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Showing 1–30 of 1709 reference entries

Mathematics & logicExamples and counterexamples

Learn what a claim covers, explain why it works, and recognise an allowed case that shows it needs changing.

MethodMathematics & logicCounting and zero

Count each object once, explain the final count, and distinguish quantity, position and identifying labels. Make zero’s role explicit.

DefinitionMathematics & logicNumber lines and order

Use number-line positions to compare whole numbers. Read the scale, recognise equality, and distinguish numerical distance from drawing size.

DefinitionMathematics & logicAddition and subtraction

Combine separate counts, remove a part, and find differences or missing parts. Connect objects, equations and number-line moves, then use tens and ones to regroup without changing value.

DefinitionMathematics & logicMultiplication and division

Count equal groups, distinguish sharing from making groups, and check with multiplication. Split calculations into parts; interpret zero and remainders in context.

DefinitionMathematics & logicFractions

Choose a whole, count equal fractional units, and connect sharing with division. Place fractions on the number line, including values beyond one, and explain what their numerator and denominator mean.

DefinitionMathematics & logic · Physics · MeasurementRatios and rates

Distinguish part-to-part ratios, fractions of a whole and rates with units. Scale quantities together, compare unit rates, and explain when a relationship is proportional. Check zero cases, starting amounts and whether a result fits its context.

DefinitionMathematics & logic · Physics · MeasurementQuantities and units

Describe a quantity as a numerical value with a unit, and distinguish its physical kind from its dimension. Build length, area, volume and rate units, check meaningful combinations, and separate exact unit relations from measurement uncertainty.

DefinitionMathematics & logic · Physics · MeasurementUnit conversions

Convert quantities with exact unit ratios, track cancellation through compound units, and apply the correct powers to area and volume. Distinguish temperature readings from differences, check dimensions and context, and retain measurement uncertainty when reporting.

MethodMathematics & logic · Physics · MeasurementEstimation and uncertainty

Distinguish exact model values, rounded approximations and estimates from measurements. Declare the rounding rule, preserve information through calculations and unit changes, and interpret resolution, precision, accuracy and the stated uncertainty type without turning a report’s digits into guaranteed bounds.

DefinitionMathematics & logicVariables and expressions

Use symbols to name numbers or quantities, describe their allowed values, and evaluate expressions by replacing every occurrence consistently. Preserve grouping, signs, fractions and units, and distinguish a calculated value from solving an equation or proving two rules equivalent.

DefinitionMathematics & logicEquality and equations

Read equality as a relationship, find the permitted values that make an equation true, and justify one-step solutions by reversible operations on both sides. Work exactly with negative numbers and fractions, recognize all-solution and no-solution cases, and check the original equation, units and context.

DefinitionMathematics & logic · PhysicsCoordinates and tables

Locate a point from an ordered pair, read named axes and their scales, and connect the same values in words, tables and plots. Keep signs, fractions and units exact, distinguish an axis point from a quadrant point, and separate selected observations from a stated rule and its domain before drawing connections.

DefinitionMathematics & logic · PhysicsSlope and straight lines

Read a straight line as a stated relationship, calculate its slope from consistent signed coordinate changes, and interpret its intersections with the axes. Keep horizontal, vertical and coincident-point cases distinct, attach units and domain conditions to context models, and use numerical scales rather than appearance alone to judge a line’s change.

DefinitionMathematics & logic · Physics · MeasurementReading graphs

Read a graph’s quantities, units, ticks and context before judging its visual impression. Calculate differences, reference-based percentages and interval rates from the represented values; distinguish legitimate zoom from distorted spacing; and keep samples, interpolation, missing information, uncertainty and causal explanations separate.

MethodPhysics · MeasurementObservables and measurement

Choose the physical quantity you want to investigate, specify the point, event or location involved, and explain how a reading relates to that quantity.

MethodPhysicsSystems and boundaries

Choose what belongs to an investigation, describe its condition at a stated time, and identify interactions with what lies outside its boundary.

MethodPhysicsModels and assumptions

Choose what a model represents, make its simplifications explicit, and check whether its predictions answer the physical question you actually asked.

MethodPhysicsHypotheses, laws and theories

Identify what a scientific statement does, examine its evidence and conditions, and avoid treating its name as a score for certainty.

MethodMathematics & logic · PhysicsApproximations

Build a useful estimate, compare a correction with a stated reference scale, and check what is lost when you leave out a term.

ApproximationMathematics & logic · PhysicsSymmetry and invariance

Specify a transformation, identify what stays unchanged, and distinguish symmetry of an object from invariance of a physical model.

Mathematical statementPhysicsConservation laws

Use a defined system and a named quantity to track transfers, distinguish changing parts from conserved totals, and check the conditions of energy and momentum balances.

ModelPhysicsEmergence and levels of description

Choose a descriptive level, connect collective quantities with component behavior, and check when a simpler physical description answers a specified question.

ModelMathematics & logic · PhysicsVectors

Describe displacement with fixed-axis components, combine compatible vectors, and use small exact examples to introduce span, independence, basis and real vector spaces.

DefinitionPhysics · Information · MeasurementScientific records

Build an inspectable physics record, test and revise a bounded cart model, preserve processing history, and plan findable data with appropriate sharing conditions.

MethodPhysicsPosition, velocity and acceleration

Describe a point's motion on a chosen straight axis, then connect signed changes, instantaneous rates and one exact set of position, velocity and acceleration graphs.

DefinitionPhysicsNewton’s laws and forces

Choose one body, identify the forces acting on it, and use their vector sum to distinguish acceleration, balanced motion and mutual interactions.

ModelPhysicsWork, energy and power

Connect force and displacement to changes in motion, follow a spring’s stored energy, and distinguish an energy transfer from its rate.

ModelPhysics · MeasurementSpeed of light in vacuum

c denotes the speed of light in vacuum. Its numerical value is fixed exactly in SI units.

DefinitionPhysics · MeasurementPlanck constant

h denotes the Planck constant. Its numerical value is fixed exactly in SI units.

Definition
ModelSource checked

Rotating shallow-water equations

Earth & cosmos

For an inviscid uniform-density layer over a flat fixed bottom, depth-averaged motion obeys u_t+u·∇_hu+fe_z×u=−g∇_hh and h_t+∇_h·(hu)=0.

Key knowledge

For an inviscid uniform-density layer over a flat fixed bottom, depth-averaged motion obeys u_t+u·∇_hu+fe_z×u=−g∇_hh and h_t+∇_h·(hu)=0. Thickness h>0 determines hydrostatic pressure gradients while horizontal flux divergence changes water storage. A reduced-gravity active layer uses its specified restoring coefficient g′ instead of free-surface gravity g.

Values and equations

u_t+u·∇_hu+fe_z×u=−g∇_hh; h_t+∇_h·(hu)=0

u
depth-averaged horizontal velocity vector m s⁻¹
h
positive layer thickness above a flat bottom m
f
local Coriolis parameter s⁻¹
g
free-surface gravitational restoring coefficient m s⁻²
e_z
upward unit vector 1
t
time s

Conditions and scope

  • Hydrostatic long horizontal scales relative to layer depth, negligible vertical velocity structure, incompressible constant density, no mass source, stress or topographic pressure term in this displayed flat-bottom model.
  • Supply initial state and boundary conditions. Variable bottom relief, wind/bottom stress, multiple layers, compressibility and short nonhydrostatic waves need explicitly modified terms; depth averaging is not a full three-dimensional closure.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. A Coriolis Tutorial, Part2 §§2.1,2.3.3 and4.1–4.3James F. Price — Woods Hole Oceanographic Institution / MIT OpenCourseWareReference guide · PDFocw.mit.eduRead source

Evidence and review

Read actual§2.1 control-volume mass equations1–8 and hydrostatic momentum equations9–16 plus reduced-gravity definition. Specialized to the flat-bottom free-surface limit, with the reduced-gravity variant separately named; no simulation or numerical-discretization claim.

2026-10-10 · Independent AI editorial review: All73 exact supplement compact statements, conditions, displayed equations and247 origin semantic links read; all109 combined original36+supplement73 author-source rows, exact locators, hypotheses, exclusions and disclosed derivations assessed. Original36 exact scientific content is independently approved in the linked receipt. All18 inherited Earth compact references and shared Math/Physics/Measurement methods were considered in the full practical Earth inventory. Targeted actual primary sections listed below; not every source body/proof or every possible university elective independently reread. AI editorial reference review, not human certification, full-degree proficiency, new experimental validation or a claim of whole combined Earth-and-Cosmos approval. Receipt: earth-completion-independent-content-inventory-review.1.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–9 of 9 relationships

Scientific relationships · 9 / 9

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Related to
    Relationship sources · 1

    Linked statements

    • For a one-dimensional incompressible reach, ∂(A+A_s)/∂t+∂Q/∂x=q_L balances conveying area A, additional nonconveying storage area A_s and discharge Q. Positive q_L adds water per unit channel length. Integrating between sections gives the rate of water-volume storage equal to upstream-minus-downstream discharge plus lateral input.

    Momentum and water continuity close a compatible depth-averaged model.

  2. Related to
    Relationship sources · 2

    Linked statements

    • For two uniform-density layers separated by z_i(y), pressure continuity across the interface relates its slope to along-front geostrophic velocities: dz_i/dy=f(ρ₁u₁−ρ₂u₂)/[g(ρ₂−ρ₁)]. Layer1 is above layer2 withρ₂>ρ₁, x runs along the front,eastward in the local convention,and z is upward. The density contrast and velocity difference jointly support a sloping interface.

    Layered dynamics provides a fuller evolving model.

  3. Derived from
    Relationship sources · 1

    Linked statements

    • For an inviscid uniform-density layer over a flat fixed bottom, depth-averaged motion obeys u_t+u·∇_hu+fe_z×u=−g∇_hh and h_t+∇_h·(hu)=0. Thickness h>0 determines hydrostatic pressure gradients while horizontal flux divergence changes water storage. A reduced-gravity active layer uses its specified restoring coefficient g′ instead of free-surface gravity g.

    The depth pressure relation follows from the stated vertical approximation.

  4. Related to
    Relationship sources · 1

    Linked statements

    • For an inviscid uniform-density layer over a flat fixed bottom, depth-averaged motion obeys u_t+u·∇_hu+fe_z×u=−g∇_hh and h_t+∇_h·(hu)=0. Thickness h>0 determines hydrostatic pressure gradients while horizontal flux divergence changes water storage. A reduced-gravity active layer uses its specified restoring coefficient g′ instead of free-surface gravity g.

    The local Coriolis term comes from a rotating reference frame.

  5. Related to
    Relationship sources · 1

    Linked statements

    • For an inviscid uniform-density layer over a flat fixed bottom, depth-averaged motion obeys u_t+u·∇_hu+fe_z×u=−g∇_hh and h_t+∇_h·(hu)=0. Thickness h>0 determines hydrostatic pressure gradients while horizontal flux divergence changes water storage. A reduced-gravity active layer uses its specified restoring coefficient g′ instead of free-surface gravity g.

    Both depth-integrated models retain mass flux divergence.

  6. Related to
    Relationship sources · 1

    Linked statements

    • For an inviscid uniform-density layer over a flat fixed bottom, depth-averaged motion obeys u_t+u·∇_hu+fe_z×u=−g∇_hh and h_t+∇_h·(hu)=0. Thickness h>0 determines hydrostatic pressure gradients while horizontal flux divergence changes water storage. A reduced-gravity active layer uses its specified restoring coefficient g′ instead of free-surface gravity g.

    Taking the curl and combining continuity gives a bounded conserved scalar.

  7. Derived from
    Relationship sources · 1

    Linked statements

    • In a smooth inviscid single-layer shallow-water model without mass exchange or external torque, q=(ζ+f)/h is materially conserved: Dq/Dt=0, with ζ=∂v/∂x−∂u/∂y. Column stretching or movement across a changing planetary f must be accompanied by compatible relative-vorticity change. Potential vorticity moves with the fluid rather than at a gravity-wave phase speed.

    Curl and thickness conservation give the stated scalar law.

  8. Derived from
    Relationship sources · 1

    Linked statements

    • Linearizing a uniform-depth rotating shallow-water layer at rest on an f-plane yields propagating roots ω²=f²+gH(k_x²+k_y²) and a balanced zero-frequency mode. The nonrotating long-wave speed is c=√(gH), and the rotational deformation length is L_d=c/|f|. Rotation and gravity therefore control different time and length limits of the same model.

    Linearizing the specified layer equations gives these modes.

  9. Related to
    Relationship sources · 1

    Linked statements

    • Differential lunar and solar gravity supplies tidal forcing, but ocean height and current respond through rotating fluid dynamics,bathymetry,coastlines and dissipation. Inertia and friction cause phase lags; basin response can be semidiurnal,mixed or diurnal. High tide therefore need not occur when the Moon is overhead,and water-level tide and tidal current are different observables.

    A forced rotating layer represents part of the basin response.

Entry detailsPermanent link to this entry ↗
Record identity
uni-earth-shallow-water-dynamics
Revision
core.1
Structured reference record
University core expansion