The application of wave-activity conservation laws to ... fileThe application of wave-activity...

The application of wave-activity conservationlaws to cloud resolving model simulations of

multiscale tropical convection

T. A. Shaw

Department of Applied Physics and Applied Mathematicsand Lamont-Doherty Earth Observatory,

Columbia University

with T. P. Lane (University of Melbourne)

What is a wave-activity conservation law?

• A wave-activity conservation law is a relation of the form:

∂A

∂t+∇ · F = 0

→ A is a quantity that is conserved in the absence of forcing

and dissipation (in contrast to disturbance energy and enstrophy).

• Useful in the study of disturbances ξ′ (waves, eddies) to somebackground flow X i.e. ξ = X + ξ′ with A ∼ O(ξ′2).

• In midlatitudes one takes ξ = X (y , z , t) + ξ′(x , y , z , t) anddefines AMx = q′2/2Qy , which satisfies the conservation law

∂AMx

∂t+∇ · F = S with FMx = −u′v ′ j +

f0N2

v ′b′ k; S =1

QyS ′q′

where Q and q′ are the background and disturbance potential

vorticity and S includes both sources and sinks S ′.

Eliassen-Palm flux divergence� �

A = wave-activity densityF = wave-activity flux

Application to large-scale circulations

• Evolution of pseudomomentum flux during a baroclinic lifecycle:

Day 1 Day 5

Held & Hoskins (1985)

∇ · FMx < 0

∇ · FMx

Day 8

AMx

∂AMx /∂t < 0

∂AMx /∂t > 0

↗

←

∇ · FMx = −∂AMx

∂t

→ Transience of pseudomomentumdensity explains flux divergence

Application to the mesoscale

• On mesoscales one can consider disturbances to a verticallydependent shear-stratified flow.

→ Variables decomposed as ξ = X (z) + ξ′(x , y , z , t).

→ Disturbance is the mesoscale convection and waves and thebackground is a shear-stratified flow.

• The 2D pseudomomentum AMx wave-activity conservation law for a[u(z), θ(z)] background state is (Scinocca & Shepherd 1992):

∂AMx

∂t+∇ · FMx = SMx where

AMx =ρ0

θzω′θ′ − 1

2

ρ20

(θz)2

(uzρ0

)z

(θ′)2; FMx

(z) = ρ0u′w ′

in the small amplitude limit (can be generalized to finite amplitude).

→ Shaw & Shepherd (2008, 2009) derived the 3D conservation lawsand showed how the conservation laws impact the resolved-scale

flow in a climate model.

gravity wave vorticity wave

Application to mesoscale momentum transfer

• Pseudomomentum transfers involve two important mesoscaleprocesses:

→ Convective momentum transport (CMT); transport associatedwith tilted convective structures (Moncrieff 1992).

→ Gravity wave momentum transport (GWMT); transportassociated with vertically propagating gravity waves.

→ Processes currently parameterized independently in climate

models, which is inconsistent.

• Here we diagnose pseudomomentum wave-activityconservation law from the cloud resolving model simulationsof Lane & Moncrieff (2008, 2010).

→ Assess the cause of the vertical momentum flux convergencei.e. transience versus source/sink non-conservative term.

Cloud resolving model simulations

• Cloud resolving model is the anelastic model of Clark et al.(1996)

→ Two-dimensional domain: 2000 km wide and 40 km deep with∆x=1 km and ∆z = 50 m to 200 m.

→ The imposed thermodynamicsounding was observed over theTiwi Islands, Australia.

→ The imposed zonal wind iseither zero (U00) or iswestward below 3 km(UMX5B).

→ Simulations run for 180 hours

-10 -5 0 5ms-1

0

5

10

15

20

280 320 360 400 440K

0

5

10

15

20

z (k

m)

θ(z) u(z)

UMX5B↘ U00

↙

Momentum transfers in unsheared background

• In U00 simulation, shallow convection develops initially andeventually self-organizes into clusters which propagate at 5m/s without a preferred direction

→ Convection is weakly organized.

-2 -1 0 1 2 3x10-3 Nm-2

0 5

10

15

20

z (k

m)

-2 -1 0 1 2 3Nm-1 s-1

0 5

10

15

20

z (k

m)

Total cloud column FMx

(z) F E(z) = p′w ′

1st EP theorem:p′w ′ = ρ0(U − c)FMx

(z)

dash-dot = 20-70hrdash = 70-120hrsolid = 120-170hr

GWMT

CMT

→

→

→

→

20hr

70hr

120hr

170hr

weakly organized

← time

← time

x (km)


• Phase speed versus height co-spectra for U00:

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

1ρ0FMx

(z)

1ρ0AMx

� Symmetric spectrum exceptbetween 120-170hr

� Flux changes sign aboveconvectively active region

� Flux broadens with height

� Pseudomomentum density AMx

changes between 10-15kmand below 2km

enhancednegativeflux

buoyancy vorticitycorrelations in a plumemodel

Lane (2008)



Time → 20-70hr 70-120hr 120-170hr

1ρ0FMx

(z)

1ρ0AMx

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

Colors indicate upward propagation

� Upward flux does not changesign below convectively activeregion

-20 -10 0 10 20-1.5-1.0

-0.5

0.0

0.5

1.01.5

x10-3

ms-1

/ms-1

-20 -10 0 10 20-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

x10-3

ms-1

/ms-1

-20 -10 0 10 20c (m/s)

-1.5-1.0

-0.5

0.0

0.5

1.0

1.5

x10-3

ms-1

/ms-1

120-170hru′w′

↙upward

↖non-upward

upward

u′w ′

4 km

7 km

20 km


• Pseudomomentum flux convergence and its contributions

-0.2 -0.1 0.0 0.1 0.2ms-1 day-1

0 5

10

15 20

z (k

m)

-0.2 -0.1 0.0 0.1 0.2ms-1 day-1

0

5

10

15 20

-0.2 -0.1 0.0 0.1 0.2ms-1 day-1

0

5

10

15 20

-4 -2 0 2 4x10-3 Nm-2

0

5

10

15 20

z (k

m)

-4 -2 0 2 4x10-3 Nm-2

0

5

10

15 20

-4 -2 0 2 4x10-3 Nm-2

0

5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

1ρ0

∂FMx(z)

∂z (solid)

1ρ0

∂AMx

∂t (dash-dot)

� Divergence/convergencepattern below 10 km

� Source/sink term dominatesvertical transfer exceptbetween 12-14 km

∂FMx(z)

∂z = SMx − ∂AMx

∂t

FMx

(z) (solid)

∫SMx dz (dashed)∫∂AMx /∂tdz(dash-dot)

� Source/sink term dominatesbut there is cancellation

� Transient term usually > 0and significant up to 20 km



-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

1ρ0SMx

- 1ρ0

∂AMx

∂t

� Divergence/convergencepattern consistent with asource between 3-10km anda sink above 10km

� Transience provides a sourceat all vertical levels

contour = 0.025 m/s/day

∂FMx(z)


∂t

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

� The convergence for c > 0below 5km is not consistentwith upward propagation

� Transience contributionconsistent with upwardpropagation



-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

1ρ0SMx

- 1ρ0

∂AMx

∂t




∂FMx(z)


∂t

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20




• Pseudomomentum flux convergence involves:

→ CMT which peaks at ±2 m/s below 8 km.

→ GWMT which peaks at ± 4-6 m/s from thesurface up to 20 km.

• Upward propagating part of spectrum is consistentbetween the two regions.

→ GWMT source is below 7 km.

• Transient signal near convective outflow from10-14 km which peaks at -4 m/s.

• Source/sink dominates below 8 km.

• Transient signal dominates aloft.

-20 -10 0 10 20-1.0

-0.5

0.0

0.5

1.0

x10-3

ms-1

/ms-1

-20 -10 0 10 20c (m/s)

-2

-1

0

1

2

x10-3

ms-1

/ms-1

Contribution to spectra

7km120-170hr

20km

solid=source/sinkdash=transience

source/sinklarge at low c

transiencedominates

� Transient contribution peaksat ± 5 m/s

� Source/sink contribution peaksbetween 2-6 m/s

Momentum transfers in the presence of low-level shear

• In the presence of low level wind shear the convectionbecomes organized; transitions from upshear propagating(positive tilt) to quasi-stationary (upright tilt) and finally tocloud clusters (negative tilt).

-10 -5 0 5 10 15x10-3 Nm-2

0 5

10

15

20

z (k

m)

-2 -1 0 1 2 3Nm-1 s-1

0 5

10

15

20

z (k

m)


(z) F E(z) − UFMx

(z)


GWMT

CMT

→

→

→

→

20hr

70hr

120hr

170hr

cloud clusters

quasi-stationary

upshear propagating

← time

← time

← time

x (km)


• Phase speed versus height co-spectra for UMX5B:

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-5 0 5x10-3 Nm-2

0 5

10 15 20

z (k

m)

-5 0 5x10-3 Nm-2

0 5

10 15 20

-5 0 5x10-3 Nm-2

0 5

10 15 20

-20 -10 0 10 20-1.5-1.0

-0.5

0.0

0.5

1.01.5

x10-3

ms-1

/ms-1

-20 -10 0 10 20c (m/s)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

x10-3

ms-1

/ms-1

Time → 20-70hr 70-120hr 120-170hr

1ρ0FMx

(z)

∫1ρ0

FMx(z)

dc

� Spectrum no longer symmetric

U00UMX5B

� Non-upward flux contribution iszero above 7km

� Upward flux contribution islarge below 10km

broader

upward↙non-upward

↘≈ 0above7km

CMT > 0GMMT > 0

CMT < 0

GMMT < 0above 3 km

↙upward

↖non-upward

upward

u′w ′ 120-170hr

4 km

20 km


• Pseudoenergy flux co-spectra for U00 & UMX5B:

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

F E(z)

F E(z)−

uFMx

(z)

� Spectrum symmetric for U00

� Flux weakens with time

� Broadening of westward partof the spectrum

� Enhanced upward propagationbetween 0-5 km

� Peak between -5 to -15 m/s

� Slight enhancement ofeastward part

��


• Pseudomomentum flux convergence and its contributions

-0.5 0.0 0.5 1.0ms-1 day-1

0 5

10

15 20

z (k

m)

-0.5 0.0 0.5 1.0ms-1 day-1

0

5

10

15 20

-0.5 0.0 0.5 1.0ms-1 day-1

0

5

10

15 20

-20 -10 0 10 20x10-3 Nm-2

0

5

10

15 20

z (k

m)

-20 -10 0 10 20x10-3 Nm-2

0

5

10

15 20

-20 -10 0 10 20x10-3 Nm-2

0

5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

1ρ0

∂FMx(z)

∂z (solid)

1ρ0

∂AMx

∂t (dash-dot)

� Transfers are much strongerin the troposphere

� Large transfers near thesurface

∂FMx(z)


∂t

FMx

(z) (solid)

∫SMx dz (dashed)∫∂AMx /∂tdz(dash-dot)

AMx1

additionalvorticityterm��

� Balance between SMx andAMx

1 transience

� Flux FMx is on the order ofAMx

2 transience



-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

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10

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0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

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-20 -10 0 10 20c (m/s)

0 5

10

15 20

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10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

1ρ0SMx

- 1ρ0

∂AMx

∂t




∂FMx(z)


∂t

enhancedwestward

enhancedeastward

enhancedwestward

enhancedeastward



-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

enhancedwestward



-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

1ρ0SMx

- 1ρ0

∂AMx

∂t




∂FMx(z)


∂t

enhancedwestward

enhancedeastward

enhancedwestward

enhancedeastward



-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

-20 -10 0 10 200

5

10

15 20

enhancedwestward


• CMT transitions from positive to negative.

→ Consistent with convective regimes(Houze 2004).

• GWMT shows similar transition (LM10).

• Both show a broadened westward spectrumbetween 10-20 m/s.

→ Transience is responsible for the broadenedspectrum

→ Associated with transient updrafts in the

organized convection.

-20 -10 0 10 20-1.0

-0.5

0.0

0.5

1.0

x10-3

ms-1

/ms-1

-20 -10 0 10 20c (m/s)

-2

-1

0

1

2

x10-3

ms-1

/ms-1

Contribution to spectra120-170hr

7km

20km

solid=source/sinkdash=transient

� Transient contribution hasa much broader westwardspectrum

� Source/sink contribution hasa weaker broadening

20-70hr: CMT > 0GMT > 0

120-170hr: CMT < 0GMT < 0

FMx(z)

> 0FMx(z)

< 0


• When the convection is organized transient updrafts moverearward in the cloud frame creating a transient signal.

FMx(z)

< 0

� Transient updrafts moverearward relative to thecore

� Spectral bias arisesbecause tilted sourceprojects onto wavespropagating in thedirection of tilt

→ Effect of transient updrafts noted by Fovell et al. (1992)

Conclusions

• Pseudomomentum wave-activity conservation law can be used tounderstand the mesoscale transfers of momentum associated withconvection and gravity waves.

→ CMT and GWMT contributions can be isolated using upwardpropagation criteria (first Eliassen-Palm theorem).

→ Can be used to isolate CMT and GWMT and their sourcesand sinks in cloud resolving model simulations.

• Low-level shear significantly impacts the CMT and GWMTspectrum both above and below the tropopause.

→ Leads to a broadened phase speed spectrum.

→ Transience from the organized convection can be connected tosignals above the tropopause and cannot be ignored.

• CMT and GWMT should not be treated independently in climatemodels.

→ GWD parameterizations should account include transient sourcesand the effects in the source region.


• Dispersion relation

(σ − Uk)2 − kUzz

k2 + m2(σ − Uk)− N2k2

k2 + m2= 0

• Long wave limit, e.g. k2 + m2 � Uzz/N2, vorticity waves:

σ ∼ Uk +kUzz

k2 + m2

• Short wave limit, e.g. k2 + m2 � Uzz/N2, gravity waves:

σ ∼ Uk +Nk

(k2 + m2)2


• Pseudoenergy

AE = |v′|2 +ρ0

2

[1

(θz)2− ρ0u

(θz)2

(Uz

ρ0

)z

](θ′2)2 +

ρ0u

θzωθ′

with

F E(z) = cpρ0π′w ′ + ρ0u

′w ′

Pseudomomentum wave-activity conservation law

• The 2D pseudomomentum AMx wave-activity conservation law for a[u(z), θ(z)] background state is:

∂AMx

∂t+∇ · FMx = 0

where

AMx =ρ0

θzω′θ′ + (ω + ω′)

∫ θ′

0

[R(θ + η)− R(θ)

]dη

−ρ0

∫ θ′

0

[S(θ + η)− S(θ)

]dη

FMx

(z) = ρ0u′w ′ + w ′AMx



-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0

5

10

15 20

z (k

m)

-20 -10 0 10 200

5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

-20 -10 0 10 20c (m/s)

0 5

10

15 20

-20 -10 0 10 200 5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

FMx

(z)

F E(z)−

uFMx

(z)

� Spectrum is symmetric

� Non-upward flux contributionis zero above 7km

� Upward flux contribution islarge below 10km


• In the presence of mid-level wind shear the convectionbecomes organized very quickly; cloud clusters (negative tilt)emerge and propagate eastward.

-40 -30 -20 -10 0 10x10-3 Nm-2

0 5

10

15

20

z (k

m)

-3 -2 -1 0 1 2 3Nm-1 s-1

0 5

10

15

20

z (k

m)


(z) F E(z) − UFMx

(z)


GWMT

CMT

→

→

→

→

20hr

70hr

120hr

170hr

cloud clusters


• Contribution of non-linear flux term to domain-averagedprofiles

-0.5 0.0 0.5 1.00 5

10 15 20

-0.5 0.0 0.5 1.00 5

10 15 20

-0.5 0.0 0.5 1.00 5

10 15 20

-0.2 -0.1 0.0 0.1 0.2ms-1 day-1

0 5

10

15 20

z (k

m)

-0.2 -0.1 0.0 0.1 0.2ms-1 day-1

0

5

10

15 20

-0.2 -0.1 0.0 0.1 0.2ms-1 day-1

0

5

10

15 20

Time → 20-70hr 70-120hr 120-170hr

∂FMx(z)

∂z

∂FMx(z)

∂z

Sources and sinks

• The source/sink terms in the wave-activity conservation lawsinvolve the source/sink terms on the mesoscale

SE = v′ · Sv + ρ0

[1

(θ0)2N2− ρ0u

[(θ0)z ]2·(uzρ0

)z

]θ′Sθ

− ρ0

Θzu ·[ω′⊥Sθ − θ′(∇× Sv)⊥

]DPx,y = − (ρ0)2

[(θ0)z ]2

(u

ρ0

)z

θ′Sθ −ρ0

Θz

[ω′⊥Sθ − θ′(∇× Sv)⊥

].

with

SE = v′ · Sv + ρ0θ′Sθ/(θ0)2N2 − u ·DPx,y

• The second term in SE is recognized as a term in the availablepotential energy budget since ρ0θ′Sθ/(θ0)2N2 = ρ0θ′w ′/θ0.

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