Overview of our Research

The research projects in our group are focused on the overarching themes of atmospheric modeling and dynamics, and we recently (2018) started exploring machine learning concepts for the atmospheric sciences and a coupled lake-atmosphere weather-modeling project that focuses on the Great Lakes region (2019 onwards). Our group is highly interdisciplinary with diverse interests in atmospheric science, applied mathematics, computational science / scientific computing and data science.
In particular, we develop new numerical methods (like Adaptive Mesh Refinement (AMR) techniques) and variable-resolution approaches for the dynamical cores of atmospheric General Circulation Models (GCMs), analyze and compare existing dynamical cores (especially their behavior at the subgrid-scale), develop simpler atmospheric models and test cases for dynamical cores that get utilized in community-wide model intercomparison projects, analyze and improve the representation of tropical cyclones and Mesoscale Convective Systems (MCSs) in global climate models, investigate stratospheric dynamics with a focus on the Quasi-Biennial Osciillation (QBO) and Sudden Stratospheric Warmings (SSWs), and provide insight into physics-dynamics coupling issues. In the past, we also designed cyber-infrastructure tools for the climate sciences. All of our research projects utilize parallel and high-performance computing architectures.

Machine Learning Concepts for the Atmospheric Sciences

Machine learning approaches and new data science algorithms are an emerging frontier for the atmospheric sciences. We explore whether newly developed physics-aware machine learning algorithms trained with model data or observations can replace subgrid-scale physical parameterizations in forecast models, such as the time-consuming solar radiation code, or the shallow or deep convection cloud schemes. A second, less aggressive approach is to utilize machine learning approaches for the estimation of uncertain parameters in physical parameterizations. We use idealized GCM model configuration to intercompare the pros and cons of various machine learning algorithms, such as linear regression, random forests, boosted forests, artificial neural networks (ANN) and deep neural networks with and without convolutions. An example of a dry simulation with NCAR’s Community Atmosphere Model version 6 (CAM6) is presented below. The figure compares the model-generated temperature tendency to the temperature tendency that was generated via a neural network (NN) emulator. The NN reproduces the temperature tendency with high precision.

Figure: Randomly chosen snapshot of the temperature tendency field (K/day) near the Earth’s surface. Left: Simulated with an idealized (Held-Suarez) version of NCAR’s Community Atmosphere Model (CAM). Right: Generated via a machine learning neural network (NN) emulator.

Reference: Limon, G. and C. Jablonowski (2019), An Assessment of Machine Learning Techniques for Replicating Physical Forcing Mechanisms in Climate Models, American Geophysical Union (AGU) Fall Meeting, Abstract A41R-2898, San Francisco, CA, USA, Dec. 9-13, 2019, available in the online archive: Earth and Space Science Open Archive, doi: 10.1002/essoar.10501799.1

We also investigate machine learning techniques, such as boosted forests (XGBoost) and Long Short-Term Memory (LSTM) Neural Networks, to predict the winter ice cover in the St. Marys River system with 7-day and 30-day lead times. The St. Marys River connects Lake Superior and Lake Huron in the Great Lakes region (see the figure below) and is an important waterway for the shipping community. See our webpage for more information on the project and the research team.

Figure: ML-learning driven predictions of the ice coverage in the St. Marys River system are explored to help the shipping community plan safe and effective operations.

Detecting Pathways in Climate Models: Project CLDERA

CLDERA (CLimate impact: Determining Etiology thRough pAthways) is funded by the Department of Energy’s (DoE) Sandia National Laboratories (2021-2024). It will enable climate attribution in currently unachievable scenarios with a novel approach to uncover and employ “pathways”, defined as the chain of physical processes between source and impacts and their spatio-temporal evolution.  Novel computational modeling techniques will fuse with large-scale observational data from the 1991 Mt. Pinatubo eruption to elucidate the dominant pathways between source and impacts.  These pathways will be crucial constraints, helping to cull the possibilities in attribution.  CLDERA is expected to be a transformative for climate statistical approaches and attribution research, and will inform national security and policy considerations of decision-makers.

Using DoE Energy Exascale Earth System Model (E3SM) v1/v2 simulations a central aim of this project is to develop a fundamentally new and blended computational and information science approach to discovering and explaining connective relationships in the form of pathways between sources and impacts. The development of computational capabilities with active and passive tracers in the atmosphere, novel software profiling applications, and new simulation strategies will combine to enable the identification of pathways due to the Mt. Pinatubo’s volcanic eruption in 1991.  These approaches, combined with anomaly detection, are expected to enable the detection of the differences in physical tendencies caused by Mt. Pinatubo. 

Detection of pathways in CLDERA via simulation and observational techniques.

Great Lakes Weather Research with Coupled Atmosphere-Lake Models

The goal of this joint project between the University of Michigan (UM), NOAA’s Great Lakes Environmental Research Laboratory (GLERL), NWS/WFO Detroit, NOAA’s ESRL/Global Systems Laboratory (GSL) and NCEP/EMC is to improve lake-effect snow and ice forecasting capabilities via advanced lake-ice-atmosphere coupling techniques for the Unified Forecast System (UFS). UFS is NOAA’s newest multi-scale modeling framework. It contains the global UFS Medium-Range Weather (UFS-MRW) application, which integrates GFDL’s FV3 dynamical core with NOAA’s GFS physics packages, as well as the UFS Short-Range Weather application (UFS-SRW, also called Rapid Refresh Forecast System RRFS). The latter is an FV3-based limited-area model which supports various GSL physics options for convection-allowing model (CAM) simulations. A missing component in UFS are lake interactions that are crucially important for accurate forecasts of extreme winter-weather precipitation events in the Great Lakes region. The project addresses this gap by (a) advancing the operational status of NOAA’s 3D lake model FVCOM and its coupling to ice in FVCOM-CICE for the Great Lakes, (b) loosely coupling FVCOM and FVCOM-CICE to the UFS high-resolution weather prediction models with a particular focus on UFS-SRW, (c) working with NWS/WFO weather forecasters on objective evaluations of the coupling capabilities via the Hydrometeorology testbed (HMT) and the participation in the Winter Weather Experiment, and (d) transitioning the modeling and coupling advances to high readiness levels in preparation for their operational use. Therefore, this project fills critical gaps in coastal hydrodynamic forecasting and provide NWS/WFO operational forecasters with improved lake-effect snowfall, precipitation, visibility and ice forecasts to protect the safety of the citizens in the Great Lakes region.

Figure: Example of a stretched, variable-resolution grid configuration (called C768s5) of the global UFS weather prediction model, which provides high-resolution grid spacings of about 3 km over the Great Lakes Region (left, inner square). Right panel depicts the forecasted precipitation (here shown as the snow water equivalent SWE) which accumulated from January 27 to February 1, 2019. This shows the ability of the variable-resolution configuration of UFS-FV3GFS to create lake-effect snowfall downwind of the Great Lakes.

The Dynamical Core Model Intercomparison Project (DCMIP-2016) and Summer School: Future-Generation Non-Hydrostatic Weather and Climate Models

DCMIP-2016 Workshop and Summer School on Future-Generation Non-Hydrostatic Weather and Climate Models

Boulder, CO, June, 6 – June, 17 2016

Organized by Paul Ullrich (University of California, Davis), Christiane Jablonowski (University of Michigan), Kevin Reed (Stony Brook University), Colin Zarzycki (NCAR), James Kent (University of South Wales, U.K.), Peter H. Lauritzen (NCAR) and Ramachandran D. Nair (NCAR)

We have organized a summer school and model intercomparison workshop with special focus on the newest non-hydrostatic global models. We invited students, postdocs and the international dynamical core modeling community to join us at the National Center for Atmospheric Research (NCAR, Boulder, CO) for 2 weeks from June/6-17/2016 for an exciting student-focused and research-driven event that has led to an unprecedented dynamical core intercomparison project. The new aspects were that DCMIP-2016 focused very strongly on idealized dynamical core test cases with simple moisture feedbacks to assess the physics-dynamics coupling and interactions. As DCMIP-2012, DCMIP-2016 has been endorsed by the WMO Working Group on Numerical Experimentation (WGNE). The summer school and model intercomparison workshop has been supported by cyberinfrastucture tools like shared workspaces via the Earth System CoG environment.

DCMIP-2016 explores new test techniques for non-hydrostatic models that are also applicable at hydrostatic scales. Examples are a new moist baroclinic wave test case without topography (applicable to both shallow-atmosphere and deep-atmosphere dynamical core equation sets) and a tropical cyclone test case of intermediate complexity that not only assesses the dynamical core but also includes a ‘simple physics’ package with a Kessler-type warm-rain scheme. These test cases let us investigate non-linear interactions and the physics-dynamics coupling. The third test is a supercell system on a reduced-size Earth that gives insight into non-hydrostatic motions at low computational cost.

Figure: Cubed-sphere and hexagonal computational grids on the sphere.

Dynamical Core Model Intercomparison Project (DCMIP-2012) and Summer School on Future-Generation Non-Hydrostatic Weather and Climate Models

DCMIP-2012 Workshop and Summer School on Future-Generation Non-Hydrostatic Weather and Climate Models

Boulder, CO, July, 30 – August, 10 2012

In the summer of 2012 we (Christiane Jablonowski, Peter Lauritzen, Paul Ullrich, Mark Taylor, Ram Nair) organized a summer school and model intercomparison workshop with special focus on non-hydrostatic global models which are under development right now. We invited students, postdoctoral researchers and the international dynamical core modeling community to join us at the National Center for Atmospheric Resaearch (NCAR, Boulder, CO) for 2 weeks from July/30-August/10/2012 for an exciting student-focused and research-driven event that will led to an unprecedented dynamical core intercomparison project, even broader in scale than our 2008 workshop. The event was endorsed by the WMO Working Group on Numerical Experimentation (WGNE). The summer school and model intercomparison workshop was also supported by newly developed cyberinfrastucture tools like shared workspaces that we have developed in collaboration with NOAA and NCAR under an NSF grant.

We explored new test techniques for non-hydrostatic models that are also applicable at hydrostatic scales. Examples are our newly developed tropical cyclone test case of intermediate complexity that not only assesses the dynamical core but also includes a ‘simple physics’ package. This lets us investigate non-linear interactions and the physics-dynamics coupling. Other test cases included ‘small Earth’ experiments that gave insight into non-hydrostatic motions at low computational cost.

Figure: Simulation of a breaking wave on a cubed-sphere grid.

2008 NCAR Advanced Study Program (ASP) Summer Colloquium

Numerical Techniques for Global Atmospheric Models

Boulder, CO, June 1-13, 2008

Organized by Peter H. Lauritzen (NCAR), Christiane Jablonowski (University of Michigan), Mark Taylor (Sandia National Laboratories) and Ramachandran D. Nair (NCAR)

The 2-week summer colloquium titled “Numerical Techniques for Global Atmospheric Models” surveyed the latest developments in numerical methods for the dynamical cores of Atmospheric General Circulation Models. The objectives of the colloquium were (1) to teach a large group of about 40 graduate students in atmospheric science and mathematics how today’s and future dynamical cores are or need to be built, (2) to invite over 10 dynamical core modeling groups to NCAR for an unprecedented student-run dynamical core intercomparison project, (3) to establish new dynamical core test cases in the community and (4) to invite keynote speakers to NCAR that give lectures on modern numerical techniques and innovative computational meshes.

We have published a book that is based on the lectures from the 2008 Summer Colloquium:
Lauritzen, P. H., C. Jablonowski, M. A. Taylor and R. D. Nair (Eds.) (2011), Numerical Techniques for Global Atmospheric Models, Lecture Notes in Computational Science and Engineering, Springer, Vol. 80, 572 pp.

Figure: Selected results of the dynamical core intercomparison project conducted during the 2008 NCAR ASP summer school. The figure shows a cross section of an advected slotted ellipse after 12 days that was transported up and down and around the sphere via an analytically prescribed 3D wind field. The initial condition and reference solution can be used to assess the characteristics of the advection schemes in the participating models denoted by the acronyms. The horizontal grid spacings are approximately 110 km with 60 levels in the vertical direction (vertical grid spacing is 250 m).

Adaptive Mesh Refinement and Variable-Resolution Grids for Atmospheric Simulations

Adaptive Mesh Refinement (AMR) techniques provide an attractive framework for atmospheric flows since they allow an improved resolution in limited regions without requiring a fine grid resolution throughout the entire model domain. The model regions at high resolution are kept at a minimum and can be individually tailored towards the research problem associated with atmospheric model simulations.

The climate system is characterized by complex nonlinear interactions over a broad range of temporal and spatial scales. Our research objective is to determine how these multi-scales interact and how to use enabling computational tools to mathematically represent scale interactions in climate models. The research focuses in particular on scale interactions in the so-called dynamical core of Atmospheric General Circulation Models (GCM). The dynamical core refers to the fluid dynamics component of a GCM and encompasses the numerical methods used to solve the equations of motion on the resolved scales. The research explores how Adaptive Mesh Refinement (AMR) and other variable resolution grid techniques allow high resolution meshes in regions of interest like the eye of a tropical cyclone or over mountainous terrain. It thereby suggests pathways how to bridge the scale discrepancies between local, regional and global phenomena, a key frontier in climate modeling.

The variable-resolution approach is focused on cubed-sphere computational meshes that have the potential to become a standard in future GCMs. Cubed-sphere grids offer an almost uniform grid point coverage on the sphere. They deliver high performance and almost perfect scaling characteristics on massively parallel computer architectures. The grid is ideally suited for local grid refinements that are based on DoE’s AMR software framework Chombo from the Lawrence Berkeley National Laboratory (LBNL). Chombo is under development by DoE’s Applied Partial Differential Equations Center (APDEC) under the leadership of our collaborator Dr. Phillip Colella. Both hydrostatic and nonhydrostatic dynamical core designs are developed, implemented and assessed in our team using high-order conservative and oscillation-free finite-volume numerical schemes. In addition, we investigate the numerical schemes in a 2d shallow-water framework that serves as an ideal testbed for 3d model developments. At a later stage our simulations will also involve an in-depth investigation of the validity of physical parameterizations at different spatial scales. The latter will be done in close collaboration with the National Center for Atmospheric Research (NCAR).


Figure: Examples of the adaptive mesh refinement and variable-resolution techniques applied to (left, first) a block-structured Finite-Volume (FV) shallow water model on a latitude-longitude grid, (middle, second) the block-structured Chombo shallow water model on the cubed-sphere grid, and (right, third) the conformal cubed-sphere grid in NCAR’s Spectral Element (SE) model. In the model FV all blocks are self-similar and contain 9×6 additional grid points. The left figure shows how the refined regions track the relative vorticity fields of a barotropic instability in the model FV (see St-Cyr et al., 2008), the middle figure depicts the vorticity field of two merging vortices after four days with the Chombo model (see Ferguson et al., 2016) and the right (third) figure illustrates a static (non-moving) variable-resolution mesh in the model CAM-SE (see e.g. Zarzycki and Jablonowski, 2014, 2015).

High-Order Finite-Volume Methods for the Dynamical Cores of GCMs

We develop, implement and test high-order finite-volume methods for the dynamical cores of atmospheric general circulation models. Our computational grid of choice is the cubed-sphere grid that is also depicted above. This project pays special attention to 3rd-order and 4th-order accurate discretizations that perform well at all spatial scales and all aspect ratios between the horizontal and vertical resolution. The methods are promising candidate for future non-hydrostatic dynamical cores with flexible computational grids. Such grids are for example the block-structured AMR grids that overlay the cubed-sphere geometry. We have developed the non-hydrostatic dynamical core ‘MCore’ in both a Cartesian channel configuration (Ullrich and Jablonowski, 2012, Mon. Wea. Rev. ) and on a spherical cubed-sphere grid (Ullrich and Jablonowski, 2012, J. Comput. Phys.).

IMEX Overview
Figure: Example simulations with our high-order finite-volume model in a Cartesian channel configuration. The figure depicts 2D and 3D idealized test cases that assess the model performance at small scales (2D thermal bubble), meso-scales (2D mountain waves) and large planetary scales (3D breaking baroclinic wave in a channel). The results are presented in the paper Ullrich and Jablonowski (2012) in the journal Monthly Weather Review.

Tropical Cyclones in General Circulation Models

This project investigates and improves the representation of tropical cyclones in the National Center for Atmospheric Research (NCAR) Community Earth System Model CESM. CESM is jointly supported by the Department of Energy (DOE) and the National Science Foundation and contains the Community Atmosphere Model CAM that offers multiple dynamical cores and physics options. In particular, CAM version 5 contains the finite-volume (FV) dynamical core, the spectral element (SE) dynamical core, and the spectral transform Eulerian (EUL) and semi-Lagriangian (SLD) dynamical cores. The most recent CESM version 2.2 with its CAM 6 atmosphere model provides access to GFDL’s cubed-sphere finite-volume dynamical core FV3 and the dynamical core of the Mpdel for Prediction Across Scales (MPAS).

The objectives of the research are to (1) simulate the evolution of an idealized, initially weak vortex into a tropical cyclone in an aqua-planet configuration of CAM, (2) investigate the sensitivity of tropical cyclone development and structure to varying resolutions and convective parameterizations within CAM, (3) explore the impact of different numerical schemes on the evolution of tropical cyclones through utilization of numerous dynamical cores available in CAM, and (4) use the knowledge gained from these simulations to project the possible effects of climate change on long-term (decadal) tropical cyclone statistics. Such process studies will reveal the impact and relative importance of the physical parameterizations and numerical schemes on the simulations of tropical cyclones in CAM, thereby providing an improved scientific basis for the projections of tropical cyclone activity under changing climate conditions.

We have developed an idealized tropical cyclone test case based on analytic initial conditions that spins up intense (up to category-5) tropical cyclones over the course of 5-10 simulation days. We have also developed a simplified physics package called ‘simple physics’ for aqua-planet studies that only contains the most basic driving mechanisms for cyclones such as surface fluxes, boundary layer diffusion and large-scale condensation. This physics package lets us define a test case of intermediate complexity.

Tropical Cyclones results
Figure: Idealized cyclone simulations with NCAR’s CAM 3.1 model in aqua-planet configuration (using the Finite Volume (FV) dynamical core). The top row displays the magnitude of the wind near the surface at the resolution 0.25 x 0.25 degrees with an initial maximum wind of 17.8 m/s. The initial radius of maximum wind is 200 km. Results for the initial vortex (day 0), and the vortex after 5 and 10 days are shown. The bottom row displays the wind magnitude for a vertical cross section through the center latitude of the vortex as a function of the radius from the center of the vortex [see also Reed and Jablonowski (Mon. Wea. Rev., 2011), Reed and Jablonowski (JAMES 2011), Reed and Jablonowski (Geophys. Lett., 2011) and Reed and Jablonowski (JAMES 2012)].

Stratospheric Dynamics: The Quasi-Biennial Oscillation (QBO) and Sudden Stratospheric Warmings (SSWs) in Atmospheric Dynamical Cores

The characteristics of the QBO and SSWs can be assessed in idealized dynamical core simulations with NCAR’s Community Atmosphere Model (CAM). The QBO is a phenomenon that takes place in the equatorial stratosphere where the zonal wind oscillates between the westward and eastward phase with a period of about 28 months. SSWs are occasional collapes of the westerly polar jet in the polar stratosphere. Both phenomena are wave driven and excellent examples of wave-mean flow interactions. The QBO is believed to be driven by gravity waves and equatorially trapped vertically propagating waves, in particular Kelvin and Mixed Rossby-Gravity waves, which act as momentum sources. In nature, these waves are e.g. triggered by tropical convection. SSWs are driven by upward propagating long Rossby waves (with wavenumber 1 or 2) which originate in the tropospheric midlatitudes.

In order to analyze the QBO behavior and the phenomena that contribute to its forcing we e.g. use the Transformed Eularian Mean (TEM) analysis as well as wavenumber-frequency assessments (Wheeler-Kiladis). Wavenumber-frequency diagrams reveal if certain wave types are present in the data, such as Kelvin or mixed Rossby-Gravity waves. Kelvin waves are expected to have periods of about 15 days, with zonal wavenumber of about 1-2, and usually observed when the mean zonal flow is easterly. Mixed Rossby-Gravity waves are expected to have a 4-5-day period, with a zonal wavenumber of around 4. They are usually observed when the mean zonal flow is westerly. The figure below illustrates such analyses for the tropical region, derived from CAM model data with the semi-Langrian dynamical core. We also apply our analysis technique to Sudden Stratospheric Warming (SSW) events to shed light on the interplay between waves and the mean flow, and investigate the impact of gravity wave drag parameterizations on the dynamics-physics interaction. These analyses reveal what the impact of the numerical dynamical core design is on the modeled stratospheric circulation.

Figure: Example of an idealized dynamical core simulation with NCAR’s semi-Lagrangian spectral transform model (driven by the so-called Held-Suarez forcing). The figure shows the zonal-mean monthly-mean zonal wind at the equator. The oscillation resembles the Quasi-Biennial Oscillation (QBO) in the stratosphere (see e.g. Yao and Jablonowski, 2013, 2015, 2016).

Dynamical Core Test Cases and Intercomparisons

The dynamical core of atmospheric General Circulation Models (GCMs) comprises the fluid dynamics component of every climate and weather forecasting model. Tests of GCMs and, in particular, tests of their dynamical cores are important steps towards future model improvements. They reveal the influence of an individual model design on climate and weather simulations and indicate whether the circulation is described representatively by the numerical approach. However, testing a global 3D atmospheric model and it dynamical core is not straightforward. In the absence of non-trivial analytic solutions, the model evaluations most commonly rely on intuition, experience and model intercomparisons.

We develop idealized test cases for dynamical cores and conduct international Dynamical Core Model Intercomparison Projects (DCMIP) (e.g. at NCAR in June 2008, DCMIP-2012 in August 2012 and DCMIP-2016 in June 2016). An example of a dynamical core test is the evolution of a baroclinic wave that is also depicted in the figure below. In addition, we work on test cases with intermediate complexity that include simple moisture feedbacks, e.g. for tropical cyclone-like simulations. Our most recent test case investigates the response of the dynamical cores to topographic forcings.

Figure: Surface pressure field at day 9 of the baroclinic instability test case simulated with 9 different dynamical cores. The tests starts with balanced initial conditions that are overlaid by a Gaussian hill perturbation. The grid imprint of the cubed-sphere and icosahedral grids can be seen in the Southern Hemisphere (GEOS-FVCUBE, GME, ICON, OLAM). Spectral ringing appears in CAM-EUL and HOMME. The baroclinic wave test is documented in Jablonowski and Williamson (QJ, 2006) and in the Jablonowski and Williamson NCAR Technical Report TN-469+STR (2006).

Subgrid-Scale Mixing in Climate Models: A Novel Look at Diffusion, Accuracy, Stability and Climate Sensitivity

It is important to assess and quantify the role of unresolved subgrid-scale mixing processes in the dynamical cores of state-of-the-art General Circulation Models (GCMs). The representation of the subgrid scale in GCMs is complex. Besides physical processes to be represented, numerical errors manifest themselves as subgrid-scale diffusion, and mixing is used to assure numerical stability and to compensate for numerical dispersion errors. The quantitative effect of physical mixing is therefore conflated with mixing processes associated with filtering, implicit and explicit diffusion and numerical errors. This raises new questions concerning the accuracy, stability and climate sensitivity of today’s climate modeling approaches.

We focus our research on the numerical schemes used in NCAR’s Community Atmosphere Model version 5 (CAM 5 and version 6 (CAM6). These two model configurations have options for several, very different dynamical cores. These are the spectral-transform Eulerian and semi-Lagrangian dynamical cores, the GFDL’s Finite Volume dynamical core on a latitude-longitude grid (FV) anf the cubed-sphere grid (FV3), the spectral element (SE) dynamics package as well as MPAS. All dynamical cores are well established and propagate resolved, long-scale, structures with credible accuracy. However, they all treat small-scales differently, from the point of view of both physical processes and numerical construction. In our research we perform a set of numerical experiments with increasing complexity. In particular, we analyze the subgrid-scale characteristics of idealized dynamical core experiments like baroclinic instability studies, perform long-term Held-Suarez climate runs with idealized forcing functions and assess aqua-planet simulations with intermediate complexity. In addition, the subgrid-scale mixing in tracer transport experiments is investigated. We develop and apply evaluation techniques and test whether the lessons learned in simplified experiments are predictors of the performance in climate models. In 2011 we published a 113-page Springer book chapter on the pros and cons of diffusion, filters and fixers in GCMs that sheds light on the many subgrid-scale mechanisms in today’s weather and climate models.

Figure: Surface pressure (hPa) and 850 hPa meridional velocity (m/s) at day 9 of the growing baroclinic wave test case of Jablonowski and Williamson (2006) from the CAM FV dynamical core at a 1×1 degree resolution (about 110 km) with 26 vertical levels. Simulations with (a,b): default second-order divergence damping, (c,d) fourth-order divergence damping, (e,f) no divergence damping. The unusual contour interval for the v wind emphasizes the very weak oscillations in (d).

Updated on January/15/2023