# publications

### Journal articles

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## 2020

- Improved bounds on horizontal convection
*Rocha, C. B.*, Bossy, T., Smith, S. G. Llewellyn, and Young, W. R.*Journal of Fluid Mechanics*2020 [abstract] [bibtex] [html]For the problem of horizontal convection the Nusselt number based on entropy production is bounded from above by CRa1/3 as the horizontal convective Rayleigh number Ra → ∞ for some constant C (Siggers et al., J. Fluid Mech., vol. 517, 2004, pp. 55–70). We re-examine the variational arguments leading to this ‘ultimate regime’ by using the Wentzel–Kramers–Brillouin method to solve the variational problem in the Ra → ∞ limit and exhibiting solutions that achieve the ultimate Ra1/3 scaling. As expected, the optimizing flows have a boundary layer of thickness ∼Ra−1/3 pressed against the non-uniformly heated surface; but the variational solutions also have rapid oscillatory variation with wavelength ∼Ra−1/3 along the wall. As a result of the exact solution of the variational problem, the constant C is smaller than the previous estimate by a factor of 2.5 for no-slip and 1.6 for no-stress boundary conditions. This modest reduction in C indicates that the inequalities used by Siggers et al. (J. Fluid Mech., vol. 517, 2004, pp. 55–70) are surprisingly accurate.

@article{Rocha_etal2020a, title = {{Improved bounds on horizontal convection}}, author = {Rocha, C. B. and Bossy, T. and Smith, S. G. Llewellyn and Young, W. R.}, journal = {Journal of Fluid Mechanics}, volume = {883}, number = {}, pages = {A41}, year = {2020}, publisher = {Cambridge University Press}, html = {https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/improved-bounds-on-horizontal-convection/23D4D31EC5E900F7BFC2E3F6355FBC36} }

- The Nusselt numbers of horizontal convection
*Rocha, C. B.*, Constantinou, N., Smith, S. G. Llewellyn, and Young, W. R.*Journal of Fluid Mechanics*2020 [abstract] [bibtex] [html]In the problem of horizontal convection a non-uniform buoyancy, bs(x,y), is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, J, defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that J · ∇bs = −κ⟨|∇b|2⟩; the overbar denotes a space–time average over the top surface, angle brackets denote a volume–time average and κ is the molecular diffusivity of buoyancy b. This connection between J and κ⟨|∇b|2⟩ justifies the definition of the horizontal-convective Nusselt number, Nu, as the ratio of κ⟨|∇b|2⟩ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of Nu over other definitions of horizontal-convective Nusselt number. We investigate transient effects and show that κ⟨|∇b|2⟩ equilibrates more rapidly than other global averages, such as the averaged kinetic energy and bottom buoyancy. We show that κ⟨|∇b|2⟩ is the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux through the top surface. This leads to an equivalent ‘surface Nusselt number’, defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy bs(x,y). In experimental situations it is easier to evaluate the surface entropy flux, rather than the volume integral of |∇b|2 demanded by κ⟨|∇b|2⟩.

@article{Rocha_etal2020b, title = {{The Nusselt numbers of horizontal convection}}, author = {Rocha, C. B. and Constantinou, N. and Smith, S. G. Llewellyn and Young, W. R.}, journal = {Journal of Fluid Mechanics}, volume = {894}, number = {}, pages = {A24}, year = {2020}, publisher = {Cambridge University Press}, html = {https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/nusselt-numbers-of-horizontal-convection/4A56213E017FE5101A520077A44CF0D8} }

## 2019

- Characterizing the transition from balanced to unbalanced motions in the southern California Current Chereskin, T. K.,
*Rocha, C. B.*, Gille, S. T., Menemenlis, D., and Passaro, M.*Journal of Geophysical Research: Oceans*2019 [abstract] [bibtex] [html]As observations and models improve their resolution of oceanic motions at ever finer horizontal scales, interest has grown in characterizing the transition from the geostrophically balanced flows that dominate at large‐scale to submesoscale turbulence and waves that dominate at small scales. In this study we examine the mesoscale‐to‐submesoscale (100 to 10 km) transition in an eastern boundary current, the southern California Current System (CCS), using repeated acoustic Doppler current profiler transects, sea surface height from high‐resolution nadir altimetry and output from a (1/48)° global model simulation. In the CCS, the submesoscale is as energetic as in western boundary current regions, but the mesoscale is much weaker, and as a result the transition lacks the change in kinetic energy (KE) spectral slope observed for western boundary currents. Helmholtz and vortex‐wave decompositions of the KE spectra are used to identify balanced and unbalanced contributions. At horizontal scales greater than 70 km, we find that observed KE is dominated by balanced geostrophic motions. At scales from 40 to 10 km, unbalanced contributions such as inertia‐gravity waves contribute as much as balanced motions. The model KE transition occurs at longer scales, around 125 km. The altimeter spectra are consistent with acoustic Doppler current profiler/model spectra at scales longer than 70/125 km, respectively. Observed seasonality is weak. Taken together, our results suggest that geostrophic velocities can be diagnosed from sea surface height on scales larger than about 70 km in the southern CCS.

@article{Chereskin2019, title = {{Characterizing the transition from balanced to unbalanced motions in the southern California Current}}, author = {Chereskin, T. K. and Rocha, C. B. and Gille, S. T. and Menemenlis, D. and Passaro, M.}, journal = {Journal of Geophysical Research: Oceans}, volume = {124}, number = {}, pages = {2088-2109}, year = {2019}, publisher = {Wiley Online Library}, html = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2018JC014583} }

- Blocking statistics in a varying climate: lessons from a ‘traffic jam’model with pseudostochastic forcing Paradise, Adiv,
*Rocha, C. B.*, Barpanda, P., and Nakamura, N.*Journal of the Atmospheric Sciences*2019 [abstract] [bibtex] [html]Recently Nakamura and Huang proposed a semiempirical, one-dimensional model of atmospheric blocking based on the observed budget of local wave activity in the boreal winter. The model dynamics is akin to that of traffic flow, wherein blocking manifests as traffic jams when the streamwise flux of local wave activity reaches capacity. Stationary waves modulate the jet stream’s capacity to transmit transient waves and thereby localize block formation. Since the model is inexpensive to run numerically, it is suited for computing blocking statistics as a function of climate variables from large-ensemble, parameter sweep experiments. We explore sensitivity of blocking statistics to (i) stationary wave amplitude, (ii) background jet speed, and (iii) transient eddy forcing, using frequency, persistence, and prevalence as metrics. For each combination of parameters we perform 240 runs of 180-day simulations with aperiodic transient eddy forcing, each time randomizing the phase relations in forcing. The model climate shifts rapidly from a block-free state to a block-dominant state as the stationary wave amplitude is increased and/or the jet speed is decreased. When eddy forcing is increased, prevalence increases similarly but frequency decreases as blocks merge and become more persistent. It is argued that the present-day climate lies close to the boundary of the two states and hence its blocking statistics are sensitive to climate perturbations. The result underscores the low confidence in GCM-based assessment of the future trend of blocking under a changing climate, while it also provides a theoretical basis for evaluating model biases and understanding trends in reanalysis data.

@article{paradise2019blocking, title = {Blocking statistics in a varying climate: lessons from a ‘traffic jam’model with pseudostochastic forcing}, author = {Paradise, Adiv and Rocha, C. B. and Barpanda, P. and Nakamura, N.}, journal = {Journal of the Atmospheric Sciences}, number = {2019}, volume = {76}, pages = {3013-3027}, year = {2019}, html = {https://doi.org/10.1175/JAS-D-19-0095.1} }

- On the steadiness and instability of the Intermediate Western Boundary Current between 24 and 18S Napolitano, D. C., Silveira, I. C. A.,
*Rocha, C. B.*, Flierl, G. R., Calil, P. H. R., and Martins, R. P.*Journal of Physical Oceanography*2019 [abstract] [bibtex] [html]The Intermediate Western Boundary Current (IWBC) transports Antarctic Intermediate Water across the Vitória–Trindade Ridge (VTR), a seamount chain at ;208S off Brazil. Recent studies suggest that the IWBC develops a strong cyclonic recirculation in Tubarão Bight, upstream of the VTR, with weak time dependency. We herein use new quasi-synoptic observations, data from the Argo array, and a regional numerical model to describe the structure and variability of the IWBC and to investigate its dynamics. Both shipboard acoustic Doppler current profiler (ADCP) data and trajectories of Argo floats confirm the existence of the IWBC recirculation, which is also captured by our Regional Oceanic Modeling System (ROMS) simulation. An ‘‘intermediate-layer’’ quasigeostrophic (QG) model indicates that the ROMS time-mean flow is a good proxy for the IWBC steady state, as revealed by largely parallel isolines of streamfunction c and potential vorticity Q; a c 2 Q scatter diagram also shows that the IWBC is potentially unstable. Further analysis of the ROMS simulation reveals that remotely generated, westward-propagating nonlinear eddies are the main source of variability in the region. These eddies enter the domain through the Tubarão Bight eastern edge and strongly interact with the IWBC. As they are advected downstream and negotiate the local topography, the eddies grow explosively through horizontal shear production.

@article{Napolitano2019, title = {{On the steadiness and instability of the Intermediate Western Boundary Current between 24 and 18S}}, author = {Napolitano, D. C. and da Silveira, I. C. A. and Rocha, C. B. and Flierl, G. R. and Calil, P. H. R. and Martins, R. P.}, journal = {Journal of Physical Oceanography}, volume = {49}, number = {}, pages = {3127-3143}, year = {2019}, publisher = {American Metereological Society}, html = {https://journals.ametsoc.org/view/journals/phoc/49/12/jpo-d-19-0011.1.xml} }

## 2018

- Stimulated generation: Extraction of energy from balanced flow by near-inertial waves
*Rocha, C. B.*, Wagner, G. L., and Young, W. R.*Journal of Fluid Mechanics*2018 [abstract] [bibtex] [html]We study stimulated generation – the transfer of energy from balanced flows to existing internal waves – using an asymptotic model that couples barotropic quasi-geostrophic flow and near-inertial waves with eimz vertical structure, where m is the vertical wavenumber and z is the vertical coordinate. A detailed description of the conservation laws of this vertical-plane-wave model illuminates the mechanism of stimulated generation associated with vertical vorticity and lateral strain. There are two sources of wave potential energy, and corresponding sinks of balanced kinetic energy: the refractive convergence of wave action density into anti-cyclones (and divergence from cyclones); and the enhancement of wave-field gradients by geostrophic straining. We quantify these energy transfers and describe the phenomenology of stimulated generation using numerical solutions of an initially uniform inertial oscillation interacting with mature freely evolving two-dimensional turbulence. In all solutions, stimulated generation co-exists with a transfer of balanced kinetic energy to large scales via vortex merging. Also, geostrophic straining accounts for most of the generation of wave potential energy, representing a sink of 10%–20% of the initial balanced kinetic energy. However, refraction is fundamental because it creates the initial eddy-scale lateral gradients in the near-inertial field that are then enhanced by advection. In these quasi-inviscid solutions, wave dispersion is the only mechanism that upsets stimulated generation: with a barotropic balanced flow, lateral straining enhances the wave group velocity, so that waves accelerate and rapidly escape from straining regions. This wave escape prevents wave energy from cascading to dissipative scales.

@article{rocha2018, title = {Stimulated generation: {Extraction} of energy from balanced flow by near-inertial waves}, author = {Rocha, C. B. and Wagner, G. L. and Young, W. R.}, journal = {Journal of Fluid Mechanics}, volume = {847}, pages = {417--451}, year = {2018}, publisher = {Cambridge University Press}, html = {https://www.cambridge.org/core/services/aop-cambridge-core/content/view/900227E2C12AA98ECEBBE64F4FF21C43/S0022112018003087a.pdf/stimulated_generation_extraction_of_energy_from_balanced_flow_by_nearinertial_waves.pdf} }

## 2017

- Small-scale open-ocean currents have large effects on wind-wave heights Ardhuin, F., Gille, S. T., Menemenlis, D.,
*Rocha, C. B.*, Rascle, N., Chapron, , Gula, J., and Molemaker, J.*Journal of Geophysical Research: Oceans*2017 [abstract] [bibtex] [html]Tidal currents and large‐scale oceanic currents are known to modify ocean wave properties, causing extreme sea states that are a hazard to navigation. Recent advances in the understanding and modeling capability of open ocean currents have revealed the ubiquitous presence of eddies, fronts, and filaments at scales 10–100 km. Based on realistic numerical models, we show that these structures can be the main source of variability in significant wave heights at scales less than 200 km, including important variations down to 10 km. Model results are consistent with wave height variations along satellite altimeter tracks, resolved at scales larger than 50 km. The spectrum of significant wave heights is found to be of the order of urn:x-wiley:21699275:media:jgrc22271:jgrc22271-math-0001 times the current spectrum, where urn:x-wiley:21699275:media:jgrc22271:jgrc22271-math-0002 is the spatially averaged significant wave height, urn:x-wiley:21699275:media:jgrc22271:jgrc22271-math-0003 is the energy‐averaged period, and g is the gravity acceleration. This variability induced by currents has been largely overlooked in spite of its relevance for extreme wave heights and remote sensing.

@article{ardhuin2017, title = {{Small-scale open-ocean currents have large effects on wind-wave heights}}, author = {Ardhuin, F. and Gille, S. T. and Menemenlis, D. and Rocha, C. B. and Rascle, N. and b. Chapron and Gula, J. and Molemaker, J.}, journal = {Journal of Geophysical Research: Oceans}, year = {2017}, publisher = {Wiley Online Library}, html = {http://onlinelibrary.wiley.com/doi/10.1002/2016JC012413/abstract} }

## 2016

- On Galerkin approximations of the surface active quasigeostrophic equations
*Rocha, C. B.*, Young, W. R., and Grooms, I.*Journal of Physical Oceanography*2016 [abstract] [bibtex] [arXiv] [html] [pdf] [code]This study investigates the representation of solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes, with particular attention to the incorporation of active surface buoyancy dynamics. This study extends two existing Galerkin approaches (A and B) and develops a new Galerkin approximation (C). Approximation A, due to Flierl, represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to Tulloch and Smith, represents the PV as a truncated Galerkin series and calculates the streamfunction that satisfies the inversion problem exactly. Approximation C, the true Galerkin approximation for the QG equations, represents both streamfunction and PV as truncated Galerkin series but does not satisfy the inversion equation exactly. The three approximations are fundamentally different unless the boundaries are isopycnal surfaces. The authors discuss the advantages and limitations of approximations A, B, and C in terms of mathematical rigor and conservation laws and illustrate their relative efficiency by solving linear stability problems with nonzero surface buoyancy. With moderate number of modes, B and C have superior accuracy than A at high wavenumbers. Because B lacks the conservation of energy, this study recommends approximation C for constructing solutions to the surface active QG equations using the Galerkin series with standard vertical modes.

@article{rocha2016a, title = {{On Galerkin approximations of the surface active quasigeostrophic equations}}, author = {Rocha, C. B. and Young, W. R. and Grooms, I.}, journal = {Journal of Physical Oceanography}, volume = {46}, number = {1}, pages = {125--139}, year = {2016}, doi = {doi:10.1175/JPO- D-15-0073.1}, html = {http://journals.ametsoc.org/doi/abs/10.1175/JPO-D-15-0073.1}, pdf = {rocha_etal2016a.pdf}, arxiv = {1504.03361}, code = {http://nbviewer.ipython.org/github/crocha700/qg_vertical_modes/blob/master/index.ipynb} }

- Mesoscale to submesoscale wavenumber spectra in Drake Passage
*Rocha, C. B.*, Chereskin, T. K., Gille, S. T., and Menemenlis, D.*Journal of Physical Oceanography*2016 [abstract] [bibtex] [html] [code]This study discusses the upper-ocean (0–200 m) horizontal wavenumber spectra in the Drake Passage from 13 yr of shipboard ADCP measurements, altimeter data, and a high-resolution numerical simulation. At scales between 10 and 200 km, the ADCP kinetic energy spectra approximately follow a k−3 power law. The observed flows are more energetic at the surface, but the shape of the kinetic energy spectra is independent of depth. These characteristics resemble predictions of isotropic interior quasigeostrophic turbulence. The ratio of across-track to along-track kinetic energy spectra, however, significantly departs from the expectation of isotropic interior quasigeostrophic turbulence. The inconsistency is dramatic at scales smaller than 40 km. A Helmholtz decomposition of the ADCP spectra and analyses of synthetic and numerical model data show that horizontally divergent, ageostrophic flows account for the discrepancy between the observed spectra and predictions of isotropic interior quasigeostrophic turbulence. In Drake Passage, ageostrophic motions appear to be dominated by inertia–gravity waves and account for about half of the near-surface kinetic energy at scales between 10 and 40 km. Model results indicate that ageostrophic flows imprint on the sea surface, accounting for about half of the sea surface height variance between 10 and 40 km.

@article{rocha2016b, title = {{Mesoscale to submesoscale wavenumber spectra in Drake Passage}}, author = {Rocha, C. B. and Chereskin, T. K. and Gille, S. T. and Menemenlis, D.}, journal = {Journal of Physical Oceanography}, volume = {46}, number = {2}, pages = {601--620}, year = {2016}, html = {http://journals.ametsoc.org/doi/abs/10.1175/JPO-D-15-0087.1}, code = {https://github.com/cesar-rocha/dp_spectra} }

- Seasonality of submesoscale dynamics in the Kuroshio Extension
*Rocha, C. B.*, Gille, S. T., Chereskin, T. K., and Menemenlis, D.*Geophysical Research Letters*2016 [abstract] [bibtex] [html] [code]Recent studies show that the vigorous seasonal cycle of the mixed layer modulates upper ocean submesoscale turbulence. Here we provide model‐based evidence that the seasonally changing upper ocean stratification in the Kuroshio Extension also modulates submesoscale (here 10–100 km) inertia‐gravity waves. Summertime restratification weakens submesoscale turbulence but enhances inertia‐gravity waves near the surface. Thus, submesoscale turbulence and inertia‐gravity waves undergo vigorous out‐of‐phase seasonal cycles. These results imply a strong seasonal modulation of the accuracy of geostrophic velocity diagnosed from submesoscale sea surface height delivered by the Surface Water and Ocean Topography satellite mission.

@article{rocha2016c, title = {{Seasonality of submesoscale dynamics in the Kuroshio Extension}}, author = {Rocha, C. B. and Gille, S. T. and Chereskin, T. K. and Menemenlis, D.}, journal = {Geophysical Research Letters}, volume = {43}, number = {21}, year = {2016}, publisher = {Wiley Online Library}, html = {http://onlinelibrary.wiley.com/doi/10.1002/2016GL071349/full}, code = {https://github.com/cesar-rocha/UpperOceanSeasonality} }

## 2014

- Vertical structure, energetics, and dynamics of the Brazil Current System at 22 S–28 S
*Rocha, C. B.*, Silveira, I. C. A., Castro, B. M., and Lima, J. A. M.*Journal of Geophysical Research: Oceans*2014 [abstract] [bibtex] [html]We use four current meter moorings and quasi‐synoptic hydrographic observations in conjunction with a one‐dimensional quasi‐geostrophic linear stability model to investigate downstream changes in the Brazil Current (BC) System between 22°S and 28°S. The data set depict the downstream thickening of the BC. Its vertical extension increases from 350 m at 22.7°S to 850 m at 27.9°S. Most of this deepening occurs between 25.5°S and 27.9°S and is linked to the bifurcation of the South Equatorial Current at intermediate depths (Santos bifurcation), which adds the Antarctic Intermediate Water flow to the BC. Geostrophic estimates suggest that the BC transport is increased by at least 4.3 Sv (∼70%) to the south of that bifurcation. Moreover, the Santos bifurcation is associated with a substantial increase in the barotropic component of the BC System. On average, the water column average kinetic energy (IKE) is 70% baroclinic to the north and 54% barotropic to the south of the bifurcation. Additionally, the BC shows conspicuous mesoscale activity off southeast Brazil. The water column average eddy kinetic energy accounts for 30–60% of the IKE. Instabilities of the mean flow may give rise to these mesoscale fluctuations. Indeed, the linear stability analysis suggests that the BC System is baroclinically unstable between 22°S and 28°S. In particular, the model predicts southwestward‐propagating fastest growing waves (∼190 km) from 25.5°S to 27.9°S and quasi‐standing most unstable modes (∼230 km) at 22.7°S. These modes have vertical structures roughly consistent with the observed eddy field.

@article{rocha2014, title = {{Vertical structure, energetics, and dynamics of the Brazil Current System at 22 S--28 S}}, author = {Rocha, C. B. and da Silveira, I. C. A. and Castro, B. M. and Lima, J. A. M.}, journal = {Journal of Geophysical Research: Oceans}, volume = {119}, number = {1}, pages = {52--69}, year = {2014}, publisher = {Wiley Online Library}, doi = {doi:10.1002/2013JC009143}, html = {http://onlinelibrary.wiley.com/doi/10.1002/2013JC009143/abstract} }

## 2013

- Traditional quasi-geostrophic modes and surface quasi-geostrophic solutions in the Southwestern Atlantic
*Rocha, C. B.*, Tandon, A., Silveira, I. C. A., and Lima, J. A. M.*Journal of Geophysical Research: Oceans*2013 [abstract] [bibtex] [html]We investigate whether the Quasi‐geostrophic (QG) modes and the Surface Quasi‐geostrophic (SQG) solutions are consistent with the vertical structure of the subinertial variability off southeast Brazil. The first‐order empirical orthogonal function (EOF) of current meter time series is reconstructed using different QG mode combinations; the first EOF is compared against SQG solutions. At two out of three moorings, the traditional flat‐bottom barotropic (BT) and first baroclinic (BC1) mode combination fails to represent the observed sharp near‐surface decay, although this combination contains up to 78% of the depth‐integrated variance. A mesoscale broad‐band combination of flat‐bottom SQG solutions is consistent with the near‐surface sharp decay, accounting for up to 85% of the first EOF variance. A higher‐order QG mode combination is also consistent with the data. Similar results are obtained for a rough topography scenario, in which the velocity vanishes at the bottom. The projection of the SQG solutions onto the QG modes confirms that these two models are mutually dependent. Consequently, as far as the observed near‐surface vertical structure is concerned, SQG solutions and four‐QG mode combination are indistinguishable. Tentative explanations for such vertical structures are given in terms of necessary conditions for baroclinic instability. “Charney‐like” instabilities, or, surface‐intensified “Phillips‐like” instabilities may explain the SQG‐like solutions at two moorings; traditional “Phillips‐like” instabilities may rationalize the BT/BC1 mode representation at the third mooring. These results point out to the presence of a richer subinertial near‐surface dynamics in some regions, which should be considered for the interpretation and projection of remotely sensed surface fields to depth.

@article{rocha2013, title = {{Traditional quasi-geostrophic modes and surface quasi-geostrophic solutions in the Southwestern Atlantic}}, author = {Rocha, C. B. and Tandon, A. and da Silveira, I. C. A. and Lima, J. A. M.}, journal = {Journal of Geophysical Research: Oceans}, volume = {118}, number = {5}, pages = {2734--2745}, year = {2013}, publisher = {Wiley Online Library}, doi = {doi:10.1002/jgrc.20214}, html = {https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/jgrc.20214} }