def rSurfPrime(th):
#Accurate approximation matrix W for approximating rSurfPrime(th):
phs = 9
pol = 7
stc = 17
if len(np.shape(th)) == 2:
tmp = th[0, :].copy()
elif len(np.shape(th)) == 1:
tmp = th.copy()
else:
raise ValueError("th should have either 1 or 2 dimensions.")
W = phs1.getPeriodicDM( z=tmp, x=tmp, m=1 \
, phs=phs, pol=pol, stc=stc, period=2*np.pi )
return W.dot(rSurf(th).T).T
def interpAngular( U0, U1, U2, U3, U4, U5, th0, th1, th2, th3, th4, th5 ) :
phsA = 9
polA = 7
stcA = 17
W1 = phs1.getPeriodicDM( period=2*np.pi, x=th1, X=th0, m=0 \
, phsDegree=phsA, polyDegree=polA, stencilSize=stcA )
W2 = phs1.getPeriodicDM( period=2*np.pi, x=th2, X=th0, m=0 \
, phsDegree=phsA, polyDegree=polA, stencilSize=stcA )
W3 = phs1.getPeriodicDM( period=2*np.pi, x=th3, X=th0, m=0 \
, phsDegree=phsA, polyDegree=polA, stencilSize=stcA )
W4 = phs1.getPeriodicDM( period=2*np.pi, x=th4, X=th0, m=0 \
, phsDegree=phsA, polyDegree=polA, stencilSize=stcA )
W5 = phs1.getPeriodicDM( period=2*np.pi, x=th5, X=th0, m=0 \
, phsDegree=phsA, polyDegree=polA, stencilSize=stcA )
U1 = np.transpose( W1.dot(np.transpose(U1)) )
U2 = np.transpose( W2.dot(np.transpose(U2)) )
U3 = np.transpose( W3.dot(np.transpose(U3)) )
U4 = np.transpose( W4.dot(np.transpose(U4)) )
U5 = np.transpose( W5.dot(np.transpose(U5)) )
return U0, U1, U2, U3, U4, U5
# Whvs = alp * dsPol.dot(Whvs) # #Complex radial HV: # dr = ( rr[2:nlv,:] - rr[0:nlv-2,:] ) / 2. # alpDrPol = alp * dr**(phs-2) # alpDrPol = alp * ((outerRadius-innerRadius)/(nlv-2)) ** (phs-2) # alpDxPol = alp * ( ( dr + ss[1:-1,:]*np.tile(dth,(nlv-2,1)) ) / 2. ) ** (phs-2) # alpDxPol = alp * ( ( (outerRadius-innerRadius)/(nlv-2) + ss[1:-1,:]*2.*np.pi/nth ) / 2. ) ** (phs-2) # alpDsPol = alp * ( ( ss[1:-1,:]*np.tile(dth,(nlv-2,1)) + np.transpose(np.tile(ds,(nth,1))) ) / 2. ) ** (phs-2) ########################################################################### #Angular PHS-FD weights arranged in a differentiation matrix: #Matrix for approximating first derivative in angular direction: Wlam = phs1.getPeriodicDM( z=th, x=th, m=1 \ , phs=phsA, pol=polA, stc=stcA, period=2*np.pi ) #Simple (and technically incorrect) angular HV: Whvlam = phs1.getPeriodicDM( z=th, x=th, m=phsA-1 \ , phs=phsA, pol=polA, stc=stcA, period=2*np.pi ) Whvlam = alpA * dth0**(phsA - 2) * Whvlam #scaled angular HV matrix # dthPol = spdiags( dth**polA, np.array([0]), len(dth), len(dth) ) # Whvlam = alpA * dthPol.dot(Whvlam) # #Complex angular HV: # alpDthPol = alpA * dth0**polA # # alpDthPol = alpA * ( np.tile(dth,(nlv-2,1)) ) ** polA ########################################################################### #Weights for interpolation to boundary (I), extrapolation to
def derivativeApproximations(x, dx, left, right, s, ds):
"""
This is where all of the weights for approximating spatial derivatives
are calculated. Also, at the end, the function outputs functions
Da(), Ds(), and HV(), which are used to approximate the corresponding
derivative of something.
"""
phs = 11 #lateral PHS exponent (odd number)
pol = 5 #highest degree polynomial in lateral basis
stc = 11 #lateral stencil size
alp = 2.**-9 * 300. #lateral dissipation coefficient
Wa = phs1.getPeriodicDM(z=x, x=x, m=1 \
, phs=phs, pol=pol, stc=stc, period=right-left) #d/da matrix
Whva = phs1.getPeriodicDM(z=x, x=x, m=6 \
, phs=phs, pol=pol, stc=stc, period=right-left)
Whva = alp * dx**5 * Whva #lateral dissipation weights
W2a = phs1.getPeriodicDM(z=x, x=x, m=2 \
, phs=phs, pol=pol, stc=stc, period=right-left)
W2a = 2.**-10. * dx * W2a
phs = 5 #vertical PHS exponent
pol = 2 #highest degree polynomial in vertical basis
stc = 5 #vertical stencil size
if phs == 7:
alp = 2.**-24. * 300.
elif phs == 5:
alp = -2.**-22. * 300. #vertical dissipation coefficient
elif phs == 3:
alp = 2.**-20 * 300.
else:
raise ValueError("Please use phs=3 or phs=5 or phs=7.")
Ws = phs1.getDM(z=s, x=s, m=1 \
, phs=phs, pol=pol, stc=stc) #d/ds matrix
Whvs = phs1.getDM(z=s[1:-1], x=s, m=phs-1 \
, phs=phs, pol=pol, stc=stc)
Whvs = alp * ds**(phs - 2) * Whvs #vertical dissipation weights
numDampedLayers = np.int(np.round(len(s) / 2.))
W2s = phs1.getDM(z=s[1:numDampedLayers], x=s, m=2 \
, phs=phs, pol=pol, stc=stc)
W2s = 2.**-10. * ds * W2s
sTop = (s[0] + s[1]) / 2.
wItop = phs1.getWeights(z=sTop, x=s[0:stc], m=0 \
, phs=phs, pol=pol) #interpolate to top boundary
wEtop = phs1.getWeights(z=s[0], x=s[1:1+stc], m=0 \
, phs=phs, pol=pol) #extrapolate to top ghost nodes
wDtop = phs1.getWeights(z=sTop, x=s[0:stc], m=1 \
, phs=phs, pol=pol) #derivative on top boundary
wHtop = phs1.getWeights(z=sTop, x=s[1:1+stc], m=0 \
, phs=phs, pol=pol) #extrapolate to top boundary
wIbot = phs1.getWeights(z=1., x=s[-1:-1-stc:-1], m=0 \
, phs=phs, pol=pol) #interpolate to bottom boundary
wEbot = phs1.getWeights(z=s[-1], x=s[-2:-2-stc:-1], m=0 \
, phs=phs, pol=pol) #extrapolate to bottom ghost nodes
wDbot = phs1.getWeights(z=1., x=s[-1:-1-stc:-1], m=1 \
, phs=phs, pol=pol) #derivative on bottom boundary
wHbot = phs1.getWeights(z=1., x=s[-2:-2-stc:-1], m=0 \
, phs=phs, pol=pol) #extrapolate to bottom boundary
#Lateral derivative on all levels:
def Da(U):
return Wa.dot(U.T).T
#Vertical first derivative on all levels:
def Ds(U):
return Ws.dot(U)
#Total dissipation on non-ghost levels:
def HV(U):
return Whva.dot(U[1:-1, :].T).T + Whvs.dot(U)
#Rayleigh-damping function for top 30 layers (mountainWaves test case):
def rayleighDamping(U):
return W2a.dot(U[1:numDampedLayers, :].T).T + W2s.dot(U)
return Wa, stc, wItop, wEtop, wDtop, wHtop, wIbot, wEbot, wDbot, wHbot \
, Da, Ds, HV, rayleighDamping, numDampedLayers
###########################################################################
s, ds, sTop, pTop, pSurf = getSvalues()
xx, ss = np.meshgrid(x, s)
###########################################################################
#All of the polyharmonic spline radial basis function weights:
phs = 11 #lateral PHS exponent (odd number)
pol = 5 #highest degree polynomial in lateral basis
stc = 11 #lateral stencil size
alp = 2.**-12. * 300. #lateral dissipation coefficient
Wa = phs1.getPeriodicDM(z=x, x=x, m=1 \
, phs=phs, pol=pol, stc=stc, period=right-left) #lateral derivative weights
Whva = phs1.getPeriodicDM(z=x, x=x, m=pol+1 \
, phs=phs, pol=pol, stc=stc, period=right-left)
Whva = alp * dx**pol * Whva #lateral dissipation weights
phs = 5 #vertical PHS exponent
pol = 3 #highest degree polynomial in vertical basis
stc = 5 #vertical stencil size
if verticalCoordinate == "pressure":
alp = -2.**-22. * 300. #vertical dissipation coefficient
else:
alp = -2.**-22. * 300. #much larger in height coordinate case?
Ws = phs1.getDM(z=s, x=s, m=1 \
, phs=phs, pol=pol, stc=stc) #vertical derivative weights
Whvs = phs1.getDM(z=s[1:-1], x=s, m=pol+1 \
, phs=phs, pol=pol, stc=stc)
Ws = phs1.getDM( x=s, X=s, m=1 \ , phsDegree=phs, polyDegree=pol, stencilSize=stc ) #Simple (but still correct with dsdr multiplier) radial HV: Whvs = phs1.getDM( x=s, X=s[1:-1], m=phs-1 \ , phsDegree=phs, polyDegree=pol, stencilSize=stc ) Whvs = alp * ds**(phs - 2) * Whvs #scaled radial HV matrix # dsPol = spdiags( ds**(phs-2), np.array([0]), len(ds), len(ds) ) # Whvs = alp * dsPol.dot(Whvs) ########################################################################### #Angular PHS-FD weights arranged in a differentiation matrix: #Matrix for approximating first derivative in angular direction: Wlam = phs1.getPeriodicDM( period=2*np.pi, x=th, X=th, m=1 \ , phsDegree=phsA, polyDegree=polA, stencilSize=stcA ) #Simple (and technically incorrect) angular HV: Whvlam = phs1.getPeriodicDM( period=2*np.pi, x=th, X=th, m=phsA-1 \ , phsDegree=phsA, polyDegree=polA, stencilSize=stcA ) # Whvlam = alpA * (ds0/innerRadius)**(phsA-2) * Whvlam #scaled angular HV dthPol = spdiags(dth**(phsA - 2), np.array([0]), len(dth), len(dth)) Whvlam = alpA * dthPol.dot(Whvlam) ########################################################################### #Weights for interpolation to boundary (I), extrapolation to #ghost-nodes (E), d/ds at boundary (D), and extrapolation to boundary(H): wIinner = phs1.getWeights(innerRadius, s[0:stcB], 0, phs, pol) wEinner = phs1.getWeights(s[0], s[1:stcB + 1], 0, phs, pol)