Richard Rotunno National Center for Atmospheric Research, USA Dynamic Mesoscale Mountain Meteorology Lecture 3: Mountain Waves
Topics Lecture 1 : Introduction, Concepts, Equations Lecture 2: Thermally Driven Circulations Lecture 3: Mountain Waves Lecture 4: Mountain Lee Vortices Lecture 5: Orographic Precipitation
Jane English Mt. Shasta Mountain Waves
Bob Houze Wave Clouds at Three Levels
Dane Gerneke Multiple Wavelength Wave Clouds
Rotor Cloud in lee of the Sierra Nevada R. Symons
Air Parcel Behavior in a Stable Atmosphere Potential Temperature z E A B
Mountain Waves under Stable Conditions
Back to Inviscid, Adiabatic Governing Equations Conservation of momentum energy mass
momentum Flow Past an Impermeable 2D Obstacle
momentum Vorticity Equation Lecture 2 2D version Baroclinicity
2D Eqns in Terms of Vorticity and Streamfunction vorticity energy Lecture 2
vorticity energy Simplify
vorticity energy 2D Linear Theory of Internal Gravity Waves combine vorticity and energy eqns Lighthill (1978, Chap. 4)
plane-wave solution relation of and to b wavenumber vector phase velocity
Example phase motion
Internal Gravity Waves: Group Velocity Form wave energy eqn For plane-wave solution wavelength average gives
Internal Gravity Waves: Group Velocity phase velocity wavenumber vector In this example Lighthill (1978, Chaps. 3-4)
momentum Linear Theory of Mountain Waves
vorticity energy Linear Theory of Mountain Waves combine
momentum Equation + Boundary Conditions Energy radiation or decay as
Example: Sinusoidal Hill Look for sol’n of the form Gov’n eqn b.c.
Case 1 Sol’n use radiation condition to indicate which sign to choose in the exponents
dispersion relation including U for steady sol’n must choose minus sign in dispersion relation and so
Doppler-shifted=Natural Freq
Case 2 Sol’n use decay condition to indicate which sign to choose in the exponents
Doppler-shifted>Natural Freq No Waves
Steady-State Linear Theory: Isolated Hill Fourier Transform
Radiation Condition for Inverse Fourier Transform Sol’n Decay Condition for F.T. of B. C. Fourier Transform Eqn
Summary - Internal gravity waves are peculiar (upward energy but downward phase propagation) -These basic ideas from IGW theory are necessary for the intepretation of mountian waves - Literature is vast: more general situations such as U=U(z), N=N(z) have been studied to understand flow response in particular atmospheric conditions