

2N

2

(∂Φ

∂z

)

2

〉

= −

〈

ρ

0

w

∂Φ

∂z

〉

+

〈

ρ

0

κJ

∂Φ

∂z

N

2

H

〉

−

〈 ρ

0

N

2

( ∂

2

Φ

∂z∂y

) (

v

∂Φ

∂z

)〉

(18)

Conversion of Eddy Energies

We notice that the first terms on the RHS of both (17) and (18) are equal and opposite, using

mass continuity, the chain rule, and the global average function (recall conversion of mean energies

done previously).

〈

ρ

0

Φ

(∂u

∂x

+

∂v

∂y

)

〉

︸

︷︷

︸

(17)

= −

〈

Φ

∂(ρ

0

w )

∂z

〉

=

〈

ρ

0

w

∂Φ

∂z

〉

︸

︷︷

︸

(18)

This term represents the transformation between eddy potential and kinetic energy.

6

Energy Quantities

Recall our four energy equations:

d

dt

〈ρ

0

¯u

2

2

〉

=

〈

ρ

0

Φ

∂¯v

∂y

〉

+

〈

ρ

0

u v

∂¯u

∂y

〉

+

〈

ρ

0

¯uF

x

〉

d

dt

〈

ρ

0

2N

2

(∂Φ

∂z

)

2

〉

= −

〈

ρ

0

¯w

∂Φ

∂z

〉

+

〈 ρ

0

κJ

N

2

H

∂Φ

∂z

〉

−

〈 ρ

0

N

2

∂Φ

∂z

∂

∂y

(

v

∂Φ

∂z

)〉

d

dt

〈

ρ

0

u

2

+ v

2

2

〉

=

〈

ρ

0

Φ

(∂u

∂x

+

∂v

∂y

)

〉

−

〈

ρ

0

u v

∂¯u

∂y

〉

+

〈

ρ

0

(

u F

x

+ v F

y

)〉

d

dt

〈

ρ

0

2N

2

(∂Φ

∂z

)

2

〉

= −

〈

ρ

0

w

∂Φ

∂z

〉

+

〈

ρ

0

κJ

∂Φ

∂z

N

2

H

〉

−

〈 ρ

0

N

2

( ∂

2

Φ

∂z∂y

) (

v

∂Φ

∂z

)〉

We define the following energy terms:

K ≡

〈

ρ

0

¯u

2

2

〉

, K ≡

〈

ρ

0

u

2

+ v

2

2

〉

,

P ≡

〈

ρ

0

2N

2

(∂Φ

∂z

)

2

〉

, P ≡

〈

ρ

0

2N

2

(∂Φ

∂z

)

2

〉

,

and the energy conversions, where [A • B] represents the conversion of energy of form A to form

B. A term found in two of the above energy equations represents a conversion between those two

forms of energy. We now define the following energy conversions:

[P • K] ≡

〈

ρ

0

Φ

∂¯v

∂y

〉

=

〈

ρ

0

¯w

∂Φ

∂z

〉

[P • K ] ≡

〈

ρ

0

Φ

(∂u

∂x

+

∂v

∂y

)

〉

=

〈

ρ

0

w

∂Φ

∂z

〉

[K • K ] ≡

〈

ρ

0

u v

∂¯u

∂y

〉

[P • P ] ≡ −

〈 ρ

0

N

2

∂Φ

∂z

∂

∂y

(

v

∂Φ

∂z

)〉

=

〈 ρ

0

N

2

v

∂Φ

∂z

( ∂

2

Φ

∂z∂y

)〉

Finally, we consider any term with diabatic heating as an energy source (R) and any term with

friction as an energy sink (ε).

R ≡

〈 ρ

0

κJ

N

2

H

∂Φ

∂z

〉

, R ≡

〈

ρ

0

κJ

∂Φ

∂z

N

2

H

〉

ε ≡

〈

ρ

0

¯uF

x

〉

, ε ≡

〈

ρ

0

(

u F

x

+ v F

y

)〉

As defined above, each energy equation can be expressed in terms of energy conversions and

source/sink terms.

7

For example, the zonal-mean equations can be rewritten as such:

d

dt

〈

ρ

0

¯u

2

2

〉

is the tendency of mean kinetic energy.

〈

ρ

0

Φ

∂¯v

∂y

〉

is positive when mean potential energy is lost and mean kinetic energy created.

〈

ρ

0

u v

∂¯u

∂y

〉

is the triad interaction between kinetic energies.

〈

ρ

0

¯uF

x

〉

is the sink term responsible for dissipation of mean kinetic energy.

dK

dt

= [P • K] + [K • K] + ε

list

The four energy equations (15), (16), (17), and (18) in terms of the quantities we just defined:

dK

dt

= [P • K] + [K • K] + ε

dP

dt

= −[P • K] + [P • P] + R

dK

dt

= [P • K ] − [K • K] + ε

dP

dt

= −[P • K ] − [P • P] + R

(19)

Adding the four equations in (19) results in an equation for the rate of change of the total

energy (kinetic plus potential).

d

dt

(K + K + P + P ) = R + R + ε + ε

The energy conversion terms all cancel and we are left with an expression of energy change

in terms of diabatics and friction. For a simple case of adiabatic inviscid flow, total energy is

conserved. For the long-term case, we can assume energy is conserved therefore the production

of potential energy by diabatics balances the dissipation of kinetic energy:

R = −ε

Where:

R is the sum of R (zonal-mean solar heating) and R (eddy diabatic heating)

ε is the sum of ε (zonal-mean fiction) and ε (eddy friction)

8

Discussion

This energy cycle is not reflective of the fundamental properties found in the atmosphere.

Given the assumptions we have made, (quasi-geosrophic flow, etc) we can only use this as

a general approximation. However, we have seen in class that this is a good first order

approximation for instability in the mid latitudes. The figure below, taken from Holton,

shows the cycle graphically.

Starting with the top left box: Mean heating at the equator and cooling at the poles generates

mean PE. A small portion of the mean PE is also transferred to the Southern Hemisphere

(represented by B(P)). The meridional gradient of zonal mean PE then leads to baroclinic

instability which results in the transport of heat to the poles.

These baroclinic eddies transform the mean PE into eddy PE. Clouds and precipitation in

the atmosphere result in a positive contribution by the eddy heating term (R ) to the eddy

PE; in a cloudless system this term would be negative.

The vertical component of the eddy motions also transform a portion of the eddy PE to eddy

KE. Eddy momentum is lost through frictional dissipation and the remaining energy is used

to maintain the mean KE.

Eddy KE is the primary source for mean KE. Mean frictional dissipation is minimal in the

system and contributes to the lost of mean KE. In the mid latitudes a small amount of KE

is converted back to PE. However, for the tropics, symmetric overturning results in mean PE

becoming mean KE.

9