Rossby Number Expansion Research Articles

Variational balance models for the three-dimensional Euler–Boussinesq equations with full Coriolis force

We derive a semi-geostrophic variational balance model for the three-dimensional Euler–Boussinesq equations on the nontraditional f-plane under the rigid lid approximation. The model is obtained by a small Rossby number expansion in the Hamilton principle, with no other approximations made. We allow for a fully non-hydrostatic flow and do not neglect the horizontal components of the Coriolis parameter; that is, we do not make the so-called “traditional approximation.” The resulting balance models have the same structure as the “L1 balance model” for the primitive equations: a kinematic balance relation, the prognostic equation for the three-dimensional tracer field, and an additional prognostic equation for a scalar field over the two-dimensional horizontal domain, which is linked to the undetermined constant of integration in the thermal wind relation. The balance relation is elliptic under the assumption of stable stratification and sufficiently small fluctuations in all prognostic fields.

A Conservative Model for Nonlinear Dynamics in a Stratified, Rotating Fluid

We present a set of equations describing the nonlinear dynamics of flows constrained by environmental rotation and stratification (Rossby numbers Ro∈[0.1,0.5] and Burger numbers of order unity). The fluid is assumed incompressible, adiabatic, inviscid and in hydrostatic balance. This set of equations is derived from the Navier Stokes equations (with the above properties), using a Rossby number expansion with second order truncation. The resulting model has the following properties: 1) it can represent motions with moderate Rossby numbers and a Burger number of order unity; 2) it filters inertia-gravity waves by assuming that the divergence of horizontal velocity remains small; 3) it is written in terms of a single function of space and time (pressure, generalized streamfunction or Bernoulli function); 4) it conserves total (Ertel) vorticity in a Lagrangian form, and its quadratic norm (potential enstrophy) at the model order in Rossby number; 5) it also conserves total energy at the same order if the work of pressure forces vanishes when integrated over the fluid domain. The layerwise version of the model is finally presented, written in terms of pressure. Integral properties (energy, enstrophy) are conserved by these layerwise equations. The model equations agree with the generalized geostrophy equations in the appropriate parameter regime. Application to vortex dynamics are mentioned.

On the Quantification of Imbalance and Inertia–Gravity Waves Generated in Numerical Simulations of Moist Baroclinic Waves Using the WRF Model

Abstract Quantification of inertia–gravity waves (IGWs) generated by upper-level jet–surface front systems and their parameterization in global models of the atmosphere relies on suitable methods to estimate the strength of IGWs. A harmonic divergence analysis (HDA) that has been previously employed for quantification of IGWs combines wave properties from linear dynamics with a sophisticated statistical analysis to provide such estimates. A question of fundamental importance that arises is how the measures of IGW activity provided by the HDA are related to the measures coming from the wave–vortex decomposition (WVD) methods. The question is addressed by employing the nonlinear balance relations of the first-order δ–γ, the Bolin–Charney, and the first- to third-order Rossby number expansion to carry out WVD. The global kinetic energy of IGWs given by the HDA and WVD are compared in numerical simulations of moist baroclinic waves by the Weather Research and Forecasting (WRF) Model in a channel on the f plane. The estimates of the HDA are found to be 2–3 times smaller than those of the optimal WVD. This is in part due to the absence of a well-defined scale separation between the waves and vortical flows, the IGW estimates by the HDA capturing only the dominant wave packets and with limited scales. It is also shown that the difference between the HDA and WVD estimates is related to the width of the IGW spectrum.

A New Surface Model for β-Plane Turbulence

Abstract This paper introduces a new numerical model for studying wave–turbulence interactions in a continuously stratified rotating flow, having a uniform potential vorticity and a rigid boundary. The meridional variation in the Coriolis parameter (β effect), a channel geometry, and the first-order nonlinear terms in a small Rossby number expansion are included into the surface quasigeostrophic dynamics. The model contains important dynamical characteristics of three-dimensional flows such as advection by ageostrophic winds, and stretching and tilting of relative vorticity. Nevertheless, it has the computational economy of two-dimensional flows. Long-term direct numerical simulations are performed for decaying turbulence arising from random initial conditions. In addition to the formation of steady zonal jets, frequently reported as a possible end state under the β effect, this model exhibits several other realistic physical effects that were lacking in the previously studied β-plane models for wave–turbulence interactions. There is a significant contrast in the spatial and time scales of the formed eddies, resulting in high meridional asymmetry of the flow. Mean surface cooling and persistent large-scale blocking eddies are observed as well. The surface potential temperature variance spectrum exhibits a well-resolved k−5/3 inertial range.

Simplification of potential vorcticity and mesoscale quasi-balanced dynamics model

The physical characteristics of mesoscale are analyzed, and results show that the unbalanced forced motion is the fundamental cause, which leads to the evolution of some important mesoscale weather systems. In this paper, an alternative asymptotic expansion method, which is quite different from the conventional Rossby-number expansion, is used to simplify the potential vorticity equation. And the quasi-balanced (QB) model based on nonlinear balance equation is derived. The QB model, which is in analogy with the quasi-geostrophic model, can describe the fundamental characteristics of the mesoscale accurately and may be used as the basis of theoretical studies on the mesoscale atmospheric dynamics.

Nonlinear balance and quasi‐geostrophic sets

It is argued that nonlinear balancing schemes obtained by Rossby number expansions usually diverge due to the inevitable presence of inertial‐gravity wave activity. This divergence undermines the theoretical basis of the slow or invariant manifold of Leith (1980) and Lorenz (1980). An alternate definition of balanced motion is proposed.

Slaving Principles, Balanced Dynamics, and the Hydrostatic Boussinesq Equations

The problem of finding higher-order generalizations of the quasigeostrophic and semigeostrophic models, that is, higher-order balanced models, is investigated systematically by using a slaving principle. It is shown that most existing balanced models are based on an underlying slave equation with varying master variables and that this slaving approach arises as an alternative to the secular conventional Rossby number expansions. While in the conventional expansions all variables are expanded in a power series in the Rossby number, the slaving approach suggests modified expansion or iteration procedures in which only the fast variables (associated with gravity modes) are expanded or iterated. The slow variables (associated with vortical modes) are now the master variables in the system; a well-known choice for large-scale extratropical flows in geophysical fluid dynamics is potential vorticity. To illustrate both the iteration and the modified expansion procedures, the slaving approach is applied to the rapidly rotating hydrostatic Boussinesq equations in a horizontally semi-infinite domain by using a Rossby number scaling. Higher-order generalizations of the quasigeostrophic equations result from modified expansions, and higher-order generalizations of the semigeostrophic equations result from nonlinear iterations. The validity of the latter for small Rossby numbers as well as small frontal parameters exemplifies the advantage nonlinear iterations have.

Extended-geostrophic Hamiltonian models for rotating shallow water motion

By using a small Rossby number expansion in Hamilton's principle for shallow water dynamics in a rapidly rotating reference frame, we derive new approximate extended-geostrophic equations with pressure, or surface height, as the dynamical variable. By introducing modeling parameters into the relation between the ageostrophic velocity and surface height, we obtain a family of models for which the functional form of the potential vorticity, conserved on fluid particles, and of the kinetic energy contribution to the globally conserved energy can be prescribed. The particular approximate equations which result when the ageostrophic velocity is chosen to be in full agreement with an asymptotic small Rossby number expansion of the rotating shallow water equations give a new model with apparently higher-order dynamical consistency than either quasigeostrophy or geostrophic momentum. The geostrophic momentum model is recovered as a special case among these new models.

Potential vorticity inversion and balanced equations of motion for rotating and stratified flows

AbstractBalanced equations of motion based on potential vorticity evolution and inversion for the shallow water and stratified primitive equations are derived and, in some shallow‐water cases, numerically tested. The schemes are based on asymptotic expansions in Rossby or Froude number, or rational scaling‐based truncations of the equations of motion, assuming that the dynamics are determined by the advection of potential vorticity. Thus, regimes of validity are rapidly rotating and/or highly stratified flow. Both new and familiar results are straightforwardly obtained, in a unified framework in both height and isentropic coordinates. For both shallow‐water and stratified equations, Rossby number expansions schemes give quasi‐geostrophy at lowest order. Both gradient‐wind balance and the nonlinear terms in the potential vorticity enter at next order. A low Froude number expansion for non‐rotating flow gives two‐dimensional flow, uncoupled in the vertical at lowest order. A single consistent inversion scheme can be derived that is valid at lowest order in Froude number for all Rossby numbers, for both shallow‐water and the stratified equations. It may be a particularly appropriate model for the atmospheric mesoscale and oceanic submesoscale, where rotation and stratification can both be important in defining balanced motion. A model is also proposed that is valid at both planetary and synoptic scales, combining the familiar planetary geostrophic and quasi‐geostrophic equations. Most of the models derived require the solution only of linear or near linear elliptic equations, possibly with varying coefficients.Numerical experiments indicate that a higher‐order inversion can be quantitatively better than quasigeostrophy, if Rossby number and divergence are sufficiently small. In some other cases, no noticeable improvement over quasi‐geostrophy is found, even when the Rossby number is quite small. However, the balanced model valid for both planetary and synoptic scales shows a significant qualitative and quantitative improvement over both planetary geostrophy and quasi‐geostrophy for large‐scale flows, and its evolution is in good agreement with a primitive equation model.

Rossby number expansions, slaving principles, and balance dynamics

AbstractWe consider the problem of constructing balance dynamics for rapidly rotating fluid systems. It is argued that the conventional Rossby number expansion—namely expanding all variables in a series in Rossby number—is secular for all but the simplest flows. In particular, the higher‐order terms in the expansion grow exponentially on average, and for moderate values of the Rossby number the expansion is, at best, useful only for times of the order of the doubling times of the instabilities of the underlying quasi‐geostrophic dynamics. Similar arguments apply in a wide class of problems involving a small parameter and sufficiently complex zeroth‐order dynamics.A modified procedure is proposed which involves expanding only the fast modes of the system; this is equivalent to an asymptotic approximation of the slaving relation that relates the fast modes to the slow modes. The procedure is systematic and thus capable, at least in principle, of being carried to any order—unlike procedures based on truncations.We apply the procedure to construct higher‐order balance approximations of the shallow‐water equations. At the lowest order quasi‐geostrophy emerges. At the next order the system incorporates gradient‐wind balance, although the balance relations themselves involve only linear inversions and hence are easily applied. There is a large class of reduced systems associated with various choices for the slow variables, but the simplest ones appear to be those based on potential vorticity.

Rossby number expansions, slaving principles, and balance dynamics

Rossby number expansions, slaving principles, and balance dynamics

Phase space density representation of inviscid fluid dynamics

A formulation of inviscid fluid dynamics based on the density F(x,v,t) in a single-particle phase space [x=(x1,x2,x3), v=(v1,v2,v3)] is presented. This density evolves in time according to a Poisson bracket of F with H(x,v,t)—a Hamiltonian in the same single-particle phase space. Compressible flows of barotropic fluid and homogeneous, incompressible flows are disscussed. The main advantage of the phase space density formulation over either Euler or Lagrange formulations is the algebraic and conceptual ease in making fully Hamiltonian approximations to the flow by altering H(x,v,t) and the Poisson brackets appropriately. The example of a shallow layer of rapidly rotating fluid where a Rossby number expansion is desired will be discussed in some detail. Changes of phase space coordinates that give an approximate H (expanded in Rossby number) and exact Poisson brackets will be exhibited. The resulting quasigeostrophic equations for F are two-dimensional partial differential equations to every order in Rossby number. The extension to multiple layers will be presented.

Quasigeostrophic energetics of open ocean regions

We present a method for local energy and vorticity analysis (EVA) of open regions of oceanic flow governed by quasigeostrophic dynamics. The purpose is to infer from real and simulated data sets the physics of synoptic/mesoscale processes, and to identify general signatures of such processes. We first derive, via a Rossby number expansion, the form of the local conservation law for quasigeostrophic energy density in terms of the geostrophic pressure field. We relate the quasigeostrophic terms to their more general form and also identify the different local ageostrophic contributions to the pressure work flux divergences. Analysis methods include time series of maps of terms, space-time integral time series, and schematic open region diagrams. Rossby wave and normal mode barotropic and baroclinic instability processes are studied in open regions, and local conversion/transport properties are defined. It is found that the instability process is indicated by both Reynolds-stress-like terms ( ΔF ϰ , ΔF A ) and ageostrophic pressure work divergence ( ΔF π a, δƒ π a ). The process of local growth of energy is indicated by the local growth of asymmetries in the divergence terms. The application of EVA to real data situations which are made self-consistent by quasigeostrophic filtering is introduced. Real data initialization of a quasigeostrophic dynamical model provides the required dynamical interpolation procedure. Finally an eddy merger event captured during a successful dynamical forecast in the California Current region (Robinson et al.) is described and interpreted via EVA.

Hamiltonian description of almost geostrophic flow

Abstract Geostrophic flow in the theory of a shallow rotating fluid is exactly analogous to the drift approximation in a strongly magnetized electrostatic plasma. This analogy is developed and exhibited in detailed to derive equations for the slow nearly geostrophic motion. The key ingredient in the theory is the isolation, to whatever order in Rossby number desired, of the fast motion near the inertial frequency. One of the remaining degrees of freedom represents a new approximate constant of the motion for nearly geostrophic flow. This is the analogue of the familiar magnetic moment adiabatic invariant in the plasma problem. The procedure is a Rossby number expansion of the Hamiltonian for the fluid expressed in Lagrangian, rather than Eulerian variables. The fundamental Poisson brackets of the theory are not expanded so desirable properties such as energy conservation are maintained throughout.

Balanced models in isentropic coordinates and the shallow water equations

It is shown that there is an appropriate family of three balanced models in isentropic coordinates and the shallow water equations. “Appropriate” means the equations include exact hydrostatic, continuity and heat equations and form a complete system; some family members are second-order accurate in a midlatitude Rossby number expansion; the models conserve at least one integral invariant analogous to the primitive equation energy and enstrophy invariants; and the terms retained in the vorticity and divergence equations are the minimum required for the accuracy of the model in an asymptotic scale analysis. We compare vorticity and divergence in physical and isentropic coordinates to illustrate an aspect of the difference in physical content between balanced models defined in the different coordinates. We show that the early balanced models of Bolin and Charney are similar to members of our family in isentropic coordinates, but fail some of our criteria for appropriateness. Finally, we discuss why we think the above criteria are the right ones for balanced model families, and compare and contrast similar criteria for other intermediate models . DOI: 10.1111/j.1600-0870.1984.tb00236.x