{"id":663,"date":"2026-08-26T14:12:34","date_gmt":"2026-08-26T14:12:34","guid":{"rendered":"https:\/\/summergeometry.org\/sgi2026\/?p=663"},"modified":"2026-08-26T16:30:34","modified_gmt":"2026-08-26T16:30:34","slug":"vortex-loops-for-jupiters-stripes","status":"publish","type":"post","link":"https:\/\/summergeometry.org\/sgi2026\/vortex-loops-for-jupiters-stripes\/","title":{"rendered":"Vortex Loops for Jupiters’ Stripes"},"content":{"rendered":"\n

Project Members<\/strong>: Aleksa Milovanovi\u0107, Reid Tang<\/p>\n\n\n\n

Mentor<\/strong>: Sadashige Ishida<\/p>\n\n\n\n

Introduction to Fluid Dynamics<\/h4>\n\n\n\n

Fluid motion in computer graphics is typically modeled by the incompressible Navier-Stokes equations:<\/p>\n\n\n\n

\u2202<\/mi>u<\/mi>\u2192<\/mo><\/mover><\/mrow>\u2202<\/mi>t<\/mi><\/mrow><\/mfrac>+<\/mo>u<\/mi>\u2192<\/mo><\/mover>\u22c5<\/mo>\u2207<\/mo>u<\/mi>\u2192<\/mo><\/mover>+<\/mo>1<\/mn>\u03c1<\/mi><\/mfrac>\u2207<\/mo>p<\/mi>=<\/mo>g<\/mi>\u2192<\/mo><\/mover>+<\/mo>\u03bd<\/mi>\u2207<\/mo>\u22c5<\/mo>\u2207<\/mo>u<\/mi>\u2192<\/mo><\/mover>,<\/mo><\/mrow>\\frac{\\partial \\vec{u}}{\\partial t} + \\vec{u} \\cdot \\nabla \\vec{u} + \\frac{1}{\\rho} \\nabla p = \\vec{g} + \\nu \\nabla \\cdot \\nabla \\vec{u},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
\u2207<\/mo>\u22c5<\/mo>u<\/mi>\u2192<\/mo><\/mover>=<\/mo>0<\/mn>,<\/mo><\/mrow>\\nabla\\cdot\\vec{u} = 0,<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where <\/math> is the velocity of the fluid, <\/math> density, <\/math> pressure, <\/math> body force and <\/math> kinematic viscosity. The first equation (momentum) is Newton’s second law applied to fluid elements, and the second one constrains the flow to be volume-preserving.<\/p>\n\n\n\n

An important concept is the material derivative<\/strong>, <\/math> which measures the rate of change of a quantity following a fluid parcel (Lagrangian) rather than at a fixed point in space (Eulerian).<\/p>\n\n\n\n

This allows us to write the momentum equation compactly as<\/p>\n\n\n\n

D<\/mi>u<\/mi>\u2192<\/mo><\/mover><\/mrow>D<\/mi>t<\/mi><\/mrow><\/mfrac>+<\/mo>1<\/mn>\u03c1<\/mi><\/mfrac>\u2207<\/mo>p<\/mi>=<\/mo>g<\/mi>\u2192<\/mo><\/mover>+<\/mo>\u03bd<\/mi>\u2207<\/mo>\u22c5<\/mo>\u2207<\/mo>u<\/mi>\u2192<\/mo><\/mover>,<\/mo><\/mrow>\\frac{D\\vec{u}}{Dt} + \\frac{1}{\\rho}\\nabla p = \\vec{g} + \\nu\\nabla\\cdot\\nabla\\vec{u},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

and defines advection s.t. <\/math> means the quantity is carried by the flow without changing.<\/p>\n\n\n\n

Vorticity<\/h4>\n\n\n\n

Vorticity<\/strong> is defined as the curl of velocity <\/math>, and measures the local rate of rotation of the fluid. For inviscid flow with no body force, taking the curl of the momentum equation gives the vorticity transport equation<\/p>\n\n\n\n

D<\/mi>\u03c9<\/mi>\u2192<\/mo><\/mover><\/mrow>D<\/mi>t<\/mi><\/mrow><\/mfrac>=<\/mo>(<\/mo>\u03c9<\/mi>\u2192<\/mo><\/mover>\u22c5<\/mo>\u2207<\/mo>)<\/mo>u<\/mi>\u2192<\/mo><\/mover><\/mrow>\\frac{D\\vec{\\omega}}{Dt} = (\\vec{\\omega}\\cdot\\nabla)\\vec{u}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where the right-hand side is the vortex-stretching term (as shown in Section 2.1, this term vanishes identically in the 2D setting used throughout this report, so vorticity there is materially conserved exactly, <\/math>); it is simply carried by the flow. This makes vorticity important for simulating turbulent and swirl-dominated phenomena, since instead of solving for pressure and velocity everywhere, one only tracks regions where vorticity is concentrated.<\/p>\n\n\n\n

The key problem is recovering the velocity field given a vorticity field. In 3D, this is posed as a least-squares problem, i.e. find <\/math> minimizing<\/p>\n\n\n\n

\u222d<\/mo>\u2016<\/mi>\u2207<\/mo>\u00d7<\/mo>u<\/mi>\u2192<\/mo><\/mover>\u2212<\/mo>\u03c9<\/mi>\u2192<\/mo><\/mover>\u2016<\/mi>2<\/mn><\/msup><\/mspace>d<\/mi>V<\/mi><\/mrow>\\iiint \\|\\nabla\\times\\vec{u} – \\vec{\\omega}\\|^2\\,dV<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

solved via calculus of variations, yielding the Poisson problem <\/math>, whose solution is the Biot-Savart law<\/strong>:<\/p>\n\n\n\n

u<\/mi>\u2192<\/mo><\/mover>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>)<\/mo>=<\/mo>\u222d<\/mo>\u2212<\/mo>\u2207<\/mo>\u03a6<\/mi><\/mrow>3<\/mn><\/msub>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>\u2212<\/mo>p<\/mi>\u2192<\/mo><\/mover>)<\/mo>\u00d7<\/mo>\u03c9<\/mi>\u2192<\/mo><\/mover>(<\/mo>p<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mspace>d<\/mi>p<\/mi>\u2192<\/mo><\/mover>,<\/mo><\/mspace>\u03a6<\/mi><\/mrow>3<\/mn><\/msub>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>)<\/mo>=<\/mo>\u2212<\/mo>1<\/mn>4<\/mn>\u03c0<\/mi>\u2016<\/mi>x<\/mi>\u2192<\/mo><\/mover>\u2016<\/mi><\/mrow><\/mfrac><\/mrow>\\vec{u}(\\vec{x}) = \\iiint -\\nabla\\Phi_3(\\vec{x} – \\vec{p})\\times\\vec{\\omega}(\\vec{p})\\,d\\vec{p}, \\quad \\Phi_3(\\vec{x}) = -\\frac{1}{4\\pi\\|\\vec{x}\\|}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
Deriving the 2D analogue<\/h5>\n\n\n\n

Jupiter’s bands and jet streams are a 2D phenomenon (a thin shell on the planet’s surface), so we re-derive the above for a 2D velocity field <\/math>, where vorticity reduces to a scalar:<\/p>\n\n\n\n

\u03c9<\/mi>=<\/mo>\u2202<\/mi>v<\/mi><\/mrow>\u2202<\/mi>x<\/mi><\/mrow><\/mfrac>\u2212<\/mo>\u2202<\/mi>u<\/mi><\/mrow>\u2202<\/mi>y<\/mi><\/mrow><\/mfrac>=<\/mo>:<\/mo><\/mtext>curl<\/mi><\/mrow>2<\/mn><\/msub>(<\/mo>u<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mrow>\\omega = \\frac{\\partial v}{\\partial x} – \\frac{\\partial u}{\\partial y} =: \\mathrm{curl}_2(\\vec{u})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

In this 2D setting <\/math> points purely out of the plane while <\/math> is purely in-plane, so the vortex-stretching term <\/math> from Section 2 vanishes identically \u2014 not as an extra assumption, but automatically, because there is no out-of-plane direction left for vorticity to be stretched or tilted into. The vorticity transport equation therefore simplifies exactly to <\/math> in 2D, which is what we use throughout.<\/p>\n\n\n\n

As in 3D, we seek <\/math> minimizing<\/p>\n\n\n\n

\u222c<\/mo>(<\/mo><\/mtext>curl<\/mi><\/mrow>2<\/mn><\/msub>(<\/mo>u<\/mi>\u2192<\/mo><\/mover>)<\/mo>\u2212<\/mo>\u03c9<\/mi>)<\/mo>2<\/mn><\/msup><\/mspace>d<\/mi>A<\/mi><\/mrow>\\iint \\big(\\mathrm{curl}_2(\\vec{u}) – \\omega\\big)^2\\,dA<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Perturbing <\/math> and using linearity of <\/math>, the stationarity condition <\/math> gives, with <\/math>,<\/p>\n\n\n\n

\u222c<\/mo>f<\/mi><\/mspace><\/mtext>curl<\/mi><\/mrow>2<\/mn><\/msub>(<\/mo>p<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mspace>d<\/mi>A<\/mi>=<\/mo>0<\/mn> <\/mtext>, for all <\/mtext>p<\/mi>\u2192<\/mo><\/mover><\/mrow>\\iint f\\, \\mathrm{curl}_2(\\vec{p})\\,dA = 0 \\ \\text{, for all } \\vec{p}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Unlike 3D, <\/math> maps a vector field to a scalar, so its adjoint maps a scalar back to a vector. Integrating by parts term by term (boundary terms vanish as fields decay at infinity),<\/p>\n\n\n\n

\u222c<\/mo>f<\/mi><\/mspace><\/mtext>curl<\/mi><\/mrow>2<\/mn><\/msub>(<\/mo>p<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mspace>d<\/mi>A<\/mi>=<\/mo>\u222c<\/mo>p<\/mi>\u2192<\/mo><\/mover>\u22c5<\/mo>\u2207<\/mo>\u27c2<\/mo><\/msup>f<\/mi><\/mspace>d<\/mi>A<\/mi>,<\/mo><\/mspace>\u2207<\/mo>\u27c2<\/mo><\/msup>f<\/mi>\u2261<\/mo>(<\/mo>\u2202<\/mi>f<\/mi><\/mrow>\u2202<\/mi>y<\/mi><\/mrow><\/mfrac>,<\/mo>\u2212<\/mo>\u2202<\/mi>f<\/mi><\/mrow>\u2202<\/mi>x<\/mi><\/mrow><\/mfrac>)<\/mo><\/mrow><\/mrow>\\iint f\\,\\mathrm{curl}_2(\\vec{p})\\,dA = \\iint \\vec{p}\\cdot\\nabla^\\perp f\\,dA, \\quad \\nabla^\\perp f \\equiv \\left(\\frac{\\partial f}{\\partial y},-\\frac{\\partial f}{\\partial x}\\right)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

so that, since this must vanish for all <\/math>,<\/p>\n\n\n\n

\u2207<\/mo>\u27c2<\/mo><\/msup>(<\/mo><\/mtext>curl<\/mi><\/mrow>2<\/mn><\/msub>(<\/mo>u<\/mi>\u2192<\/mo><\/mover>)<\/mo>)<\/mo>=<\/mo>\u2207<\/mo>\u27c2<\/mo><\/msup>\u03c9<\/mi><\/mrow>\\nabla^\\perp\\big(\\mathrm{curl}_2(\\vec{u})\\big) = \\nabla^\\perp\\omega<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The same vector identity used in 3D,<\/p>\n\n\n\n

\u2207<\/mo>\u00d7<\/mo>\u2207<\/mo>\u00d7<\/mo>u<\/mi>\u2192<\/mo><\/mover>=<\/mo>\u2212<\/mo>\u2207<\/mo>\u22c5<\/mo>\u2207<\/mo>u<\/mi>\u2192<\/mo><\/mover>+<\/mo>\u2207<\/mo>(<\/mo>\u2207<\/mo>\u22c5<\/mo>u<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mrow>\\nabla\\times\\nabla\\times\\vec{u} = -\\nabla\\cdot\\nabla\\vec{u} + \\nabla(\\nabla\\cdot\\vec{u})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

restricted to 2D becomes <\/math>. With the constraint <\/math>, we obtain a Poisson problem:<\/p>\n\n\n\n

\u2212<\/mo>\u2207<\/mo>\u22c5<\/mo>\u2207<\/mo>u<\/mi>\u2192<\/mo><\/mover>=<\/mo>\u2207<\/mo>\u27c2<\/mo><\/msup>\u03c9<\/mi><\/mrow>-\\nabla\\cdot\\nabla\\vec{u} = \\nabla^\\perp\\omega<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The 2D fundamental solution of the Laplacian is<\/p>\n\n\n\n

\u03a6<\/mi><\/mrow>2<\/mn><\/msub>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>)<\/mo>=<\/mo>1<\/mn>2<\/mn>\u03c0<\/mi><\/mrow><\/mfrac>log<\/mi>\u2061<\/mo><\/mrow>\u2016<\/mi>x<\/mi>\u2192<\/mo><\/mover>\u2016<\/mi><\/mrow>\\Phi_2(\\vec{x}) = \\frac{1}{2\\pi}\\log\\|\\vec{x}\\|<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

satisfying <\/math>, verified by the divergence theorem on the unit circle:<\/p>\n\n\n\n

\u222e<\/mo>\u2207<\/mo>\u03a6<\/mi><\/mrow>2<\/mn><\/msub>\u22c5<\/mo>n<\/mi>^<\/mo><\/mover><\/mspace>d<\/mi>\u2113<\/mi>=<\/mo>\u222e<\/mo>1<\/mn>2<\/mn>\u03c0<\/mi><\/mrow><\/mfrac><\/mspace>d<\/mi>\u2113<\/mi>=<\/mo>1<\/mn><\/mrow>\\oint \\nabla\\Phi_2\\cdot\\hat{n}\\,d\\ell = \\oint \\frac{1}{2\\pi}\\,d\\ell = 1<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The solution to the Poisson problem, via convolution with <\/math>:<\/p>\n\n\n\n

u<\/mi>\u2192<\/mo><\/mover>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>)<\/mo>=<\/mo>\u222c<\/mo>\u03a6<\/mi><\/mrow>2<\/mn><\/msub>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>\u2212<\/mo>p<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mspace>\u2207<\/mo>\u27c2<\/mo><\/msup>\u03c9<\/mi>(<\/mo>p<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mspace>d<\/mi>p<\/mi>\u2192<\/mo><\/mover><\/mrow>\\vec{u}(\\vec{x}) = \\iint \\Phi_2(\\vec{x} – \\vec{p})\\,\\nabla^\\perp\\omega(\\vec{p})\\,d\\vec{p}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Integrating by parts once again (moving <\/math> off <\/math> onto <\/math>) gives the 2D Biot-Savart law<\/strong> in the form:<\/p>\n\n\n\n

u<\/mi>\u2192<\/mo><\/mover>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>)<\/mo>=<\/mo>\u222c<\/mo>\u03c9<\/mi>(<\/mo>p<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mspace>\u2207<\/mo>\u27c2<\/mo><\/msup>\u03a6<\/mi><\/mrow>2<\/mn><\/msub>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>\u2212<\/mo>p<\/mi>\u2192<\/mo><\/mover>)<\/mo><\/mspace>d<\/mi>p<\/mi>\u2192<\/mo><\/mover><\/mrow>\\vec{u}(\\vec{x}) = \\iint \\omega(\\vec{p})\\,\\nabla^\\perp\\Phi_2(\\vec{x}-\\vec{p})\\,d\\vec{p}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Discretizing <\/math> as a sum of Dirac deltas collapses the integral to a sum:<\/p>\n\n\n\n

u<\/mi>\u2192<\/mo><\/mover>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>)<\/mo>=<\/mo>\u2211<\/mo>i<\/mi><\/munder><\/mrow>\u0393<\/mi><\/mrow>i<\/mi><\/msub>2<\/mn>\u03c0<\/mi><\/mrow><\/mfrac>(<\/mo>x<\/mi>\u2192<\/mo><\/mover>\u2212<\/mo>x<\/mi>\u2192<\/mo><\/mover>i<\/mi><\/msub>)<\/mo>\u27c2<\/mo><\/msup><\/mrow>\u2016<\/mi>x<\/mi>\u2192<\/mo><\/mover>\u2212<\/mo>x<\/mi>\u2192<\/mo><\/mover>i<\/mi><\/msub>\u2016<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac><\/mrow>\\vec{u}(\\vec{x}) = \\sum_i \\frac{\\Gamma_i}{2\\pi} \\frac{(\\vec{x} – \\vec{x}_i)^\\perp}{\\|\\vec{x} – \\vec{x}_i\\|^2}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Since inviscid vorticity is materially conserved, each point vortex simply moves with the velocity induced by all other vortices at its own location (the self-term is excluded), giving the point vortex dynamics equation<\/strong>:<\/p>\n\n\n\n

d<\/mi>x<\/mi>\u2192<\/mo><\/mover>i<\/mi><\/msub><\/mrow>d<\/mi>t<\/mi><\/mrow><\/mfrac>=<\/mo>\u2211<\/mo>j<\/mi>\u2260<\/mo>i<\/mi><\/mrow><\/munder><\/mrow>\u0393<\/mi><\/mrow>j<\/mi><\/msub>2<\/mn>\u03c0<\/mi>\u2016<\/mi>x<\/mi>\u2192<\/mo><\/mover>i<\/mi><\/msub>\u2212<\/mo>x<\/mi>\u2192<\/mo><\/mover>j<\/mi><\/msub>\u2016<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>(<\/mo>\u2212<\/mo>(<\/mo>y<\/mi>i<\/mi><\/msub>\u2212<\/mo>y<\/mi>j<\/mi><\/msub>)<\/mo>,<\/mo><\/mspace>x<\/mi>i<\/mi><\/msub>\u2212<\/mo>x<\/mi>j<\/mi><\/msub>)<\/mo><\/mrow>\\frac{d\\vec{x}_i}{dt} = \\sum_{j\\neq i}\\frac{\\Gamma_j}{2\\pi\\|\\vec{x}_i-\\vec{x}_j\\|^2}\\big(-(y_i-y_j),\\,x_i-x_j\\big)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

This is the equation we implemented and verified on various examples (e.g. leapfrogging etc.) before extending it to the sphere.<\/p>\n\n\n\n

Implementation and Validation in the Plane<\/h4>\n\n\n\n

The point vortex dynamics equation above was implemented in Houdini using a custom SOP Solver: each timestep, a VEX wrangle reads the previous frame’s point cloud (position and circulation <\/math> per point), evaluates the Biot\u2013Savart sum, and advances each point’s position. Before trusting this on any nontrivial configuration, it was validated against two exactly-solvable two-vortex cases: equal circulations <\/math>, where both vortices orbit a common, stationary centroid on a circular path, and opposite circulations <\/math>, where the pair instead translates together in a straight line at constant speed. Both matched their analytic period\/speed formulas.<\/p>\n\n\n\n

As a further, qualitative check, a classical four-vortex leapfrogging<\/strong> configuration (two counter-rotating pairs, arranged so the trailing pair periodically overtakes the leading pair) was simulated and reproduced the expected periodic behavior.<\/p>\n\n\n\n

\"\"
Trajectories of the four-vortex leapfrogging configuration at three points in the cycle (frames 20, 80, 140). Each mark traces the recent path of one vortex; the pairs periodically exchange places as they orbit and overtake one another.<\/figcaption><\/figure>\n\n\n\n

A Vortex-Ring Model of Kelvin\u2013Helmholtz Instability<\/h4>\n\n\n\n

To move from the plane to Jupiter’s spherical atmosphere, the point vortex equation was extended to the unit sphere <\/math> by replacing the planar kernel with the spherical Biot\u2013Savart law and integrating positions via the exponential map, <\/math>, so that points remain exactly on the sphere regardless of step size. Because <\/math> has no boundary, the total vorticity must also integrate to zero, <\/math> \u2014 a constraint with no analogue in the plane, enforced here by construction (equal numbers of positive and negative vortices, or an explicit normalization step subtracting the mean circulation from every point).<\/p>\n\n\n\n

On this spherical setup, a shear layer was modeled as a vortex sheet: a ring of point vortices along a line of latitude, one sign on each side of the interface, with a small sinusoidal perturbation added to seed the instability. With no additional forcing, mutual induction alone causes the shear layer to roll up into a periodic array of vortices, as expected from classical vortex-sheet theory.<\/p>\n\n\n\n

\"\"
A single vortex-sheet interface at three stages: the initial perturbed line (frame 50), partial roll-up into discrete billows (frame 250), and a more developed, mixed state (frame 450).<\/figcaption><\/figure>\n\n\n\n

This was then generalized to several alternating-sign rings restricted to a narrower latitude band, approximating Jupiter’s alternating belt\/zone structure. Passive marker particles (zero circulation, contributing nothing to the velocity field but advected by it) were added and colored with a Jupiter-like palette for visualization.<\/p>\n\n\n\n

\"\"
Multi-band configuration (M=3) at frames 20, 250, and 500. Marker particles, seeded uniformly, are progressively advected and mixed along the band-induced shear flow.<\/figcaption><\/figure>\n\n\n\n

Modeling the Coriolis Effect<\/h4>\n\n\n\n

To incorporate planetary rotation, the momentum equation was written in a frame rotating with angular velocity <\/math>. Taking the curl and combining terms gives the vorticity equation with a rotational source,<\/p>\n\n\n\n

D<\/mi>\u03c9<\/mi><\/mrow>D<\/mi>t<\/mi><\/mrow><\/mfrac>=<\/mo>\u2212<\/mo>2<\/mn>u<\/mi>\u2192<\/mo><\/mover>\u22c5<\/mo>\u03a9<\/mi><\/mrow>\u2192<\/mo><\/mover>,<\/mo><\/mrow>\\frac{D\\omega}{Dt} = -2\\vec{u}\\cdot\\vec{\\Omega},<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

in place of the inviscid <\/math> used above. This was implemented by updating each vortex’s circulation every step, <\/math>, so that circulation is no longer materially conserved. This is a first, direct discretization of the effect; a more careful treatment (in particular, its interaction with the <\/math> constraint over long integration times) is left for future work.<\/p>\n\n\n\n

Discussion and Limitations<\/h4>\n\n\n\n

The point-vortex approach reproduces known analytic solutions in the plane, the classical leapfrogging trajectory, and qualitatively correct Kelvin\u2013Helmholtz roll-up on the sphere, both for a single interface and for a multi-band configuration resembling Jupiter’s belts and zones. The Coriolis implementation here is a direct, unrefined discretization rather than a fully validated one, and the multi-band results are qualitative: parameters such as band spacing, perturbation amplitude, and desingularization radius were chosen for a visually clear roll-up rather than calibrated against a specific analytic growth rate or against Jupiter’s actual physical parameters. Extending the integrator beyond forward Euler, and validating the Coriolis term against a known conserved quantity, are the natural next steps.<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Project Members: Aleksa Milovanovi\u0107, Reid Tang Mentor: Sadashige Ishida Introduction to Fluid Dynamics Fluid motion in computer graphics is typically modeled by the incompressible Navier-Stokes equations: where is the velocity of the fluid, density, pressure, body force and kinematic viscosity. The first equation (momentum) is Newton’s second law applied to fluid elements, and the second […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[110],"tags":[],"ppma_author":[55,17],"class_list":["post-663","post","type-post","status-publish","format-standard","hentry","category-research"],"authors":[{"term_id":55,"user_id":0,"is_guest":1,"slug":"cap-aleksa-milovanovic","display_name":"aleksa.milovanovic","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/?s=96&d=mm&r=g","author_category":"","first_name":"","last_name":"","user_url":"","job_title":"","description":""},{"term_id":17,"user_id":0,"is_guest":1,"slug":"cap-reidtang","display_name":"reidtang","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/?s=96&d=mm&r=g","author_category":"","first_name":"","last_name":"","user_url":"","job_title":"","description":""}],"_links":{"self":[{"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/posts\/663","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/comments?post=663"}],"version-history":[{"count":5,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/posts\/663\/revisions"}],"predecessor-version":[{"id":1184,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/posts\/663\/revisions\/1184"}],"wp:attachment":[{"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/media?parent=663"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/categories?post=663"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/tags?post=663"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2026\/wp-json\/wp\/v2\/ppma_author?post=663"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}