SCIENCE · INVESTIGATE AND EXPLAIN

Integrate glacier mass balance over an elevation-area distribution

A glacier gains mass through accumulation and loses it through ablation. The equilibrium line separates locally positive and negative annual surface balance for a stated interval, but glacier-wide balance also depends on how much area occupies each elevation band. The line’s height alone is not a complete mass budget.

Learning outcome: Hold the balance relation fixed while changing the highest-band area. Retain every band contribution and compare total with specific mean. Explain why an unweighted mean of the four band values is not generally correct.

Starting recap

Preserve flux, storage, geometry and source boundaries. Distinguish actual discharge records from original process models and conditional interpretations.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

At ELA 1800 m and areas 1,2,3,2 km², local balances are −1.6,−0.6,+0.4,+1.4 m water-equivalent/year. The integrated balance is +1.2 million m³ water/year and area mean +0.15 m/year; equivalent ice gain is about 1.33333 million m³/year.

Open the experiment

Predict

Can two glaciers with the same equilibrium-line elevation have different integrated mass balances?

Your prediction and reason

Experiment

Hold the balance relation fixed while changing the highest-band area. Retain every band contribution and compare total with specific mean. Explain why an unweighted mean of the four band values is not generally correct.

Integrate elevation-dependent glacier balance over unequal areas

Worked teaching model · synthetic values · Original elevation bands1000,1500,2000,2500 m with areas1,2,3,b km². Annual balance0.002(z−ELA) m water-equivalent/year. Integrate over area; convert water-equivalent volume to ice using1000/900. No actual glacier calibration or fixed future area prediction.

Equilibrium-line elevation: 1800 m Highest elevation-band area: 2 km²

Reset keeps your prediction, captures and practice history.

Glacier-wide balance weights each elevation band by area, not just by the sign of its local balance.

Accumulation and ablation vary spatially; equilibrium-line elevation alone does not uniquely determine integrated balance without hypsometry and a balance relation. Fixed areas and linear gradients are illustrative, not a forecast of evolving glacier geometry.

Complete conditional ledger with quantity-specific units
Quantity	Value	Unit
Total glacier area	8	km²
Integrated water-equivalent volume balance	1200000	m³ water/year
Area-mean specific balance	0.15	m water-equivalent/year
Equivalent ice-volume balance	1333333.33	m³ ice/year
Area above equilibrium line	5	km²

Elevation 1000 m: 1 km² at balance -1.6 m water-equivalent/year

Elevation 1500 m: 2 km² at balance -0.6 m water-equivalent/year

Elevation 2000 m: 3 km² at balance 0.4 m water-equivalent/year

Elevation 2500 m: 2 km² at balance 1.4000000000000001 m water-equivalent/year

Elevation (m), local balance (m water-equivalent/year)
x	y
1000	-1.6
1500	-0.6
2000	0.4
2500	1.4

Glacier-wide balance weights each elevation band by area, not just by the sign of its local balance.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the process and evidence

2. Connect mechanism and boundary

The original bands are centered at 1000,1500,2000 and 2500 m, with areas 1,2,3 and b km². Specific annual balance is stipulated as 0.002(z−ELA) m water-equivalent. Multiply each band’s balance by its area in square meters and sum; dividing by total area gives the mean specific balance.

Water-equivalent units express mass using water density, not actual snow or ice thickness. For a supplied ice density 900 kg/m³, V_ice = (1000/900) × V_water-equivalent: the ice volume is larger for the same mass. Mixing water and ice thickness without density conversion violates the mass interpretation.

The linear balance relation and fixed areas are original teaching assumptions. Real accumulation, melt, refreezing, debris and changing geometry vary spatially. A growing high-elevation area can alter total balance without moving the equilibrium line; a future geometry prediction requires an evolving ice-flow model rather than repeated static multiplication.

3. Reconstruct the worked packet

4. Predict and investigate

Predict first: Can two glaciers with the same equilibrium-line elevation have different integrated mass balances?

5. Test a changed case

Apply the evidence to a changed case

At ELA 2000 m with areas 1,2,3,2 km², find integrated and mean balance.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Balances are −2,−1,0,+1 m water-equivalent/year. Weighted volume is −2 million m³ water/year; mean is −0.25 m/year. The band exactly at ELA contributes zero.

6. Retain a defensible process notebook

Preserve first predictions, complete source/units, geometry, flux and storage boundaries, calculations and competing interpretations. Only the explicitly labeled USGS discharge archive is actual measured-derived source data; all other numerical packets and diagrams are invented. Public practice does not certify independent reasoning, actual field competence or a safe environmental decision.

Mechanism or dataset reference, checked 30 October 2026. Original prose, calculations, diagrams and questions; source observations retain agency and provenance.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

At ELA 2000 m with areas 1,2,3,2 km², find integrated and mean balance.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Infer stream exchange from a shared hydraulic-head datum Next: Distinguish downstream ice flow from retreat and thinning

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.