Appendix A
Hydrodynamic equations, basic approximations
In Eulerian flow description we look at the generic variable describing the fluid property per


unit mass    x, y, z, t which
Lagrangian fluid particle is
d 
full differential along the trajectory of




dt 
dx 
dy 
dz .
t
x
y
z
Eulerian current components are defined as time derivative of the coordinates
u, v, w   dx , dy , dz  .
xt , yt , zt  of passing Lagrangian fluid particles
 dt dt dt 
The material
d

dt
time derivative is defined as




u
v
w
t 
x
y
z




local time derivative
advective derivative
Hydrodynamic (HD) and thermodynamic (TD) equations originate from the consevation laws
for mass, 3 momentum components, heat and salt .
Generic conservation equation of variable  in the fluid volume V surrounded


d
D
 dD n is written in a form
D
by a permeable surface
with a differential element
 






dV



v
 dD   q dV



t
V
D
V










change of total content
in a volume
advective fluxes
throughthe volume boundaries
net production
in a volume
The second term is converted by the Gauss divergence theorem

  v  dD      v dV    u    v    w  dV . Omitting the integrals we


D
V
  x
V
y
z




d











v

q





v
 q
obtain
or

t
dt
For the mass conservation we take   1 and q  0 , then



d
    v   0
   v  0 .
or
t
dt
d
 
0
 v    0
Incompressibility
or
dt
t
 u v w
v 


0
leads to non-divergence of velocity
x y z



continuityequation
.
Lecture notes by Jüri Elken
Page 1
Advection-diffusion equation of a property  is derived with an assumption that

 
diffusive flux  m   dD is added to the advective flux     v  dD . Then for
imcompressible fluid we obtain
  2  2  2   




u
v
w
  m  2  2  2   P  D
t
x
y
z
y
z 
 x
where
q

,
 P  D is production minus destruction per unit volume and  m is molecular
diffusion coefficient.
Turbulent motions are treated by Reynolds decomposition into a mean,
regular component and irregular pulsation component
  0,
     ,
  .
Averaging the advection-diffusion equation we obtain
  2  2  2




u
v
w
  m  2  2  2
t
x
y
z
y
z
 x
  
  P  D 





  u    v   w  
x
y
z

,
turbulent fluxes
where additional "turbulent terms" appear due to non-linearity of advection.
Turbulent fluxes are usually parameterised by gradient approach, assuming they
act like diffusion or viscosity whereas horizontal and vertical turbulence coefficients
have different value
u     



; v     
; w    
x
y
z
and generally are not constants. Since turbulent diffusion is much greater than molecular, the
latter can be neglected. We will reach the
advection-diffusion equation of turbulent motions





 



u
v
w


 
 
 
P 

D
t
x
y
z x
x y
y 
z

z net production





















local
change
advective change
Lecture notes by Jüri Elken
horizontalturbulent
diffusion
vertical turbulent
diffusion
Page 2
Equations of motion or momentum equations follow the Second Newton Law

d
 v    F
dt

F
where
is the sum of mass forces and surfacial forces (stresses) acting on a fluid
element of unit mass.
In the Newtonian fluid the normal
stress (acting across an area in the fluid) is expressed
by pressure gradient  p and tangential stresses (acting parallel to an area in

the fluid) are expressed by viscosity   m v  where  m is the molecular viscosity
coefficient.
We will reach then Navier-Stokes




  v   v
   v   m




t 

 advective momentum mass
local momentum
change
change
force
equations of motion





p





v
m
 

pressure
gradient
force
tangential stress
due to friction
In geophysics the main



mass forces are:
effective gravity force, gravity of Earth acting together with centrifugal force
perpendicular to the geopotential surface (undisturbed sea level);
Coriolis force deflecting motion in the Northern Hemisphere to the right due to Earth
rotation;
tidal forces acting periodically due to gravity forces of Moon and Sun on rotating
Earth.
Cartesian coordinate system is used in f - plane approximation
for medium-scale studies. Common axis orientation is east (x), north (y) and up (z).
undisturbed
sea level
y - axis
z - axis

f - plane
Mass forces on a f -plane:

mg
 0, 0,  g 
gravity

Coriolis force

mc

  fv,  fu, 0 
where

effective
gravity
Lecture notes by Jüri Elken
f  2 sin 
is
Coriolis parameter
 - rotation speed of Earth
 - geographical lattitude of coordinate
origin
Page 3
Equations used in 3D circulation models
Horizontal momentum equations
u
u
u
u
1 p  u  u  u
u
 v  w  fv  
 
 
 
t
x
y
z
 x x x y y z z
v
v
v
v
1 p  v  v  v
 u  v  w  fu  
     
t
x
y
z
 y x x y y z z
Vertical momentum equation in hydrostatic approximation
p
  g
z
Continuity equation
u v w


0
x y z
Advection-diffusion equation for temperature
T
T
T
T

T 
T 
T
u
v
w
 T
 T
 T
t
x
y
z x
x y
y z z
Advection-diffusion equation for salinity
S
S
S
S 
S 
S 
S
u
v
w
 S
 S
 S
t
x
y
z x
x y
y z z
Equation of state
   T , S , p
Boundary conditions
1) at lateral walls and bottom:
absence of flow and turbulent flux across the "closed" boundary
S
0
n
T
0
n
un  0
2) at "open" boundaries: prescribed values or gradients,
constituting "remote forcing"
3) forcing at sea surface
wind stress
x  
u
z
y  
heat and salt flux
FQ   T
T
z
Lecture notes by Jüri Elken
FS   S
v
z
S
z
Page 4
APPENDIX B. Shallow water equations
When the depth is much less than horizontal scale of motions
~
H
  ~  1 and water is vertically well-mixed, fluid motion is well described by
L


vertically integrated hydrodynamic equations in x, y, t coordinates. We integrate over the
whole depth range of water thickness h from the bottom z   H to the surface elevated from
h  H 
2D shallow water equations.
the rest by  ,
and obtain
Basic variables are:
1) Water level  that is related to pressure by integrating the hydrostatic equation
~x, y, t  x, y, t 
px, y, z, t   pa x, y, t   g~x, y, t  z  g
2) Depth-averaged velocity components and constituents
(temperature, salinity, density, subctance concentration etc)

1
u~   u dz
h H

1
v~   v dz
h H

1
~    dz
h H
2D equations are: Continuity equation



 hu~  hv~  0
t
x
y



change
of volume
per unit area
divergence of transport
Horizontal momentum equations
 ~  ~ 2  ~~
 gh 2 ~ ~ ~
~
h u  h u  hu v  f hv   gh

 hu   Fx  Bx
t
x
y
x 2 ~ x
 ~  ~~  ~ 2
 gh 2 ~ ~ ~
~
hv  hu v  hv  
f hu   gh
 ~
 hv  


t

x

y

y
2


y
 
horizontal

 Coriolis










force
local
momentum
change
advective momentum
change
barotropic
pressure
gradient
baroclinic
pressure
gradient
friction
Fy  B y



forcing stress and
bottom frictionstress
Advection-diffusion equation
~ ~
 ~  ~ ~  ~ ~  ~ ~  ~ ~
h  h u  h v   h
 h
 h
P

D


t x
y
x
x y
y



  net production

local
change
advective change
Equation of state
Lecture notes by Jüri Elken

horizontalturbulent diffusion
~ ~ 
~   T , S
Page 5
Boundary conditions, forcing and dissipation
Isolating conditions at the non-permeable coastline:
u~n  0
free-slip conditions
normal velocity is set to zero
or
non-slip conditions
u~  v~  0

0
n
both velocity components are set to zero
(require accounting horizontal friction)
absence of normal constituent fluxes
Forcing (energy input) is done by:
Wind stress, by quatratic drag formulae from the wind speed U W ,VW 


Fx  ~a c d U W U W2  VW2

Fy  ~a c d VW U W2  VW2

,
Tidal forces, periodic functions depending on geographic coordinates
Wave drift (transformation of wind waves), discussion will follow in next sections
Remote forcing through "open" boundaries:
1) prescribe inflow and constituents of river discharge
2) prescribe exchange with the adjacent sea area (an item of art!!!)
 use larger scale model to get dynamically consistent currents and water level,
avoid artificial effects
 at inflow areas specify the properties of imported water
Energy dissipation is done by bottom friction:
Quadratic approach is used in numerical models
Bx  cb u~ u~ 2  v~ 2
,
By  cb v~ u~ 2  v~ 2
coefficient cb depend on Reynolds number, bottom roughness etc
g  n2
is frequently used, where n  10 2 is Manning
h7 3
bottom friction coefficient and  is unit conversion constant
in engineering applications cb 
~
Bx  rhu
Linear approach
,
is used sometimes for simplicity of analytical solutions
and
B y  rhv~
horizontal friction.
Lecture notes by Jüri Elken
Page 6