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- Author or Editor: Qin Xu x
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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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Abstract
The time solution of a transverse inviscid circulation in a basic flow possessing either symmetric instability (SI) or conditional symmetric instability (CSI) is studied. By making the assumption that the circulation pattern is arbitrary but invariable with times, it can be proven that (i) the time evolution of a SI circulation integral is periodic and similar to that of a large amplitude pendulum started from the unstable equilibrium position, (ii) the same feature exists for a CSI circulation integral only when the mass center of the circulation loop is just at the mean (latent) heating level, (iii) a CSI circulation will spin up like a forced large amplitude pendulum (or oscillate like a damping pendulum) if its mass center is above (or below) the mean heating level.
Abstract
The time solution of a transverse inviscid circulation in a basic flow possessing either symmetric instability (SI) or conditional symmetric instability (CSI) is studied. By making the assumption that the circulation pattern is arbitrary but invariable with times, it can be proven that (i) the time evolution of a SI circulation integral is periodic and similar to that of a large amplitude pendulum started from the unstable equilibrium position, (ii) the same feature exists for a CSI circulation integral only when the mass center of the circulation loop is just at the mean (latent) heating level, (iii) a CSI circulation will spin up like a forced large amplitude pendulum (or oscillate like a damping pendulum) if its mass center is above (or below) the mean heating level.
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.
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.
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.
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.
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.
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.
