Connect transport, grid cells, and unresolved processes.
By the end: Explain why finer resolution helps without making a model exact.
Weather models combine mass conservation, momentum changes, energy changes, and water in different forms. Pressure differences accelerate air; rotation deflects motion; heating and cooling affect density and circulation. Moisture changes phase, exchanging latent heat with the surroundings.
Advection means transport by moving air. The small lab transports a passive substance around eight equal cells on a ring. Each step transfers a fraction C of a cell value into its next neighbor. There is no chemistry, source, sink, pressure, or real geography in this model.
A balance starts with an amount inside a chosen region, rather than with a picture of a weather pattern. Over one interval, the new amount equals the old amount plus arrivals minus departures, with any production or removal counted separately. For a passive substance, production and removal are absent by assumption. Wind can change where the substance is without changing how much exists. This distinction matters because a forecast can conserve its total substance while putting most of that substance in the wrong place at the forecast horizon.
Consider a separate four-cell ring with averages 2, 6, 0, and 0 arbitrary concentration units. Let the cells have equal width and let C be 0.5. Each cell retains half its old average and receives half its left neighbor's old average. Cell one receives from cell four, so its new value is 1. Cell two becomes 4, cell three becomes 3, and cell four stays zero. The sum remains eight. All four calculations must use the old values, not values already updated during the same step.
cᵢ is a cell-average concentration in arbitrary concentration units; C = 0.5. The four equal cells form a periodic ring.
A concentration sum is proportional to total amount because the cell volumes are equal.
| Cell | Retained + received | New average |
|---|---|---|
| 1 | 1 + 0 | 1 |
| 2 | 3 + 1 | 4 |
| 3 | 0 + 3 | 3 |
| 4 | 0 + 0 | 0 |
To see why update order matters, suppose cell two mistakenly receives half of cell one's new value. It would receive 0.5 rather than 1, giving 3.5 rather than 4. A left-to-right computer loop that overwrites its inputs can therefore change the intended balance. The correct conceptual procedure keeps the old profile intact until every new cell has been calculated. This is an arithmetic dependency, not a claim that atmospheric transport happens simultaneously at a computer's update instants. The interval represents continuous transport through a discrete approximation.
Concentration and amount are different quantities. If each equal cell has volume V, its amount is concentration times V, and the domain amount is V times the sum of concentrations. Comparing sums is sufficient only because the common volume factor cancels. With unequal volumes, a simple unweighted sum can change even when the physical amount stays constant. For example, two cells of volumes one and three with concentrations four and zero contain four amount units, whereas concentrations zero and four contain twelve. The same concentration sum does not establish conservation.
The four-cell example isolates boundary transfers from sources. A chemical reaction that destroys the substance would require a removal term. Evaporation adds water vapor while reducing liquid water elsewhere, so a balance for vapor alone differs from a balance for all water phases together. Similarly, energy can enter a region through radiation rather than through moving air. Choosing the conserved quantity and drawing its boundary come before writing the update. Otherwise a missing transfer may look like unexplained numerical loss even when it belongs in a different physical account.
| Concentrations | Cell volumes | Total amount |
|---|---|---|
| 4, 0 | 1, 3 | 4 × 1 + 0 × 3 = 4 |
| 0, 4 | 1, 3 | 0 × 1 + 4 × 3 = 12 |
| 2, 2 | 1, 3 | 2 × 1 + 2 × 3 = 8 |
The ring boundary is a modeling choice with a visible consequence. Material leaving its final cell reappears in its first cell; it does not disappear or encounter an external reservoir. This makes the total especially easy to check and avoids prescribing an incoming concentration. A regional atmospheric model instead needs information at its outer boundary, and errors there can enter the interior. Our ring cannot demonstrate that dependence. Its useful result is narrower: paired outgoing and incoming contributions preserve a closed-domain total for this particular update rule.
Cell values represent averages over space. The fraction C equals speed times time step divided by cell width for this one-direction scheme. For C between zero and one, the new average is a nonnegative mixture of two old averages. Total amount is conserved because each outgoing contribution enters another cell.
At C = 1 the profile shifts by one cell per step. At C = 0.5 it spreads across cells. That broadening is numerical diffusion here, not an explicit physical mixing process. Conservation alone does not guarantee accurate shape or timing.
The transport fraction C connects a numerical update to physical scales. Imagine a teaching strip with cell width two kilometers, constant rightward speed one kilometer per hour, and update interval one hour. The air travels one kilometer per update, so C is one half. Doubling the interval would make C equal one on the same grid. These invented scales explain the ratio; the interactive ring itself supplies no kilometer or hour labels. Its eight updates must not be reported as eight hours of a real atmospheric forecast without an added scale assumption.
This paper scale uses u = 1 km/hour, Δt = 1 hour, and cell width Δx = 2 km; the interactive ring itself has no physical time scale.
C is dimensionless: half a cell width is traveled in this stipulated update.
A cell average does not specify a substance's position inside the cell. Take one cell of width two kilometers and average concentration four. A uniform concentration of four across the cell has that average. Concentration eight over its left half and zero over its right half has the same average. Their total amounts agree, but their short-term departures through the right edge can differ under rightward motion. The model's single stored number cannot distinguish these states. Refining the grid may retain part of that missing information, but cannot recover it from the old average alone.
| Interval | Distance at 1 km/hour | C for 2 km cells |
|---|---|---|
| 0.5 hour | 0.5 km | 0.25 |
| 1 hour | 1 km | 0.5 |
| 2 hours | 2 km | 1 |
For a passive translation at constant speed, an exact continuous profile moves without changing shape. The two-neighbor scheme usually changes the sampled shape because it distributes a cell's content between retained and transferred portions. Starting from a pulse of eight in one cell, one half-cell update gives two values of four. A second gives values two, four, and two. This widening occurs with no prescribed physical diffusion coefficient. It is a property of the numerical approximation. Calling it atmospheric mixing would assign a physical mechanism that this calculation never included.
The special case C equal one helps separate translation from widening. Every new cell then copies its left neighbor, so a grid-aligned profile shifts exactly one cell each update on this periodic grid. C equal zero leaves the profile unchanged. Intermediate C values use mixtures instead of pure copies. This does not mean a forecast should choose its time interval solely to make C one everywhere. Real atmospheric speeds vary across space and evolve, and several coupled processes constrain acceptable intervals. The special exact shift belongs to this simplified constant-speed transport problem.
The supported range zero through one also explains nonnegativity. Both mixing weights are then nonnegative and sum to one, so each new value lies between the two old values used to form it. Outside that range this argument fails. With C equal 1.25, an old local value eight and left value zero would produce negative two. That unsupported counterexample is algebra, not another available lab setting. Smaller intervals can keep this transport ratio within range, but other numerical methods have their own restrictions and cannot inherit this particular condition without analysis.
Average concentration, arbitrary units
Cell number on periodic ring
The plot below uses the separate two-kilometer, one-hour assumptions over four elapsed hours. Its peak falls even though its total remains eight. Height therefore cannot stand in for amount, and a lower maximum need not imply chemical destruction. The plotted lines join cell averages for visual comparison, not resolved concentrations between cell centers. On the ring, spatial adjacency also joins the final cell back to the first, although a flat horizontal axis cuts that connection. Interpretation needs both the storage convention and the periodic boundary, not only the drawn profile.
Changing C restarts the profile and playback. Reset retains C. Equal cells form a ring; cell 8 sends material to cell 1. Total sum remains 8. Values are arbitrary concentration units. Lines join cell averages; no within-cell structure is resolved.
Completed updates: 0. Total: 8.000. The frame changes only at a completed update.
| Update | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| 0 | 0.00 | 0.00 | 8.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 1 | 0.00 | 0.00 | 4.00 | 4.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 2 | 0.00 | 0.00 | 2.00 | 4.00 | 2.00 | 0.00 | 0.00 | 0.00 |
| 3 | 0.00 | 0.00 | 1.00 | 3.00 | 3.00 | 1.00 | 0.00 | 0.00 |
| 4 | 0.00 | 0.00 | 0.50 | 2.00 | 3.00 | 2.00 | 0.50 | 0.00 |
| 5 | 0.00 | 0.00 | 0.25 | 1.25 | 2.50 | 2.50 | 1.25 | 0.25 |
| 6 | 0.13 | 0.00 | 0.13 | 0.75 | 1.88 | 2.50 | 1.88 | 0.75 |
| 7 | 0.44 | 0.06 | 0.06 | 0.44 | 1.31 | 2.19 | 2.19 | 1.31 |
| 8 | 0.88 | 0.25 | 0.06 | 0.25 | 0.88 | 1.75 | 2.19 | 1.75 |
| Part of a cell update | Contribution |
|---|---|
| Retained locally | (1 − C) × old local value |
| Received from left | C × old left value |
| Periodic boundary | Last cell feeds first cell |
The existing lab offers No movement, Half a cell, and One cell presets for C equal zero, one half, and one. Every choice restarts the initial eight-cell pulse and resets playback. Reset keeps the chosen C. The comparison therefore holds the initial profile fixed while changing the transfer fraction. It does not continue material from the previous preset. Playback reveals completed updates only; the elapsed animation duration has no physical interpretation. The static trace provides the same computed cell averages without requiring the motion to be watched.
For Half a cell, inspect the first two completed updates before the profile wraps around the ring. The initial cell containing eight becomes four after one update and two after two updates. Its right neighbor becomes four and then four again, because the second update combines two separate contributions. The next cell becomes two. Summing just the old peak would miss the transferred material. Summing every cell checks a domain balance, whereas checking the distribution across cells asks a separate accuracy question that the total cannot answer.
| C | Updates | Nominal displacement |
|---|---|---|
| 0 | 4 | 0 cell widths |
| 0.5 | 4 | 2 cell widths |
| 1 | 2 | 2 cell widths |
| 1 | 4 | 4 cell widths |
A useful concentration diagnostic is the sum of squared cell averages. It is not the total amount. For the pulse, this diagnostic starts at sixty-four. After one half-cell update it is thirty-two, and after two it is twenty-four. The decreasing value records a less concentrated profile while the ordinary sum stays eight. No universal atmospheric law requires this particular diagnostic to decrease for every modeled variable. Here it quantifies the broadening caused by this averaging scheme and gives a numerical meaning to the visible fall in the peak.
The three presets do not represent equal physical displacement after the same number of updates. After four updates their nominal displacements are zero, two, and four cell widths. Comparing their profiles at update four therefore combines differences in movement with differences in numerical broadening. To compare half-cell and one-cell transport over two cell widths instead, use four updates of the former and two of the latter. Under a fixed speed and cell width those choices correspond to different step durations but equal elapsed physical time. The lab labels updates, so that interpretation must be supplied explicitly.
Over two cell widths, the one-cell calculation retains the pulse value eight in a single shifted cell. The half-cell calculation spreads it into five contributions with values 0.5, 2, 3, 2, and 0.5 before periodic overlap matters. Both sums are eight, but the maximum differs. This comparison separates a conservation check from a shape check using the same starting information and nominal travel distance. It does not establish which operational atmospheric algorithm is best; the lab contains only one unusually simple rule and a few transfer fractions.
Sum of squared cell averages, squared arbitrary units
Completed updates, not physical seconds
Eight updates also make the periodic domain important. At C equal one, an initial pulse returns to its initial cell after eight whole-cell shifts. That return is the consequence of the ring's eight-cell circumference, not a reversal of wind or evidence of an oscillating atmospheric process. At C equal one half, eight updates correspond to four cell widths of nominal travel while the pulse has broadened. Compare those outcomes using displacement and boundary conditions. Similar-looking pictures at selected updates need not describe the same transport history or the same physical elapsed interval.
Cloud droplets, turbulent eddies, and small terrain features cannot all be explicitly resolved in a global forecast. A parameterization estimates their aggregate effect on resolved variables. It is a physical or statistical model of missing scales, not merely a display smoothing filter.
Smaller cells can resolve more structure but cost more computation and often require smaller time steps. They do not remove observation errors, boundary errors, or all model bias. Real forecast evaluation must test whether the extra detail improves relevant outcomes, not just whether the maps look sharper.
A grid average removes information even before a time-stepping method introduces error. This matters when a physical response is nonlinear. Suppose an invented local response equals the square of concentration, with units chosen so the numerical rule is meaningful. Two equal subregions have concentrations zero and four. The average concentration is two, whose squared value is four. The average of the actual squared values is eight. Computing the response from the grid average would underestimate this constructed response by half. Averaging inputs and evaluating a nonlinear process need not commute.
A parameterization addresses a missing aggregate effect rather than guessing every hidden microscopic state. In the square-response example, knowing only the mean leaves the response undetermined. Knowing the spread helps because the mean square equals the square of the mean plus the variance. The uniform subgrid state with values two and two has zero variance and mean square four. The contrasting state zero and four has variance four and mean square eight. Both share the same resolved mean, so an extra assumption or extra information is necessary to choose between their aggregate effects.
Two equal subregions have concentrations 0 and 4 arbitrary units. The invented response is concentration squared, not a cloud law.
V is the equally weighted concentration variance. Squaring the mean alone gives only 4.
| Subregion concentrations | Mean concentration | Mean squared response |
|---|---|---|
| 2, 2 | 2 | 4 |
| 1, 3 | 2 | 5 |
| 0, 4 | 2 | 8 |
Cloud formation is not the square-response law, but it raises the same logical issue. A model cell can contain variation in humidity and temperature that its stored averages do not describe. An aggregate cloud estimate needs assumptions about that unresolved variation and its effect on resolved heating and moisture. Such assumptions can produce systematic error even when the resolved transport conserves its chosen totals. Increasing numerical precision would not supply the missing physical relationship. More digits preserve the selected calculation; they do not establish that the selected relationship describes the atmosphere.
Resolution changes what remains unresolved. In a one-dimensional teaching strip, suppose we reduce cell width from two kilometers to half a kilometer while keeping domain length and constant speed fixed. There are four times as many cells. Keeping the same C requires an interval four times shorter, so the same forecast duration needs four times as many updates. The product is sixteen times as many cell-updates for this particular one-dimensional comparison. This counts arithmetic work units, not wall-clock runtime, memory traffic, or an operational forecast budget. Different refinement patterns give different factors.
Finer cells can reduce transport broadening for a fixed physical problem while exposing physical structures that the coarser model could not represent. Those are different reasons for changed results. A clean numerical experiment can prescribe a known translated profile and compare cell averages at the same elapsed time. A forecast comparison instead needs observations and an agreed target, such as a regional rainfall total or event timing. A more detailed field can score worse if its features are misplaced. Visible structure alone cannot distinguish improved physical representation from a new source of error.
| Quantity | Coarse strip | Fine strip |
|---|---|---|
| Cell width | 2 km | 0.5 km |
| Cells per fixed length | N | 4N |
| Steps per fixed horizon | M | 4M |
| Cell-updates | NM | 16NM |
The physical-balance approach therefore supplies checks with different jobs. A closed passive-transport calculation should preserve its total, and the supported averaging update should preserve nonnegative inputs. A translated-profile comparison can then measure shape and location error. Agreement with those checks still leaves untested forcing, unresolved processes, and incoming boundary information in a weather application. The ring makes those omissions explicit enough to reason about them. It teaches how a model can satisfy an exact discrete accounting identity while remaining an imperfect approximation to a continuous process over a chosen forecast horizon.
Four cells contain 8, 0, 0, 0. With C = 0.25, what is the next profile and total?
Keep three quarters and transfer one quarter rightward.
| Cell | Before | Retained | Incoming | After |
|---|---|---|---|---|
| 1 | 8 | 6 | 0 | 6 |
| 2 | 0 | 0 | 2 | 2 |
| 3 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 |
The next values are 6, 2, 0, 0 and the total remains eight. The first cell retains six and sends two into the second. Equal cell widths let us compare sums directly.
Halve grid width in each of three spatial dimensions and halve the time step. Roughly how many cell-updates cover the same domain and time?
Count the extra cells, then the extra steps.
| Factor | Multiplier |
|---|---|
| Cells along each axis | 2 |
| Total cells | 2 × 2 × 2 = 8 |
| Steps over the same time | 2 |
| Cell-updates | 8 × 2 = 16 |
There are about eight times as many cells and twice as many time steps, giving sixteen times as many cell-updates. Actual runtime also depends on algorithms, communication, hardware, and which dimensions are refined.
Selected reference passages support the definitions and modeling distinctions below. The examples, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Atmospheric dynamics, physical processes, model limitations, and probabilistic forecasting. No current operational resolution or skill claim is inferred.
Initial-condition and model perturbations. Operational configurations change; the twelve invented rain members are not ECMWF output.