Trapped lee wave drag in

two-layer atmospheres

Miguel A. C. Teixeira1

José L. Argaín2

Pedro M. A. Miranda3

1Department of Meteorology, University of Reading,

Reading, UK

2Department of Physics, University of Algarve,

Faro, Portugal

3Instituto Dom Luiz (IDL), University of Lisbon,

Lisbon, Portugal

2

Trapped lee waves

Energy Flux

dz

dgN

0

2

2/1

2

2

2

2 1

dz

Ud

UU

Nl

Non-hydrostatic

Scorer parameter

Rotors

3

Mountain wave drag

2

0

2

04

lhUD

2D bell shaped ridge: Linear, hydrostatic, non-rotating, constant l limit:

2

0

)/(1 ax

hh

It is known that drag decreases as flow becomes more nonhydrostatic (la

decreases) → this would suggest that trapped lee waves (which are highly

nonhyrsotatic) would produce little drag

However, trapped lee waves exist due to energy trapping in a layer or

interface: wave reflections and resonance → may lead to drag amplification

Aim here is to calculate trapped lee wave drag and untrapped mountain

wave drag in two-layer atmospheres

Important for drag parametrization schemes in global climate and weather

prediction models

4

Linear theory

0ˆ)(ˆ 22

2

2

wkldz

wd

2-layer atmospheres

dkezkwzxw ikx),(ˆ),(

•Boussinesq approximation

•Linearization, 2D flow

•Inviscid, nonrotating, stationary flow

hiUkzw ˆ)0(ˆ

Taylor-Goldstein equation

Waves propagate energy upward or

decay as z w p

Boundary conditions:

continuous at z=H

U

Nl 11

U

Nl 2

2

Scorer (1949) Vosper (2004)

Case 1 Case 2

0

*ˆ)0(ˆIm4)0( dkhzpkdxx

hzpD

Gravity

wave drag p determined from

solutions for w

0

gg

5

Propagating wave drag

Trapped lee wave drag

2 2 22 1 2

1 0 2 2 2 2

1 1 2 10

ˆ| |4

cos ( ) sin ( )

lk h m m

D U dkm m H m m H

Hkn

knkmkhUD

j

jj

j

j )(1

)()(|)(ˆ|4

2

2

2

122

0

2

2

|k|<l2

l2<|k|<l1

Drag normalized by Depends on: 2

01

2

004

hlUD

Hl1 12 / ll al1

Case 1

Propagating wave drag

Trapped lee wave drag

|k|<l2

|k|>l2

2

022

2

22

2

222

01

)(sinh)()sinh()cosh(

))((|ˆ|4

l

dkkHHmkHFrkHkH

kHHmhkUD

Case 2

HknFrFrHknHknHHk

HkHknFrkhkUD

LLLL

LLLL

)()(1)()(

)()()(ˆ

42

22

2

1

2

2

22

2

22

2

2

0

2

2

Drag normalized by 2

02

2

004

hlUD

Depends on: Hl2 Hg

UFr

al2

6 6

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

1.2

l2/(2)

l2/(2)

l2/(2)

jl 2

/(2

)

l1H/

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

NL

l1H/

(a)

Case 1 2.0/ 12 ll

These

quantities do

not depend

on

al1

Numerical simulations 02.001 hl

NL

jl2/(2)

Drag maxima

coincide with

establishement

of new trapped

lee wave

modes

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

FLEX

D/D0

D1/D

0

D2/D

0

D/D

0

l1H/

(a)

110l a

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

FLEX

D/D0

D1/D

0

D2/D

0

(b)

D/D

0

l1H/

15l a

21 DDD

7 7

Case 1 2.0/ 12 ll

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

FLEX

D/D0

D1/D

0

D2/D

0

(c)

D/D

0

l1H/

12l a

-10 -5 0 5 10 15 20 25 30 35 40 45 50

x/a

(c)

0

1

2

3

4

5

6

7

8

l 1z/

-10 -5 0 5 10 15 20 25 30 35 40 45 50

x/a

(d)

0

1

2

3

4

5

6

7

8

l 1z/

08.0/ 12 DD5.1/1 Hl

Propagating waves

dominate

7.1/1 Hl 06.0/ 21 DD

Trapped lee waves

dominate

D2/D1 increases as l2a

decreases

w from numerical simulations

1 2l a

10-1

100

101

0

1

2

3

4

5

D1/D

0

D2/D

0

D/D0

FLEX

(b)

D/D

0

Fr

10-1

100

101

0

1

2

3

4

5

D1/D

0

D2/D

0

D/D0

FLEX

(c)

D/D

0

Fr

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0 l

2H=0.5

l2H=1

l2H=2

Ll 2

/(2)

Fr

Case 2

22l a

jl2/(2) This

quantity

does not

depend on

al2

01.002 hl

5.02 Hl

Numerical simulations

8

21l a

Drag maxima

occur for

Fr 1

D2/D1

increases as

l2a decreases 10

-110

010

10

1

2

3

4

5

D1/D

0

D2/D

0

D/D0

FLEX

(d)

D/D

0

Fr

20.5l a

0 0.4 0.8 1.2 1.6 2 2.4

(l2H)-1

(b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fr

Case 2

D2/D0

Regime diagram of Vosper (2004)

9

• Rotors occur mostly for large j

• There is some correlation between D2 and rotor formation

• This connection must be explored for Case 1 as well

0 0.4 0.8 1.2 1.6 2 2.4

(l2H)-1

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fr

jl2/(2)

5.002 hl• Nonlinear ( ) numerical simulations including friction

685.12 al

10

Summary

• Trapped lee wave drag is calculated explicitly for first time for two-layer atmospheres of Scorer (1949) and Vosper (2004).

• Trapped lee wave drag is given by closed analytical expressions.

• Waves trapped within layer may have multiple modes, while waves trapped at temperature inversion may only have single mode.

• Due to resonant amplification, trapped lee wave drag may be comparable to drag associated with waves propagating in stable upper layer, and higher than single-layer, hydrostatic reference value.

• Drag maxima coincide with establishment of trapped lee wave modes (as H increases). For waves trapped at an inversion, drag is maximized for Fr 1.

• D2/D1 increases as l2a decreases. Trapped lee wave drag seems to be maximized for l2a = O(1): wavelength of trapped lee waves matches mountain width.

• Trapped lee wave drag correlates fairly well with occurrence of rotors for Case 2 (still have to check Case 1).

Teixeira, Argain and Miranda (2013a), QJRMS, in press (Early view papers)

Teixeira, Argain and Miranda (2013b), JAS, in press (Early online releases)