August 28, 2013 John P. McHugh University of New Hampshire 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 Hawaii Balloon 2, first day

McHugh, Dors, Jumper, Roadcap, Murphy, and Hahn, JGR, 2008 Experiments over Hawaii Balloon 2, second day

McHugh, Jumper, Chen, PASP, 2008 Experiments over Hawaii Balloon 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, 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-like equations, coupled through the linear interfacial conditions and the (nonlinear) mean flow. and on z=0.

The wave-induced mean flow is

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:

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

Contours of horizontal velocity: closeup of tropopause region.

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

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

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