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Qin Xu

Abstract

The generalized adjoint is extended to special situations in which the concerned switches are triggered simultaneously by more than one threshold condition. It is shown that the involved threshold conditions can be combined into a single threshold condition represented by the envelope of the involved threshold surfaces in the space constituted by model variables and time. This envelope is piecewise smooth, and the concerned switch point is a nonsmooth point on the envelope at the intersection of the involved threshold surfaces. When the concerned switch point is perturbed along different sectors (formed by the involved threshold surfaces) on the envelope surface, different threshold conditions should be used in the tangent linear and adjoint matching conditions. This complicates the tangent linearization and the backward adjoint integration. Four basic types of situations are examined and illustrated by using simple examples, showing how the generalized adjoint can be extended to these complicated situations.

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Qin Xu

Abstract

As an extension of CCI (conditional convective instability) theory, three-dimensional, deep and steady convective waves in a barotropic basic flow with an embedded moist zone were studied via perturbation methods. The basic state is assumed to be slightly supercritical to the gravest two-dimensional CCI mode which gives the cross-sectional structure for the leading order solution. Along the basic flow the inviscid solution is a solitary wave, manifesting an energy conservation of O(ε3). Far upstream (downstream), the exponential growth (decay) of the perturbation is similar to the linear, growing (decaying) CCI mode, except that the time variation now is related to the spatial variation through the basic flow advection. At the wave peak, the vertical displacement is “overshot” in that the buoyancy is maximally decreased (increased) in the moist ascent (dry descents) of the deep convection due to the strong warming effect and expansion of the moist zone in the upper levels, which tends to reverse the process after the wave peak.

In the presence of viscosity, the wave energy is no longer conserved and the solution along the moist zone is analogous to an undular bore in open channel flows; i.e., a wave head followed by a damping wave train with a transition to a new steady state. However, unlike undular bores, the convective waves are three-dimensional. On the upstream side of the wave head, there is strong moist ascent. In the wave head region, the horizontal flow is cyclonic (anticyclonic) around a pressure low (high) in the lower (upper) levels, which induces a lower-(upper-) level jet on the right (left) flank of the basic flow. These features bear resemblances to the gross structures of mature mesoscale convective complexes (MCCs). The theoretical results suggest that a supercritical CCI environment with a wide (meso-β scale) low-level moist inflow is necessary for the occurrence of MCCs, and the gross structure of a MCC may be largely controlled by the competition between the inflow advection and turbulent (including cloud mixing) dissipation.

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Qin Xu

Abstract

By treating the latent heating as an energy source which is implicitly related to the motion field, the existence of steady nonlinear circulations in a flow susceptible to Conditionally Symmetric Instability (CSI) is studied. Steady viscous symmetric circulations are shown to be unique and asymptotically stable, when the latent heat sources are weak and insensitive to the motion perturbations.

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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

Diagnoses are presented of the three-dimensional vertical circulation for a coupled cold-warm frontal system in an idealized moist semi-geostrophic (SG) baroclinic wave. The vertical circulation is computed in SG space where the solution corresponds to its quasi-geostrophic (QG) counterpart in physical space. It is shown for this QG solution that (i) the vertical motion is enhanced in the moist region due to small moist stability and strong local geostrophic forcing; (ii) the cyclonic/anti-cyclonic ageostrophic wind vorticity is induced locally by the geostrophic wind deformation/rotation; (iii) the along-flow/cross-flow ageostrophic wind divergence is associated with the along-flow curvature/speed variation of the system-relative geostrophic wind.

When the solution is transformed back to physical space, the vertical motion is dramatically enhanced and concentrated along the cold and warm fronts in the moist region due to time-integrated ageostrophic wind convergence and ageostrophic feedback to the forcing. It is found that the cross-cold front circulation is strong, narrow, and deep while the cross-warm front circulation is relatively weak, broad, shallow and very slantwise with the circulation center displaced toward the cold air. But, the along-warm front circulation is stronger than the along-cold front circulation, so the vertical circulation is less two-dimensional and more complex in the warm-frontal region. Dynamical factors responsible for the differences between the cold and warm frontal circulations are examined in detail.

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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

It is shown analytically and graphically that when parameterized on/off switches are triggered at discrete time levels by a threshold condition in a numerical model, the model solution is not continuously dependent on the initial state. Consequently, the response function and costfunction contain small zigzag discontinuities; their gradients contain delta functions and thus are not good approximations of the original continuous gradients. The problem is caused by the traditional time discretization and cannot be solved by the conventional treatment of on/off switches. To solve the problem, the traditional time discretization is modified with the switch time determined by interpolation as a continuous function of the initial state. With this modification, the response function and costfunction become continuous in the space of the initial state and their gradients can be accurately computed by the generalized adjoint.

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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

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

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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