# Presentation: 243238

## Advection and Modelling

- John Finnigan
- CSIRO Atmospheric Research

## Contents

- The integral mass balance-eddy flux, advection and storage
- Reynolds decomposition-dealing with the ‘mean flow’ deterministically
- Examples of modelled flow in a canopy on a hill
- The momentum field
- The scalar field-photosynthesis
- Diabatic effects-stable flows in the canopy
- Treating advection as an inverse problem
- Conclusions

## The Integral Mass balance in a control volume

- Horizontally Homogeneous Flow

- Complex Topography (or short averaging time)

*x* axis parallel to

**Notes:**
Aligning the x axis with the mean wind aligns the notional control volume

## The Integral Mass balance

- The instantaneous mass balance at a point (Cartesian coordinates)
- The time averaged, integral mass balance in a control volume
- The integral mass balance in horizontally homogeneous terrain

**Notes:**
The connection between surface exchange and eddy flux is achieved by integrating over a conceptual control volume.

We are using Gauss’s theorem  and we need to measure or estimate all the fluxes ax the walls of the control volume

The shape of the control volume is determined by the coordinate system in which it is written

## The division between the deterministic flow and the eddy flux

- Formally, this is a function of the averaging time we choose.  All the covariance must be accounted for either deterministically as ‘Advection’ or statistically as ‘Eddy Flux’.
- We look for physical grounds for a logical division, however.

**Notes:**
It is the total covariance that appears in the conservation equation, not the ‘eddy covariance’.  We split the flow so we can define coordinates using the ‘mean flow’.  We also want to treat the ‘turbulent flow statistically and the mean flow deterministically.

## Evidence for a Spectral Gap in PBL is equivocal

- ASL Energy spectrum at 40m Ayotte et al (2001)

- Upper Troposphere Energy Spectrum  Nastrom and Gage (1985)

**Notes:**
Early reports of a spectral gap were much exaggerated (Mark Twain). 

We shall see below that variations in u, temporal or spatial,  are translated into variations in <wc>. It is not clear how we deal with a continuous spectrum of LF contributions to the covariance

We can be confident, however, that flow perturbations caused by surface features have a spatially deterministic component.  We will focus on this for most of the rest of this talk.

## Deterministic Advection

- Once the averaging time *T* is chosen, and coordinates rotated so that           over *T* then all remaining inhomogeneity in the flow must be treated as deterministic.
- On a single tower, only the terms in black can be measured.  To close the mass balance, we must add more information

**Notes:**
For simplicity, the equation is written for two dimensions only.

We will return to comment briefly on coordinate rotation at the end.

## Deterministic Advection: Adding Information-Some Strategies

**Assume flow is horizontally homogeneous. If it isn’t, we measure not       but**
**Assume                                .  The vertical advection can be measured on a single tower but, in general, there is no physical justification for this assumption. **
**Measure the advection directly.  **
**Use what we know about flow in topography to supply the missing information in a systematic way.  To do this we need to model flow over topography covered with a canopy.**

**Notes:**
Method 2 is Xuhui Lee’s correction..  A fourth strategy is to measure advection directly as KT Paw U has done at the Wind River Crane and Aubinet and Heinesch.    The results can be very noisy.

## Linear Analytic model of flow in a tall canopy on a low hill  (Finnigan and Belcher, 2002)

**Some basic physics of flow over hills**
**Pressure field scales on hill steepness and ‘outer’ velocity *****U******0***
**Pressure gradient varies on scale *****L***
**Turbulent stress is only important in the momentum balance in a thin layer near the surface**

**Notes:**
Strategy 3 requires a more appropriate statement of the mass balance for complex terrain, ie. A flow and transport model.  Next few slides describe a new model by Finnigan and Belcher.

## Linear Analytic model of flow in a tall canopy on a low hill  (Finnigan and Belcher, 2002)

**We add a foliage drag term to the momentum equation in the lowest layer of the model.**
**Solution proceeds by doing a scale analysis of the equations of motion in each layer, retaining the dominant terms and solving for that layer**
**Solutions are then matched asymptotically between layers**

## Linear Analytic model of flow in a tall canopy on a low hill  (Finnigan and Belcher, 2002)

**Velocity perturbations in the deep canopy are proportional to (minus) the pressure gradient but just above the canopy they are proportional to (minus) the pressure.  **

**Notes:**
Pressure gradient passes undiminished through canopy so as the mean velocity faklls exponentially, the perturbation must grow to balance dp/dx until it exceeds the mean and a non-linear treatment is necessary

## Velocity Perturbation Profiles with and without a Canopy

*Δ**u/Usc  vs  Z/l*           *H/L*=0.1  *Lc*=(*ca*)-1=10m
- Finnigan and Belcher (2002)

**Notes:**
Flow pert above canopy peaks near hill top.  Flow in deep canopy peaks 1⁄2 way up and goes through zero on top of hill, reversing in lee.  Strong shear and shear stress develops between these layers

## Global behaviour 1: Velocity Profiles on a rough hill:The limit of the canopy solution is not a constant z0

**In the limit of                                           We expect to recover the ‘rough hill’ case where the effective z****0**** is constant.  In fact the canopy solution does not converge to this limit.  Implication is that constant z****0**** may be an incorrect concept in flows with pressure gradients.**

**Notes:**
In the deep canopy on the lee side, the pressure driven perturbation can reverse the flow.  No matter how small the hill slope there is a canopy deep enough to allow perturbation in the lee.

## Global behaviour 2: Microscale dynamics of the canopy affect macroscale properties like topographic drag

- Because the flow in the lower canopy peaks halfway up the hill, the peak in velocity perturbation moves upstream compared to flow on a rough hill without a tall canopy.
- The ‘out of phase’ component of the pressure field follows the velocity perturbation, becoming more out of phase so that pressure drag increases.

## Total Velocity Profiles with and without a Canopy

- (*U**B**+**Δ**u*)/*Usc*  vs  *Z/l *          *H/L*=0.1  *Lc*=(*ca*)-1=10m

**Notes:**
In the deep canopy on the lee side, the pressure driven perturbation can reverse the flow.  No matter how small the hill slope there is a canopy deep enough to allow perturbation in the lee.

## Streamlines showing separation within the canopy

## Preliminary WT Modelling (Hughes and Finnigan, 2002)

**Notes:**
The hill slope is too small to provoke separation if it were covered with short roughness.  We have added a deep canopy within which a separation region should appear according to Finnigan and Belcher, (2002)

## Separation occurs within the canopy at the foot of the hill

**Notes:**
Flow viz shows no separation above canopy but a reversed flow region close to sfce in downstream trough.

## Separation region determined by pitot traverses and flow visualization

**Notes:**
Pitot traverses indicate sepn region

## The windfield model allows us to derive an analytic scalar flow and transport model that produces criteria for topographic advection at a given site

**Notes:**
The model results can be used to provide a simple test to check whether you should worry about advection at your site.  Plugging in parameters for our Tumbarumba site we see that the advection terms are of order 20% of the average eddy flux <w’c’> while the variation of <w’c’> over the hill is ~4% of the mean.

## More information from the analytic scalar model:   Variations in advection and flux over the hill.   (Canopy Sink)

- Red:
- Blue
- Green

- Hill shape

- Fractional error in S

- Hill shape

**Notes:**
We can use the model to compute the advection and flux terms over the hill- left hand curves.  The right hand curve is the %ge error we would see in computed surface exchange if we ignore advection as a function of tower position over the hill.

## To deal with more realistic forcing and source behaviour we need a numerical approach (Katul, Finnigan, Leuning and Belcher, 2002)

- 1-equation with 3 unknowns:

- Parameterize using first order closure for the eddy flux and ecophysiological models for the source/sink at leaf level

## Leaf level source/sink is described by Farquhar-Leuning Model

**Scalar mass balance can now be solved for CO2**
** ****concentration**

**(Light Limited)**

**(Rubisco Limited)**

**(semi-empirical)**

**Leaf Conductance:**

**Leaf Photosynthesis:**

**Notes:**
Where lamda is the CO2 compensation point, Cs is the mean surface CO2 concentration which can be related to using ,Ds  is the mean surface vapor pressure deficit,a1 ,Do , and are constants that vary among species.  The term in equation (5) is also related to using the Farquhar et al. (1980) and Collatz et al. (1991) models, whose canonical form is given by
where is the leaf absortivity for PAR, PAR is the photosynthetically active radiation, is the maximum quantum efficiency, and is the PAR irradiance, is the maximum catalytic capacity of Rubisco, and are the Michaelis constants for CO2 fixation and O2 inhibition with respect to CO2, and is the oxygen concentration, and is the dark respiration of the foliage.   The latter constants vary with temperature as described in Appendix B (see e.g., Lai et al., 2000). 

## Slide 24

**Notes:**
Effect of hill on driving variables-wind field and PAR

## Estimation of light within the canopy

- 08:30 am

- 11:30 am

**To solve for leaf photosynthesis, **
**photon flux density (Qp(x,z)) is required:**

**Simplified radiative transfer scheme that accounts for: **
**horizontal variation of incident radiation -  (Raupach et al., 1992) **
**vertical light attenuation - (Campbell and Norman, 1998)**

**Notes:**
Ecophysiological parameters, LAI dynamics, and drag coefficients
taken from Katul et al. (2000) and Katul and Albertson (1998) and correspond to Duke forest data

Meteorological drivers: sample run - DOY = 156

## Comparison between numerical and analytic solutions (Katul, Finnigan, Leuning and Belcher, 2002)

**Notes:**
As a dual check on the numerics and the analytic approximations we see the reversed flow region reproduced in the canopy

## Slide 27

**Notes:**
The expected patterns of velocity perturbation are clear and we also see the weak radiation forcing

Impact of photosynthesis S is relatively small but large impact on eddy flux Fc=<w’c’>

## Profiles of model output at three positions, halfway up the hill, x=-L, the hill crest, x=0 and halfway down the hill, x=L

**Profiles of model output at three positions, halfway up the hill, x=-L, the hill crest, x=0 and halfway down the hill, x=L**
**Lower bc is a constant concentration of 600ppm.  Results are essentially unaffected if we use a flux lower bc at the soil.**

**Notes:**
Figure 7: The effect of the hill on the profiles of mean velocity components (and ), turbulent diffusivity (), effective conductance (stomatal and aerodynamic), CO2 concentration (), and turbulent fluxes () at three positions on the hill surface (upwind or x=-100 m, crest or x=0, and downwind or x=100 m) and for time of day = 14:30. 

## Components of the scalar conservation budget at the same three positions on the hill

**Components of the scalar conservation budget at the same three positions on the hill**
**Most individual terms and the sum of the advection and horizontal flux divergence terms are much bigger than the photosynthesis, Sc**

**Notes:**
Figure 8: The effect of the hill on the components of the mean scalar concentration budget at the three positions (upwind, crest, downwind) shown in Figure 7.

## Slide 30

**Notes:**
These are the terms in the integrated budget.
We see a small net decrease in NEE relative to flat ground but of course we have not included important effects like temp and moisture availability perturbations

The actual effect of the hill on photosynthesis is small  ~H/L but the error if advection is ignored is up to a factor of 5 depending on tower position. 
1) Advective terms are (individually) of the same order of magnitude 
   as photosynthesis.  Fc is of opposite sign to Sc true on hill top and uptake is supplied by convergence (advection)

2) Within the deeper canopy layers, the sum of these advective terms
     is small (partial cancellation).

3) The forest on a hill (~20%) seems to increase photosynthesis 
     by ~ 10%.

4)  Leaf photosynthesis is either limited by Rubisco 
    (temperature dependent) or light (but in a linear manner).  Hence,  
    the effect of the hill on total stand photosynthesis through the 
     radiation is small.    

## Interim conclusions 1

- The radiative perturbations to photosynthesis are smaller than those caused by the wind and turbulence perturbations
- wind and turbulence perturbations produce ‘true’ perturbations in NEE ~H/L
- However, the hill flow produces very large advective flux divergence and horizontal gradients in Fc(x,z) that would severely compromise single tower measurements near the hill crest

## Preliminary WT results: constant flux from the (soil) surface

**Notes:**
Agema senses heated sfce ~150 w/m2 constant flux.

## Buoyancy driven flow-the nighttime CO2 problem

- We do not yet have comparable models of this situation but there are some preliminary things we can say about it
- Onset of stable stratification in an established hydrodynamic hill flow-when to throw the u* switch
- Gravity currents on very stable nights

## Buoyancy adds extra forcing terms to the momentum equation

- Top of the cool layer
-                           is hg

**Notes:**
As drawn, the thermal wind is in the same direction as the downslope forcing theta hat is the depth integrated potential temp perturbation.

## Tumbarumba-view from the tower

**Notes:**
Not a flat site.   H/L~0.1

## Tumbarumba-a relatively open canopy

- We do not see the unstable profile in the trunk space often observed at night in closed canopies on level ground

## Tumbarumba Temperature Profiles indicate a gravity Current

- Weak turbulence, gravity driven flow

**Notes:**
 if we ignore the thermal wind and assume steady state with no background wind we obtain a flow velocity that approximately matches measurements 

Time to attain steady state can be predicted.  It depends, inter alia on cooling rate.  A steady state may not be attainable.

Inverse Fr number is large indicating buoyancy dominance

6 days data in ??

When the flow reaches steady state the downslope component and the drag are in balance.

## CO2 profiles show large concentrations near the soil but storage doesn’t balance soil respiration

- Advection driven by the gravity current is the prime suspect

## Early Wind Tunnel results suggest the Froude number criterion is useful as a signal of the onset of diabatically forced flow near the canopy

**Notes:**
With Fr-1~10-20 anabatic flow reverses hydrodynamic flow in the lee

## surface atmosphere exchange as an inverse problem

- Direct measurement of surface exchange-faces crippling sampling problems-chambers and cuvettes
- Aerodynamic methods exploit atmospheric mixing as an averaging operator but the number of terms we can measure will always be much less than is needed to close the mass balance in most conditions and especially so in complex terrain
- Merging measurements and verified knowledge via data assimilation is a common strategy in sister disciplines
- We need to capture our ‘verified knowledge’ as appropriate statements of the mass balance

## Data Assimilation into ‘appropriate’ flow and transport models

- Make a first estimate of *S* (assume 1-D flow perhaps)
- Given *S*, the Flow and transport model predicts all the terms on the LHS

- Adjust *S* until the mean square difference between model predictions of I,III and IV and measurements of I, III and IV is minimized
- At this point *S* is the best estimate of surface exchange that is compatible with both the measurements and the prior knowledge built into the model

**Notes:**
If we can write a flow and transport model for a given site, we can systematically adjust for advection using data assimilation methods.  This slide indicates the basic philosphy of data assimilation.

## Data assimilation as a strategy

- We require models that work in problem areas:
- Complex surfaces
- Complex topography
- With diabatic forcing
- At patch and BL budget scales
- Producing and testing the models is the forward problem
- Assimilating measurements into the models is the inverse problem

## Issues

- Coordinate frame of the model implies a vector basis
- Measurements are made in a basis defined by (for example) planar fit
- In general these two bases are not the same and the model output or data will need to be transformed so both are in a common frame.
- ***How do we manage data assimilation so that more uncertainty (systematic bias) is removed by using a model that describes advection than is added by uncertainty in model parameterisation***

## To cope systematically with advection requires a Program with a series of linked elements

- Produce and verify models for critical *conditions-the forward problem*
- Produce the data sets necessary to verify the models
- Investigate optimal data assimilation schemes
- How will we know we have improved matters?  By comparison of ecologically similar sites on flat and complex terrain? ...Other strategies?

## Conclusions

- Modelling flow over a canopy-covered hill has revealed strong symmetry-breaking effects in the flow
- Although influence on true NEE is small, these translate to large advective flux divergences that must be taken into account or *inferred* NEE may be severely compromised
- First steps in understanding gravity driven nighttime flows appear promising
- To cope systematically with advection we must stop regarding it as a correction and realise that aerodynamic approaches to measurement of land-atmosphere exchange are classic inverse problems
- This realisation dictates a research program that the community should seriously consider