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Qin Xu

Abstract

Liny's formula for eddy viscosity in the presence of purely convective instability is extended by including both the buoyancy and inertial contributions to the turbulence energy in a slantwise convection of moist symmetric instability (MSI). When convective available potential energy is small or not observed but MSI is strong for a frontal rainband, a large value of eddy viscosity is estimated from the extended formula.

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Qin Xu

Abstract

Symbolic operations are used together with delta functions to derive the generalized adjoint method for physical processes that contain first-order discontinuities caused by parameterized on/off switches with zero-order discontinuities in the source term. Generalized adjoint solutions are obtained analytically for simple heuristic examples and verified by direct perturbation analyses. Errors due to the conventional treatment with the “classic” adjoint method (which ignores the variation of the switch point) are quantified and found to be significant. The classic adjoint method encounters more serious problems when the parameterized process causes on/off oscillations in a numerical integration of the equation. In the limit of a vanishing computational time step, the on/off oscillations approach a marginal state that can be well treated by the generalized adjoint method. It is found that the marginal state imposes a constraint on the perturbation.

Three basic issues are raised and addressed concerning whether and how discontinuous on/off switches may affect (i) the existence of adjoint and gradient, (ii) the nonlinearity and sensitivity, and (iii) the bifurcation properties. It is found that the gradient becomes discontinuous and has a regular (or singular) jump at a non-bifurcated (or bifurcated) branch point but still can be correctly computed by the generalized adjoint. Unless the switch is branched at a bifurcation point, its nonlinearity is local and lower by a half-order than the quadratic nonlinearity. The linear sensitivity of the solution to the initial state will be reduced (or enhanced) by a discontinuous switch if the perturbation is reduced (or amplified) by the switch.

Smoothing modifications of switches with their jumps fitted by continuous functions are examined for their effectiveness in making the switches suitable for the classic adjoint method. It is found that fitting the jump with a continuous function of time (control variable) cannot (can) make the switch suitable for the classic adjoint method.

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Qin Xu

Abstract

A hybrid intermediate model, called the semibalance model, is derived from a single truncation of the vector vorticity equation with a balanced vorticity approximation that neglects the advection and stretching–tilting of the unbalanced secondary flow vorticity. This approximation applies not only to straight fronts, like the semi-geostrophy (SG), but also to highly curved fronts and vortices in which the balanced leading-order velocity and unbalanced secondary vorticity are nearly parallel with slow spatial variations along the front or vortex flow. The semibalance model is similar to the balance equations based on momentum equations (BEM) except that the leading-order flow is nonlinearly balanced and the secondary circulation is not free of vertical vorticity. As in BEM, the truncated potential vorticity in the semibalance model is more accurate than in SG, and the problem with spurious high-frequency oscillation in BEM is eliminated in the semibalance model. The potential vorticity in the semibalance model is not only conserved but also “invertible,” so the semibalance dynamics can be examined through “potential vorticity thinking.” In this sense, the semibalance model combines the advantages of SG and BEM.

Diagnostic equations for the secondary circulation are derived. The associated boundary value problem is shown to be well posed in iterative form, provided the leading-order potential vorticity is positive and, thus, the flow is inertially and convectively stable. Methods for the numerical solution of the semibalance model are presented. Under more restrictive conditions, the semibalance model reduces to the quasi-balance and bilinear quasi-balance models. Through the semibalance and quasi-balance models, the geostrophic-type and balanced-type models are connected.

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Qin Xu

Abstract

By use of an energy integral based on generalized energy conservation, previous analytical results for linear CSI (Conditional Symmetric Instability) with a uniform basic state are extended to linear CSI with a nonuniform basic state in which the stratification and shear of the basic flow are functions of space.

The generalized energy conservation is similarly found for nonlinear inviscid SI (Symmetric Instability). As this energy conservation is used to study SI, it is proved that a) the linear theory fails to predict the stability in certain cases where the basic state is transitional between stability and instability; b) the initial growth of the SI perturbations can be fairly well approximated by linear theory, but the longtime nonlinear evolutions will be bounded energetically. The upper bound of available SI energy is estimated with considerations more rigorous than that of the Lagrangian parcel dynamics A further extension of the energetics to CSI shows that the nonlinear evolution of a CSI circulation may energetically depend on the precipitation in a complicated way.

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Qin Xu

Abstract

It is shown that the classic normal modes for symmetric perturbations in a layer of vertically sheared basic flow can be classified into three types: paired growing and decaying modes, paired slowly propagating modes, and paired fast propagating modes. In the limit of vanishing growth rate (or frequency), the paired growing and decaying (or slowly propagating) modes degenerate into paired stationary and linearly growing modes. Degeneracies occur only on a discrete set (which is infinite but has zero measure) in the wavenumber space when the Richardson number is smaller than one. A nonmodal solution is not affected by the degenerate modes unless the solution is horizontally periodic and contains wavenumbers at which the degeneracy occurs. The classic modes and degenerate modes form a complete set in the sense that they contain all the wavenumbers and thus can construct any admissible nonmodal solutions.

The mode structures are analyzed by considering the slopes of their slantwise circulations relative to the absolute-momentum surface and isentropic surface of the basic state in the vertical cross section perpendicular to the circulation bands. The cross-band streamfunction component modes are shown to be orthogonal between different pairs (measured by the subspace inner product associated with the cross-band kinetic energy). The full-component modes, however, are nonorthogonal (measured by the full-space inner product associated with the total perturbation energy), and two paired modes have exactly opposite polarization relationships between the cross-band motion and its driving inertial–buoyancy force (associated with the along-band velocity and buoyancy perturbations). These properties have important implications for the nonmodal growths examined in Part II.

In association with the complete set of normal modes, a complete set of adjoint modes is derived along with the biorthogonal relationships between the normal modes and adjoint modes. By using the biorthogonality, nonmodal solutions can be conveniently constructed from the normal modes for the initial value problem.

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Qin Xu

Abstract

By combining the two Q-vector component equations with the third quasigeostrophic (QG) diagnostic equation (the vertical ageostrophic vorticity equation) a complete set of QG diagnostic equations is formed in a three-dimensional vector form with the ageostrophic pseudovorticity vector on the left-hand side and a newly defined geostrophic forcing vector (the C vector) on the right-hand side. The horizontal projection of the C vector is a rotated Q vector (by 90° to the right). The vertical C-vector component is proportional to the Gaussian curvature of the geopotential surface of constant pressure. Since C-vector streamlines can be viewed as ageostrophic pseudovortex lines, ageostrophic circulations can be easily inferred through three-dimensional “vorticity thinking,” which considers both the boundary effect and moist processes. The C vector is interpreted physically in terms of generation of Coriolis force curl and buoyancy curl due to the geostrophic advection alone. The basic techniques and possible merits of C-vector analyses are explored with simple examples. The C-vector concept is shown to be useful not only for qualitative analyses but also for quantitative computations of three-dimensional ageostrophic circulations. In particular, the pseudorotational part of the ageostrophic wind can be obtained from a convolution integral of the Green's function and C vector.

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Qin Xu

Abstract

The dynamics of cold air damming are examined analytically with a two-layer steady state model. The upper layer is a warm and saturated cross-mountain (easterly or southeasterly onshore) flow. The lower layer is a cold mountain-parallel (northerly) jet trapped on the windward (eastern) side of the mountain. The interface between the two layers represents a coastal front—a sloping inversion layer coupling the trapped cold dome with the warm onshore flow above through pressure continuity.

An analytical expression is obtained for the inviscid upper-layer flow with hydrostatic and moist adiabatic approximations. Blackadar's PBL parameterization of eddy viscosity is used in the lower-layer equations. Solutions for the mountain-parallel jet and its associated secondary transverse circulation are obtained by expanding asymptotically upon a small parameter proportional to the square root of the inertial aspect ratio—the ratio between the mountain height and the radius of inertial oscillation. The geometric shape of the sloping interface is solved numerically from a differential-integral equation derived from the pressure continuity condition imposed at the interface.

The observed flow structures and force balances of cold air damming events are produced qualitatively by the model. In the cold dome the mountain-parallel jet is controlled by the competition between the mountain-parallel pressure gradient and friction: the jet is stronger with smoother surfaces, higher mountains, and faster mountain-normal geostrophic winds. In the mountain-normal direction the vertically averaged force balance in the cold dome is nearly geostrophic and controls the geometric shape of the cold dome. The basic mountain-normal pressure gradient generated in the cold dome by the negative buoyancy distribution tends to flatten the sloping interface and expand the cold dome upstream against the mountain-normal pressure gradient (produced by the upper-layer onshore wind) and Coriolis force (induced by the lower-layer mountain-parallel jet). It is found that the interface slope increases and the cold dome shrinks as the Froude number and/or upstream mountain-parallel geostrophic wind increase, or as the Rossby number, upper-layer depth, and/or surface roughness length decrease, and vice versa. The cold dome will either vanish or not be in a steady state if the Froude number is large enough or the roughness length gets too small. The theoretical findings are explained physically based on detailed analyses of the force balance along the inversion interface.

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Qin Xu

Abstract

Modal and nonmodal growths of nonhydrostatic symmetric perturbations in an unbounded domain are examined in comparison with their hydrostatic counterparts. It is shown that the modal growth rate is a function of a single internal parameter s, the slope of the cross-band wave pattern. The maximum nonmodal growth of total perturbation energy norm is produced, also as a function of s, by an optimal combination of one geostrophic neutral mode and two paired nongeostrophic growing and decaying (or propagating) modes in the unstable (or stable) region. The hydrostatic approximation inflates the maximum modal growth rate significantly (or boundlessly) as the basic-state Richardson number Ri is small (or → 0) and inflates the maximum nonmodal growth rate significantly (or boundlessly) as |s| is large (or → ∞).

Inside the unstable region, the maximum nonmodal growth scaled by the modal growth is a bounded increasing function of growth time τ but reduces to 1 at (Ri, s) = (¼, −2) where the three modes become orthogonal to each other. Outside the unstable region, the maximum nonmodal growth is a periodic function of τ and the maximum growth time τm is bounded between ¼ and ½ of the period of the paired propagating modes. The scaled maximum nonmodal growth reaches the global maximum at s = −Ri−1 ± Ri−1(1 − Ri)1/2 (the marginal-stability boundary) for any τ if Ri ≤ 1, or at s = −1 ± (1 − Ri−1)1/2 for τ = τm if Ri > 1. When the neutral mode is filtered, the nonmodal growth becomes nongeostrophic and smaller than its counterpart growth constructed by the three modes but still significantly larger than the modal growth in general. The scaled maximum nongeostrophic nonmodal growth reaches the global maximum at s = −Ri−1 ± Ri−1(1 − Ri)1/2 for any τ if Ri ≤ 1, or at s = −Ri−1/2 for τ = τm if Ri > 1. Normalized inner products between the modes are introduced to measure their nonorthogonality and interpret their constructed nonmodal growths physically.

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Qin Xu

Abstract

A viscous semigeostrophic model is developed and used to study the formation and evolution of frontal rainbands in association with the dry and moist geostrophic potential vorticity (GPV) anomalies. The numerical results show that when moist GPV (MGPV) becomes negative in the saturated region (but the flow is still stable to viscous symmetric perturbations), banded substructures can be generated internally by a positive feedback between the moist circulation bands and geostrophic forcing anomalies in association with the generation of mesoscale GPV anomalies. In addition to the previous diagnostic results for idealized forcings, the new aspect here is that the large-scale moist ascent evolves into much finer multiple moist bands as soon as the positive feedback begins to generate banded substructures in the forcing and GPV fields. When MGPV is positive, multiple rainbands can only be generated externally by preexisting GPV or MGPV anomalies. These rainbands can be self-maintained by a weak feedback between the vertical motion and warming anomalies that operates in a partially saturated layer between the maximum and minimum levels of the undulated cloud-base boundary in association with the preexisting GPV or MGPV anomalies. The bands are seen as weak cores of upward motion surrounded by the large-scale moist ascent, rather than separated by mesoscale dry subsidences as in the case of negative MGPV.

As the negative MGPV area diminishes (mainly due to the boundary MGPV flux) and the GPV anomalies are lifted into the saturated region, the later evolution of the bands is largely controlled by the Lagrangian advection and eddy dissipation. As the geostrophic confluence flow squeezes (stretches) the bands toward (along) the front, the fine structures of GPV anomalies are smoothed by eddy viscosity, and multibands gradually “merge” into a larger single band of moist ascent. Rainbands produce not only horizontal-mean positive (negative) GPV anomalies in the lower (upper) levels but also significant mesoscale GPV anomalies in the horizontal. Boundary-layer processes can produce either positive or negative GPV flux, depending on the boundary conditions. In general, positive (negative) GPV flux is produced when warm (cold) air moves over a cold (warm) surface. The complex and yet somewhat subtle feature of GPV flux near the surface front is discussed in detail.

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Qin Xu

Abstract

The Sawyer–Eliassen (S–E) equation for frontal circulations forced by a geostrophic stretching deformation is extended to include the effects of both negative moist potential vorticity (MPV) and eddy viscosity. Since the moist (precipitation) region depends on the vertical motion and thus needs to be solved together with the frontal circulation, the extended S–E equation is a nonlinear, elliptic, partial differential equation of sixth order. When MPV is positive and viscosity is negligible, this equation degenerates into the conventional S–E equation. The existence, uniqueness and stability of the solutions of the extended S–E equation in the presence of negative MPV (but still stable to viscous moist symmetric perturbations) are examined both analytically and numerically.

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