Lateral and Upper Boundary
Conditions
8 October 2012
Thematic Outline of Basic Concepts
• An introduction to lateral boundary condition (LBC)
formulations.
• Contributing factors to forecast errors resulting from
the LBCs.
• Methods for isolating forecast errors from the LBCs.
• Considerations related to the placement and
treatment of a model’s upper boundary.
Lateral Boundary Conditions
• There are two types of LBC formulations…
– Open/free
– Periodic/cyclic
• Open LBCs: variables’ values specified on LBs by data
on a grid larger than that of the simulation.
– Such data may come from a synthesis or syntheses of
observations or from another model’s forecast fields.
– Horizontal grid spacing of LBC data is typically coarser than
that to be obtained from your model simulation.
Lateral Boundary Conditions
• Two approaches for open LBCs…
– One-way nest: an outer domain specifies LBCs for an inner
domain; no interaction as each domain is run separately.
– Two-way nest: an outer domain specifies LBCs for an inner
domain; both domains run concurrently and can interact.
• Outermost domain of a limited-area model requires
LBCs and is an example of a one-way nest.
• Nested domains within a limited-area model are
typically handled using interactive two-way nests.
Lateral Boundary Conditions
• Meteorological info must be able to enter into each
domain along the boundaries with small distortion.
• Inertia-gravity waves must be able to propagate out
from the domain without being reflected inward.
• What about outward transfer of meteorological info?
– One-way nest: not permitted
– Two-way nest: permitted and beneficial if interactivity
does not bring about artificial numerical issues
Lateral Boundary Conditions
• LBCs are obtained via interpolation of the coarser
outer domain data to the inner domain’s grid.
• This typically occurs within a few points of the LBs to
smooth the transition from one domain to another.
• After filtering smaller scales, interpolation is also
used to communicate info from the inner to the
outer domain along the LBs if a two-way nest is used.
Lateral Boundary Conditions
• If a one-way nest is used, a damping or absorption
mechanism is applied to handle waves propagating
toward the LBs.
• This mechanism has the added benefit of damping
spurious phenomena that may arise as the inner
domain forecast strays from that on the LBs.
• Damping methods are covered with upper BCs.
Lateral Boundary Conditions
• Consideration: update frequency of LBCs
– Two-way nest: ∆t of outer domain
• Roughly once every three-five inner domain time steps.
• Solution near boundaries thus remains close to the LBC data.
– One-way nest: ∆t of data product driving simulation
• This time step is often large (1-6+ hr).
• Solution near boundaries can stray substantially from LBCs.
• This highlights two key points…
– One source of LBC-related forecast error.
– Benefit of damping (as a smoothing operator) near LBs.
Lateral Boundary Conditions
• Why would we ever want to use a one-way nest?
– Running domains successively allows coarse-grid forecast
data to be available sooner than if run concurrently.
– Available resources may not permit concurrent integration,
forcing each simulation to be run successively.
– Running a global simulation as our outermost domain is
often not the best use of computational resources.
• The error minimization by doing so does not outweigh the added
computational expense by doing so (with many inner nests).
• Otherwise, for most applications, use a two-way nest.
Lateral Boundary Conditions
• Periodic LBCs: features that exit the domain on one
end enter back in at the other end.
• Commonly used in idealized modeling.
• Can be applied to either horizontal direction.
– If applied to both in a 2-D model: doubly periodic LBCs
– If applied to one in a 2-D model: rigid boundaries are
typically used for the other LB (channel model config)
Lateral Boundary Conditions
• Integration carried over j = 2 to j = jmax-1.
• With a 2nd order centered finite differencing scheme,
this allows ending points to be handled elegantly.
– Alternative: applying a lower-order scheme at the ends.
– Higher-order schemes require even more overlap at ends.
LBC-Related Forecast Error
• The specification of the LBCs is a notable contributor
to forecast error on the interior of the domain.
• Even if the LBC formulation has all of the desired
properties noted earlier, there are still many ways in
which it can negatively impact forecast skill.
• We’ll now step through a number of possible causes
of LBC-related forecast error.
LBC-Related Forecast Error
1. LBC Data Resolution
– Typically on a coarser horizontal and vertical grid than
that of the model simulation (necessitates interpolation).
– Errors can arise as the model spins-up the smaller scale
structures not present in the interpolation.
– As discussed earlier, the temporal resolution of LBC data
is also typically much coarser than the model’s ∆t.
– This can cause unrealistic gradients to occur near the LBs
as the forecast strays from the LBCs.
LBC-Related Forecast Error
2. LBC Data Quality
– LBCs typically obtained from model syntheses of
observations or model forecast fields.
– Even with perfect observations, model syntheses of
observations can deviate from the true atmospheric state.
– Such syntheses also do not resolve all observed scales.
– Model forecast fields can depart significantly from reality.
– All of these factors can impact forecast quality.
LBC-Related Forecast Error
3. Scale Interactivity
– Reiteration of a one- vs. two-way nesting difference.
– For one-way nests, specified LBCs exert a domain-scale
control on the inner domain forecast.
– However, the inner domain forecast (on smaller scales)
cannot feed back to and update the LBCs.
– This has been shown to degrade forecast quality as
compared to otherwise identical two-way nested cases.
LBC-Related Forecast Error
4. Noise Generation
– Primarily an issue with one-way nests.
– The solution near the boundaries can deviate from that
along the LBs themselves.
– This can lead to the generation of spurious, transient
inertia-gravity waves near the LBs.
– At best, this can complicate forecast interpretation; at
worst, it can negatively impact forecast skill.
– Typically addressed using a damping layer.
LBC-Related Forecast Error
5. Physical Process Parameterization Inconsistencies
– Parameterization packages differ substantially from one
another in how any process is mimicked in the model.
– This can lead to meaningful differences in how certain
meteorological phenomena are represented.
•
Ex: thunderstorms, cold pools, and interdependent fields.
– If different parameterizations are used on the coarse and
fine meshes, errors can arise as a feature propagates from
one mesh to the other.
•
Adjustment from one parameterization to another; balance issue.
LBC-Related Forecast Error
6. Numerical Wave Dispersion Issues
– For a given finite differencing scheme, numerical wave
dispersion depends upon wavelength.
– Dispersion is typically greatest for smaller wavelengths.
– Wavelength depends upon the horizontal grid increment
∆x, where L = n∆x.
– As you move from a coarse to a fine mesh (or vice versa),
the representation of a given wave will change.
•
•
Coarse -> fine will improve its representation.
Fine -> coarse will degrade its representation.
– This can impact wave dispersion near the LBs, potentially
leading to forecast error growth (on one or both domains)
LBC-Related Forecast Error
• We now wish to describe ways in which LBC-related
forecast errors can be isolated and identified.
• To do so, there are four ways described by the text…
–
–
–
–
Domain configuration studies
Mesoscale predictability studies
Adjoint sensitivity studies
Big Brother-Little Brother experiments
Domain Configuration Studies
• Run a model forecast over some area using an inner
mesh, whether using one- or two-way nesting.
• Next, run an identical model forecast, except using
only one larger mesh (e.g., covering the area of the
outer mesh used for the first forecast).
• Differences between the simulations are attributable
to one or more issues related to LBC formulation.
Domain Configuration Studies
Large 1-Domain Case Minus Nested 2-Domain Case
p at z = 6 km, ∆p = 1 hPa
Domain Configuration Studies
• Differences propagate into the domain primarily…
– On the northern and western boundaries, as the synopticscale flow is primarily out of the west.
– In the middle latitudes, where meteorological conditions
are active (cyclones, fronts, etc.).
• Locations furthest from the LBs are ‘protected’ from
LBC-related error for the longest amount of time.
– Enlargen the domain  protect areas within the middle of
the domain from LBC-related error for a longer duration.
Domain Configuration Studies
• Similar differences noted whether outer domain in
two-domain case has coarser (shown) or equal (not
shown) resolution to that of the inner domain.
– In other words, no resolution dependence – differences
are nearly entirely attributable to LBC-related errors.
• There is also little sensitivity to the method used to
prescribe LBCs, at least for this experiment.
(of 500 hPa height)
Domain Configuration Studies
Solid: 1-Domain 2.5° Case Versus 1-Domain 5° Case
Dashed: 1-Domain 2.5° Case Versus 2-Domain 2.5° Case with 5° LBCs
Dotted: 1-Domain 2.5° Case Versus 2-Domain 2.5° Case with 2.5° LBCs
Domain Configuration Studies
• Through t=30 h, solid line remains below the dotted
and dashed lines.
– The two 1-domain simulations are more alike than the 1domain 2.5° simulation is like its nested 2.5° counterparts!
• Differences largely grow with time for all cases.
– Exception: dotted line (LBC errors largest at earlier times
due to transient damped waves generated early at LBs?).
• Compared to observations, LBC-related error
increased total forecast error by ~50% after 24 h.
Domain Configuration Studies
Multiple Domain Size Experiment
How does LBC-related forecast error impact forecast skill over the central US?
(RMSE of 500 hPa height)
Domain Configuration Studies
Multiple Domain Size Experiment
Increased proximity of lateral boundaries to region of interest  degraded skill!
Domain Configuration Studies
• Errors grow with time in all cases, but do so most
rapidly for the smallest domain(s).
• Note that similar results were obtained for a summer
(shown) and winter (not shown) case examined.
• How are these manifest in the atmospheric fields?
Domain Configuration Studies
• (b): Note implied large-scale control by coarse LBCs.
• This results in an artificially smooth jet streak.
Left: outermost domain, Right: innermost domain
Displayed: 12-h Forecast of 250 hPa isotachs (m s-1)
Mesoscale Predictability Studies
• Example: assess the relative control of LBCs vs. initial
conditions (ICs) upon mesoscale forecast evolution.
• Figure 3.45 – identical LBCs, different ICs (at t = 96 h)
• Figure 3.46 – different LBCs, identical ICs (at t = 6 h)
• Both: control minus perturbed ∆(500 hPa height)
Mesoscale Predictability Studies
Displayed: 96-h Sensitivity to Perturbed ICs (Identical LBCs)
Mesoscale Predictability Studies
• After 96 h, substantial differences are noted
primarily in the northeastern portion of the domain.
• Synoptic-scale westerly flow advects in identical LBC
info from west->east over time, reducing differences.
• Despite this, however, there remains some memory
of the ICs even through 96 h.
Mesoscale Predictability Studies
Displayed: 6-h Sensitivity to Perturbed LBCs (Identical ICs)
Mesoscale Predictability Studies
• After 6 h, differences are largest along the LBs but
have also propagated inward.
• The rapid inward propagation of differences due to
LBC formulation is associated with transient high
frequency modes.
• What does the corresponding 96-h difference field
look like?
Mesoscale Predictability Studies
Left: 6-h Sensitivity to Perturbed LBCs (Identical ICs)
Right: 96-h Sensitivity to Perturbed LBCs (Identical ICs)
LBC-related differences grow with time and exceed those from IC differences by 96 h!
Adjoint Sensitivity Studies
• Adjoint operators provide a quantitative measure of
the impact on some part of a forecast by some small
and/or arbitrary perturbations (LBCs, ICs, etc.).
• Often used to assess forecast sensitivity for a
meteorological event to IC perturbations.
• Used as a means of targeting specific regions for
observation gathering during field programs.
Adjoint Sensitivity Studies
Step 4: Conduct Non-Linear Model Forecast
Step 3: Acquire Observations
Step 2: Integrate Adjoint Model
Step 1: Integrate Forward Linear (Often Dry) Model
ti = 0
ta = 36 h
ti = initial time, ta = adjoint/analysis time, tv = verifying time
tv = 84 h
Adjoint Sensitivity Studies
• Despite linearity assumption underlying the method,
adjoint sensitivity studies are widely-used.
• With ensemble covariance methods, adjoints offer
powerful information about IC-related uncertainty.
• We’ll discuss related issues more in Chapter 6.
• Here, we apply the method to examine sensitivity
related to LBCs vs. that related to ICs.
Adjoint Sensitivity Studies
Left: Sensitivity to Perturbed 400 hPa v (ICs) at 0 h
Right: Sensitivity to Perturbed 400 hPa v (LBCs) at 0 h
Adjoint Sensitivity Studies
• How sensitive is the 72-h ζ to 0-h IC and LBC
perturbations in the meridional wind at 400 hPa?
– Note: ζ considered over a finite volume given by the
circular area in (a) on the previous slide.
• Displayed: value of the sensitivity metric, scaled to a
contour interval 4x larger in (b) than in (a).
• Take-home: sensitivity in 72-h ζ to initial LBCs is
much greater than to ICs.
Adjoint Sensitivity Studies
• What is the sensitivity in the 72-h ζ to LBCs versus
interior conditions at later times?
• Table 3.2: more sensitive to LBCs through 48 h, more
sensitive to interior conditions thereafter.
– It takes ~24 h for most LB info to reach the area of interest.
Thus, there is lesser sensitivity to LBCs after 48 h.
– Interior conditions prior to 48 h come and go through the
domain well before the forecast time of 72 h.
– Thus, sensitivity is a function of both lead time and the
domain’s predominant information propagation velocity.
Big Brother-Little Brother Experiments
• Similar construction to the 2-domain, 2.5° simulation
with 2.5° LBC case described earlier in this lecture.
• Figure 3.48 in the text provides an example of
differences from such an experiment.
• Note, however, that these are somewhat muted
given the spatial averaging applied to a local field.
– Location error could be (very) large within the domain!
Practical Recommendations
• How can we minimize the deleterious impacts of LBC
error on a forecast?
• Section 3.5.4 of the text describes nine ways of doing
so; here, these are briefly summarized.
• 1) Use a buffer zone between the feature or region of
interest and the lateral boundaries.
– Constraints: simulation length; computational power
– Controllable or evaluable by: model users
Practical Recommendations
• 2) Minimize interpolation error along LBs.
– Use the best-quality, highest-resolution, highest-frequency
data available to you for your LBCs!
– Controllable or evaluable by: model users
• 3) Use a consistent model configuration between
inner and outer meshes.
– Where feasible, do not change physical parameterizations
(for example) between the two meshes.
– Controllable or evaluable by: model users
Practical Recommendations
• 4) Use a well-tested, well-construed LBC formulation
– Typically not user-selectable; read the documentation!
– Controllable or evaluable by: model developers
• 5) Account for effects of data assimilation on LBCs
– If a pre-forecast period is used to assimilate data, IC and
LBC quality may be improved, but pre-forecast LBC error
may reach the domain interior by the start of the forecast.
– We’ll discuss this in more detail in Chapter 6.
– Controllable or evaluable by: model users
Practical Recommendations
• 6) Know how the feature you are simulating may be
impacted by LBC data and/or errors.
– If a feature is locally-driven (e.g., a seabreeze or mountain
circulation), LBCs may not impact it substantially.
– Controllable or evaluable by: model users
• 7) Avoid strong forcing along the LBs.
– Don’t place LBs through sharply sloped terrain or regions
with quasi-stationary sharp gradients (e.g., Gulf Stream).
– Controllable or evaluable by: model users
Practical Recommendations
• 8) Utilize two-way/interactive nesting if possible.
– Benefits from doing so generally outweigh added
computational expense (except for outermost domain).
– Controllable or evaluable by: model developers and users
• 9) Use sensitivity studies to examine LBC influences.
– In other words, perform tests to evaluate all of the above.
– Don’t assume that what is true for one case or event holds
true for all cases or events.
– Controllable or evaluable by: model users
Upper Boundary Conditions
• The model atmosphere does not extend to infinity;
rather, it must be cut off at some altitude.
• This altitude should be above features of
meteorological interest.
– Commonly set in the stratosphere or near the stratopause.
– Longer time scale processes dominate the stratosphere, so
climate-scale simulations should use a higher bound.
– Above this level, the only input of interest is from incoming
solar radiation, which we typically parameterize.
Upper Boundary Conditions
• With this in mind, how do we represent the upper
boundary of the model?
• Key issue: how to handle the transfer of energy by
gravity waves upward and out of the domain?
– A minor issue if the upper bound is set high enough; a
more significant issue if not.
– Outgoing longwave radiation may be important, but this is
also typically parameterized.
Upper Boundary Conditions
• Many ways of representing the upper boundary…
• Method #1: Rigid Lid
– Model is capped at some specified altitude; energy
reaching this lid is reflected downward.
• Method #2: Free Surface
– Treats the model atmosphere and higher altitudes as two
distinct, non-mixing fluids; also reflects energy downward.
Upper Boundary Conditions
• Method #3: Absorption/Damping Layer
– Incorporated with the rigid lid or free surface methods.
• Damping layer characteristics…
– Placed just below top of the model
– Applies a diffusion/damping operator to selected vertical
levels in order to dampen upward-propagating energy
– Must be relatively thick to mitigate the development of
large vertical gradients and wave reflection issues
– Can dampen to a pre-defined reference state (Rayleigh
damping) or to one defined by the model atmosphere
Upper Boundary Conditions
Rigid Lid + Damping Layer
Contoured:
w (white – up,
shaded – down)
Rigid Lid Without Damping Layer
Reflects westerly
flow over an
isolated
topographical
feature.
Errors in bottom panel result from wave reflection from upper boundary.
Errors in top panel result from wave reflection from lateral boundary.
Upper Boundary Conditions
• Method #4: Radiative Boundary Condition
– Mimics the effects of wave energy propagating upward
and out of the domain at the top of the model.
• General recommendation: choose an upper
boundary sufficiently high to mitigate these issues.
Practical Summary of Model Setup
• Section 3.8 of the text lists seven factors that should
be kept in mind when configuring a model.
• In actuality, there are many more factors that one
must consider before running a model simulation.
– Dynamically-based (all of Chapter 3 and more)
– And many more (physics, initialization, etc.)
• A listing of the factors we’ve discussed follows.
Practical Summary of Model Setup
• What form of the primitive equations to use?
• Map projection: function of latitude
• Model formulation: grid-point, spectral, finite
volume, finite element, etc.
• If grid-point, what grid structure to use?
• Horizontal and vertical grid spacing
Practical Summary of Model Setup
• Horizontal grid structure: staggering, nesting,
variable resolution, location of boundaries, etc.
• Vertical grid structure: staggering, choice of
coordinate, location of model top, etc.
• Temporal differencing schemes: explicit schemes,
implicit methods, hybrid schemes
Practical Summary of Model Setup
• Spatial differencing schemes: Eulerian, Lagrangian,
semi-Lagrangian
• Truncation error: order of finite difference approx.
• Linear stability: wavelength & Courant # dependent
• Numerical wave dispersion: also wavelength and
Courant # dependent
Practical Summary of Model Setup
• Aliasing and non-linear instability
• Utility and formulation of numerical diffusion
• Lateral boundary conditions: data source,
interpolation methods, placement, etc.
• Upper boundary conditions: methods, location
• What is conserved and is it sufficient?