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