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The equations of motion and their numerical solutions I

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High resolution in-situ observations show large small-scale variability of ... Use high resolution (10km in horizontal) simulation of flow over Scandinavia ... – PowerPoint PPT presentation

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Title: The equations of motion and their numerical solutions I


1
The equations of motion and their numerical
solutions I
by Nils Wedi (2006) contributions by Mike Cullen
and Piotr Smolarkiewicz
2
Why this presentation ?
  • There is NO perfect model description for either
    atmosphere or ocean (so far?)

3
Introduction
  • Observations
  • What do we solve in our models ?
  • Some properties of the equations we solve
  • What do we (re-)solve in our models ?
  • How do we treat artificial and/or natural
    boundaries and how do these influence the
    solution ?

4
Introduction
  • Review and comparison of a few distinct modelling
    approaches for atmospheric and oceanic flows
  • Highlighting the modelling assumptions,
    advantages and disadvantages inherent in the
    different modelling approaches
  • Highlighting issues with respect to upper and
    lower boundary conditions

5
Observations
  • Much information about the solution of the
    equations can be deduced from observations.
  • Satellite pictures show large-scale structures,
    but closer examination reveals more and more
    detail.
  • High resolution in-situ observations show large
    small-scale variability of comparable amplitude
    to underlying larger scale variations

6
Satellite picture of N. Atlantic
Large-Scale Frontal systems on scale of
N.Atlantic. Cellular convection S. of Iceland. 2d
vortex shedding S. of Greenland.
7
Enlarged picture over U.K.
More detail … Gravity-wave train running SW-NE
over UK.
8
Balloon measurements of static stability
Stratification S Brunt-Väisällä frequency
N Potential temperature ?
Idealized reference or initial state in
numerical models
9
Time series of wind speed from anemograph
High resolution observations are still above the
viscous scale! However, only spatially and
temporally averaged values suitable for
numerical solution.
10
Observed spectra of motions in the atmosphere
Spectral slope near k-3 for wavelengths gt500km.
Near k-5/3 for shorter wavelengths. Possible
difference in larger-scale dynamics.
No spectral gap!
11
Scales of atmospheric phenomena
Practical averaging scales do not correspond to a
physical scale separation. If equations are
averaged, there may be strong interactions
between resolved and unresolved scales.
12
Observations of boundary layers the tropical
thermocline
M. Balmaseda
13
Observations of boundary layers EPIC - PBL over
oceans
M. Koehler
14
Averaged equations
  • The equations as used in an operational NWP model
    represent the evolution of a space-time average
    of the true solution.
  • The equations become empirical once averaged, we
    cannot claim we are solving the fundamental
    equations.
  • Possibly we do not have to use the full form of
    the exact equations to represent an averaged
    flow, e.g. hydrostatic approximation OK for large
    enough averaging scales in the horizontal.
  • The sub-grid model represents the effect of the
    unresolved scales on the averaged flow expressed
    in terms of the input data which represents an
    averaged state.
  • The average of the exact solution may not look
    like what we expect, e.g. since vertical motions
    over land may contain averages of very large
    local values.
  • Averaging scale does not correspond to a subset
    of observed phenomena, e.g. gravity waves are
    partly included at T511, but will not be properly
    represented.

15
Demonstration of averaging
  • Use high resolution (10km in horizontal)
    simulation of flow over Scandinavia
  • Average the results to a scale of 80km
  • Compare with solution of model with 40km and 80km
    resolution
  • The hope is that, allowing for numerical errors,
    the solution will be accurate on a scale of 80km
  • Compare low-level flows and vertical velocity
    cross-section, reasonable agreement

Cullen et al. (2000) and references therein
16
Mean-sea-level pressure
17
Satellite picture
VIS 18/03/1998 1300
18
High resolution numerical solution
  • Test problem is flow at 10 ms-1 impinging on
    Scandinavian orography
  • Resolution 10km, 91 levels, level spacing 300m
  • No turbulence model or viscosity, free-slip lower
    boundary
  • Semi-Lagrangian, semi-implicit integration scheme
    with 5 minute timestep
  • Errors in flow Jacobian are the errors of the
    Lagrangian continuity equation integrated over a
    timestep contour interval is 3.

19
Low-level flow 10km resolution
20
Cross-section of potential temperature
21
x-y errors in the inverse flow Jacobian
In IFS
22
Flow Jacobian in T799 L91
Remember Contour line 1 is the correct
answer!
23
y-z errors in the inverse flow Jacobian
24
Low level flow- 40km resolution
25
Low level flow-10km resolution averaged to 80km
26
x-z vertical velocity - 40km resolution
27
Vertical velocity - 10km resolution averaged to
80km
28
Conclusion
  • Averaged high resolution contains more
    information than lower resolution runs
  • The ratio of comparison was found approximately
    as
  • dx (averaged resol) ?dx (lower resol) with ?
    1.5-2

29
Vertical velocity cross-section ECMWF operational
run
Observations show mountain wave activity limited
to neighbourhood of mountains. Operational
forecast (T511L60) shows unrealistically large
extent of mountain wave activity. Other tests
show that the simulations are only slightly
affected by numerical error. The idealised
integrations suggest that the predictions
represent the averaged state well (despite
hydrostatic assumption). The real solution is
much more localised and more intense.
30
What is the basis for a stable numerical
implementation ?
  • A Removal of fast - supposedly insignificant -
    external and/or internal acoustic modes (relaxed
    or eliminated), making use of infinite sound
    speed (cs??) from the governing equations BEFORE
    numerics is introduced.
  • B Use of the full equations WITH a semi-implicit
    numerical framework, reducing the propagation
    speed (cs ?0) of fast acoustic and buoyancy
    disturbances, retaining the slow
    convective-advective component (ideally)
    undistorted.
  • C Split-explicit integration of the full
    equations, since explicit NOT practical (100
    times slower)
  • ? Determines the choice of the numerical scheme

Rõõm (2001)
31
Choices for numerical implementation
  • Avoiding the solution of an elliptic equation
  • fractional step methods Skamrock and Klemp
    (1992) Durran (1999)
  • Solving an elliptic equation
  • Projection method Durran (1999)
  • Semi-implicit Durran (1999) Cullen et.
    Al.(1994)
  • Preconditioned conjugate-residual solvers or
    multigrid methods for solving the resulting
    Poisson or Helmholtz equations Skamarock et. al.
    (1997)

32
Split-explicit integration
Skamarock and Klemp (1992) Durran (1999) Doms
and Schättler (1999)
33
Semi-implicit schemes
  • (i) coefficients constant in time and
    horizontally (hydrostatic models Robert et al.
    (1972), ECMWF/Arpege/Aladin NH)
  • (ii) coefficients constant in time Thomas (1998)
    Qian, (1998) see references in Bénard (2004)
  • (iii) non-constant coefficients Skamarock et. al.
    (1997), (UK Met Office NH model)
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