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3.2 Phenomenology of Turbulence

3.2.6 Quasigeostrophic Turbulence

We are now in a position to discuss quasigeostrophic turbulence, which is the turbulence of large-scale atmospheric and oceanic motions. The continuously-stratified QGPV equation is Dq Dt = 0, (3.27) where q =2ψ+f+ ∂ ∂z f2 0 N2 ∂ψ ∂z (3.28) is the three-dimensional quasigeostrophic potential vorticity. If we compare Eqs 3.27 and 3.15, we immediately see that the QGPV equation is isomorphic to the two-dimensional vorticity equation. Furthermore, these equations are the same in the absence of rotation (f) and stratification (vertical derivative term). Two-dimensional turbulence phenomenol- ogy, therefore, paints a picture of geostrophic turbulence, but it is not a complete picture. In this section, we shall consider the effects of rotation and stratification separately.

Beta-Plane Turbulence

Two-dimensional turbulence on a beta plane creates large scale zonally elongated struc- tures (Rhines, 1975; Vallis and Maltrud, 1993; Frederiksen et al., 1996). A lucid discussion of this phenomenon is given by Vallis (2006). Consider the barotropic vorticity equation on the beta-plane, without topography:

∂ζ

∂t +J(ψ, ζ+βy) = 0. (3.29)

In spectral space, we have from Eq. 2.48 (in the absence of topography):

∂ζk ∂t = X p X q δ(k+p+q)K(k,p,q)ζpζ−q−iωkζk, (3.30)

where ωk = −βkk2x is the Rossby-wave frequency. Now, in the absence of the non-linear terms, the solution to Eq. 3.30 will consist of Rossby waves; that is ζk(t) = expiωkt. On

the other hand, as k→ ∞, we have ωk→0, so the Rossby waves are confined to the large

scales. We therefore expect that the small scales are dominated by turbulent eddies. If we assume, then, that at large scales Rossby waves dominate, and at small-scales, turbulence dominates, what is the scale at which they are equally important? A simple estimate of this is given by Vallis (2006). The time-scale of the Rossby waves is given by the inverse

§3.2 Phenomenology of Turbulence 27

Rossby-wave frequency

Tr =

k

β. (3.31)

The turbulence time-scale is estimated by

Tt= 1

U k, (3.32)

which corresponds to a linearly-falling eddy turnover time. Equating Eqs. 3.31 and 3.32, we obtain the beta scale, also sometimes known as the Rhines scale:

Lβ =

s

U

β, (3.33)

whereLβ = k1β. A typical value forβ is 10−11 m−1s−1. For the atmosphere,U ≈10 ms−1, giving Lβ ≈ 1000 km, while for the ocean, U ≈ 0.1 ms−1, giving Lβ ≈ 100 km. These values are very rough estimates, but they do seem to imply that the Rhines scale is close to the radii of deformation of both the atmosphere and the ocean. The Rhines scale is usually interpreted as the scale where the inverse cascade of energy is arrested by the beta effect, although ‘arrested’ is misleading terminology as discussed further below.

The analysis above assumes that the Rossby wave frequency is isotropic. In reality, the Rossby wave frequency is highly anisotropic, and thus a better estimate for the time scale of the Rossby waves would be

Tr =

k2

x+k2y

βkx

. (3.34)

Equating this to Eq. 3.32, Vallis (2006) obtains (kx)β = r β U cos 3 2 θ (ky)β = r β U sinθcos 1 2 θ, (3.35) where θ = arctanky kx

. Plotting ky against kx results in a ‘dumb-bell’ shape with cen- ter at the origin (kx = ky = 0). The inside of the dumbbell describes the region where Rossby waves dominate whereas the outside is dominated by turbulence. Energy injected at an intermediate scale, outside the dumbbell, will cascade upscale as demanded by the quasi two-dimensional nature of Eq. 3.29. Moreover, if there is not sufficient large-scale dissipation, energy will continue to cascade upscale in an anisotropic fashion. The flux of energy will ‘avoid’ the inside of the dumbbell, but can continue upscale (towards lower wavenumber) by going through thekx= 0 (vertical) axis. In other words, the turbulence will create zonally-elongated structures. Sukoriansky et al. (2006) have questioned the existence of a spectral separation between large-scale Rossby waves and small-scale turbu- lence. This suggests a different interpretation of the Rhines scale; however, the creation of zonal jets by ‘turbulence anisotropization’ as described above remains undisputed.

28 Quasigeostrophic Turbulence

Topographic Turbulence

The interaction between turbulence and large scale topography can create large scale mean fields, which arise spontaneously even if there are no mean fields initially. Bretherton and Haidvogel (1976) used numerical simulations to show that an initially random field over topography eventually becomes correlated with the topography. Salmon et al. (1976) used the methods of equilibrium statistical mechanics to show that a mean state proportional to the topography emerges in an inviscid unforced barotropic fluid. A more extensive review of the literature is given by Holloway (1986).

It is, however, possible to understand this phenomenon using more simplistic arguments as follows. Consider the form drag equation for barotropic flow over topography, Eq. 2.57, written as ∂U ∂t =T, (3.36) where T = 1 S Z S hvdS (3.37)

is the form drag on the zonal flow, U, defined as,

U = 1

S

Z

S

udS (3.38)

and the meridional flow is defined as

V = 1

S

Z

S

vdS. (3.39)

Now, suppose initially there is no mean meridional circulation; that is V = 0. However, we imagine that turbulent (transient) eddies exist so that v is not zero everywhere. It is not hard to see then from the definition, Eq. 3.37, that T = 0 only if h has no spatial variation; otherwise, in general, T 6= 0 even ifV = 0. Thus, the zonal flow will experience form drag even if there is no mean eddy field initially; this form drag is purely a result of the interactions between transient eddies and the topography.

Stratified Two-level Turbulence

We now consider the effects of stratification. The basic ideas originate from the works of Charney (1971), Rhines (1977), and Salmon (1978). The two-level QGPV equations have two quadratic invariants just like the two-dimensional vorticity equation. The quantities conserved are the total energy and potential enstrophy. The total energy is inherently a three-dimensional quantity as it contains the coupling between the two levels (the potential energy). This means that to extend the arguments for two-dimensional turbulence to QG turbulence, we must work in three-dimensional wavenumber space (which is nevertheless severely truncated in the vertical for the two-level problem). Hence, for example, anal- ogously to the two-dimensional case, the energy is preferentially cascaded towards lower three-dimensional wavenumber, which implies not only larger scales, but also barotropiza- tion. This is because barotropic flow can be thought of as having a vertical wavenumber of zero. It turns out that there are three distinct regimes for QG phenomenology: scales near the deformation scale, scales larger than the deformation scale, and scales smaller than the deformation scale. These ideas are explored in more detail in Appendix D.