Waves and resonance in free-boundary baroclinic instability

2001 ◽  
Vol 428 ◽  
pp. 387-408 ◽  
Author(s):  
P. RIPA
Keyword(s):  
Free Boundary ◽  
Potential Vorticity ◽  
Baroclinic Instability ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Meridional Gradient ◽  
Instability Problem ◽  
Total Potential ◽  
Wave Resonance ◽  
Basic Shear

Eady's model of baroclinic instability has been generalized by including β (the meridional gradient of planetary potential vorticity) while assuming that total potential vorticity is uniform. Moreover, the problems of Eady and of Phillips have been enriched by including a fixed topography or a free boundary (which implies a flow-dependent geostrophic topography). The most general cases (with β, fixed topography and a free boundary) of both problems are shown to have nearly identical stability properties, mainly determined by two Charney numbers: the planetary one and a topographic one. The question of whether this generalized baroclinic instability problem can be described by wave resonance or component ‘resonance’ is addressed. By waves are meant physical modes, which could freely propagate by themselves but are effectively coupled by an independent basic shear, producing the instability. Components, on the other hand, are mathematical modes for which the shear is also crucial for their existence, not just for their coupling, hence the quotation marks around ‘resonance’. In this paper it is shown that both scenarios, components ‘resonance’ and waves resonance, cast light on the free-boundary baroclinic instability problem by providing explanations of the instability onset (at minimum shear) and maximum growth rate cases, respectively. The importance of the mode pseudomomentum for the fulfillment of both mechanisms is also stressed.

Ageostrophic instability of ocean currents

1982 ◽  
Vol 117 ◽  
pp. 343-377 ◽  
Author(s):  
R. W. Griffiths ◽  
Peter D. Killworth ◽  
Melvin E. Stern
Keyword(s):  
Potential Vorticity ◽  
Lower Layer ◽  
Rotation Period ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Inertial Effects ◽  
Mean Flow ◽  
Special Cases ◽  
Free Streamlines ◽  
A Current

We investigate the stability of gravity currents, in a rotating system, that are infinitely long and uniform in the direction of flow and for which the current depth vanishes on both sides of the flow. Thus, owing to the role of the Earth's rotation in restraining horizontal motions, the currents are bounded on both sides by free streamlines, or sharp density fronts. A model is used in which only one layer of fluid is dynamically important, with a second layer being infinitely deep and passive. The analysis includes the influence of vanishing layer depth and large inertial effects near the edges of the current, and shows that such currents are always unstable to linearized perturbations (except possibly in very special cases), even when there is no extremum (or gradient) in the potential vorticity profile. Hence the established Rayleigh condition for instability in quasi-geostrophic models, where inertial effects are assumed to be vanishingly small relative to Coriolis effects, does not apply. The instability does not depend upon the vorticity profile but instead relies upon a coupling of the two free streamlines. The waves permit the release of both kinetic and potential energy from the mean flow. They can have rapid growth rates, the e-folding time for waves on a current with zero potential vorticity, for example, being close to one-half of a rotation period. Though they are not discussed here, there are other unstable solutions to this same model when the potential vorticity varies monotonically across the stream, verifying that flows involving a sharp density front are much more likely to be unstable than flows with a small ratio of inertial to Coriolis forces.Experiments with a current of buoyant fluid at the free surface of a lower layer are described, and the observations are compared with the computed mode of maximum growth rate for a flow with a uniform potential vorticity. The current is observed to be always unstable, but, contrary to the predicted behaviour of the one-layer coupled mode, the dominant length scale of growing disturbances is independent of current width. On the other hand, the structure of the observed disturbances does vary: when the current is sufficiently narrow compared with the Rossby deformation radius (and the lower layer is deep) disturbances have the structure predicted by our one-layer model. The flow then breaks up into a chain of anticyclonic eddies. When the current is wide, unstable waves appear to grow independently on each edge of the current and, at large amplitude, form both anticyclonic and cyclonic eddies in the two-layer fluid. This behaviour is attributed to another unstable mode.

Instabilities of a Baroclinic, Double Diffusive Frontal Zone

2008 ◽  
Vol 38 (4) ◽  
pp. 840-861 ◽  
Author(s):  
W. D. Smyth
Keyword(s):  
Growth Rate ◽  
Frontal Zone ◽  
Baroclinic Instability ◽  
Flux Ratio ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Variable Diffusivity ◽  
The Stability ◽  
Parameter Values ◽  
Double Diffusive

Abstract The linear theory of double diffusive interleaving is extended to take account of baroclinic effects. This study goes beyond previous studies by including the possibility of modes with nonzero tilt in the alongfront direction, which allows for advection by the baroclinic frontal flow. This requires that the stability equations be solved numerically. The main example is based on observations of interleaving on the lower flank of Meddy Sharon, but a range of parameter values is covered, leading to conclusions that are relevant in a variety of oceanic regimes. The frontal zone is treated as infinitely wide with uniform gradients of temperature, salinity, and alongfront velocity. The stationary, vertically symmetric interleaving mode is shown to have maximum growth rate when its alongfront wavenumber is zero, providing validation for previous studies in which this property was assumed. Besides this, there exist two additional modes of instability: the ageostrophic Eady mode of baroclinic instability and a mode not previously identified. The new mode is oblique (i.e., it tilts in the alongfront direction), vertically asymmetric, and propagating. It is strongly dependent on boundary conditions, and its relevance in the ocean interior is uncertain as a result. Effects of variable diffusivity and buoyancy flux ratio are also considered.

Updraft/Downdraft Constraints for Moist Baroclinic Modes and Their Implications for the Short-Wave Cutoff and Maximum Growth Rate

2005 ◽  
Vol 62 (12) ◽  
pp. 4450-4458 ◽  
Author(s):  
Pablo Zurita-Gotor
Keyword(s):  
Growth Rate ◽  
Vertical Structure ◽  
Baroclinic Instability ◽  
Maximum Growth ◽  
Static Stability ◽  
Short Wave ◽  
Maximum Growth Rate ◽  
Unstable Solutions ◽  
Baroclinic Modes

Abstract This paper examines the dynamics of moist baroclinic modes, based on the idealized model of moist baroclinic instability devised by Emanuel et al. These authors found that the finite static stability along the downdraft prevents the explosive short-wave cyclogenesis of the zero stratification limit in the moist problem, and allows only moderate (order 2) changes in the growth rate and short-wave cutoff, even when the moist static stability vanishes. To understand the limiting role of the dry static stability, a constraint is derived in this paper that relates the updraft and downdraft structures. This constraint is based on continuity and implies that a bulk wavenumber (defined in the paper) scales as the relevant deformation radius in each region. Because neutral solutions are separable, the vertical structure can be encapsulated in terms of a single, equivalent wavenumber based on the downdraft width. This allows an interpretation of the results in terms of the equivalent dry mode. As the ratio between moist and dry static stability decreases, the downdraft width takes an increasingly larger fraction of the total wavelength. In the limit of moist neutrality all the wavelength is occupied by the downdraft, so that the short-wave cutoff is halved. The vertical phase tilt makes unstable solutions nonseparable, and prevents defining an equivalent wavenumber in that case. However, the constraint between the bulk wavenumbers still applies. As the moist stability is reduced, the updraft solution becomes more suboptimal; in the limit of moist neutrality, the updraft wavenumber equals the short-wave cutoff. This provides a bound to the maximum growth rate in the moist problem, which is in agreement with the results of Emanuel et al.

Analogies and differences between the stability of an isolated pancake vortex and a columnar vortex in stratified fluid

2016 ◽  
Vol 796 ◽  
pp. 732-766 ◽  
Author(s):  
Eunok Yim ◽  
Paul Billant
Keyword(s):  
Reynolds Number ◽  
Aspect Ratio ◽  
Froude Number ◽  
Gravitational Instability ◽  
Baroclinic Instability ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Shear Instability ◽  
Axisymmetric Mode ◽  
Columnar Vortex

In order to understand the dynamics of pancake shaped vortices in stably stratified fluids, we perform a linear stability analysis of an axisymmetric vortex with Gaussian angular velocity in both the radial and axial directions with an aspect ratio of ${\it\alpha}$. The results are compared to those for a columnar vortex (${\it\alpha}=\infty$) in order to identify the instabilities. Centrifugal instability occurs when $\mathscr{R}>c(m)$ where $\mathscr{R}=ReF_{h}^{2}$ is the buoyancy Reynolds number, $F_{h}$ the Froude number, $Re$ the Reynolds number and $c(m)$ a constant which differs for the three unstable azimuthal wavenumbers $m=0,1,2$. The maximum growth rate depends mostly on $\mathscr{R}$ and is almost independent of the aspect ratio ${\it\alpha}$. For sufficiently large buoyancy Reynolds number, the axisymmetric mode is the most unstable centrifugal mode whereas for moderate $\mathscr{R}$, the mode $m=1$ is the most unstable. Shear instability for $m=2$ develops only when $F_{h}\leqslant 0.5{\it\alpha}$. By considering the characteristics of shear instability for a columnar vortex with the same parameters, this condition is shown to be such that the vortex is taller than the minimum wavelength of shear instability in the columnar case. For larger Froude number $F_{h}\geqslant 1.5{\it\alpha}$, the isopycnals overturn and gravitational instability can operate. Just below this threshold, the azimuthal wavenumbers $m=1,2,3$ are unstable to baroclinic instability. A simple model shows that baroclinic instability develops only above a critical vertical Froude number $F_{h}/{\it\alpha}$ because of confinement effects.

scholarly journals Optimizing of Microalgae Scenedesmus sp. Biomass Production in Wet Market Wastewater Using Response Surface Methodology

2021 ◽  
Vol 13 (4) ◽  
pp. 2216
Author(s):  
Najeeha Mohd Apandi ◽  
Mimi Suliza Muhamad ◽  
Radin Maya Saphira Radin Mohamed ◽  
Norshuhaila Mohamed Sunar ◽  
Adel Al-Gheethi ◽  
...  
Keyword(s):  
Response Surface Methodology ◽  
Response Surface ◽  
Biomass Yield ◽  
Biomass Productivity ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Aeration Rate ◽  
Composition Analysis ◽  
Wet Market ◽  
Central Composite

The present study aimed to optimize the production of Scenedesmus sp. biomass during the phycoremediation process. The biomass productivity was optimized using face centred central composite design (FCCCD) in response surface methodology (RSM) as a function of two independent variables that included wet market wastewater concentrations (A) with a range of 10% to 75% and aeration rate (B) with a range of 0.02 to 4.0 L/min. The results revealed that the highest biomass productivity (73 mg/L/d) and maximum growth rate (1.19 day−1) was achieved with the 64.26% of (A) and 3.08 L/min of (B). The GC-MS composition analysis of the biomass yield extract revealed that the major compounds are hexadecane (25%), glaucine (16.2%), and phytol (8.33%). The presence of these compounds suggests that WMW has the potential to be used as a production medium for Scenedesmus sp. Biomass, which has several applications in the pharmaceutical and chemical industry.

A note on growth curves of rabbit lines selected on growth rate or litter size

1993 ◽  
Vol 57 (2) ◽  
pp. 332-334 ◽  
Author(s):  
A. Blasco ◽  
E. Gómez
Keyword(s):  
Growth Rate ◽  
Litter Size ◽  
Growth Curves ◽  
Live Weight ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Standard Errors ◽  
Adult Weight ◽  
Weight Growth ◽  
Using Data

Two synthetic lines of rabbits were used in the experiment. Line V, selected on litter size, and line R, selected on growth rate. Ninety-six animals were randomly collected from 48 litters, taking a male and a female each time. Richards and Gompertz growth curves were fitted. Sexual dimorphism appeared in the line V but not in the R. Values for b and k were similar in all curves. Maximum growth rate took place in weeks 7 to 8. A break due to weaning could be observed in weeks 4 to 5. Although there is a remarkable similarity of the values of all the parameters using data from the first 20 weeks only, the higher standard errors on adult weight would make 30 weeks the preferable time to take data for live-weight growth curves.

scholarly journals Jet Formation and Evolution in Baroclinic Turbulence with Simple Topography

2010 ◽  
Vol 40 (2) ◽  
pp. 257-278 ◽  
Author(s):  
Andrew F. Thompson
Keyword(s):  
Southern Ocean ◽  
Potential Vorticity ◽  
Baroclinic Instability ◽  
Length Scale ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Jet Formation ◽  
Meridional Transport ◽  
Jet Behavior ◽  
Jet Variability

Abstract Satellite altimetry and high-resolution ocean models indicate that the Southern Ocean comprises an intricate web of narrow, meandering jets that undergo spontaneous formation, merger, and splitting events, as well as rapid latitude shifts over periods of weeks to months. The role of topography in controlling jet variability is explored using over 100 simulations from a doubly periodic, forced-dissipative, two-layer quasigeostrophic model. The system is forced by a baroclinically unstable, vertically sheared mean flow in a domain that is large enough to accommodate multiple jets. The dependence of (i) meridional jet spacing, (ii) jet variability, and (iii) domain-averaged meridional transport on changes in the length scale and steepness of simple sinusoidal topographical features is analyzed. The Rhines scale, ℓβ = 2πVe/β, where Ve is an eddy velocity scale and β is the barotropic potential vorticity gradient, measures the meridional extent of eddy mixing by a single jet. The ratio ℓβ /ℓT, where ℓT is the topographic length scale, governs jet behavior. Multiple, steady jets with fixed meridional spacing are observed when ℓβ ≫ ℓT or when ℓβ ≈ ℓT. When ℓβ < ℓT, a pattern of perpetual jet formation and jet merger dominates the time evolution of the system. Zonal ridges systematically reduce the domain-averaged meridional transport, while two-dimensional, sinusoidal bumps can increase transport by an order of magnitude or more. For certain parameters, bumpy topography gives rise to periodic oscillations in the jet structure between purely zonal and topographically steered states. In these cases, transport is dominated by bursts of mixing associated with the transition between the two regimes. Topography modifies local potential vorticity (PV) gradients and mean flows; this can generate asymmetric Reynolds stresses about the jet core and can feed back on the conversion of potential energy to kinetic energy through baroclinic instability. Both processes contribute to unsteady jet behavior. It is likely that these processes play a role in the dynamic nature of Southern Ocean jets.

Reassessment of Maximum Growth Rates for C3 and C4 Crops

1978 ◽  
Vol 14 (1) ◽  
pp. 1-5 ◽  
Author(s):  
J. L. Monteith
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Growing Season ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Crop Growth ◽  
Close Inspection ◽  
Similar Difference ◽  
Photosynthetic Efficiencies

SUMMARYFigures for maximum crop growth rates, reviewed by Gifford (1974), suggest that the productivity of C3 and C4 species is almost indistinguishable. However, close inspection of these figures at source and correspondence with several authors revealed a number of errors. When all unreliable figures were discarded, the maximum growth rate for C3 stands fell in the range 34–39 g m−2 d−1 compared with 50–54 g m−2 d−1 for C4 stands. Maximum growth rates averaged over the whole growing season showed a similar difference: 13 g m−2 d−1 for C3 and 22 g m−2 d−1 for C4. These figures correspond to photosynthetic efficiencies of approximately 1·4 and 2·0%.

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