Tsunami speed variations in the deep ocean caused by seawater density stratification is investigated using a newly developed propagator matrix method that is applicable to seawater with depth-variable sound speeds and density gradients. For a 4 km deep ocean, the total tsunami speed reduction is 0.44% compared with incompressible homogeneous seawater; two thirds of the reduction is due to elastic energy stored in the water and one third is due to water density stratification mainly by hydrostatic compression. Tsunami speeds are computed for global ocean density and sound speed profiles, and characteristic structures are discussed. Tsunami speed reductions are proportional to ocean depth with small variations, except in warm Mediterranean seas. The impacts of seawater compressibility and the elasticity effect of the solid earth on tsunami traveltime should be included for precise modeling of transoceanic tsunamis.
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 Recent tsunami observations in the deep ocean, such as the Deep-ocean Assessment of Reporting of Tsunamis stations on the deep ocean floor [Wei et al., 2008; Kusumoto et al., 2011; Fujii and Satake, 2013], tsunami sensors attached to the deep ocean bottom cables, and GPS buoys continuously recording sea surface elevations [Kato et al., 2011], have accumulated unequivocal evidence that tsunami traveltime delays compared with the linear long-wave tsunami simulations occur during tsunami propagation in the deep ocean. The delay is up to 2% of the tsunami traveltime.
Watada et al. [2011, 2012] investigated the cause of the delay using the normal mode theory of tsunamis [Ward, 1980; Okal, 1982] and attributed the delay to the compressibility of seawater, the elasticity of the solid earth, and the gravitational potential change associated with mass motion during the passage of tsunamis. The normal mode theory has been applied to earth models with a compressible homogeneous ocean layer. Okal  obtained an asymptotic formula of the effect of seawater compressibility on the tsunami propagation speed. Tsai et al.  gave a back-of-an-envelope estimate of the tsunami speed reduction caused by the elastic solid earth and compressible seawater and found a factor inconsistency in the estimates of the effect of the seawater compressibility by Okal  and by Tsai et al. .
 Tsunami speed is affected by the seawater compressibility in two ways. The gravity potential energy of elevated and depressed seawater is not only converted to kinetic energy but also stored and released as an elastic energy in the seawater by the fluctuating hydrodynamic pressure. For a given input of gravity potential energy, smaller available tsunami kinetic energy for tsunami motion in the compressible water compared with the incompressible water results in a smaller wave frequency and hence lower tsunami speed. Real seawater is inevitably density stratified by the hydrostatic compression. The long-wave speed of a density-stratified fluid is always slower than that of a homogeneous fluid, as we see in the application section. Density stratification is also controlled by the temperature and salinity vertical profiles. The normal mode approach [e.g., Ward, 1980; Okal, 1982; Watada et al., 2011, 2012] has overlooked the stratification of seawater, and Tsai et al.  assumed an adiabatic density profile of seawater. It is desirable to discuss the compressibility effect and density stratification effect on tsunami speed separately.
 This paper focuses on tsunami speed reductions caused by the density stratification and the compressibility of seawater. Starting from a propagator matrix of a single-fluid layer with a constant sound speed and a density scale height, a dispersion relation of water waves is computed for a density-stratified ocean represented by a stack of thin water layers each with a constant density gradient and a sound velocity. The analytic forms of the dispersion relation for singe- and two-layer models are compared with the known dispersion relations of compressible and incompressible water cases. Finally, based on the ocean grid model of the World Ocean Atlas 2009 [Boyer et al., 2009] (hereinafter WOA09), global distribution of tsunami speed perturbations and the impacts of the variability of ocean structure on the global distribution of tsunami speed perturbations are discussed.
 The equations of motion of a compressible inviscid fluid are found in e.g., section 6.14 of Gill  and equation 2.10 of Watada . The x and z axes are taken along the horizontal and vertical upward positive directions, respectively. The ocean bottom is at z=0. ρ, p, (ux,uz), (vx,vz) represent density, pressure, fluid displacement, and velocity, respectively, and g is constant gravity downward (Figure 1. The subscript l denotes lth layer, and the subscript o denotes the background equilibrium state, the superscript ′ and prefix δ denote Eulerian and Lagrangian perturbations, respectively. For brevity, the subscript l is omitted when the layer is obvious and is used when needed. u, v, and the variables with ′ and δ are assumed small, and higher order terms are neglected.
 Elimination of density perturbation terms in the linearized momentum equation, the mass conservation equation, and the adiabatic equation of states, and the use of the Lagrangian pressure perturbation yield the set of equations
where the density-scaled velocities and the pressure perturbation are defined by
Uz and Vz are related by and P=δP+gUz, and cs is the sound velocity. ρo and cs are dependent on z only. Γ(z) and the buoyancy frequency N(z) are defined by
Assuming a constant scale height for density and a constant cs within the lth layer, N2 and Γbecome constant and the fluid density is expressed by
where the subscripts t and b denote quantities at the top and bottom of the layer, respectively, and dl denotes the thickness of the layer. Adopting a plane wave solution for the density-scaled variables of the form exp(i(kx+mz−ωt)), the differential equations (2) and (3) are expressed by
By the density scaling in equation (4), the coefficient matrices E and F are independent on z and the dispersion relation between k,m, and ω is computed from det∥EF∥= det∥E∥ det∥F∥= det∥E∥=0. δP, P, Vx, and Vz share the dispersion relation that
 Because m2 is a constant real number for a given (ω,k) in the lth layer, R(z), the dependence of the normalized vertical velocity Vz on z satisfies the differential equation
where M is defined by M2=−m2, and its solution has the form
where C and D are constants to be determined from boundary conditions. Note that in the case of m2<0, cos(mz) should be interpreted as and sin(mz) as .
 From two sets of Vz(z) and δP(z) at z=zl−1 and zl, C and D are eliminated and A(dl), the propagator matrix from z=zl−1 to z=zl=zl−1+dl of a vector o(z)T=(vz(z),δp(z))T, where the superscript T denotes the transpose of a matrix, is obtained (detailed steps are in the supporting information) as
where the (i,j) matrix element of matrix A(dl) is given. If M2=−m2>0,
and if m2=−M2>0,
, , c(z)= cosh(Mz), s(z)= sinh(Mz), e(z)= sin(mz), f(z)= cos(mz), and is the water density at the midpoint of the lth layer. If the layer density is constant, factors are replaced by 1. The propagator matrix, which was also developed for the atmospheric waves by Harkrider , has the same characteristics with det∥A(z)∥=1.
 Vertically non-uniform stratification of seawater is modeled as stacked multiple thin layers, each with a constant-scale height and a constant sound speed. A density jump can exist between the layers if ρol(zl)≠ρol+1(zl). In that case, δp but not p′ is continuous at the boundary z=zl; hence, o(zl) is always continuous across the internal boundaries at zl. BL, the propagator matrix from the bottom to the top of L-stacked layers, is computed as
where z0 and zL are the bottom and top of L layers, respectively.
 The boundary conditions are vz(z0)=0 at the rigid bottom and δp(zL)=0 at the free surface. Thus, the dispersion relation between the horizontal wave number k and the angular frequency ω for a given layered structure (ρobl,Hl,csl,dl,l=1,⋯,L) is expressed by bL22(ω,k)=0, where bLij represents the (i,j) matrix element of 2×2 matrix BL. The propagator matrix BL is applicable to all types of linear water waves including acoustic waves, surface gravity waves, and internal gravity waves.
3.1 Single Layer
 The dispersion relation of a single layer (L=1) using aLij, the (i,j) matrix element of 2 × 2 matrix AL, is
which is rewritten explicitly as
where Cp is the phase velocity. The newly obtained dispersion equation (25) expresses how the density stratification and compressibility alter the dispersion relation of water waves of a single layer. Note that in a single-layer case, it is the density scale height and not the absolute density that contributes to the dispersion relation. When M2<0, replacing M by m and tanh by tan gives the correct dispersion relation. Assuming a homogeneous () incompressible () single layer, ω2=gk tanh(kd) is confirmed.
 To evaluate the effects of the density stratification and the compressibility of seawater separately, four single-layer problems, in which parameters H and cs are constant (Table 1, are examined (Figure 2. Assuming adiabatic density stratification of compressible water, i.e., N=0, the scale height is estimated as . For a 4 km deep ocean with g = 9.822 m/s2 and H = 229.1 km, the density increases by from the bottom to the top of the ocean. When the ocean layer has uniform density and a constant sound velocity (case B), the dispersion relation equation (25) reduces to
where . Assuming long-wave Md≪1 and knowing that the tsunami of the Earth's ocean satisfies , the dispersion relation for waves with horizontal wavelength becomes (details are in the supporting information)
When water is stratified adiabatically with a uniform sound velocity (case D), gH is equal to and the dispersion relation equation (25) reduces to
where . Again, with the long-wave and assumptions, the dispersion relation for waves with horizontal wavelength λ≫ 200 km becomes (details are in the supporting information)
Table 1. Tsunami Speed of Single-Layer Ocean Modelsa
aDepth averaged densities are the same.
bEvaluated at a wavelength of 8000 km as a long-wave limit.
cSuperadiabatic density profile which does not exist stably in nature.
 Case B is used for normal mode tsunami computation by Ward , Okal , and Watada and Kanamori . Case D is equivalent to the ocean assumed by Tsai et al. . The difference of the estimates of the tsunami speed dependency on the seawater sound velocity, or incompressibility, between Okal  and Tsai et al.  originates from the assumption of the reference density profile of the water layer. Okal  obtained for a homogeneous compressible ocean and Tsai et al.  obtained for an adiabatically stratified compressible ocean. Their asymptotic tsunami speed expressions are identical to mine only when the tsunami period is much larger than 1000 s or the wavelength is much longer than 200 km, which is a stronger condition than the long-wave condition kd≪1.
3.2 Two Layers
 The dispersion relation of a compressive fluid with two layers of different densities (L=2) is
which is rewritten explicitly, assuming incompressible homogeneous water, as
This is equivalent to equation 17 in section 231 of Lamb . The dispersion relation can be rewritten as
Assuming long waves (x≪1), tanh(x) is approximately x, coth(x) approximately 1/x, sinh(x) approximately x, and cosh(x) is approximately 1, equation (32) approaches
which is equivalent to equation 48 in p. 219 of Stokes  and equation 6.2.14 of Gill  for long waves in a two-layer fluid. The solution, which corresponds to external and internal waves (+ and −, respectively), is
In a layered fluid, usually α=ρ2/ρ1≤1 and 1−α≪1, and two long-wave modes approach
Imamura and Imteaz  obtained the external and the internal wave phase velocities which are expressed with α and β:
Note that α and β in Imamura and Imteaz  are defined differently from the definitions here. Equation (38) is in error, which is easily checked, for example when β = 1, the tsunami speed p+ should be 1, not α.
3.3 Multiple Layers
 As tests of the code, a density-stratified incompressible layer with a scale height H (case C in Table 1 is emulated by stacked multiple homogeneous incompressible layers (case A), and a density-stratified compressible layer (case D) by stacked multiple homogeneous compressible water layers (case B). In both cases, the computed dispersion relations of 32 layers are identical to the single-layer dispersion relation within the accuracy of numerical errors (Figures are in the supporting information). In the analysis of realistic density profiles, because N2 determined from is less accurate than ρo measurement, a staircase representation of the density gradient with many homogeneous multiple layers is preferred to a fewer thick layers with density gradients.
4.1 Global Variations in Tsunami Speed Reduction
 Seawater density and compressibility are controlled by the pressure, salinity, and temperature of the ocean [Talley et al., 2011]. WOA09 gives salinity and in situ temperature profiles at 1°× 1° time-averaged grid points. The TEOS-10 toolbox [McDougall and Barker, 2011] converts a WOA09 profile to in situ density and sound velocity profiles. WOA09 unevenly covers all the oceans (Figure 3. WOA09 defines 33 depth grids; the grid spacing becomes coarser with depth (every 500 m after 2000 m) toward the deepest grid at 5500 m. Local ocean models are artificially truncated at the maximum grid depth where salinity and temperature data are listed in WOA09 and do not represent the ocean profiles down to the real bottom. Tsunami speed, defined as the phase velocity at a wavelength of 8000 km in the dispersion curve, has been computed at each surface grid having a depth profile deeper than 2500 m (Figure 4c). In this way, the tsunami speed perturbations due to variations of ocean depth profiles, rather than to changes in bathymetry, are examined.
 Equation (37) shows that tsunami speeds in oceans with warmer and deeper (up to a half of the depth) layers are slower than incompressible long-wave speed . The Mediterranean Sea is characterized by warm water at all depths, which results in higher sound velocities and smaller tsunami speed reductions (Figures 4a and 4b). The North Atlantic near Greenland is characterized by nearly constant cold temperature over the entire water column, similar to uniform water (Figure 4c). Adiabatically stratified cold dense water is the easiest to sink. In fact, off the coast of Greenland in the North Atlantic is the sinking point of the great ocean conveyor ocean circulation model [Broecker, 2010].
 Given that the two vertical columns have identical sound speed, tsunami speed is larger when the potential density is constant, i.e., when the buoyancy frequency is zero throughout the water column and therefore the tsunami speed is insensitive to the absolute magnitude of water density. The western Pacific Ocean near Taiwan, which is characterized by a 1 km thick warm top layer, exhibits global maximum tsunami speed reduction in 4000 m deep oceans (Figure 4a). The difference of the tsunami speeds between near Taiwan and near Greenland is due to the thick surface warm layer near Taiwan. The difference of the tsunami speeds between near Greenland and in the Mediterranean Sea is due to temperature, hence the sound speed, differences extending throughout the water columns (Figure 4c).
5 Discussion and Conclusion
 The analytic forms of the dispersion relation for single- and two-layer models are compared with the known dispersion relations for compressible and incompressible water. The dispersion relation of water waves for an ocean layer found in sections 54–57 of Eckart , which is obtained under the condition of Γ=0 in equation (10), is extended to Γ≠0 and multiple-layer cases. Panza et al.  gave a propagator matrix for seawater composed of homogeneous layers. For a 4 km deep compressible stratified ocean model, the total tsunami speed reduction is expected to be 0.44% from the tsunami speed in incompressible homogeneous seawater; 0.29% is due to the elastic energy stored in compressible water, and 0.15% is due to the density stratification mainly by the hydrostatic compression. Note that the differences in the phase velocity reduction between compressible and incompressible models are evidenced in a slight difference of the corner periods of the dispersion curves (Figure 2, bottom left).
 The propagator matrix method has been applied to the real ocean profiles deeper than 2500 m compiled in WOA09, and tsunami speeds in the deep oceans have been computed. An expression for the tsunami speed reduction of a given depth has been obtained as ΔV/V=a∗(depth, m)+b (Figure 4b), where a=1.00×10−6 m−1 for an average 4000 m deep ocean (except in the Mediterranean Sea), and a=9.63×10−7 m−1 for the Mediterranean Sea. The depth coefficient a has been previously estimated by Okal  as and by Tsai et al.  as . In the deep ocean, the tsunami speed perturbation from the average tsunami speed due to natural variations in the vertical structure of the ocean is usually ±0.01%. An exceptionally diminished tsunami speed reduction of −0.05%, i.e., faster than the global average, is found in the warm Mediterranean seas.
 Seawater compressibility affects tsunami speeds through its effects on density stratification and elastic energy stored in seawater. These effects, in addition to the solid earth elasticity effect and the gravitational potential perturbation effect [Watada et al., 2011, 2012], should be included for the precise estimates of the traveltime of transoceanic tsunamis. Local variations in the seawater properties of deep ocean water have negligible impacts on tsunami speeds, and their effects are not likely to be observed.
 The Editor thanks Victor Tsai and an anonymous reviewer for assistance in evaluating this manuscript.