Abstract
Due to their mathematical tractability,two-dimensional (2D) fluid equations are often used by mathematiciansas a model for quasi-geostrophic (QG) turbulence in the atmosphere,using Charney's 1971 paper as justification. Superficially, 2D and QGturbulence both satisfy the twin conservation of energy and enstrophyand thus are unlike 3D flows, which do not conserve enstrophy. Yet QGturbulence differs from 2D turbulence in fundamental ways, which arenot generally known. Here we discuss ingredients missing in 2Dturbulence formulations of large-scale atmospheric turbulence. Weargue that there is no proof that energy cannot cascade downscale inQG turbulence. Indeed, observational evidence supports a downscaleflux of both energy and enstrophy in the mesoscales.It is suggested that the observed atmospheric energy spectrum isexplainable if there is a downscale energy cascade of QG turbulence, butis inconsistent with 2D turbulence theories, which require an upscaleenergy flux. A simple solved example isused to illustrate some of the ideas discussed.
Highlights
Two-dimensional incompressible fluid flows satisfy the following equation in the absence of forcing and dissipation: ∂ ω + J(ψ, ω) = 0, (1)∂t where ω = ∇2ψ ≡ ∂2∂x2 + ∂y2 ψ is the vertical component of vorticity, (x, y) are the horizontal coordinates, and ψ is the streamfunction for the horizontal velocities (u, v) = − ∂ ∂y ψ, ∂ ∂x ψ J(A, B) ≡
Atmospheric data are routinely analyzed and presented in 2D-like quantities, treating each atmospheric layer as if it were a 2D fluid and lumping baroclinic terms as forcing (Boer and Shepherd, 1983; Straus and Ditlevsen, 1999; Lindborg, 1999). Much of this perspective has its origin in a Note by Charney (1971), which laid the foundation for QG turbulence
The isotropy in the three stretched coordinates is termed “Charney isotropy” by McWilliams et al (1999)]. It is this isomorphism between QG and 2D flows which prompted Charney to conclude that an energy cascade to small scales is impossible in QG turbulence, borrowing a proof from Fjørtoft (1953) on 2D turbulence
Summary
Two-dimensional incompressible fluid flows satisfy the following equation in the absence of forcing and dissipation:. Large-scale atmospheric motion is confined in a shallow layer of fluid, whose horizontal dimension is of the order of the radius of the earth (a = 6, 400km), while. Most of the kinetic energy of large-scale motion is contained in the horizontal velocities, the vertical velocity, w, being smaller by at least a factor of the aspect ratio: δ = H/a ∼ 10−3. This fact is often used by mathematicians to proclaim, naively as it turns out, that large-scale geophysical flows should be “quasi two-dimensional”, and could be modeled by 2D equations such as Eq(1). Dt g which says that in a stably stratified atmosphere, sinking air (w < 0) is adiabatically warmed
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