On the Three Dimensional Mass-Weighted Isentropic Time Mean Equation for Rossby Waves

The mass-weighted isentropic zonal mean (Z-MIM) equations derived by T. Iwasaki are powerful tools for diagnosing meridional circulation and wave-mean interaction, especially for the lower boundary and unstable waves. Recently, some studies have extended the equations to three dimensions by using the time mean instead of the zonal mean. However, the relation between wave activity flux and residual mean flow (not mass-weighed mean flow) is unclear. In the present study, we derive the threedimensional (3D) wave activity flux and residual mean flow for Rossby waves on the mass-weighted isentropic time mean equations. Next, we discuss the relation between the obtained formulae and 3D transformed Eulerian-mean (TEM) equations. (Citation: Kinoshita, T., K. Takaya, and T. Iwasaki, 2019: On the three dimensional mass-weighted isentropic time mean equation for Rossby waves. SOLA, 15, 193−197, doi:10.2151/sola. 2019-035.)


Introduction
Transformed Eulerian-mean (TEM) equations are widely used to diagnose wave propagation and wave-mean interaction in meridional cross-section (e.g., Andrews andMcIntyre 1976, 1978). However, TEM equations [including three dimensional (3D) TEM equations derived by many studies] have limitations in terms of expressing the lower boundary (Tanaka et al. 2004) and unstable waves, such as baroclinic instability waves (Noda 2014). The mass-weighted isentropic zonal mean (Z-MIM) equations derived by Iwasaki (1989Iwasaki ( , 1990 are equations that overcome these limitations. Recently, Kinoshita et al. (2016) extended wave activity flux on Z-MIM equations to 3D, and Kanno and Iwasaki (2018) derived mass-weighted isentropic time mean (T-MIM) meridional circulation. However, the relation between 3D wave activity flux and T-MIM meridional circulation has not been fully understood. Also, the difference between 3D-TEM and T-MIM equations has not been investigated.
As a challenge to understand the 3D structure of wave-mean interaction by using the T-MIM framework, the purpose of the study is to derive the 3D residual mean flow and wave activity flux for Rossby waves on T-MIM equations. The derived 3D residual mean flow and wave activity flux are compared with those derived in the 3D-TEM equations (Plumb 1986). In Section 2, the 3D wave activity flux and residual mean flow are derived by using a quasi-geostrophic potential vorticity equation on T-MIM equations. The comparison between the derived formulae and those derived in the 3D-TEM equations is shown in Section 3. Conclusions are given in Section 4.

Quasi-geostrophic potential vorticity equation
To derive the 3D wave activity flux and corresponding residual mean flow for Rossby waves on T-MIM equations, a quasigeostrophic potential vorticity equation on the T-MIM framework is derived. First, we assume that nonconservative terms including diabatic heating are negligible. The quasi-geostrophic potential vorticity equation for isentropic coordinates was derived by Berrisford et al. (1993), which is written as follows; where u and v are the zonal and meridional geostrophic flows, respectively, f 0 is a constant Coriolis parameter, β is a beta effect, M is a Montgomery stream function, ψ is a stream function, Ñ is a horizontal partial differential operator on the isentropic surface, p 0 (θ) is a reference pressure, ρ 0 is a reference density, θ is a potential temperature. The suffixes x, y, and θ denote respective partial derivatives on zonal, meridional, and vertical directions. Here, we define the mass-weighted time mean as ¯ * and its deviation as ¢. From the difference between quasi-potential vorticity equation (1) and that of mass-weighted time mean, the quasi-geostrophic potential vorticity equation for perturbation is written as follows.
where N 2 (θ) is the static stability, and we assume that second derivatives of reference variables are negligibly small. Note that the following equation shows the definition of N 2 (θ) is the same as that of Kinoshita et al. (2016).
where p S is a surface pressure, R is a gas constant for dry air, κ º R /C p and C p is a specific heat at a constant pressure.

3D wave activity flux
In this section, we derive the 3D wave activity flux by using the potential enstrophy equation in the T-MIM framework. Taking q¢ ´ (2) and using the mass-weight time mean yields Then, we obtain the relation Thus, the derived 3D wave activity flux can describe Rossby wave propagation.

3D residual mean flow
Next, the residual mean flow in T-MIM equations is derived by using the obtained 3D wave activity flux. The 3D wave activity flux (6) is substituted into the zonal and meridional momentum equations of the T-MIM framework where u a º u du and v a º v dv are zonal and meridional ageostrophic flows, respectively, and u d and v d are zonal and meridional flows, respectively. The obtained flow is similar to the 3D residual mean flow derived by Plumb (1986). Note that the obtained flow corresponds to 3D residual mean flow in the T-MIM framework, which is shown in Appendix. However, the residual flow is different from the T-MIM flow of Kanno and Iwasaki (2018). Now, we examine this difference in more detail. When the deviation from the unweighted isentropic time-mean state is defined as ², the T-MIM flow is rewritten Here, we use (1) and the relation When the pseudo momentum for Rossby waves is defined as A º A q q ≡ ′ ∇ σ 2 2 * * , where σ º -g -1 p θ , p is the pressure and g is the magnitude of the gravity acceleration, (4) is rewritten as; When we use the assumptions that the derivative of time-mean state has an amplitude O (α), which is slowly varying time-mean state, and horizontal divergence of horizontal geostrophic flows for perturbations is the second order of perturbation and negligibly small (u¢ x + v¢ y = O (α 2 )), the third term of the left-hand side of (5) is rewritten as Note that the term M R is rewritten as follows by using the assumption that the T-MIM basic state satisfies u * This 3D wave activity flux is similar to that of Plumb (1986) and is equal to the product of the pseudo momentum and group velocity. The latter is shown in the following calculations. For a perturbation, the form of a plane wave is considered: where k, l and m are the zonal, meridional and vertical wavenumbers, respectively, and ω is the ground-based angular frequency.
Substituting (8) The intrinsic group velocity is written as where K = (k, l, f 0 N (θ) -1 m) T is the wavenumber vector. Next, the pseudo momentum and 3D wave activity flux M R are written in terms of ψ¢ as 195 SOLA, 2019, Vol. 15, 193−197, doi:10.2151 Note that ¢¢ ¢¢ u σ σ is a temporal correlation of mass-weight perturbations and velocity perturbations, which indicates the eddycorrelated mass transport velocity and is the so-called bolus velocity (e.g., Rhines 1982;Gent et al. 1995;Lee et al. 1997;Mc Dougall and McIntosh 1996;McDougall and McIntosh 2001;Aiki et al. 2015;Kanno and Iwasaki 2018). When we use the assumption that the difference between ψ¢ and ψ² is the second order of perturbation and negligibly small (

Relation between T-MIM and 3D-TEM equations
To examine the relation between 3D wave activity flux and residual mean flow for Rossby waves in TEM equations and those derived in this study, we perform the following calculations. First, the term σ can be rewritten as follows when σ depends on only the vertical axis of the log-pressure height coordinate.
where z * and z are the geometric and log-pressure height, respectively, ρ 0 is the basic density, and θ 0 is the reference potential temperature. By referencing Iwasaki (1992) and using relation (16), the zonal component of 3D wave activity flux in T-MIM equations (7) is rewritten as where __ (z) is the time mean on the log-pressure height coordinate and † is its deviation. Note that the perturbation difference between A¢ and A † is −( and p θ » p 0θ are used in the first line. N 2 º (g/θ 0 ) θ 0z is the static stability in the logpressure height coordinate. Thus, 3D wave activity flux in T-MIM equations corresponds to that in 3D-TEM equations derived by Plumb (1986) although the horizontal component of 3D wave activity flux must be multiplied by θ 0z . By using these calculations, the 3D residual mean flow in T-MIM equations is rewritten as follows

Summary and discussions
In this study, we formulated 3D wave activity flux and residual mean flow for Rossby waves on T-MIM equations. By examining the relation between 3D wave activity flux and residual mean flow in 3D-TEM equations and those derived in this study, it was found that 3D wave activity flux and residual mean flow in T-MIM equations correspond to those derived by Plumb (1986). Comparison between 3D residual mean flow and T-MIM velocity (Kanno and Iwasaki 2018) suggested that the horizontal component of bolus velocity is different from that of QG-Stokes correction. Note that 3D wave activity flux of this study agrees with that of Kinoshita et al. (2016), which is derived by using a unified dispersion relation and polarization relation, when the quasi-