## 1. Introduction

*β*-plane vorticity equation with a background mean shear

*γ*:

*ξ*(

*x*,

*y*,

*t*) is spatially homogenous and white noise in time and is characterized more precisely in section 2. Drag, with coefficient

*μ*, is the dissipative mechanism.

Our main concern is the eddy transport of momentum, potential vorticity, and tracer in the ultimate statistically steady state of the model (1). Despite the evident importance of this linear problem, to our knowledge, it has only been discussed previously by Farrell and Ioannou (1993) and Bakas and Ioannou (2013) for the case of weak background shear. The problem is, however, closely related to the initial value problem for the evolution of linearized disturbances on an unbounded viscous Couette flow, first considered by Kelvin (1887) and Orr (1907). A main result of these early studies is that an initial sinusoidal disturbance with crests “leaning” into the shear is amplified for some time. Various aspects of the Kelvin–Orr initial value problem, such as the inclusion of planetary vorticity gradient *β* and a spectrum of initial disturbances, were subsequently discussed by Rosen (1971), Tung (1983), Boyd (1983), and Shepherd (1985). The transient amplification of Kelvin and Orr is now understood as one consequence of the nonnormality of the linear vorticity equation in the energy norm (Farrell 1982).

*c*′(

*x*,

*y*,

*t*) satisfying the linearized tracer equation

*β*

_{c}is the large-scale tracer gradient. For simplicity, we assume that the scalar damping rate

*μ*is the same as that of the vorticity. The tracer

*c*′ differs from the vorticity

*ζ*′ because there is no stochastic forcing in (2). Instead, scalar fluctuations are created by the eddy velocity

*υ*′ stirring the mean gradient

*β*

_{c}.

*σ*,

*κ*

_{e}, and other quadratic statistics, such as the eddy kinetic energy and enstrophy, and the anisotropy of the velocity field.

The correlation function formalism—introduced in sections 2 and 4—is an economical framework for analysis of the statistically steady flow. Rather than solving (1) and (2) explicitly and then averaging the solution, one averages at the outset, and then solves steady deterministic equations that directly provide *σ* and *κ*_{e}.

Farrell and Ioannou (1993) have previously discussed the statistically steady flow corresponding to (1) with *β* = 0. Farrell and Ioannou (1993) use the eddy kinetic energy (rather than *σ* and *κ*_{e}) as the main statistical descriptor of the flow and they emphasize viscosity (rather than Ekman drag) as the dissipative mechanism. In a related geophysical context, the tracer equation has recently been considered (with *γ* = 0) by Ferrari and Nikurashin (2010) and Klocker et al. (2012).

The forcing and drag terms in (1) incorporate the effects of two different processes, which we characterize as “external” and “internal.” External processes, such as small-scale convection in planetary atmospheres (Smith 2004; Scott and Polvani 2007) or baroclinic instability (Williams 1978), are often modeled as a stochastic driving agent combined with a damping term representing Ekman friction. On the other hand, internal nonlinear interactions—that is, *J*(*ψ*′, *ζ*′)—are sometimes represented using a stochastic turbulence model (DelSole 2001). This is the interpretation of *ξ* and *μ* in the studies of Farrell and Ioannou (2003, 2007), Ferrari and Nikurashin (2010), and Klocker et al. (2012). As suggested by fluctuation–dissipation arguments, the turbulence model has a stochastic forcing term and eddy damping; the combination ensures energy conservation. In section 2, we introduce a forcing that is distributed anisotropically around a circle in wavenumber space.

Section 2 contains a description of the forcing structure and symmetries of the correlation functions. Section 3 shows that (1) and (2) have a statistical Galilean symmetry implying that all quadratic statistics, in particular *σ* and *κ*_{e}, are independent of *y*. Section 4 summarizes the quadratic power integrals that follow from taking quadratic averages of (1) and (2). These power integrals are used to obtain some simple and general bounds on *σ* and *κ*_{e}. Analytic expressions for *σ* and *κ*_{e} are presented in sections 5 and 6. Section 7 is a conclusion and discussion of the results. Technical details are relegated to the appendixes.

## 2. Correlation functions and statistical symmetries

*ξ*(

*x*,

*y*,

*t*) in (1) is temporal white noise, with a two-point, two-time correlation function

**x**=

**x**

_{1}−

**x**

_{2}.

We do not assume that the forcing is isotropic: Ξ(**x**) might depend on the direction of the two-point separation **x** = (*x*, *y*). One motivation for examining the effect of anisotropy is that in many studies of zonal jets on the *β* plane (Vallis and Maltrud 1993; Smith 2004) and on the sphere (Williams 1978; Scott and Polvani 2007; Showman 2007), the small-scale forcing used to drive the jets is assumed to be isotropic, even though the physical processes that the forcing models, such as baroclinic instability in the ocean and moist convection in planetary atmospheres, are typically not isotropic (Arbic and Flierl 2004; Li et al. 2006).

### a. A remark on scale separation and homogeneity in y

*γy*in (1), can be interpreted as a local approximation to a mean flow

*U*(

*y*) that is slowly varying relative to the eddy scale and to the scale of Ξ(

**x**). At a particular position

*y*, the mean flow is

*γ*=

*U*′(0). However, the constant

*U*(0) has no physical consequences in this model: one can move the origin of the coordinate system with

*U*(0) from the problem. This removal hinges on the spatial homogeneity of the statistical properties of

### b. Statistical properties of the solution

*x*

_{1},

*y*

_{1}) and likewise for

*x*and

*y*are the components of the two-point separation

**x**=

**x**

_{1}−

**x**

_{2}. In (7), we have anticipated that statistical properties of the solution are spatially homogeneous so that the correlation functions

**x**=

**x**

_{1}−

**x**

_{2}. The correlation functions are connected by the biharmonic equation

### c. Exchange symmetries

**x**, and ensures that the spectrum,

### d. Reflexion symmetry

*x*, then the correlation function has a second symmetry

*u*′,

*υ*′) → (−

*u*′,

*υ*′) and therefore

*σ*→ −

*σ*. If the statistics of the forcing

*ξ*also obey (15), then (17) is a statistical symmetry of (1) and therefore

But the symmetry (15) is not compulsory; for example, the single-wave forcing of Ferrari and Nikurashin (2010) and Klocker et al. (2012) does not satisfy (15). However we make the assumption that the forcing is reflexionally symmetric and we proceed confining attention to *ξ* with statistics obeying (15). As a consequence of this restriction, *σ*(*γ*), calculated explicitly in section 5, satisfies (18).

### e. The stochastic forcing

*x*,

*y*) =

*r*(cos

*θ*, sin

*θ*),

*k*

_{f}is the “forced wavenumber,” and

*J*

_{m}(

*z*) is the Bessel function of order

*m*. The corresponding spectrum is

*p*,

*q*) =

*k*(cos

*ϕ*, sin

*ϕ*). The forcing is concentrated on a circle with radius

*k*

_{f}in wavenumber space. To ensure that the spectrum is nonnegative, the anisotropy parameter

*α*must satisfy −1 ≤

*α*≤ 1. Figure 1 shows model correlation functions and forcing obtained by varying

*α*in (19).

## 3. Statistical Galilean invariance

The linearized vorticity equation in (1), with the rapidly decorrelating forcing in (5), has a form of statistical Galilean invariance. To explain this, consider two observers—one of whom is stationary and at the origin of the (*x*, *y*, *t*) coordinate system in (1). The other observer is at *y* = *b* and moves “with the mean flow,” at speed *γb* along the axis of *x* relative to the first. Because of the rapid temporal decorrelation of the forcing *ξ*, these two observers see statistically identical versions of the problem (1). Thus all zonally averaged quantities are independent of *y*. This simple argument allows us to anticipate some curious aspects of the detailed calculations that follow in section 5.

Notice that if the forcing has a nonzero temporal decorrelation time then the statistical properties of *ξ* are different in the two frames of reference, and consequently the problem is no longer statistically Galilean invariant (or even Galilean invariant). If there is a nonzero decorrelation time, then averaged quantities do depend on *y*. A clear example is steady forcing, such as *ξ* = cos*k*_{f}*x* used by Manfroi and Young (1999). In the frame of the observer at *y* = *b*, this forcing is periodic in time. In this example, the forcing breaks Galilean invariance because there is a special frame in which the forcing is steady (or has the longest decorrelation time in the stochastic case).

*ξ*(

*x*,

*y*,

*t*) in (1), then this Galilean transformation is a statistical symmetry. And indeed, because of the

*δ*(

*t*

_{1}−

*t*

_{2}) correlation in (5), this is the case.

*y*, despite the explicit

*y*dependence in (1) and (2). As an application of this result, the eddy vorticity flux is related to the Reynolds stress by the Taylor identity

*y*, it follows that the statistically steady solution of (1) must have

*βy*is stirred by eddies [but see the discussion surrounding (45)].

## 4. Power integrals

### a. Enstrophy

*ζ*′ and zonally averaging. Using (24), the result is

*β*gradient, there is a strict balance in (25) between local eddy enstrophy production on the right-hand side and enstrophy dissipation by drag on the left-hand side.

### b. Energy

*ψ*′ and zonally averaging. Again, because of statistical Galilean symmetry, zonally averaged quantities, such as

*y*and one finds

*ε*in (27) is the same as

*ε*in (19) and (20). A more detailed discussion of this aspect can be found in Srinivasan and Young (2012).

### c. Tracer variance

*c*′ and zonally averaging:

### d. Covariance of tracer and vorticity

### e. A bound on the Reynolds stress

### f. Bounds on the eddy diffusivity

*κ*

_{e}is obtained by combining the covariance integral in (29) with the Cauchy–Schwarz inequality for

The four power integrals, and the ensuing bounds on *σ* and *κ*_{e}, provide important and general connections between quadratic statistics characterizing the main properties of the flow. However, these relations are unclosed and to make further progress, we consider the dynamics of correlation functions.

## 5. Reynolds stress and anisotropy

^{1}of (38), at zero separation.

*β*: the

*β*term that would appear in the left-hand side of (38), on performing the replica trick mentioned above, is

The term above is zero because, owing to the homogeneity property of Ψ in section 2a,

*β*; that is, anisotropic Rossby wave propagation does not affect the vorticity correlation function

*x*,

*y*) nor the Reynolds stress in (40). Thus, all results in this section, which follow from the solution of (38) alone, apply to

*β*-plane flows, even though the parameter

*β*does not appear.

### a. Reynolds stress

*F*

_{1}can be expressed in terms of the exponential integral (see appendix C) and is shown in Fig. 2.

Plot of *F*_{1} in (42) as a function of *γ*/*μ*. The dashed curve is the approximation *F*_{1}(*γ*/*μ*) = 4*μ/γ* − 8*πμ*^{2}/*γ*^{2} + *O*[*μ*^{3}/*γ*^{3} ln(*μ*/*γ*)].

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

Plot of *F*_{1} in (42) as a function of *γ*/*μ*. The dashed curve is the approximation *F*_{1}(*γ*/*μ*) = 4*μ/γ* − 8*πμ*^{2}/*γ*^{2} + *O*[*μ*^{3}/*γ*^{3} ln(*μ*/*γ*)].

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

Plot of *F*_{1} in (42) as a function of *γ*/*μ*. The dashed curve is the approximation *F*_{1}(*γ*/*μ*) = 4*μ/γ* − 8*πμ*^{2}/*γ*^{2} + *O*[*μ*^{3}/*γ*^{3} ln(*μ*/*γ*)].

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

We emphasize the linear dependence of *σ* in (41) on *α*. In particular, if *α* = 0 (isotropic forcing), there is no Reynolds stress. This recapitulates the result that anisotropic forcing, or initial conditions, is essential to the generation of nonzero Reynolds stress (Kraichnan 1976; Shepherd 1985; Farrell and Ioannou 1993; Holloway 2010; Cummins and Holloway 2010; Srinivasan and Young 2012).

*γ*and thus the concept of an eddy viscosity is not generally useful. Instead, there is a nonlinear, and nonmonotonic, stress–strain relation encoded in

*F*

_{1}. However in the weak-shear limit,

*γ*/

*μ*≪ 1, the integral in (42) simplifies and the Reynolds stress is then

*ν*

_{e}is determined by

*α*, with

*α*> 0 being the antifrictional case. A negative viscosity in the weak-shear limit, with the same form as (43), was also found by Bakas and Ioannou (2013) using a forcing function that is similar to the

*α*= 1 case in this paper.

*γ*/

*μ*≫ 1, reducing to

^{2}But as we show in the next section, a complicating factor is the dependence of the kinetic energy density on the shear.

Another interpretation of (44) is that if *α* = ±1, then the Reynolds stress bound in (30) is an asymptotic equality as *γ*/2*μ* → ∞. One might say that the *γ*^{−1} dependence in (44) is the strongest possible Reynolds stress that can be achieved, consistent with the energy power integral (26) and the associated bound (30). Notice that (44) was obtained with the anisotropic ring forcing in (19), but the bound (30), which makes no assumptions about the structure of the forcing, indicates that *σ* ∝ *γ*^{−1} is a general result in the strongly sheared limit.

### b. The vorticity flux of a slowly varying mean flow

*γ*as the shear of a slowly varying

*U*(

*y*), then a nonzero

*σ*is the function in (41). Then using the Taylor identity (23), one has

*σ*′ is the derivative with respect to

*γ*.

### c. Eddy kinetic energy and enstrophy

**x**= 0 and is simply

*γ*/

*μ*(nor on

*β*).

*μE*′/

*ε*as a function of

*γ*/

*μ*. The antifrictional case is

*α*= +1, with 2

*μE*′/

*ε*< 1; that is, the eddy kinetic energy is depleted below the unsheared value by transfer to the large-scale shear flow. In the frictional case (

*α*= −1), the eddy kinetic energy is enhanced by transfer from the mean flow: the energy level approaches twice that of the isotropically forced flow as the shear increases.

(a) The nondimensional eddy kinetic energy as a function of *γ*/*μ*, calculated from (48). (b) The nondimensional meridional velocity variance as a function of *γ*/*μ*.

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The nondimensional eddy kinetic energy as a function of *γ*/*μ*, calculated from (48). (b) The nondimensional meridional velocity variance as a function of *γ*/*μ*.

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The nondimensional eddy kinetic energy as a function of *γ*/*μ*, calculated from (48). (b) The nondimensional meridional velocity variance as a function of *γ*/*μ*.

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

*α*= −1) than in the antifrictional flow (

*α*= +1). Specifically, if

*γ*/

*μ*→ ∞ then, using (C5), the eddy kinetic energy is

*α*= 1 is special: only in this case does the

*E*′ vanish in the strong-shear limit. The relatively energetic

*α*= −1 eddies are inefficient at forming the requisite correlation to produce a Reynolds stress. This motivates further examination of the anisotropy of the eddies.

### d. Velocity anisotropy

*F*

_{1}is in (42) and

*F*

_{2}as a function of the nondimensional shear

*γ*/

*μ*, and Fig. 3b shows the variation of the mean-square meridional velocity with

*γ*/

*μ*.

(a) The function *F*_{2}(*γ*/*μ*) defined in (51). The dashed curve is the asymptotic approximation (16*μ*^{2}/*γ*^{2})[ln(*γ*/4*μ*) − *γ*_{E}], where *γ*_{E} = 0.577 21… is Euler’s constant. (b) The index aniso in (52), with *α* = −1, 0, and 1.

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The function *F*_{2}(*γ*/*μ*) defined in (51). The dashed curve is the asymptotic approximation (16*μ*^{2}/*γ*^{2})[ln(*γ*/4*μ*) − *γ*_{E}], where *γ*_{E} = 0.577 21… is Euler’s constant. (b) The index aniso in (52), with *α* = −1, 0, and 1.

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The function *F*_{2}(*γ*/*μ*) defined in (51). The dashed curve is the asymptotic approximation (16*μ*^{2}/*γ*^{2})[ln(*γ*/4*μ*) − *γ*_{E}], where *γ*_{E} = 0.577 21… is Euler’s constant. (b) The index aniso in (52), with *α* = −1, 0, and 1.

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

In the large-shear limit, the case *α* = −1 in Fig. 4b rapidly tends to isotropy (*α* = +1 also tends to zero, but much more slowly than *α* = −1). This is consistent with the earlier result that the amplitude of the Reynolds stress in (44) is the same for *α* = +1, as for *α* = −1, despite the great difference in the energy level of the two flows as *γ*/*μ* → ∞. In other words, with *α* = −1, the eddies are energetic but almost isotropic and are therefore not very efficient at producing a nonzero Reynolds stress.

### e. Tenacity of isotropy

Figure 4b shows that if the forcing is isotropic (*α* = 0), then the flow is also isotropic; that is, if the flow is isotropically forced, then neither the mean shear nor the *β* effect induces anisotropy of the eddies. Moreover, if the forcing is anisotropic, then the effect of shear is to make the flow more isotropic: in both Fig. 4a and 4b, the index of flow anisotropy approaches zero monotonically as *γ*/*μ* increases. We cannot provide an intuitive explanation of this result.

For a recent discussion of isotropy in the context of fully nonlinear sheared turbulence, see Cummins and Holloway (2010): a main point is that nonlinear eddy–eddy interactions also decrease anisotropy. We summarize all these results by saying that isotropy is tenacious.

## 6. Eddy diffusivity

*κ*

_{e}(

*α*,

*β*,

*γ*,

*μ*). Using the replica trick, one can obtain evolution equations for the tracer correlation functions defined in (9) and (10). Combining (1) and (2) one has

*Q*is obtained by

*P*→

*Q*and (

*x*,

*y*) → −(

*x*,

*y*) in (55). After solving (55), the tracer diffusivity defined in (4), is obtained as

### a. The case γ = β = 0

*β*=

*γ*= 0, then we do not need the complicated expressions for

*κ*

_{e}above: cancel ∇

^{2}in (55) and then take an

*x*derivative to obtain

*κ*

_{υ}in (33). Notice that the upper bound on

*κ*

_{e}in (32) is too generous by a factor of 2 relative to (61). Using results from section 5, the eddy diffusivity in (61) can also be written as

*κ*

_{e}on the anisotropy

*α*reflects that of

### b. The suppression factor

*υ*′, times the mixing length

*κ*

_{e}in terms of

*κ*

_{υ}and Ferrari and Nikurashin’s (2010) suppression factor

*S*as

*S*= 1. But the effect of nonzero

*β*and

*γ*is usually to make

*κ*

_{e}less than

*κ*

_{υ}.

### c. The case γ = 0

*γ*= 0. With no mean shear, the phase in (60) simplifies to

*χ*=

*ωt*, where

*γ*= 0 tracer diffusivity obtained from the integral in (58) is then

*β*, and Fig. 5b shows the factor

*S*in (70). Increasing

*β*reduces both measures of the tracer diffusivity.

(a) The nondimensional *γ* = 0 tracer diffusivity in (67) as a function of

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The nondimensional *γ* = 0 tracer diffusivity in (67) as a function of

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The nondimensional *γ* = 0 tracer diffusivity in (67) as a function of

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

The dependence of *κ*_{e} on *α* in Fig. 5a is intuitive: in Fig. 1f, *α* > 0 forces meridionally elongated eddies resulting in enhanced diffusive fluxes in the *y* direction. The difference between *α* = 1 and *α* = −1 is a factor of 3 in diffusivity at *α* is reduced as

### d. Comparison with Klocker et al. (2012)

*β*

^{−2}suppression of transport in (73) is via the mechanism of Ferrari and Nikurashin (2010) and Klocker et al. (2012): nonzero

*β*enables Rossby wave propagation so that eddies drift relative to the mean flow. We have used the anisotropic ring forcing in (20), whereas Klocker et al. force a single wave. To fully explain the connection, we briefly consider the single-wave forcing of Klocker et al. with correlation function

*μc*′ in (2) by

*μ*

_{c}

*c*′. Ferrari and Nikurashin (2010) and Klocker et al. (2012) take

*μ*

_{c}= 0. With

*γ*≠ 0, this change complicates the expression for the diffusivity in (58). But, for comparison with Klocker et al. (2012), we restrict attention to

*γ*= 0. Then there is only a minor modification in the tracer correlation equation [see (55)] and the diffusivity formula in (D11): every 2

*μ*term is just replaced by

*μ*+

*μ*

_{c}. In particular, the kernel

*μ*+

*μ*

_{c})/2

*μ*and the diffusivity then evaluates to

*μ*=

*μ*

_{c}, then the expression above has the same form as the anisotropic ring diffusivity in (70); if

*μ*

_{c}= 0, then the expression in (78) is identical to (20) in Klocker et al. (2012). Further, in the limit of

*β*→ ∞, the general result

*κ*

_{e}∝

*β*

^{−2}in (73) is recovered by using

*U*is the background mean flow in the upper layer of their equivalent barotropic model (the lower layer is quiescent) and

*L*

_{d}is the deformation length. Because the potential vorticity gradient,

*c*

_{R}and therefore suppresses

*κ*

_{e}. By comparison, in (79), our

*c*

_{R}does not depend on a background mean flow. In both models, it is the meridional potential vorticity gradient,

*β*in (79) and

*c*

_{R}=

*c*−

*U*and the associated suppression of

*κ*

_{e}. (Note that the Doppler-shifted phase speed

*c*is the observed zonal speed of eddies, as seen, for example, in satellite altimetry.)

### e. The case β = 0

With *β* = 0, we evaluate the integrals for *κ*_{e} in (58) and (59) numerically. Figure 6a shows *κ*_{e}(*α*, 0, *γ*, *μ*) as a function of *γ*/*μ*. In Fig. 6b, we express the diffusivity in terms of *S* in (64). The three curves are much closer together in Fig. 6b than in Fig. 6a and therefore the variation in *κ*_{e} with *α* and *γ*/*μ* is due mainly to variation in

(a) The nondimensional tracer diffusivity as a function of *γ*/*μ* with *β* = 0 and different values of *α*. (b) The corresponding suppression factor defined in (64).

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The nondimensional tracer diffusivity as a function of *γ*/*μ* with *β* = 0 and different values of *α*. (b) The corresponding suppression factor defined in (64).

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The nondimensional tracer diffusivity as a function of *γ*/*μ* with *β* = 0 and different values of *α*. (b) The corresponding suppression factor defined in (64).

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

The case *α* = −1 in Fig. 6b shows a slight enhancement of *κ*_{e} above *κ*_{υ}. Thus, in some cases at least, shear can enhance eddy diffusivity, so that *S* is slightly greater than 1. This weak effect is due to the Kelvin–Orr mechanism: *α* = −1 loads the forcing variance deep in Farrell and Ioannou (1993)’s favorable sector of the wavenumber plane. The diffusivity in (58)–(60) is given by a weighted time integral of the *υ*′^{2} associated with a sheared wave. Apparently, this time integral is not necessarily bounded above *κ*_{υ} (though it is by 2*κ*_{υ}).

### f. Large shear

*ϕ*=

*π*/2, and the integrals can be evaluated approximately (see appendix E). In this large-shear limit, the eddy diffusivity is

*B*

_{1}is

*κ*

_{e}using (58).

(a) The function *γ*/*μ* = 500 (dashed curve) with the approximate expression in (83) (solid curve).

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The function *γ*/*μ* = 500 (dashed curve) with the approximate expression in (83) (solid curve).

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The function *γ*/*μ* = 500 (dashed curve) with the approximate expression in (83) (solid curve).

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

*S*in the form

*S*∝

*γ*

^{−1}; that is, large shear suppresses eddy diffusivity and in the large-shear limit, the

*γ*

^{−1}dependence is the same as the earlier result for the Reynolds stress in (44). Second, the effect of anisotropy on the diffusivity is completely included in

*S*is independent of

*α*. Limiting forms of the suppression factor in (86) for large and small

*α*= 0), with the asymptotic forms displayed in (87).

(a) The numerically computed nondimensional tracer diffusivity as a function of *γ*/*μ*, with different values of *β*, and *α* = 0. Also plotted are the large *γ* asymptotes for *κ*_{e} for

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The numerically computed nondimensional tracer diffusivity as a function of *γ*/*μ*, with different values of *β*, and *α* = 0. Also plotted are the large *γ* asymptotes for *κ*_{e} for

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

(a) The numerically computed nondimensional tracer diffusivity as a function of *γ*/*μ*, with different values of *β*, and *α* = 0. Also plotted are the large *γ* asymptotes for *κ*_{e} for

Citation: Journal of the Atmospheric Sciences 71, 6; 10.1175/JAS-D-13-0246.1

## 7. Discussion and conclusions

The model (1) and (2) has a special status as an analytically tractable example whose solution sheds light on eddy transport of momentum, vorticity, and tracer. To be sure, the model is linear and, unless one has strong faith in stochastic turbulence models, the results might therefore apply only in the case of weak, externally forced eddies in a strong mean flow. We caution also that the Kelvin–Orr mechanism is quite special to the infinite shear flow *U* = *γy*: at first, a wave “leaning into the shear” gains energy from the mean. Ultimately, the energy is returned as the shear tilts the wave into the unfavorable quadrant; that is, *U* has no discrete shear modes that serve as a repository for eddy energy. The next step is to consider the eddy diffusivity and Reynolds stresses of more structured shear flows.

Key results for (1) and (2) detailed in this paper emphasize the dependence of the statistical properties of the solutions of the linear vorticity equation [see (1)] and the scalar equation [see (2)] on the spatial structure of the forcing *ξ* and the shear *γ*. However, the role of *β* is peculiar: a great and unexpected simplification is that the eddy kinetic energy level and the Reynolds stress *σ* are independent of *β*. But *σ* is a nonlinear and nonmonotonic function of the *γ*. Thus, while it is sensible to define an eddy diffusivity according to (4), one cannot define an analogous eddy viscosity because *σ* is not linearly proportional to *γ*. Thus, our result for *σ* in (41) provides an explicit analytic example of Dritschel and McIntyre (2008)’s “antifriction” (as opposed to negative eddy viscosity).

The spatial structure of *ξ* is characterized by the anisotropy parameter *α* in (19). The Reynolds stress is found to be directly proportional to *α*, so “frictional” and “antifrictional” stresses are obtained when *α* is negative and positive, respectively. And if the forcing is isotropic, then the Reynolds stress is identically zero. When *γ* is weak, the Reynolds stress is proportional to *γ*. Thus, in this special case, one can identify an effective viscosity *ν*_{e} whose sign is opposite to that of *α*. The expression for *ν*_{e} in (43) connects with a similar result obtained by Bakas and Ioannou (2013) for a forcing function resembling our *α* = 1: in this case *ν*_{e} < 0.

In general, the most important determinants of the tracer eddy diffusivity *κ*_{e} are the meridional kinetic energy *μ*. With *β* = *γ* = 0, the diffusivity is precisely *κ*_{e} is smaller than *β* or *γ* are nonzero. In other words, both *γ* and *β* suppress eddy diffusivity. If *γ* = 0, then the suppression due to *β* is a consequence of propagation of Rossby waves relative to a background mean flow. The suppression of diffusivity due to *β* (or more generally, any background potential vorticity gradient) has been discussed previously by Klocker et al. (2012), and their results can be interpreted as a special case of ours with *γ* = 0. Strong shear also causes the diffusivity to decrease as *γ*^{−1}, and this inverse proportionality mirrors the *γ*^{−1} variation of the Reynolds stress for large *γ*.

We caution against summarizing the results above by saying that “mean flow suppresses eddy diffusivity.” The mean flow is *γy* and “mean-flow suppression” invites the incorrect conclusion that *κ*_{e} would decrease as |*y*| increases. Instead, fundamentally because of the Galilean invariance in section 3, *κ*_{e} is independent of *y*. The mean-flow suppression explained in Klocker et al. (2012) and Ferrari and Nikurashin (2010) is caused by the relative motion of eddies with respect to the mean flow. However, this relative motion is due to a nonzero potential vorticity gradient, which in the case of Klocker et al. (2012) includes both *β* and a term resulting from the baroclinic shear of the mean flow. If a barotropic mean flow *U*(*y*) has *U*_{yy} ≠ 0, then the background potential vorticity gradient is modified to *β* − *U*_{yy}, and it is this total gradient (rather than just *β*) that is relevant for eddy suppression. Thus, it is not the mean flow directly, but rather the contribution of the mean flow to the PV gradient that results in suppression of diffusivity.

*β*plane. Our results pile on more: although the passive scalar eddy diffusivity is nonzero, the vorticity flux on the left-hand side of (89) is zero. Moreover, in general agreement with Prandtl’s views, there is a nonzero momentum flux that is, painfully for Taylor, independent of the mean potential vorticity gradient

*β*. Thus, in the model solved here,

*β*is an important control on passive scalar transport, but it is irrelevant for momentum transport.

## Acknowledgments

This work was supported by the National Science Foundation under Award OCE1057838. The authors thank Michael McIntyre and Ryan Abernathey for useful discussions.

## APPENDIX A

### A Bound on Eddy Diffusivity

*p*+

*q*+

*r*= 1. Completing the square involving

*c*′, assuming that

*p*< 0, and then dropping the squared term (which has the same sign as

*p*) gives the inequality

*κ*

_{υ}and

*κ*

_{ζ}are defined in (33) and (35). Minimizing the right-hand side of (A2) over

*q*and

*r*, we find

*p*= −1. The smallest value of the right-hand side of (A2) produces the best upper bound on

*κ*

_{e}, which is the result in (36).

## APPENDIX B

### Details of the Solution

#### a. A polar representation of

*p*=

*k*cos

*ϕ*and

*q*=

*k*sin

*ϕ*. The anisotropic ring forcing in (12) has this form. Because of the exchange symmetry [see (11)], only even terms appear within the sum on the right-hand side of (B4). And because of the assumed reflexion symmetry in (15) there are no sin2

*nϕ*terms in (B4).

#### b. The Reynolds stress

*ϕ*integrals using (B16) below, one finds

*T*

_{n}is the Chebyshev polynomial of order

*n*.

When the forcing *n* = 1 term in (B8) is nonzero, and the *k* integral is trivial. The expression for *n* = 1.

Notice that the isotropic part of the spectrum [i.e.,

#### c. Anisotropy

*J*

_{2}(

*p*,

*q*), we compute

*B*is defined in (B20). Using

#### d. Two angular integrals

*A*

_{n}(

*t*) can be evaluated using the method of residues:

*n*≥ 1, real and imaginary parts of (B15) are separated as

*U*

_{n−1}is the modified Chebyshev polynomial.

## APPENDIX C

### Properties of *F*_{1} and *F*_{2}

*F*

*F*

*F*

_{1}in (42) and

*F*

_{2}in (51) can be written compactly in terms of the exponential integral

*m*, in the main text we use the more natural nondimensional group

*γ*/

*μ*.) We record some useful approximations. If

*γ*/

*μ*→ ∞ then

## APPENDIX D

### Details of the Solution for *κ*_{e}

*H*, the Fourier transform of (55) is

*p*and

*q*. The ensuing triple integral is disentangled by changing variables in the wavenumber integrals from (

*p*,

*q*) to

*χ*is evaluated explicitly in (60). The kernel in (D8) has the symmetry

## APPENDIX E

### Tracer Diffusivity in the Limit *γ***/***μ* **→ ∞**

*p*and

*q*integrals in (58) over the right half plane

*p*> 0, and then multiply by 2. In polar coordinates, we therefore limit attention to −

*π*/2 <

*ϕ*<

*π*/2, so that the arctan(

*q*/

*p*) =

*ϕ*. As

*γ*/

*μ*→ ∞ and

*ϕ*=

*π*/2. Indeed, in the distinguished limit

*γ*/

*μ*→ ∞, with

*γ*becomes large, the arctangent above approaches a discontinuous step function with a jump at

*t*=

*t*

_{*}. In this limit, the function cos

*χ*(

*t*) in (59) is constant on either side of the jump at

*t*

_{*}. This observation enables one to easily perform the integral in (60) with the result

*O*(

*γ*

^{−1}).

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^{1}

Note that (38) can be written identically as

^{2}

In the solution in appendix B, the sheared wavenumber is