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## Abstract

If the momentum, energy and circulation of a fluid in a periodic, quasi-geostrophic, *β*-plane channel are specified, then there is a minimum enstrophy implied. This minimum enstrophy flow is obtained using the calculus of variations and is found to be also a solution of the quasi-geostrophic equations. It is either a parallel flow or a finite-amplitude Rossby wave, depending on the aspect ratio of the channel and the amount of energy and momentum within it. The most geophysically relevant case is a channel whose zonal length is substantially greater than its meridional breadth. In this instance the form of the minimum enstrophy solution is decided by the ratio of the energy to the squared momentum. When this parameter is below a Critical value one has a parallel flow, while if this value is exceeded, the minimum enstrophy, solution is a Rossby wave.

Heuristic arguments based on the enstrophy cascade in two-dimensional turbulence suggest a “selective decay hypothesis”. This is that scale-selective dissipation will decrease the enstrophy more rapidly than the energy, momentum and circulation. If this is the case, then the system should approach the minimum enstrophy solution.

## Abstract

If the momentum, energy and circulation of a fluid in a periodic, quasi-geostrophic, *β*-plane channel are specified, then there is a minimum enstrophy implied. This minimum enstrophy flow is obtained using the calculus of variations and is found to be also a solution of the quasi-geostrophic equations. It is either a parallel flow or a finite-amplitude Rossby wave, depending on the aspect ratio of the channel and the amount of energy and momentum within it. The most geophysically relevant case is a channel whose zonal length is substantially greater than its meridional breadth. In this instance the form of the minimum enstrophy solution is decided by the ratio of the energy to the squared momentum. When this parameter is below a Critical value one has a parallel flow, while if this value is exceeded, the minimum enstrophy, solution is a Rossby wave.

Heuristic arguments based on the enstrophy cascade in two-dimensional turbulence suggest a “selective decay hypothesis”. This is that scale-selective dissipation will decrease the enstrophy more rapidly than the energy, momentum and circulation. If this is the case, then the system should approach the minimum enstrophy solution.

## Abstract

Zonostrophic instability leads to the spontaneous emergence of zonal jets on a *β* plane from a jetless basic-state flow that is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal mean and eddy equations, and neglecting the eddy–eddy interactions, defines the quasilinear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system.

Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. The authors characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of 6 smaller than that for QL zonostrophic instability.

Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors until the flow matures into a state with jets that are significantly broader than the linear prediction but have spacing similar to NL jets.

## Abstract

Zonostrophic instability leads to the spontaneous emergence of zonal jets on a *β* plane from a jetless basic-state flow that is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal mean and eddy equations, and neglecting the eddy–eddy interactions, defines the quasilinear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system.

Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. The authors characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of 6 smaller than that for QL zonostrophic instability.

Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors until the flow matures into a state with jets that are significantly broader than the linear prediction but have spacing similar to NL jets.

## Abstract

The Reynolds stress induced by anisotropically forcing an unbounded Couette flow, with uniform shear *γ*, on a *β* plane, is calculated in conjunction with the eddy diffusivity of a coevolving passive tracer. The flow is damped by linear drag on a time scale *μ*
^{−1}. The stochastic forcing is white noise in time and its spatial anisotropy is controlled by a parameter *α* that characterizes whether eddies are elongated along the zonal direction (*α* < 0), are elongated along the meridional direction (*α* > 0), or are isotropic (*α* = 0). The Reynolds stress varies linearly with *α* and nonlinearly and nonmonotonically with *γ*, but the Reynolds stress is independent of *β*. For positive values of *α*, the Reynolds stress displays an “antifrictional” effect (energy is transferred from the eddies to the mean flow); for negative values of *α*, it displays a frictional effect. When *γ*/*μ* ≪ 1, these transfers can be identified as negative and positive eddy viscosities, respectively. With *γ* = *β* = 0, the meridional tracer eddy diffusivity is *υ*′ is the meridional eddy velocity. In general, nonzero *β* and *γ* suppress the eddy diffusivity below *γ* varies as *γ*
^{−1} while the suppression due to *β* varies between *β*
^{−1} and *β*
^{−2} depending on whether the shear is strong or weak, respectively.

## Abstract

The Reynolds stress induced by anisotropically forcing an unbounded Couette flow, with uniform shear *γ*, on a *β* plane, is calculated in conjunction with the eddy diffusivity of a coevolving passive tracer. The flow is damped by linear drag on a time scale *μ*
^{−1}. The stochastic forcing is white noise in time and its spatial anisotropy is controlled by a parameter *α* that characterizes whether eddies are elongated along the zonal direction (*α* < 0), are elongated along the meridional direction (*α* > 0), or are isotropic (*α* = 0). The Reynolds stress varies linearly with *α* and nonlinearly and nonmonotonically with *γ*, but the Reynolds stress is independent of *β*. For positive values of *α*, the Reynolds stress displays an “antifrictional” effect (energy is transferred from the eddies to the mean flow); for negative values of *α*, it displays a frictional effect. When *γ*/*μ* ≪ 1, these transfers can be identified as negative and positive eddy viscosities, respectively. With *γ* = *β* = 0, the meridional tracer eddy diffusivity is *υ*′ is the meridional eddy velocity. In general, nonzero *β* and *γ* suppress the eddy diffusivity below *γ* varies as *γ*
^{−1} while the suppression due to *β* varies between *β*
^{−1} and *β*
^{−2} depending on whether the shear is strong or weak, respectively.

## Abstract

The authors study the stability of a barotropic sinusoidal meridional flow on a *β* plane. Because of bottom drag and lateral viscosity, the system is dissipative and forcing maintains a basic-state velocity that carries fluid across the planetary vorticity contours; this is a simple model of forced potential vorticity mixing. When the Reynolds number is slightly above the stability threshold, a perturbation expansion can be used to obtain an amplitude equation for the most unstable disturbances. These instabilities are zonal flows with a much larger length scale than that of the basic state.

Numerical and analytic considerations show that random initial perturbations rapidly reorganize into a set of fast and narrow eastward jets separated by slower and broader regions of westward flow. There then follows a much slower adjustment of the jets, involving gradual meridional migration and merger. Because of the existence of a Lyapunov functional for the dynamics, this one-dimensional inverse cascade ultimately settles into a steady solution.

For a fixed *β,* the meridional separation of the eastward jets depends on the bottom drag. When the bottom drag is zero, the process of jet merger proceeds very slowly to completion until only one jet is left in the domain. For small bottom drag, the steady-state meridional separation of the jets varies as (bottom drag)^{−1/3}. Varying the nondimensional *β* parameter can change the instability from supercritical (when *β* is small) to subcritical (when *β* is larger). Thus, the system has a rich phenomenology involving multiple stable solutions, hysteritic transitions, and so on.

## Abstract

The authors study the stability of a barotropic sinusoidal meridional flow on a *β* plane. Because of bottom drag and lateral viscosity, the system is dissipative and forcing maintains a basic-state velocity that carries fluid across the planetary vorticity contours; this is a simple model of forced potential vorticity mixing. When the Reynolds number is slightly above the stability threshold, a perturbation expansion can be used to obtain an amplitude equation for the most unstable disturbances. These instabilities are zonal flows with a much larger length scale than that of the basic state.

Numerical and analytic considerations show that random initial perturbations rapidly reorganize into a set of fast and narrow eastward jets separated by slower and broader regions of westward flow. There then follows a much slower adjustment of the jets, involving gradual meridional migration and merger. Because of the existence of a Lyapunov functional for the dynamics, this one-dimensional inverse cascade ultimately settles into a steady solution.

For a fixed *β,* the meridional separation of the eastward jets depends on the bottom drag. When the bottom drag is zero, the process of jet merger proceeds very slowly to completion until only one jet is left in the domain. For small bottom drag, the steady-state meridional separation of the jets varies as (bottom drag)^{−1/3}. Varying the nondimensional *β* parameter can change the instability from supercritical (when *β* is small) to subcritical (when *β* is larger). Thus, the system has a rich phenomenology involving multiple stable solutions, hysteritic transitions, and so on.

## Abstract

Observations from a wide variety of instruments and platforms are used to validate many different aspects of a three-dimensional mesoscale simulation of the dynamics, cloud microphysics, and radiative transfer of a cirrus cloud system observed on 26 November 1991 during the second cirrus field program of the First International Satellite Cloud Climatology Program (ISCCP) Regional Experiment (FIRE-II) located in southeastern Kansas. The simulation was made with a mesoscale dynamical model utilizing a simplified bulk water cloud scheme and a spectral model of radiative transfer. Expressions for cirrus optical properties for solar and infrared wavelength intervals as functions of ice water content and effective particle radius are modified for the midlatitude cirrus observed during FIRE-II and are shown to compare favorably with explicit size-resolving calculations of the optical properties. Rawinsonde, Raman lidar, and satellite data are evaluated and combined to produce a time–height cross section of humidity at the central FIRE-II site for model verification. Due to the wide spacing of rawinsondes and their infrequent release, important moisture features go undetected and are absent in the conventional analyses. The upper-tropospheric humidities used for the initial conditions were generally less than 50% of those inferred from satellite data, yet over the course of a 24-h simulation the model produced a distribution that closely resembles the large-scale features of the satellite analysis. The simulated distribution and concentration of ice compares favorably with data from radar, lidar, satellite, and aircraft. Direct comparison is made between the radiative transfer simulation and data from broadband and spectral sensors and inferred quantities such as cloud albedo, optical depth, and top-of-the-atmosphere 11-µm brightness temperature, and the 6.7-µm brightness temperature. Comparison is also made with theoretical heating rates calculated using the rawinsonde data and measured ice water size distributions near the central site. For this case study, and perhaps for most other mesoscale applications, the differences between the observed and simulated radiative quantities are due more to errors in the prediction of ice water content, than to errors in the optical properties or the radiative transfer solution technique.

## Abstract

Observations from a wide variety of instruments and platforms are used to validate many different aspects of a three-dimensional mesoscale simulation of the dynamics, cloud microphysics, and radiative transfer of a cirrus cloud system observed on 26 November 1991 during the second cirrus field program of the First International Satellite Cloud Climatology Program (ISCCP) Regional Experiment (FIRE-II) located in southeastern Kansas. The simulation was made with a mesoscale dynamical model utilizing a simplified bulk water cloud scheme and a spectral model of radiative transfer. Expressions for cirrus optical properties for solar and infrared wavelength intervals as functions of ice water content and effective particle radius are modified for the midlatitude cirrus observed during FIRE-II and are shown to compare favorably with explicit size-resolving calculations of the optical properties. Rawinsonde, Raman lidar, and satellite data are evaluated and combined to produce a time–height cross section of humidity at the central FIRE-II site for model verification. Due to the wide spacing of rawinsondes and their infrequent release, important moisture features go undetected and are absent in the conventional analyses. The upper-tropospheric humidities used for the initial conditions were generally less than 50% of those inferred from satellite data, yet over the course of a 24-h simulation the model produced a distribution that closely resembles the large-scale features of the satellite analysis. The simulated distribution and concentration of ice compares favorably with data from radar, lidar, satellite, and aircraft. Direct comparison is made between the radiative transfer simulation and data from broadband and spectral sensors and inferred quantities such as cloud albedo, optical depth, and top-of-the-atmosphere 11-µm brightness temperature, and the 6.7-µm brightness temperature. Comparison is also made with theoretical heating rates calculated using the rawinsonde data and measured ice water size distributions near the central site. For this case study, and perhaps for most other mesoscale applications, the differences between the observed and simulated radiative quantities are due more to errors in the prediction of ice water content, than to errors in the optical properties or the radiative transfer solution technique.