Formation of 2d and 3d Circulation Responding to Unsteady Wave Forcing in the Middle Atmosphere

Lagrangian-mean meridional circulation in the middle atmosphere is important for the earth climate because it globally transports minor species such as ozone and changes the temperature structure through adiabatic heating/cooling associated with its vertical branches. This meridional circulation is mainly driven by remote redistribution of the momentum by atmospheric waves. In previous studies such as Haynes et al. (1991), a steady-state assumption has been frequently used for the analysis of the wave-induced meridional circulation. In general, however, the wave forcing is not constant. Thus, the induced circulation can vary in time. When a stratospheric sudden warming occurs, for example, wave forcing time scale may be so short that behaviors of the resultant circulation differ from those expected under the steady-state assumption, as was discussed through a numerical model (Matsuno 1971) and through satellite observations (Labitzke 1972). The purpose of this study is to theoretically examine the response of meridional circulation to unsteady wave forcing. So as to examine and understand such transient and initial behaviors of the circulation, primitive equations are used, allowing us to treat time-evolution of not only a sow variable (linearized PV) describing balanced flow but also two fast variables (horizontal divergence and ageostrophic vorticity) describing gravity waves (GW) and slaved ageostrophic flow. First, the response to the zonal-mean unsteady wave forcing is examined. In the second part, the three-dimensional response to a zonally-nonuniform and unsteady forcing is examined using a balance equation which is derived in this study. As we are interested in large-scale atmospheric responses to forcing, it can be assumed that the Rossby number is sufficiently small and response to the response can be described as a linear response. Thus, Green’s function, which is a response to the delta function, is used to analytically obtain the evolution of meridional circulation. Responses to a wave forcing in the zonal momentum equation are examined in detail, while responses to a diabatic heating in the thermodynamic equation and to wave forcing in the meridional momentum equation are briefly discussed. In the first part of this study, the zonally-mean response is examined using two-dimensional equations. The solution of meridional circulation responding to the steady forcing restricted to a finite latitude and height region takes a form of composed of two vertically aligned cells (Figure 1). For forcing taking the shape of a step function in time, large-scale gravity waves with broader range of frequencies are radiated and the quasi-steady meridional circulation finally remains (Figure 2). The quasi-steady meridional circulation patterns accord well with the steady state solution for a steady forcing. Note that this steady state corresponds to the initial response of the DC solution. Thus, the steady state response in this study is different from the Downward Control (DC) principle. The transient process to the steady state examined in this study is regarded as the rapid change to the balanced circulation in the real atmosphere, which process is important when the time scale of the wave forcing is small. The time scale needed for meridional circulation formations depends on the aspect ratio (i.e., latitudinal to vertical lengths) of the wave forcing, as is consistent with a theoretical expectation of the dimensional analysis. In addition, we found that the group emitted of emitted GW determines the time scale of circulation formation. A case in which the magnitude of forcing gradually change over time is also examined. When the forcing time scale is longer than the inertial period, the response does not include GW radiation but is composed only of meridional circulation that slowly changes following time-varying forcing. The distribution ratio of the wave forcing to zonal-wind acceleration and Coriolis torque is also investigated. Assuming a sinusoidal shape for the wave forcing, this distribution is completely determined by the aspect ratio of the wave forcing. However, in general, a wave forcing is composed of a wide range of spatial wavelengths. Thus, we investigate the distribution for the spatially isolated wave forcing quantitatively using Green’s function method. In the second part of this study, the response to the three-dimensional forcing is examined. In this case, it is expected that Rossby waves are radiated as a transient response because of the beta effect. So as to focus only on the Rossby wave response, governing equations are derived following the method of balance equations used by Leith (1980). For the steady forcing case with the beta effect, the geostrophic flow becomes zonally asymmetric and has large magnitudes to the west of the forcing (Figure 3). For the step-function forcing, Rossby waves are radiated as a transient response as expected. Rossby waves having smaller zonal wavenumbers radiated earlier from the forcing region (Figure 4). Time period needed to reach the steady state strongly depends on the strength of linear relaxation. From these theoretical investigations, the following picture is obtained for the formation of the circulation by the unsteady forcing. For the wave forcing with time scales faster than the inertial period, the fast variable contains radiating gravity waves as an earlier transient response, and reaches the steady state with a two-celled structure in the vertical in a time scale of the energy propagation of gravity waves. The conventionally-used steady state assumption is appropriate only when the forcing time scale is slower than the inertial period. On the other hand, the time scale of the slow variable is much longer than that of the fast variables and is determined by the strength of linear relaxation. As a transient response of the slow variable, Rossby waves are radiated from the zonally-nonuniform forcing and the circulation changes slowly depending on the linear relaxation.