# Introduction to Geophysical Fluid Dynamics

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```					 Introduction to
Geophysical Fluid
Dynamics

by
Sybren Drijfhout
Fluid or gas?

• Geophysical fluids are: Ocean, atmosphere,ice and
(solid) earth

• All 3 can be deformed by (pressure) forces and flow

• For large scale motion the discrete molecular nature of
the atmosphere (gas) can be ignored

• The geophysical nature of these fluids is that they flow
on a rotating sphere and they are layered
The Continuum hypothesis

Why do we need models?
Because it’s all so damned complicated

 There are > 1040 molecules in atmosphere and ocean

 In GFD the discrete nature of molecules is neglected

 We consider a continuum of microscopically large particles

 Yet macroscopically small; local thermodynamical equilibrium

 The governing equations represent a mean field theory

 Everything flows: Liquid, gas, stone and time
The Continuum

 Various physical quantities characterize the state of the
fluid (pressure, density, temperature, velocity, etc.)

 They depend on 4 dimensionally independent
properties: length, time, mass and temperature

 They have unique values in the fluid continuum and are
continuous in space and time, as well as their derivates
Fundamental Laws
 Fluid motion is governed by 3 conservation principles: Newton’s
second law of motion, mass conservation (equation of continuity)
(and tracer conservation) and the thermodynamic energy equation

 No general, analytical solution exists. Myriads of motions are
allowed for, with different first order balances. Fluid motion is
turbulent, nonlinear and chaotic

~1000 km
~100 km
~1 km      ~10 km       HYDROSTATIC
~20 m       ~100 m
NON-HYDROSTATIC

Solution: reduce the range over which the equations are valid by
approximation
Approximations

 GFD is based on a systematic simplification of the
fundamental equations: approximate balances are
treated as physical laws

 Simplifications are based on scale/dimensional analysis;
different length-scales imply different balances

 Terms are neglected on the basis of:
a) Magnitude of field variables
b) Amplitude of fluctuations
c) Characteristic length and time scales
Equation of motion

 Fundamental equation is Newton’s equation for a
continuous medium (Navier-Stokes)

 Transformed to a coordinate frame that is fixed with the
rotating earth; Coriolis force, curvature terms

d *V           1
 2  V   p  g  F  0
dt            

* denotes inclusion of curvature terms
Scaling the horizontal momentum equation

du                                                                                                                      uw                            uv tan                                              1 p
 2v sin                                 2w cos                                                                                                                                                   Fx
dt                                                                                                                       r                                r                                                 x

dv                                                                                                                  vw                              u 2 tan                                              1 p
 2u sin                                                                                                                                                                                             Fy
dt                                                                                                                   r                                   r                                                 y

U2                                                                                                                   UW                                   U2                                          P                              KU
f 0U                                          f 0W
L                                                                                                                     a                                   a                                           L                              L2

104                               103                                         106                                   108                               105                                       103                            1010
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

U ~ 10                                               W ~ 102                                            L ~ 106                                             P /  ~ 10 3                                                 K ~ 10
Geostrophic balance
1 p                                 1 p
fv                                fu  
 x                                  y

2
U /L    U
Ro               0.1
f 0U   f0 L
Ageostrophic motion

du                              dv
 fva   Fx                     fua  Fy
dt                              dt
Hierarchy of models

• Navier-Stokes equations use continuum approximation

• Euler equations use the incompressible approximation:
d
 0,   U  0
dt
• Primitive equations use the hydrostatic approximation:
e.g.              p
 g  0
z
and traditional approximation: r = a; and some Coriolis terms
1          1
• Boussinesq equations use:         p         p
           0

• Shallow water equations linearize in h/L about a basic state

• QG vorticity equations: Beta-plane and geostrophic balance

• But still we need computers to solve these systems!
What is a model and why
do we need models?

• The governing equations are impossible to solve: approximate and idealized
sets of equations are necessary

• When making approximations we are guided by 2 principles
a) we want to choose the simplest model possible
b) the model must be efficient in terms of computer demands

• Models are only simplified representations (maps) of reality; a tool to interpret
the world

• The main purpose of a model is not to fit the data but to sharpen the questions

• All models are wrong, but some are useful
Shallow-water Equations
  gh

du        
 fv     0
dt        x

dv        
 fu     0
dt        y

d     u v
 (  )  0
dt     x y
Rossby waves
Linear, large-scale: no divergence, transverse motion:
The effect of changing Coriolis drives changes in flow and wave motion


 fv     0
x
v 
   0
t y
 2v
Eliminate  :             v  0
xt

Solution of wave equation:          v  v0 exp[ ik ( x  ct )]


Phase speed:        c
k2
Rossby waves in
the atmosphere
Rossby waves   in the ocean
Gravity Waves
Linear, fast: no rotation, small scale: east-west motion, no
u 
   0
t x
    u
    0
t    x
 2u    2u
Eliminate one variable:            2  0
t 2
x
Solution of wave equation:    u  u( x  ct )   ( x  ct )

Phase speed:      c 
Gravity waves in the atmosphere
Vorticity and Divergence

In place of the horizontal eqs. of motion we can use a vorticity
and divergence eq. obtained by the vector operations:

k  x           and                

Gravity waves are filtered out when the divergence tendency

d                     d
0              (
dt
 f  u   2  0   )
dt

by demanding                             f 0   2 
Potential Vorticity

The vorticity equation is then scaled to yield the
quasi-geostrophic system:

v  k           and            2

In this system potential vorticity is a conserved quantity

dq                                          f 02  
0           with       q   2  f      2
 N z  
z         
dt

  f 02                             f 02
→
1
q   f 
2
 N z 
 2                                2 2
z                               L2
H N

HN       g' H          Scale at which rotation becomes equally
L                       important as buoyancy effects
f0      f0

Lat  300 km                         Loc  30 km
The 2-layer QG model
dq d  2        f 02  
    f   2
 N z   Forc  Diss
                     becomes:
dt dt        z         

   1   1   2
                        
 t  y x  x y    1  F1 ( 1  2 )  y 
                    

1 4            Y X 
  1  T 
 x  y 

Re                     

   2      2   2

 t  y x  x y    2  F2 ( 1  2 )  y  
                    
1
 4 2
Re
Nondimensional parameters
UL                       0L
Re                  T 
AH                      0 D1U 2

0L  2
1     E
Ro

AH
            
U         Ro       Re  0 L3

2 2
f L                  f L 2 2
F1    0
F2     0
g ' D1               g ' D2
Message
 In GFD we obtain exact solutions from approximate
equations

 Physical intuition is needed to develop the pertinent model
and adequate set of equations for each type of motion

 Dimensional and scale analysis come from turbulence theory

 Simple physical concepts bridge the gap between the
equations and the actual flows

 Lacking a completely formal theory, each flow, being
different and unique, deserves its own model
Building the bridge between the math and the flow

 2                                   w
   k    (   f 0 )  f 0
2

t                                    z

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