The corresponding working quasilinear wave equation for the barotropic azonal stream function Ψm ′ of the forced waves with m = 6, 7, and 8 (m waves) with nonzero right - hand side (forcing + eddy friction) yields (34) u˜ ∂ ∂ x (∂ 2Ψm ′ ∂ x2 + ∂ 2Ψm ′ ∂ y2) + β˜ ∂ Ψm ′ ∂ x = 2Ω sin ϕ cos2 ϕT˜u˜ ∂ Tm ′ ∂ x − 2Ω sin ϕcos2 ϕHκu˜ ∂ hor, m ∂ x − (kha2 + kzH2)(∂ 2Ψm ′ ∂ x2 + ∂ 2Ψm ′ ∂ y2), [S3] where x = aλ and y = a ln -LSB-(1 + sin ϕ) / cos ϕ] are the coordinates of the Mercator projection of Earth's sphere, with λ as the longitude, H is the characteristic value of the atmospheric density vertical scale, T˜ is a
constant reference temperature
at the EBL, Tm ′ is the m component of azonal temperature
at this
level, u˜ = u ¯ / cos ϕ, κ is the ratio of the zonally averaged module of the geostrophic
wind at the top of the PBL to that
at the EBL (53), hor, m is the m component of the large - scale orography height, and kh and kz are the horizontal and vertical eddy diffusion coefficients.