Does Peat Creep?

Peatlands are dynamic, deformable porous media storing over 30% of global soil carbon. Researchers in the MATHPEAT Network are currently investigating the evidence for peat creep, how it should be represented mathematically, and what its consequences may be for peatland age-depth structure and long-term carbon storage.

The model below is deliberately simple. It is not a full 3D landscape-scale creep model: creep mostly redistributes peat locally. Here we ask what a single representative column would look like under different vertical creep profiles and depth-scaling assumptions if unresolved downslope or marginal creep eventually exports material from that column.

Simple Column Model & Creep Formulations

Let $M(a)$ be cumulative peat dry mass per unit area at cohort age $a$, measured in $\mathrm{kg\,m^{-2}}$. The estimated peat depth is

\[H(a) = \frac{M(a)}{\rho_b},\]

where fixed dry bulk density is $\rho_b = 100\,\mathrm{kg\,m^{-3}}$.

Without creep, the classic accumulation-decay balance (Clymo 1984) is

\[\frac{dM}{da} = A - \alpha M, \qquad M(0)=0,\]

where $A$ is the accumulation flux (default $60\,\mathrm{g\,m^{-2}\,yr^{-1}}$) and $\alpha$ is the first-order decay rate (default $10^{-4}\,\mathrm{yr^{-1}}$).

Two Independent Creep Choices

The creep model separates two assumptions that are often mixed together:

  1. Vertical velocity profile: both profile choices restrict lateral creep to an active upper layer of thickness $H_{\text{active}}$:
    • Uniform active layer: peat above $H_{\text{active}}$ moves downslope at one speed. Below that depth the peat is stable. This is a surface-driven lid, the sort of profile suggested by near-surface drivers and by a nearly buoyant saturated column. It is not the plug that rides above a basal shear zone in gravity-driven viscoplastic flow.
    • Linear shear-zone profile: speed is maximum at the surface and decreases linearly to zero at depth $H_{\text{shear}}$. Below this depth, the peat is stable.
  2. Depth scaling of measured speed: the user sets the present-day surface creep speed measured at the final simulated depth. The model then reconstructs what that speed would have been when the peat column was shallower using one of four simple hypotheses: constant, proportional to $H$, proportional to $H^2$, or proportional to $1/H$.

Creep Velocity Profiles: Plug vs. Shear Flow

Uniform active layer Active Layer (Creeping) Stable Layer (Stationary) H_active u = u_surf Linear Shear Flow Active Layer (Creeping) Stable Layer (Stationary) H_active u decreases to 0

The Reverse-Engineering Challenge (Equifinality)

In field peatland science, scientists extract a physical peat core, measure its depth $H$, and perform Radiocarbon ($^{14}\text{C}$) dating to get age $a$ at depth $H$.

The Challenge: A given observed age-depth profile could be fitted by TWO completely different mechanisms:

  1. Classic Clymo Model (Decay Only): Fitted with a lower surface accumulation rate $A_{\text{Clymo}}$ or higher decay rate $\alpha_{\text{Clymo}}$.
  2. Peat Creep Model (Decay + Export): Driven by a higher true surface accumulation rate $A_{\text{Creep}}$, standard decay $\alpha_{\text{Creep}}$, and active lateral creep $u_{\text{meas}}$.

If an exporting column is fitted with only Clymo’s equation, the fit underestimates the true carbon input ($A$) and reads lateral loss as decomposition. A column that receives peat from upslope does the opposite. This page only treats net export.

Interactive Age-Depth & Cumulative Mass Model

Clymo / Yu Model (Decay Only) Teal Line
Present-day surface peat input flux
60 g/m²/yr
First-order catotelm decay rate
1.0e-4 /yr
Past cohort input $A(a) = A_0 e^{+\beta a}$. Positive $\beta$: higher input in the past.
0.0e+0 /yr
Creep Model (Decay + Export) Dashed Red Line
Choose where in the peat column lateral motion is active
How today's measured surface speed is reconstructed for shallower past peat
True surface input flux
60 g/m²/yr
True catotelm decay rate
1.0e-4 /yr
Present-day surface speed at final simulated depth
0.50 cm/yr
Depth limit for lateral motion
1.0 m
Clymo / Yu Depth (10,000 yrs)
-
With Creep Depth (10,000 yrs)
-
Depth Discrepancy
-
Derived Coeff
-

Estimated Depth Profile ($H$, meters)

Cumulative Dry Mass Profile ($M$, kg/m²)

Integrated Lateral Volume Flux ($Q$, m²/yr)

🔍 Step-by-Step Equifinality Challenge (Try It!)

Peatland science and inverse modeling are plagued by unidentifiability. Can you distinguish a creeping bog from climate change? Try this:

  1. Set up an exporting column: Leave the profile on Uniform active layer and the speed law on Constant with depth. Set $A_{\text{Creep}}$ to 90 g/m²/yr and $u_{\text{meas}}$ to 2.00 cm/yr, then drag the active-depth slider to its right-hand end so the whole column creeps. The red curve ends near 2.85 m. The same input with no creep would end near 5.69 m. Export length is fixed at $L = 100\,\mathrm{m}$, so the loss rate is $u/L = 2\times 10^{-4}\,\mathrm{yr^{-1}}$.
  2. Scenario A: Ignored creep (Clymo fit): Keep $\beta = 0$. Set $A_0$ to 45 g/m²/yr. The teal curve ends at about 2.84 m, against 2.85 m for the creeping column. The accumulation slider moves in steps of 5 g, so this is as close as it goes. The fitted input is half the true input.
  3. Scenario B: Creep read as a Yu trend: Set $A_0$ back to 90 g/m²/yr and $\beta$ to -2.0e-4 /yr. For this one closure the teal and red curves overlie: the age profile is Clymo's curve with decay rate $\alpha + u/L$.
    The catch is narrower than a general creep law. A linear shear profile, an active layer thinner than the column, or a surface speed that scales with depth does not overlie a single $\beta$. A column that gains peat from upslope would be read as the opposite trend.

Mathematical, Physical & Peatland Science Foundations

1. Peatland Science & Eulerian vs. Lagrangian Views

Standard 1D peat accumulation models (such as Clymo 1984) split the peat column into an upper oxic layer (acrotelm) and a lower water-saturated anoxic layer (catotelm). Organic matter enters the catotelm at surface accumulation flux $A$ ($\text{g m}^{-2}\text{yr}^{-1}$) and undergoes slow anaerobic decay at a first-order rate $\alpha$ ($\text{yr}^{-1}$).

Under steady-state conditions over 10,000 years:

2. Temporal Variations: The Yu et al. (2003) & Belyea & Malmer (2004) Model

Clymo’s model assumes a constant peat addition rate $A$ through time. To allow for climate-driven or developmental variations, the model by Yu et al. (2003) and Belyea & Malmer (2004) permits the initial accumulation rate of cohorts to vary exponentially back through calendar time. Expressed in terms of cohort age $a$ today, the remaining mass of a cohort of age $a$ today is $m(a) = A_0 e^{-(\alpha - \beta) a}$.

Integrating this from age $0$ to $a$ today yields the cumulative peat mass profile:

\[M(a) = \frac{A_0}{\alpha - \beta} \left( 1 - e^{-(\alpha - \beta) a} \right), \qquad \text{for } \alpha \neq \beta\]

and $M(a) = A_0 a$ when $\alpha = \beta$.

Here, $A_0$ is the present-day surface accumulation flux, $\alpha$ is the decomposition rate, and $\beta$ controls the temporal accumulation change. A cohort of age $a$ was added at rate $A_0 e^{+\beta a}$. This age profile has the same shape as Clymo’s constant-accumulation equation with $\alpha_{\text{eff}} = \alpha - \beta$. The calendar-time mass of the whole column is a different curve. If the input at the start of growth is $p$ and then $A(t) = p e^{-\beta t}$, today’s input is $A_0 = p e^{-\beta T}$ and

\[M(t) = \frac{p}{\alpha - \beta}\left(e^{-\beta t} - e^{-\alpha t}\right).\]

A positive $\beta$ represents a peatland where initial accumulation was higher in the past (decreasing towards the present, resulting in a thicker bottom). A negative $\beta$ represents a peatland where initial accumulation was lower in the past (increasing towards the present, resulting in a thinner bottom).

3. Mathematical Derivation of the Creep Closure

Column mass conservation is $\partial M/\partial t + \partial_x(\bar u M) = A - \alpha M$, with $\bar u = Q/H$ the mean lateral speed. This demo replaces the divergence by export over a fixed length $L = 100\,\mathrm{m}$:

\[\frac{dM}{dt} = A - \alpha M - \frac{\rho_b Q}{L}, \qquad M(0) = 0.\]

$Q$ is the integrated volume flux defined below, so the export term is $\rho_b Q/L$ and not $\rho_b u_{\text{surf}} H/L$. Peat below the active layer is not exported. The integrator gives each cohort its own speed $u(z)$ and removes it at rate $u(z)/L$, which sums to the same $\rho_b Q/L$. The plotted creep curve is the cumulative mass above age $a$ in the finished column. $u_{\text{meas}}$ is the surface speed at the final depth $H_{\text{final}}$; earlier surface speeds follow the selected scaling of that value.

Vertical Profile Choices

Both profiles confine creep to an upper layer of thickness $H_{\text{active}}$:

Integrated Lateral Flux

Integrating the lateral velocity profile $u(z)$ over the depth of the peat column gives the integrated lateral volume flux $Q$ (in $\mathrm{m^2\,yr^{-1}}$):

\[Q = \int_0^H u(z) \, dz\]

We can analyze this integral in two distinct ways:

  1. Spatial Cumulative Flux (Present Day):
    • Mathematics: This is an indefinite integral over depth $z$ evaluated at the final present time ($t = T_{\text{final}}$): \(Q_{\text{spatial}}(z) = \int_0^z u(T_{\text{final}}, s) \, ds \qquad \text{for } z \in [0, H_{\text{final}}]\) where $z = H(a)$ is the depth of the cohort of age $a$ today.
    • Physics: It answers: “In the modern peat column today, how much lateral creep flux is carried by the upper layers down to depth $z$?” Because the surface ($z = 0$, cohort age 0) has a thickness of 0, this cumulative value starts at $0$ today and increases to the total column flux at the bottom of the core (age 10,000).
  2. Historical Total Flux (Over Time):
    • Mathematics: This is a definite integral over the entire peat thickness $H(t)$ at varying historical times $t \in [0, T_{\text{final}}]$: \(Q_{\text{historical}}(t) = \int_0^{H(t)} u(t, z) \, dz\) where $t = T_{\text{final}} - a$ is the historical time when the oldest cohort was $a$ years old.
    • Physics: It answers: “At historical time $t$, what was the total lateral flux through the column?” The value at age 0 is today’s flux, and the value at 10,000 years is the start of growth. For a constant or thickening-driven surface speed the present flux is the largest. A resistance-limited speed, $u \propto 1/H$, can be larger while the column is still thin.

For the two profile types under an active layer of depth $H_{\text{active}}$ (where $u_{\text{surf}}$ is the surface speed):

This lateral volume flux can be converted to a lateral dry mass flux (in $\mathrm{kg\,m^{-1}\,yr^{-1}}$) by multiplying by the constant bulk density: $F_{\text{lateral}} = Q \cdot \rho_b$.

Depth-Scaling Choices for the Measured Surface Speed

The depth-scaling selector asks how the present-day measured speed should be extrapolated backward when the peat column was thinner:

These are not competing claims about the true law; they are deliberately simple closures for exploring how different physical hypotheses alter the inferred age-depth profile.

4. The Equifinality & Inverse Problem Challenge

When analyzing an observed radiocarbon-dated core from a field site, scientists face an inverse problem (fitting parameters from measured age-depth points).

If an exporting column is fitted without creep:

The alias is exact for one closure only. A uniform active layer that fills the column, at constant surface speed, loses a fraction $u/L$ of its mass per year. The age profile is Clymo’s curve with rate $\alpha + u/L$, which is the Yu profile at the true $A$ and $\alpha$ with $\beta = -u/L$.

Creep read as a Yu trend

A shear profile, an active layer thinner than the column, or a surface speed that depends on $H$ changes shape through time. A single $\beta$ can approximate the final depth and still miss the profile. The sign is also local. Export, as at an extending crest or an open margin, looks like a negative $\beta$. Import, as in a compressing margin, looks like a positive $\beta$. This demo has no import.

Where a core is taken

Cores are often taken in the deepest peat. That does not by itself select the column that lost the most mass. Where the centre extends and the margin compresses, the deepest remaining peat can be the place creep has thickened.

References