The document discusses equations of motion used in weather forecasting and climate change studies. It begins with an introduction to geophysical fluid dynamics and the distinguishing effects of rotation and stratification. It then outlines the basic equations of motion, including conservation of momentum, mass, energy, and state. It describes how these equations are solved on grids using numerical models. It discusses the challenges of modeling processes at different spatial scales from synoptic to urban. It also addresses challenges in tropical weather prediction and how dynamical prediction of weather over South Asia has improved.

1. The equations of motion for
weather forecasting
through climate change studies
K. Ashok
Centre for Earth, Ocean & Atmospheric Sciences
University of Hyderabad
ashokkarumuri@uohyd.ac.in

2. Introduction: Geophysical Fluid Dynamics
• The effects of rotation and those of stratification Distinguish the GFD.
• The rotation of the earth around its axis introduces Coriolis force, which
adds a certain amount of rigidity.
• Stratification, the other distinguishing attribute of geophysical fluid
dynamics, arises because naturally occurring flows typically involve
fluids of different densities (e.g., warm and cold air masses, fresh and
saline waters). Here, the gravitational force is of great importance, for it
tends to lower the heaviest fluid and to raise the lightest. Under
equilibrium conditions, the fluid is stably stratified, Fluid motions,
however, disturb this equilibrium. Small perturbations generate internal
waves.
• Large perturbations, especially those maintained over time, may cause
mixing and convection. For example, the prevailing winds in our
atmosphere are manifestations of the planetary convection driven by
the pole-to-equator temperature difference

3. The start: equations of motion:
• Conservation of momentum
(F=ma)

Du
Dt
 fv  

x
 Fx

Dv
Dt
 fu  

y
 Fy

0  

p

RT
p
x
y
z
Hydrostatic balance
(neglect vertical accelerations)
• Conservation of mass
(no thermonuclear reactions)

d
dt
   
r
u  0

u
x

v
y


p
 0
 = dp/dt
= pressure velocity

T
dt
 u
T
dx
 v
T
dy
 (
T
p

RT
pcp
) 
J
cp
• Conservation of energy
(ditto)
• Equation of state
(a gas)

p  RT
* For an ocean model, you have also an equation for salinity.

4. Prognostic/Diagnostic Equations
Equations (1a), (1b), (3), and (4) are
called prognostic equations because
time changes in forecast variables
(u, v, T, and q) are determined
explicitly using dynamic forcing
equations.
In equations (2) and (5), the
remaining variables ( w and z) are
determined from the prognostic
variables. Because they do not
calculate time changes directly, they
are known as diagnostic equations.

5. What is a Model?
•Take the equations of fluid mechanics and
thermodynamics that describe atmospheric
processes.
•Convert them to a form where they can be
programmed into a large computer.
•Solve them so that this software representation of the
atmosphere evolves within the computer.
•This is called a “model” of the atmosphere

6. Solve equations on a sphere
Solve in the vertical Solve in the horizontal
grid
Spherical harmonics

7. Not just the atmosphere…

8. One-Dimensional
Climate Model

9. Three-Dimensional
Climate Models
(GCM)

10. Governing equations
• An example of one momentum equation:
1-d wind accelerated by only the pressure gradient force
x
p
Dt
Du



1

Computers cannot analytically solve even this
very simple equation!

11. Integration of the equations
1
1 1
2
k k k k
ki i i i
i
U U U U
U
t x

 
  
  
  
U U
U
t x
  
  
  
Nonlinear advection

12. Weather Prediction by Numerical Process
Lewis Fry Richardson 1922

13. Weather Prediction by Numerical Process
Lewis Fry Richardson 1922
•Predicted:
145 mb/ 6 hrs
•Observed:
-1.0 mb / 6 hs
ps
t

14. In 1928, the German mathematicians Courant,
Friedrichs, and Lewy systematically studied how to
solve partial derivative equations by using finite
differences and specified the constraints to comply
with when performing discretization (Courant et
al., 1928), the CFL criteria.
In 1939, the Swede Carl Gustav Rossby showed that
the absolute vorticity conservation equation
provided a correct interpretation of the observed
displacement of atmospheric
centres of action (Rossby, 1939).

15. First Successful Numerical Weather
Forecast: March 1950
• Grid over US
• 24 hour, 48 hour forecast
• 33 days to debug code and do the forecast
• Led by J. Charney (far left) who figured out
the quasi-geostrophic equations
Tropical Weather Prediction is all together extremely challenging…a
different level

16. Scales of processes/models
•Jet streams
•High and low
pressure centers
•Troughs and
ridges
•Fronts
•Thunderstorms
•Convective complexes
•Tropical storms
•Land/sea breezes
•Mountain/valley breezes
•Downslope wind storms
•Gap flows
•Cold air damming
•Nocturnal low-level jets
•Lake-effect snow bands
Synoptic MesoGlobal
•Long waves
•El Nino
•Street-canyon flows
•Channeling around
buildings, wakes
•Vertical transport on
upwind and warm
faces of buildings
•Flow in subway
tunnels
Urban

17. How the Model Forecasts
Time 
Temperature
T now
(observed)
X
X
Model-calculated T
changes
X
X
X
X

18. Integration of the equations
1
1 1
2
k k k k
ki i i i
i
U U U U
U
t x

 
  
  
  
U U
U
t x
  
  
  
Choose time step based on expected wind
speeds and grid spacing
k
i
x
t
U

 
Nonlinear advection Time step

19. Horizontal grid structures
MM5 and others WRF and others
From Randall (1994)

20. Unstructured: Omega Model
From Boybeyi et al. (2001)

21. For example
•The use of sigma coordinates, rather than pressure or height, avoids complications
that arise when the pressure or height surfaces intersect the ground, especially in
mountainous areas
sigma coordinate is defined by = p/ps

22. Purpose of modelling
•Forecasting weather
•Extended & Seasonal prediction
•Ocean weather
•Climate change
•Applications
•Understanding of physical and dynamical
mechanical Processes
•Theoretical studies with simplified processes

23. Modelling Philosophy
The MOST important principle of modelling,
and challenge is ..

24. Sources of model error
• Numerics/numerical schemes
• Physics (radiation, turbulence, moist processes)
• Initial conditions - define the atmosphere’s current state…the starting point
• Lateral boundary conditions - define the atmosphere’s state at domains’
edges
• Lower boundary conditions – conditions at Earth’s surface

25. MM5: leapfrog (t) and 2nd-order
centered (x)
Runge-Kutta (t) and 6th-order
centered (x)

26. “Nested” grids
• Grids can be telescoped, or nested, to zoom in on a small area
Large grid-point spacing – say 90 km
30 km
10 km

27. Physical Process representations: Parameterizations
• Parameterizations approximate the bulk effects of physical
processes that are too small, too complex, or too poorly
understood to be explicitly represented
Bauer et al., Nature 2015
Physical process representations

28. Data Assimilation injects observed data sets intermittently in to
evolving model forecast to constrain its evolution towards
observations
Figures from

29. Verification of model skill

30. Example of error
growth

32. P Bauer et al. Nature 525, 47-55 (2015) doi:10.1038/nature14956
Key challenge areas for NWP in the future.
• Advances in forecast skill will come
from scientific and technological
innovation in computing
• The representation of physical
processes in parameterizations,
• Coupling of Earth-system
components,
• The use of observations with
advanced data assimilation
algorithms, and the consistent
description of uncertainties through
ensemble methods and how they
interact across scales.
• The ellipses show key phenomena
relevant for NWP as a function of
scales between 10-2 and 104 km
resolved in numerical models and the
modelled complexity of processes
characterizing the small-scale flow
up to the fully coupled Earth system.
• The boxes represent scale-
complexity regions where the most
significant challenges for future
predictive skill improvement exist.
• The arrow highlights the importance
of error propagation across
resolution range and Earth-system
components.

33. TROPICAL WEATHER PREDICTION
• IS CHALLENGING OWING TO THE DIFFERENT DYNAMICS
• FOR EXAMPLE, HUGE MERIDIONAL TEMPERATURE GRADIENTS
PROVIDE THE ENERGY FOR MID-LATITUDE SYSTEMS. HUGE
PRESSURE/TEMPERATURE GRADIENTS. SIMPLE PHYSICS
• UNLIKE THIS, IN THE TRPOPICS, THE ENERGY HAS TO BE DIABATIC IN
NATURE, MAINLY FROM CONVECTION
• ~CLOUDS OF 1-10 KM HORIZONTAL SCALE PROVIUDE ENERGY TO
DRIVE A 600 KM SIZED TROPICAL CYCLONE.
• LACK OF OBSERVATIONS, RESOURCES, AND COMPUTING POWER.

34. Normal Conditions
El Niño-Southern Oscillation, and its types
Ashok et al., J. Geophys. Res., 2007;
Ashok and Yamagata, Nature New & Views, 2009

35. Operational MME seasonal prediction of the ENSOs,
types and impacts
Jeong et al., Clim. Dyn. JGR (2012)
As the Chief of operations,
introduced operational 6-month lead
MME seasonal prediction at the APCC
for ENSO types as well as climate
(Jeong et al., JGR 2012).
Also introduced a new MME method, named
as “Climate Filter” based multi-model MME
(Lee et al., JGR 2011), and a global drought
monitoring product (Sohn et al., Int. J.
Climatol. 2011)

36. Dynamical Prediction on weather scales in the monsoon region has
improved substantially
Fig. MME forecasts track based on different initial
conditions for the tropical cyclone HUDHUD, from NWP REPORT
ON CYCLONIC STORMS OVER THE NORTH INDIAN OCEAN DURING 2014, Kotal et al.,
NCMRWF rep., 2015
Total precipitation [mm/day] for (a) CMORPH over 28–29 July 2010 and (b)
ECMWF ensemble mean of the forecast initialized four days previously (July
24, 2010) for the same time period. White contour shows 20 mm/day.
ECMWF 15‐day forecast of the precipitation [mm/day] in the red rectangle
(Figure 1a) initialized on July (c) 22nd, and (d) 24th, 2010. Black dashed line
shows the ensemble mean. Colored shading depicts the probability of
precipitation rate based on the 51 ensemble members. Dark blue line
represents the observed CMORPH precipitation averaged for the same
Region (Webster et al., 2010, GRL)

37. Dynamicalweather
prediction for the Indian
sub-continent has
improved substantially
Fig. MME forecasts track based on different initial conditions for the tropical cyclone HUDHUD,
from NWP REPORT ON CYCLONIC STORMS OVER THE NORTH INDIAN OCEAN DURING 2014, Kotal et al., NCMRWF rep., 2015

38. Diagnosis methods
• Statistics help as powerful data reduction methods - linear and non-
linear (e.g. linear regression, EOF, SOM, etc.).
• Graphics/Math softwares – GrADs, Ferret, MATLAB, NCL, IDL, Vis5D,
etc.

39. 44
JJA Feb IC 0.50 MAR IC 0.42
Apr IC 0.25 May IC 0.24
• Feb IC exhibits maximum skill
at seasonal time scales viz.
0.55 for JJAS and 0.5 for JJA.
• This translates to the spatial
pattern as well, with Feb IC
exhibiting the pattern
reasonably.
Courtesy: Suryachandra Rao, IITM
Hindcast seasonal prediction correlation skills for summer monsoon rainfall (1983-
2010)

40. Aerosols affecting the Climate variability…
Fadnavis et al. Climate Dynamics, (2017)
TOMS Aerosol
index (1978 -
2005)
MISR AOD
2000 - 2010
Climatology El Niño years
ECHAM5-HAMMOZ
 TOMS AI, and MISR AOD averaged
for the April and May months show
relatively high AOD loading can be
clearly visible over the IGP.
 The ECHAM5-HAMMOZ model
could reproduce higher than
climatology during El-Nino over the
IGP.
Distribution of April-May average (a) TOMS aerosols index (AI) climatology 1978-2005, (b) TOMS aerosols index
during El Niño years. MISR aerosols optical depth (AOD), (c) climatology (2000-2010) (d) El Niño years, ECHAM5-
HAMMOZ simulated AOD obtained from (e) CTR_aeron (f) ElNiño_aero experiments

41. Aerosol impact on precipitation
Climatology El Niño
El Nino producing
less precipitation
is captured by the
model
Distribution of mean seasonal mean precipitation (mm/day) from (a) CTR_aero (b) ElNiño_aero. Aerosol-
induced changes in seasonal mean precipitation as obtained from (c) difference between CTR_aero and
CTR_aeroOFF (d) difference between ElNiño _aero and ElNiño _aeroOFF experiments.
Inclusion of aerosols in the
model experiment reduces
the severity of drought
during El Niño
Fadnavis et al. Climate Dynamics, (2017)

42. 47
(a) 101-year running mean anomalies of ISMR (mm/day); The MWP & LIA period are shown in red & blue
boxes, respectively. (b) Linear trend lines of the area-averaged ISMR during LM, as simulated by the nine
PMIP3 models.
• A statistically significant (at 0.10 levels) but the moderate decreasing trend in area-averaged ISMR in four models,
and a weaker decreasing trend in four more models throughout the LM, in agreement with findings from several
proxy records.
• A wet and warm Indian monsoon during MWP, and a cool and dry LIA.
Tejavath et al., 2018

43. HISTORICALSIMULATIONS
FUTUREPROJECTIONS
Soraisam et
al., 2017

44. Nino3 SST
Precipitation
(5N-35N; 65E-95E)
Lagged correlation between ISMR and
Nino3 SST in the preceding/following months
(Swapna et al., BAMS, 2015)
ENSO-Monsoon relationship
Indian (land + ocean)
Precipitation
IITM ESM: IITM contributing to the CMIP6 with an
improved version.
Improved Monsoon-ENSO
links are crucial o climate
scales.

45. - A combination of the BoB warming, a Mega El Niño, and
unplanned development
Images from the internet
Diagnosis of the Chennai extreme
Rainfall event, Dec. 2015: A perfect
Extreme event
Boyaj et al.; Clim. Dyn. (2017)

46. The extreme rainfall event in Chennai, Dec 2015
Boyaj et al., Climate Dynamics, (2017)
Daily accumulated rainfall (mm/day) from the TRMM
The Mega El Niño, Dec 2015 SSTA (°C)
• Did the Mega El Niño
and the BoB warming
trend contribute to the
extreme Chennai rainfall
event?
• Our experiments with
WRF model (30 km)
answer this query.
Time series area average (85E to 95E, 80N to 15N) of BoB SST anomalies

47. Coupling atmospheric models
with special-application models
• Transport and diffusion models
• Sound-propagation models
• Ocean wave models
• Ocean circulation models
• Parachute-drift models

48. Acknowledgments of source information
• Fundamentals of Numerical Weather Prediction..Jean Coffier
• Geophysical Fluid Dynamics: Cushman et al.
• Dynamic Meteorology.. Holton
• Introduction to Theoretical Meteorology.. Hess
• Numerical weather and Climate Prediction..T.T. Warner
• Research Papers
• Drs. Suraychandra Rao, Pattanaik, Sahai, etc.
• Internet material: L. Bengtson, M. Cresswell, Warner, Inez Fung,
Aaron Donohoe, etc.