Vollenweider Phosphorus Load Limit

Invert a caller Vollenweider-form steady retention relation for target areal external phosphorus load; not an authoritative load limit or site-specific ecological guarantee.

Description

A steady phosphorus loading model links lake concentration to external load, depth, residence time and assumed retention.

A steady phosphorus loading model links lake concentration to external load, depth, residence time and assumed retention. Inverting a supplied Vollenweider-form relation gives the load corresponding to a chosen concentration target.^[1]

The tool returns permittedArealExternalLoadGramsPerM2Year. The computation uses supplied observations and parameters rather than acquiring measurements.

When to use Vollenweider Phosphorus Load Limit

Use this for the following task: Invert a caller Vollenweider-form steady retention relation for target areal external phosphorus load; not an authoritative load limit or site-specific ecological guarantee. Prepare observations and calibration parameters on the same unit and reporting basis.

How to use Vollenweider Phosphorus Load Limit and understand its results

Prepare the measured quantities and supplied model parameters. Match the units and align paired lists before calculating.
Enter targetPhosphorusMgPerM3, meanDepthM, hydraulicResidenceYears, retentionMultiplier, retentionExponent according to the field table, then run the tool.
Read permittedArealExternalLoadGramsPerM2Year. Use the calculation below to interpret the selected mean or the model’s signs, units and comparison flags.
Review the supplied assumptions and calibration before comparing or reporting the result.

Input field	Meaning	Value type and bounds	Requirement
targetPhosphorusMgPerM3	Target phosphorus mg per m3	number	Required
meanDepthM	Mean depth m	number	Required
hydraulicResidenceYears	Hydraulic residence years	number	Required
retentionMultiplier	Retention multiplier	number	Required
retentionExponent	Retention exponent	number	Required

Supply positive target concentration, depth and residence years, with nonnegative retention multiplier and exponent. Evaluate the inverse relation and convert milligrams to grams.

load g/(m²·year)=(target mg/m³×depth/residence)×(1+multiplier×residence^exponent)/1000

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
permittedArealExternalLoadGramsPerM2Year	Permitted areal external load grams per m2 year	number

json

{
  "input": {
    "targetPhosphorusMgPerM3": 10,
    "meanDepthM": 5,
    "hydraulicResidenceYears": 4,
    "retentionMultiplier": 1,
    "retentionExponent": 0.5
  },
  "output": {
    "permittedArealExternalLoadGramsPerM2Year": 0.0375
  }
}

Limitations and assumptions

The field named permittedArealExternalLoadGramsPerM2Year is a model result rather than an authoritative limit. Internal loading, seasonal variation and site-specific ecological response are excluded.

Numerical inputs must be finite. Input-shape and calculation checks do not establish measurement quality or the validity of a supplied calibration.

Alternative or Complementary methods

Choose Chlorophyll Target Load Back Calculation Tool when you need its operation: Compose inversion of a supplied chlorophyll/P power calibration with a steady Vollenweider-form loading model; excludes internal loads and does not guarantee a management outcome. Choose Phosphorus Retention Coefficient for the complementary operation: Close matched phosphorus inflow/outflow/storage budget to apparent retention; negative retention or ratios above one expose unmodeled exchange and are not clamped.

Background reference: Water Quality, Hydrology, and Response to Changes in Phosphorus Loading of Nagawicka Lake.

References

Water Quality, Hydrology, and Response to Changes in Phosphorus Loading of Nagawicka Lake — U.S. Geological Survey

Similar or alternative tools

Chlorophyll Target Load Back Calculation Tool
Compose inversion of a supplied chlorophyll/P power calibration with a steady Vollenweider-form loading model; excludes internal loads and does not guarantee a management outcome.
Phosphorus Retention Coefficient
Close matched phosphorus inflow/outflow/storage budget to apparent retention; negative retention or ratios above one expose unmodeled exchange and are not clamped.