The creation of magnetic structures from velocity shear

1
The creation of magnetic structures from
velocity shear (Constraints for the solar
dynamo) Nic Brummell Applied Mathematics,
University of California Santa Cruz Geoff
Vasil, Kelly Cline JILA/APS, University of
Colorado Lara Silvers, Mike Proctor DAMTP,
University of Cambridge, UK
UCSD Feb 2009
2
We are Sun worshippers!

• Lots of fun things far away, but can we even
  understand our nearest and dearest star?
• Only 93.5 million miles away (150 mill km) gt
  easy to observe!
• Run-of-the-mill, average star
• G2 5000-6000 deg K
• 15 million degs K in core
• 2 million degs K in corona
• 4.5 billion years old
• 5.5 billion years to go!
• Mainly hydrogen
• Fusion in core
• When hydrogen runs out burns helium -gt red
  giant -gt white dwarf
• 1.4 million km diameter ( 100 x Earth)

Looks pretty boring in white light, BUT .
3
The Sun
4
We are Sun worshippers!
5
Major puzzle solar magnetic activity cycle
Solar magnetic activity is VARIABLE but
remarkably ORDERED.
How does it work? A DYNAMO! But how does the
dynamo work?
6
Observations exterior
Look at sun in different ways, see different
scales.
Really big stuff Flares, Coronal
holes, CMEs 700Mm Days, months
Giant cells 200 Mm 10-20 days
Supergranulation 30-50 Mm 20 hours
Mesogranulation 7-10 Mm 2 hours
Granulation 1-2 Mm 5 mins
Even smaller stuff Intergranular
lanes, magnetic bright points, diffusion
TURBULENT!
7
Theories driven by observations
Things we know surprisingly well -- solar
interior structure (thanks to helioseismology)
Solid body rotation in the core
Interface layer the TACHOCLINE
Differential rotation in the convection zone
8
Large-scale dynamo Theory
Existing poloidal field is stretched by
latitudinal (and/or radial) differential rotation
into the toroidal direction an ?-like mechanism
Strong toroidal field rises due to magnetic
buoyancy and, under the influence of global
rotation, twists into the poloidal direction an
a-like mechanism
9
Strong toroidal field rises as structures?
MAGNETIC BUOYANCY the standard explanation

• Magnetic field exerts a magnetic pressure
  (Lorenz force JxB can be split into pressure and
  tension
  )
• Concentrated B contributes to the total
  pressure
• Isothermal pressure balance implies density
  lower in tube

Pg densitytemperature
Outside Pt Pg1
Inside Pg2 Pt-Pm Pm B2 gt Pg2 lt
Pg1 densityinside lt densityoutside
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Magnetic buoyancy instabilities 101
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Magnetic buoyancy instabilities 101
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Magnetic buoyancy instabilities 101
End result Magnetic GRADIENTS that are
important! (gradients in the direction of gravity)
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Standard concept of the Omega effect
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Large-scale dynamo intuitive picture

1. Generation/shredding of magnetic field
2. Transport of magnetic field from CZ into
   tachocline -- TURBULENT TRANSPORT OF B ?
3. W effect Conversion of poloidal field to
   toroidal field -- DIFFERENTIAL ROTATION --
   ORIGIN ?
4. Formation of structures and magnetic buoyant rise
   -- MAGNETIC BUOYANCY / SHEAR ?
5. a mechanism Regeneration of poloidal field
   -- ROTATION, TURBULENT u, b CORRLNS??
6. Recycling/breakdown of field
7. Emergence of structures

STUDY ELEMENTS OF THE DYNAMO
15
Heavily dependent on the tachocline

• Boundary layer between
• the convection zone (differentially rotating)
• the radiative interior (solid body rotation)
• Layer (plus sub-layers?) of
• strong radial shear
• weaker latitudinal shear
• Almost certainly threaded with magnetic field
• weak poloidal?
• strong toroidal?

• We wish to examine, via fully nonlinear MHD
  simulations
• the interaction of velocity shear and weak
  background (poloidal) magnetic fields.
• the production of strong (toroidal) magnetic
  fields by the shear
• magnetic structures produced
• any magnetic buoyancy instabilities of the
  magnetic configuration
• the nonlinear evolution of the system

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So how does it work?
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Local simulations of elements of the dynamo
e.g. Brummell, Clune Toomre, 2002

• Full compressible MHD (poloidal/toroidal)
• DNS
• Cartesian
• Pseudospectral / finite-difference
• Semi-implicit
• HIGHER Rq, Re, Pe, LOWER Pr than global sims
  (resolves from diffusive scale UP)

Thermal diffusivity kk(z) ( not
k(r,Tx,y,z) ) Ck(layer1)/Ck(layer2)(m21)/(m1
1) Stiffness, S (m2-mad)/(mad-m1)
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So how do we solve these?
Blue Gene/L Fastest machine in the world!
213,000 cpus 596 Tflops peak, 478 sustained
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The ?mega effect
Interaction between shear (gradients) in the
azimuthal velocity and the poloidal magnetic
field produces strong toroidal magnetic field.
TWO POSSIBILITIES
(the usual)
(the not-so-usual)
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Good reasons for the not-so-usual
But really interested in gradients for magnetic
buoyancy ??????t BT BP dO , dO
rsin(?) ?O Differentiate MDI observational
results

• Therefore reasons
• Radial shear is stronger
• Need radial gradients for magnetic buoyancy
• Some radial component likely
• Horizontal scale selection - better option?

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Case (a) Latitudinal shear
(a) Latitudinal shear
Bf

q

latitude
Bf
f
longitude
Bq
dVf/dq
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Model Localised latitudinal velocity shear

• Mimic some properties of the tachocline
• Use a convectively stable layer
• Force a velocity shear in both the vertical
  (z) and one horizontal (y) direction.
• e.g. U(y,z) f(z) cos(2 p y/ym)
• where f(z) is a polynomial function chosen to
  confine the shear to a particular layer between
  zu and zl (and to be sufficiently continuous)
• Shear flow is hydrodynamically stable
• Then add an initial magnetic field
• B0 (0, By , 0) with By 1

Add term in the equations that induces desired
flow in absence of magnetic effects
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Induction of strong toroidal field by shear
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Increasing Rm Magnetic buoyancy instability

• A more interesting movie!
• Instability
• Cyclic activity
• Two out-of-phase sequences of identical but
  oppositely-directed magnetic structures.
• Broken the y-reflectional symmetry

25
Latitudinal shear conclusions
First demonstration that buoyantly-rising strong
(toroidal) magnetic structures can be
spontaneously generated by the action of shear on
a background (poloidal) magnetic field. BUT
Radial gradients stronger gt stronger toroidal
field? Currently the structures have scale of the
latitudinal shear BIG (too big) SO Try
radial gradients?
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So consider Case (b) Radial shear
One might expect
Yes! but turns out that this is much harder to
make work than you might think!
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Radial shear Magnetic buoyancy
Magnetic buoyancy instability volume rendering
of abs(B) 7683 200,000 cpu hours
yikes! Measured Taylor microscale Reynolds
number 30
Pm0.625
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Radial shear Looks good, but problems
Bx
z
Time
Not much flux transport. Inefficient
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Radial shear Problems
The forcing required to make this work is HUGE!
Target velocity of forcing is about Mach 15!
Yikes! If you run the same case without magnetic
field, keeping the same large forcing, get some
serious Kelvin-Helmholtz turbulence! WHY SUCH A
LARGE FORCING? RE-CAST THIS QUESTION
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Is the tachocline this strongly maintained?

• Is it maintained by external processes
  independent of tachocline dynamics? (USUAL POV)
• Or must maintaining processes take into account
  internal dynamics? (UNUSUAL POV)
• Must the tachocline be STRONGLY MAINTAINED
  against the tachocline dynamics?
• In other words is the shear we oberve the target
  shear of a a particular forcing or a modified
  shear returned at the end of a complex nonlinear
  process?
• Two simple 1-D analytic models to try and answer
  this question
• Completely unmaintained shear
• Weakly maintained shear
• If these dont work in some sense, then answer is
  shear is strongly-maintained.
• We evaluate under what conditions a magnetic
  buoyancy instability can manifest itself

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Stability condition
Condition for instability (Newcomb 1961)
around a mechanical equiilibrium
where N is the Brunt-Vaisalla frequency
can be recast as
Thermodynamics Adiabatic gt N N0, const.
(Reasonable but well come back to this)

• Note that NONE of this is really valid in our
  case, of course!
• No mechanical equilibrium background state
  evolves!
• Shear flow on top
• BUT look to see if this is even CLOSE to being
  satisfied as an indication of LIKELIHOOD of
  magnetic buoyancy

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Model 1

• No maintenance of the shear (slow rundown)
• things happen fast so can ignore complex
  nonlinear back reactions on the shear
• maintainence of shear independent of tachocline
  dynamics
• Examine time-dependent buildup of the toroidal
  field
• No forcing, linearisation

Induction eqn stretching production of tor field
by shear on pol field
Momentum eqn linear back reaction on the shear
Easy assume background density const (but dont
have to) Initial conditions Bx0 and a given
profile uU0(z) gt solutions
gt Alfven wave dynamics
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Alfven wave dynamics
Analytical constant wave speed
U(t,z)
Initial condition U0 tachocline-like
jump, Magnitude ?U0, width ?z
Bx(t,z)
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Full 3D simulations Failures (
We have plenty of examples of the types of
dynamics we describe Forcing not large
enough System heads to diffusively-balanced
state plus Alfven waves.
U
Bx
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Amplitude of Bx

• Bx grows but saturates.
• Really is just superposition of two waves, and
  only grows when they are interfering.
• Bx grows whilst interference is in shear zone,
  then halts in the constant parts of U0
• i.e. Bx grows until
  i.e. grows for Alfven crossing time tA across
  half layer.
• Two phases of dynamics
• Growth (actually interfering waves) for t lt tA
• Bx grows linearly like
  and u U0
• Wave propagation for t gt tA with
• Can Bx get big enough for magnetic buoyancy
  instability?

Linear growth up to a saturation max value
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Model 1 Instability?
Estimate gradients of Bx from this amplitude
simply as (Bx(max)-0)/?z/2 Differentiate Plug
into mag buoy instability condition (adiabatic
version) or Richardson number of forced flow,
Ri lt ?z/H Small!! Need highly shear unstable
original flow for possibility of mag buoy
instabilities!! Rule of thumb Ri lt 1/4 gt
turbulent shear Tachocline Ri 1000? Maybe NO
maintenance is not enough! Try some weak
maintenance? gt Model 2
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Model 2
Weak maintenance of the shear Examine
time-independent state weakly forced state
Momentum
Induction
Integrate, combine and figure out some bounds
gt estimate of max amplitude of toroidal field
Pm magnetic Prandtl number
Magnetic energy bounded by the kinetic energy of
the the forcing (but with Pm factor) Stability?
Ri/Pm lt ?z/H or Ri lt Pm . ?z/H EVEN
WORSE!
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Conclusions
UNDER THE ASSUMPTIONS -- WEAK OR NO MAINTENANCE
OF SHEAR gt SHEAR INDEPENDENT OF TACHOCLINE
DYNAMICS then to get magnetic buoyancy
instabilities NEED -- large forcing (large
hydrodynamically unstable) -- or high Pm We have
confirmed these simple ideas with massive
numerical simulations! (buoyancy found at small
Pm for large forcing, or at large Pm for small
forcing) Ultimate
conclusion The maintenance of the tachocline
needs to know about tachocline dynamics. In other
words, we have to do a more self-consistent
problem.
Pm4000 ?U01.2
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Radial shear Problem avoidance

• To avoid these theoretical restrictions, need
• Large forcing
• Large magnetic Prandtl number
• Both work
• Seen already
• Large Pm

Pm 4000
40
Radial shear Thoughts

• Action of radial shear on radial field --
  conclusions
• For magnetic buoyancy to occur, must stretch the
  field fast enough that Alfven waves cannot
  radiate the magnetic energy away.
• Simple models, confrmed by numerical experiments,
  show that this requires a very strong
  initial/target shear -- one that is probably
  hydrodynamically unstable (or a high Pm which is
  not valid for the Sun).
• The models assumed that the tachocline dynamics
  and the processes that maintained the tachocline
  itself were independent. At this stage we
  concluded that this must not be true and a more
  holistic, more complicated model of tachocline
  dynamics was needed.
• We had pondered one quirky caveat in this work
  however, and had only speculated on its effect.
  That was the possibility of a DOUBLE-DIFFUSIVE
  INSTABILITY.

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Double-diffusive instabilities 101
In the presence of TWO components that affect the
density, can have an instability even if the
overall density stratification is convectively
stable. e.g. ocean water warm salty water
overlying cooler fresh water IF the
diffusivity of the stabilising component is much
LARGER than that of the stabilising component,
can have instability Parcel perturbed upwards
should return downwards due to temperature
deficit but this diffuses away quickly, and so
continues upwards due to relative freshness.
warm salty
cool fresh
Overall density stratification stable
Thermal stratification stabilising
Salinity stratification destabilising
42
Double-diffusive magnetic buoyancy
In the earlier work , we assumed that the
thermodynamics adjusted adiabatically. If we
assumed that the thermal diffusivity was
sufficiently large, then we could re-do the work
assuming that the adjustment was isothermal. Now
we have to include the diffusivities via the
ratio (Note not Pm!) This is
equivalent to looking for double-diffusive
instabilities in this system.
Adiabatic
Isothermal Clearly, if ? is small,
then this inequality is more likely satisfied.
43
Double-diffusive instability
Two simulations differing only in the thermal
diffusivity ? and therefore ? Fixed magnetic and
viscous diffusivity (gt Prandtl number
changes) ?????????????? Lower ??? is unstable!
44
Double-diffusive instability
Fluctuations are induced in B and w
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Double-diffusive instability
Added complexity Since this is not a bifurcation
about a static equilibrium but rather one about a
dynamically-evolving background state, there is a
condition on the growth rate as well as just a
Rayleigh number criterion (RatotRaT-?RaB) Growt
h rate must be faster than the evolution of the
background for instability to manifest. Background
evolves on the Alfven timescale of the
background field, so now dynamics depends on
background field strength! Instability occurs
weaker and higher up.
We are still working on the nonlinear evolution
of this instability
46
Conclusions

• Solar boundary layer -- the tachocline --
  essential for solar dynamics.
• Dynamics of the tachocline very mysterious.
• In terms of generating buoyant magnetic
  structures
• TYPE 1 Latitudinal shear can produce buoyant
  structures but of the wrong scales.
• TYPE 2 Radial shear struggles to produce buoyant
  structures at solar parameters by regular
  magnetic buoyancy instabilities.
• This is due to the fact that a system laced
  radially with magnetic field can radiate away
  magnetic energy through Alfven waves. This will
  occur always be the case because it is unlikely
  that you can have Type 1 alone (no poloidal
  field)
• Differences between Type 1 and Type 2? Layers vs
  finite structures? etc
• Is the tachocline shear strongly maintained
  against internal processes?
• TYPE 3 Double-diffusive instead of regular
  magnetic buoyancy instabilities CAN occur at
  solar parameter. It remains to be seen if this
  is a viable mechanism of magnetic transport.
• Are any of these mechanisms strong enough to
  transport magnetic field through a convection
  zone?

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The End
48
Latitudinal shear AND radial field

• Very similar dynamics except for tilted geometry
• Equilibria are equilibria.
• Instabilities still occur.
• So why do layers behave differently?

By1, Bz0 (no vertical field)
By1, Bz1 (vertical field)
49
Layer dynamics
Probably due to the fact that layers CANNOT have
the following sort of behaviour
No horizontal pressure gradients no poloidal
flows driven in layers. Wide but finite layers?
Ends rise first, waves in middle?
50
Wide but finite layers

• 8x wider shear than previous latitudinal shear
  case
• Shows some signs of what we expect
• Ends rise first
• Middle spreads by waves
• More work to come

51
Another escape route?
A completely different type of instability
DOUBLE-DIFFUSIVE INSTABILITIES For the work we
presented so far, arguments were made under an
adiabatic parcel argument in general. Adiabatic
gt thermally-sealed. Opposite extremely
isothermal gt infinitely leaky (in heat)
52
High Pm, low forcing
Pm4000 ?U01.2
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Radial shear
Why was such a large forcing necessary? Initial
forcing of shear r dt uo
F Initial induction of Bx dt Bx0
Bz dz u0 Back-reaction on shear r
dtu1 dx(p a Bx0Bx0) Bz dz Bx0 m dzz u0
F Question Does magnetic buoyancy act before
magnetic tension destroys the shear?
magnetic tension
magnetic pressure
54
Radial shear
Answer Only if the forcing is LARGE r dtu1
dx(p a Bx0Bx0) Bz dz Bx0 m dzz u0
F Usually balance diffusion and forcing so that
in the absence of magnetic fields, a steady
target velocity is produced.
X
X

• r dtu1 dx(p a Bx0Bx0) Bz dz Bx0 m dzz
  u0 F
• Here, must also balance magnetic tension with a
  larger forcing otherwise
• tension acts back on the shear before the
  magnetic pressure becomes significant
• dynamics are dominated by Alfven waves

dt Bxn Bz dz un r dtun Bz dz Bxn F
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Radial shear Problems
Want to work out effectively whether it is
possible to amplify the horizontal field Bx fast
enough to go buoyantly unstable before Alfven
waves radiate away the magnetic energy. i.e. you
have an Alfven crossing time of the layer to get
Bx as big as possible! Theory Unforced case
(rundown problem from an initial velocity shear
profile)
For instability, we need (ballpark based on
Newcomb condition) dPM/dz PN0(z)2/g
magnetic pressure, PM Bx(z)2/2 N0(z)
buoyancy frequency of the initial background
profile The best that unforced shear can do after
roughly one Alfven time is Bx Bz tA ?zu
Afven time, tA ?z/vA Therefore, for
instability, require ?z (?zu(z))2
N0(z)2 HP ?z/HP Ri i.e.
for ?z small, Ri must be small gt SHEAR FLOW IS
(HYDRODYNAMICALLY) UNSTABLE!
56
Radial shear Problems
Theory Forced case (forcing set up to balance
dffusion in absence of B)
There are some steady solutions Bz?zBx µ?zzu
-µ?zzu0 0 Bz?zu ??zzBx
0 Combining these gives Q Bx - ?zzBx (Bz/?)
?zu0 Q (Bz)2/?µ In terms of the
velocity jump ?U ? ?zu0 dz and the magnetic
Prandtl number sM ?/? there is a sharp
bound sup(PM) sM ?0 ?U2/8 Buoyancy
needs ?z/HP Ri/sM Even worse for solar
values since sM is small, Ri must be even
smaller.
57
Radial shear Thoughts

• Action of radial shear on radial field --
  conclusions
• For magnetic buoyancy to occur, must stretch the
  field fast enough that Alfven waves cannot
  radiate the magnetic energy away.
• Simple models, confrmed by numerical experiments,
  show that this requires a very strong
  initial/target shear -- one that is probably
  hydrodynamically unstable.
• In the case where magnetic buoyancy is induced,
  simulations show that the presence of magnetic
  field suppresses the hydrodynamic instability, so
  that the resultant MHD velocity shear is
  reasonable. Adding the magnetic field back into
  the hydrodynamic simulation seems to return the
  system to the same MHD state.
• In the case where magnetic buoyancy is not
  induced, a diffusive balance results.
• These results are true for forced AND slow
  rundown (very low diffusivity) cases.
• Two big questions remain
• Even when buoyancy does occur, it is inefficient
  at transporting magnetic flux vertically. Why?
• The latitudinal shear problem worked just fine.
  Would adding vertical field to this problem ruin
  things?

58
Radial shear Problem avoidance

• To avoid these theoretical restrictions, need
• Large forcing
• Large magnetic Prandtl number
• Both work
• Seen already
• Large Pm

Pm 4000
59
Latitudinal shear AND radial field

• Very similar dynamics except for tilted geometry
• Equilibria are equilibria.
• Instabilities still occur.
• So why do layers behave differently?

By1, Bz0 (no vertical field)
By1, Bz1 (vertical field)
60
Layer dynamics
Probably due to the fact that layers CANNOT have
the following sort of behaviour
No horizontal pressure gradients no poloidal
flows driven in layers. Wide but finite layers?
Ends rise first, waves in middle?
61
Wide but finite layers

• 8x wider shear than previous latitudinal shear
  case
• Shows some signs of what we expect
• Ends rise first
• Middle spreads by waves
• More work to come

62
Conclusions

• Solar boundary layer -- the tachocline --
  essential for solar dynamics.
• Dynamics of the tachocline very mysterious (Why
  is it so thin? Is it turbulent? -- Much more to
  come from Steve Tobias later).
• In terms of generating buoyant magnetic
  structures
• Radial shear acting on radial field is much less
  efficient than latitudinal shear acting on
  latitudinal field
• This is due to the fact that a system laced
  radially with magnetic field can radiate away
  magnetic energy through Alfven waves.
• Horizontal field gt waves confined to structures
  gt energy not radiated away?
• Furthermore, the dynamics of layers are very
  different from the dynamics of tubes
• No poloidal flows
• Layers gt tethering of structures by magnetic
  tension
• What is the role of waves in the tachocline?
• (Important for angular momentum transport too)

63
The End
64
Latitudinal shear with radial field

• Very similar dynamics except for tilted geometry
• Equilibria are equilibria.
• Instabilities still occur.
• So why do layers behave differently?

By1, Bx0 (old)
By1, Bx1 (new)
65
Latest 1D shear (Grad student Geoff Vasil)
Structures have twist
66
Radial shear Magnetic buoyancy
High enough forcing strong enough magnetic
layer created such that magnetic buoyancy can
act.
However, things are much more complicated and
turbulent than previous case
67
Alfven wave dynamics
Initial condition U0 tachocline-like jump,
jump ?U0, width ?z
Bx(t,z)
U(t,z)
Analytical constant wave speed
Toy Calculation with density stratification