Exact nonlinear mountain waves propagating upwards

Abstract

We derive an exact solution to the nonlinear governing equations for mountain 
waves in the material (Lagrangian) framework. The explicit specification of the 
individual particle paths enables a detailed study of the flow consisting 
of oscillations superimposed on a mean current propagating upwards.

Exact Solutions Modelling Nonlinear Atmospheric Gravity Waves

“…The second scenario modelled in (ii), and described in Sect. 4.2 , generalises a recently derived solution [ 6 ] which models upward-propagating mountain waves by incorporating a variable transverse current. Upward propagating mountain waves feature a vertical velocity component which is significant, and they can occur where there is no thermal inversion layer above the mountain peak, and when the wind speed also does not increase significantly with altitude.…”

“…The applicability of the inviscid governing equations ( 2.1 )–( 2.4 ) to modelling atmospheric waves, and in particular mountain waves, is discussed in greater detail in [ 6 , 28 , 29 , 32 , 34 ]. The Euler equations given in ( 2.1 ) correspond to an f -plane approximation in the equatorial region whereby the and axes point in the longitudinal and latitudinal directions, respectively.…”

“…These are gravity atmospheric waves resulting leeward from the crest of a mountain range whereby the air, which experiences a relatively uniform orographic lift up the mountain slope, develops a complicated downstream flow pattern corresponding to oscillations in the flow velocity, air temperature, and atmospheric pressure, see [ 12 , 21 , 25 , 27 , 28 , 34 ]. The orographic lifting of air particles can (outside regions of active precipitation—see the discussions in [ 6 , 18 ]) be reasonably considered a dry adiabatic process and so we set in ( 2.6f ) for the atmospheric wave motions being considered in this paper. As mentioned above, the compatibility condition ( 2.6f ) then ensures that the prescribed atmospheric flow is consistent with both the first and second law of thermodynamics.…”

“…[ 20 , 22 , 23 ]). An exact nonlinear Lagrangian solution for incompressible inviscid air flow on an interface between two regions of constant density was derived in [ 24 ], while recently in [ 6 ] this work was significantly generalised to accommodate a continuous density variation as well as a tilted direction of wave propagation.…”

“…In this paper we generalise the two-dimensional flow in [ 6 ] to three dimensions by introducing variable transverse currents into the exact solution, and further generalise the structural complexity of the governing equations by incorporating the Coriolis effects of the Earth’s rotation. We consider waves whose physical scale is such that we may neglect the Earth’s sphericity, but we aim to retain the effects of the Earth’s rotation by incorporating Coriolis forces in an f -plane approximation.…”