Non-modal baroclinic instability

A class of non-linear instabilities of a vertically sheared zonal flow is discussed. This is a type of baroclinic instability that lies outside the purview of a linear eigenmode analysis of baroclinic instability problems. The form taken by the instability is that of an ensemble of three neutral Rossby waves whose amplitudes are slowly modified by their mutual non-linear interactions. For a triad of small amplitude, these interactions introduce a weak, vertical variation of phase to the structure of the individual waves. This allows the generation of rectified heat fluxes and an exchange of energy with the mean flow.This instability exhibits explosive growth and spans a range of horizontal wavenumbers that exceeds the range that is unstable in the corresponding linear model. It is shown that the type of instability discussed can only occur when the model used admits unstable eigenmodes as well as neutral Rossby waves.The mechanism for the non-linear instability discussed here is believed to be fairly general and should exist also in the context of a horizontally sheared flow where it would take the form of a barotropic instability.     

An experimental study of nonlinear baroclinic instability and mode selection in a large basin

This paper reports on two-layer rotating liquid experiments designed to study the behavior of non-linear baroclinic waves under conditions where the Rossby radius of deformation R_d is much smaller than the geometric length scale L imposed by the size of the laboratory apparatus. The apparatus is constructed to consistently simulate f-plane dynamics. When F = L²/R_d² > > 1, it is found that the unstable waves first encountered as friction is decreased have high frequencies, in accord with linear theory. As the friction parameter $\text{Q = 0.7 E}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/R}{}_0$ (where E is the Ekman number and R₀ the Rossby number) is further decreased into the non-linear region, singlewave amplitude vacillation is observed. Generally, as Q decreases lower frequencies (and low wavenumbers) dominate the response, which ultimately becomes turbulent at values of Q of the order 0.1. This is contrary to the result expected from an extrapolation of linear theory. Further observations show that the finite-amplitude state is not unique: multi-equilibria are possible depending on the initial conditions.     

发展方程的非线性计算不稳定问题

进一步讨论了有关非线性不稳定的一些问题,其主要内容有： １．考察了有代表性的三类发展方程,指出其对应的差分格式是否出现非线性计算不稳定,与原微分方程解的性质密切相关。 ２．进一步讨论了带周期边条件的守恒型差分格式的非线性计算稳定性问题,总结了克服非线性不稳定的有效措施。 ３．以非线性平流方程为例,着重分析了带非周期边条件的非守恒差分格式的非线性计算稳定性问题,给出了判别其计算稳定性的“综合分析判别法”。     

斜压不稳定理论的发展历程分析

斜压不稳定理论是中高纬度地区天气尺度扰动生成和发展的机制,是继长波理论之后大气动力学上的又一重大进展,对现代天气预报具有重要的指导意义。按照斜压不稳定理论发展的时间脉络,阐述了赵九章、查尼和伊迪在斜压不稳定方面开展的研究工作,分析了3位科学家各自的研究特点及历史贡献。赵九章1946年发表在《Journal of Meteorology》上的论文,最早提出了"斜压不稳定"这一概念,给出了不稳定的临界波长,并阐述了在不稳定扰动情况下能量的转换,以及不稳定波对大气环流带来的可能影响,尽管得出的不稳定临界波长与观测差别较大,但其对波-流相互作用的讨论在当时是超前的。查尼于1947年采用滤波和尺度分析等方法,将大气扰动方程简化为一个可以求解的系统,推导出大气稳定状态的判据,建立了斜压不稳定理论,其结果与实况比较接近;并据此把准地转模式成功应用于数值天气预报实践中,促使数值天气预报获得首次成功。1949年,伊迪在查尼研究工作的基础上通过合理的简化方法,得到了更为简洁的模型。最后,通过对比他们的研究思路,重点分析了赵九章未能使得斜压不稳定理论提前一年建立的原因:由于其研究思路始终局限在大气水平运动上,忽略了斜压系统发展中散度项的贡献,因而未能抓住天气系统发生、发展的本质,致使其最终与斜压不稳定理论的成功建立失之交臂,其论文本身存在的一些亮点也因此被后人忽视。      

A note on the temporal evolution of taylor columns over topography

The formation of two regions of vorticity, one cyclonic and one anticyclonic, after the initiation of flow is discussed. In particular, the linearized case of elongated topography is examined. The non-linear interaction of the two vortices, once formed, is discussed using a simple model.     

On the evolution of finite-amplitude disturbance to the barotropic and baroclinic quasigeostrophic flows

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Nonlinear meridional scale interaction in baroclinic instability

Baroclinic instability of a zonal flow with latitudinal structure is examined using a nonlinear quasi-geostrophic, two-level β-plane model. An initially small perturbation with the structure of the linearly most unstable mode is allowed to grow to finite amplitude through nonlinear interaction. Because of latitudinal asymmetries of the basic zonal flow, a spectrum of meridional modes is generated in the perturbation. The time evolution of zonal wind and perturbation meridional structures, and their Fourier meridional mode spectra are examined. The radius of deformation is an important meridional scale in both the zonal flow and perturbation. This is especially true during the barotropic decay phase of the baroclinic wave. Time series of energy conversion terms show there is no energy accumulation.     

Bounds on the eigenvalues of the planetary-scale baroclinic instability problem

In the framework of the linear baroclinic instability problem for planetary geostrophic flows, bounds on the related complex eigenvalues are deduced. The main feature of this result is the independence of these bounds from the latitude and the density stratification, so that it generalizes the results of previous work on the subject.     

Loop Current Eddy formation and baroclinic instability

The formation of three Loop Current Eddies, Ekman, Franklin, and Hadal, during the period April 2009 through November 2011 was observed by an array of moored current meters and bottom mounted pressure equipped inverted echo sounders. The array design, areal extent nominally 89° W to 85° W, 25° N to 27° N with 30–50 km mesoscale resolution, permits quantitative mapping of the regional circulation at all depths. During Loop Current Eddy detachment and formation events, a marked increase in deep eddy kinetic energy occurs coincident with the growth of a large-scale meander along the northern and eastern parts of the Loop Current. Deep eddies develop in a pattern where the deep fields were offset and leading upper meanders consistent with developing baroclinic instability. The interaction between the upper and deep fields is quantified by evaluating the mean eddy potential energy budget. Largest down-gradient heat fluxes are found along the eastern side of the Loop Current. Where strong, the horizontal down-gradient eddy heat flux (baroclinic conversion rate) nearly balances the vertical down-gradient eddy heat flux indicating that eddies extract available potential energy from the mean field and convert eddy potential energy to eddy kinetic energy.     

The effects of latitudinal asymmetries on baroclinic instability

Abstract

Baroclinic instability of zonal flows with different latitudinal structures is examined, using a linear, quasi‐geostrophic two‐level ß‐plane model. The flows have different amounts of skew, with respect to the channel centre, at different vertical levels. The results are interpreted in terms of the instability of the baroclinic components of the zonal flows. Because of the presence of latitudinal asymmetries, a spectrum of meridional modes is generated in the perturbation. In general, the meridional spectrum has two peaks: a primary peak at the planetary basic flow scale, and a secondary peak near the radius of deformation. As neutral stability is approached, the latter scale becomes more important, i.e. there is a tendency for more small‐scale structure near neutral stability. The perturbation zonal scale is close to the radius of deformation. The eddy amplitudes and momentum fluxes are also examined. The case that best applies to the atmosphere is also discussed.     

Barotropic and baroclinic instability in rotating stratified fluids

The linear normal mode instabilities of a parallel shear flow which varies both vertically (z) and meridionally (y) in a quasigeostrophic, rotating, stratified fluid are considered. The β effect (variation of Coriolis parameter with y) is included. Both two-layer and continuous fluids are treated. Attention is concentrated on the types of instability possible for a given shear flow. It is found that the instability can be described adequately by three nondimensional parameters: Λ, the ratio of the horizontal length scale of the shear to the internal deformation radius: δ, which is either the ratio of layer depths in the two-layer fluid or the fractional depth of variation of the stratification in the continuous fluid; and β, suitably nondimensionalized.Asymptotic analyses, confirmed by direct numerical solutions, are performed for conditions in which various parameters become large or small. The β effect is essentially quantitative, whereas Λ and δ define the type of instability as barotropic (if the kinetic energy of the mean flow feeds the growing perturbations), baroclinic (if the available potential energy of the mean flow feeds the perturbations) or mixed (a combination of the two).The case of large Λ (the most relevant for oceanographic applications) is treated in detail. It is shown that y-independent problems have only limited relevance. For a fixed deformation radius and the y scale of the mean flow increasing without limit, the asymptote is not the case of no y variation in the mean flow.     

On the onset of entrainment instability over the ocean

The marine atmospheric boundary layer is characterized by cool temperatures and high humidity. Clouds are observed over most of the oceans. It is generally accepted that these overcast cloud decks break up into scattered fragments due to cloud-top entrainment instability. That is, if the air above the boundary layer is sufficiently cool and dry relative to cloud top, the buoyancy flux will be directed upwards and entrainment can occur freely.A boundary-layer model is used to test the sensitivity of the model atmosphere to the various processes which promote the onset of cloud-top entrainment instability. It is found that the transition from a solid cloud deck to scattered cumulus clouds depends on a rate process. The cloud cover is sensitive to mesoscale variations in sea surface temperature only if the cloud-top inversion is sufficiently weak.     

Dynamics of nonlinear baroclinic Ekman boundary layer

By the geostrophic momentum approximation, the wind structure and vertical motion within the non-linear baroclinic Ekman layer matching with the surface layer are determined. A comparison of the Ekman solution with the classical one is made. It is demonstrated that the contributions of baroclinity, stratification and nonlinear effects to the wind profile within the layer are all of definite importance.     

On the enhanced smoothing over topography in some mesometeorological models

An equation is derived for the components of the horizontal (turbulent) frictional force in the -coordinate system with special attention to mesometeorological flow models. The starting point is the horizontal equation of motion in its flux-form in the -system in which we replace (following Reynolds' procedure) the velocity components u,v and % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbai % aaaaa!37B8! \[ \dot \sigma \] aswell as other relevant quantities by terms of the form u = + u,..., = ± + % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbai % Gbauaaaaa!37C3! \[ \dot \sigma ' \] , etc. ( = time average of u; u = fluctuating part of u.) Next, the equation is averaged with respect to time and terms which we believe are small in mesometeorological flows, are neglected. On expressing by an appropriate expression that involves w, the result shows the appearance of two new terms which, have not been considered previously in the published literature. While the expression earlier used in the literature involved the -derivative of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaace% WG1bGbauaaceWG3bGbauaaaaaaaa!380B!\[\overline {u'w'} \] alone, the new terms add the -derivatives of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaace% WG1bGbauaadaahaaWcbeqaaiaaikdaaaaaaaaa!37EC!\[\overline {u'^2 } \] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaace% WG1bGbauaaceWG2bGbauaaaaaaaa!380A!\[\overline {u'v'} \] for the x-component of the force, and the -derivatives of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaace% WG2bGbauaadaahaaWcbeqaaiaaikdaaaaaaaaa!37ED!\[\overline {v'^2 } \]} and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaace% WG1bGbauaaceWG2bGbauaaaaaaaa!380A!\[\overline {u'v'} \] for the y-component, where and are the slopes of the -surfaces in the x- and y-directions, respectively. Further, a few numerical simulations of the sea-breeze over topography are carried out with and without the correction terms. It is shown that when corrections terms are not included the effective smoothing is stronger above the sloping regions and may amount to as high as 50 percent of the convergence with slopes of ~.04. The ìnclusìon of the new terms does not lead to any special computational difficulties and for that reason there is no compelling reason to neglect them, all the more so because, as is shown, the addition of the new terms results in a consistent apportioning of the degree of horizontal diffusion.On leave from CIMMS, Norman, OK.Now visiting Dept. of Met., Helsinki, Finland.     

Nonlinear spatial baroclinic instability in slowly varying zonal flow

The linear and weakly nonlinear dynamics of long, low frequency, spatially growing baroclinic waves embedded in slowly varying zonal flow on a β-plane channel are examined in a continuous model of the atmosphere. For a basic state jet flow possessing a locally unstable region, the nonlinear solution yields a maximum amplitude that is located near the region of maximum baroclinicity and substantially upstream of the maximum amplitude obtained from linear theory. The difference between the linear and nonlinear solutions is due to the time-averaged wave fluxes becoming large enough in the nonlinear problem to stabilize the flow prior to reaching the location (jet center) where the basic state baroclinicity and locally computed linear growth rate are maximized.     

A note on baroclinic instability of a source-sink driven flow

The stability properties of a source-sink driven flow in a two-layer, rotating fluid is considered. A steady flow in an annulus has been determined previously (Maeland, 1982) by Ekman-layer dynamics at the interface of the two layers; in this paper, the stability will be studied by linear, inviscid theory. For a given configuration, the two parameters which govern the stability of the basic flow are the internal Froude numbers of each layer. The growth rates for the instabilities are calculated by a simple numerical method (matrix method).     

A fluid experiment of large-scale topography effect on baroclinic wave flows

The effects of topography on baroclinic wave flows are studied experimentally in a thermally driven rotating annulus of fluid.Fourier analysis and complex principal component (CPC) analysis of the experimental data show that, due to topographic forcing, the flow is bimodal rather than a single mode. Under suitable imposed experimental parameters, near thermal Rossby number ROT = 0.1 and Taylor number Ta = 2.2 × 107, the large-scale topography produces low-frequency oscillation in the flow and rather long-lived flow pattern resembling blocking in the atmospheric cir-culation. The ‘blocking’ phenomenon is caused by the resonance of travelling waves and the quasi-stationary waves forced by topography.The large-scale topography transforms wavenumber-homogeneous flows into wavenumber-dispersed flows, and the dispersed flows possess lower wavenumbers.     

On the parameterization of drag over small-scale topography in neutrally-stratified boundary-layer flow

Predictions of the surface drag in turbulent boundary-layer flow over two-dimensional sinusoidal topography from various numerical models are compared. For simple 2D terrain, the model results show that the drag increases associated with topography are essentially proportional to (slope)² up to the steepness at which the flow separates. For the purposes of boundary-layer parameterisation within larger-scale models, we propose a representation of the effects of simple 2D topography via an effective roughness length, z ₀ ^eff. The form of the varation of z ₀ ^eff with terrain slope and topographic wavelength is established for small slopes from the model results and a semi-empirical formula is proposed.     

Eddy fluxes and mean flow tendencies in open-ocean baroclinic instability

Linear normal-mode instabilities of a mean flow varying over many internal radii of deformation are considered. Attention is devoted not to the instabilities themselves, but to the second-order effects of such instabilities on the mean flow. These are derived analytically from solutions by Killworth. The effects on symmetric profiles are very different from those on asymmetric profiles. Specifically, the tendencies are much larger in the latter case with the strongest tendency being on the barotropic part of the mean flow; this tendency apparently attempts to make the profile more symmetric.     

On the role of criticality in coastal flows over irregular bottom topography

A simple expansion technique is developed for predicting the response of a horizontally sheared oceanic coastal flow over uniform coastal topography to variations in that bottom topography. The technique is illustrated by application to supercritical and subcritical flows. The response of the supercritical flow is local, while that of the subcritical flow extends downstream to be eventually attenuated by frictional damping. However, the response of both flow types is weak unless the flow is nearly critical.