August 28, 2013

John P. McHughUniversity of New Hampshire

john.mchugh@unh.edu

Internal waves, mean flows, and turbulence at the tropopause

Is there more turbulence at the tropopause altitude?

Observations say ‘maybe’ Tropopause has a sudden change

in N, suggesting a wave effect Recent results point to three or

more possible explanations

Schematic of Earth’s vertical temperature profile

McHugh, Dors, Jumper, Roadcap, Murphy, and Hahn, JGR, 2008

Experiments over HawaiiBalloon 2, first day

McHugh, Dors, Jumper, Roadcap, Murphy, and Hahn, JGR, 2008

Experiments over HawaiiBalloon 2, second day

McHugh, Jumper, Chen, PASP, 2008

Experiments over HawaiiBalloon 1, first day

• Boussinesq• Two-layers with constant N• Periodic side boundaries• Uniform wave amplitude• Velocity is continuous at the

interface

First reason: nonlinear wave behavior near the tropopause

Uniform (Stokes) waves

• First few harmonics are scattered by the interface.• Remaining harmonics are evanescent in both layers.• Wave behavior at the interface is ‘Stokesian’.• Wave-generated mean flow is not local to the interface.• JAS, 66, 2009.

Results

Direct numerical solution of wave packets

Reason 2: Wave induced mean flow at the tropopause

Simulations

• Inviscid Anelastic equations• Spectral filter with p = 15• Two layers of constant N• Periodic boundaries on the sides• Damping (Rayleigh) layer at the top• Spectral method in space• TCFD, v 22, 2008

Primary Results of DNS

• Wave packet creates a localised mean flow (jet) at the interface.

• If wave amplitude is high enough, this mean flow exceeds horizontal wave speed and waves overturn below interface.

Why does this jet form? Consider a simpler model.

Reason 3: Mean flow velocity gradients at the tropopause

• NLS amplitude equations• Two layers with constant N• Periodic side boundaries• Wave amplitude varies vertically• Paper being revised for JFM• Grimshaw and McHugh, to appear in QJRMS.

Overall, have three nonlinear Schrodinger-likeequations, coupled through the linear interfacialconditions and the (nonlinear) mean flow.

and on z=0.

0221 22

12222

12' RInkRInkiRciRcR gg

0221 22

12222

12' IRnkRInkiIciIcI gg

0221 22

22' TTnkiTciTcT gg

JITKIR ,

The wave-induced mean flow is

0,2

2

2

0,

22

22

*2

21

22

2

21

2*2 11

zTNc

u

RINc

u

RINc

u

zeueuuu

p

pi

pm

znii

zniim

4.01 kn

4.01 kn

4.01 kn

4.01 kn

4.01 kn

21 kn

21 kn

21 kn

21 kn

21 kn

Results from amplitude equations

• Either frequency modulation or the oscillating mean flow may form a 'jet' underneath the mean interface.

• Mean flow is discontinous at the mean position of the interface (this feature was missing in DNS).

• Frequency modulation appears to be the stronger feature but is not significant in large amplitude waves.

Unsteady flow past an obstacle

McHugh and Sharman, QJRMS, 2012.

Numerical simulations

• Witch of Agnesi mountain shape• Linear bottom boundary condition• No rotation• Mountain is introduced gradually• 2nd order finite difference• Arakawa C grid• Leap frog method for time stepping• Typical case is U=10 m/s, H=1000,• A=1000m, N2/N1=2

Unsteady mountain waves

U= 10 m/s, H=1000m, NH/U=1

Mean flow• Need a mean definition that is analogous to the periodic

case• But no scale separation between waves and wave-induced

mean flow• Average over the computational domain depends strongly

on domain length.• Finally:

dxuuA

u 021

U= 10 m/s, H=1000m, NH/U=1

dxuuA

u 021

dxuuA

u 021

Contours of horizontal velocity: closeup of tropopause region.

U= 5 m/s, H=500m, NH/U=1

dxuuA

u 021

dxuuA

u 021

Conclusions for mountain wave case

• An upstream wave-induced mean flow usually forms above the mean position of the tropopause

• A counter flow forms beneath the mean tropopause, not present in the periodic simulations

• Mean flow remains in the steady mountain wave flow, and is different than the flow determined with the steady equations directly

• The combination of upstream and downstream flow at the tropopause suggests a higher likelihood of breaking there, or perhaps even a circulation

Some concluding remarks

• The tropopause region is complex when the tropopause is sharp

• Probably need a two-layer simulation (DNS) that allows slip to get the correct mean flow at the tropopause

• If N is constant, then wave amplitude may be unity and the dispersive term doesn't exist. What happens to the jet?

• Still cannot completely explain the observations over Hawaii

END