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George H. Bryan

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A spurious updraft pattern has been documented in some numerical simulations of squall lines. The pattern is notable because of a regular, repeating pattern of updrafts and downdrafts that are three–six grid lengths wide. This study examines the environmental and numerical conditions that lead to this problem. The spurious pattern is found only in simulations of upshear-tilted convective systems. Furthermore, the pattern coincides with deep (2–3 km) and wide (5–20 km) moist absolutely unstable layers (MAULs)—saturated layers of air that are statically unstable. In this physical environment, small-scale perturbations grow rapidly. The necessarily imperfect numerical schemes of the model introduce spurious small-scale perturbations into the MAULs, and these perturbations amplify owing to the unstable stratification. Some techniques are investigated that diffuse the perturbations or minimize their introduction in the statically unstable flow.

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George H. Bryan

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A set of approximate equations for pseudoadiabatic thermodynamics is developed. The equations are derived by neglecting the entropy of water vapor and then compensating for this error by using a constant (but relatively large) value for the latent heat of vaporization. The subsequent formulations for entropy and equivalent potential temperature have errors that are comparable to those of previous formulations, but their simple form makes them attractive for use in theoretical studies. It is also shown that, if the latent heat of vaporization is replaced with a constant value, an optimal value should be chosen to minimize error; a value of 2.555 × 106 J kg−1 is found in tests herein.

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George H. Bryan

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Using numerical simulations, this study examines the sensitivity of hurricane intensity and structure to changes in the surface exchange coefficients and to changes in the length scales of a turbulence parameterization. Compared to other recent articles on the topic, this study uses higher vertical resolution, more values for the turbulence length scales, a different initial environment (including higher sea surface temperature), a broader specification of surface exchange coefficients, a more realistic microphysics scheme, and a set of three-dimensional simulations. The primary conclusions from a recent study by Bryan and Rotunno are all upheld: maximum intensity is strongly affected by the horizontal turbulence length scale lh but not by the vertical turbulence length scale lυ, and the ratio of surface exchange coefficients for enthalpy and momentum, Ck/Cd, has less effect on maximum wind speed than suggested by an often-cited theoretical model. The model output is further evaluated against various metrics of hurricane intensity and structure from recent observational studies, including maximum wind speed, minimum pressure, surface wind–pressure relationships, height of maximum wind, and surface inflow angle. The model settings lh ≈ 1000 m, lυ ≈ 50 m, and Ck/Cd ≈ 0.5 produce the most reasonable match to the observational studies. This article also reconciles a recent controversy about the likely value of Ck/Cd in high wind speeds by noting that simulations in a study by Emanuel used relatively large horizontal diffusion and low sea surface temperature. The model in this study can produce category 5 hurricanes with Ck/Cd as low as 0.25.

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George H. Bryan and Richard Rotunno

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Several studies have shown that the intensity of numerically simulated tropical cyclones can exceed (by 50%) a theoretical upper limit. To investigate the cause, this study evaluates the underlying components of Emanuel’s commonly cited analytic theory for potential intensity (herein referred to as E-PI). A review of the derivation of E-PI highlights three primary components: a dynamical component (gradient-wind and hydrostatic balance); a thermodynamical component (reversible or pseudoadiabatic thermodynamics, although the pseudoadiabatic assumption yields greater intensity); and a planetary boundary layer (PBL) closure (which relates the horizontal gradients of entropy and angular momentum at the top of the PBL to fluxes and stresses at the ocean surface). These three components are evaluated using output from an axisymmetric numerical model. The present analysis finds the thermodynamical component and the PBL closure to be sufficiently accurate for several different simulations. In contrast, the dynamical component is clearly violated. Although the balanced portion of the flow (υg, to which E-PI applies) appears to also exceed E-PI, it is shown that this difference is attributable to the method used to calculate υg from the model output. Evidence is shown that υg for a truly balanced cyclone does not exceed E-PI. To clearly quantify the impact of unbalanced flow, a more complete analytic model is presented. The model is not expressed in terms of external conditions and thus cannot be used to predict maximum intensity for a given environment; however, it does allow for evaluation of the relative contributions to maximum intensity from balanced and unbalanced (i.e., inertial) terms in the governing equations. Using numerical model output, this more complete model is shown to accurately model maximum intensity. Analysis against observations further confirms that the effects of unbalanced flow on maximum intensity are not always negligible. The contribution to intensity from unbalanced flow can become negligible in axisymmetric models as radial turbulence (i.e., viscosity) increases, and this explains why some previous studies concluded that E-PI was an accurate upper bound for their simulations. Conclusions of this study are also compared and contrasted to those from previous studies.

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George H. Bryan and Hugh Morrison

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Idealized simulations of the 15 May 2009 squall line from the Second Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX2) are evaluated in this study. Four different microphysical setups are used, with either single-moment (1M) or double-moment (2M) microphysics, and either hail or graupel as the dense (rimed) ice species. Three different horizontal grid spacings are used: Δx = 4, 1, or 0.25 km (with identical vertical grids). Overall, results show that simulated squall lines are sensitive to both microphysical setup and horizontal resolution, although some quantities (i.e., surface rainfall) are more sensitive to Δx in this study. Simulations with larger Δx are slower to develop, produce more precipitation, and have higher cloud tops, all of which are attributable to larger convective cells that do not entrain midlevel air. The highest-resolution simulations have substantially more cloud water evaporation, which is partly attributable to the development of resolved turbulence. For a given Δx, the 1M simulations produce less rain, more intense cold pools, and do not have trailing stratiform precipitation at the surface, owing to excessive rainwater evaporation. The simulations with graupel as the dense ice species have unrealistically wide convective regions. Comparison against analyses from VORTEX2 data shows that the 2M setup with hail and Δx = 0.25 km produces the most realistic simulation because (i) this simulation produces realistic distributions of reflectivity associated with convective, transition, and trailing stratiform regions, (ii) the cold pool properties are reasonably close to analyses from VORTEX2, and (iii) relative humidity in the cold pool is closest to observations.

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George H. Bryan and Richard Rotunno

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An axisymmetric numerical model is used to evaluate the maximum possible intensity of tropical cyclones. As compared with traditionally formulated nonhydrostatic models, this new model has improved mass and energy conservation in saturated conditions. In comparison with the axisymmetric model developed by Rotunno and Emanuel, the new model produces weaker cyclones (by ∼10%, in terms of maximum azimuthal velocity); the difference is attributable to several approximations in the Rotunno–Emanuel model. Then, using a single specification for initial conditions (with a sea surface temperature of 26°C), the authors conduct model sensitivity tests to determine the sensitivity of maximum azimuthal velocity (υ max) to uncertain aspects of the modeling system. For fixed mixing lengths in the turbulence parameterization, a converged value of υ max is achieved for radial grid spacing of order 1 km and vertical grid spacing of order 250 m. The fall velocity of condensate (Vt) changes υ max by up to 60%, and the largest υ max occurs for pseudoadiabatic thermodynamics (i.e., for Vt > 10 m s−1). The sensitivity of υ max to the ratio of surface exchange coefficients for entropy and momentum (CE/CD) matches the theoretical result, υ max ∼ (CE/CD)1/2, for nearly inviscid flow, but simulations with increasing turbulence intensity show less dependence on CE/CD; this result suggests that the effect of CE/CD is less important than has been argued previously. The authors find that υ max is most sensitive to the intensity of turbulence in the radial direction. However, some settings, such as inviscid flow, yield clearly unnatural structures; for example, υ max exceeds 110 m s−1, despite a maximum observed intensity of ∼70 m s−1 for this environment. The authors show that turbulence in the radial direction limits maximum axisymmetric intensity by weakening the radial gradients of angular momentum (which prevents environmental air from being drawn to small radius) and of entropy (which is consistent with weaker intensity by consideration of thermal wind balance). It is also argued that future studies should consider parameterized turbulence as an important factor in simulated tropical cyclone intensity.

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Richard Rotunno and George H. Bryan

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This study considers a two-layer fluid with constant density in each layer connected by a layer of continuously varying density for flows past topography in which hydraulic jumps with lee vortices are expected based on shallow-water theory. Numerical integrations of the Navier–Stokes equations at a Reynolds number high enough for a direct numerical simulation of turbulent flow allow an examination of the internal mechanics of the turbulent leeside hydraulic jump and how this mechanics is related to lee vortices. Analysis of the statistically steady state shows that the original source of lee-vortex vertical vorticity is through the leeside descent of baroclinically produced spanwise vorticity associated with the hydraulic jump. This spanwise vorticity is tilted to the vertical at the spanwise extremities of the leeside hydraulic jump. Turbulent energy dissipation in flow through the hydraulic jump allows this leeside vertical vorticity to diffuse and extend downstream. The present simulations also suggest a geometrical interpretation of lee-vortex potential-vorticity creation, a concept central to interpretations of lee vortices based on the shallow-water equations.

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George H. Bryan and Richard Rotunno

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Using a time-dependent axisymmetric numerical model, the authors evaluate whether high-entropy air near the surface in hurricane eyes can substantially increase hurricanes’ maximum intensity. This local high-entropy anomaly is ultimately created by surface entropy fluxes in the eye. Therefore, simulations are conducted in which these surface fluxes are set to zero; results show that the high-entropy anomaly is eliminated, yet the axisymmetric tangential wind speed is only slightly weakened (by ∼4%, on average). These results contradict the hypothesis that transport of high-entropy air from the eye into the eyewall can significantly increase the maximum axisymmetric intensity of hurricanes. In fact, all simulations (with or without high-entropy anomalies) have an intensity that is 25–30 m s−1 higher than Emanuel’s theoretical maximum intensity. Further analysis demonstrates that less then 3% of the total surface-entropy input to the hurricane comes from the eye, and therefore the total magnitude of entropy transport between the eye and eyewall is a negligible component of the entropy budget of the simulated hurricanes. This latter finding is consistent with a cursory comparison with observations.

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Richard Rotunno and George H. Bryan

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In this study the authors analyze and interpret the effects of parameterized diffusion on the nearly steady axisymmetric numerical simulations of hurricanes presented in a recent study. In that study it was concluded that horizontal diffusion was the most important control factor for the maximum simulated hurricane intensity. Through budget analysis it is shown here that horizontal diffusion is a major contributor to the angular momentum budget in the boundary layer of the numerically simulated storms. Moreover, a new scale analysis recognizing the anisotropic nature of the parameterized model diffusion shows why the horizontal diffusion plays such a dominant role. A simple analytical model is developed that captures the essence of the effect. The role of vertical diffusion in the boundary layer in the aforementioned numerical simulations is more closely examined here. It is shown that the boundary layer in these simulations is consistent with known analytical solutions in that boundary layer depth increases and the amount of “overshoot” (maximum wind in excess of the gradient wind) decreases with increasing vertical diffusion. However, the maximum wind itself depends mainly on horizontal diffusion and is relatively insensitive to vertical diffusion; the overshoot variation with vertical viscosity mainly comes from changes in the gradient wind with vertical viscosity. The present considerations of parameterized diffusion allow a new contribution to the dialog in the literature on the meaning and interpretation of the Emanuel potential intensity theory.

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George H. Bryan and Richard Rotunno

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This study presents analytic results for steady gravity currents in a channel using the deep anelastic equations. Results are cast in terms of a nondimensional parameter H/H 0 that relates the channel depth H to a scale depth H 0 (the depth at which density goes to zero in an isentropic atmosphere). The classic results based on the incompressible equations correspond to H/H 0 = 0. For cold gravity currents (at the bottom of a channel), assuming energy-conserving flow, the nondimensional current depth h/H is much smaller, and nondimensional propagation speed C/(gH)1/2 is slightly smaller as H/H 0 increases. For flows with energy dissipation, C/(gH)1/2 decreases as H/H 0 increases, even for fixed h/H. The authors conclude that as H/H 0 increases the normalized hydrostatic pressure rise in the cold pool increases near the bottom of the channel, whereas drag decreases near the top of the channel; these changes require gravity currents to propagate slower for steady flow to be maintained. From these results, the authors find that steady cold pools have a likely maximum depth of 4 km in the atmosphere (in the absence of shear). For warm gravity currents (at the top of a channel), h/H is slightly larger and C/(gH)1/2 is much larger as H/H 0 increases. The authors also conduct two-dimensional numerical simulations of “lock-exchange flow” to provide an independent evaluation of the analytic results. For cold gravity currents the simulations support the analytic results. However, for warm gravity currents the simulations show unsteady behavior that cannot be captured by the analytic theory and which appears to have no analog in incompressible flow.

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